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MathematicsCalc is arbitrary precision C-like arithmetic system that is a calculator, an algorithm prototyper and mathematical research tool. Calc comes with a rich set of builtin mathematical and programmatic functions
Mastering Essential Math Skills This exercise book is an excellent resource to practice and review math skills youŽll need to establish a strong foundation and smooth transition into Algebra and other higher math courses. Workbooks are available for 4th – 5th grade and middle school / high school. Investing $10K in 2014 Looking to invest in 2014? This EBook will show you how to create a simple portfolio that helps you invest successfully in any year
MERLOT Search - category=2526 A search of MERLOT materialsCopyright 1997-2014 MERLOT. All rights reserved.Fri, 14 Mar 2014 02:15:08 PDTFri, 14 Mar 2014 02:15:08 PDTMERLOT Search - category=2526 4434Population Modeling Applet In this applet, the user applies Euler's Method to modeling population growth using the Malthus exponential model and the Verhulst constrained growth model. After finding the Euler solution, the user can check the solution with the Adaptive Euler Approximation or with a slope field. Also, the user can enter an exact solution obtained from separating variables (or whatever) and again check the Euler solution graphically.Math Warehouse This site has has interactive explanations and simulations of math from alegrbra to trigonometry. Just click the "interactive" tab on the top left menu and you can choose different simulations. It includes, the complete definition of parabolas, reaching beyond the ability to graph into the realm of why the graph appears as it does. It also has vivid descriptions of angles including circle angles for geometry. It also has calculators for principal nth roots, gdc, matrices, and prime factorization. It's definitely worth checking out. Quote from site: "A parabola is actually a locus of a point and a line. The point is called the focus and the line the directrix. That means that all points on a parabola are equidistant from the focus and the directrix. To change the equation and the graph of the interactive parabola below just click and drag either the point A, which is the focus, or point B, which controls the directrix." This is an interactive site that allows people to change the graph to understand why directrix and focus dictate parabolic graphs. Cyberkidz educational games Cyberkidz is an educational platform for boys and girls in the age of 4 till 12 years. By playing the educational games, children will practice subjects they learn in elementary school (Preschool, Kindergarten, Grades 1-5).Electronic Journals Library Numerical Methods in Engineering course This web page shows links to lectures for a course on Numerical Methods in Engineering taught by the author in the Spring Semester of 2009. Click on the lecture links for class notes, Matlab scripts and functions, and assignments. Subjects covered: vectors and matrices in Matlab, graphics in Matlab, programming, numerical linear algebra, solution to equations, numerical integration, data fitting, and ordinary differential equations. Measuring Biodiversity across North America Through this series on measuring biodiversity students will conduct investigations based on their own questioning, they will develop a methodology, collect and analyze data, test hypotheses, and communicate results. Each example given is a model for analysis with step-by-step procedures for investigating categories of questions on biodiversity and the inherent value of the different landscapes of North America.Goals: As a result of completing an investigation into the biodiversity of North American Mammals, all students should develop an understanding of the following. The concept of biodiversity, and ways to measure the diversity of organisms The role of taxonomy in assessing biodiversity Associations between the distribution of organisms and environments How to plan and conduct an investigationIn addition, students should become more familiar with the mammal communities and ecoregions in their residential areas, the biomes and ecoregions across North America, and practice independent inquiry about the natural world.A+Click Math Skill Self-Study Tests for Grade 1 to 12Adding apples and oranges To calculate the value of an apple and an orange from 2 purchases.About mental arithmetic, with a pre-algebra tool introducing the Gaussian elimination.In the mirror site, there's the Android 2.2 (and up) version of this program.Adding apples oranges and pears To calculate the value of apple, orange, pear from 3 purchases.About mental arithmetic, with a pre-algebra tool introducing the Gaussian elimination.Adding apples oranges pears and lemons To calculate the value o apple, orange, pear and lemon from 4 purchases.About mental arithmetic, with a pre-algebra tool introducing the Gaussian elimination.
Algebra The Boundless Algebra textbook is a college-level, introductory textbook that covers the important subject of Algebra -- one of the basic building blocks of studies in higher mathematics. Boundless ... This textbook covers: The Building Blocks of Algebra Graphs, Functions, and Models Functions, Equations, and Inequalities Polynomial and Rational Functions Exponents and Logarithms Systems of Equations and Matrices Conic Sections Sequences, Series and Combinatorics We create our textbooks by finding the best content from open educational libraries, government resources, and other free learning sites. We then tie it all together with our proprietary process, resulting in great textbooks. Stop lugging around heavy, expensive, archaic textbooks. Get your Boundless alternative today and see why students at thousands of colleges and universities are getting smart and going Boundless. Boundless is better than your assigned textbook. ii SmartNotes It's like your professor summarized the readings for you. Flashcards Flashcards are a great way to study key terms, concepts and more. Quizzes When you feel ready, you can quiz yourself to see how much you know. Instant search Chapters, key term definitions, and anything else at your fingertips. Highlights Highlight key points and key terms so you can come back to them later. Notes Add notes to your highlights to make them even more meaningful. Boundless goes beyond a traditional textbook. Way beyond. Chapter 1 The Building Blocks of Algebra algebra/ Real Numbers: Basic Operations The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division. KEY POINTS • A real number is a value that represents a quantity along a continuous line. Real numbers can be thought of as points on an infinitely long line called the number line or real line. • The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division. • Arithmetic operations are performed according to a specific hierarchy or order, not from left to right. A real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Real numbers can be thought of as points on an infinitely long line called the number line (real line), where the points corresponding to integers are equally spaced as shown in Figure 1.1. The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division. Arithmetic operations are performed according to a specific hierarchy or order, not from left to right. Addition and Subtraction Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers into a single number. Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series. Addition is commutative and associative, so the order in which the terms are added does not affect their sum. The identity element of addition is 0; that is, adding zero to any number yields that same number. Subtraction is the inverse of addition; it finds the difference between two numbers. As such, taking a number x, adding b to it and subsequently subtracting b from it affords the same number x. Subtraction is neither commutative nor associative. 5 Real numbers can be thought of as points on an infinitely long number line. Figure 1.1 Real Numbers Multiplication and Division Multiplication also combines two numbers into a single number, the product. Multiplication is best viewed as a simplification of many additions. For example the product of x and y is the sum of x written out y times. Multiplication is commutative and associative, and its identity is 1. That is, multiplying any number by 1 yields that same number. Division is the inverse of multiplication. Thus, taking a number x and multiplying it by b and then dividing it by b results in the same number x. Division is neither commutative nor associative. Source: algebra/real-numbers/real-numbers-basic-operations/ CC-BY-SA Boundless is an openly licensed educational resource Interval Notation Intervals notation uses parentheses and brackets to describe sets of real numbers and their endpoints. KEY POINTS • A real interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. • The interval of numbers between a and b, including a and b, is often denoted [a, b]. The two numbers are called the endpoints of the interval. • To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. One may use an infinite endpoint to indicate that there is no bound in that direction. • An open interval does not include its endpoints, and is indicated with parentheses. A closed interval includes its endpoints, and is denoted with square brackets. A real interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0 and 1, as well as all numbers between 6 them. Other examples of intervals are the set of all real numbers and the set of all negative real numbers. The interval of numbers between a and b, including a and b, is often denoted [a, b]. The two numbers are called the endpoints of the interval. To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. One may use an infinite endpoint to indicate that there is no bound in that direction. For example, (0, +∞) is the set of all positive real numbers, and (−∞, +∞) is the set of real numbers. An open interval does not include its endpoints, and is indicated with parentheses. For example (0,1) means greater than 0 and less than 1. A closed interval includes its endpoints, and is denoted with square brackets. For example [0,1] means greater than or equal to 0 and less than or equal to 1 (Figure 1.2). An interval is said to be left-bounded or right-bounded if there is some real number that is, respectively, smaller than or larger than all its elements. An interval is said to be bounded if it is both left- and right-bounded, and is said to be unbounded otherwise. Intervals that are bounded at only one end are said to be half- bounded. The empty set is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are also commonly known as finite intervals. Source: algebra/real-numbers/interval-notation/ CC-BY-SA Boundless is an openly licensed educational resource 7 Representation on real number line. Figure 1.2 Intervals Equations, Inequalities, Properties An equation states that two expressions are equal, while an inequality relates two different values. KEY POINTS • An equation is a mathematical statement that asserts the equality of two expressions. • An inequality is a relation that holds between two values when they are different. • The notation a ≠ b means that a is not equal to b. It does not say that one is greater than the other, or even that they can be compared in size. If one were to compare the size of the values, the notation a < b means that a is less than b, while the notation a > b means that a is greater than b. Equations An equation is a mathematical statement that asserts the equality of two expressions. This is written by placing the expressions on either side of an equals sign (=), for example: x + 3 = 5 asserts that x + 3 is equal to 5 (Figure 1.3). Equations often express relationships between given quantities—the knowns—and quantities yet to be determined—the unknowns. By convention, unknowns are denoted by letters at the end of the alphabet, x,y, z, w, …, while knowns are denoted by letters at the beginning, a, b, c, d, …Inequalities An inequality is a relation that holds between two values when they are different. The notation a ≠ b means that a is not equal to b. 8 Illustration of a simple equation as a balance. x, y, and z are real numbers, analogous to weights. Figure 1.3 Equation as a Balance It does not say that one is greater than the other, or even that they can be compared in size. In either case, a is not equal to b. These relations are known as strict inequalities. To compare the size of the values, there are two types of relations: • The notation a < b means that a is less than b. • The notation a > b means that a is greater than b. The notation a < b may also be read as "a is strictly less than b". In contrast to strict inequalities, there are two types of inequality relations that are not strict: • The notation a ≤ b means that a is less than or equal to b (or, equivalently, not greater than b, or at most b). • The notation a ≥ b means that a is greater than or equal to b (or, equivalently, not less than b, or at least b) Source: algebra/real-numbers/equations-inequalities-properties/ CC-BY-SA Boundless is an openly licensed educational resource Introduction to Absolute Value The absolute value may be thought of as the distance of a real number from zero (or the non-negative value without regard to its sign). KEY POINTS • The absolute value of a number may be thought of as its distance from zero along the real number line, and more generally, the absolute value of the difference of two real numbers is the distance between them. • The absolute value \| a \| of a real number a is the non-negative value of a without regard to its sign. Namely, \| a \| = a for a positive a, \| a \| = −a for a negative a, and \| 0 \| = 0. • The absolute value of a is always either positive or zero but never negative. The absolute value of a number may be thought of as its distance from zero (Figure 1.4). In mathematics, the absolute value (or modulus) \| a \| of a real number a is the non-negative value of a 9 The absolute value of a real number may be thought of as its distance from zero. Figure 1.4 Absolute Value without regard to its sign (Figure 1.5). Namely, \| a \| = a for a positive a, \| a \| = −a for a negative a, and \| 0 \| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts. Terminology and Notation The term "absolute value" has been used in this sense since at least 1806 in French and 1857 in English. The notation \| a \| was introduced by Karl Weierstrass in 1841. Other names for absolute value include "the numerical value" and "the magnitude." Definition and Properties For any real number a, the absolute value or modulus of a is denoted by \| a \| (a vertical bar on each side of the quantity) and is defined as \| a \| = a for a greater than or equal to 0, and \| a \| = −a for a < 0. For instance, if a = -3, \| -3 \| = 3 = −(-3). The double negative yields a positive number. As can be seen from the above definition, the absolute value of a is always either positive or zero but never negative. From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line, and more generally, the absolute value of the difference of two real numbers is the distance between them. Source: algebra/real-numbers/introduction-to-absolute-value/ CC-BY-SA Boundless is an openly licensed educational resource 10 The graph of y = \|x\|. The graph is symmetric for both negative and positive values of x. Figure 1.5 Absolute Value Integer Exponents An exponent, written bn , indicated multiplying b times itself n times, so b3 is b·b·b. KEY POINTS • Exponentiation is a mathematical operation, written as bn , involving two numbers, the base b and the exponent (or index or power) n. When n is a positive integer, exponentiation corresponds to repeated multiplication. • The expression b2 = b·b is called the square of b because the area of a square with side-length b is b2 . The expression b3 = b·b·b is called the cube, because the volume of a cube with side-length b is b3 . • Some observations may be made about exponents. Any number raised by the exponent 1 is the number itself. Any nonzero number raised by the exponent 0 is 1. These equations do not decide the value of 00 . Raising 0 by a negative exponent would imply division by 0, so it is undefined. Exponentiation is a mathematical operation, written as bn , involving two numbers, the base b and the exponent (or index or power) n. When n is a positive integer, exponentiation corresponds to repeated multiplication; in other words, a product of n factors, each of which is equal to b (the product itself can also be called power) (Figure 1.7). Similarly, multiplication by a positive integer corresponds to repeated addition (Figure 1.6). The exponent is usually shown as a superscript to the right of the base. The exponentiation bn can be read as: b raised to the n-th power, b raised to the power of n, b raised by the exponent of n, or most briefly as b to the n. Some exponents have their own pronunciation. For example, b2 is usually read as b squared and b3 as b cubed. It is also often common to see bn represented as b^n. Exponentiation is used pervasively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography. 12 Taking b to the n power, as shown, is equivalent to multiplying b times itself an n number of times. Figure 1.7 Exponent Exponentiation is related to multiplication in that multiplication of b times n is equivalent to adding b together n number of times. Figure 1.6 Multiplication Background and Terminology The expression b2 = b ⋅ b is called the square of b because the area of a square with side-length b is b2 . The expression b3 = b ⋅ b ⋅ b is called the cube, because the volume of a cube with side-length b is b3 . So 32 is pronounced "three squared", and 23 is "two cubed." The exponent says how many copies of the base are multiplied together. For example, 35 = 3·3·3·3·3 = 243. The base 3 appears 5 times in the repeated multiplication, because the exponent is 5. Here, 3 is the base, 5 is the exponent, and 243 is the power or, more specifically, the fifth power of 3, 3 raised to the fifth power, or 3 to the power of 5. The word "raised" is usually omitted, and very often "power" as well, so 35 is typically pronounced "three to the fifth" or "three to the five." Integer Exponents The exponentiation operation with integer exponents requires only elementary algebra. Positive Integer Exponents Formally, powers with positive integer exponents may be defined by the initial condition b1 = b and the recurrence relation bn+1 = bn ⋅ b From the associativity of multiplication, it follows that for any positive integers m and n bm+n = bm ⋅ bn Arbitrary Integer Exponents For non-zero b and positive n, the recurrence relation from the previous subsection can be rewritten as bn = bn+1 b By defining this relation as valid for all integer n and nonzero b, it follows that b0 = b1 b = 1 b−1 = b0 b = 1 b and more generally, 13 b−n = 1 bn for any nonzero b and any nonnegative integer n (and indeed any integer n). The following observations may be made: • Any number raised by the exponent 1 is the number itself. • Any nonzero number raised by the exponent 0 is 1; one interpretation of these powers is as empty products. • These equations do not decide the value of 00 . • Raising 0 by a negative exponent would imply division by 0, so it is left undefined. The identity bm+n = bm ⋅ bn initially defined only for positive integers m and n, holds for arbitrary integers m and n, with the constraint that m and n must both be positive when b is zero. Source: algebra/exponents-scientific-notation-order-of-operations/integer- exponents/ CC-BY-SA Boundless is an openly licensed educational resource 14 Scientific Notation Scientific notation expresses a number as a· 10b , where a has one digit to the left of the decimal. KEY POINTS • Scientific notation is a way of writing numbers that are too big or too small to be conveniently written in decimal form. • In normalized scientific notation, the exponent b is chosen so that the absolute value of a remains at least one but less than ten (1 ≤ \|a\| < 10). Following these rules, 350 would always be written as 3.5× 102 . • Most calculators present very large and very small results in scientific notation. Because superscripted exponents like 107 cannot always be conveniently displayed, the letter E or e is often used to represent "times ten raised to the power of" (which would be written as "x 10b "). Standard Form to Scientific Form Very large numbers such as 43,000,000,000,000,000,000 (the number of different possible configurations of Rubik's cube) and very small numbers such as 0.000000000000000000000340 (the mass of the amino acid tryptophan) are extremely inconvenient to write and read. Such numbers can be expressed more conveniently by writing them as part of a power of 10. To see how this is done, let us start with a somewhat smaller number such as 2480. The last form in Figure 1.8 is called the scientific form of the number. There is one nonzero digit to the left of the decimal point and the absolute value of the exponent on 10 records the number of places the original decimal point was moved to the left. If instead we have a very small number, such as 0.00059, we instead move the decimal place to the right, as in the following: 0.00059 = 5.9 10000 = 5.9 104 = 5.9 ⋅ 10−4 15 In standard form, the number is written out as you are accustomed to, the ones digit to the farthest to the right (unless there is a decimal), then the tens digit to the left of the ones, and so on. In scientific notation, a number in standard notation with one nonzero digit to the left of the decimal is multiplied by ten to some power, as shown. Figure 1.8 Standard Form Versus Scientific Form There is one nonzero digit to the left of the decimal point and the absolute value of the exponent of 10 records the number of places the original decimal point was moved to the right. Writing a Number in Scientific Notation To write a number in scientific notation: • Move the decimal point so that there is one nonzero digit to its left. • Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved. Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right. A number written in scientific notation can be converted to standard form by reversing the process described above. Normalized Scientific Notation Any given number can be written in the form of a× 10b in many ways; for example, 350 can be written as 3.5× 102 or 35× 101 or 350× 100 . In normalized scientific notation, the exponent b is chosen so that the absolute value of a remains at least one but less than ten (1 ≤ \|a\| < 10). Following these rules, 350 would always be written as 3.5× 102 . This form allows easy comparison of two numbers of the same sign in a, as the exponent b gives the number's order of magnitude. In normalized notation, the exponent b is negative for a number with absolute value between 0 and 1 (e.g., negative one half is written as −5× 10−1 ). The 10 and exponent are usually omitted when the exponent is 0. Note that 0 cannot be written in normalized scientific notation since it cannot be expressed as a× 10b for any non-zero a. Normalized scientific form is the typical form of expression of large numbers for many fields, except during intermediate calculations or when an unnormalised form, such as engineering notation, is desired. Normalized scientific notation is often called exponential notation—although the latter term is more general and also applies when a is not restricted to the range 1 to 10 (as in engineering notation for instance) and to bases other than 10 (as in 315 × 220 ). E Notation Most calculators and many computer programs present very large and very small results in scientific notation. Because superscripted exponents like 107 cannot always be conveniently displayed, the letter E or e is often used to represent "times ten raised to the power of" (which would be written as "x 10b ") and is followed by the value of the exponent. Note that in this usage the character e is not related to the mathematical constant e or the exponential function ex (a confusion that is less likely with capital E), and though it stands for 16 exponent, the notation is usually referred to as (scientific) E notation or (scientific) e notation, rather than (scientific) exponential notation. The use of this notation is not encouraged by publications. Source: algebra/exponents-scientific-notation-order-of-operations/scientific- notation--2/ CC-BY-SA Boundless is an openly licensed educational resource Order of Operations Order of Operations is a way of evaluating expressions with more than one operation and it governs precedence in mathematical operations. KEY POINTS • In order to communicate using mathematical expressions there must be a logical and agreed upon order of operations so that each expression can be written unambiguously. • The order of operations, or precedence, used throughout mathematics, science, technology and many computer programming languages is expressed here: 1) terms inside parentheses or brackets 2) exponents and roots 3) multiplication and division 4) addition and subtraction. • Mnemonics are often used to help students remember the rules. In the U.S. the acronym PEMDAS is common. It stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. PEMDAS is often expanded to "Please Excuse My Dear Aunt Sally". Order of Operations is a way of evaluating expressions with more than one operation. These rules govern precedence in mathematical operations, such as Figure 1.9. 17 For example, when faced with 4 + 2 ⋅ 3, how do you proceed? There are two apparent options: 4 + 2 ⋅ 3 = (4 + 2) ⋅ 3 4 + 2 ⋅ 3 = 6 ⋅ 3 4 + 2 ⋅ 3 = 18 OR 4 + 2 ⋅ 3 = 4 + (2 ⋅ 3) 4 + 2 ⋅ 3 = 4 + 6 4 + 2 ⋅ 3 = 10 Which one is the correct order of operations in which to solve the problem? In order to communicate using mathematical expressions there must be a logical and agreed upon order of operations so that each expression can be written unambiguously. For the above example, all mathematicians agree the correct answer is 10. The key question is: what is the order upon which mathematicians have agreed? The order of operations, or precedence, used throughout mathematics, science, technology, and many computer programming languages is expressed here: Order of Operations 1. terms inside parentheses or brackets 2. exponents and roots 3. multiplication and division 4. addition and subtraction Theserules means that if a mathematical expression is preceded by one operator and followed by another, the operator higher on the list should be applied first. The commutative and associative laws of addition and multiplication allow terms to be added in any order and factors to be multiplied in any order, but mixed operations must obey the standard order of operations. It is helpful to treat division as multiplication by the reciprocal (multiplicative inverse) and subtraction as addition of the opposite (additive inverse). Thus 3/4 = 3 ÷ 4 = 3 ⋅ 1 4 ; in other words the quotient of 3 and 4 equals the product of 3 and ¼. Also 3 − 4 = 3 + (−4); in other words the difference of 3 and 4 equals the sum of positive three and negative four. With this understanding, The four most common mathematical operators. Figure 1.9 Common mathematical operators 18 think of 1 − 3 + 7 as the sum of 1, negative 3, and 7, and add in any order: (1 − 3) + 7 = − 2 + 7 = 5 and in reverse order (7 − 3) + 1 = 4 + 1 = 5. The important thing is to keep the negative sign with the 3. Mnemonics Mnemonics are often used to help students remember the rules, but the rules taught by the use of acronyms can be misleading. In the United States the acronym PEMDAS is common. It stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. PEMDAS is often expanded to "Please Excuse My Dear Aunt Sally" with the first letter of each word creating the acronym PEMDAS. These mnemonics may be misleading when written this way, especially if the user is not aware that multiplication and division are of equal precedence, as are addition and subtraction. Using any of the above rules in the order "addition first, subtraction afterward" would also give the wrong answer. 10 − 3 + 2 The correct answer is 9, and not 5, which we get when we add 3 and 2 first to get 5,and then subtract it from 10 to get the final answer of 5, which is best understood by thinking of the problem as the sum of positive ten, negative three, and positive two. 10 + (−3) + 2 An alternative way to write the mnemonic is: P E MD AS Or, simply as PEMA, where it is taught that multiplication and division inherently share the same precedence and that addition and subtraction inherently share the same precedence. This mnemonic makes the equivalence of multiplication and division and of addition and subtraction clear. Source: algebra/exponents-scientific-notation-order-of-operations/order-of- operations/ CC-BY-SA Boundless is an openly licensed educational resource 19 Polynomials: Introduction, Addition, and Subtration A polynomial is a finite expression containing constants and variables connected only through basic operations of algebra. KEY POINTS • A polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. • A polynomial is either zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a constant (called the coefficient of the term) and a finite number of variables (usually represented by letters) raised to whole number powers. • Polynomials can be added or subtracted using the associative law of addition (grouping all their terms together into a single sum), possibly followed by reordering, and combining of like terms. A polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, x2 − x /4 + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not. Why? Because its second term involves division by the variable x (4/x), and its third term contains an exponent that is not a non-negative integer (3/2). Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences. They are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science. A graph of a polynomial is shown in Figure 1.10. raised to whole number powers. The value of a variable's exponent is called the degree of that variable. The degree of the term is the sum of the degrees of the variables in that term, and the 21 Graph of a polynomial. Figure 1.10 Polynomial degree of a polynomial is the largest degree of any one term. If a variable is written without an exponent, you are to assume the degree of that variable is 1. A constant is a term with no variable or degree. The commutative law of addition can be used to rearrange terms into any preferred order. In polynomials with one variable, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". Terms that are similar, meaning they contain the same variables of the same degree, can be combined by adding the coefficients. Source: algebra/working-with-polynomials/polynomials-introduction- addition-and-subtration/ CC-BY-SA Boundless is an openly licensed educational resource Multiplication and Special Products To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial. KEY POINTS • To multiply a polynomial by a monomial, multiply every term of the polynomial by the monomial and then add the resulting products together. • To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial. • Three binomial products occur so frequently in algebra that we designate them as special binomial products. These special products can be shown as the squares of a binomial (a + b)2 and (a − b)2 and as the sum and difference of two terms: (a + b)(a − b). Multiplication Multiplying a polynomial by a monomial is a direct application of the distributive property. a(b + c) = ab + ac 22 The distributive property suggests the following rule: to multiply a polynomial by a monomial, multiply every term of the polynomial by the monomial and then add the resulting products together. An example is shown in Figure 1.11. To multiply a polynomial by a polynomial, we have, by the distributive property: (a + b)(c + d) = (a + b)c + (a + b)d = ac + bc + ad + bd For convenience, we will use the commutative property of addition to write this expression so that the first two terms contain a and the second two contain b: (a + b)(c + d) = ac + ad + bc + bd This method is commonly called the FOIL method. • F - First terms • O - Outer terms • I - Inner terms • L - Last terms Special Products Three binomial products occur so frequently in algebra that we designate them as special binomial products. These special products can be shown as the squares of a binomial (a + b)2 and (a − b)2 and as the sum and difference of two terms: (a + b)(a − b). There are two simple rules that allow us to easily expand (multiply out) these binomials. They are well worth memorizing, as they will save a lot of time in the future. To square a binomial: 1. Square the first term. 2. Take the product of the two terms and double it. 3. Square the last term. 4. Add the three results together. (a + b)2 = a2 + 2ab + b2 (a − b)2 = a2 − 2ab + b2 23 An example of multiplying a polynomial by a monomial. Figure 1.11 Polynomial Multiplication To expand the sum and difference of two terms: 1. Square the first term and square the second term. 2. Subtract the square of the second term from the square of the first term. (a + b)(a − b) = a2 − b2 These are just simplified rules of FOIL. If you do forget these specific rules, you can go ahead and use the FOIL method. Source: algebra/working-with-polynomials/multiplication-and-special- products/ CC-BY-SA Boundless is an openly licensed educational resource Dividing Polynomials Polynomial long division is a method for dividing a polynomial by another polynomial of the same or lower degree. KEY POINTS • Polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. • Polynomial long division can be done easily by hand, because it separates an otherwise complex division problem into smaller ones. • It is easiest to perform polynomial division when you have a term for each degree power, even if one is just a place holder. For example, if you have a polynomial that looks like: x3 − 12x2 + 0x − 42. Polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree. This method is a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones. 24 For example, find the quotient and the remainder of the division of x3 − 12x2 − 42,, the dividend, by x - 3, the divisor. The dividend is first rewritten as follows: x3 − 12x2 + 0x − 42 The quotient and remainder can then be determined as follows: • Divide the first term of the dividend by the highest term of the divisor (meaning the one with the highest power of x, which in this case is x): x3 ÷ x = x2 . • Multiply the divisor by the result just obtained (the first term of the eventual quotient): x2 ⋅ (x − 3) = x3 − 3x2 . • Subtract the product just obtained from the appropriate terms of the original dividend (being careful that subtracting something having a minus sign is equivalent to adding something having a plus sign): (x3 − 12x2 ) − (x3 − 3x2 ) = − 12x2 + 3x2 = − 9x2 . • Repeat the previous three steps. • Repeat step 4. This time, there is nothing to "pull down". The calculated polynomial is the quotient, and the number left over (−123) is the remainder: x3 − 12x2 − 42 = (x − 3)(x2 − 9x − 27) − 123 Another example of polynomial long division is shown in Figure 1. 12. Source: algebra/working-with-polynomials/dividing-polynomials/ CC-BY-SA Boundless is an openly licensed educational resource This image shows an example of a polynomial long division. Figure 1.12 Polynomial Long Division 25 Greatest Common Factor and Factoring by Grouping Factoring Trinomials of the Form: ax2 + bx + c; Perfect Squares Trinomials of the Form: ax2 + bx + c Where a is not Equal to 1 Special Factorizations and Binomials Solving Quadratic Equations By Factoring Section 4 Factoring 26 Greatest Common Factor and Factoring by Grouping Factoring by grouping divides the terms in a polynomial into groups, which can be factored using the greatest common factor. KEY POINTS • Factorization or factoring is the decomposition of an object, for example, a number or a polynomial, into a product of other objects, or factors, which when multiplied together give the original. • The greatest common factor of two polynomials is a polynomial, of the highest possible degree, that evenly divides each of the two original polynomials. • Factoring by grouping is done by placing the terms in the polynomial into two or more groups, where each group can be factored by a known method. The results of these factorizations can sometimes be combined to make an even more simplified expression. Factorization or factoring is the decomposition of an object, for example, a number or a polynomial, into a product of other objects, or factors, which when multiplied together give the original. As an example, the number 15 factors as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to "basic building blocks", such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an "expanded" polynomial, written as just a sum of terms. The relationship between the two processes can be seen in Figure 1.13. 27 A visual illustration of the polynomial x2 + cx + d = (x + a)(x + b) where a plus b equals c and a times b equals d. Figure 1.13 Factorization Greatest Common Factor The greatest common divisor (GCD), also known as the greatest common factor (GCF), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4. This notion can be extended to polynomials. In algebra, the greatest common divisor of two polynomials is a polynomial, of the highest possible degree, that evenly divides each of the two original polynomials. Factoring by Grouping A way to factor some polynomials is factoring by grouping. For those who like algorithms, "factoring by grouping" may be the best way to approach factoring a trinomial, as it takes the guess work out of the process. Factoring by grouping is done by placing the terms in the polynomial into two or more groups, where each group can be factored by a known method. The results of these factorizations can sometimes be combined to make an even more simplified expression. For example, to factor the polynomial 4x2 + 20x + 3yx + 15y Group similar terms, (4x2 + 20x) + (3yx + 15y) Factor out the greatest common factor, 4x(x + 5) + 3y(x + 5) Factor out binomial (x + 5)(4x + 3y) Source: algebra/factoring/greatest-common-factor-and-factoring-by- grouping/ CC-BY-SA Boundless is an openly licensed educational resource 28 Factoring Trinomials of the Form: ax2 + bx + c; Perfect Squares The polynomial ax2 + bx + c can be factored using a variety of methods, including trial and error. KEY POINTS • Some trinomials, known as perfect square trinomials, can be factored into two equal binomials. factor as the square of a binomial. To recognize them, look for whether (1) the first and last terms are perfect squares, and (2) the middle term is divisible by 2, and when halved, equals the product of the terms that when squared produce the first and last terms. Factoring Trinomials The polynomial ax2 + bx + c (like the one graphed below) can be factored using a variety of methods. One such method is trial and error (Figure 1.14). Ultimately, the trinomial should be factored in the form (px + q)(rx + s), where p, q, r, and s are constants, and x is a variable. Using trial and error, we can find values for each of the constants, using the FOIL method to determine whether the constants used produce the trinomial ax2 + bx + c. We know that the product of px and rx must equal a x2 . Additionally, the sum of products pxs and qrx must equal bx. Finally, the product of q and s must equal c. Perfect Squares Some trinomials, known as perfect square trinomials, can be factored into two equal binomials. For example: a2 + 2ab + b2 = (a + b)2 and a2 − 2ab + b2 = (a − b)2 Perfect square trinomials always factor as the square of a binomial. 29 Graph of a trinomial a x2 + bx + c . Figure 1.14 Polynomial To recognize a perfect square trinomial, look for the following features: 1. The first and last terms are perfect squares. 2. The middle term is divisible by 2. For example, x2 − 10x + 25 can be identified as a perfect square because x2 is the square of x, and 25 is the square of 5. The middle term (-10x) is divisible by 2 (equalling -5x). Given that the coefficient of x2 is 1, we know that the factored form will be (x + a)(x + b), where a and b are to-be-determined coefficients. We need xb+ax to equal -10x, and ab to equal 25. Filling in -5 for a and b, we find a plausible solution that reads (x-5) (x-5), or (x − 5)2 . This is a perfect square. Source: algebra/factoring/factoring-trinomials-of-the-form-ax-2-bx-c-perfect- squares/ CC-BY-SA Boundless is an openly licensed educational resource Trinomials of the Form: ax2 + bx + c Where a is not Equal to 1 Two methods for factoring polynomials ax2 + bx + c are the trial and error method and the collect and discard method. KEY POINTS • Two methods for factoring polynomials ax2 + bx + c are the trial and error method and the collect and discard method. • For the trial and error method, we look for some factors of the first and last terms. Our goal is to choose the proper combination of factors of the first and last terms that yield the middle term. • The collect and discard method requires less guessing than the trial and error method. Factoring Trinomials of the Form ax2 + bx + c We can easily factor trinomials of the form ax2 + bx + c (Figure 1. 15) by finding the factors of the constant c that add to the coefficient of the linear term b, as shown in the following example: 30 Factor ax2 + 4x + 21. The third term of the trinomial is −21. We seek two numbers whose product is −21 and sum is -4. Clearly, the required numbers are −7 and +3. 4x − 21 = (x − 7)(x + 3) The problem of factoring the polynomial , ax2 + bx + c, a ≠ 1, becomes more involved. Two methods for factoring these polynomials are the trial and error method and the collect and discard method. Each method produces the same result, and you should select the method you prefer. The trial and error method requires some educated guesses, while the collect and discard method requires less guessing. Trial and Error Method For the trial and error method, consider a product. Examining a trinomial, we look for some factors of the first and last terms. Our goal is to choose the proper combination of factors of the first and last terms that yield the middle term. Notice the middle term comes from the sum of the outer and inner products in the multiplication of the two binomials. This fact provides us a way to find the proper combination. Look for the combination that when multiplied and then added yields the middle term. Collect and Discard Method For the collect and discard method, consider the polynomial 6x2 + x − 12. We begin by identifying a and c. In this case, a = 6 and c = −12. We start out as we would with a=1. 6x2 + x − 12 → (6x + k)(6x + m) where k and m are constants. Now, compute : Find the factors of −72 that add to 1, the coefficient of x, the linear term. The factors are 9 and −8. Include these factors in the parentheses. 6x2 + x − 12 → (6x + 9)(6x − 8) But we have included too much. We must eliminate the surplus. 31 A graph of a trinomial a x2 + bx + c with a = 1. Figure 1.15 Polynomial Factor each parentheses. 6x2 + x − 12 → 3(2x + 3) ⋅ 2(3x − 4) Discard the factors that multiply to a = 6. In this case, 3 and 2. We are left with the proper factorization. 6x2 + x − 12 = (2x + 3)(3x − 4) Source: algebra/factoring/trinomials-of-the-form-ax-2-bx-c-where-a-is-not- equal-to-1/ CC-BY-SA Boundless is an openly licensed educational resource Special Factorizations and Binomials By simplifying the FOIL method, you can save a lot of time when factoring for the difference of two square and perfect square trinomials. KEY POINTS always factor as the square of a binomial. • To recognize a perfect square trinomial, look for: (1) The first and last terms are perfect squares: a2 + 2ab + b2 (2) The middle term is divisible by 2: a2 + 2ab + b2 (3)The middle term, once divided by 2, is a product of the square roots of the first and last terms: a2 + 2ab + b2 . The Difference of Two Squares Recall that when we multiplied together the two binomials (a + b) and (a − b), we obtained the product a2 − b2 : This is obtained by 32 using the FOIL method, and then adding like terms. Remember, FOIL means First, Outer, Inner, Last. (a + b)(a − b) = (a a) + (a * −b) + (b * a) + (b * −b) = a2 − ab + ab − b2 = a2 − b2 Since we know that (a + b)(a − b) = a2 − b2 , we need only turn the equation around to find the factorization form: a2 − b2 = (a + b)(a − b) Note: a perfect square is a term that is the square of another term. 4 is a perfect square of 2, 9 is a perfect square of 3, a2 is a perfect square of a The factorization form says that we can factor a2 − b2 , the difference of two squares, by finding the terms that produce the perfect squares and substituting these quantities into the factorization form. When using real numbers (as we are), there is no factored form for the sum of two squares. That is, using real numbers, a2 + b2 cannot be factored. Perfect Square Trinomials Recall the process of squaring a binomial, which is done using the FOIL method: (a + b)2 = a2 + 2ab + b2 (a − b)2 = a2 − 2ab + b2 Perfect square trinomials (Figure 1.16) always factor as the square of a binomial. To recognize a perfect square trinomial, look for the following features: 1. The first and last terms are perfect squares: a2 + 2ab + b2 2. The middle term is divisible by 2: a2 + 2ab + b2 3. The middle term, once divided by 2, is a product of the square roots of the first and last terms: a2 + 2ab + b2 In other words, factoring a perfect square trinomial amounts to finding the terms that, when squared, produce the first and last terms of the trinomial, and substituting into one of the formula a2 + 2ab + b2 = (a + b)2 a2 − 2ab + b2 = (a − b)2 A visual illustration of (a + b)2 = a2 + 2ab + b2 Figure 1.16 Perfect Square Trinomial 33 EXAMPLES Factor x2 − 16. This example is pretty straightforward. Let's look at the first term in the expression first: x2 , it is really easy to see that this is a perfect square of x. Now let's look at the second term: 16, this is a perfect square of 4 so now we have x _ 16. The last piece of the puzzle is to find the sign. We can see that in the original expression, that the sign before the 16 is negative. We can try the equation (x − 4)2 : (x − 4)(x − 4) = x2 − 8x + 16 This is obviously not the answer we are looking for, so let's try another approach. We know that we need the 16 to be negative, and the only way to make that happen is by:(x-4)(x+4). Before submitting anything, always check your answer. When you try this equation, you get: (x − 4)(x + 4) = x2 − 4x + 4x − 16 = x2 − 16. It works! As you do more examples, you will start to be able to recognize patterns without having to go through so many steps. Factor x2 + 6x + 9 In the text, we outlined three steps to factoring out a trinomial. Let's start with the first step: 1. The first and last terms are perfect squares: x2 and 9 are both perfect squares, of x and 3 respectively. Check! 2. The middle term must be divisible by 2:6x/2 = 3x. Check! 3. The middle term, once divided by 2, is a product of the square roots of the first and last terms. Let's take the roots of the first two terms that we found in step 1 and find their product. x times 3 = 3x. Now let's multiply this by 2: 2(3x) = 6x. Check! So now we put all this information together and.... (x + 3)2 . Don't forget to check this work! (x + 3)2 = (x + 3)(x + 3) = x2 + 3x + 3x + 9 = x2 + 6x + 9 Source: algebra/factoring/special-factorizations-and-binomials/ CC-BY-SA Boundless is an openly licensed educational resource 34 Solving Quadratic Equations By Factoring A quadratic eq. has two answers, both of which will correctly solve the equation; remember the FOIL method, and this will be much easier. KEY POINTS • A quadratic equation is a polynomial equation of the second degree. A general quadratic equation can be written in the form: ax2 + bx + c = 0. • It is a lot easier to solve a quadratic equation if you remember the FOIL method, and try and reverse this process. • When you plug either of your answers back into the original equation, they both must work. This might seem like a lot of work right now, but after trying a few examples, it will be a lot easier to quickly recognize patterns and do it much more quickly. The key is to practice, practice, practice. A quadratic equation is a polynomial equation of the second degree. A general quadratic equation can be written in theSome plots of quadratic functions are shown in Figure 1.17. The constants a, b, and c are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. The term "quadratic" comes from quadratus, which is the Latin word for "square". Quadratic equations can be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula. To solve quadratic equations by factoring, lets remember the FOIL property. FOIL is an acronym for the order in which to multiply out a function. It stands for First, Outer, Inner, Last. Before we jump into factoring out a quadratic equation, lets try to FOIL a function: (x − 3)(x − 4) = 1x2 − 4x − 3x + 12 = 1x2 − 7x + 12 = 0 Remembering this, lets think about how we need to go about factoring out a quadratic equation. Lets do this by using the same example: x2 − 7x + 12 = 0 1. The first step is to set up the format: ( )( ) = 0 2. Now lets look at how to fill in the variable slots in this format. To do this, we need to take the square root of the first term, 35 or find two terms that will multiply together to the first term of our quadratic. In our example, it is simple, the square root of x2 is x, so lets input this into our format: (x )(x ) = 0 3. Now, lets look at the constants to put in our equation. We need to have two numbers that (1) add up to the coefficient of our middle term and (2) multiple together to give us the same product of our last term. A good way to do this is make a list of all the numbers whose product is the last term, and make a note of what they add up to. In this example, our last term is 12, so lets make a list of all the numbers that multiply up to 12: 112; 26; 3* 4. We can easily see that the only option here that adds up to 7 is 3+4. So lets put this into our equation: (x 3)(x 4) = 0 4. The last thing to do is input the signs to our equation. A good thing to remember is that two negatives make a positive. So if our last term has a positive sign, then both of the signs in our factored equation are the same. If the last term is has a negative sign, the one of the equations will be negative, and one will be positive. Since we have a positive last term, we know that our equation is going to have the same signs in both terms. Since the middle term is negative, we know that both of our equations will have a negative sign. So now we can complete the factored equation: (x − 3)(x − 4) = 0 5. Now we really need to multiply this back out, and make sure it makes sense. ALWAYS CHECK YOUR WORK! When factored out, this returns the original equation, so we know it is correct. 6. Now, to solve the equation, set each term equal to zero: x-3=0 and x-4=0. So, x=3 and x=4 When you plug either of these terms back into the original equation, they both work. This might seem like a lot of work right now, but after trying a few examples, it will be a lot easier to quickly Plots of quadratic function a x2 + bx + c, varying each coefficient separately. Figure 1.17 Plots of Quadratic Equations 36 recognize patterns and do it much more quickly. The key is to practice, practice, practice. Source: algebra/factoring/solving-quadratic-equations-by-facotring/ CC-BY-SA Boundless is an openly licensed educational resource 37 Domain of a Rational Expression A rational expression cannot have a denominator of zero, so its domain is all numbers that do not cause the denominator to equal zero. KEY POINTS • A rational expression is the quotient of two polynomials. It can be expressed as. • A rational expression's domain is set such that the denominator cannot equal zero. Therefore, Q(x) ≠ 0. • To determine the domain of a rational expression, set the denominator equal to zero, then solve for x. All values of x except for those that satisfy Q(x)=0 are the domain of the expression. A rational expression is one which can be written as the ratio of two polynomial functions. Despite being called a rational expression, neither the coefficients of the polynomials nor the values taken by the function are necessarily rational numbers. In the case of one variable, x, an expression is called rational if and only if it can be written in the form P(x) Q(x) where P(x) and Q(x) are polynomial functions in x and Q(x) is not the zero polynomial (Q(x)=0). The domain of a rational expression of is the set of all points for which the denominator is not zero, where one assumes that the fraction is written in its lower degree terms, that is, P(x) and Q(x) have several factors of the positive degree. If the denominator of the equation becomes equal to zero, the Examples The rational expression x3 − 2x 2(x2 − 5) is not defined at x2=5, so it is not defined at x = ± 5 Therefore its domain is all numbers not equal to the square root of five or the negative square root of five. The rational expression 39 x2 − 2 x is not defined at x=0, again because we would have to divide by 0. Therefore, the domain of this expression is all numbers not equal to zero. For the expression x2 + 2 x2 + 1 has a domain that includes all rational numbers. However, if x is equal to the square root of negative one, an irrational number, then the quotient is equal to zero. Therefore, the square root of negative one is outside the domain of this expression. EXAMPLE What is the domains of x2 − 2 x ? One way to determine this is to look at it graphically, and we can see that the graph is discontinuous at x=0, indicating that the domain is all numbers other than x=0. This makes sense, because at x=0 we would have to divide by zero, which is undefined. Source: algebra/rational-expressions/domain-of-a-rational-expression/ CC-BY-SA Boundless is an openly licensed educational resource 40 To determine the domain of this function, we can graph it and look for where the function appears to go to infinity. Indeed, at x=0 the denominator will equal zero, and this is therefore outside of the domain of the function. Figure 1.18 (x2 − 2)/x Simplifying, Multiplying, and Dividing A rational expression can be treated like a fraction, and can be manipulated via multiplication and division. KEY POINTS • A rational expression is a quotient of two polynomials, of the form P(x)/Q(x). • Rational expressions can often be simplified by removing terms that can be factored out of the numerator and denominator. These can either be numbers or functions of x. • A rational expression can also be multiplied and divided, just like a normal fraction. When multiplying two rational expressions together, multiply the numerator of each together, then the denominator of each together. Sometimes, it is possible to simplify them after multiplying them together. Just like a fraction involving numbers, a fraction involving polynomials (a rational expression) can be simplified, multiplied, and divided. The rules for performing these operations often mirror the rules for simplifying, multiplying, and dividing fractions, though instead, we are now factoring out polynomial expressions. Simplifying a Rational Expression As a first example, the rational expression 3x3 x can be simplified by canceling out one factor of x in the numerator and denominator, which gives the expression 3x2 . The domain of the equation, however, does not include x, as this would cause division by 0 in the original equation. We can see that the latter form is a simplified version of the former graphically, as they appear the same (Figure 1.19). 41 Both 3x2 and 3x3 /x Figure 1.19 Graph to Illustrate Simplification A more complicated example, x2 + 5x + 6 2x2 + 5x + 2 must first be factored to provide the expression (x + 2)(x + 3) (2x + 1)(x + 2) , which, after canceling the common factor of (x+2) from both the numerator and denominator, gives the simplified expression x + 3 2x + 1 which is a simplified form of the expression shown above. Multiplying and Dividing Rational Expressions Just like fractions, rational expressions can be multiplied and divided. First, we will multiply by whole numbers, then we will multiply one rational expression by another. For example, the rational expression x2 + 3 2x − 3 can be multiplied by the fraction 2 3 to provide 2(x2 + 3) 3(2x − 3) , which can be multiplied through to give 2x2 + 6 6x − 9 . In this example, we multiplied the numerators together and the denominators together, but we did not multiply the numerator by the denominator or vice-versa. If we want to multiply two rational expressions together, the rules are the same, but the operations are typically somewhat more complicated. As an example, we will look at the expression x + 1 x − 1 and multiply it by the expression x + 2 x + 3 . The product of these two, initially, is (x + 1)(x + 2) (x − 1)(x + 3) , which can be written out as x2 + 3x + 2 x2 + 2x − 3 . Notice that this expression cannot be simplified further. Source: algebra/rational-expressions/simplifying-multiplying-and-dividing/ CC-BY-SA Boundless is an openly licensed educational resource 42 Adding and Subtracting With Like and Unlike Denominators Adding and subtracting rational expressions follows all of the same rules as adding and subtracting fractions. KEY POINTS • Always factor rational expressions before doing anything else. • When two rational expressions are to be added or subtracted, they must be multiplied by a constant [in form f(x)/f(x)] so that they both have the same denominator. • Once two rational expressions have the same denominator, the numerators can be added or subtracted together, leaving the denominator alone. Then simplify. Adding and subtracting fractions should be a familiar process, and we will use this concept as a lead-in to start discussing the addition and subtraction of rational expressions. 1 2 + 1 3 = 3 6 + 2 6 = 5 6 The key is finding the least common denominator: the smallest multiple of both denominators. Then you rewrite the two fractions with this denominator. Finally, you add the fractions by adding the numerators and leaving the denominator alone. But how do you find the least common denominator? Consider this problem: 5 12 + 7 30 = ? You could probably find the least common denominator if you played around with the numbers long enough. Here we will show you a systematic method for finding least common denominators—a method that works with rational expressions just as well as it does with numbers. We start, as usual, by factoring. For each of the denominators, we find all the prime factors, the prime numbers that multiply to give that number. 5 2 ⋅ 2 ⋅ 3 + 7 2 ⋅ 3 ⋅ 5 If you are not familiar with the concept of prime factors, it may take a few minutes to get used to. 2×2×3 is 12, broken into its prime factors: that is, it is the list of prime numbers that multiply to give 12. Similarly, the prime factors of 30 are 2×3×5. 43 Why does that help? Because 12=2×2×3, any number whose prime factors include two 2s and one 3 will be a multiple of 12. Similarly, any number whose prime factors include a 2, a 3, and a 5 will be a multiple of 30. See Figure 1.20 for further details. The least common denominator is the smallest number that meets both these criteria: it must have two 2s, one 3, and one 5. Hence, the least common denominator must be 2×2×3×5=60, and we can finish the problem like this. 5 2 ⋅ 2 ⋅ 3 + 7 2 ⋅ 3 ⋅ 5 = 5 2 ⋅ 2 ⋅ 3 ⋅ 5 5 + 7 2 ⋅ 3 ⋅ 5 ⋅ 2 2 = 25 60 + 14 60 = 39 60 = 13 20 This may look like a very strange way of solving problems that you have known how to solve since the third grade. However, you should spend a few minutes carefully following that solution, focusing on the question: why is 2×2×3×5 guaranteed to be the least common denominator? Because once you understand that, you have the key concept required to add and subtract rational expressions. Addition and Subtraction of Rational Expressions When applying this strategy to rational expressions, first look at the denominators of the two rational expressions and see if they are the same. If they are the same, then simply add or subtract the numerators from each other, leaving the denominator the same. If the two denominators are different, however, then a strategy similar to the one shown above is applicable. For example if you wanted to perform the subtraction 3 x2 + 12x + 36 − 4x x3 + 4x2 − 12x we begin rational expression problems of this type by factoring! The denominators then become 3 (x + 6)2 − 4x x(x + 6)(x − 2) The least common denominator must have two (x+6) s, one x, and one (x−2). We then rewrite both fractions with the common denominator, giving 3(x)(x − 2) (x + 6)2(x)(x − 2) − 4x(x + 6) x(x + 6)2(x − 2) Subtracting fractions is easy when you have a common denominator! It is best to leave the bottom alone, since it is factored. The top, however, consists of two separate factored pieces, Finding the prime factors of the denominators of two fractions enables us to find a common denominator. Figure 1.20 Prime Factors of Fractions 44 and will be simpler if we multiply them out so we can combine them. 3(x)(x − 2) − 4x(x + 6) x(x − 2)(x + 6)2 After multiplying out, we obtain 3x2 − 6x − (4x2 + 24x) x(x − 2)(x + 6)2 A common student mistake here is forgetting the parentheses. The entire second term is subtracted; without the parentheses, the 24x ends up being added. −x2 − 30x x(x − 2)(x + 6)2 Almost done! But finally, we note that we can factor the top again. If we factor out an x it will cancel with the x in the denominator. −x − 30 (x − 2)(x + 6)2 The problem is long, and the math is complicated. So after following all the steps, it is worth stepping back to realize that even this problem results simply from the two rules we started with. First, always factor rational expressions before doing anything else. Second, follow the regular processes for fractions: in this case, the procedure for subtracting fractions, which involves finding a common denominator. After that, you subtract the numerators while leaving the denominator alone, and then simplify. Source: algebra/rational-expressions/adding-and-subtracting-with-like-and- unlike-denominators/ CC-BY-SA Boundless is an openly licensed educational resource 45 Complex Rational Expressions A complex fraction is one where the numerator, denominator, or both are fractions. These fractions can contain variables, constants or both. KEY POINTS • A few examples of complex fractions, or complex rational expressions can be found below: 1 − 1 x 1 − 1 x2 . • Now, before we can solve these complex rational expressions, we first want to simplify them as much as possible. The best way to do this, is to get rid of the fractions in the numerator, denominator, or both if at all possible. The way this is done is by using simple algebraic techniques. • One of the techniques that can be used is called the Combine- Divide Method. (1)Combine the terms in the numerator. (2)Combine the terms in the denominator. (3)Divide the numerator by the denominator. We know from previous sections, that a simple fraction is in the form P Q , where Q does not equal zero. The reason for this is simple, if the denominator were 0, the fraction would not a defined, meaning it would not be a real number. A complex fraction is one in which the numerator, denominator, or both are fractions. These fractions can contain variables, constants or a mixture of both. A few examples of complex fractions, or complex rational expressions can be found below: 8 15 2 3 and 1 − 1 x 1 − 1 x2 . Now, before we can solve these complex rational expressions, we first want to simplify them as much as possible. The best way to do this, is to get rid of the fractions in the numerator, denominator, or both if at all possible. The way this is done is by using simple algebraic techniques. One of the techniques that can be used is called the Combine-Divide Method. 1. Combine the terms in the numerator. 2. Combine the terms in the denominator. 3. Divide the numerator by the denominator. 46 Lets try to apply this to the first example complex fraction we saw: 8 15 2 3 Since there are no terms to combine in either the numerator or denominator, lets go right to step 3. Divide the numerator by the denominator: 8 15 ÷ 2 3 . From previous sections, we know that dividing by a fraction is the same as multiplying by the inverse of that fraction, which would turn this expression into: 8 15 * 3 2 = 4 5 * 1 1 = 4 5 Source: algebra/rational-expressions/complex-rational-expressions/ CC-BY-SA Boundless is an openly licensed educational resource Solving Equations with Rational Expressions; Problems Involving Proportions Rational expressions, like proportions, are extremely useful applications of algebra, that can be solved using simple algebraic techniques. KEY POINTS • A rational equation means that you are setting two rational expressions equal to each other. Proportions are perfect examples of a rational expression. Even if they look different, they can be simplified down into the same expression: ( 2 4 ) = ( 1 2 ). • If you have a rational equation where the denominators are the same, then the numerators must be the same. This in turn suggests a strategy: find a common denominator, and then set the numerators equal using algebraic techniques. • Remember, all normal algebraic rules apply to solving rational equations. Such as, you still can not divide by 0. 47 A rational equation means that you are setting two rational expressions equal to each other. The goal is to solve for x; that is, find the value(s) that make the equation true. Suppose you are told that: x 5 = 3 5 If you think about it, the x in this equation has to be a 3. That is to say, if x=3 then this equation is true; for any other x value, this equation is false.If this is not so apparent to you, you can always solve it the old fashioned way, by working it out. Start by isolating the variable you are solving for: x = ( 3 5 )3 which simplifies down to x = 3 This leads us to a very general rule: If you have a rational equation where the denominators are the same, then the numerators must be the same. This in turn suggests a strategy: find a common denominator, and then set the numerators equal. For example, consider the rational equation 3 x2 + 12x + 36 = 4x x3 + 4x2 − 12x by factoring the denominators,we find that we must multiply the left side of the equation by x(x − 2) x(x − 2) and the right side of the equation by x + 6 x + 6 , giving—since the denominators are equal, we can now assume the numerators are equal, so we know that 3(x)(x − 2) = 4x(x + 6) or, multiplied out, that 3x2 − 6x = 4x2 + 24x What we're dealing with, in this case, is a quadratic equation. As always, move everything to one side, giving x2 + 30x = 0 and then factor. A common mistake in this kind of problem is to divide both sides by x; this loses one of the two solutions. x(x − 30) = 0 Two solutions to the quadratic equation. However, in this case, x=0 is not valid, since it was not in the domain of the original right-hand fraction. (Why?) So this problem actually has only one solution, x=– 30.This is shown in Figure 1.21. 48 As always, it is vital to remember what we have found here. We started with the equation 3(x)(x − 2) (x + 6)2(x)(x − 2) = 4x(x + 6) x(x + 6)2(x − 2) . We have concluded now that if you plug x=–30 into that equation, you will get a true equation (you can verify this on your calculator). For any other value, this equation will evaluate false. EXAMPLE When given the rational equation: ( a b ) = ( c d ) This can be solved by either finding a common denominator, or by setting it up like: ad = cb and then solving it algebraically. Source: algebra/rational-expressions/solving-equations-with-rational- expressions-problems-involving-proportions/ CC-BY-SA Boundless is an openly licensed educational resource To determine the solutions to the equation x(x-30)=0, we can graph it and look for where the dependent variable crosses the x-axis. We find this is true at 0 and 30. Figure 1.21 Graphical determination of the solutions of x(x-30)=0 49 Simplifying Expressions Both radicals and exponents can be simplified to a basic expression by understanding the rules of working with each. KEY POINTS • A simplified radical adheres by the following: There is no factor of the radicand that can be written as a power greater than or equal to the index, there are no fractions under the radical sign, and there are no radicals in the denominator. • Multiplication and division of exponents of the same base can be simplified using addition and subtraction. • Raising a power to a power is simplified by muliplication. In mathematics, the nth root of a number x is a number r that, when raised to the power of n, equals x: rn = x where n is the degree of the root. A root of degree 2 is called a square root (Figure 1.22) and a root of degree 3, a cube root. Roots of higher degrees are referred to using ordinal numbers, as in fourth root, twentieth root, etc. A radical expression is said to be in simplified form if: 1. There is no factor of the radicand that can be written as a power greater than or equal to the index. 2. There are no fractions under the radical sign. 3. There are no radicals in the denominator. For example, to write the radical expression 32 5 in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it: 32 5 = 16 ⋅ 2 5 = 4 2 5 51 The graph of the function, made up of half a parabola with a vertical directrix. Figure 1.22 Square root Next, there is a fraction under the radical sign, which we change as follows: 4 2 5 = 4 2 5 Finally, we remove the radical from the denominator as follows: 4 2 5 = 4 2 5 ⋅ 5 5 = 4 10 5 For simplifying exponents, just follow the rules of exponents and get down to the most basic expression. Multiplying Powers with the Same Base am ⋅ an = am+n am means that you have a factor of m times. If you add n more factors of a then you have n+m factors of a. Dividing Powers with the Same Base am an = am−n In the same way that am ⋅ an = am+n , because you are adding on factors of a, dividing is taking away factors of a. If you have n factors of a in the denominator, then you can cross out n factors from the numerator. If there were m factors in the numerator, now you have m-n factors in the numerator. Raising a Power to a Power (an )m = an⋅m If you think about an exponent as telling you that you have so many factors of the base, then (an )m means that you have factors m of an . So you have m groups of an and each one of those has n groups of a. So you have m groups of n groups of a. So you have n ⋅ m groups of a, or an⋅m . Products Raised to Powers (ab)n = an bn You can multiply numbers in any order you please. Instead of multiplying together n factors equal to ab, you could multiply all of the a s together, multiply all the b s together, then finish by multiplying an times bn . Source: algebra/radical-notation-and-exponents/simplifying-expressions/ CC-BY-SA Boundless is an openly licensed educational resource 52 Adding, Subtracting, and Multiplication Radicals and exponents have particular requirements for addition and subtraction while multiplication is carried out more freely. KEY POINTS • To add radicals, the radicand (the number that is under the radical) must be the same for each radical. • Subtraction follows the same rules as addition: the radicand must be the same. • Multiplication of radicals simply requires that we multiply the term under the radical signs. • When multiplying exponents, the bases must be the same then we just add together the exponents. am means that you have a factor of m times. If you add n more factors of a then you have n+m factors of a. Roots are the inverse operation for exponents. An expression with roots is called a radical expression. It's easy, although perhaps tedious, to compute exponents given a root. For instance 7777 = 4949 = 2401. So, we know the fourth root of 2401 is 7, and the square root of 2401 is 49. What is the third root of 2401? Finding the value for a particular root is difficult. This is because exponentiation is a different kind of function than addition, subtraction, multiplication, and division. Let's go through some basic mathematical operations with radicals and exponents. Addition To add radicals, the radicand (the number that is under the radical) must be the same for each radical, so, a generic equation will look like Figure 1.23. Figure 1.24 shows some examples of radical manipulations. Let's plug some numbers in place of the variables: 3 + 2 3 = 3 3 53 The generic equation for adding radicals Figure 1.23 Addition of Radicals Subtraction Subtraction follows the same rules as addition: a b − c b = (a − c) b For example: 3 3 − 2 3 = 3 Multiplication Multiplication of radicals simply requires that we multiply the variable under the radical signs. a ⋅ b = a ⋅ b When multiplying exponents, the bases must be the same then we just add together the exponents. am means that you have a factor of m times. If you add n more factors of a then you have n+m factors of a. am ⋅ an = am+n Some examples with real numbers: 3 ⋅ 6 = 18 This equation can actually be simplified further; we will go over simplification in another section. Let's throw in some variables to demonstrate that the concept carries over: 2x ⋅ 3x2 = 6x3 Here we incorporated what we have learned about radicals and exponents; multiply the terms that are under the radical, in the case of the exponents, which have a the same base, x, we add them together. 54 Examples of additions of radicals, subtractions of radicals, and combinations of the two. Figure 1.24 Examples of additions of radicals Source: algebra/radical-notation-and-exponents/adding-subtracting-and- multiplication/ CC-BY-SA Boundless is an openly licensed educational resource Solving Problems with Radicals Roots are written using a radical sign, and a number denoting which root to solve for. When none is given, it is an implied square root. KEY POINTS • Roots are usually written using the radical symbol, but can also be written by raising the number to a fraction. Then, the root is the inverse of the raised power. Like this: x = x 1 2. • To solve an equation with a radical: isolate the radical on one side of the equation, get rid of your radical, solve the remaining equation. • To eliminate a square root, square the radical, to eliminate a cubed root, cube the radical - don't forget to do the exact same thing to the other side of the equation! Roots are written using a radical sign. If there is no denotation, it is implied that you are finding the square root. Otherwise, a number will appear denoting which root to solve for. Any expression containing a radical is called a radical expression. The best way to solve an equation, is to start by simplifying it as much as possible. You want to start by getting rid of the radical. Do 55 Source: algebra/radical-notation-and-exponents/solving-problems-with- radicals/ CC-BY-SA Boundless is an openly licensed educational resource Rationalizing Denominators or Numerators Root rationalization is a process by which roots in the denominator of an irrational fraction are eliminated. KEY POINTS • Both the numerator and the denominator can be rationalized but it is more common to rationalize the denominator. • To rationalize the denominator, multiply both the numerator and denominator by the radical in the denominator. • You may not to be to rationalize both the numerator and the denominator. Obviously, with a graphing calculator, we can solve almost any expression, but it is always good to be able to do these things by hand before we start just plugging things in. Calculators are great, but if you plug something in wrong, you won't be able to recognize that the answer is incorrect if you aren't able to do it by hand as well. It is good to practice these so that you can start to recognize patterns and understand what it is you are doing as opposed to just trusting a device. 57 Make sure to use these properties when solving an equation. Figure 1.25 Properties of Equality A lot of times in mathematics, we are given terms in the form of fractions that have radicals in the denominator. When we are given expressions like this, before we start to solve it, it is easiest to write them in the most simplified way possible. You want to start by taking the radicals OUT of the denominator. This can be done using simple, basic, algebraic techniques. We know that what ever we do to one side of an algebraic equation, we must also do to the other side. This same principal can be applied to fractions. What ever we do to the numerator, we must also do to the denominator, and visa versa. As many things are, this is easier to show than the explain. Lets start with the following example: You are given this fraction, 10 a * a a = a * 10 a 2 = 10 * a a A more complicated example of denominator rationalization is shown in Figure 1.26. Source: algebra/radical-notation-and-exponents/rationalizing-denominators- or-numerators/ CC-BY-SA Boundless is an openly licensed educational resource 58 A more complicated example of denominator rationalization Figure 1.26 Rationalization of a Denominator Rational Exponents Exponents are shorthand for repeated multiplication and can be used to express an n-th root of a number: b is a number x such that xn = b. KEY POINTS • The number in larger font is called the base. The number in superscript (that is, the smaller number written above) is called the exponent. • If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to xn = b. This solution is called the principal n-th root of b. It is denoted n√b, where √ is the radical symbol; alternatively, it may be written b1/n . • A power of a positive real number b with a rational exponent m/n in lowest terms satisfies b m n = (bm ) 1 n = n bm . Exponents are a shorthand used for repeated multiplication. Remember that when you were first introduced to multiplication it was as a shorthand for repeated addition. For example, you learned that: 4 × 5 = 5 + 5 + 5 + 5. The expression "4 × " told us how many times we needed to add. Exponents are the same type of shorthand for multiplication. Exponents are written in superscript after a regular-sized number. For example: 23 = 2 x 2 x 2. The number in larger font is called the base. The number in superscript (that is, the smaller number written above) is called the exponent. The exponent tells us how many times the base is multiplied by itself. In this example, 2 is the base and 3 is the exponent.The expression 23 is read aloud as "2 raised to the third power", or simply "2 cubed". Here are some other examples: 6 × 6 = 62 (This would read aloud as "six times six is six raised to the second power" or more simply "six times six is six squared".) 7 × 7 × 7 × 7 = 74 (This would read aloud as "seven times seven times seven times seven equals seven raised to the fourth power". There are no alternate expression for raised to the fourth power. It is only the second and third powers that usually get abbreviated because they come up more often. When it is clear what is being talked about, people often drop the words "raised" and "power" and might simply say "seven to the fourth".) Rational Exponents A rational exponent is a rational number that can be used as another way to write roots. An n-th root of a number b is a number x such that xn = b. If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to xn = b. This solution is called the principal n-th root of b. It is denoted n√b, where √ is the 59 radical symbol; alternatively, it may be written b1/n . For example: 41/2 = 2, 81/3 = 2. When one speaks of the n-th root of a positive real number b, one usually means the principal n-th root. • If n is even, then xn = b has two real solutions; • If b is positive, which are the positive and negative nth roots. • The equation has no solution in real numbers if b is negative. • If n is odd, then xn = b has one real solution. • The solution is positive if b is positive and negative if b is negative. Rational powers m/n, where m/n is in lowest terms, are positive if m is even, negative for negative b if m and n are odd, and can be either sign if b is positive and n is even. (−27)1/3 = − 3, (−27)2/3 = 9, and 43/2 has two roots 8 and −8. Since there is no real number x such that x2 = − 1, the definition of bm/n when b is negative and n is even must use the imaginary unit i. A power of a positive real number b with a rational exponent m/n in lowest terms satisfies b m n = (bm ) 1 n = n bm where m is an integer and n is a positive integer. Examples of exponents graphed can be seen in this figure Figure 1. 27. 60 Graphs of y = bx for various bases b: base 10 (green), base e (red), base 2 (blue), and base ½ (cyan). Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x=1, the y- value equals the base because any number raised to the power of 1 is the number itself. Figure 1.27 Exponential Graph Source: algebra/radical-notation-and-exponents/rational-exponents/ CC-BY-SA Boundless is an openly licensed educational resource Complex Numbers We say that all numbers of the form a +bi, where a and b are any real numbers, is the set of complex numbers. KEY POINTS • x2 = − 1 results in a "number" that would not be a real number, referred to as an imaginary number. • When working with imaginary numbers, both -i and i have equal standing: −1 = ± i. • Complex numbers can be added, subtracted, multiplied, and divided (except by 0). Complex numbers are the extension of the real numbers, i.e., the number line, into a number plane. They allow us to turn the rules of plane geometry into arithmetic. Complex numbers have fundamental importance in describing the laws of the universe at the subatomic level, including the propagation of light and quantum mechanics. They also have practical uses in many fields, including signal processing and electrical engineering. Currently, one is able to solve many different kinds of equations for x, such as x + 7 = 12 or 2x = 4. In each of these cases the solution for is a real number. However, there is no real number x that satisfies the equation x2 = − 1, since the square of any real number is non- 61 negative. Conceptually, it would be nice to have some kind of number to be the solution of x2 = − 1. This "number" would not be a real number, however, and is referred to as an imaginary number. Then the real number system is extended to accommodate this special number. It turns out that there will be two imaginary solutions of the equation. One of them will be called i, following the normal rules for arithmetic, the other solution is -i. One may be inclined to say that −1 = i. However, that would be incorrect, because in words this says that "the square root of -1 is i", but there is no basis for preferring i over -i (or vice versa) as the square root of -1. Rather, the two square roots have equal standing. All numbers of the form a +bi, where a and b are any real numbers, are a set of complex numbers, and denoted as set C. The real numbers R may be considered to be the subset of complex numbers C = {a +bi} for which b = 0. Complex numbers can be added, subtracted, multiplied, and divided (except by 0). For a negative root like −4, split the number into two parts such that one part is −1 like −4 = 4 ⋅ −1 = 4 ⋅ −1 which leads to 2i. More examples of complex numbers can be seen in Figure 1.28. Source: algebra/radical-notation-and-exponents/complex-numbers/ CC-BY-SA Boundless is an openly licensed educational resource Examples of Complex Numbers Figure 1.28 Complex Numbers 62 Radical Functions An expression with roots is called a radical function, there are many kinds of roots, square root and cube root being the most common. KEY POINTS • Roots are the inverse operation for exponents. If rn = x. • If the square root of a number is taken, the result is a number which when squared gives the first number. • The cube root is the number which, when cubed, or multiplied by itself and then multiplied by itself again, gives back the original number. • If a root of a whole number is squared root, which is not itself the square of a rational number, the answer will have an infinite number of decimal places.. Such a number is described as irrational. Roots are the inverse operation for exponents. An expression with roots is called a radical expression. It's easy, although perhaps tedious, to compute exponents given a root. For instance 7 ⋅ 7 ⋅ 7 ⋅ 7 = 49 ⋅ 49 = 2401. If fourth root of 2401 is 7, and the square root of 2401 is 49, then what is the third root of 2401? Finding the value for a particular root is difficult. This is because exponentiation is a different kind of function than addition, subtraction, multiplication, and division. When graphing functions, expressions that use exponentiation use curves instead of lines. Using algebra will show that not all of these expressions are functions and that knowing when an expression is a relation or a function allows certain types of assumptions to be made. These assumptions can be used to build mental models for topics that would otherwise be impossible to understand. For now, deal with roots by turning them back into exponents. If a root is defined as the nth root of X, it is represented as n x = r. Get rid of the root by raising the answer to the nth power, i.e. rn = x. Square root If the square root of a number is taken, the result is a number which when squared gives the first number. This can be written symbolically as: x = y if y2 = x. In the series of real numbers y2 ≥ 0, regardless of the value of y. As such, when x < 0 then x cannot be defined. Such examples of square roots can be seen in Figure 1.29. 63 Cube roots Roots do not have to be square. The cube root of a number ( 3 ) can also be taken. The cube root is the number which, when cubed, or multiplied by itself and then multiplied by itself again, gives back the original number. For example, the cube root of 8 is 2 because 2 ⋅ 2 ⋅ 2 = 8, or 3 8 = 2. Other roots There are an infinite number of possible roots all in the form of n a which corresponds to a 1 n, when expressed using exponents. If n a = b then bn = a. The only exception is 0. 0 a is undefined, as it corresponds to a 1 0, resulting in a division by zero. Even if attempting to discover the 0th root of 1, no progress will be made, as practically any number to the power of zero equals 1, leaving only an undefined result. Irrational numbers If a root of a whole number is squared root, which is not itself the square of a rational number, the answer will have an infinite number of decimal places. Such a number is described as irrational and is defined as a number which cannot be written as a rational number: a b , where a and b are integers. However, using a calculator can approximate the square root of a non-square number: 3 = 1.73205080757 The result of taking the square root is written with the approximately equal sign because the result is an irrational value which cannot be written in decimal notation exactly. Writing the square root of 3 or any other non-square number as 3 is the simplest way to represent the exact value. Irrational numbers also appear when attempting to take cube roots or other roots. However, they are not restricted to roots, and may also appear in other mathematical constants (e.g. π, e, φ, etc.). Source: algebra/radical-notation-and-exponents/radical-functions/ CC-BY-SA Boundless is an openly licensed educational resource Examples of square roots Figure 1.29 Square Root 64 Linear and Quadratic Equations Equations often express relationships between given quantities; two types of equations are linear and quadratic. KEY POINTS • A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. Linear equations can have one or more variables. • Using the laws of elementary algebra, linear equations can be rewritten into several different forms, such as general, slope- intercept, and point-slope form. • A quadratic equation is a univariate polynomial equation of the second degree. The constants a, b, and c are respectively called the quadratic coefficient, the linear coefficient, and the constant term (or free term). Equations often express relationships between given quantities (the knowns) and quantities yet to be determined (the unknowns). By convention, unknowns are denoted by letters at the end of the alphabet, x, y, z, w, …, while knowns are denoted by letters at the beginning, a, b, c, d, ….simultaneous equations, or system of equations, multiple equations are given with multiple unknowns. A solution to the system is an assignment of values to all the unknowns so that all of the equations are true. Two kinds of equations are linear and quadratic. Linear Equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. Linear equations can have one or more variables. Linear equations do not include exponents. A common form of a linear equation in the two variables x and y is: y = mx + b where m and b designate constants. The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane. In this particular equation, the constant m determines the slope or gradient of that line, and the constant term b determines the point at which the line crosses the y-axis, otherwise known as the y-intercept. Since terms of linear equations cannot contain products of distinct or equal variables, nor any power (other than 1) or other function of a variable, equations 66 involving terms such as xy, x2 , y1/3 , and sin(x) are nonlinear. An example of a graphed linear equation is presented in Figure 1.30. Using the laws of elementary algebra, linear equations can be rewritten into several different forms. A few of these forms are below. These equations are often referred to as the "equations of the straight line." In what follows, x, y, t, and θ are variables, m is the slope, and b is the y-intercept. General (or Standard) Form Ax + By = C where A and B are both not equal to zero. The equation is usually written so that A ≥ 0, by convention. If A is nonzero, then the x- intercept, or the x-coordinate of the point where the graph crosses the x-axis (where y is zero), is C/A. If B is nonzero, then the y- intercept, or the y-coordinate of the point where the graph crosses the y-axis (where x is zero), is C/B, and the slope of the line is −A/B. Slope–Intercept Form y = mx + b This can be seen by letting x = 0, which immediately gives y = b. Vertical lines, having undefined slopes, cannot be represented by this form. Point–Slope Form y − y1 = m(x − x1) where (x1, y1) is any point on the line. The point-slope form expresses the fact that the difference in the y-coordinate between two points on a line (that is, y − y1) is proportional to the difference in the x-coordinate (that is, x − x1). Quadratic Equations A quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in theThe constants a, b, and c are respectively called the quadratic 67 Graph sample of linear equations. Figure 1.30 Linear Function Graph coefficient, the linear coefficient, and the constant term (or free term). The term "quadratic" comes from quadratus, which is Latin for "square." Quadratic equations can be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula. Examples of graphed quadratic equations can be seen in Figure 1.31. A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real. Having ax2 + bx + c = 0 the roots are given by the quadratic formula x = −b ± b2 − 4ac 2a where the symbol "±" indicates that both x = −b + b2 − 4ac 2a x = −b − b2 − 4ac 2a are solutions of the quadratic equation. Source: algebra/basics-of-equation-solving/linear-and-quadratic-equations/ CC-BY-SA Boundless is an openly licensed educational resource 68 Plots of the real-valued quadratic function a x2 + bx + c, varying each coefficient separately. Figure 1.31 Plots of Quadratic Equations Fractions A fraction represents a part of a whole and consists of an integer numerator and non-zero integer denominator. KEY POINTS • Addition and subtraction require like quantities, a common denominator, in order to add or subtract fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. • Multiplication requires multiplying the numerators and then the denominators. A shortcut is to use cancellation by reducing to the lowest number prior to multiplication. • To divide a fraction by a whole number, either divide the numerator by the number, if it goes evenly into the numerator, or multiply the denominator by the number. A fraction represents a part of a whole or, more generally, any number of equal parts. A common or vulgar fraction, such as 1/2, 8/5, 3/4, consists of an integer numerator and a non-zero integer denominator—the numerator representing a number of equal parts and the denominator indicating how many of those parts make up a whole. An example can be seen in Figure 1.32, where a cake is divided into quarters. The set of all numbers which can be expressed in the form a/b, where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. The test for whether a number is a rational is that it can be written in that form (i.e., as a common fraction). However, the word fraction is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions, and expressions that contain irrational numbers, such as √2/2 and π/4. Addition The first rule of addition is that only like quantities can be added; for example, various quantities of quarters. Unlike quantities, such as adding thirds to quarters, must first be converted to like quantities as described below: Imagine a pocket containing two quarters, and another pocket containing three quarters; in total, 69 A cake with one fourth removed. The remaining three fourths are shown. Dotted lines indicate where the cake may be cut in order to divide it into equal parts. Each fourth of the cake is denoted by the fraction ¼. Figure 1.32 Quarters of a Cake there are five quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows: 2 4 + 3 4 = 5 4 = 1 1 4 To add fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easy to work out the chosen type of fraction to which to convert. Simply multiply together the two denominators, bottom number, of each fraction. For adding quarters to thirds, both types of fraction are converted to twelfths, thus: 1 3 + 1 4 = 1 * 4 3 * 4 + 1 * 3 4 * 3 = 4 12 + 3 12 = 7 12 This method can be expressed algebraically: a b + c d = ad + cd bd This method always works, but sometimes there is a smaller denominator, or a least common denominator, that can be used. For example, to add 3 4 and 5 12 the denominator 48 can be used (the product of 4 and 12), but the smaller denominator 12 may also be used, being the least common multiple of 4 and 12. Subtraction The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. For instance, 2 3 − 1 2 = 2 * 2 3 * 2 − 1 * 3 2 * 3 = 4 6 − 3 6 = 1 6 Multiplication To multiply fractions, multiply the numerators and multiply the denominators. Thus: 2 3 × 3 4 = 6 12 A short cut for multiplying fractions is called "cancellation". In effect, one reduces the answer to lowest terms during multiplication. For example: 2 3 × 3 4 = 1 1 × 1 2 = 1 2 A two is a common factor in both the numerator of the left fraction and the denominator of the right and is divided out of both. Three is 70 a common factor of the left denominator and right numerator and is divided out of both. To multiply a fraction by a whole number, place the whole number over one and multiply.This method works because the fraction 6/1 means six equal parts, each one of which is a whole. When multiplying mixed numbers, it's best to convert the mixed number into an improper fraction. Division To divide a fraction by a whole number, either divide the numerator by the number, if it goes evenly into the numerator, or multiply the denominator by the number. For example, 10 3 ÷ 5 = 2 3 which also equals: 10 3 * 5 = 10 15 = 2 3 To divide a number by a fraction, multiply that number by the reciprocal of that fraction. Thus, 1 2 ÷ 3 4 = 1 2 × 4 3 = 4 6 = 2 3 . Source: algebra/basics-of-equation-solving/fractions/ CC-BY-SA Boundless is an openly licensed educational resource 71 Simplifying Algebraic Expressions Algebraic expressions may be simplified, based on the basic properties of arithmetic operations. KEY POINTS • Added terms are simplified using coefficients: 2x2 + 3ab − x2 + ab = x2 + 4ab. • Multiplied terms are simplified using exponents: x × x × x = x3 . • Using distributive law, brackets can be multiplied out: 6x5 + 3x2 = 3x2 (2x3 + 1). Algebraic notation follows certain rules and conventions, and has its own terminology. For example, look at Figure 1.33, as you can see, the expression consists of an exponent, coefficients, terms, operators, constants and variables. A coefficient is a numerical value which multiplies a variable (the operator is omitted). A term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators. Letters represent variables and constants. By convention, letters at the beginning of the alphabet are typically used to represent constants, and those toward the end of the alphabet are used to represent variables. They are usually written in italics. Algebraic operations work in the same way as arithmetic operations, such as addition, subtraction, multiplication, division and exponentiation and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a coefficient is used. Usually terms with the highest power (exponent), are written on the left. When a coefficient is one, it is usually omitted. Likewise when the exponent (power) is one. When the exponent is zero, the result is always 1. However, being undefined, should not appear in an expression, and care should be 72 1 – Exponent (power), 2 – Coefficient, 3 – term, 4 – operator, 5 – constant, x,y – variables Figure 1.33 Algebraic Notation taken in simplifying expressions in which variables may appear in exponents. Now that we understand each of the components of the expression, let's look at how we simplify them. Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation). Let's look at each individually: • Added terms are simplified using coefficients. For example, x + x + x can be simplified as 3x (where 3 is the coefficient). • Multiplied terms are simplified using exponents. For example, x × x × x is represented as x3 . • Like terms are added together. For example, 2x2 + 3ab − x2 + ab is written as: x2 + 4ab, because the terms containing are added together, and, the terms containing are added together. • Brackets can be "multiplied out", using distributivity. For example, x(2x + 3) can be written as (x × 2x) + (x × 3) which can be written as: 2x2 + 3x • Expressions can be factored. For example, 6x5 + 3x2 , by dividing both terms by 3x2 can be written as: 3x2 (2x3 + 1). Source: algebra/basics-of-equation-solving/simplifying-algebraic-expressions/ CC-BY-SA Boundless is an openly licensed educational resource 73 Additive and Multiplicative Properties of Equality The additive and multiplicative properties of equalities are common ways used to solve equations. KEY POINTS • Equations often express relationships between given quantities, the knowns, and quantities yet to be determined, the unknowns. • The Additive Property of Equality states that any real number can be added to both sides. • The Multiplicative Property of Equality states that any real number can be multiplied to both sides. An equation, in a mathematical context, is generally understood to mean a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign (=), for example x + 3 = 5 asserts that x+3 is equal to 5. The = symbol was invented by Robert Recorde (1510–1558), who considered that nothing could be more equal than parallel straight lines with the same length. Equations often express relationships between given quantities, the knowns, and quantities yet to be determined, the unknowns (Figure 1.34). By convention, unknowns are denoted by letters at the end of the alphabet, x, y, z, w, …, while knowns are denoted by letters at the beginning, a, b, c, d, …. The process of expressing the unknowns in terms of the knowns is called solving the equation. In an equation Animation illustrating Pythagoras' rule for a right-angle triangle, which shows the algebraic relationship between the triangle's hypotenuse, and the other two sides. Figure 1.34 Equation Solving 74 If an equation in algebra is known to be true, the following operations may be used to produce another true equation: • Any real number can be added to both sides (the Additive Property of Equality). • Any real number can be subtracted from both sides. • Any real number can be multiplied to both sides (the Multiplicative Property of Equality). • Any non-zero real number can divide both sides. The Additive Property If a = b then a + c = b + c The Multiplicative Property If a = b then a ⋅ c = b ⋅ c Source: algebra/basics-of-equation-solving/additive-and-multiplicative- properties-of-equality/ CC-BY-SA Boundless is an openly licensed educational resource 75 Percent and Mixture Problem Solving Percentages are used to express how large/small one quantity is relative to another quantity. KEY POINTS • Mixture problems may have a mix of two or more things requiring you to find the percentage of one of those things. • The first quantity of a percentage usually represents a part of, or a change in, the second quantity, which should be greater than zero.The percent value is computed by multiplying the numeric value of the ratio by 100. • Due to inconsistent usage, it is not always clear from the context what a percentage is relative to. When speaking of a "10% rise" or a "10% fall" in a quantity, the usual interpretation is that this is relative to the initial value of that quantity. Percentages In mathematics, a percentage is a number or ratio as a fraction of 100. It is often denoted using the percent sign (Figure 1.35) or the abbreviation "pct." For example, 45% (read as "forty-five percent") is equal to 45/100, or 0.45. A related system which expresses a number as a fraction of 1,000 uses the terms "per mil" and "millage." Percentages are used to express how large/small one quantity is relative to another quantity. The first quantity usually represents a part of or a change in the second quantity, which should be greater than zero. For example, an increase of $0.15 on a price of $2.50 is an increase by a fraction of 0.15/2.50 = 0.06. Expressed as a percentage, this is therefore a 6% increase. Although percentages are usually used to express numbers between zero and one, any ratio can be expressed as a percentage. For instance, 111% is 1.11 and −0.35% is −0.0035. Although this is technically inaccurate as per the definition of percent, an alternative wording in terms of a change in an observed value is "an increase/decrease by a factor of..." Mixture Problems Mixture problems may have a mix of two or more things requiring you to find the percent of one of those things. The percent value is computed by multiplying the numeric value of the ratio by 100. For example, to find the percentage of 50 apples out of 1,250 apples, first compute the ratio 50/1250 =.04, and then multiply by 100 to obtain 4%. The percent value can also be found by multiplying first, The sign for percent (which means of 100). Figure 1.35 Percent Sign 76 so in this example the 50 would be multiplied by 100 to give 5,000, and this result would be divided by 1,250 to give 4%. To calculate a percentage of a percentage, convert both percentages to fractions of 100, or to decimals, and multiply them. For example, 50% of 40% is:(50/100) × (40/100) = 0.50 × 0.40 = 0.20 = 20/100 = 20%. It is not correct to divide by 100 and use the percent sign at the same time. (E.g. 25% = 25/100 = 0.25, not 25% / 100, which actually is (25/100) / 100 = 0.0025. A term such as (100/100)% would also be incorrect, as this would be read as (1) percent even if the intent was to say 100%.) The easy way to calculate addition in percentage (discount 10% + 5%): For example, if a department store has a "10% + 5% discount," the total discount is not 15%. However, whenever we talk about a percentage, it is important to specify what it is relative to, i.e. what is the total that corresponds to 100%. The following problem illustrates this point: In a certain college 60% of all students are female, and 10% of all students are computer science majors. If 5% of female students are computer science majors, what percentage of computer science majors are female? We are asked to compute the ratio of female computer science majors to all computer science majors. We know that 60% of all students are female, and among these 5% are computer science majors, so we conclude that (60/100) × (5/100) = 3/100 or 3% of all students are female computer science majors. Dividing this by the 10% of all students that are computer science majors, we arrive at the answer: 3%/10% = 30/100 or 30% of all computer science majors are female. This example is closely related to the concept of conditional probability. Due to inconsistent usage, it is not always clear what a percentage is relative to. When speaking of a "10% rise" or a "10% fall" in a quantity, this is usually relative to the initial value of that quantity. For example, if an item is initially priced at $200 and the price rises 10% (an increase of $20), the new price will be $220. Note that this final price is 110% of the initial price (100% + 10% = 110%). Source: algebra/basics-of-equation-solving/percent-and-mixture-problem- solving/ CC-BY-SA Boundless is an openly licensed educational resource 77 The Cartesian System Equations and their Solutions Graphing Equations Distance Formula and Midpoints of Segments Circles Section 1 Introduction to Graphing 79 The Cartesian System The Cartesian coordinate system is used to specify points on a graph by showing their absolute distances from two axes. KEY POINTS • The Cartesian coordinate system is a 2-dimensional plane with a horizontal axis, known as the x-axis, and a vertical axis, known as the y-axis. • A Cartesian coordinate system specifies each point uniquely in a plane with a pair of numerical coordinates, which are the signed distances from the point to the two axes. • Points are represented by an ordered pair (x, y), where the x- coordinate is the point's distance from the y axis, and the y- coordinate is the distance from the x-axis. Named for "the father of analytic geometry," 17th-century French mathematician René Descartes, the Cartesian coordinate system is a 2-dimensional plane with a horizontal axis and a vertical axis used for graphing. Both axes extend to infinity, but in graphs only segments of them are drawn, and sometimes arrows are used to indicate the infinite length. The horizontal axis is known as the x- axis and the vertical axis is known as the y-axis. The point where the axes intersect is known as the origin. Cartesian coordinates are the foundation of analytic geometry and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more. A familiar example is the graph of a function, which you will learn in subsequent chapters. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering, and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design, and other geometry- related data processing. Coordinates of a Point A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to the two axes. Each point can be represented by an ordered pair (x, y), where the x-coordinate is the point's distance from the y axis, and the y-coordinate is the distance from the x-axis. The origin where the two axes meet is thus (0, 0). When the coordinates are integer numbers, they can be easily found on a graph by looking at the numbers on the axes. The ordered pair (3, 5) represents a point three units to the right of the origin and five units upwards from it. On the x-axis, numbers increase toward the right and decrease toward the left. On the y- 80 axis, numbers increase going upward and decrease going downward. The intersection of the two axes splits the coordinate system into four quadrants, typically labelled I, II, III, and IV, starting from the upper right and continuing counter- clockwise, as seen in Figure 2.1. Any point in the first quadrant has both positive x and y coordinates. Points in the second quadrant have negative x and positive y coordinates. The third quadrant has both negative x and y coordinates, and the fourth quadrant contains points with positive x and negative y coordinates. Thus we know that the point (3, 5) is in the first quadrant. Source: models/introduction-to-graphing/the-cartesian-system/ CC-BY-SA Boundless is an openly licensed educational resource 81 The four quadrants of a Cartesian coordinate system. The arrows on the axes indicate that they extend forever in their respective directions (i.e. infinitely). Figure 2.1 2D Cartesian Coordinates Equations and their Solutions Solutions to equations can be plotted on graphs for an easy visualization of how the function behaves. KEY POINTS • To solve an equation is to find what values (numbers, functions, sets, etc.) fulfill a condition stated in the form of an equation (two expressions related by equality). • In mathematics, the graph of a function f is the collection of all ordered pairs (x, f(x)). • Once a function has been graphed, solutions to any particular x or y value are readily available by looking at the graph. In mathematics, to solve an equation is to find what values (numbers, functions, sets, etc.) fulfill a condition stated in the form of an equation (two expressions related by equality). These expressions contain one or more unknowns, which are free variables for which values are sought that cause the condition to be fulfilled. To be precise, what is sought are often not necessarily actual values, but, more in general, mathematical expressions, or a visualization of the solution. Using Graphs In mathematics, the graph of a function f is the collection of all ordered pairs (x, f(x)). In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is sometimes referred to as curve sketching. Say the graph of a function is {(1, a), (2, d), (3, c)}. You can then plot these points on a Cartesian plot as a graph. Now you know if you ever get the value x=1, you can look at the graph and see that the value is a, x=2 and you get d. What if x = 4? In this situation, there is no value on the graph for which x = 4, and so there is no solution at this point. The graph of the cubic polynomial on the real line f(x) = x3 − 9x is {(x, x3-9x) : x is a real number). So how does this help us? With this information we can plot specific points by choosing an x value, finding the corresponding y value, and graphing it. The set of 82 This is the graph of the function f (x) = x3 − 9x Figure 2.2 Graph of a Function infinite points is the curve that defines the function. If this set is plotted on a Cartesian plane, the result is Figure 2.2. Once we have a graph, we can then look at the graph to see how the function changes as x changes, as well as choose any x value and find its y value along the graph. Source: models/introduction-to-graphing/equations-and-their-solutions/ CC-BY-SA Boundless is an openly licensed educational resource Graphing Equations Equations can lead to very interesting graphs, such as circles, ellipses, parabolas, hyperbolas, sine waves, and much more! KEY POINTS • Graphs are important tools to visualizing equations. • To graph an equation, choose a value for either x or y, solve for the variable you didn't choose to get a point, plot the point, and repeat until you have enough points to draw the graph by connecting the points. • Graphs can be closed, like a circle, or open, like a parabola. Now that we know what equations are, how do we go about visualizing them? With graphs! For an equation with two variables, x and y, we need a graph with two axes, an x axis and a y axis. The x-axis will be a horizontal line, the y-axis a vertical line, and where the two cross is the origin. Let's look at the equation y = 2x. To begin graphing this equation, all you need to do is choose an x or y value, plug it in, solve for the unknown variable, and plot the point on the graph. For instance, if you choose x = 1, then y = 2, and we plot the point (1, 2). If x = 0, y = 0, and we can plot this point as well (0, 0). We could keep doing this with a lot of points, 83 and if we used an infinite amount of points, we would get a line that passes through all the points. Obviously we can't plot an infinite amount of points, but if we plot just enough to see the overall shape of the graph of the equation, then we can "connect the dots" and visualize the equation. A line is pretty boring, so let's look at some other equations. We won't go into detail as to why these equations are what they are, as that will be covered in subsequent sections. First let's look at the equation x2 + y2 = 100. Before we tell you what this type of equation actually is, let's choose some points and plot them. Easiest would be to choose x = 0, thus we're left with y2 = 100, and y2 = 100, and finally y = ± 10 So we plot (0, 10) and (0, -10). Now let's choose another easy point, y = 0. By the same calculation as above we get the ordered pairs (10, 0) and (-10, 0). Plot these as well. We still don't have enough points to really see what's going on, so let's choose some more. How about x = 6. Solve for y: 36 + y2 = 100, then y2 = 64, and finally y = ± 8. So that's two new points (6, 8) and (6, -8). We get similar results with x = -6, to get (-6, 8) and (-6, -8). You may begin to see where this is going. With a few more points you can show that (8, 6), (8, -6), (-8, 6) and (-8, -6) are also all part of the equation. Now you can begin seeing that what we're drawing is a circle with radius 10 (Figure 2. 3). Let's try another example. This time let's use the equation y = − (x2 ) + 9. Again, let's plug in some numbers and begin plotting points. Let's try x = 0, we then get y=9, so let's plot (0, 9). Now x = 1, and you get 8. The same goes with x = -1, so plot the two points (1, 8) and (-1, 8). Next try x = ± 2, and you get the two ordered pairs (2, 5) and (-2, 5). A few more calculations gets us a few more ordered pairs to plot (3, 0), (-3, 0), (4, -7), (-4, -7). Connect these points with the best curve you can, and you'll discover you've drawn a parabola (Figure 2.4). 84 This is a circle that's graphed from the equation x2 + y2 = 100 Figure 2.3 Circle As you can see from the two examples, some equations are closed, e.g. the circle, and some are open, e.g. the parabola. Source: models/introduction-to-graphing/graphing-equations/ CC-BY-SA Boundless is an openly licensed educational resource Distance Formula and Midpoints of Segments The distance formula and the midpoint formula give us the tools to find important information about two points. KEY POINTS • The Pythagorean Theorem is a very powerful theorem relating the lengths of the three sides of a triangle. This theorem tells us that if c is the hypotenuse and a and b are the other two sides, c2 = a2 + b2 . • Using the Pythagorean Theorem and two points (x1, y1) and (x2, y2), we can derive the distance formula: d = (x2 − x1)2 + (y2 − y1)2 . • The midpoint of a line segment given by two points (x1, y1) and (x2, y2) is ( x1 + x2 2 , y1 + y2 2 ). The Distance Formula Let's say you have two points, for example (2, 4) and (5, 8) and want to know the distance between these two points. You could always take a ruler and attempt to find the distance based based on the units of the graph, but this is not accurate, nor will you always have access to a ruler. A much more accurate way to get the distance 85 This is a graph of a parabola given by the equation y = − (x2 ) + 9. Figure 2.4 Parabola Graph between two points is to use geometry and the Pythagorean Theorem. This theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Figure 2.5). Let's say that there are two dots on a coordinate plane. How would you find the distance between the two without a ruler? Hint: draw a right triangle. Suppose you have two points, (x1, y1) and (x2, y2), and suppose that the length of the straight line between them is d. You can derive the distance formula by noticing that you can follow the following path between any two points to obtain a right triangle: start at point one, change x (keep y constant) until you're directly above or below point two, and then alter y and keep x constant until you're at point two. Using the Pythagorean Theorem and Figure 2.6, it becomes evident that d2 = (x2 − x1)2 + (y2 − y1)2 22 and solving for d by taking the square root of both sides, we have the full distance formula: d = (x2 − x1)2 + (y2 − y1)2 Using this formula and our original two points (2, 4) and (5, 8), we can plug those values into x1, x2, y1 and y2 to get the distance. d = (5 − 2)2 + (8 − 4)2 = 32 + 42 = 25 = 5 Midpoint of a Line Segment In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints. If you have two points, (x1, y1) and (x2, y2), the midpoint of the segment connecting the two points can be found with the formula ( x1 + x2 2 , y1 + y2 2 ) 86 The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Figure 2.5 Pythagorean Theorem The distance formula between two points, (x1, y1) and (x2, y2), shown as the hypotenuse of a right triangle Figure 2.6 Distance Formula By looking at each coordinate, you can see that the x coordinate is halfway between x1 and x2, and the y coordinate is halfway between y1 and y2 (Figure 2.7). Using our example points from above, (2, 4) and (5, 8), and the midpoint formula, we see that the midpoint of the line connecting these two points is ( 2 + 5 2 , 4 + 8 2 ) = ( 7 2 ,6) Source: models/introduction-to-graphing/distance-formula-and-midpoints-of- segments/ CC-BY-SA Boundless is an openly licensed educational resource Circles The equation for a circle is just an extension of the distance formula. KEY POINTS • A circle is defined as the set of points that are a fixed distance from a center point. • The distance formula can be extended directly to the definition of a circle by noting the radius is simply the distance between the center of a circle and the edge. • The general equation for a circle, centered at (a, b), with radius r is the set of all points (x, y) such that (x − a)2 + (y − b)2 = r2 . A circle is defined as the set of points that are a fixed distance from a center point. This definition is expressed in the way we draw circles. We pick a point as the center and then use some mechanical means to rotate a drawing utensil around that point. Of course our drawings are always approximations of the shape we think of as a circle. Drawings are only as accurate as the hardness and thickness of our tools will allow. For instance if you blow (Figure 2.8) up large enough on your computer screen you might see it is composed of adjacent colored squares and rectangles on your screen. If you used software that maintained the scale of the points as you magnified 87 The equation for a midpoint of a line segment with end points (x1, y1) and (x2, y2) Figure 2.7 Midpoint of a Line Segment the circle then eventually it would start to look like a line. This is the reason that we perceive roads as being straight even though we know the Earth is round. Since we know a circle is the set of points a fixed distance from a center point, let's look at how we can construct a circle in a Cartesian coordinate plane with variables x and y. To find a formula for this, suppose that the center is the point (a,b). According to the distance formula, the distance c from the point (a,b) to any other point (x, y) is c2 = (x − a)2 + (y − b)2 . The radius of the circle, r, is the distance between the center of the circle and any point along the edge. Therefore, the general equation for a circle, centered at (a, b), with radius r is the set of all points (x, y) such that (x − a)2 + (y − b)2 = r2 . Source: models/introduction-to-graphing/circles/ CC-BY-SA Boundless is an openly licensed educational resource Graphed is a circle with radius 10 centered on the origin. Figure 2.8 Circle 88 Functions and Their Notation A function maps a set of inputs onto permissible outputs, and each input maps onto one and only one output. KEY POINTS • Functions are relations between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. • Typically functions are named with a single letter, most commonly f, g and h. A function takes the form f(x) for one input variable, but it can take any number of variables, e.g. f(x, y, z). • Functions can be thought of as a machine in a box open on two ends. You put something in one end, something happens to it in the middle, and something pops out the other end. What Are Functions? In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2 . Functions are typically named with a single letter, most typically f, so we'll call this function f. The output of a function f corresponding to an input x and is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. The input variable(s) are sometimes referred to as the argument(s) of the function. In the case of a function with just one input variable, the input and output of the function can be expressed as an ordered pair, ordered so that the first element is the input, the second the output. In the example above, f(x) = x2 , we have the ordered pair (−3, 9). If both the input and output are real numbers, this ordered pair can be viewed as the Cartesian coordinates of a point on the graph of the function. Another commonly used notation for a function is f:X→Y, which reads as saying that f is a function that maps values from the set X onto values of the set Y. Functions As a Box Functions are often described as a machine in a box open on two ends. You put something in one end, something happens to it in the middle, and something pops 90 A function f takes an input x and returns an output f(x). One metaphor describes the function as a "machine" or "black box" that for each input returns a corresponding output. Figure 2.9 Function Machine out the other end. The function is the machine inside, and it's defined by what it does to whatever you give it (Figure 2.9). Let's say the machine has a blade that slices whatever you put into it in two and sends one half out the other end. If you put in a banana, you'd get back half a banana. If you put in an apple, you'd get back half an apple (Figure 2.10). You may wonder what happened to the other half of the piece of fruit, but since this is algebra, the things that go in and come out of functions will be numbers, so the box simply fills up with numbers and will not break. Let's define the function to take what you give it and cut it in half, that is, divide it by two. If you put in 2, you'd get back 1. If you put in 57, you'd get back 28.5. The function machine allows us to alter expressions. In this example, f(x) = 1 2 x. Functions As a Relation Functions can also be thought of as a subset of relations. A relation is a connection between numbers in one set and numbers in another (Figure 2.11). In other words, each number you put in is associated with each number you get out. The difference is that in a function, every input number is associated with exactly one output number, whereas in a relation, an input number may be associated with multiple or no output numbers. This is an important fact about functions that cannot be stressed enough: every possible input to the function must have one and only one output. All functions are relations, but not all relations are functions. Source: models/functions-an-introduction/functions-and-their-notation/ CC-BY-SA Boundless is an openly licensed educational resource 91 This shows some of the potential input to output of a function. For instance, -1 and 1 both map to the value 1, 7 to 49, and 0.5 to 0.25. The function is f(x)=xx. Figure 2.11 Mapping of a Function This shows a function that takes a fruit as input and releases half the fruit as output. Figure 2.10 Fruit Halving Function Graphing Functions Graphs are a great visual representation of functions, showing the relation between the input and the output as a line or curve. KEY POINTS • Functions have an independent variable and a dependent variable. Typically x is the independent variable and y the dependent variable. • As you choose any valid value for the independent variable, the dependent variable is determined by the function and you will always get the same result. • To graph a function, choose some values for the independent variable, x, plug them into the function to get a set of ordered pairs (x, y), and plot these on the graph. Then connect the points to best match how the points are arranged on the graph. Make sure you have enough points. Independent and Dependent Variables Functions have an independent variable and a dependent variable. When we look at a function such as f(x) = 1 2 x, we call the variable that we are changing, in this case x, the independent variable. We assign the value of the function to a variable, in this case y, that we call the dependent variable. The reason that we say that x is independent is because we can pick any value for which the function is defined, in this case the real numbers R, as an input into the function. Once we pick the value of the independent variable the same result will always come out of the function. We say the result is assigned to the dependent variable, since it depends on what value we placed into the function. It should be noted that you can switch the dependent and independent variable. Do this by equating y with our function y = 1 2 x, then 2y = 2( 1 2 x), then 2y = x, and so g(y) = 2y. The independent variable is now y and the dependent variable x. Graphing Functions Let's start with a rather simple function, f(x) = 5 − 5 2 x. How can we begin graphing this function? Start by choosing a few values for the independent variable, x. Say 0 and 2. Next, plug these values into the function, and we get the ordered pairs (0, 5) and (2, 0). Now this function is that of a line, so simply connect the two points, extend them in either direction past the points to infinity, and we have our graph (Figure 2.12). Now let's choose a more interesting function, f(x) = x3 − 9x. 92 Again, start by choosing a few values for the independent variable, x. This function is a little more complicated than a line, so we'll need to choose some more points. Say x = {0, ± 1, ± 2, ± 3, ± 4}. Next, plug these values into the function, f(x) = x3 − 9x, to get a set of ordered pairs, in this case we get the set of ordered pairs: {(−4, − 28), (−3,0)(−2,10), (−1,8), (0,0), (1, − 8), (2, − 10), (3,0), (4,28)}. Next place these points on the graph, and connect them as best as possible with a curve. If you don't have enough points to be sure about what the graph should look like, simply calculate some more! The graph for this function is (Figure 2.13). Source: models/functions-an-introduction/graphing-functions/ CC-BY-SA Boundless is an openly licensed educational resource 93 This is the graph of the function f (x) = x3 − 9x. Figure 2.13 Graph of a Function This is a graph of the line with function f (x) = 5 − 2.5x. Figure 2.12 Graph of a Line Finding Domains of Functions The domain of a function is the set of all possible input values that produce some output value. KEY POINTS • Given a function f:X→Y, the set X is the domain of f, and the set Y is the codomain of f. • The domain of a function f is all of the values for which the function is defined. For instance, x is not defined when x<0. • To find the domain of a function f, you must find the values for which f is not defined. What Is the Domain of a Function? The domain of a function is the set of "input" or argument values for which the function is defined. The domain is shown in the left oval in (Figure 2.14). The function provides an "output" or value for each member of the domain. For instance, the domain of f(x) = x2 is the set of all real numbers, R, as every real number you put into f will give an output, namely x2 . Given a function f:X→Y, the set X is the domain of f; the set Y is the codomain of f. In the expression f(x), x is the argument, and f(x) is the value. One can think of an argument as an input to the function and the value as the output. It is important to note that not all functions have the set of real numbers as their domain. For instance, the function f(x) = 1 x is not defined for x=0, because you cannot divide a number by 0. In this case, the domain of f is the set of all real numbers except 0. That is, x ≠ 0. So the domain of this function is R − {0}. What about the function f(x) = x? In this case, the square root of a negative number is not defined, and so the domain is the set of all real numbers with x ≥ 0. 94 The oval on the left is the domain of the function f, and the oval on the right is the range. Figure 2.14 Mapping of a Function Finding the Domain of a Function In order to find the domain of a function, if it isn't stated to begin with, we need to look at the function definition to determine what values are disallowed. For instance, we know that you cannot take the square root of a negative number, and you cannot divide by 0. With this knowledge in hand, let's find the domain of the function f(x) = 1 x − 1 − 2 + x. First, we know we cannot divide by 0, so any value of x that causes a division by 0 is disallowed in the domain. In this example, this occurs when x − 1 − 2 = 0. Solving for x, this happens is when x=5, so we know that x ≠ 5. We also know we can't take the square root of a negative number. This means that x − 1 > 0 or x > 1. So this function's domain is the set of all real numbers such that x > 1 and x ≠ 5. Therefore, to find what values are not in the domain, you must find the values where the function (or parts of it) is not defined. Source: models/functions-an-introduction/finding-domains-of-functions/ CC-BY-SA Boundless is an openly licensed educational resource Visualizing Domain and Range All values in the domain are mapped onto values in the range that consist of one of four types of relations. KEY POINTS • Values in the domain map onto values in the range. • There are four types of relations between the domain and range, one-to-one, one-to-many, many-to-one, and many-to- many. • The horizontal and vertical line test can help determine the type of relation between the domain and range. Domain and Range As stated in a previous section, the domain of a function is the set of 'input' numbers for which the function is defined. The domain is part of the definition of a function. In most algebra formulas, x is usually the variable associated with domain. For example, the domain of the function f(x) = x is x ≥ 0. The range of a function is the set of results, or solutions, to the equation for a given input. A true function only has one result for every domain. In most algebra formulas, y is usually the variable 95 associated with Range. As such, it can also be expressed f(x), which says that its value is a function of x. For instance, the function f(x) = x2 has a range of f(x) ≥ 0, because the square of a number is always positive. In taking both domain and range into account, a function is any mathematical formula that produces one and only one result for each input. Hence, it can be said that in a valid function, domain (x) and range (y) have a many to one correspondence so that every given domain value has one and only one range value as a result, but not necessarily vice versa. This makes sense since results can repeat, but inputs cannot. Types of Relations There are four types of relations, one-to-one, one-to-many, many- to-one, and many-to-many. These types describe how inputs and outputs are related. 1. One-to-one means that every input has one unique output, and every output has one unique input. This kind of relation is a function. An example of this relation is f(x) = ± 1. 2. Many-to-one means that multiple inputs can map onto the same output, but each input still has only one output. This kind of relation is a function, as each input only has one output. An example of a many-to-one function is f(x) = x2 , as with both x=-1 and x=1, f(x) = 1. (Figure 2.15) is also an example of a many-to-one relation. 3. Many-to-many means that multiple inputs map onto the same output, and inputs have multiple outputs. This relation is not a function. An example of this relation is f(x) = ± x2 , for instance, x = 1 and x = − 1 has f(x) = ± 1.Figure 2.16 is also an example of a many-to-many relations. The line test is a test to see the type of relation. The horizontal line test checks if multiple input values have the same output value. To perform a horizontal line test take a horizontal line and if it intersects with two or more points, then the test 'fails'. The vertical line test checks if a single input value has multiple output values. To perform a vertical line test take a vertical line and if it intersects with two or more points, then the test 'fails'. One-to-one has both horizontal and vertical line tests passing. One-to-many has the 96 This is a many-to- many visualization of a domain and range. Because it is many-to-many it is not a valid function. Figure 2.15 Many to Many Domain and Range horizontal test passing and the vertical test failing. Many-to-one has the horizontal test failing and the vertical test passing. Many-to- many has both tests failing. An important result of these tests is that any graph where the vertical line test fails, then the graph is not a function. Visualizing Domain and Range The domain and range can be visualized by Figure 2.15 and Figure 2.16. One can also use a graph, such as the graph for f(x) = x2 in Figure 2.17. It is important to note that a graph is not always perfect, if for instance you are zoomed in on the graph where all values are non-negative, however for larger values of x, the graph turns around and becomes negative. Source: models/functions-an-introduction/visualizing-domain-and-range/ CC-BY-SA Boundless is an openly licensed educational resource 97 Using graphs, you can visualize the domain and range, noting that the range is always a non-negative number. Figure 2.17 Domain and Range Visualization This is a many-to- one visualization of a domain and range. Because it is many-to-one, this is a valid function. Figure 2.16 Mapping of a Function The Linear Function f(x) = mx + b and Slope The linear function f(x) = mx + b defines a line in a 2D coordinate system with slope m and y-intercept b. KEY POINTS • A linear function is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. The typical linear function in slope-intercept form is y = mx+b where m and b are constants. • The y-intercept, b, is the y-coordinate of the location where the line crosses the y-axis, which can be found by setting x to 0. • The slope, m, is the change in the vertical distance of a line on a coordinate plane over the change in horizontal distance. The slope of the line is m = y2 − y1 x2 − x1 . What is a Linear Function? A linear function is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. Linear equations can have one or more variables. Linear equations occur with great regularity in applied mathematics. A common form of a linear function in the independent variable x and the dependent variable y is y = mx + b where m and b are constants. The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane. In this particular equation, the constant m determines the slope or gradient of that line, and the constant term b determines the point at which the line crosses the y-axis, otherwise known as the y- intercept. (Figure 2.18) shows a graph of a couple of linear functions. The y-intercept and the Slope The y-intercept, which is b in the standard form, is the y-coordinate of the location where the line crosses the y-axis. This can be seen by letting x = 0, which immediately gives y = b. It may be helpful to think about this in terms of y = b + mx; where the line passes through the point (0, b) and extends to the left and right at a slope of m. Vertical lines having undefined slope, cannot be represented by this form. Vertical lines have the form x = c for some constant c. It's also possible to find the x-intercept of the function, the x- coordinate where the line crosses the x-axis, but it isn't immediately clear from the intercept-slope form. To find the x-intercept, simple 98 solve for x to get x = y − b m = y m − b m and so you can see that by letting y = 0, the x-intercept is therefore − b m . The slope, m, is the change in the vertical distance of a line on a coordinate plane over the change in horizontal distance. In other words, it is the "rise" over the "run" or the steepness of a line. Lines that have the same slope are parallel lines, and these lines will never touch or cross. Slope is computed by measuring the change in vertical distance divided by the change in horizontal difference, in order to compute this you'll need two points on the line, say (x1, y1) and (x2, y2), such that x2 > x1. From these two points, and the knowledge that the slope is the change in the vertical distance over the change in the horizontal distance, it's easy to see the slope m = y2 − y1 x2 − x1 . Finding the Intercept and Slope The linear function won't always be in slope-intercept form, or sometimes you'll need to find the function from a pair of points. If you have a function that is not in the slope-intercept form, to get it into that form, simply solve for y. For example, if you have the function 3y − 15 6 = 2x solve for y by first multiplying both sides by 6 to get 3y − 15 = 12x, add 15 to both sides for 3y = 12x + 15, and finally divide both sides by 3 to put it into slope-intercept form y = 4x + 5, which tells us the y-intercept is 5, and the slope is 4. Another possibility is that you are simply given two points, say (1, 3) and (4, 0). First find the slope from the slope equation m = 0 − 3 4 − 1 = −3 3 = − 1. Now that we know the slope, choose one of the two points and plug it into the slope-intercept form. Let's choose (1, 3), plug this in to get 3 = − 1 × 1 + b, and solve for b to get An example of two linear functions, y = -x + 5 and y = 0.5x+2. Both functions graph as straight lines, but their y-intercepts and slopes are determined through their functional terms. The positive (red) and negative (blue) slope terms change the orientation of the lines relative to each other. Figure 2.18 Linear Function Graph 99 b = 4. Therefore the slope-intercept form for the general function is y = − 1x + 4. Source: models/functions-an-introduction/the-linear-function-f-x-mx-b-and- slope/ CC-BY-SA Boundless is an openly licensed educational resource Applications of Linear Functions and Slope Linear functions apply to real world problems that involve a constant rate. KEY POINTS • If you know a real-world problem is roughly linear, such as the distance you travel when you go for a jog, all you need is two points and you can graph the function and make some assumptions as to what happens beyond the two points, so long as you maintain the same rate. • The slope of a function is the same as the rate of change for the dependent variable. For instance, if you're graphing distance vs. time, then the slope is how fast your distance is changing with time, or in other words, your speed. • When checking where two linear functions intersect, set them equal to each other and solve for the dependent variable, x. Let's say that one day you decide to start training for a marathon. You start at your house, stretch, and look at your watch. It reads 6:00 pm. You plug in your headphones and begin to run around town. After a while, you realize you can't run anymore, and look at your watch. It reads 7:30 pm, and you're 7.5 miles from home. How fast was your average speed over the course of the run? 100 Our two variables are time and distance, and you have the data for two separate points. The first point is at your house, where your watch read 6:00, let's call this the beginning time and set it to 0. So our first point is (0, 0) as we hadn't run any distance yet. Let's think about our time in hours. So our second point is 1.5 hours later, and we ran 7.5 miles. So our second point is (1.5, 7.5). Our speed is simply the slope of the line connecting the two points. The slope, given by m = y2 − y1 x2 − x1 becomes m = 7.5 1.5 = 5 miles per hour. To graph this line, we need the y-intercept and the slope. We have just calculated the slope, but what's the y-intercept? Since we started at (0, 0), we can see that the y-intercept is 0. So our final function is y = 5x. With this new function, we can now answer some more questions. How many miles had we run after the first half hour? If we kept running at the same pace for a total of 3 hours, how many miles will we have run? 26 = 5x, so x = 26 5 = 5.2 hours. There are many such applications for linear equations. Anything that involves a constant rate of change can be nicely represented with a line with the slope. Indeed, so long as you have just two points, if you know the function is linear, you can graph it and begin asking questions! Just make sure what you're asking and graphing makes sense. For instance, in the marathon example, the domain is really only x ≥ 0, since it doesn't make sense to go into negative time and lose miles! Trains are just one example of things that can be used with linear functions. For instance, to see when two trains travelling at constant rates towards each other meet is a simple linear function. Figure 2.19 Trains Used in Applications 101 Let's look at another application. Two trains start 200 miles apart. They are travelling towards each other. The first train is travelling 40 miles per hour, while the second train is travelling 60 miles per hour. When and where do they meet? For simplicity we should put one of the trains starting at (0, 0), and so the other train must start at either (0, 200) or (0, -200) as it is 200 miles away. We'll choose (0, 200). The first train is travelling 40 miles per hour towards the other train, which is in the positive y direction, so the slope is positive 40. The second train is travelling towards the first at 60 miles per hour, which is in the negative y direction, so the slope is negative 60. Now we have the equations for both of the functions. Let f(x) = 40x represent the first train and g(x) = − 60x + 200 represent the second train. To find when they meet, simply set them equal to each other and solve for x. 40x = − 60x + 200 100x = 200 x = 2 They meet after 2 hours of travel. To find out where they meet, plug x=2 into one of the equations, say f(x). Thus f(2) = 40 × 2 = 80 so they meet 80 miles away from where the first train started, or 120 miles from where the second train started. This can also be viewed by graphing the two equations and seeing their intersection (Figure 2.20). Source: models/functions-an-introduction/applications-of-linear-functions- and-slope/ CC-BY-SA Boundless is an openly licensed educational resource The intersection of f(x) = 40x and g(x) = -60x + 200 can be viewed graphically. This can be used to determine where they will cross. Figure 2.20 Two Trains Graphically 102 Slope-Intercept Equations The slope-intercept form of a line, gives you the information to construct a quick and easy line using the slope, m, and the y-intercept, b. KEY POINTS • The slope-intercept form of a line is given by y = mx + b where x and y are variables, and m and b are constants. The constant m is the slope, and b is the y-intercept. • The constant b is known as the y-intercept, when x=0, y = b, and the point (0, b) is the unique member of the line where the y-axis is 'intercepted' or crossed. • The x-intercept, when y = 0, x=-b/m, assuming m is non- zero, and the point (-b/m, 0) is the unique member of the line where the x-axis is 'intercepted' or crossed. • The constant m is known as the slope. Slope is the measure of how much a line moves up or down related to how much it moves left to right. • If you have two points (x1, y1) and (x2, y2), where y2 − y1 x2 − x1 . One of the most common representations for a line on a graph is with the slope-intercept form. Such an equation is given by y = mx + b, where x and y are variables and m and b are constants. The constant m is known as the slope and b is known as the y- intercept. Note that if m is 0, then y = b, and thus y is a horizontal line. Note that this equation does not allow for vertical lines, since that would require that m be infinite, though a vertical line can be define by x = c for some constant c. Intercepts The constant b is known as the y-intercept. The equation y = f(x) = mx + b has an infinite number of solutions. Points will be mapped with independent variable x on the horizontal axis, and y on the vertical. By assigning x to a value and evaluating, you get a single point (x, y). When x=0, y = b, and the point (0, b) is the unique member of the line where the y-axis is 'intercepted' or crossed. There is also an x-intercept, but it is not immediately clear from the equation. To find the x-intercept, set y = 0 and solve for x to get x = − b m , giving the point (− b m ,0) and this is the unique point where the x-axis is 'intercepted'. Exceptions to these are the horizontal and vertical lines. Horizontal lines given by y = b can intercept the x-axis 0 or infinite times. It intersects an infinite number of times if b = 0, and 0 times otherwise. Similarly the vertical line given by x = c can intercept the 104 y-axis 0 or infinite times. It intersects an infinite number of times if c = 0, and 0 times otherwise. Slope The constant m is known as the slope. Slope is the measure of how much a line moves up or down related to how much it moves left to right. Slope is computed by measuring the change in vertical distance divided by the change in horizontal difference, i.e. m = Δy Δx = rise run . So if there are two points (x1, y1) and (x2, y2) where x1 ≠ x2, then m = y2 − y1 x2 − x1 If a line goes up from left to right, then the slope has to be positive. For example, a slope of ¾ would have a "rise" of 3, or go up 3; and a "run" of 4, or go right 4. Both numbers in the slope are either negative or positive in order to have a positive slope. If a line goes down from left to right, then the slope has to be negative. For example, a slope of -3/4 would have a "rise" of -3, or go down 3; and a "run" of 4, or go right 4. Only one number in the slope can be negative for a line to have a negative slope. There are two special circumstances, no slope and slope of zero. A horizontal line has a slope of 0 and a vertical line has an undefined slope. Horizontal lines have the form y = b, where b is a constant. Vertical lines have the form x = c, where c is a constant. Figure 2.21 shows numerous different lines with the same y- intercept at y=1, and a multitude of differing slopes. Two separate line equations with the same slope are parallel lines. For example y = 2x+1 and y = 2x - 1 are parallel lines. Source: models/modeling-equations-of-lines/slope-intercept-equations/ CC-BY-SA Boundless is an openly licensed educational resource 105 This is a graph of multiple lines with the same y- intercept, 1, and many differing slopes. Figure 2.21 Differing Slopes Point-Slope Equations The point-slope equation is another way to represent a line; to use the point-slope equation, only the slope and a single point are needed. KEY POINTS • The point-slope equation is given by (x1, y1) is any point on the line, and m is the slope of the line. • The point-slope equation requires that there are at least one point and the slope. If there are two points and no slope, the slope can be calculated from the two points. Then use one of the two points in the final equation. • The point-slope and slope-intercept equations are equivalent, and with a little bit of algebraic manipulation it can be shown that given a point y1 − mx1. The point-slope equation is another way of describing the equation of a line. The point-slope form is great if you have the slope and only one point, or if you have two points and do not know what the y-intercept is. Given a slope, m, and a point (x1, y1), the point-slope equation is given by y − y1 = m(x − x1). Changing between point-slope and slope-intercept forms is a simple task. If an equation is in a slope-intercept form, such as y = mx + b, to switch to point-slope form only one point is needed (x1, y1) that satisfies the slope-intercept form. By choosing either an x value or a y value, solving for the other variable, and plugging that information into the point-slope equation, using the same m, will get the other equation. To show that these two equations are equivalent, choose a generic point (x1, y1). Plug in the generic point into the equation. The equation is now, y1 = mx1 + b, giving us the ordered pair, (x1, mx1 + b). Then plug this point into the point-slope equation to get: • y − (mx1 + b) = m(x − x1) • y − mx1 − b = mx − mx1 • y − mx1 + mx1 − b = mx − mx1 + mx1 • y − b = mx • y − b + b = mx + b • y = mx + b Therefore, the two equations are equivalent so long as any point on the line is chosen. 106 To reverse this process, from point-slope to slope-intercept follows a similar process. To switch from point-slope to slope-intercept, solve for y. For instance, in the generic case y − y1 = m(x − x1), solve for y and multiplying m throughout. The equation ends up as y = mx − mx1 + y1. If one rearranges equation and puts in parentheses, the result is y = mx + (y1 − mx1). This equation looks very similar to slope-intercept, and indeed y1 − mx1 is a constant, and is the y-intercept of the equation. It is interesting to note that, if the point chosen for the point-slope form has x1 =0, then the equation comes out to be y − y1 = mx. If we add y1 to both sides, slope-intercept form is obtained. Thus, the slope-intercept form is just a basic case of the point-slope form with x1 = 0. Examples If there is a single point (2, 1) and the slope is known to be -4. The point-slope equation for this line is y − 1 = − 4(x − 2). To switch this equation into slope-intercept form: • y − 1 = − 4(x − 2) • y − 1 = − 4x + 8 • y = − 4x + 9 If there are two points, (2, 6) and (7, -1), first find the slope by remembering that given two points the equation of the slope is m = y2 − y1 x2 − x1 . Therefore, the slope of this line is m = −1 − 6 7 − 2 = −7 5 . Now choose any of the two points, such as (2, 6). Plug those points into the point-slope equation to get y − 6 = −7 5 (x − 2). Source: models/modeling-equations-of-lines/point-slope-equations/ CC-BY-SA Boundless is an openly licensed educational resource 107 This is the graph of the line y-5=-2.5(x-0) Figure 2.22 Graph of a Line Parallel and Perpendicular Lines Parallel lines never intersect; perpendicular lines intersect at right angles. KEY POINTS • Parallel lines have the same slope. A line is technically parallel with itself. • Perpendicular lines slopes are negative reciprocals of each other. This means that one has m and the other -1/m. • The distance between two parallel lines d = \|b2 − b1 \| m2 + 1 . • If the line f(x) is perpendicular to g(x), then f(x) is perpendicular to all lines that are parallel with g(x), and all lines parallel with f(x) are perpendicular to g(x). • If line f(x) is perpendicular to g(x), and g(x) is perpendicular to h(x), then h(x) and f(x) are parallel. Two lines in a plane that do not intersect or touch at a point are called parallel lines. The parallel symbol is ∥. For example, given two lines f(x) = m1x + b1 and g(x) = m2x + b2, f(x) ∥ g(x) states that the two lines are parallel to each other. Given two parallel lines f(x) and g(x), the following is true: 1. Every point on f(x) is located at exactly the same minimum distance from g(x). 2. Line f(x) is on the same plane as g(x) but does not intersect g(x), even assuming that the two lines extend to infinite in either direction. In 2D, two lines are parallel if they have the slope. Recall that the slope intercept form y = mx + b and the point slope form y − y1 = m(x − x1) both contain information about the slope, namely the constant m. If two lines, say f(x) = mx + b and g(x) = nx + c, are parallel, then n must equal m. 108 These are two parallel lines: f(x)=2x+3 and g(x)=2x-1. Figure 2.23 Parallel Lines For example, in Figure 2.23, the two lines f(x) = 2x + 3 and g(x) = 2x − 1 are parallel since they have the same slope, m = 2. Perpendicular Lines Two lines are perpendicular to each other if they form congruent adjacent angles -- in other words, they are perpendicular if the angles at their intersection are right angles of 90 degrees (Figure 2. 24). The perpendicular symbol is ⊥. For example given two lines, f(x) = m1x + b1 and g(x) = m2x + b2, f(x) ⊥ g(x) states that the two lines are perpendicular to each other. For two lines in a 2D plane to be perpendicular, their slopes must be negative reciprocals of one another. This means that if the slope of one line is m, then the slope of its perpendicular line is −1 m . In other words, the two slopes multiplied together must equal -1. For example, in Figure 2.24, the two lines are f(x) = 3x − 2 and g(x) = −1 3 x + 1. Note that perpendicular lines have slopes that are negative reciprocals of each other. Additionally, if line f(x) is perpendicular to g(x), then f(x) is perpendicular to all lines that are parallel to g(x), and all lines parallel with f(x) are perpendicular to g(x). If line f(x) is perpendicular to g(x), and g(x) is perpendicular to h(x), then h(x) and f(x) are parallel. However, this fact is only true in two dimensions! Distance Between Two Parallel Lines Let's say we want to know the distance between two parallel lines y = mx + b1 and y = mx + b2. The shortest distance between these two lines is the perpendicular line between them, thus any line with slope −1 m . We'll choose the perpendicular line to be y = −x m . To find where the perpendicular line and the first parallel line intersect, set them equal to each other and solve for x to get: • mx1 + b1 = −x1 m • m2 x1 + mb1 = − x1 • m2 x1 + x1 = − mb1 109 These two lines are perpendicular, where one line has f(x)=3x-2 and the other is g(x)=(-1/3)x +1. Note their slopes are negative reciprocals of each other. Figure 2.24 Perpendicular Lines Linear Mathematical Models Linear mathematical models take real world applications and describe them with lines. KEY POINTS • A mathematical model describes a system using mathematical concepts and language. • Linear mathematical models can be described with lines. For instance, a car going 50 mph, has traveled a distance represented by y = 50x, where x is time in hours and y is miles. • Real world applications can be modeled with multiple lines. If two trains travel toward each other—one at 50 mph and the other at 80 mph—and they initially start at 350 miles away, the two lines are y=50x and y=350-80x. The point where the two lines intersect is the point where the trains meet. A mathematical model is a description of a system using mathematical concepts and language. Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, and meteorology) and engineering disciplines (such as computer science and artificial intelligence), but also in the social sciences (such as economics, psychology, sociology, and political science). Physicists, engineers, statisticians, operations research analysts, and economists use mathematical models most extensively. A model may help explain a system, study the effects of different components, and make predictions about behavior. Dead Reckoning Many everyday activities require the use of mathematical models, perhaps unconsciously. An example is seen in predicting the position of a vehicle based on its initial position, direction, and speed of travel. This is known as "dead reckoning" when used more formally (Figure 2.25). This type of mathematical modeling does not necessarily require formal mathematics; animals have been shown to use dead reckoning. This is an example of dead reckoning, which uses a linear model to predict how far one has traveled over time. Figure 2.25 Dead Reckoning 111 Example 1 One difficulty with mathematical models lies in translating the real world application into an accurate mathematical representation. Linear models can be used for problems concerned with straight lines, such as how far a vehicle has traveled from its initial position, moving at 50 mph. This can be easily modeled with a line y = 50x, where x is in hours. For instance, if you travel for 3.5 hours, then: y = 50(3.5) = 175 miles Example 2 It's also possible to model multiple lines at once. For instance, you have two trains, A and B. At the beginning, the A and B are 325 miles away from each other. Train A is traveling towards B at 50 miles per hour; B is traveling towards A at 80 miles per hour. First let's model the position of the two trains, using the initial location of A as the reference point. The origin (0, 0) is where A starts. B is 325 miles away from A at the beginning, so its position is (0, 325). Since A is traveling towards B, which has a greater y value, A's slope must be positive, equal to its speed, 50. B is traveling towards A, which has a lesser y value, giving B a negative slope: -80. The two lines are thus yA = 50x and yB = − 80x + 325 . This model allows for several observations—the two trains will meet where the two lines intersect. Setting yA = yB we have 50x = − 80x + 325. Solving for x gives us x =2.5. The two trains meet after 2.5 hours. To find where this is, plug 2.5 into either equation. Plugging it into the first equation gives us 50(2.5) = 125, which means it meets after A travels 125 miles. For multiple observations that are roughly linear, you can use linear regression to create a line that best fits the data. This line allows linear modeling to provide new observations about the data. Source: models/modeling-equations-of-lines/linear-mathematical-models/ CC-BY-SA Boundless is an openly licensed educational resource 112 Fitting a Curve Curve fitting with a line attempts to draw a line so that it "best fits" all of the data. KEY POINTS • Curve fitting is useful for finding a curve that best fits the data, so you can see how the data is roughly spread out as well as make observations about data you don't have. • Linear regression attempts to graph a line that best fits the data available. • Ordinary least squares approximation is a type of linear regression that minimizes the sum of the squares of the difference between the approximated value (from the line), and the actual value. • The slope of the line that approximates n data points is given by m = ∑ n i=1 xiyi − 1 n ∑ n i=1 xi ∑ n j=1 yj ∑ n i=1 (x2 i ) − 1 n (∑ n i=1 xi)2 . • The y-intercept of the line that approximates n data points is given by b = 1 n n ∑ i=1 yi − m 1 n n ∑ i=1 xi = ¯y − m ¯x. Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables. Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a greater degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data. In this section, we will only be fitting lines to data points, but it should be noted that one can fit polynomial functions, circles, piecewise functions, and any number of functions to data and it is a heavily used topic in statistics. Linear Regression Linear regression is an approach to modeling the linear relationship between a dependent variable y and an independent variable x. Under linear regression, a line of the form y = mx + b is found that "best fits" the data. The simplest and perhaps most common linear regression model is the ordinary least squares approximation. This approximation attempts to minimize the sums of the squared distance between the line and every point. Suppose there are n points, our equation for 113 • n ∑ i=1 yi = 0 + 0 + 1 + 2 + 1 + 2.5 + 3 + 4 = 13.5 Therefore the numerator in the slope equation is 57 − 1 8 (20)(13.5) = 23.25. n ∑ i=1 (x2 i ) = 1 + 0 + 1 + 4 + 9 + 16 + 25 + 36 = 92 So the denominator is 92 − 1 8 (20)2 = 92 − 50 = 42 and the slope is 23.25 42 ≈ 0.554. Now for the y-intercept, b, we have ¯x = 20 8 = 2.5 and ¯y = 13.5 8 = 1.6875. Therefore b ≈ 1.6875 − 0.554(2.5) = 0.3025. Our final equation is therefore y = 0.554x + 0.3025, and this line is graphed, along with the points, in Figure 2.27. As shown, the approximation is quite good. Least squares can sometimes fail, for instance if we have a point that is far away from the approximating line, then it will skew the results and make the line much worse. For instance, let's say in our original example, instead of the point (-1, 0), we have (-1, 6). We won't go through all the calculations again, but simply state that if one does the same thing as above with the new point, it ends up as m ≈ 0.0536 and b ≈ 2.3035, to get the new equation y = 0.0536x + 2.3035, shown in Figure 2.28. As can be seen, this new line does not fit the data well due to the outlier. Indeed, trying to fit linear models to data that is quadratic, cubic, or anything non- linear, or data with many outliers and errors can result in bad approximations. Here is the approximated line given the new outlier point at (-1, 6). Figure 2.28 Outlier Approximated Line Here is a line used by the least squares approximation. The line equation is y = 0.554x+0.3025. Figure 2.27 Least Squares Fit Line 115 Source: models/modeling-equations-of-lines/fitting-a-curve/ CC-BY-SA Boundless is an openly licensed educational resource Fitting a line to a curve using linear regression. Figure 2.29 Line Fitting 116 Increasing, Decreasing, and Constant Functions Functions can either be constant with a single value, increasing as x increases, or decreasing as x increases. KEY POINTS • A constant function is a function whose values do not vary, regardless of the input into the function. • An increasing function is one where for every x1 and x2 that satisfies x2 > x1, then f(x2) ≥ f(x1). If it is strictly greater than, then it is strictly increasing. • A decreasing function is one where for every x1 and x2 that satisfies x2 > x1, then f(x2) ≤ f(x1). If it is strictly less than, then it is strictly decreasing. Constant Functions In mathematics, a constant function is a function whose values do not vary—they are constant. For example, the function f(x) = 4 (Figure 2.30) is constant since f maps any value to 4. More formally, a function f : A → B is a constant function if f(x) = f(y) for all values of x and y in A. Every empty function is constant, vacuously, since there are no values of x and y in A for which f(x) and f(y) are different when A is the empty set. In polynomial functions, a non-zero constant function is called a polynomial of degree zero. A function is said to be "identically zero" if it takes the value 0 for every argument; it is trivially a constant function. A composite function is an application of one function to the results of another. Composing two functions—f(x) and g(x) for example—is written as either f(g(x)) or: f ∘ g(x) We won't go into details in this section, but say you have two functions: f(x) = 2x − 2 and g(x) = 3x − 1 The composite f(g(x)) would be: 118 A constant function f(x)=4 Figure 2.30 Constant Function f(g(x)) = f ∘ g(x) = f(3x − 1) = 2(3x − 1) − 2 Several key points about constant functions with respect to composite functions are: • f : A → B is a constant function. • For all functions g, h : C → A, f ∘ g = f ∘ h. • The composition of f with any other function is also a constant function. Increasing and Decreasing Functions An increasing function is a function such for every x1 and x2 that satisfies x2 > x1, then f(x2) ≥ f(x1), if the equality is strictly greater than (not greater than or equal to), then the function is "strictly increasing." A decreasing function is a function such that for every x1 and x2 that satisfies x2 > x1, then f(x2) ≤ f(x1). If the equality is strictly less than (not less than or equal to), then the function is "strictly decreasing." In terms of the linear function f(x) = mx + b, if m is positive, the function is increasing, if m is negative, it is decreasing, and if m is zero, the function is a constant function. 119 This is an increasing linear function, f(x)=2x-2. Figure 2.31 Increasing Linear Function This is the natural logarithm, or ln(x), and is an increasing function. Figure 2.33 Natural Logarithm This is the function f (x) = − 5x3 − 1, and is a decreasing function. Figure 2.32 A Cubic Function Source: models/functions-revisited/increasing-decreasing-and-constant- functions/ CC-BY-SA Boundless is an openly licensed educational resource Relative Minimums and Maximums Relative minima and maxima are points of the smallest and greatest values in their neighborhoods respectively. KEY POINTS • Minima and maxima are collectively known as extrema. • A function has a global (or absolute) maximum point at x∗ if f(x∗) ≥ f(x) for all x. Similarly, a function has a global (or absolute) minimum point at x∗ if f(x∗) ≤ f(x) for all x. • A real-valued function f defined on a real line is said to have a relative maximum at the point x∗, if there exists some ε > 0 such that f(x∗) ≥ f(x) when \|x − x∗\| < ε. Similarly, a function has a relative minimum at x∗, if f(x∗) ≤ f(x) when \|x − x∗\| < ε. • Functions don't necessarily have extrema in them. For example any line, f(x) = mx+b where m and b are constants, does not have any extrema, be they local or global. In mathematics, the maximum and minimum of a function, known collectively as extrema, are the largest and smallest value that a function takes at a point either within a given neighborhood (local or relative extremum) or within the function domain in its entirety (global or absolute extremum). 120 A real-valued function, f, defined on a real line, is said to have a relative maximum point at the point x∗, if there exists some ε > 0 such that f(x∗) ≥ f(x) when \|x − x∗\| < ε. The value of the function at this point is called the maximum of the function. Similarly, a function has a relative minimum point at x∗, if f(x∗) ≤ f(x) when \|x − x∗\| < ε. The value of the function at this point is called the minimum of the function. That is to say, a point is a relative maximum if there is no point an infinitesimally small distance away to the left or right that is greater than this point. Similarly, a point is a relative minimum if there is no point an infinitesimally small distance away to the left or right that is less than this point. A function has a global (or absolute) maximum point at x∗ if f(x∗) ≥ f(x) for all x. Similarly, a function has a global (or absolute) minimum point at x∗ if f(x∗) ≤ f(x) for all x. Global extrema are also relative extrema. Functions may not have any extrema in them, such as the line y = x. This line increases towards infinite and decreases towards negative infinite, and has no relative extrema. Why should we care about minima and maxima? They're used heavily in optimization problems and artificial intelligence where, 121 This graph has examples of all four possibilities: relative maximum and minimum, and global maximum and minimum. Figure 2.36 Examples of Relative and Global Extrema This function, f(x) = x2, has a global (and also relative) minimum at x=0. Figure 2.34 Global and Relative Minimum Example In this function, f(x) = (x3)/3-x, there is a relative maximum at x=-1 and a relative minimum at x=1. Figure 2.35 Relative Minimum and Maximum Example given a number of constraints on resources, we want the best use of our resources. For instance, we may want to maximize our profits given the items we can make and our resources. In artificial intelligence, we may want to discover what the least costly plan of action to take is for a robot. Ideally you'd want to find the global minima for the plans. However, because there is not unlimited time to identify the right plan, artificial intelligence often simply finds the local minima. EXAMPLE The function f(x) = x2 has a global (and relative) minimum at x=0. The function f(x) = x3 3 − x has a local maximum at x=-1 and a local minimum at x=1. The aforementioned minima and maxima can be discovered using calculus and identifying where the first derivative crosses 0. However, because this is algebra and not calculus, we will not describe how to find a maxima and minima. That being said, it is important to know what they are and how they are applied. Source: models/functions-revisited/relative-minimums-and-maximums/ CC-BY-SA Boundless is an openly licensed educational resource 122 Piecewise Functions A piecewise function is a function defined by multiple subfunctions that are each applied to separate intervals of the input. KEY POINTS • Piecewise functions are defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. • The absolute value \|x\| is a very common piecewise function. For a real number, its value is -x when x < 0 and x when x ≥ 0. • Piecewise functions may have horizontal or vertical gaps (or both) in their functions. A horizontal gap means that the function is not defined for those inputs. • An open circle at the end of an interval in one of the subdomains means that the end point is not included in the interval, i.e. strictly less than or strictly greater than. A closed circle means the end point is included. In mathematics, a piecewise-defined function, also called a piecewise function, is a function which is defined by multiple subfunctions, each subfunction applying to a certain interval of the main function's domain, a subdomain. Piecewise is actually a way of expressing the function, rather than a characteristic of the function itself, but with additional qualification, it can describe the nature of the function. Notation and Interpretation Piecewise functions are defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. Crucially, in most settings, there must only 123 The absolute value is a piecewise function where f(x) = x when x >= 0 and f(x) = -x when x < 0. Figure 2.38 Absolute Value For this piecewise function that defines f(x) = -1 when x <= 0, and f(x) = x^2 when x >= 1. Figure 2.37 Gap in the Input be a finite number of subdomains, each of which must be an interval, in order for the overall function to be called "piecewise". For example, consider the piecewise definition of the absolute value function: x = { −x, if x < 0 x, if x ≥ 0 For all values less than zero, the first function (−x) is used, which negates the sign of the input value, making negative numbers positive. For all values of x greater than or equal to zero, the second function (x) is used, which evaluates trivially to the input value itself (Figure 2.38). Jumps Piecewise functions can have jumps in either the input or the output. If there are gaps in the input, then the function is not defined over those input values. For example, the piecewise function: f(x) = { −x, if x ≤ 0 x2 , if x ≥ 1 , x is not defined between 0 and 1, and there is a noticeable visible gap between the two parts of the function Figure 2.37. Therefore the domain of this function is x ≤ 0 and x ≥ 1. Piecewise functions can also have vertical gaps. An example of this is the piecewise function: f(x) = { x2 , if x < 1 −(x − 2)2 + 3, if x ≥ 1 , has a vertical jump at x = 1. Since 12 = 1 and −(1 − 2)2 + 3 = 2, not the open circle as on the first equation, x2 . The open circle means that it does not include that value, this is because the domain on the first function is x < 1, strictly less than, therefore x does not equal 1. However, x can equal 0.9, or 0.99, or 0.9999, and so on, so there has to be some representation that it goes up to 1, but never actually equals 1. A closed circle means that point is included, and the second part of this equation does have a closed circle associated with it. Source: models/functions-revisited/piecewise-functions/ CC-BY-SA Boundless is an openly licensed educational resource 124 There is a vertical jump in the piecewise function at x=1. Figure 2.39 Vertical Jump of a Piecewise Function Sums, Differences, Products, and Quotients Adding, subtracting, multiplying, and dividing equations requires one to follow some key steps that can simplify the problem. KEY POINTS • Adding and subtracting equations involves grouping like terms and carrying out basic arithmetic. • Multiplying and dividing monomials applies to both like and unalike terms. • Multiplying binomials and trinomials follows the FOIL method. Adding and Subtracting Functions Terms in algebraic equations are separated by + or –. For instance, in the equation y = x + 5, there are two terms, while in the equation y = 2x2 , there is only one term. In adding equations, it is important to collect like terms to simplify the expression. "Like terms" are those that have the same kind of variable. For instance, take two equations: f(x) = x + 5 and g(x) = 2x − 3 By adding these two equations together, we get: f(x) + g(x) = h(x) = x + 5 + 2x − 3 We then collect like terms. In this case, "x" and "2x" are like terms, as are "5" and "-3." The result is: h(x) = 3x + 2 Subtracting two equations follows similar logic, except a negative sign is applied to an entire equation. In this case, the -3 of g(x) becomes +3. f(x) − g(x) = h(x) = x + 5 − (2x − 3) = x + 5 − 2x + 3 h(x) = − x + 8 It is important to remember to only add together like terms. For instance, x2 cannot be added to x because they are not the same term, although they use the same variable. Multiplying and Dividing Functions Adding and subtracting functions is quite straightforward, as shown. While adding and subtracting equations only affects like terms, multiplying and dividing functions affects all terms equally. 126 It is easiest to start with monomials. A monomial equations has one term; a binomial has two terms; a trinomial has three terms. f(x) = 3 and g(x) = x f(x) g(x) = 3 * x = 3x In this case, two unlike terms (3 and x) could be multiplied. If the terms both contain the same variables, their exponents are added together and their multipliers are multiplied. f(x) = 3x and g(x) = 2x3 f(x) * g(x) = (3x) * (2x3 ) = 6x4 When there are multiple variables, the two exponents are added separately and the variables remain next to each other. f(x) = − 8ab and g(x) = 9a3 b f(x) * g(x) = − 36a4 b2 Multiplying binomials and trinomials is more complicated, and follows the FOIL method. FOIL is a mnemonic for the standard method of multiplying two binomials; the method may be referred to as the FOIL method. FOIL is an acronym for the four terms of the product: • First ("first" terms of each binomial are multiplied together) • Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second) • Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second) 127 A color-coded diagram of the FOIL method using both numbers and variables. Figure 2.40 FOIL Method Diagram • Last ("last" terms of each binomial are multiplied) The general form is shown in Figure 2.40 and is diagrammed in Figure 2.41. Dividing equations uses similar theory as multiplying, since division is the equivalent of multiplying by the inverse. Source: models/algebra-of-functions/sums-differences-products-and- quotients/ CC-BY-SA Boundless is an openly licensed educational resource Difference Quotients The difference quotient is used in algebra to calculate the average slope between two points but has broader effects in calculus. KEY POINTS • Two points on a graph may give different results when plugged into a function. The difference between two distinct points is known as their Delta (Δ) P, as is the difference in their function result. • The function difference between two point values divided by the point difference is known as the difference quotient. • The difference quotient is essentially the average slope of a function between two points. • In calculus, the difference quotient is used to calculate the derivative when the difference between the two points is infinitesimally small. The primary vehicle of calculus and other higher mathematics is the function. Its "input value" is its argument, usually a point ("P") expressible on a graph. The difference between two distinct points, themselves, is known as their Delta (Δ) P, as is the difference in their function result, the particular notation being determined by the direction of formation: 128 The general form of the FOIL method using only variables as the potential multipliers. Figure 2.41 General Form of the FOIL Method • Forward difference: ΔF(P) = F(P + ΔP) − F(P) • Central difference: δF(P) = F(P + 1/2ΔP) − F(P − 1/2ΔP) • Backward difference: ∇F(P) = F(P) − F(P − ΔP) The function difference divided by the point difference is known as the difference quotient, attributed to Isaac Newton. It is also known as Newton's quotient: ΔF(P) ΔP = F(P + ΔP) − F(P) ΔP = [∇F(P + ΔP)]ΔP The difference quotient is the average slope of a function between two points. However, it is important to note it is not necessarily to actual slope of the curve, as can be visually seen in Figure 2.42. To show how the above equation can be written as the average slope, put it into more familiar terms. The points can be represented represent as (P, F(P)) and (P+ΔP, F(P+ΔP)) as (x, f(x)) and (x+h, f(x +h)). Remember, the formula for slope is m = y2 − y1 x2 − x1 This becomes: m = f(x + h) − f(x) (x + h) − x Which simplifies to: m = f(x + h) − f(x) h If you look carefully, this formula is the same as F(P + ΔP) − F(P) ΔP , just written differently. Applications in Calculus If \|ΔP\| is finite, meaning measurable, then ΔF(P) is known as a finite difference, with specific denotations of DP and DF(P). This is the case for all algebraic applications. If \|ΔP\| is infinitesimal, an infinitely small amount usually expressed in standard analysis as a limit, then ΔF(P) is known as an 129 The difference quotient can be used to calculate the average slope (here, represented by a straight line) between two points P and Q. Figure 2.42 Difference Quotient Chart infinitesimal difference, with specific denotations of dP and dF(P). In this case, the difference quotient is know as a derivative, a useful tool in calculus. Source: models/algebra-of-functions/difference-quotients/ CC-BY-SA Boundless is an openly licensed educational resource Composition of Functions and Decomposing a Function Functional composition allows for the application of one function to another; this step can be undone by using functional decomposition. KEY POINTS • Functional composition applies one function to the results of another. • Functional decomposition resolves a functional relationship into its constituent parts so that the original function can be reconstructed from those parts by function composition. • Decomposition of a function into non-interacting components generally permits more economical representations of the function. Function Composition Function composition is the application of one function to the results of another. For instance, the functions f: X → Y and g: Y → Z can be composed by computing the output of g when it has an argument of f(x) instead of x. Intuitively, if z is a function g(y) and y is a function f(x), then z is a function of x. 130 Thus, one obtains a composite function g ∘ f: X → Z defined by (g ∘ f )(x) = g( f(x)) for all x in X. The notation g ∘ f is read as "g circle f", or "g composed with f", "g after f", "g following f", or just "g of f". The composition of functions is always associative. That is, if f, g, and h are three functions with suitably chosen domains and codomains, then f ∘ (g ∘ h) = ( f ∘ g) ∘ h, where the parentheses serve to indicate that composition is to be performed first for the parenthesized functions. Since there is no distinction between the choices of placement of parentheses, they may be safely left off. Functional Decomposition Functional decomposition broadly refers to the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recomposed) from those parts by function composition. In general, this process of decomposition is undertaken either for the purpose of gaining insight into the identity of the constituent components (which may reflect individual physical processes of interest), or for the purpose of obtaining a compressed representation of the global function; a task which is feasible only when the constituent processes possess a certain level of modularity (i.e., independence or non-interaction). For a multivariate function y = f(x1, x2, . . . , xn), functional decomposition generally refers to a process of identifying a set of functions { g1, g2, …, gm } such that f(x1, x2, . . . , xn) = ϕ(g1(x1, x2, . . . , xn), g2(x1, x2, . . . , xn), . . . , gm(x1, x2, . . . , xn)) where φ is some other function. Thus, we would say that the function f is decomposed into functions {g1, g2, …, gm}. As illustrated in Figure 2.43, this process is intrinsically hierarchical in the sense that we can (and often do) seek to further decompose the functions into a collection of constituent functions { h1, h2, …, hp } such that gi(x1, x2, . . . , xn) = γ(h1(x1, x2, . . . , xn), h2(x1, x2, . . . , xn), . . . , hp(x1, x2, . . . , xn)) In general, functional decompositions are worthwhile when there is a certain "sparseness" in the dependency structure i.e. when constituent functions are found to depend on approximately disjointed sets of variables. Also, decomposition of a function into An example of a sparsely connected dependency structure. Direction of causal flow is upward. Figure 2.43 A Chow-Liu Tree 131 Symmetry Two objects are symmetric if one object is obtained from the other by a transformation. KEY POINTS • If one object is obtained from the other by an invariant transformations, the two objects are symmetric to each other. • The value of the output is invariant under permutations of variables in the case of symmetric functions. • A binary operation is commutative if the operator, as a function of two variables, is a symmetric function. Two objects are symmetric to each other with respect to the invariant transformations if one object is obtained from the other by one of the transformations. It is an equivalence relation. In the case of symmetric functions, the value of the output is invariant under permutations of variables. These permutations form the symmetric group. Symmetric Functions From the form of an equation, one may observe that certain permutations of the unknowns result in an equivalent equation. In that case, the set of solutions is invariant under any permutation of the unknowns in the group generated by the aforementioned permutations. For example: (a − b)(b − c)(c − a) = 10; for any solution (a,b,c), permutations (a b c) and (a c b) can be applied, giving additional solutions (b, c, a) and (c, a, b). a2 c + 3ab + b2 c remains unchanged under interchanging of a and b. For a sphere, if φ is the longitude, θ the colatitude, and r the radius, as illustrated in Figure 2.44, then the great-circle distance is given by: d(θ1, ϕ1, θ2, ϕ2) = rcos−1 (cosθ1cosθ2 + sinθ1sinθ2cos(ϕ1 − ϕ2)). Some symmetries cleared from the problem can be verified in the formula; the distance is invariant under: • adding the same angle to both longitudes • interchanging longitudes and/or interchanging latitudes • reflecting both colatitudes in the value 90° In Algebra A relation is symmetric if, and only if, the corresponding boolean- valued function is a symmetric function. A binary operation is commutative if the operator, as a function of two variables, is a 134 symmetric function. Symmetric operators on sets include the union, intersection, and symmetric difference. A symmetric matrix, seen as a symmetric function of the row and column number, is an example of a symmetric function in algebra. Another example is the second order partial derivatives of a smooth function, seen as a function of the two indexes. Source: models/transformations/symmetry/ CC-BY-SA Boundless is an openly licensed educational resource 135 In this sphere, φ is the longitude and θ the colatitude. The radius is the green dotted line. The notation of \|0⟩, \|1⟩, and \|ψ⟩ may be unfamiliar (this is Bra-Ket notation), but none of the lengths depicted by 0, 1 and ψ is essential to the discussion of symmetry. Figure 2.44 Sphere Even and Odd Functions Functions that have an additive inverse can be classified as odd or even depending on their symmetry properties. KEY POINTS • The parity of a function does not necessarily reveal whether the function is odd or even. • An even function is symmetric about the y-axis. • An odd function is symmetric with respect to 180-degree rotation about the origin. Functions can be classified as "odd" or "even" based on their composition. These labels correlate with symmetry properties of the function. The terms "odd" and "even" can only be applied to a limited set of functions. For a function to be classified as one or the other, it must have an additive inverse. Therefore, it must have a number that, when added to it, equals 0. Oftentimes, the parity of a function will reveal whether it is odd or even. For example, the function f(x)=x2 is even and has an exponent (2) that is an even integer. This does not apply in every instance, however. For example, f(x)=\|x\| has an exponent that is of an odd integer, but is also an even function. Even Functions Even functions are algebraically defined as functions such that the following relationship holds for all values of x: f(x) = f(−x) Functions that satisfy the requirements of being even are symmetric about the y-axis. Therefore, a reflection about the y-axis produces no change in the points on the graph. Examples of even functions include: x2, x4, \|x\|, and cos(x) (Figure 2. 45). 136 y=x^2 is an even function; the order of x is even (2), and it is symmetrical with respect to the vertical axis. Figure 2.45 Graph of y=x^2 Odd Functions Odd functions are algebraically defined as functions such that the following relationship holds for all values of x: −f(x) = f(−x) This relationship can also be expressed as: f(x) + f(−x) = 0 Functions that satisfy the requirements of being odd are symmetric with respect to rotation about the origin. In other words, rotating the graph 180 degrees about the point of origin results in the same, unchanged graph. Examples of odd functions include: x, x3, x5, and sin(x) (Figure 2. 46). Source: models/transformations/even-and-odd-functions/ CC-BY-SA Boundless is an openly licensed educational resource 137 y=x^3 is an odd function; the order of x is odd (3), and it is symmetrical when rotated 180 degrees about the origin. Figure 2.46 Graph of y=x^3 Transformations of Functions Transformations alter a function while maintaining the original characteristics of that funcction. KEY POINTS • Transformations are ways that a function can be adjusted to create new functions. • Transformations often preserve the original shape of the function. • Common types of transformations include rotations, translations, reflections, and scaling (also known as stretching/shrinking). A transformation could be any function mapping a set, X, on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure. Examples include translations, reflections, rotations, and scaling. These can be carried out in Euclidean space, particularly in dimensions 2 and 3. They are also operations that can be performed using linear algebra and described explicitly using matrices. A translation, or translation operator, is an affine transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, or as the shifting of the origin of the coordinate system. In other words, if v is a fixed vector, then the translation Tv will work as Tv(p) = p + v. A graphical representation of vertical translations can be viewed in Figure 2.47. A reflection is a map that transforms an object into its mirror image. In geometry a "mirror" is a hyperplane of fixed points. For example, a reflection of the small English letter p in respect to a vertical line would look like q. In order to reflect a planar figure one needs the "mirror" to be a line (axis of reflection or axis of 138 The black curve indicates the original function while the red and blue curves represent two vertical translations of that function. Figure 2.47 Basic Quadratic Equation symmetry), while for reflections in the three-dimensional space one would use a plane (the plane of reflection or symmetry) for a mirror. A rotation is a transformation that is performed by "spinning" the object around a fixed point known as the center of rotation. You can rotate your object at any degree measure but 90° and 180° are two of the most common. Uniform scaling is a linear transformation that enlarges or diminishes objects. The scale factor is the same in all directions; it is also called a homothety or dilation. The result of uniform scaling is similar (in the geometric sense) to the original. Scaling can also be referred to as "stretching" or "shrinking" a function. Source: models/transformations/transformations-of-functions/ CC-BY-SA Boundless is an openly licensed educational resource Translations A translation of a function is a shift in one of the cardinal directions; it is represented by adding or subtracting from either y or x. KEY POINTS • A translation is a function that moves every point a constant distance in a specified direction. • A vertical translation is generally given by the equation y = f(x) + b. These translations shift the whole function up or down the y-axis. • A horizontal translation is generally given by the equation y = f(x−a). These translations shift the whole function side to side on the x-axis. A translation moves every point in a function a constant distance in a specified direction. In algebra, this essentially manifests as a vertical or horizontal shift of a function. A translation can be interpreted as the addition of a constant vector to every point or as shifting the origin of the coordinate system. Let's use the basic quadratic function to explore translations. In Figure 2.48, we can see three functions. The function in black is the 139 untransformed function f(x)=x2. The blue function has been translated up by 2. The red function has been translated down by 2. The equation for the blue line is: y = f(x) + 2 = x2 + 2 The equation for the red line is: y = f(x) − 2 = x2 − 2 These transformations are fairly straightforward. If a positive number is added, the function shifts up by the same amount. If a positive number is subtracted, the function shifts down by the same amount. In general, a vertical translation is given by the equation: y = f(x) + b Continuing with the basic quadratic function, lets look at horizontal translations as in Figure 2.49. Again, the basic black function is f(x) = x2 . The red function has been shifted to the left by 2 and has the equation: 140 In this function f(x) = x^2, the basic function is black. The blue and red functions are representations of the function translated up or down by two on the y-axis. Figure 2.48 Basic Quadratic Equation Vertical Translations In this function f(x) = x^2, the basic function is black. The blue and red functions are representations of the function translated up or down by two on the x-axis. Figure 2.49 Basic Quadratic Equation Horizontal Translations y = f(x + 2) = (x + 2)2 The blue function has been shifted to the left by 2 and has the equation: y = f(x − 2) = (x − 2)2 The general equation for a horizontal translation is given by: y = f(x − a) Note that the general form is x–a. When a is positive, the function is shifted to the right. When a is negative the shift is to the left. Source: models/transformations/translations/ CC-BY-SA Boundless is an openly licensed educational resource Reflections Reflections are a type of transformation that move an entire curve such that its mirror image lies on the other side of the y- or x-axis. KEY POINTS • A reflection swaps all of the x or y values across the y- or x- axis, respectively. It can be visualized by imagining a mirror lies across that axis. • A vertical reflection is given by the equation y = −f(x) and results in the curve being "reflected" across the x-axis. • A horizontal reflection is given by the equation y = f(−x) and results in the curve being "reflected" across the y-axis. Reflections are another type of transformations that can be done with functions. A reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points. This set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. In algebra, reflections are generally done over either the x- or y-axis. Figure 2.50 shows two types of reflections of an exponential function. A vertical reflection is a reflection across the x-axis as 141 shown below by the red function. The blue function has been reflected horizontally across the y-axis. A vertical reflection is given by the equation y = − f(x) In this general equation, all y values are switched to their negative counterparts while the x values remain the same. The result is that the curve is flipped over the x-axis. A horizontal reflection is given by the equation y = f(−x) The third type of reflection is a reflection across the line y = x. This reflection has the effect of swapping the variables x and y, which is exactly what happens in the case of an inverse function. In Figure 2. 50, the red and blue curves are the inverse of each other. Source: models/transformations/reflections/ CC-BY-SA Boundless is an openly licensed educational resource The black line represents the original function, while the red and blue lines are vertical and horizontal reflections, respectively. Figure 2.50 Reflected Functions 142 Stretching and Shrinking Stretching and shrinking refer to transformations that alter how compact a function looks in the x or y direction. KEY POINTS • When by either f(x) or x is multiplied by a number, functions can "stretch" or "shrink" vertically or horizontally, respectively, when graphed. • In general, a vertical stretch is given by the equation y = pf(x). If p>1, the graph stretches upward and downward. If p<1, the graph shrinks. • In general, a horizontal stretch is given by the equation y = f(x/q). If q>1, the graph shrinks horizontally, becoming more compact. If q<1, the graph stretches horizontally. In algebra, equations can be stretched horizontally or vertically along an axis by multiplying either x or y by a number, respectively. By multiplying f(x) by a number greater than one, all the y values of an equation increase. This leads to a "stretched" appearance in the vertical direction. If f(x) is multiplied by a value less than one, all the y values of the equation decrease, leading to a "shrunken" appearance in the vertical direction. Alternatively, if only x is multiplied, the graph stretches or shrinks in the horizontal direction. For examples, we will use the basic trignometric function f(x) = sin(x), which is black in the two graphs in Figure 2.51. Stretches can be a bit confusing with linear or quadratic functions, but they are much more straight forward with the sine function. The red function in Figure 2.51 has been stretched (dilated) vertically by a factor of 3 and follows the equation: y = 3f(x) = 3 sin x. In general a vertical stretch is given by the equation: y = pf(x) Here, the basic sine function is shown in black while the red function is stretched vertically by a factor of 3. Figure 2.51 Sine Function Stretching Vertically 143 If p is larger than 1, the function gets "taller." If p is smaller than 1, the function gets "shorter." The blue function in Figure 2.52 has been been stretched horizontally by a factor of 3 and has the equation: y = f(3x) = sin(3x). In general, a horizontal stretch is given by the equation: y = f(x /q) In the example above, q = 1/3. When q is larger than 1, the function will get "longer" and when q is smaller than 1, the function will "squish." Source: models/transformations/stretching-and-shrinking/ CC-BY-SA Boundless is an openly licensed educational resource Here, the basic sine function is shown in black while the blue function is stretched horizontally by a factor of 3. Figure 2.52 Sine Function Stretching Horizontally 144 Linear Equations and their Applications Linear equations are those with one or more variables of the first order. KEY POINTS • Linear equations can be expressed in the form: Ax+By+Cz +...=D. • Linear equations can contain one or more variables; it's possible for such an equation to include an infinite number of variables. • Linear equations can be used to solve for unknowns in any relationship in which all the variables are first order. 7 shelves? 2 * 10 + 7 * 5 = 55 Also to make it total from the 45 feet you also need 10 for a top and bottom.. So the actual answer should be 3 shelves. 2 * 10 for the sides 2 * 5 for the top and bottom and 3 * 5 for the shelves A linear equation is an algebraic equation that is of the first order —that is, an equation in which each term is either a constant or the product of a constant and a variable raised to the first power. Linear equations are commonly seen in two dimensions, but can be represented with three, four, or more variables. There is in fact a field of mathematics known as linear algebra, in which linear equations in up to an infinite number of variables are studied. Linear equations can therefore be expressed in general (standard) form as: ax + by + cz + . . . = d where a, b, c, and d are constants and x, y, and z are variables. Note that there can be infinitely more terms. This is known as general (or standard) form. Applications of Linear Equations Linear equations can be used to solve many problems, both everyday and technically specific. Consider, for example, a situation in which one has 45 feet of wood to use for making a bookcase. If the height and width are to be 10 feet and 5 feet, respectively, how many shelves can be made between the top and bottom of the frame? To solve this equation, we can use a linear relationship: Nv + Mh = 45 147 where v and h respectively represent the length in feet of vertical and horizontal sections of wood. N and M represent the number of vertical and horizontal pieces, respectively. Knowing that there will be only two vertical pieces, this formula can be simplified to: 2 ⋅ 10 + M ⋅ 5 = 45 Solving for M, we find that there is enough material for 5 shelves. Similarly, we can use linear equations to solve for the original price of an item that is on sale. For example, consider an item that costs $24 when on a 40% discount. If the original price is x, we can write the following relationship: x − 0.4 ⋅ x = 24 Solving for x, we find that the original price was $40. Using similar models we can solve equations pertaining to distance, speed, and time (Distance=SpeedTime); density (Density=Mass/ Volume); and any other relationship in which all variables are first order. Source: and-inequalities/linear-equations-and-functions/linear-equations- and-their-applications/ CC-BY-SA Boundless is an openly licensed educational resource 148 These linear equations can represent the trajectories of two vehicles. If the drivers want to designate a meeting point, they can algebraically find the point of intersection of the two functions. Figure 3.1 Linear Function Graph Zeros of Linear Functions A zero, or x-intercept, is the point at which a linear function's value will equal zero. KEY POINTS • A zero is a point at which a function's value will be equal to zero. Its coordinates are (x, 0), where x is equal to the zero of the graph. • Zeros can be observed graphically or solved for algebraically. • A linear function can have zero, one, or infinitely many zeros. If the function is a horizontal line (slope=0), it will have no zero unless its equation is y=0, in which case it will have infinitely many. If the line is non-horizontal, it will have one zero. A linear function, when graphed, is a straight line. It may or may not have an independent variable with a nonzero coefficient. If it does, that variable is of the first order; that is, the variable's exponent is equal to 1. An x-intercept, or zero, is a property of many functions. This is a point at which the function crosses the x-axis. Thus, that point will have the value (x, 0), where x is the zero. All lines will have one zero, provided they are functions of x to the first power, with the x-term having a nonzero coefficient. If the x- term has a zero coefficient, the line is horizontal and does not have an x-intercept unless the equation is y=0. If the equation is y=0, then all real values of x are valid x-intercepts. Zeros can be observed graphically (Figure 3.2). Additionally, they can be computed algebraically. Linear equations can be expressed in many forms, some of which are more explicit than others in revealing the x- and y-intercepts. The general (standard) form for a line is: Ax + By = C 149 Graphically, it can be observed that the equation y=-x +5 has a zero at x=5, and that y=1/2x+2 has a zero at x=-4 Figure 3.2 Linear Function Graph where x and y are variables and A, B, and C are constants. In this form, the ratio of C/A is equal to the x-intercept (zero), so long as A is not equal to 0. If A is 0, the line is horizontal. Lines can also be expressed in intercept form: x a + y b = 1 In this case, the x-intercept (zero) is a and the y-intercept is b. Another common form is slope-intercept form: y = mx + b where x and y are variables, m is the slope of the line and b is the y- intercept. To find the x-intercept, set y=0 (as it is at the point of the zero) and solve for x. Rearranging the formula gives: x = −b m Thus, the x value at the zero will be equal to the quotient of the opposite of the y-intercept and the slope. Source: and-inequalities/linear-equations-and-functions/zeros-of-linear- functions/ CC-BY-SA Boundless is an openly licensed educational resource 150 Formulas and Problem Solving Linear equations can be used to solve many everyday and technically specific problems. KEY POINTS • A linear equation can be used to solve any problem that includes constants and variable(s) of first order. • A linear equation can be solved for any one variable provided that the values of all other variables are known. • Linear equations can be used to calculate tip, cost of goods, velocity, simple interest, and many more variables. Linear equations can be used to solve many practical and technical problems. Such an equation may include many variables so long as all are of the first order, and the value of any one variable can be calculated if the values of all the other variables are known. For example, one can use a linear equation to determine the amount of interest accrued on a home equity line of credit after a given amount of time. Consider the hypothetical situation in which you need money to make home improvements and can open a $20,000 credit line at an interest rate of 2.5% per year. You plan to pay off the debt in its entirety within 15 months. To find out how much it will cost you can use following formula: I = P ⋅ r ⋅ T Where I is interest, p is the principal amount loaned ($20,000), r is the interest rate (2%, or 0.02) per year, and T is the number of years elapsed (15 months will be 1.25 years). Plugging the known values into the above formula, we can determine that you will pay $500 in interest. There are many other common formulas that can be used for everyday computations. Some have more variables than others, but none has a variable of order higher than one. Let's take a few examples of other linear equations, namely velocity, gratuity (tip), and cost of purchased goods: V = d T The formula relating velocity (V), distance (d), and time (T). G = c ⋅ r (Figure 3.3) The formula relating gratuity (G), cost (c), and desired percent gratuity (r, expressed as a decimal). A ⋅ x + B ⋅ y + C ⋅ z + . . . = T 151 Where A, B and C represent the quantities of goods that cost x, y and z, respectively. This could be expanded or contracted relative to the number of different items purchased. T represents the total cost of goods purchased. Source: and-inequalities/linear-equations-and-functions/formulas-and- problem-solving/ CC-BY-SA Boundless is an openly licensed educational resource In the above plot, the dependent variable (y) represents gratuity (tip) as a function of cost of the bill (x) before gratuity. The tip rate is 18%. Figure 3.3 Gratuity as a function of bill price 152 The Complex-Number System A complex number has the form a + bi, where a and b are real numbers and i is the imaginary unit. KEY POINTS • A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. • The real number a of the complex number z = a + bi is called the real part of z, and the real number b is often called the imaginary part. • The real part is denoted by Re(z) or ℜ(z), and the imaginary part b is denoted by Im(z) or ℑ(z). A complex number is a number that can be put in the form a + bi, where a and b are real numbers and i is called the imaginary unit, where i2 = −1. In this expression, a is called the real part and b the imaginary part of the complex number. Complex numbers extend the idea of the one-dimensional number line to the two- dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) as shown in the figure below. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with only real numbers (Figure 3. 4). Complex numbers are used in many scientific fields, including engineering, electromagnetism, quantum physics, and applied mathematics, such as chaos theory. Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow for solutions to certain equations that have no real solution. For example, the equation (x + 1)2 = − 9 154 A complex number can be visually represented as a pair of numbers (a,b) forming a vector on the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and "i" is the imaginary unit, satisfying i^2 = −1. Figure 3.4 Complex number illustration has no real solution, since the square of a real number is either 0 or positive. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the imaginary unit i where i2 = -1, so that solutions to equations like the preceding one can be found. In this case, the solutions are −1 ± 3i. In fact, not only quadratic equations, but all polynomial equations in a single variable can be solved using complex numbers. Another is example is the complex number −3.5 + 2i. It is common to write a for a + 0i and bi for 0 + bi. Moreover, when the imaginary part is negative, it is common to write a − bi with b > 0 instead of a + (−b)i, for example 3 − 4i instead of 3 + (−4)i. The real number a of the complex number z = a + bi is called the real part of z, and the real number b is often called the imaginary part. By this convention, the imaginary part is a real number – not including the imaginary unit: hence b, not bi, is the imaginary part. The real part is denoted by Re(z) or ℜ(z), and the imaginary part b is denoted by Im(z) or ℑ(z). Some authors write a + ib instead of a + bi (scalar multiplication between b and i is commutative). A real number a can usually be regarded as a complex number with an imaginary part of zero, that is to say, a + 0i. A pure imaginary number is a complex number whose real part is zero, that is to say, of the form 0 + bi. Source: and-inequalities/complex-numbers/the-complex-number-system/ CC-BY-SA Boundless is an openly licensed educational resource 155 Addition and Subtraction, and Multiplication Complex numbers are added by adding the real and imaginary parts; multiplication follows the rule i2 = − 1. KEY POINTS • Complex numbers are added by adding the real and imaginary parts of the summands. That is to say: (a + bi) + (c + di) = (a + c) + (b + d)i. • Similarly, subtraction is defined by (a + bi) - (c + di) = (a - c) + (b - d)i. • The multiplication of two complex numbers is defined by the following formula: (a + bi)(c + di) = (ac - bd) + (bc + ad)i. Addition and Subtraction Complex numbers are added by adding the real and imaginary parts of the summands. That is to say: (a + bi) + (c + di) = (a + c) + (b + d)i Similarly, subtraction is defined by: (a + bi) − (c + di) = (a − c) + (b − d)i Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram, three of whose vertices are O, A, and B (as shown in Figure 3.5). Multiplication The multiplication of two complex numbers is defined by the following formula: (a + bi)(c + di) = (ac − bd) + (bc + ad)i In particular, the square of the imaginary unit is -1: 156 Addition of two complex numbers can be done geometrically by constructing a parallelogram. Figure 3.5 Addition of complex numbers i2 = i ⋅ i = − 1 The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of the imaginary unit. Indeed, if i is treated as a number so that di means d time i, the above multiplication rule is identical to the usual rule for multiplying the sum of two terms. (a + bi)(c + di) = ac + bci + adi + bidi (by the distributive law) = ac + bidi + bci + adi (by the commutative law of addition) = ac + bdi2 + (bc + ad)i (by the commutative law of multiplication) = (ac − bd) + (bc + ad)i (by the fundamental property of the imaginary unit) Source: and-inequalities/complex-numbers/addition-and-subtraction-and- multiplication/ CC-BY-SA Boundless is an openly licensed educational resource Complex Conjugates and Division The complex conjugate of x + yi is x - yi, and the division of two complex numbers can be defined using the complex conjugate. KEY POINTS • The complex conjugate of the complex number z = x + yi is defined to be x - yi. It is denoted z. • The division of two complex numbers is defined in terms of complex multiplication and real division. Where at least one of c and d is non-zero: (a + bi)/(c + di) = (ac + bd)/(c2 + d2) + [(bc - ad)/(c2 + d2)]i. • Division can be defined in this way because of the following observation: (a + bi)/(c + di) = [(a + bi)(c - di)]/[(c + di)(c - di)] = (ac + bd)/(c2 + d2) + [(bc - ad)/(c2 + d2)]i. As shown earlier, c - di is the complex conjugate of the denominator c + di. Conjugation The complex conjugate of the complex number z = x + yi is defined as x - yi. It is denoted z. Geometrically, z is the "reflection" of z about the real axis (as shown in the figure below). 157 Specifically, conjugating twice gives the original complex number: z** = z (Figure 3.6). The real and imaginary parts of a complex number can be extracted using the conjugate, respectively: Re(z) = (1/2)(z + z) Im(z) = (1/2i)(z − z) Moreover, a complex number is real if and only if it equals its conjugate. Conjugation distributes over the standard arithmetic operations: (z + w)* = z* + w* (zw)* = zw (z /w)* = z/w The reciprocal of a nonzero complex number z = x + yi is given by 1/z = z/zz = z/(x2 + y2 ) Division The division of two complex numbers is defined in terms of complex multiplication (described above) and real division. Where at least one of c and d is non-zero: (a + bi) (c + di) = (ac + bd) (c2 + d2) + (bc − ad) (c2 + d2) i Division can be defined in this way because of the following observation: (a + bi) (c + di) = (a + bi)(c − di) (c + di)(c − di) = (ac + bd) (c2 + d2) + (bc − ad) (c2 + d2) i As shown earlier, c - di is the complex conjugate of the denominator c + di. Neither the real part c nor the imaginary part d of the denominator can be equal to zero for division to be defined. 158 Geometric representation of z and its conjugate in the complex plane. Figure 3.6 Complex Conjugate Quadratic Equations and Quadratic Functions Equations that are quadratic are of the second order, and have the form f(x) = ax2 + bx + c. KEY POINTS • A quadratic function is of the form: f(x) = ax2 + bx + c where a is a nonzero constant, b and c are constants of any value, and x is a variable. A quadratic equation is a quadratic function set equal to 0. • The shape of a quadratic function is known as a parabola. • The solution(s) to a quadratic equation is/are known as its zero(s), or root(s). A quadratic function is of the form: f(x) = ax2 + bx + c where a, b, and c are constants and x is a variable. The constants b and c can take any value, but a cannot be equal to 0. The single defining feature of quadratic functions is that they are of the second order (degree). Therefore, in all quadratic functions the highest exponent of x in a nonzero term is equal to 2. The shape of a quadratic function is called a parabola. If a is positive, the shape resembles a U; if a is negative, the U is flipped upside-down. Quadratic functions can be expressed in many different forms. The form written above is standard form. Additionally: f(x) = a(x − x1)(x − x2) is known as factored form, where x1 and x2 are the zeros, or roots, of the equation. These are x values at which the function crosses the y- axis (and thus y equals 0). Also: f(x) = a(x − h)2 + k 161 The function f(x)=x^2-x-2 is quadratic; it has a parabolic shape and is of the second order. Figure 3.7 Polynomial is known as the vertex form, where h and k are respectively the coordinates of the vertex, the point at which the function reaches either its maximum (if a is negative) or minimum (if a is positive). A quadratic equation is a specific case of a quadratic function, with the function set equal to 0: ax2 + bx + c = 0 As in a quadratic function, x is a variable, a, b, and c are constants, and a cannot equal 0. With a, b, and c known, a quadratic equation can be solved for x, such solutions are known as zeros. There are several ways of finding x, which depending on the equation can have 0, 1, or 2 values, but these methods will be discussed later. Source: and-inequalities/quadratic-equations-functions-and-applications/ quadratic-equations-and-quadratic-functions/ CC-BY-SA Boundless is an openly licensed educational resource Completing the Square Completing the square is a common method for solving quadratic equations: 0 = a(x − h)2 + k. KEY POINTS • In the form, a(x − h)2 + k, h and k are constants and should be found. • Once a quadratic polynomial is in the form: 0 = a(x − h)2 + k, one can solve for two values of x (using the positive and negative square roots). • When a parabola is plotted, h is the x-coordinate of the axis of symmetry, and k is the minimum value (or maximum value, if a < 0) of the quadratic function. Along with factoring and using the quadratic formula, completing the square is a common method for solving quadratic equations. Consider a generic quadratic equation: 0 = ax2 + bx + c "Completing the square," one can convert that equation to the form: 0 = a(x − h)2 + k where one must find constants h and k. 162 This method is meant to be used when the quadratic function is not a perfect square; the value of k is meant to adjust the function to compensate for the difference between the expanded a(x-h)2 and the quadratic function ax2+bx+c. There is a simple formula in elementary algebra for computing the square of a binomial: (x + p)2 = x2 + 2px + p2 For example: (x + 3)2 = x2 + 6x + 9 (p = 3) (x − 5)2 = x2 − 10x + 25 (p = − 5) In any perfect square, the number p is always half the coefficient of x, and the constant term is equal to p2. Consider the following quadratic polynomial: x2 + 10x + 28 This quadratic is not a perfect square, since 28 is not the square of 5: (x + 5)2 = x 2 + 10x + 25 However, it is possible to write the original quadratic as the sum of this square and a constant: x2 + 10x + 28 = (x + 5)2 + 3 Thus, -5 is equal to h and 3 is equal to k. Knowing that: (x + 5)2 + 3 = x2 + 10x + 28 = 0 We can solve for x: (x + 5)2 = 3 x + 5 = − 3 and x + 5 = 3 Thus: x = − 3 − 5 and x = 3 − 5 Let's go through the steps of completing the square another quadratic equation: 3x2 + 12x + 27 = 3(x2 + 4x + 9) Factor out the coefficient a, and then complete the square for the resulting monic polynomial: = 3[(x + 2)2 + 5] 163 = 3(x + 2)2 + 15 Which brings us back to the form: a(x − h)2 + k where h=-2 and k=15. Graphical Representation The graph of any quadratic function is a parabola in the xy-plane. Given a quadratic polynomial of the form: a whose vertex is at (h, 0), as shown in Figure 3.8. In contrast, the graph of the function f(x) + k = x2 + k is a parabola shifted upward by k whose vertex is at (0, k), as shown in Figure 3.9. Combining both horizontal and vertical shifts yields f(x − h) + k = (x − h)2 + k is a parabola shifted to the right by h and upward by k whose vertex is at (h, k), as shown in Figure 3.10. 164 Graphs of quadratic functions shifted upward by k = 0, 5, 10, and 15. Figure 3.9 K values in Quadratic Functions Graphs of quadratic functions shifted to the right by h = 0, 5, 10, and 15. Figure 3.8 H values in Quadratic Functions Source: and-inequalities/quadratic-equations-functions-and-applications/ completing-the-square/ CC-BY-SA Boundless is an openly licensed educational resource The Quadratic Formula The zeros of a quadratic equation can be found not only through factoring, but by solving what is known as the quadratic formula. KEY POINTS • The quadratic formula is: x = −b ± b2 − 4ac 2a where a, and b are the coefficients of the x2 and x terms, respectively, in a quadratic equation, and c is the value of the equation's constant. • If the discriminant (b2-4ac) is equal to zero, both values of x obtained will be the same, real number. • If the discriminant is positive, both roots will be distinct and real. • If the discriminant is negative, there will be two imaginary roots with i coefficients. Recall that a quadratic equation essentially reveals one or two points of a quadratic function. Whereas a quadratic function will have the form: f(x) = ax2 + bx + c 165 Graphs of quadratic functions shifted upward and to the right by 0, 5, 10, and 15. Figure 3.10 Quadratic functions at varying h and k values with infinite solutions possible for f(x) and x, a quadratic equation will have the form: 0 = ax2 + bx + c In a quadratic equation, there is only one unknown value (x), not two (f(x) and x). As such, one can solve for x, which happens to be the value of the x-intercept(s), also known as roots or zeros. If f(x)=0 at the vertex of the parabola, there will be one value of x that fits; otherwise, there will be two. Finding X X can be found not only by factoring, but also with use of the quadratic formula. That is: x = −b ± b2 − 4ac 2a where the symbol ± indicates there will be two solutions, one that involves adding the square root of the discriminant (b2-4ac), and the other found by subtracting said square root. The resulting x values (zeros) may or may not be distinct, and may or may not be real. Determining values If the discriminant is equal to zero, both values of x obtained will be the same, real number. If the discriminant is positive, both roots 166 Note that the function with discriminant (Δ) value greater than 0 crosses the x-axis twice, explaining the two distinct values of zeros obtained from the quadratic formula. When the discriminant value is 0, there is only one point at which the function touches the x-axis; hence there being only one calculable root. And for discriminants less than 0, the functions never touch the x-axis at all, resulting in no real roots, only imaginary ones. Figure 3.11 Parabolas of differing discriminant values will be distinct and real. If the discriminant is negative, there will be two imaginary roots, defined by: −b 2a + i −(b2 − 4ac) 2a and −b 2a − i −(b2 − 4ac) 2a All the possibilities concerning number of solutions to a quadratic equation can be explained by examining the graphs of parabolas with different discriminant values (Figure 3.11). Source: and-inequalities/quadratic-equations-functions-and-applications/the- quadratic-formula/ CC-BY-SA Boundless is an openly licensed educational resource The Discriminant The discriminant of a polynomial is a function of its coefficients that reveals information about the polynomial's zeros, or roots. KEY POINTS • Δ = b2 − 4ac is the formula for a quadratic expression's discriminant, where Δ is the discriminant, and a, b and c are coefficients from the parent function. • If Δ is greater than 0, the polynomial has two real, distinct zeros. If Δ is equal to 0, the polynomial has only one real zero. And if Δ is less than 0, the polynomial has no real zeros, only two distinct complex zeros. • A zero is the x value whereat the function crosses the x-axis. That is, it is the x-coordinate at which the function's value equals 0. The discriminant of a polynomial is a function of its coefficients that reveals information about the polynomial's zeros, or roots. A zero is the x value whereat the function crosses the x-axis. That is, it is the x-coordinate at which the function's value equals 0. The concept of a discriminant can be applied to many different orders of polynomial; depending on degree, the discriminant can be calculated with differing equations. In this case, we will discuss its 167 application to those polynomials of the second order (quadratics). Recall the form of a quadratic polynomial: ax2 + bx + c where a, b and c are constants (a must be nonzero), and x is a variable. The discriminant (Δ) of the above polynomial can be calculated based on the equation: Δ = b2 − 4ac If Δ is greater than 0, then the polynomial has two real, distinct zeros. If Δ is equal to 0, then the polynomial has only one real zero. And if Δ is less than 0, then the polynomial has no real zeros; it has only two distinct complex zeros. Example Consider, for example, the quadratic function: f(x) = x2 − x − 2 Using 1 as the value of a, -1 as the value of b, and -2 as the value of c, the discriminant of this function can be determined as follows: Δ = (−1)2 − 4 ⋅ 1 ⋅ (−2) Δ = 9 Because Δ is greater than 0, the function has two distinct, real zeros. Checking graphically, we can confirm this is true; the zeros of the function can be found at x=-1 and x=2 (Figure 3.12). Source: and-inequalities/quadratic-equations-functions-and-applications/the- discriminant/ CC-BY-SA Boundless is an openly licensed educational resource 168 The discriminant of f(x)=x^2-x-2 is 9. Because the value is greater than 0, the function has two distinct, real zeros. The graph of f(x) shows that it clearly has two roots: the function crosses the x-axis at x=-1 and x=2. Figure 3.12 Polynomial Reducing Equations to a Quadratic Quartic equations with no terms of odd-degree can be reduced to quadratics; the square roots of their solutions solve the parent equation. KEY POINTS • A biquadratic equation (quartic equation with no terms of odd-degree) has the form 0 = ap2 + bp + c (where p=x2). • The values of p can be found by graphing, factoring, completing the square, or using the quadratic formula. Their square roots (positive and negative) are the values of x that satisfy the original equation. • Higher-order equations can be solved by a similar process that involves reducing their exponents. The requirement is that there are two terms of x such that the ratio of the highest exponent of x to the lower is 2:1. Quartic (fourth-degree) equations can be very difficult to solve. In some special situations, however, they can be more manageable. If, for example, a quartic equation is biquadratic—that is, it includes no terms of an odd-degree— there is a quick way to find the zeroes of a quartic function by reducing it into a quadratic form. Consider a quartic function with no odd-degree terms, which therefore has the form: 0 = ax4 + bx2 + c Substituting p in place of x2 (and thus p2 in place of x4), this can be reduced to an equation of lower degree: 0 = ap2 + bp + c This quadratic equation can be solved for p by any of a number of methods (by graphing, factoring, completing the square, or by using the quadratic formula). Once values of p are found, each positive value thereof can be used to find two values of x as such: x = p As with every square root, that of p will have two (positive and negative) values. Example Consider, for example, the quartic equation: 0 = x4 − 12x2 + 20 We can substitute p in place of x2: 169 0 = p2 − 12p + 20 And solve for p using the quadratic formula: p = 12 ± (−12)2 − 4 ⋅ 1 ⋅ 20 2 ⋅ 1 p = 2 and p = 10 Knowing that p=x2, we can use each value of p to solve for two values of x: x = ± 2 and x = ± 10 A similar procedure can be used to solve higher-order equations. The requirement is that there are two terms of x such that the ratio of the highest exponent of x to the lower is 2:1. Source: and-inequalities/quadratic-equations-functions-and-applications/ reducing-equations-to-a-quadratic/ CC-BY-SA Boundless is an openly licensed educational resource This equation is biquadratic and has three distinct zeroes that can be found by reducing the terms to quadratic form and finding their square roots. Figure 3.13 Graph of f(x)=x^4-2x^2 170 Applications and Problem Solving Quadratic relationships between variables are commonly found in physical sciences, engineering, and elsewhere. KEY POINTS • The Pythagorean Theorem, c2 = a2 + b2 relates the length of the hypotenuse (c) of a right triangle to the lengths of its legs (a and b). • Problems involving gravity and projectile motion are typically dependent upon a second-order variable, usually time or initial velocity depending on the relationship. • Coulomb's Law, which relates electrostatic force, charge amount and distance between two charged particles, has a second-order dependence on the separation of the particles. Solving for either charge results in a quadratic function. Quadratic relationships between variables are commonly found in physical sciences, engineering, and elsewhere. Perhaps the most universally used example of quadratic relationships in problem solving concerns right triangles. The Pythagorean Theorem The Pythagorean Theorem is used to relate the three sides of right triangles. It states: c2 = a2 + b2 According thereto, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the two legs (a and b) of the triangle. This has been proven in many ways, among the most famous of which was devised by Euclid (Figure 3.14). Euclid used this diagram to explain how the sum of the squares of the triangle's smaller sides (pink and blue) sum to equal the area of the square of the hypotenuse. Figure 3.14 Euclid's Proof of the Pythagorean Theorem 171 Gravity and Projectile Motion Most all equations involving gravity include a second-order relationship. Consider, for example, the equation relating gravitational force (F) between two objects to the masses of each object (m1 and m2) and the distance between them (r): F = G m1m2 r2 The shape of this function is not a parabola, but becomes such a shape if rearranged to solve for m1 or m2. The maximum height of a projectile launched directly upwards can also be calculated from a quadratic relationship. The formula relates height (h) to initial velocity (v0) and gravitational acceleration (g): h = v2 0 2g The same maximum height of a projectile launched directly upwards can be found using the time at the projectile's peak (th): h = v0th 1 2 gt2 h Substituting any time (t) in place of th leaves the equation for height as a function of time. Electrostatic Force The form of the equation relating electrostatic force (F) between two particles, the particles' respective charges (q1 and q2), and the distance between them (r) is very similar to the aforementioned formula for gravitational force: F = q1q2 4πϵ0r2 This is known as Coulomb's Law. Solving for either charge results in a quadratic equation where the charge is dependent on r2. Source: and-inequalities/quadratic-equations-functions-and-applications/ applications-and-problem-solving--2/ CC-BY-SA Boundless is an openly licensed educational resource 172 Quadratic Functions of the Form: f(x) = a(x − h)2 + k The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis. KEY POINTS • If the quadratic function is set equal to zero, then the result is a quadratic equation. • The solutions to the equation are called the roots of the equation. • The coefficient a controls the speed of increase (or decrease) of the quadratic function from the vertex. • The coefficients b and a together control the axis of symmetry of the parabola. • The coefficient b alone is the declivity of the parabola as y- axis intercepts. • The coefficient c controls the height of the parabola, or more specifically, it is the point where the parabola intercepts the y-axis. A quadratic function can be expressed in three formats: f(x) = ax2 + bx + x is called the standard form. f(x) = a(x − x1)(x − x2) is called the factored form, where x1 and x2 are the roots of the quadratic equation (used in logistic map). f(x) = (x − h)2 + k is called the vertex form, where h and k are the x and y coordinates of the vertex, respectively. To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots x1 and x2. To convert the standard form to vertex form, one needs a process called completing the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand, and/or distribute the factors. Looking at a general case in which b=0. Figure 3.15 shows the dependency on the quadratic form. 174 This is a standard polynomial, with f(x) plotted against x, for f(x)=a2 for various values of a. Figure 3.15 Standard Polynomial In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form f(x) = ax2 + bx + x to the form a( . . . f(x) = (x − h)2 + k and one must find h and k. In analytic geometry, the minimum value graph of any quadratic function is a parabola in the xy-plane. Given a quadratic polynomial of the form (x − h)2 + k or a and whose vertex is at (h, 0), as shown in Figure 3.16. In contrast, the graph of the function f(x) + k = x2 + k is a parabola shifted upward by k and whose vertex is at (0, k), as shown in Figure 3.17. Combining both horizontal and vertical shifts yields f(x − h) + k = (x − h)2 + k, a parabola shifted to the right by h and upward by k and whose vertex is at (h, k), as shown in the bottom (Figure 3.18). 175 The graph of the function ƒ(x − h) = ( x − h)2 is a parabola shifted to the right by h, whose vertex is at (h, 0). Figure 3.16 Graphs of quadratic functions shifted to the right by h = 0, 5, 10, and 15. Source: and-inequalities/graphs-of-quadratic-functions/quadratic-functions- of-the-form-f-x-a-x-h-2-k/ CC-BY-SA Boundless is an openly licensed educational resource Quadratic Functions of the Type f(x) = ax2 + bx + c Where a is not Equal to 0 A quadratic function, in mathematics, is a polynomial function of the form y = ax2 + bx + c. KEY POINTS • The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis. • If the quadratic function is set equal to zero, then the result is a quadratic equation. • The solutions to the equation are called the roots of the equation. The adjective quadratic comes from the Latin word quadrātum ("square"). A term like x2 is called a square in algebra because it is the area of a square with side x. In general, a prefix quadr(i)- indicates the number 4. Examples are quadrilateral and quadrant. Quadratum is the Latin word for square because a square has four sides. A quadratic function in the form: 176 The graph of the function ƒ(x) + k = x 2 + k is a parabola shifted upward by k, whose vertex is at (0, k). Figure 3.17 Graphs of quadratic functions shifted upward by k = 0, 5, 10, and 15. The graph of the function ƒ(x − h) + k = (x − h)2 + k is a parabola shifted to the right by h and upward by k andwhose vertex is at (h, k). Figure 3.18 Graphs of quadratic functions shifted upward and to the right by 0, 5, 10, and 15. f(x) = ax2 + bx + x is in standard form. Regardless of the format, the graph of a quadratic function is a parabola (as shown in Figure 3.19). • If a>0 (or is a positive number), the parabola opens upward. • If a<0 (or is a negative number), the parabola opens downward. The coefficient a controls the speed of increase (or decrease) of the quadratic function from the vertex. A larger, positive a makes the function increase faster and the graph appear more closed. The coefficients b and a together control the axis of symmetry of the parabola (also the x-coordinate of the vertex), which is at x = − b/2a. The coefficient b alone is the declivity of the parabola as y-axis intercepts. The coefficient c controls the height of the parabola, or more specifically, it is the point where the parabola intercepts the y-axis. A parabola is a conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. Another way to generate a parabola is to examine a point (the focus) and a line (the directrix). The locus of points in that plane that are equidistant from both the line and point is a parabola. In algebra, parabolas are frequently encountered as graphs of quadratic functions, such as y = x2 The vertex of a parabola is the place where it turns; hence, it's also called the turning point. If the quadratic function is in vertex form, the vertex is (h, k). Source: and-inequalities/graphs-of-quadratic-functions/quadratic-functions- of-the-type-f-x-ax-2-bx-c-where-a-is-not-equal-to-0/ CC-BY-SA Boundless is an openly licensed educational resource 177 where a=1, b=-1, c=-2 Figure 3.19 A graphed quadratic equation Applications Graphs display complete pictures of quadratic functions and from them one can easily find critical values of the function by inspection. KEY POINTS • If several key points on a function are desired, it can become tedious to calculate each algebraically. • Rather than calculating each key point of a function, one can find these values by inspection of its graph. • Graphs of quadratic functions can be used to find key points in many different relationships, from finance to science and beyond. Given the algebraic equation for a quadratic function, one can calculate any point on the function, including critical values like minimum/maximum and x- and y-intercepts. These calculations can be more tedious than is necessary, however. A graph contains all the above critical points and more, and is essentially a clear and concise representation of a function. If one needs to determine several values on a quadratic function, glancing at a graph is quicker than calculating several points. Consider the function: F(x) = − x2 10 + 50x − 750 Suppose this models the profit (f(x)) in dollars that a company earns as a function of the number of products (x) of a given type that are sold, and is valid for values of x greater than or equal to 0 and less than or equal to 500. If one wanted to find the number of sales required to break even, the maximum possible loss (and the number of sales required for this loss), and the maximum profit (and the number of sales required for this profit), one could calculate algebraically or simply reference a graph (Figure 3.20). By inspection, we can find that the maximum loss is $750, which is lost at both 0 and 500 sales. Maximum profit is $5500, which is 178 Critical points of the function can be determined by inspection. This can be less time- consuming than performing several calculations. Figure 3.20 Profit versus sales achieved at 250 sales. The break-even points are between 15 and 16 sales, and between 484 and 485 sales. The above example pertained to business sales and profits, but a similar model can be used for many other relationships in finance, science and otherwise. For example, the reproduction rate of a strand of bacteria can be modeled as a function of differing temperature or pH using a quadratic functionality. Source: and-inequalities/graphs-of-quadratic-functions/applications/ CC-BY-SA Boundless is an openly licensed educational resource 179 Rational Equations A rational equation is when two rational expressions are set equal to each other and x values are found that make the equation true. KEY POINTS • You can have multiple values that will satisfy the equations. It is important to check your work to make sure that all the values you find all work for the equations. • If you have a rational equation where the denominators are the same, then the numerators must be the same.This gives us a strategy: Find a common denominator, and then set the numerators equal. • It is important to write each equation in its simplest form. This means to factor each problem out as much as you possibly can before starting. A rational equation means that you are setting two rational expressions equal to each other. The goal is to find the x value or values that make the equation true. Suppose you are told that: x 8 = 3 8 We can use simple algebra to solve this equation. There are several ways to do this. One way is to isolate the variable on one side. And because we are familiar with basic algebra, we know that what you do to one side, you must also do to the other side. ( 8 1 ) x 8 = 3 8 * ( 8 1 ) The 8 on both sides will cancel out the denominator in the expressions and return this: x = 3 You can also come to this conclusion by deductive reasoning. If you think about it, the x in this equation has to be a 3. That is to say, if x=3 then this equation is true; for any other x value, this equation is false. This leads us to a very general rule about rational equations: If you have a rational equation where the denominators on either side of the equation are the same, then their respective numerators must be the same value, even though they might be expressed differently.This suggests a strategy: Find a common denominator, and then set the numerators equal. To start, we rewrite both fractions with their common denominator. You can find the common denominator by first factoring each expression, and then looking to see what each must be multiplied by 181 to be equal to one another. This is easier illustrated as an example. Lets start with this equation: 3 x2 + 12x + 36 = 4x x3 + 4x2 − 12x Now we factor the denominators: 3 (x + 6)2 = 4x x(x + 6)(x − 2) The next step is to make both denominators similar to each other: (x)(x − 2) * [ 3 (x + 6)2 ] = [ 4x x(x + 6)(x − 2) ] * (x + 6), since the denominators are equal, we can now assume the numerators are equal. 3x(x − 2) = 4x(x + 6) We then multiply it out: 3x2 − 6x = 4x2 + 24x Now, we have simplified this into a basic quadratic equation. It is always easiest to deal with equations by isolating the variables to one side: 0 = x2 + 30x and then factor: 0 = x(x + 30) Two solutions to the quadratic equation, x=0 and x=-30. However, in this case, x=0 is not valid, why is this? Well, lets look again at the original expression of the equation: 3 x2 + 12x + 36 = 4x x3 + 4x2 − 12x Now, lest plug in x=0: 3 02 + 12 * 0 + 36 = 4 * 0 03 + 4 * 02 − 12 * 0 3 36 = 0 0 We know that you can not divide a number by 0 or have any other number divide a 0, so this answer is invalid. Now, why don't we try and plug in x=-30 182 3 (−30)2 + 12 * (−30) + 36 = 4 * (−30) (−30)3 + 4 * (−30)2 − 12 * (−30) 3 900 − 360 + 36 = −120 −27000 + 3600 + 360 3 576 = −120 −23040 If you were to divide both the numerator and denominator by 40, you will see that the terms are equal. So this problem actually has only one solution, x=–30. As this example shows, it is important to always check your work. If you graphed the two function, the two graphs would intersect at one point only: the point when x=–30. Source: and-inequalities/further-equation-solving/rational-equations/ CC-BY-SA Boundless is an openly licensed educational resource 183 Figure 3.21 A graph of 3x^2-6x and 4x^2+24x We can see that they intersect at both x=-30 and x=0. However, from our original equation, we know that x=0 is outside of the domain, and so is not a valid solution. Radical Equations Equations involving radicals are often solven by moving the radical to one side then squaring both sides. KEY POINTS • When solving equations that involve radicals, begin by asking: is there an x under the square root? The answer to this question will determine the way you approach the problem. • If there is not an x under the square root—if only numbers are under the radicals—the problem can be solved much the same way as if it had no radicals. However, if there is an x under a square root, then move everything except the radical to one side, then square both sides. • Squaring both sides can introduce false answers—so it is important to check the answers after solving! When solving equations that involve radicals, begin by asking: is there an x under the square root? The answer to this question will determine the way the problem is approached. If there is not an x under the square root—if only numbers are under the radicals—the problem can be solved much the same way as if it had no radicals. Example 1: Radical Equation with No Variables Under Square Roots Sample problem: no variables under radicals. 2x + 5 = 7 − 3x 1. Get everything with an x on one side, everything else on the other 2x + 3x = 7 − 5 2. Factor out the x x( 2 + 3) = 2 3. Then divide to solve for x x = 2 2 + 3 The key thing to note about such problems is that both sides of the equation do not have to be squared. 2 may look complicated, but it is just a number—it can be found on a calculator—it functions in the equation just the way that the number 10, or 1 3 , or π would. If there is an x under the square root, the problem is must be approached differently. Both sides have to be squared to get rid of 184 So the algebra yielded two solutions: 1/4 and –1. Checking, however, it is discovered that only the first solution is valid. This problem demonstrates how important it is to check solutions whenever squaring both sides of an equation. Looking at the equation graphically is another way to check algebraic solutions, as seen in Figure 3.22, it only crosses the y-axis once. If variables under the radical occur more than once, then this procedure must be done multiple times. Each time, isolate a radical and then square both sides. Source: and-inequalities/further-equation-solving/radical-equations/ CC-BY-SA Boundless is an openly licensed educational resource This graph shows where the given equation is equal to zero. This is equivalent to showing where (x+2)^(1/2)+3x=5x+1. Figure 3.22 Graph of (x+2)^(1/2)-2x-1 186 Equations with Absolute Value To solve an equation with an absolute value, first move the absolute value to one side, then solve for the positive and negative cases. KEY POINTS • Absolute value is one of the simplest functions—and paradoxically, one of the most problematic. On the face of it, nothing could be simpler: it just means "whatever comes in, a positive number comes out". • In order to solve an absolute value equation: 1) Isolate the absolute value algebraically. 2) Think through the problem. That is to say, set the absolute value term equal to the other side of the equation, then the opposite of the other side (-1 times it). 3) Finally, do more algebra to isolate x. • Absolute values are always positive. An absolute value equation can have, at most, two solutions. Absolute value is one of the simplest functions—and paradoxically, one of the most problematic. At face value, nothing could be simpler: absolute value simply means "for any input, a positive value is returned." It can also be thought of the distance a number is from 0, as shown in Figure 3. 23. 5 = 5 and −5 = 5 But consider these three equations. They look very similar—only the number changes—but the solutions are completely different. Equation 1: x = 10 x=10 works, as does x=-10, therefore our solution is x = ± 10 Equation 2: x = − 10 Now, x=10 doesn't work, and neither does x=-10. This equations has no solutions because absolute values are never negative! Equation 3: x = 0 Now x=0 is the only solution. Figure 3.23 Absolute Value Both 3 and -3 are the same distance from 0, so the absolute value of both is 3. 187 We see that the first problem has two solutions, the second problem has no solutions, and the third problem has one solution. This is only one example of how absolute value equations may become confusing—and how you can solve them if you think more freely than with memorized rules. For more complicated problems, follow a three-step approach. 1. Isolate the absolute value algebraically. 2. Think the problem through like the simpler problems above. That is to say, set the absolute value term equal to the other side of the equation, and the opposite of the other side (-1 times it) 3. Isolate the value of x. Most problems with this type of equation do not occur in the first and third step. Also they do not occur because students try to think it through incorrectly (second step). The problems often occur because students try to take "shortcuts" to avoid the second step entirely. Example: Solving An Absolute Value Equation (No Variable on the Other Side) Solve the following equation for x: 3 2x + 1 − 7 = 5 1. Algebraically isolate the absolute value 3 2x + 1 = 12 2x + 1 = 4 2. Think! For the moment, forget about the quantity 2x+1; just think of it as something. The absolute value of "something" is 4. So, in analogy to what we did before, the "something" can either be 4, or –4. So that gives us two possibilities 2x + 1 = 4 2x + 1 = − 4 3. Algebraically solve both equations for x: 2x = 3 or 2x = − 5 x = 3/2 or x = − 5/2 Therefore, this problem has two answers, x=3/2 and x=-5/2 Source: and-inequalities/further-equation-solving/equations-with-absolute- value/ CC-BY-SA Boundless is an openly licensed educational resource 188 Linear Inequalities Expressions that are designated as less than, greater than, less that or equal to or greater than or equal to, it is a linear inequality. KEY POINTS • When two expressions are connected by < , > , ≤ , ≥ sign, we have an inequality. For inequalities that contain variable expressions, you may be asked to solve the inequality for that variable. This just means that you need to find the values of the variable that make the inequality true. • A linear inequality is solved very similarly to how we solve equal functions. The difference the answers are:- ≥,contains values equal and greater than the found solution. • Linear inequalities are commonly written out as a0 + a1x1 + a2x2 +... + anxn < 0. When two linear (first-order) expressions are not equal, but are designated as less than, greater than, less that or equal to or greater than or equal to, it is called a linear inequality. These can be denoted by, respectively: < , > , ≤ , ≥ . For inequalities that contain variable expressions, you may be asked to solve the inequality for that variable. This just means that you need to find the values of the variable that make the inequality true. A linear inequality is a type of inequality which involves a linear function, or a first order function. Remember that when you solve a linear equation, there is usually one value that makes the equation true. But when you solve an inequality, there can be many values that make the statement true. Look at this inequality: x>4. The solution to this inequality includes every number that is greater than 4 as shown in Figure 3.24. When operating in terms of real numbers, linear inequalities are the ones written in the forms f(x) < b where f(x) is a linear function in real numbers, and b is a constant real number. The above are commonly written out as a0 + a1x1 + a2x2 + . . . + anxn < 0 190 Solutions to x>4 are graphed on the number line. Figure 3.24 Inequality Sometimes they may be written out in the form a1x1 + a2x2 + . . . + anxn < b Here x1, x2,... xn are called the unknowns, a0, a1, a2,... an are called the coefficients, and b is the constant term. A linear inequality looks like a linear equation, with the inequality sign replacing the equal sign. A system of linear inequalities is a set of linear inequalities in the same variables. Not all systems of linear inequalities have solutions. EXAMPLE Try and solve this inequality: 7x + 3 + x ≤ 1 + 4x − 10. The first step, as in most equations, is to simplify the problem by isolating the variables to one side: 7x + x − 4x ≤ 1 − 10 − 3 4x ≤ − 12 x ≤ − 3 This means that x is less than, or equal to -3. Meaning that: x = − 3, − 4, − 5,..., − ∞ Source: and-inequalities/working-with-linear-inequalities/linear-inequalities/ CC-BY-SA Boundless is an openly licensed educational resource Compound Inequalities Another type of inequality is the compound inequality, a compound inequality is of the form: a < x < b. KEY POINTS • Another type of inequality is the compound inequality. A compound inequality is of the form: a < x < b. • There are two statements in a compound inequality. The first statement is a < x. The next statement is x < b. When we read this statement, we say "a is less than x," then continue saying "and x is less than b". • An example of a compound inequality is: 4 < x < 9. In other words, x is some number strictly between 4 and 9. Another type of inequality is the compound inequality. A compound inequality is of the form: a < x < b There are actually two statements here. The first statement is a < x. The next statement is x < b. When reading this statement, the forms say "a is less than x," then continue saying "and x is less than b." Just by looking at the inequality, it can be seen that the number x is between the numbers a and b. The compound inequality a < x < b 191 indicates "betweenness." Without changing the meaning, the statement a < x can be read as x > a. Thus, the form a < x < b can be read as "x is greater than a and at the same time is less than b." For example: 1. 4 < x < 9. x is some number strictly between 4 and 9. The numbers 4 and 9 are not included, so we use open circles at these points (Figure 3.25). 2. 2 < z < 0. z is some number between -2 and 0. 3. 1 < x + 6 < 8. The expression x + 6 represents some number strictly between 1 and 8. Consider problem 3 above, 1 < x + 6 < 8. The statement says that the quantity x + 6 is between 1 and 8. This statement will be true for only certain values of x. To solve for these values, subtract 6 from all three parts of the inequality to yield -5 < x < 2. Thus, if x is any number strictly between -5 and 2, the statement 1 < x + 6 < 8 will be true. Source: and-inequalities/working-with-linear-inequalities/compound- inequalities/ CC-BY-SA Boundless is an openly licensed educational resource 192 Solutions to the compound inequality 4 < x < 9 are indicated on the number line. Figure 3.25 4 < x < 9 Inequalities with Absolute Value Inequalities with absolute values can be solved by considering the absolute value as the distance from 0 on the number line. KEY POINTS • Problems involving absolute values and inequalities can be approached in at least two ways. • Inequalities with absolute values can be solved by trial-and- error. • Another way to solve inequalities with absolute values is to think of the absolute value as representing distance from 0 and then finding the values that satisfy that condition. Consider the following inequality that includes an absolute value: \|x\| < 10 Having seen that the solution to ∣x∣=10 is x=±10, many students answer this question x<±10. However, this is wrong. Here are two different, perfectly correct, ways to look at this problem. 1. What numbers work? 4 works. -4 does too. 13 doesn't work. How about -13? No: If x=-13, then ∣x∣=13, which is not less than 10. By playing with numbers in this way, you should be able to convince yourself that the numbers that work must be somewhere between -10 and 10. This is one way to approach finding the answer. 2. The other way is to think of absolute value as representing distance from 0. ∣5∣ and ∣−5∣ are both 5 because both number are 5 away from 0. In this case, ∣x∣<10 means "the distance between x and 0 is less than 10" - in other words, you are within 10 units of zero in either direction. Once again, we conclude that the answer must be between -10 and 10. This answer can be visualized on the number line as shown in Figure 3.26, in which all numbers whose absolute value is less than 10 are highlighted. It is not necessary to use both of these methods; use whichever method is easier for you to understand. More complicated absolute value problems 193 All numbers whose absolute value is less than 10; −10 Figure 3.26 Solution to ∣x∣<10 should be approached using the same steps as the equations discussed above: algebraically isolate the absolute value and then algebraically solve for x. Source: and-inequalities/working-with-linear-inequalities/inequalities-with- absolute-value/ CC-BY-SA Boundless is an openly licensed educational resource Solving Problems with Inequalities Inequalities can be solved by similar methods to linear equations, except that multiplication by negative numbers reverses the inequality. KEY POINTS • Inequalities can be solved by basically the same methods as linear equations with the exception that multiplication by a negative number reverses the direction of the inequality. • Let a, b, and c represent real numbers and assume that a < b. Then, if a < b, a + c < b + c and a − c < b − c; if any real number is added to or subtracted from both sides of an inequality, the sense of the inequality remains unchanged. • If c is a positive real number, then if a < b, ac < bc and ac < bc. If both sides of an inequality are multiplied or divided by the same positive number, the sense of the inequality remains unchanged. • While, if c is a negative real number, then if a < b, ac > bc and ac > bc. If both sides of an inequality are multiplied or divided by the same negative number, the inequality sign must be reversed in order for the resulting inequality to be equivalent to the original inequality. 194 We have discovered that an equation is a mathematical way of expressing the relationship of equality between quantities. However, not all relationships need be relationships of equality. Certainly the number of human beings on earth is greater than 20. Also, the average American consumes less than 10 grams of vitamin C every day. These types of relationships are not relationships of equality but, rather, relationships of inequality. A linear inequality is a mathematical statement that one linear expression is greater than or less than another linear expression. The following notation is used to express relationships of inequality: • > Strictly greater than • < Strictly less than • ≥ Greater than or equal to • ≤ Less than or equal to Note that the expression x > 12 has infinitely many solutions. Any number strictly greater than 12 will satisfy the statement. Some solutions are: 13, 15, 90, 12.1, 16.3, and 102.51. A linear equation, we know, may have exactly one solution, infinitely many solutions, or no solution. Speculate on the number of solutions of a linear inequality. (Hint: Consider the inequalities x < x−6 and x ≥ 9.) A linear inequality may have infinitely many solutions or no solutions. Inequalities can be solved by basically the same methods as linear equations. Let a, b, and c represent real numbers and assume that a < b. Then, if a < b, • a + c < b + c and a − c < b − c. If any real number is added to or subtracted from both sides of an inequality, the sense of the inequality remains unchanged. 195 Miah was asked to find the values of x that make this inequality true: 2x + 1 ≤ 7. Figure 3.27 Solving Linear Inequalities • If c is a positive real number, consider then if a < b, ac < bc and ac < bc. If both sides of an inequality are multiplied or divided by the same positive number, the sense of the inequality remains unchanged. • If c is a negative real number, then if a < b, ac > bc and ac > bc. If both sides of an inequality are multiplied or divided by the same negative number, the inequality sign must be reversed (change direction) in order for the resulting inequality to be equivalent to the original inequality. Source: and-inequalities/working-with-linear-inequalities/solving-problems- with-inequalities/ CC-BY-SA Boundless is an openly licensed educational resource 196 The Leading-Term Test Analysis of a polynomial reveals whether the function will increase or decrease as x approaches positive and negative infinity. KEY POINTS • Properties of the leading term of a polynomial reveal whether the function increases or decreases continually as x values approach the left and to the right. All polynomial functions of first or higher order either increase or decrease indefinitely as x values grow larger and smaller. It is possible to determine the end behavior of a polynomial function without using a graph. Consider the polynomial function: f(x) = anxn + an−1xn−1 + . . . + a1x + a0 anxn is called the leading term, while an is known as the leading coefficient. The properties of the leading term and leading coefficient indicate whether f(x) increases or decreases continually as x values approach the left and to the right. Consider the polynomial: f(x) = 2x3 − 3x2 − 3x + 2 In the leading term, a equals 2, and n equals 3. Because n is odd and a is positive, the graph declines to the left and inclines to the right (Figure 4.1). 199 Source: rational-functions/polynomial-functions-and-models/the-leading- term-test/ CC-BY-SA Boundless is an openly licensed educational resource Finding Zeros of Factored Polynomials The factored form of a polynomial reveals its zeros, which are defined as points where the function touches the x axis. KEY POINTS • A polynomial function may have zero, one, or many zeros. • All polynomial functions of positive, odd order have at least one zero, while polynomial functions of positive, even order may not have a zero. • Regardless of odd or even, any polynomial of positive order can have a maximum number of zeros equal to its order. The factored form of a polynomial can reveal where the function crosses the x axis. An x value at which this occurs is called a "zero" or "root." Consider the factored function: f(x) = (x − a1)(x − a2) . . . (x − an) Each value a1, a2, and so on is a zero. 200 Figure 4.1 Graph of cubic polynomial A polynomial function may have zero, one, or many zeros. All polynomial functions of positive, odd order have at least one zero, while polynomial functions of positive, even order may not have a zero. Regardless of odd or even, any polynomial of positive order can have a maximum number of zeros equal to its order. For example, a cubic function can have as many as three zeros, but no more. This is known as the fundamental theorem of algebra. Consider the function: f(x) = x3 + 2x2 − 5x − 6 This can be rewritten in the factored form: f(x) = (x + 3)(x + 1)(x − 2) Replacing x with a value that will make (x+3), (x+1), or (x-2) will result in f(x) being equal to zero. Thus, the zeroes for f(x) are at x=-3, x=-1, and x=2. This can also be shown graphically Figure 4.2. Source: rational-functions/polynomial-functions-and-models/finding-zeros-of- factored-polynomials/ CC-BY-SA Boundless is an openly licensed educational resource 201 f(x) crosses the x axis at x=-3, x=-1, and x=2. Figure 4.2 Graph of f(x)=(x+1)(x+3)(x-2) Introduction: Polynomial and Rational Functions and Models Functions are commonly used in fitting data to a trend line and differ in terms of accuracy and ease of use. KEY POINTS • Researchers will often collect many discrete samples of data, relating two or more variables, without knowing the mathematical relationship between them. Curve fitting is used to create trend lines intended to fill in the points between and beyond collected data points. • Polynomial functions are easy to use for modeling but ill- suited to modeling asymptotes and some functional forms, and they can become very inaccurate outside the bounds of the collected data. • Rational functions can take on a much greater range of shapes and are more accurate both inside and outside the limits of collected data than polynomial functions. However, rational functions are more difficult to use and can include undesirable asymptotes. Polynomial and rational functions are often used in statistical modeling. For a set of data, such functions can be used to create a trend line that relates discrete findings on two or more axes. A polynomial function has the following form: y = anxn + an−1xn−1 + . . . + a2x2 + a1x + a0 where n is an integer greater than or equal to 0, x and y are variables, and an,an-1, a2, a1, and a are constants. The value of n defines the degree of the polynomial. If n=0, the function is a constant; if n=1, the function is a line; if n=2, the function is quadratic; if n=3, the function is cubic, and so on. The form of the polynomial function is very versatile and can be used to represent not only complex but also simple functions. Consider the following function: y = x5 + 3x in which n equals 5 and an equals 1. Because there are no terms for x4, x3, and x2, the values of an-1, an-2, and a2 are equal to 0. The value of a1 is equal to 3. Polynomial functions are very simple in form and easy to use, but they have limitations with regard to statistical modeling. They can take on only a limited number of shapes and are particularly ill- 202 suited to modeling asymptotes. They also make for trend lines that become increasingly unreliable as they extend further beyond the limits of collected data. A rational function is the ratio of two polynomial functions and has the following form: y = anxn + an−1 + xn−1 + . . . + a2x2 + a1x + a0 bmxm + bm−1 + xm−1 + . . . + b2x2 + b1x + b0 Here, n and m define the degrees of the numerator and denominator, respectively, and together, they define the degree of the polynomial. For example, if n=2 and m=1, the function is described as a quadratic/linear rational function. Rational functions are more complex in form than polynomial functions, but they have an advantage in that they can take on a much greater range of shapes and can effectively model asymptotes. They are also more accurate than polynomial functions both inside and outside the limits of collected data. However, rational functions sometimes include undesirable asymptotes that can disrupt an otherwise smooth trend line. Source: rational-functions/polynomial-functions-and-models/introduction- polynomial-and-rational-functions-and-models/ CC-BY-SA Boundless is an openly licensed educational resource Polynomial curves generated to fit points (black dots) of a sine function: The red line is a first degree polynomial; the green is a second degree; the orange is a third degree; and the blue is a fourth degree. Figure 4.3 Curve Fitting 203 Basics of Graphing Polynomial Functions A polynomial function in one real variable can be represented by a graph. KEY POINTS • The graph of the zero polynomial f(x) = 0 is the x-axis. • The graph of a degree 1 polynomial (or linear function) f he graph of any polynomial with degree 2 or greater f (x) = a0 + a1x + a2x2 + . . . + anxn , where an ≠ 0 and n ≥ 2 is a continuous non-linear curve. In mathematics, a polynomial is an expression of finite length constructed from variables (also called indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. However, the division by a constant is allowed, because the multiplicative inverse of a non zero constant is also a constant. For example, x2 − x/4 + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x (4/x), and also because its third term contains an exponent that is not a non-negative integer (3/2). The term "polynomial" can also be used as an adjective, for quantities that can be expressed as a polynomial of some parameter, as in polynomial time, which is used in computational complexity theory. Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences. They are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science. They are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings, a central concept in abstract algebra and algebraic geometry. A typical graph of a polynomial function is shown in Figure 4.4. 205 A graph of a polynomial function of degree 3 Figure 4.4 A graph of a polynomial of degree = 3 (usually represented by letters), also called indeterminates, raised to whole number powers. The exponent on a variable in a term is called the degree of that variable in that term; the degree of the term is the sum of the degrees of the variables in that term, and the degree of a polynomial is the largest degree of any one term. Since x = x1, the degree of a variable without a written exponent is one. A term with no variables is called a constant term, or just a constant; the degree of a (non zero) constant term is 0. The coefficient of a term may be any number from a specified set. If that set is the set of real numbers, we speak of "polynomials over the reals." Other common kinds of polynomials are polynomials with integer coefficients, polynomials with complex coefficients, and polynomials with coefficients that are integers modulo of some prime number p. In most of the examples in this section, the coefficients are integers. A polynomial function in one real variable can be represented by a graph. • The graph of the zero polynomial f(x) = 0 is the x-axis. • The graph of a degree 0 polynomial f(x) = a0, where a0 ≠ 0, is a horizontal line with y-intercept a0 • The graph of a degree 1 polynomial (or linear function) f The graph of a degree 3 polynomial f(x) = a0 + a1x + a2x2 + a3x3 , where a3 ≠ 0, is a cubic curve. 206 Quadratic equations of different orders Figure 4.5 Graphical quadratic equations • The graph of any polynomial with degree 2 or greater f(x) = a0 + a1x + a2x2 + . . . + anxn , where an ≠ 0 and n ≥ 2 is a continuous non-linear curve. • The graph of a non-constant (univariate) polynomial always tends to infinity when the variable increases indefinitely (in absolute value). • Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior. Figure 4.5 shows a series of graphs of polynomials. Note: the degree of the polynomial in these cases can be assessed by counting the number of times the function graphically crosses the x axis. Source: rational-functions/graphing-polynomial-functions/basics-of- graphing-polynomial-functions/ CC-BY-SA Boundless is an openly licensed educational resource The Intermediate Value Theorem For each value between the bounds of a continuous function, there is at least one point where the function maps to that value. KEY POINTS • Simply stated, the Intermediate Value Theorem points out that: if the plotted route between points A and C is smooth and continuous between point A to point C, you will have to pass through all points "B" on the journey, as long as they are on the plotted route. • The Intermediate Value Theorem capitalizes on the completeness of functions of real numbers. • Functions containing irrational roots do not meet the requirements of the Intermediate Value Theorem. Simply stated, this theorem points out that, if the plotted route between points A and C is smooth and continuous between point A to point C, you will have to pass through all points "B" on the journey, as long as they are on the plotted route. In a clearer example, when your car accelerates from 0 to 100 mph, at some point, even for a microsecond, your car is traveling at 21.5 mph. 207 When you bike between points X and Z, and your path follows a semicircular route, you will bike through any point on a semicircle connecting points X and Z. Stated in the language of algebra supported by Figure 4.6: If f is a real-valued continuous function on the interval [a, b], and u is a number between f(a) and f(b), then there is a c ∈ [a, b] such that f(c) = u. This can be clearly seen by following the curve from point b. It is frequently stated in the following equivalent form: Suppose that f : [a, b] → R is continuous and that u is a real number satisfying f(a) < u < f(b) or f(a) > u > f(b). Then for some c ∈ [a, b], f(c) = u. In other words, regardless of whether f(a)>f(b) or f(b)>f(a), f(c) lies between them. The theorem depends on (and is actually equivalent to) the completeness of the real numbers. It is false for the rational numbers Q. For example, the function f(x) = x2 − 2 for x ∈ Q satisfies f(0) = −2 and f(2) = 2. However there is no rational number x such that f(x) = 0, because √2 is irrational. All polynomial functions, such as Figure 4.7 f(x) = x3 + x2 − 5x + 11 are continuous - there are no singularities or discontinuities. However, in equations of the form f(x) = (2x − 1) (x + 2) a problem occurs when x=-2, as seen in Figure 4.8. The function is defined for all real numbers x ≠ −2 and is continuous at every such point. The question of continuity at x = −2 does not arise, since x = 208 In plotting a continuous and smooth function between two points, all points on the function between the extremes are described and predicted by the Intermediate Value Theorem. Figure 4.6 The Intermediate Value Theorem A graphed third- order equation meeting the requirements of the Intermediate Value Theorem Figure 4.7 A Continuous Function −2 is not in the domain of f. There is no continuous function F: R → R that agrees with f(x) for all x ≠ −2. If f is continuous on [a, b] and f(a) and f(b) differ in sign, then, at some point c in [a, b], f(c) must equal zero. Source: rational-functions/graphing-polynomial-functions/the-intermediate- value-theorem/ CC-BY-SA Boundless is an openly licensed educational resource 209 A discontinuity occurs when x = -2: the function is not defined at x=-2. Figure 4.8 Graphing a rational function Division and Factors Polynomial long division functions similarly to long division, and if the division leaves no remainder, then the divisor is called a factor. KEY POINTS • Dividing one polynomial by another can be achieved by using long division. The rules for polynomial long division are the same as the rules learned for long division of integers. • The four steps of long division are divide, multiply, subtract, and bring down. • After completing polynomial long division, it is good to check the answers, either by plugging in a number or by multiplying the quotient times the divisor to get the dividend back. Simplifying, multiplying, dividing, adding, and subtracting rational expressions are all based on the basic skills of working with fractions. Dividing polynomials is based on an even earlier skill, one that pretty much everyone remembers with horror: long division. To refresh one's memory, try dividing 745 3 by hand. The answer should end up as something that looks something like Figure 4.9. Therefore, it can be concluded that 745/3 is 248 with a remainder of 1, or, to put it another way, 745/3 =248 + 1/3. Long division is a skill that many may have decided they could forget, since calculators perform this task much faster. However, long division comes roaring back, because here is a problem that a calculator cannot solve: 6x3 − 8x2 + 4x − 2 2x − 4 . This problem is solved in a very similar way as the previous problem. Start by rewriting the problem in standard long division form (A). Follow along the text with the graphic Figure 4.10. Then divide the first time to get 3x2 (B). Why 3x2? This comes from the question: "How many times does 2x go into 6x3" or, to put the same question another way: "What would I multiply 2x by, in order to get 6x3?" This is comparable to the first step in the long division problem: "What do I multiply 3 by, to get 7?" Now (C), multiply the 3x2 times the (2x–4) and get 6x3–12x2. Then subtract this from the line above it. The 6x3 terms cancel—that shows the right term was picked above. Note that careful work must 211 The long division is shown here explicitly to serve as a refresher for more complicated long division of polynomials. Figure 4.9 745 divided by 3 be done with the signs, –8x2–(–12x2) gives positive 4x2. Next, bring down the 4x, as shown in (D). All four steps of long division are now complete—divide, multiply, subtract, and bring down. At this point, the process begins again, with the question "How many times does 2x go into 4x2?" (E) is not the next step. This is merely what the process looks like after all the steps have been finished. It is a good idea to go through the problem once more to check the work. Therefore, it can be concluded that 6x3 − 8x2 + 4x − 2 2x − 4 is 3x2 + 2x + 6 with a remainder of 22, or, to put it another way, 3x2 + 2x + 6 + 22 2x − 4 . If a polynomial can be divided by another equation and have no remainder, then the equation that was divided by is called a factor. In this case, a remainder will not be written, as the divisor divided evenly into the dividend. Be sure to check the answers after doing these types of problems, either by plugging in numbers, or by multiplying the divisor by the quotient to see if the dividend can be gotten back! Source: rational-functions/polynomial-division-the-remainder-and-factor- theorems/division-and-factors/ CC-BY-SA Boundless is an openly licensed educational resource 212 For explanations of each step, see the text. Figure 4.10 Polynomial long divion The Remainder Theorem and Synthetic Division Synthetic division is a technique for dividing a polynomial and finding the quotient and remainder. KEY POINTS • Synthetic division is most commonly applied when dividing by a monomial such as x-a. • The most useful aspects of synthetic division are that it allows one to calculate without writing variables and uses fewer calculations. • In algebra, synthetic division is a method of performing polynomial long division, with less writing and fewer calculations. In algebra, the polynomial remainder theorem or little Bézout's theorem is an application of polynomial long division. It states that the remainder of a polynomial f(x) divided by a linear divisor (x − a) is equal to f(a). For example, take f(x) = x3 − 12x2 − 42 and divide it by x − 3. This gives the quotient x2 − 9x − 27 and the remainder -123. Therefore, f(3) = − 123. We can check this by plugging 3 into the equation above, which gives 33 − 12(3)2 − 42 : 27 − 12(9) − 42 27 − 108 − 42 −123 In order to find the remainder so as to use the remainder theorem, one must first perform division. One way of doing polynomial division is to use synthetic division. In algebra, synthetic division is a method of performing polynomial long division, with less writing and fewer calculations. It is mostly taught for division by binomials of the form x − a, but the method generalizes to division by any monic polynomial. The most useful aspects of synthetic division are that it allows one to calculate without writing variables and uses fewer calculations. As well, it takes significantly less space than long division. Most importantly, the subtractions in long division are converted to additions by switching the signs at the very beginning, preventing sign errors. Synthetic division for linear denominators is also called division through Ruffini's rule. The first example is synthetic division with only a monic linear denominator x − a : x3 − 12x2 − 42 x − 3 213 The steps of synthetic division are outlined in Figure 4.11. The text for each step follows. A. Write the coefficients of the polynomial to be divided at the top (the zero is for the unseen 0x). Next negate the coefficients of the divisor: −1x + 3. B. Write in every coefficient of the divisor but the first one on the left. C. Note the change of sign from −3 to 3. "Drop" the first coefficient after the bar to the last row. D. Multiply the dropped number by the number before the bar, and place it in the next column. E. Perform an addition in the next column. F. Repeat the previous two steps to obtain the values shown. Count the terms to the left of the bar. Since there is only one, the remainder has degree zero. Mark the separation with a vertical bar, as shown: 1 -9 -27 \| -123 The terms are written with increasing degree from right to left beginning with degree zero for both the remainder and the result. x3 − 12x2 − 42 x − 3 = x2 − 9x − 27 − 123 x − 3 Source: rational-functions/polynomial-division-the-remainder-and-factor- theorems/the-remainder-theorem-and-synthetic-division/ CC-BY-SA Boundless is an openly licensed educational resource 214 Follow along each step in the text. Figure 4.11 The steps of synthetic division Finding Factors of Polynomials Finding factors of polynomials is important, since it is always best to work with the simplest version of a polynomial. KEY POINTS • Factoring is a critical skill in simplifying functions and solving equations. • There are four types of factoring shown which are "pulling out" common factors, factoring perfect squares, the difference between two squares, and then how to factor when the other three techniques are not applicable. • The first step should always be "pulling out" common factors. Even if this does not factor out the polynomial completely, this will make the rest of the process much easier. When using multiplication things are put together. When factoring, things are pulled apart. Factoring is a critical skill in simplifying functions and solving equations. There are four basic types of factoring. In each case, it is beneficial to start by showing a multiplication problem—then show how to use factoring to reverse the results of that multiplication. "Pulling Out" Common Factors This type of factoring is based on the distributive property, which states: 2x(4x2 − 7x + 3) = 8x3 − 14x2 + 6x When factoring, this property is done in reverse. Therefore, starting with an expression such as the one above it can be noted that every one of those terms is divisible by 2. Also, every one of those terms is divisible by x. Hence, one can "factor out," or "pull out," 2x. 8x3 − 14x2 + 6x = 2x(? − ? + ?) For each term, it can be shown what happens when that term is divided by 2x. For instance, if 8x3 is divided by 2x, then the answer is 4x2. Doing this process for each term, the result is: 8x3 − 14x2 + 6x = 2x(4x2 − 7x + 3) For many types of problems, it is easier to work with this factored form. As another example, consider 6x+3. The common factor is 3. When factoring 3 from 6x, 2x is left. When factoring 3 out of the 3, only 1 remains. 6x + 3 = 3(2x + 1) There are two key points to understand about this kind of factoring: 215 1. This is the simplest kind of factoring. Whenever trying to factor a complicated expression, always begin by looking for common factors that can be pulled out. 2. The factor must be common to all the terms. For instance, 8x3−14x2+6x+7 has no common factor, since the last term, 7, is not divisible by 2 or x. Factoring Perfect Squares The second type of factoring is based on the "squaring" formulae: (x + a)2 = x2 + 2ax + a2 (x − a)2 = x2 − 2ax + a2 For instance, if the problem is x2+6x+9, then one may recognize the signature of the first formula: the middle term is three doubled, and the last term is three squared. Thus, this simplifies to (x+3)2. Attempt to notice patterns, and the problems will become easier. x2 + 10x + 25 = (x + 5)2 x2 + 2x + 1 = (x + 1)2 If the middle term is negative, then the second formula is: x2 − 8x + 16 = (x − 4)2 x2 − 14x = 49 = (x − 7)2 This type of factoring only works in this specific case: the middle number is something doubled, and the last number is that same value squared. Furthermore, although the middle term can be either positive or negative, the last term cannot be negative. This is because if a negative is squared, the answer is positive. The Difference Between Two Squares The third type of factoring is based on the third of the basic formulae: (x + a)(x − a) = x2 − a2 This formula can be run in reverse whenever subtracting two perfect squares. For instance, if there is x2−25, it can be seen that both x2 and 25 are perfect squares. Therefore it can factor as (x+5)(x −5). Other examples include: Example 1 x2 − 64 = (x + 8)(x − 8) 16y2 − 49 = (4y + 7)(4y − 7) 2x2 − 18 = 2(x2 − 9) = 2(x + 3)(x − 3) 216 Note that, in the last example, the first step is done by pulling out a 2, and there are two perfect squares left. This follows the rule: always begin by pulling out common factors before trying anything else. It is also important to note that the sum of two squares cannot be factored. x2+4 is a perfectly good function, but it cannot be factored. Brute Force, Old-Fashioned, Bare-Knuckle, No-Holds-Barred Factoring In this case, the multiplication that is being reversed is just FOIL, a mnemonic for multiplying two binomials, reminding: First, Outside, Inside, Last, as shown here: Figure 4.12. For instance, consider: (x + 3)(x + 7) = x2 + 3x + 7x + 21 The 3 and 7 added to yield the middle term,10, and multiplied to yield the final term, 21. This can be generalized as: Therefore, if you are given a problem such as the one above to factor, look for two numbers that add up to 10, and multiply to 21.There are a lot of pairs of numbers that add up to 10, but relatively few that multiply to 21, so start by looking for factors of 21. The example sections provides more information. Source: rational-functions/polynomial-division-the-remainder-and-factor- theorems/finding-factors-of-polynomials/ CC-BY-SA Boundless is an openly licensed educational resource start by multiplying the First terms, then the Outside terms, then the Inside terms, and finally the Last terms. Often, the outside and inside terms can eventually be added together. It is important to understand this method, in order to be able to perform it in reverse. Figure 4.12 FOIL Method Diagram 217 The Fundamental Theorem of Algebra Finding Polynomials with Given Zeroes Zeroes of Polynomial Functions with Real Coefficients Rational Coefficients Integer Coefficients and the Rational Zeroes Theorem The Rule of Signs Section 4 Zeroes of Polynomial Functions and Their Theorems 218 The Fundamental Theorem of Algebra The fundamental theorem states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. KEY POINTS • The fundamental theorem of algebra states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero imaginary part. • Equivalently (by definition), the fundamental theorem states that the field of complex numbers is algebraically closed. • The fundamental theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division. The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero imaginary part. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division. In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients. Complex-Analytic Proof Find a closed disk D of radius r centered at the origin such that \| p(z)\| > \|p(0)\| whenever \|z\| ≥ r. The minimum of \|p(z)\| on D, which must exist since D is compact, is therefore achieved at some point z0 in the interior of D, but not at any point of its boundary. The Maximum modulus principle (applied to 1/p(z)) implies then that p(z0) = 0. In other words, z0 is a zero of p(z). 219 In mathematics, the maximum modulus principle Figure 4.13 in complex analysis states that if f is a holomorphic function, then the modulus cannot exhibit a true local maximum that is properly within the domain of f. In other words, either f is a constant function, or, for any point z0 inside the domain of f there exist other points arbitrarily close to z0 at which \|f \| takes larger values. Source: rational-functions/zeroes-of-polynomial-functions-and-their- theorems/the-fundamental-theorem-of-algebra/ CC-BY-SA Boundless is an openly licensed educational resource Finding Polynomials with Given Zeroes To construct a polynomial from given zeroes, set x equal to each zero, move everything to one side, then multiply each resulting equation. KEY POINTS • A polynomial constructed from n roots will have degree n or less. That is to say, if given three roots, then the highest exponential term needed will be x3 . • Each zero given will end up being one term of the factored polynomial. After finding all the factored terms, simply multiply them together to obtain the whole polynomial. • Because a polynomial and a polynomial multiplied by a constant have the came roots, every a polynomial is constructed from given zeroes the general solution includes a constant, shown here as c. One type of problem is to generate a polynomial from given zeroes. Several provisions are worth noting during the excursion into this territory. Remember that the degree of a polynomial, the highest exponent, dictates the maximum number of roots it can have. Therefore, if given two zeroes then a polynomial of second 220 A plot of the modulus of cos(z) (in red) for z in the unit disk centered at the origin (shown in blue). As predicted by the fundamental theorem, the maximum of the modulus cannot be inside of the disk (so the highest value on the red surface is somewhere along its edge). Figure 4.13 The Maximum Modulus Principle degree only needs constructed, that is, one with highest exponential term x2 . One thing to remember, when starting with the zeroes and creating a polynomial from them, an unknown constant must always be added. For example, if given x=a and x=b as constants, then the resulting initial terms would be a constant c times the two equation that give zeroes at the appropriate place: c(x − a)(x − b) Multiplied out, this gives: cx2 − c(a + b)x + abc. As an example, find a polynomial with zeroes x=-2 and x=-9. Simply manipulating these equations so that all terms are on one side of the equation, the results are x+2=0 and x+9=0. These two terms make up the factored polynomial, giving c(x+2)(x+9)=0. Notice that a c term as a constant is included, as any constant multiplied by this equation will still give zero at the appropriate places. Multiplying out, the result is c(x2 + 11x + 18). To see the effect of this constant more clearly, look at the graph of (x2 + 11x + 18) and 2(x2 + 11x + 18) (Figure 4.14). Notice how the intercepts do not change, even though it has been multiplied by a different constant. This is because any number times 0 still equals 0. As another example, find a polynomial with given zeroes x=1 and x=12. Again, manipulate each of these equations until zero is on one side, giving x-1=0 and x-12=0. Entering these two conditions, the result is c(x-1)(x-12) as the factored polynomial. Multiplying out, the answer is c(x2 − 13x + 12). Source: rational-functions/zeroes-of-polynomial-functions-and-their- theorems/finding-polynomials-with-given-zeroes/ CC-BY-SA Boundless is an openly licensed educational resource 221 Notice how the intercepts do not change, even when we multiply the function by a constant. Figure 4.14 A graph of (x2+11x+18) and 2(x2+11x+18). Zeroes of Polynomial Functions with Real Coefficients A root, or zero, of a polynomial function is a value that can be plugged into the function and yield zero. KEY POINTS • Real numbers include all the rational and irrational numbers. For example: −5,4/3, √2 are all real numbers. • If given the function, f(x) = 0 is a root of the function. For this reason, roots are often referred to as a zero of the function. • There are many ways to find the roots of a polynomial. If one is confident factoring out polynomials into their simplest forms, its roots can usually be found by inspection. However, if one is not confident, or it is a tricky polynomial, the quadratic equation can be used. The zero of a function, f(x), refers to the value or values of x that will result in the function equaling zero, f(x) = 0. These are often called the roots of the function. There are many methods to find the roots of a function. One can simply factor out many functions and determine by inspecting their roots. For example, if given the function: f(x) = x2 − x − 2, one can factor that out: f(x) = (x + 1)(x − 2). By using basic algebraic knowledge, it is known that if one uses f(−1) or f(2) the result will be 0. Therefore both -1 and 2 are roots of the function. This section specifically deals with polynomials that have real coefficients. A real number is any rational or irrational number, such as -5, 4/3, or even √2. An example of a non-real number would be √-1. Even though all polynomials have roots, not all roots are real numbers. Some roots can be complex, but no matter how many of the roots are real or complex, there are always as many roots as there are powers in the function. This is demonstrated with a quadratic equation in Figure 4.15. While there are many types of polynomials and many ways to find their roots, one type of method is explored below. As a reminder, a quadratic function has the following form: ax2 + bx + c This type of polynomial can be solved, meaning its roots can be found, using this equation: x = −b ± b2 − 4ac 2a 222 More complicated equations also exist for the higher functions, such as cubic and quartic functions, though their expressions are beyond the scope of this atom. Source: rational-functions/zeroes-of-polynomial-functions-and-their- theorems/zeroes-of-polynomial-functions-with-real-coefficients/ CC-BY-SA Boundless is an openly licensed educational resource Rational Coefficients Polynomials with rational coefficients should be treated and worked the same as other polynomials. KEY POINTS • In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. • A real number that is not rational is called irrational. Irrational numbers include √2, π, and e. • Polynomials with rational coefficients can be treated just like any other polynomial, just remember to utilize all the properties of fractions necessary during your operations. In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. Since q may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q (or Unicode ℚ). It was thus named in 1895 by Peano after quoziente, Italian for "quotient". The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any 223 Plots of real-valued quadratic function ax2 + bx + c, varying each coefficient separately. Figure 4.15 Plots of Quadratic Equations repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for binary, hexadecimal, or any other integer base. A real number that is not rational is called irrational. Irrational numbers include √2, π, and e. The decimal expansion of an irrational number continues forever without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Zero divided by any other integer equals zero. Therefore zero is a rational number, but division by zero is undefined. The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, the adjective rational often means that the underlying field considered is the field Q of rational numbers. Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a "polynomial over the rationals". However, rational function does not mean the underlying field is the rational numbers, and a rational algebraic curve is not an algebraic curve with rational coefficients. Finding Zeroes of a Polynomial with Rational Coefficients Polynomials with rational coefficients can be treated just like any other polynomial, just remember to utilize all the properties of fractions necessary during your operations. Multiplying fractions a/ b times c/d gives (ac)/(bd), whereas if one wanted to add a/b plus c/d, first convert them into ad/bd and cb/db, giving (ad+cb)/(db). For example, the polynomial 2x2 9 + 7x 3 + 6 can be factored to give ( x 3 + 2)( 2x 3 + 3). By setting each term to zero, it can be found that the zeros for this equation are x=-6 and x=-9/2. This matches what is observed graphically, as shown in Figure 4.16. Source: rational-functions/zeroes-of-polynomial-functions-and-their- theorems/rational-coefficients/ CC-BY-SA Boundless is an openly licensed educational resource 224 We can graph this equation, and in doing so see where it intercepts the y axis, as a means of checking our solutions to this problem. Figure 4.16 Graph of (2x^2)/9+7x/3+6 Integer Coefficients and the Rational Zeroes Theorem Each solution to a polynomial, expressed as x=p/q, which satisfies p and q are integer factors of a0 and an, respectively. KEY POINTS • In algebra, the Rational Zeros Theorem (also known as Rational Root Theorem, or Rational Root Test) states a constraint on rational solutions (or roots) of the polynomial equation anxn + an−1xn−1 + . . . + a0 = 0 with integer coefficients. • If a0 and an are non-zero, then each rational solution x, when written as a fraction x = p/q in lowest terms (i.e., the greatest common divisor of p and q is 1), satisfies 1) p is an integer factor of the constant term a0, and 2) q is an integer factor of the leading coefficient an. • A proof can be derived by first moving the constants to one side, factoring and multiplying by qn . Then a generalized form of Euclid's lemma states that p divides a0. The proof for q is similar. In algebra, the Rational Zeros Theorem (or Rational Root Theorem, or Rational Root Test) states a constraint on rational solutions (also known as zeros, or roots) of the polynomial equation: anxn + an−1xn−1 + . . . + a0 = 0 With integer coefficients an, an-1,... a0. If a0 and an are nonzero, then each rational solution x, when written as a fraction x = p/q in lowest terms, satisfies: • p is an integer factor of the constant term a0. • q is an integer factor of the leading coefficient an. An Elementary Proof Let P(x) = anxn + an−1xn−1 +... + a1x + a0 for some a0,..., an ∈ Z, and suppose P(p/q) = 0 for some coprime p, q ∈ Z: P( p q ) = an( p q )n + an−1( p q )n−1 + . . . + a1( p q ) + a0 = 0 If we shift the constant term to the right hand side, factor a p and multiply by qn, we get: p(an pn−1 + an−1qpn−2 + . . . + a1qn−1 ) = − a0qn 225 We see that p times the integer quantity in parentheses equals −a0qn, so p divides a0qn. But p is coprime to q and therefore to qn, so by (the generalized form of) Euclid's lemma, or first theorem, it must divide the remaining factor a0 of the product. If we instead shift the leading term to the right hand side and multiply by qn, we get: q(an−1pn+1 + an−2qpn−2 + . . . + a0qn−1 = − an pn And for similar reasons, we can conclude that q divides an. Example For example, every rational solution of the equation 3x3 − 5x2 + 5x − 2 = 0 (shown graphically as: Figure 4.17) must be among the numbers symbolically indicated by: ± 1,2 1,3 Which gives the list of possible answers: 1, − 1,2, − 2, 1 3 , − 1 3 , 2 3 , − 2 3 These root candidates can be tested using the Horner's method (for instance). In this particular case, there is exactly one rational root. If a root candidate does not satisfy the equation, it can be used to shorten the list of remaining candidates. For example, x = 1 does not satisfy the equation as the left hand side equals 1. This means that substituting x = 1 + t yields a polynomial in t with constant term 1, while the coefficient of t3 remains the same as the coefficient of x3. Applying the Rational Root Theorem thus yields the following possible roots for t: t = ± 1 1,3 226 One can also use the Rational Zeros Theorem to narrow down the candidates for solutions, then look to see which one is represented by the graphical form of the equation. Figure 4.17 Graph of 3x^3-5x^2+5x-2 Therefore, x = 1 + t = 2,0, 4 3 , 2 3 Root candidates that do not occur on both lists are ruled out. The list of rational root candidates has thus shrunk to just x = 2 and x = 2/3. Source: rational-functions/zeroes-of-polynomial-functions-and-their- theorems/integer-coefficients-and-the-rational-zeroes-theorem/ CC-BY-SA Boundless is an openly licensed educational resource The Rule of Signs The rule of signs gives an upper bound number of positive or negative roots of a polynomial. KEY POINTS • The rule of signs gives us an upper bound number of positive or negative roots of a polynomial. It is not a complete criterion, meaning that it does not tell the exact number of positive or negative roots. • The rule states that if the terms of a polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or is less by a multiple of 2. • As a corollary of the rule, the number of negative roots is the number of sign changes after multiplying the coefficients of odd-power terms by −1 [f(-x)], or fewer than it by a multiple of 2. The rule of signs, first described by René Descartes in his work La Géométrie, is a technique for determining the number of positive or negative real roots of a polynomial. 227 The rule gives us an upper bound number of positive or negative roots of a polynomial. However, it does not tell the exact number of positive or negative roots. Positive Roots In order to find the number of positive roots in a polynomial with only one variable, you must first arrange the polynomial by descending variable exponent. For example, x2+x3+x would have to be written x3+x2+x. Then, you must count the number of sign differences between consecutive nonzero coefficients. This number, and any number less than it by a multiple of 2, is your number of positive roots. It is important to note that multiple roots of the same value should be counted separately. Negative Roots Finding the negative roots is similar to finding the positive roots. The difference is, you must start by finding the coefficients of odd power (x3 or x5, not x2 or x4). Once you have located them, multiply each by -1. Then the procedure is the same, count the number of sign changes between consecutive nonzero coefficients. This number, and any number less than it by a multiple of 2, is your number of positive roots. Again it is important to note that multiple roots of the same value should be counted separately. This can also be done by taking the function, f(x), and substituting the x for -x, f(-x). The reason we only bother to change the sign of the odd power coefficients, is because if we substitute in -x in an even power, it will just become a positive again. For example: (-x)3= (-x)(-x)(-x) = (-x)3(-x)2= (-x)(-x) = (x)2 The negative signs cancel out. By only multiplying the odd powered coefficients by -1, we are essentially saving ourselves a step. Example The polynomial: f(x) = + x3 + x2 − x − 1 This function has one sign change between the second and third terms (the sequence of pairs of successive signs is ++, +−, −−). Therefore it has exactly one positive root. Note: Don't forget that the first term HAS A SIGN, which, in this case, is positive. Be sure to remember to change its sign when looking for the negative roots. Next, we move on to finding the negative roots. To do this we apply what we learned from the rule of signs. Change the exponents of the odd powered coefficients. Once you have done this, you have obtained the second polynomial and are ready to find the number of negative roots. This second polynomial is shown below: f(−x) = − x3 + x2 + x − 1 228 This polynomial has two sign changes (the sequence of pairs of successive signs is −+, ++, +−), so we know that it has at most two negative roots. From the rule of signs, we learned that the number or roots of either sign is the number of sign changes, or a multiple of two less than that. So this polynomial can have either 2 or 0 negative roots. We can validate this graphically, as shown in Figure 4.18. So in this example the roots are 1, -1 and -1. This can be checked by factoring the polynomial: f(x) = (x + 1)(x + 1)(x − 1), which simplifies into: f(x) = (x + 1)2 (x − 1) so the roots are −1,-1 and 1. Complex Roots A polynomial of the nth degree has exactly n roots. The minimum number of complex roots is equal to: n − (p + q) n = total number of roots in a polynomial; p = the maximum number of positive roots; q = the maximum number of negative roots. A simple example is: x2 + b To find the positive roots we count the sign changes. For this example, we will assume that b>0. This means that there are no positive roots. So p = 0. Now we look for negative roots. Since there are no odd powered coefficients, there are no changes to be made before looking for sign changes, therefore there are no negative roots. So, q = 0. Now we apply the complex root equation: n-(p+q): 2-(0+0) = 2. There are 2 complex roots. We can graphically see there are two solutions to this polynomial. This still fits with the rule of signs, as -1 is a negative root twice in the equation. Figure 4.18 Plot of x^3+x^2-x-1 229 Finding the Domain of a Rational Function The domain of a rational function f(x) = P(x)/Q(x) is the set of all points x for which the denominator Q(x) is not zero. KEY POINTS • A rational function is any function which can be written as the ratio of two polynomial functions. • The domain of f(x) = P(x)/Q(x) is the set of all points x for which the denominator Q(x) is not zero, where one assumes that the fraction is written in its lower degree terms, that is, P and Q have several factors of the positive degree. • Domain restrictions can be determined by setting the denominator equal to zero and solving. The Rational Function A rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials, nor the values taken by the function, are necessarily rational numbers. In the case of one variable, x, a function is called a rational function if, and only if, it can be written in the form: f(x) = P(x)/Q(x) where P and Q are polynomial functions in x and Q is not the zero polynomial. The domain of f is the set of all points x for which the denominator Q(x) is not zero, where one assumes that the fraction is written in its lower degree terms; that is, P and Q have several factors of the positive degree. Every polynomial function is a rational function with Q(x) = 1. The Domain of a Rational Function The denominator of the fraction cannot equal zero. Domain restrictions can be determined by setting the denominator equal to zero and solving. For example, the domain of y = 1/x is comprised of all values of x for which x is not equal to zero. The domain of the rational function (x2 − 3x − 2)/(x2 − 4) includes all x not equal to +2 or -2 as can be seen in Figure 4.19. An additional example is the domain of (x + 3)/(x2 + 2). The domain of this function is all real numbers, since for x2 + 2 to equal 0, x2 would need to equal -2, and this condition cannot be satisfied by a real number. 232 Source: rational-functions/rational-functions/finding-the-domain-of-a- rational-function/ CC-BY-SA Boundless is an openly licensed educational resource Asymptotes An asymptote is a line such that the distance between a curve and the line approaches zero as both tend to infinity. KEY POINTS • An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. • There are potentially three kinds of asymptotes: horizontal, vertical and oblique asymptotes. • A rational function has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes. In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. There are potentially three kinds of asymptotes: horizontal, vertical and oblique asymptotes. For curves given by the graph of a function y = ƒ(x), horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. Vertical 233 The domain of this function includes all x not equal to +2 or -2. Figure 4.19 Example of a Rational Polynomial asymptotes are vertical lines near which the function grows without bound. A Simple Example Consider the graph of the equation y = 1/x shown to the right. The coordinates of the points on the curve are of the form (x, 1/x) where x is a number other than 0. For example, the graph contains the points (1, 1), (2, 0.5), (5, 0.2), (10, 0.1),... As the values of x become larger and larger, say 100, 1000, 10,000..., putting them far to the right of the illustration, the corresponding values of y,. 01,.001,.0001,..., become infinitesimal relative to the scale shown. But no matter how large x becomes, its reciprocal 1/x is never 0, so the curve never actually touches the x-axis. Similarly, as the values of x become smaller and smaller, say.01,.001,.0001,..., making them infinitesimal relative to the scale shown, the corresponding values of y, 100, 1000, 10,000..., become larger and larger. So the curve extends farther and farther upward as it comes closer and closer to the y-axis. Thus, both the x and y-axes are asymptotes of the curve. These ideas are part of the basis of concept of a limit in mathematics. Asymptotes of functions Asymptotes can be classified into horizontal, vertical and oblique asymptotes depending on orientation. Examples are shown in Figure 4.20. Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. As the name indicate they are parallel to the x-axis. Functions may lack horizontal asymptotes on either or both sides, or may have one horizontal asymptote that is the same in both directions. Vertical asymptotes are vertical lines (perpendicular to the x-axis) near which the function grows without bound. A common example of a vertical asymptote is the case of a rational function at a point x such that the denominator is zero and the numerator is non-zero. Oblique asymptotes are diagonal lines so that the difference between the curve and the line approaches 0 as x tends to +∞ or −∞. When a linear asymptote is not parallel to the x- or y-axis, it is called an oblique asymptote or slant asymptote. 234 The graph of a function with a horizontal, vertical, and oblique asymptote. Figure 4.20 Asymptotes Asymptotes of many elementary functions can be found without the explicit use of limits (although the derivations of such methods typically use limits). Asymptotes of Rational Functions A rational function has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes. The degree of the numerator and degree of the denominator determine whether or not there are any horizontal or oblique asymptotes. When the numerator of a rational function has degree exactly one greater than the denominator, the function has an oblique (slant) asymptote. The asymptote is the polynomial term after dividing the numerator and denominator. This phenomenon occurs because when dividing the fraction, there will be a linear term, and a remainder. The vertical asymptotes occur only when the denominator is zero. If both the numerator and denominator are zero, the multiplicities of the zero are compared. Source: rational-functions/rational-functions/asymptotes/ CC-BY-SA Boundless is an openly licensed educational resource Solving Problems with Rational Functions In the case of rational functions, the x-intercepts exist when the numerator is equal to 0. KEY POINTS • A rational function is defined as the ratio of two real polynomials with the condition that the polynomial in the denominator is not a zero polynomial. • The x-intercepts, also known as zeros of the function or real roots, can be more than one x-intercept. On graphs, x- intercepts are points where a graph intersects the x-axis. Thus, x-intercepts are x-values for which the function has a value of zero. • In the case of rational functions, the x-intercepts exist when the numerator is equal to 0. In the case of rational functions, the x-intercepts exist when the numerator is equal to 0. For f(x) = P(x)/Q(x), if P(x) = 0, then f(x) = 0. A rational function is defined in similar fashion as a rational number is defined in terms of numerator and denominator. It is defined as the ratio of two real polynomials with the condition that the polynomial in the denominator is not a zero polynomial. 235 f(x) = P(x)/Q(x) An example of a rational functions is: f(x) = (x + 1)/(2x2 − x − 1) The x-intercepts are also known as zeros of the function or real roots. There can be more than one x-intercept. On graphs, x- intercepts are points where a graph intersects the x-axis. Thus, x- intercepts are x-values for which the function has a value of zero: f(x) = 0 In the case of rational functions, the x-intercepts exist when the numerator is equal to 0. For f(x) = P(x)/Q(x), if P(x) = 0, then f(x) = 0. Example 1 Find the x-intercepts of the function f(x) = 1/x. Here, the numerator is 1 and cannot be zero. Thus, this function does not have any x- intercepts. Example 2 Find the x-intercepts of the function (x2 − 3x + 2)/(x2 − 2x − 3). The numerator of this rational function can be factored: x2 − 3x + 2 = (x − 1)(x − 2). This polynomial equals 0 when x = 1 or 2. The x-intercepts can thus be found at 1 and 2. Example 3 Find the roots of (x3 − 2x)/(2x2 − 10). Equating the numerator to 0, it can be seen that the roots of this function are 0 and the positive and negative square roots of 2. These values can also be identified from Figure 4.21. Source: rational-functions/rational-functions/solving-problems-with-rational- functions/ CC-BY-SA Boundless is an openly licensed educational resource This function has three x-intercepts. Figure 4.21 Example of a Rational Function 236 Polynomial Inequalities Polynomials can be expressed as inequalities, the solutions for which can be determined from the polynomial's zeros. KEY POINTS • To solve a polynomial inequality, first rewrite the polynomial in factored form to find its zeros. • For each zero, input the value of the zero in place of x in the polynomial. Determine the sign (positive or negative) of the polynomial as it passes the zero in the rightward direction. • Determine the intervals between these roots which satisfy the inequality. Like any other function, a polynomial may be written as an inequality, giving a large range of solutions. The best way to solve a polynomial inequality is to find its zeros. At these points, the polynomial's value goes from negative to positive or positive to negative. The easiest way to find the zeros of a polynomial is to express it in factored form. Consider the polynomial inequality: x3 + 2x2 − 5x − 6 > 0 This can be expressed as the product of these three terms: (x − 2)(x + 1)(x + 3) > 0 The three terms reveal zeros at x=-3, x=-1, and x=2. We know that the lower limit of the inequality crosses the x axis at each of these x values, but now have to determine which direction (positive or negative) it takes at each crossing. x+3>0 for x>-3 x+1>0 for x>-1 x-2>0 for x>2 Thus, as the polynomial crosses the x axis at x=-3, the term (x+3) equals 0, becoming positive to the right. At the same time, (x+1) and (x-2) are negative. The product of a positive and two negatives is positive, so we can conclude that the polynomial becomes positive as it passes x=-3. The next zero is at x=-1. From the paragraph above, we know that the polynomial is positive as it approaches its next zero, but we can use the same reasoning for proof. At x=-1, (x+1) equals 0, becoming positive to the right. The term (x+3) is positive, while (x-2) is negative. The product of two positives and a negative is negative, so we can conclude that the polynomial becomes negative as it passes x=-1. 238 The same process can be used to show that the polynomial becomes positive again at x=2. Recalling the initial inequality, we can now determine the solution of exactly where the polynomial is greater than 0. Because there is no zero to the left of x=-3, we can assume that the polynomial is negative for all x values -∞ to -3. The polynomial is positive from x=-3 to x=-1 before becoming negative once more. It becomes positive at x=2, and because there are no more zeros to the right, we can assume the polynomial remains positive as x approaches ∞ (Figure 4.22). Thus, the solution is: (-3, -1), (2,∞) For inequalities that are not expressed relative to zero, expressions can be added or subtracted from each side to take it into the desired form. Source: rational-functions/inequalities/polynomial-inequalities/ CC-BY-SA Boundless is an openly licensed educational resource 239 This polynomial has four roots. It is positive in three segments and negative in two. If it were a polynomial inequality with the condition that all values are greater than 0, the two negative segments would be removed. Figure 4.22 A fourth-degree polynomial Rational Inequalities Rational inequalities can be solved much like polynomial inequalities. KEY POINTS • First factor the numerator and denominator polynomial to reveal the zeros in each. • Substitute x with a zero (root) to determine whether the rational function is positive or negative to the right of that point. Repeat for all zeros. • The intervals that satisfy the inequality symbol will be the answer. Note that for any ≥ or ≤, the interval will only be closed to include the zero if the zero is found in the numerator. If the zero is found in the denominator, that point is undefined, and cannot be included in the solution. As with solving polynomial inequalities, the first step to solving rational inequalities is to find the zeros. Because a rational expression consists of the ratio of two polynomials, the zeroes will be needed for both. The zeros in the numerator are x values at which the rational inequality crosses from negative to positive or from positive to negative. The zeros in the denominator are x values are at which the rational inequality is undefined, the result of dividing by 0 (Figure 4.23). Consider the rational inequality: x2 + 2x − 3 x2 − 4 > 0 This equation can be factored to give: (x + 3)(x − 1) (x + 2)(x − 2) ≥ 0 240 For x values that are zeros for the numerator polynomial, the rational function overall is equal to zero. For x values that are zeros for the denominator polynomial, the rational function is undefined, with a vertical asymptote forming instead. Figure 4.23 Example of a rational polynomial The numerator has zeros at x=-3 and x=1. The denominator has zeros at x=-2 and x=2. As x crosses rightward past -3, (x+3) becomes positive. At that same point, (x-1), (x+2), and (x-2) are all negative. The product of a positive and three negatives is negative, so the rational expression becomes negative as it crosses x=-3 in the rightward direction. The same process can be used to determine that the rational expression is positive after passing the zero at x=-2, is negative after passing x=1, and is positive after passing x=2. Thus we can conclude that for x values on the open interval from -∞ to -3, the rational expression is negative. From -3 to -2, it is positive; from -2 to 1 it is negative; from 1 to 2 it is positive, and from 2 to ∞ it is negative. Because the inequality is written as ≥ 0 as opposed to > 0, we will need to evaluate the x values at zeros to determine whether the function is defined. In the case of x=-2 and x=2, the rational function has a denominator equal to 0 and becomes undefined. In the case of x=-3 and x=1, the rational function has a numerator equal to 0, which makes the function overall equal to 0, making it inclusive in the solution. Thus, the full solution is: [−3, − 2), [1,2) Source: rational-functions/inequalities/rational-inequalities/ CC-BY-SA Boundless is an openly licensed educational resource 241 Direct Variation When two variables change proportionally, or are directly proportional, to each other, they are said to be in direct variation. KEY POINTS • The ratio of variables in direct variation is always constant • Direct variation between variables is easily modeled using a linear graph. • The equation relating directly varying variables to a constant can be rearranged to slope-intercept form. When two variables change proportionally to each other, they are said to be in direct variation. This can also be called directly proportional. For example, a toothbrush costs $2. Purchasing five toothbrushes would cost $10; purchasing 10 toothbrushes would cost $20. No matter how many toothbrushes purchased, the ratio will always remain: $2 per toothbrush. Thus we can say that cost varies directly as the value of toothbrushes. Direct variation is easily illustrated using a linear graph. Knowing that the relationship between two variables is constant, we can show their relationship as (Figure 4.24): y x = k where k is a constant known as the constant of proportionality. This can be rearranged to slope-intercept format: y = kx In this case, the y-intercept is equal to 0. Revisiting the example with toothbrushes and dollars, we can define the x axis as number of toothbrushes and the y axis as number of dollars. Doing so, the variables would abide by the relationship: y x = 2 243 The line y=kx is an example of direct variation between variables x and y. For all points on the line, y/ x=k. Figure 4.24 Direct Variation Any augmentation of one variable would lead to an equal augmentation of the other. For example, doubling y would result in the doubling of x. Source: rational-functions/variation-and-problem-solving/direct-variation/ CC-BY-SA Boundless is an openly licensed educational resource Inverse Variation Indirect variation is used to describe the relationship between two variables when their product is constant. KEY POINTS • The ratio of variables in direct variation is always constant. • Direct variation between variables is depicted by an hyperbola. • The equation relating indirectly varying variables to a constant can be rearranged to hyberbolic form. Inverse variation is the opposite of direct variation. In the case of inverse variation, the increase of one variable leads to the decrease of another. Consider a car driving on a flat surface at a certain speed. If the driver shifts into neutral gear, the car's speed will decrease at a constant rate as time increases, eventually coming to a stop. Inverse variation can be illustrated, forming a graph in the shape of a hyperbola (Figure 4.25). Knowing that the relationship between the two variables is constant, we can show that their relationship is: yx = k 244 where k is a constant known as the constant of proportionality. Note that as long as k is not equal to 0, neither x nor y can ever equal 0 either. We can rearrange the above equation to place the variables on opposite sides: y = k /x Revisiting the example of the decelerating car, let's say it starts at 50 miles per hour and slows at a constant rate. If we define y as its speed in miles per hour, and x as time, the relationship between x and y can be expressed as: y = 50/x Note that realistically, other factors (e.g., friction), will influence the rate of deceleration. Other constants can be incorporated into the equation for the sake of accuracy, but the overall form will remain the same. Source: rational-functions/variation-and-problem-solving/inverse-variation/ CC-BY-SA Boundless is an openly licensed educational resource 245 This hyperbola shows the indirect variation of variables x and y. Figure 4.25 Indirect variation Combined Variation Combined variation describes the relationship between three or more variables that vary directly and inversely with one another. KEY POINTS • There must be a minimum of three related variables for their relationship to be one of combined variation. • Among the three or more related variables, one must directly vary with another and inversely vary with a third in order for the relationship to be one of combined variation. • An example of combined variation in the physical world is the Combined Gas Law, which relates pressure, temperature, volume, and moles (amount of molecules) of a gas. Combined variation is used to describe the relationship between three or more variables that vary directly and inversely with one another. Before go deeper into the concept of combined variation, it is important to first understand what direct and inverse variation mean. Direct and Inverse Variation Simply put, two variables are in direct variation when the same thing that happens to one variable happens to the other. If x and y are in direct variation, and x is doubled, then y would also be doubled. The two variables may be considered directly proportional. Two variables are said to be in inverse variation, or are inversely proportional, when an operation of change is performed on one variable and the opposite happens to the other. For example, if x and y are inversely proportional, if x is doubled, then y is halved. Combined Variation To have variables that are in combined variation, the equation must have variables that are in both direct and inverse variation, as shown in the example below. Consider the equation: z = k( x y ) where x, y, and z are variables and k is a constant known as the proportionality constant. In this example, z varies directly as x and inversely as y. Given values for any three of x, y, z, and k, the fourth can be found by substitution. For example, if z=12, x=4 and y=2, we can solve for k: 246 12 = k 4 2 k = 6 Practical Application A practical example of combined variation is the Combined Gas Law, which relates the pressure (p), volume (v), moles (n), and temperature (T) of a sample of gas: PV = nRT where R is a constant (Figure 4.26). Solving for P, we can determine the variation of the variables. P = nRT V In the above equation, P varies directly with n and T, and inversely with V. Thus, pressure increases as temperature and moles increase. What's more, pressure decreases as volume increases. Source: rational-functions/variation-and-problem-solving/combined- variation/ CC-BY-SA Boundless is an openly licensed educational resource A constant amount of gas will exert pressure that varies directly with temperature. In this illustration, volume is held constant by an increased mass weighing down the lid of the container. If not for that extra mass, the lid would raise, increasing the volume and relieving the pressure. Figure 4.26 Illustration of Gay-Lussac's Law, derived from the Combined Gas Law 247 Inverses Logarithm reverses exponentiation. The complex logarithm is the inverse function of the exponential function applied to complex numbers. KEY POINTS • In trigonometric functions, a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of −1 denotes the inverse function. • An inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa (i.e., ƒ(x)=y, and g(y)=x). • The logarithm to base b is the inverse function of f(x) = bx: logb (b)x = x logb(b) = x. • The natural logarithm ln(x) is the inverse of the exponential function ex: b = elnb . In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa (i.e., ƒ(x)=y, and g(y)=x). More directly, g(ƒ(x))=x means that g(x) composed with ƒ(x) leaves x unchanged. A function ƒ that has an inverse is called invertible; the inverse function is then uniquely determined by ƒ and is denoted by ƒ−1 (read f inverse, not to be confused with exponentiation). Exponential Notation Placing an integer superscript after the name or symbol of a function, as if the function were being raised to a power, commonly refers to repeated function composition rather than repeated multiplication. Thus f 3(x) may mean f(f(f(x))); in particular, f −1(x) usually denotes the inverse function of f. Iterated functions are of interest in the study of fractals and dynamical systems. A special syntax applies to the trigonometric functions: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of −1 denotes the inverse function. That is, sin2x is just a shorthand way to write (sin x)2 without using parentheses, whereas sin−1x refers to the inverse function of the sine, also called arcsin x. There is no need for a shorthand for the reciprocals of trigonometric functions since each has its own name and abbreviation; for example, 1/(sin x) = (sin x)−1 = csc x. A similar convention applies to logarithms, where log2x usually means (log x)2, not log log x. 250 Exponents and Logarithms The formula for the logarithm of a power says in particular that for any number x: logb(b)x = x logb(b) = x In prose, taking the x-th power of b and then the base-b logarithm gives back x. Conversely, given a positive number y, the formula says that first taking the logarithm and then exponentiating gives back y. Thus, the two possible ways of combining (or composing) logarithms and exponentiation give back the original number. Therefore, the logarithm to base b is the inverse function of f(x) = bx. Inverse functions are closely related to the original functions. Their graphs correspond to each other upon exchanging the x- and the y-coordinates (or upon reflection at the diagonal line x = y), as shown here (Figure 5.1). A point (t, u = bt) on the graph of f yields a point (u, t = logbu) on the graph of the logarithm and vice versa. As a consequence, logb(x) diverges to infinity (gets bigger than any given number) if x grows to infinity, provided that b is greater than one. In that case, logb(x) is an increasing function. For b < 1, logb(x) tends to minus infinity instead. When x approaches zero, logb(x) goes to minus infinity for b > 1 (plus infinity for b < 1, respectively). In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function applied to complex numbers. The natural logarithm ln(x) is the inverse of the exponential function ex. It is defined for b > 0, and satisfies: b = elnb If bx is to preserve the logarithm and exponent rules, then one must have: bx = (elnb ) x = ex⋅lnb for each real number x. This can be used as an alternative definition of the real number power bx. 251 The graph of the logarithm function logb(x) (blue) is obtained by reflecting the graph of the function bx (red) at the diagonal line (x = y). Figure 5.1 Logarithm Function Source: logarithms/inverse-functions/inverses/ CC-BY-SA Boundless is an openly licensed educational resource One-to-One Functions A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its codomain. KEY POINTS • A one-to-one function has an inverse function, if a function is not one-to-one it cannot have an inverse function as it yields multiple outputs. • Domain restriction can allow a function to become one-to- one, as in the case of f(x) = x2 . • An easy way to check if a function is a one-to-one function is by graphing it and then performing the horizontal line test. A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its co- domain. In other words, every element of the function's co-domain is mapped to by at most one element of its domain. If, in addition, all of the elements in the co-domain are in fact mapped to by some element of the domain, then the function is said to be bi-jective. An injective function is also said to be a one-to-one function. Occasionally, an injective function from X to Y is denoted f: X ↣ Y, using an arrow with a barbed tail. The set of injective functions from X to Y may be denoted YX using a notation derived from that 252 used for falling factorial powers, since if X and Y are finite sets with respectively m and n elements, the number of injections from X to Y is nm. A function f that is not injective is sometimes called many-to- one. However, this terminology is also sometimes used to mean "single-valued", i.e., each argument is mapped to at most one value. This is the case for any function, but is used to stress the opposition with multi-valued functions, which are not true functions. An easy way to check if a function is a one-to-one is by graphing it and then performing the horizontal line test. If any horizontal line intersects the graph in more than one point, the function is not injective. To see this, note that the points of intersection have the same y-value, because they lie on the line, but different x values, which by definition means the function cannot be injective. The exponential function exp : R → R defined by exp(x) = ex is injective, but not surjective as no real value maps to a negative number. The natural logarithm function ln : (0, ∞) → R defined by x ↦ ln x is injective. The exponential function, f(x) = x2 , without any domain restriction, is not one-to-one. It forms a parabola and fails the horizontal line test (Figure 5.3). Since it is not a one-to-one function, it cannot have an inverse. Looking at another exponential function: f(x) = 10x (Figure 5.2), it is a one-to-one function and therefore, has an inverse which is: f −1 (x) = log(x). Source: logarithms/inverse-functions/one-to-one-functions/ CC-BY-SA Boundless is an openly licensed educational resource 253 This function forms a parabola and therefore fails the the horizontal line test. Figure 5.3 Graph of f(x)=x^2 f(x)=10^x is a one- to-one function and is shown here with its inverse, f(x)=10^x. Figure 5.2 f(x)=10^x and its inverse f(x)=10^x Finding Formulas for Inverses To find the inverse function, switch the x and y values, and then solve for y. KEY POINTS • An inverse function reverses the inputs and outputs. • To find the inverse formula of a function, write it in the form of y and x, switch y and x, and then solve for y. • Some functions have no inverse function, as a function cannot have multiple outputs. Inverse function, f−1(x), is defined as the inverse function of f(x) if it consistently reverses the f(x) process. That is, if f(x) turns a into b, then f−1(x) must turn b into a. More concisely and formally, f−1(x) is the inverse function of f(x) if f( f −1 (x)) = x. Finding an Inverse Function In general, given a function, how do you find its inverse function? Remember that an inverse function reverses the inputs and outputs. When we graph functions (Figure 5.4), we always represent the incoming number as x and the outgoing number as y. So to find the inverse function, switch the x and y values, and then solve for y. Example Building and Testing an Inverse Function Find the inverse function of : f(x) = 2x a.: Write the function as: y = 2x b.: Switch the x and y variables: x = 2y c.: Solve for y: log2 x = log2 2y log2 x = ylog2 2 log2 x = y So: f −1 (x) = log2 (x) 254 The graphs of y = ƒ(x) and y = ƒ–1(x). The dotted line is y = x. Figure 5.4 Graph of the Inverse Test to make sure this solution fills the definition of an inverse function. a.: Pick a number, and plug it into the original function. 2→f(x)→4. b.: See if the inverse function reverses this process. 4→f−1(x)→2. ✓ Some functions have no inverse function because of the rule of consistency. For instance, consider the function y=x2. This function takes both 3 and –3 and turns them into 9. No problem: a function is allowed to turn different inputs into the same output. However, what does that say about the inverse of this particular function? In order to fulfill the requirement of an inverse function, it would have to take 9, and turn it into both 3 and –3, which is the one and only thing that functions are not allowed to do. Hence, the inverse of this function would not be a function at all! Source: logarithms/inverse-functions/finding-formulas-for-inverses/ CC-BY-SA Boundless is an openly licensed educational resource Composition and Composite Functions A composite function represents, in one function, the results of an entire chain of dependent functions. KEY POINTS • The composition of functions is always associative. That is, if f, g, and h are three functions with suitably chosen domains and codomains, then f ∘ (g ∘ h) = (f ∘ g) ∘ h, where the parentheses serve to indicate that composition is to be performed first for the parenthesized functions. • Functions can be inverted and then composed, giving the notation of: ( f ′∘ g′)(x). • Functions can be composed and then inverted, yielding the following notation: ( f ∘ g)′(x). In mathematics, function composition is the application of one function to the results of another Figure 5.5. For instance, the functions f: X → Y and g: Y → Z can be composed by computing the output of g when it has an argument of f(x) instead of x. Intuitively, if z is a function g of y and y is a function f of x, then z is a function of x. Thus one obtains a composite function g ∘ f: X → Z defined by (g ∘ f)(x) = g(f(x)) for all x in X. The notation g ∘ f is read 255 as "g circle f", or "g composed with f", "g after f", "g following f", or just "g of f". The composition of functions is always associative. That is, if f, g, and h are three functions with suitably chosen domains and codomains, then f ∘ (g ∘ h) = (f ∘ g) ∘ h, where the parentheses serve to indicate that composition is to be performed first for the parenthesized functions. Since there is no distinction between the choices of placement of parentheses, they may be safely left off. The functions g and f are said to commute with each other if g ∘ f = f ∘ g. In general, composition of functions will not be commutative. Commutativity is a special property, attained only by particular functions, and often in special circumstances. Let's look at it in a basic sense: You are working in the school cafeteria, making peanut butter sandwiches for today's lunch. The more classes the school has, the more children there are. The more children there are, the more sandwiches you have to make. The more sandwiches you have to make, the more pounds (lbs) of peanut butter you will use. The more peanut butter you use, the more money you need to budget for peanut butter...and so on. Each sentence in this little story is a function. Mathematically, if c is the number of classes and h is the number of children, then the first sentence asserts the existence of a function h(c).The principal walks up to you at the beginning of the year and says "We're considering expanding the school. If we expand to 70 classes, how much money do we need to budget? What if we expand to 75? How about 80?" For each of these numbers, you have to calculate each number from the previous one, until you find the final budget number. But going through this process each time is tedious. What you want is one function that puts the entire chain together: "You tell me the number of classes, and I will tell you the budget." This is a composite function—a function that represents in one function the results of an entire chain of dependent functions. Since such chains are very common in real life, finding composite functions is a very important skill. 256 g ∘ f, the composition of f and g. For example, (g ∘ f)(c) = #. Figure 5.5 Composition of Functions Restricting Domains Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse. KEY POINTS • f −1 (x) is defined as the inverse function of f(x) if it consistently reverses the f(x) process. • Informally, a restriction of a function f is the result of trimming its domain. • f(x) = x2 ,without any domain restriction, does not have an inverse function as it fails the horizontal line test. f−1(x) is defined as the inverse function of f(x) if it consistently reverses the f(x) process. That is, if f(x) turns a into b, then f−1(x) must turn b into a. More concisely and formally, f−1(x) is the inverse function of f(x) if f( f −1 (x)) = x. Informally, a restriction of a function f is the result of trimming its domain. More precisely, if S is any subset of X, the restriction of f to S is the function f\|S from S to Y such that f\|S(s) = f(s) for all s in S. If g is a restriction of f, then it is said that f is an extension of g. The overriding of f: X → Y by g: W → Y (also called overriding union) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y. Its graph is the set-theoretical union of the graphs of g and f\|X W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an associative operation; it has the empty function as an identity element. If f\|X ∩ W and g\|X ∩ W are pointwise equal (e.g., the domains of f and g are disjoint), then the union of f and g is defined and is equal to their overriding union. This definition agrees with the definition of union for binary relations. Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse. For example, f(x) = x2 , without any domain restriction, This function fails the horizontal line test and therefore does not have an inverse. Figure 5.6 Graph of y=x^2 258 does not have an inverse function as it fails the horizontal line test Figure 5.6. But if we restrict the domain to be x>0 then we find that it passes the horizontal line test and therefore has an inverse function. Source: logarithms/inverse-functions/restricting-domains/ CC-BY-SA Boundless is an openly licensed educational resource 259 Basics of Graphing Exponential Functions The exponential function y = a ⋅ bx ] is a function that will remain proportional to its original value when it grows or decays. KEY POINTS • If the base, b, is greater than 1, then the function increases exponentially at a growth rate of b. This is known as exponential growth. • If the base, b, is less than 1 (but greater than 0) the function decreases exponentially at a rate of b. This is known as exponential decay. • If the base, b, is equal to 1, then the function trivially becomes y = a. An exponential function is: • A function that has exponents • Is proportional to its original value whenever a quantity grows. This means that no matter what value is plugged in for the variable, it stays proportional to its original value. • Is proportional to its original value whenever it decays. This means that if you were to take its derivative, its value would still be proportional to the original. This can be shown with the function ex . The derivative of ex returns the same function, therefore it is always proportional to the original value. Another example is the function y = 2x . A graph of an exponential function becomes a curved line that steadily gets steeper, like in Figure 5.7. The basic formula for an exponential function is y = a ⋅ bx where the constant a is the initial value or y-intercept of the function (0, a). A good way to start a plot of the function is to choose numbers to plug in. For example, the function y = 2x , choosing the x values 0, 1, 261 This is the graph of the exponential function 2^x. Figure 5.7 An Exponential Function 2, 3, 4, and 5 you get the set of ordered pairs (0, 1), (1, 2), (2, 4), (3, 8), (4, 16), (5, 32). As you connect the points you will notice a smooth curve, as in Figure 5.7. You can see in Figure 5.8 that exponential functions will grow very rapidly compared to polynomials. If b is greater than 1, then it is known as exponential growth. If b is less than 1, but greater than 0, then it is known as exponential decay. The difference between the two can be seen by Figure 5.9, where the function y = 2x is growing exponentially, and increasing rapidly as x increases, whereas y = ( 1 2 )x is decaying exponentially, and decreases as x increases. In this case, the two are mirrors of each other across the y axis. Note that while neither function will ever reach 0, it will get infinitely close to 0 on one side. The line that the function approaches is referred to as the asymptote. The asymptote can be horizontal, vertical or oblique, depending on the function. If b = 1, then the function becomes y = a, which is a horizontal line. Source: logarithms/graphing-exponential-functions/basics-of-graphing- exponential-functions/ CC-BY-SA Boundless is an openly licensed educational resource 262 This is a graph that shows how the exponential function will eventually grow much faster than any polynomial function. Figure 5.8 Exponential Functions vs. Others Graphed are two functions, one with exponential growth, the other exponential decay. The exponential growth increases rapidly as x increases, and exponential decay decreases rapidly as x increases. Figure 5.9 Exponential Growth and Decay Problem Solving Graphically solving problems with exponential functions allows visualization of sometimes complicated interrelationships. KEY POINTS • Exponential functions are used in numerous applications, such as Biology, Economics, Finance, and many more. • To model a general problem that you know is an exponential function, you only need two points to solve for the constants a and b. • In the exponential function, y = a ⋅ bx , the constant a represents the initial value of the function, as at x = 0 the y value is a. The constant b is the rate of growth (or decay if less than 1) of the function. The exponential function has numerous applications. Biology, Economics, and Finance are only just a few of the applications that use exponential functions. Applications in Biology In Biology, the number of microorganisms in a culture will increase exponentially until an essential nutrient is exhausted. Typically the first organism splits into two sister organisms, who then each split to form four, who split to form eight, and so on, which is the simple function f(t) = 2t/x where x is the time it takes for the organism to split (Figure 5.10). With this function, you can figure out how many organisms you will have after a certain amount of time. You can also determine how long it would take to reach a certain number of organisms with the help of the logarithm function, log(x). 263 This is a graph of an exponential function that is a representation of microorganism growth. In this case, the Y axis can be taken as the bacterial count, and the X axis is time. The resulting curve is nonlinear with respect to time -- it is exponential. Figure 5.10 An Exponential Function A virus (for example SARS, or smallpox) typically will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people. Human population, if the number of births and deaths per person per year were to remain at current levels, is another example. According to the United States Census Bureau, over the last 100 years (1910 to 2010), the population of the United States of America is exponentially increasing at an average rate of one and a half percent a year (1.5%). This means that the doubling time of the American population (depending on the yearly growth in population) is approximately 50 years. A related constant is half- life, which is the amount of time required for a process to consume half the original quantity, or the time it takes a decaying substance to drop to half its original value. Applications in Economics Economic growth is expressed in percentage terms, implying exponential growth. For example, U.S. GDP per capita has grown at an exponential rate of approximately two percent per year for two centuries. Applications in Finance Compound interest at a constant interest rate provides exponential growth of the capital. Pyramid schemes or Ponzi schemes also show this type of growth resulting in high profits for a few initial investors and losses among great numbers of investors. Solving General Problems In general, a real world application that can be modeled with an exponential function is one that will continuously increase at an increasing rate based on the current value. For instance, in the microorganisms example, if the current value is 8, then the next time step will give 16, whereas if the current level is 1024, the next time step gives you 2048. To figure out the values used to model a certain function, you need either an initial point (at x = 0), and any other point, or two general points if you do not have an initial point. For instance, let us say you have the two points (0, 4) and (3, 108). With the initial point (0, 4), you can plug it into the equation to get that a = 4. Now we know that the function takes the form y = 4bx , and we need to next find b. With the second point we find that 108 = 4b3 , and b3 = 27, which means that b = 3. We now have the full equation y = 4 ⋅ 3x . Now let us say that we have two general points, say (2, 12) and (5, 96). Plugging the first point into the equation gives us 12 = a ⋅ b2 , or a = 12 b2 . Now with the second equation we have 96 = a ⋅ b5 , plugging in the value of a gives us 96 = 12b5 b2 , or b3 = 8, and so b = 2. Plugging 264 that back into the equation for a gives us a = 3. We now have our final equation of y = 3 ⋅ 2x . Source: logarithms/graphing-exponential-functions/problem-solving/ CC-BY-SA Boundless is an openly licensed educational resource e The number e is an important mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm, ln(x). KEY POINTS • The number e is defined in many ways, one of which is lim n→∞ (1 + 1 n )n , which was discovered through the idea of compound interest. • The natural logarithm, written f(x) = ln(x), is the power to which e must be raised to obtain x. • The constant can be defined in many ways; for example, e is the unique real number such that the derivative (slope of the tangent line) of the function ex . • More generally, an account that starts at $1 and offers an annual interest rate of R will, after t years, yield eRt dollars with continuous compounding. The number e is an important mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm. It is lim n→∞ (1 + 1 n )n , an expression that arises in the study 265 of compound interest, and can also be calculated as the sum of the infinite series: e = 1 + 1 1! + 1 2! + 1 3! + 1 4! + . . . . The logarithm is the power to which a given base number must be raised in order to obtain x. For the natural logarithm, written ln(x), this is the power to which e must be raised to obtain x. For example, ln(e) = 1 and ln(1) = 0. The constant can be defined in many ways; for example, e is the unique real number such that the derivative (slope of the tangent line) of the function f(x) = ex is ex . Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, e is not to be confused with γ—the Euler– Mascheroni constant, sometimes called simply Euler's constant. The number e is also known as Napier's constant, but Euler's choice of this symbol is said to have been retained in his honor. The number e is of eminent importance in mathematics, alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant π, e is irrational: it is not a ratio of integers; and it is transcendental: it is not a root of any non-zero polynomial with rational coefficients. Compound Interest Jacob Bernoulli discovered this constant by studying a question about compound interest: An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year? The area between the x-axis and the graph y = 1/x, between x = 1 and x = e is 1. Figure 5.11 The Constant e. 266 If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00×1.52 = $2.25 at the end of the year. Compounding quarterly yields $1.00×1.254 = $2.4414..., and compounding monthly yields $1.00×(1+1/12)12 = $2.613035... If there are n compounding intervals, the interest for each interval will be 100%/ n and the value at the end of the year will be $1.00×(1 + 1/n)n. Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger n and, thus, smaller compounding intervals. Compounding weekly (n = 52) yields $2.692597..., while compounding daily (n = 365) yields $2.714567..., just two cents more. The limit as n grows large is the number that came to be known as e; with continuous compounding, the account value will reach $2.7182818.... More generally, an account that starts at $1 and offers an annual interest rate of R will, after t years, yield eRt dollars with continuous compounding. (Here R is a fraction, so for 5% interest, R = 5/100 = 0.05). Source: logarithms/graphing-exponential-functions/e/ CC-BY-SA Boundless is an openly licensed educational resource Graphs of Exponential Functions, Base e The function f(x) = ex is a basic exponential function with some very interesting properties. KEY POINTS • The function f(x) = ex is the only function whose derivative is itself. In other words, the slope of a tangent line to the graph at any point is equal to the y-value at that point. • The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value. • The exponential function can be characterized as the power series: ex . • d dx ex = ex The basic exponential function, sometimes referred to as the exponential function, is f(x) = ex , where e is the number (approximately 2.718281828) such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in 267 the dependent variable. The function is often written as exp(x), especially when it is impractical to write the independent variable as a superscript. The exponential function is widely used in physics, chemistry, and mathematics. The graph of y = ex is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis, but can get arbitrarily close to it for negative x; thus, the x-axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y-coordinate at that point. The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value. One such situation is continuously compounded interest, which is exactly what led Jacob Bernoulli in 1683 to the number now known as e. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function. Formal Definition The exponential function ex can be characterized in a variety of equivalent ways. In particular it may be defined by the following power series: ex This can be seen in Figure 5.12. As you add more and more of the power series, you get closer and closer to the exponential function. The function may also be defined as the following limit: ex = lim n→∞ (1 + x n )n The Graph The graph of f(x) = ex , as shown in Figure 5.13, is like any other exponential function, where it has important values (0, 1) and (1, e). There are some very interesting properties about the exponential function's graph that no other function has. The main property of This is an animation showing the exponential function (in blue) and the first n+1 terms of the power series (in red) that make up the exponential function. Figure 5.12 The Exponential Function as an Infinite Series 268 note is that the slope of the tangent line to the graph at any point is equal to the y-value at that point, as seen in Figure 5.14. Or, as stated before, the derivative of ex is ex . Source: logarithms/graphing-exponential-functions/graphs-of-exponential- functions-base-e/ CC-BY-SA Boundless is an openly licensed educational resource The graph of the natural exponential function with base e, y=ex. Figure 5.13 The Exponential Function The tangent to the exponential function with base e at any point is equal to the y-value at that point. Figure 5.14 Tangents to the Exponential Function 269 Logarithmic Functions The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. KEY POINTS • The inverse of the logarithmic operation is exponentiation. • The logarithm is commonly used in many fields: that with base 2 in computer science, that with base e in pure mathematics, and that with base 10 in natural science and engineering. • The logarithm of a product is the sum of the logarithms of the factors. In its simplest form, a logarithm is an exponent. Taking the logarithm of a number, one finds the exponent to which a certain value, known as a base, is raised to produce that number once more. Logarithms have the following structure: logb(x) = c where b is known as the base, c is the exponent to which the base is raised to afford x. Consider the following logarithm, for example: log3(243) = 5 The left side of the equation states that the right will be the exponent to which 3 is raised to yield 243 and indeed, 35=243. The explanation of the previous example reveals the inverse of the logarithmic operation: exponentiation. Starting with 243, if we take its logarithm with base 3, then raise 3 to the logarithm, we will once again arrive at 243. Logarithms are most commonly used to simplify complex calculations that involve high-level exponents. In science and engineering, the logarithm with base 10 is commonly used. In pure mathematics, the logarithm with base e (≈ 2.718) is often applicable. In computer science, the binary logarithm, with base 2, is commonly used. In natural science and engineering, the logarithm with base 10 is frequently used. In chemistry, for example, pH and pKa are used to simplify concentrations and dissociation constants, respectively, of high exponential value. The purpose is to bring wide-ranging values into a more manageable scope. A dissociation constant may be smaller than 271 1010, or higher than 10-50. Taking the logarithm of each brings the values into a more comprehensible scope (10 to -50) (Figure 5.15). Source: logarithms/graphing-logarithmic-functions/logarithmic-functions/ CC-BY-SA Boundless is an openly licensed educational resource Special Logarithms Any number can be used as the base of a logarithm but certain bases (10, e, and 2) have more widespread applications than others. KEY POINTS • Logarithms with base equal to 10 are called common logarithms. They are most applicable in physical and natural sciences and engineering. • Logarithms with base equal to e are called natural logarithms. They are most applicable in pure mathematics. • Logarithms with base equal to 2 are called binary logarithms. They are most applicable in computer science. Not all logarithms are equally useful in practice. Some, depending on their base, are more useful than others in a given field of study. Out of the infinite number of possible bases, three stand out as particularly useful. Logarithms with base 10 are called common logarithms. They are named so because they are widely used, and in fact the common logarithm of x can be denoted as: log(x) 272 The function's slope decreases with increasing x, thus containing its vertical growth. More importantly, the scale is greatly reduced, allowing for easy visualization of large values of y. In many cases, by taking the logarithms of exponential equations, the equations become linear, and plotting the appropriate logarithms can help solve for roots, slopes, and intercepts. Figure 5.15 Graph of Binary Logarithm Note that the base is not specified; it is implied. Common logarithms are often used in physical and natural sciences and engineering. For example, the magnitude of an earthquake (M) can be determined based on the logarithm of an intensity measurement from a seismograph (I): M = log( I I0 ) where I0 is a constant. Logarithms with base e are called natural logarithms. They are so commonly used that they are have a unique symbol. The natural logarithm with base x is denoted as: ln(x) Like common logarithms, natural logarithms are not expressed with a base. The base of e is implicit in their symbol (Figure 5.16) Natural logarithms are often found in physical sciences and pure math. For example, the entropy (S) of a system can be calculated from the natural logarithm of the number of possible microstates (W) the system can adopt: S = k ln(W ) where k is a constant. Logarithms with base 2 are called binary logarithms. The binary logarithm of x is commonly written as: ld(x) or lb(x) Binary logarithms are useful in any application that involves the doubling of a quantity, and particularly in computer science with the use of integral parts. Source: logarithms/graphing-logarithmic-functions/special-logarithms/ CC-BY-SA Boundless is an openly licensed educational resource 273 The natural logarithm is the logarithm with base equal to Euler's number, e. Figure 5.16 Graph of the Natural Logarithm Converting Between Exponential and Logarithmic Equations Logarithmic and exponential forms are closely related, and an equation in either form can be freely converted into the other. KEY POINTS • The logarithmic and exponential operations are inverses. • If given an exponential equation, one can take the natural logarithm to isolate the variables of interest, and vice versa. • Converting from logarithmic to exponential form can make for easier equation solving. Because logarithmic and exponential functions are inverses of one another, one can be converted into the form of the other. Logarithmic equations have the form: y = logb(x) where b is a base (of defined value) and x and y are the independent and dependent variables, respectively. Exponential equations have the form: x = by where x and y are the dependent and independent variables, respectively, and b is of defined value (Figure 2). If b is the same in both the above equations, the x and y from the logarithmic equation are respectively equal to the x and y in the exponential equation. Thus, if given the exponential equation: 43 = 64 One can convert it to logarithmic form: lob4(64) = 3 And if given the logarithmic equation: log7(2401) = 4 One can convert to exponential form: 74 = 2401 Conversion from logarithmic to exponential form can help one solve otherwise difficult equations. 274 Natural Logarithms The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828. KEY POINTS • The natural logarithm is the logarithm with base equal to e. • Also known as Euler's number, e is an irrational number that often appears in natural relationships in pure math and science. • The number e and the natural logarithm have many applications in calculus, number theory, differential equations, complex numbers, compound interest, and more. The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. The natural logarithm is the logarithm with base equal to e (Figure 5. 18). Also known as Euler's number, e is an irrational number representing the limit of: (1 + 1/n)n as n approaches infinity. In other words, e is the sum of 1 plus 1/1 plus 1/(12) plus 1/(123), and so on. The number e has many applications in calculus, number theory, differential equations, complex numbers, compound interest, and more. It also is extremely useful as a base in logarithms; so useful that the logarithm with base e has its own name (natural logarithm) and symbol. Here is the proper notation for the natural logarithm of x: ln(x) The natural logarithm is so named because unlike 10, which is given value by culture and has minimal intrinsic use, e is an extremely interesting number that often "naturally" appears, especially in calculus. The inverse of the natural log appears, for example, upon differentiating a logarithm of any base: d dx logb(x) = 1 x ln(b) Outside of calculus, the natural logarithm can be used to relate 1, e, i, and π, four of the most important numbers in mathematics: ln(−1) = iπ 276 Source: logarithms/graphing-logarithmic-functions/natural-logarithms/ CC-BY-SA Boundless is an openly licensed educational resource Changing Logarithmic Bases The base of a logarithm can be converted to another value through a simple, one-step process. KEY POINTS • If a logarithm and/or its base are not whole numbers, evaluation can be near-impossible. • One can change the base of a logarithm by expressing it as the quotient of two logarithms with a common, same base. • Changing a logarithm's base to 10 makes it much simpler to evaluate; it can be done on a calculator. So long as a logarithm, its base, and the number upon which it operates are all whole numbers, one can evaluate logarithms manually with minimal difficulty. When decimals are involved, however, it can become exceedingly difficult to evaluate a logarithm. Let's consider: log4(9) We can easily determine that the above will simplify to a number between one and two, because 41 = 4 and 42 = 16. The exact value, however, is not so easily determined. 277 The function slowly grows to positive infinity as x increases and rapidly goes to negative infinity as x approaches 0 ("slowly" and "rapidly" as compared to any power law of x); the y-axis is an asymptote. Figure 5.18 Graph of the Natural Logarithm Not all calculators have logarithm functions and, those that do almost always have a base of 10. Fortunately, any logarithm can be converted into a logarithm of equal value with a different base. The formula for this transformation is: loga(x) = logb(x) logb(a) where a is the original base and b is the desired base. Revisiting the example above, we can change the base from 4 to 10, which can be input into a calculator. log4(9) = log10(9) log10(4) The left side of the equation is extremely difficult to calculate manually, and would be impossible to find on most calculators. However, the quotient of logarithms with base equal to 10 can easily be found on a scientific calculator (Figure 5.19). Source: logarithms/graphing-logarithmic-functions/changing-logarithmic- bases/ CC-BY-SA Boundless is an openly licensed educational resource 278 This function can be expressed as the quotient of log (x) and log (2), both with a base of 10. Figure 5.19 Graph of Binary Logarithm Graphs of Logarithmic Functions Logarithms can be graphed manually or electronically with points generally determined via a calculator or table. KEY POINTS • The logarithmic graph is similar in shape to the square root graph, but with a vertical asymptote as x approaches 0 from the right. • With use of defined numbers, logarithmic graphs can be shifted horizontally and/or vertically. • When graphing logarithms, it is usually advisable to use several x values between the vertical asymptote and the next integer. After the first integer, it is usually best to make points less and less frequently. At first glance, the graph of the logarithmic function can easily be mistaken for that of the square root function. Both the square root and logarithmic functions have a domain limited to x values greater than 0. However, the logarithmic function has a vertical asymptote descending towards negative ∞ as x approaches 0, whereas the square root reaches a minimum y value of 0. The logarithmic graph begins with a steep climb after x=0, but stretches more and more horizontally, its slope ever-decreasing with increasing x. Thus, the graph of the logarithmic function shows, as expected, an inverse relationship to the graph of the exponential. Graphing logarithmic functions can be performed manually or with a calculator. In either case, except if graphing a natural logarithm on a calculator, one must first convert the logarithm's base to 10. This is easily done using a table or an electronic calculator, converting the logarithm to its original form, then taking the log in base 10. Base 10 is simple to understand in that the numbering system is based on 10, so working with powers of 10 is "almost" second nature. The logarithm to base b = 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the constant e (≈ 2.718) as its base. Its use is widespread in pure mathematics, especially calculus. The binary logarithm uses base b = 2 and is prominent in computer science. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly 279 adopted by navigators — important in its own right — that the logarithm of a product is the sum of the logarithms of the factors: The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century. Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel is a logarithmic unit quantifying sound pressure and voltage ratios. In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They describe musical intervals, appear in formulae counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting. Once the logarithm is converted to a base of 10, one can create a table of points using x values and calculating their corresponding y values. Generally it is best to use several x values between 0 and 1 (0.125, 0.25, 0.5, and 1) to capture the asymptotic rise, and then double x values (2, 4, 8, 16) for every point thereafter. Graphs of logarithmic functions can be shifted horizontally or vertically by incorporating other values inside and outside the log brackets (Figure 5.20): y = log(x + a) where a is a certain defined number, shifts the graph horizontally. If a is a positive number, the graph shifts to the left; if a is negative, the graph shifts to the right. Such a shift would change the x values of interest in determining points to graph. 280 All logarithmic functions will have the same general shape, but their graphs will vary depending on base and any coefficients inserted into the equation. In this figure, red is ln(x) (which is log base e), green is log(x) base 10, purple is log(x) base 1.7. Figure 5.20 Logarithms y = log(x) + a where a is a certain defined number, shifts the graph vertically. If a is positive, the graph shifts upward; if a is negative, the graph shifts downward. Source: logarithms/graphing-logarithmic-functions/graphs-of-logarithmic- functions/ CC-BY-SA Boundless is an openly licensed educational resource Solving Problems with Logarithmic Graphs Some functions with rapidly changing shape are best plotted on a scale that increases exponentially; such scales make up logarithmic graphs. KEY POINTS • Logarithmic graphs use logarithmic scales, in which the values differ exponentially. For example, instead of including marks at 0, 1, 2, and 3, a logarithmic scale may include marks at 0.1, 1, 10, and 100, each an equal distance from the previous and next. • Logarithmic graphs allow one to plot a very large range of data without losing the shape of the graph. • Logarithmic graphs make it easier to interpolate in areas that may be difficult to read on linear axes. For example, if the plot of y=x5 is scaled to show a very wide range of y values, the curvature near the origin may be indistinguishable on linear axes. It is much clearer on logarithmic axes. Many mathematical and physical relationships are functionally dependent on high-order variables. 281 Consider the Stefan-Boltzmann law, which relates the power (j) emitted by a black body to temperature (T). J = σT4 . On a standard graph, this equation can be quite unwieldy. The fourth-degree dependence on temperature means that power increases extremely quickly. The fact that the rate is ever-increasing (and steeply so) means that changing scale is of little help in making the graph easier to interpret. For very steep functions, it is possible to plot points more smoothly while retaining the integrity of the data: one can use a graph with a logarithmic scale (Figure 5.21). The primary difference between the logarithmic and linear scales is that, while the difference in value between linear points of equal distance remains constant (that is, if the space from 0 to 1 on the scale is 1cm on the page, the distance from 1 to 2, 2 to 3, etc., will be the same), the difference in value between points on a logarithmic scale will change exponentially. A logarithmic scale will start at a certain power of 10, and with every unit will increase by a power of 10. Thus, if one wanted to convert a linear scale (with values 0-5) to a logarithmic scale, one option would be to replace 0, 1, 2, 3, 4, and 5 with 0.001, 0.01, 0.1, 1, 10, and 100, respectively. Between each major value on the logarithmic scale, the hashmarks become increasingly closer together with increasing value. For example, in the space between 1 and 10, the 8 and 9 are much closer together than the 2 and 3. The advantages of using a logarithmic scale are twofold. Firstly, doing so allows one to plot a very large range of data without losing the shape of the graph. Secondly, it allows one to interpolate at 282 The graphs of functions f(x)=10^x, f(x)=x, and f(x)=log(x) on four different coordinate plots. Note how each function changes shape on each set of coordinates. Figure 5.21 Logarithmic Scale any point on the plot, regardless of the range of the graph (Figure 5.23). Similar data plotted on a linear scale is less clear (Figure 5.22). A key point about using logarithmic graphs to solve problems is that they expand scales to the point at which large ranges of data make more sense. In logarithms, the product of numbers is the sum of their logarithms. In the equation mentioned above, plotting j vs. T would generate the expected curve, but the scale would be such that minute changes go unnoticed and the large scale effects of the relationship dominate the graph—it is so big that the "interesting areas" won't fit on the paper on a readable scale. Taking logarithms, however, one ends with logj = 4 * log(σT ) = 4 * logσ + 4logT. This is the equation of a straight line with a slope of logT and a "y intercept" of 4 log σ. Plotting a straight line as indicated simplifies the interpretation. Source: logarithms/graphing-logarithmic-functions/solving-problems-with- logarithmic-graphs/ CC-BY-SA Boundless is an openly licensed educational resource Notice how values of y less than 10 are indistinguishable. Figure 5.22 Points of log(y) on a Linear Scale Both plots capture y well for their respective ranges, but note how easily distinguishable the points are in both the lower and higher areas. Figure 5.23 Graph of log(y) on a Semi-Log Scale 283 Logarithms of Products Logarithms of Powers Logarithms of Quotients Solving General Problems with Logarithms and Exponents Simplifying Expressions of the Form logaax and a(logax) Section 4 Properties of Logarithmic Functions 284 Logarithms of Products A useful property of logarithms states that the sum of two logarithms of factors is equal to the logarithm of the factors' product. KEY POINTS • The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. • The logarithm of a product is the sum of the logarithms of the factors. In addition, the sum of the logarithms of two number is equal to the logarithm of the product of those two numbers. • The product rule does not apply when the base of the two logarithms are different. The shape of the graphs that logarithms take can be viewed in Figure 5.24. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. Logarithms were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition, because of the fact that the logarithm of a product is the sum of the logarithms of the factors: logb(xy) = logb(x) + logb(y) Of course, the reverse is true also. The sum of two logarithms of factors is equal to the logarithm of the factors' product. This is a very useful property of logarithms, because it can sometimes simplify more complex equations. However, it is also important to note that this property is only true when the two logarithms share the same base. An example of this would be: logb(x) + logc(y) = logb(xy) This equation is not true, of course, in a situation such as b = c. 285 The graphs of logarithms of different bases have the same general shape but different curvatures. Figure 5.24 Logarithms Source: logarithms/properties-of-logarithmic-functions/logarithms-of- products/ CC-BY-SA Boundless is an openly licensed educational resource Logarithms of Powers A simplifying principle of logarithms is that the logarithm of the p-th power of a number is p times the logarithm of the number. KEY POINTS • The logarithm of a product is the sum of the logarithms of the factors. • An exponent, p, signifies that a number is being multiplied by itself p number of times. Because the logarithm of a product is the sum of the logarithms of the factors, the logarithm of a number, a, to an exponent, p, is the same as the logarithm of a added together p times. • The logarithm of a added together p times is the same as p ⋅ logb(a), where b is an arbitrary base. The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3,written: log10(1000) = 3. Here, I have chosen to write the 10, to show where you would denote the base, however if you are working with a base of 10, you do not have to denote that. If it is left blank, you can assume you are looking for the base10. This is true because: 286 103 = 10 10 * 10 = 1000. More generally, if x = by, then y is the logarithm of x to base b, and is written logbx. Logarithms are used commonly to measure earthquakes, distances of stars, economics, and throughout the scientific world. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were adopted by navigators that the logarithm of a product is the sum of the logarithms of the factors: logb(ac) = logb(a) + logb(c) The logarithm of the p-th power of a number is p times the logarithm of the number itself. logb(a)p = p * logb(a) This can be proved by using the fact that the log of a product is equal to the sum of the logs of the two factors. Because an exponent just indicates the the number is multiplied by itself many times, we can write: logb(a)p = logb(a * a * a * ⋯ * ap) Where a is multiplied by itself p times. This equation can be written by following the product rule: logb(a) + logb(a) + logb(a) + ⋯ + logb(ap) Where logba is added together with itself p times. This, of course, by the simple rules of algebra, can be written as: p * logb (a) Thus, we have demonstrated the rule of the logarithm of powers. 287 The graph of a common logarithm takes a characteristic shape. Figure 5.25 Graph of Common Logarithm Source: logarithms/properties-of-logarithmic-functions/logarithms-of- powers/ CC-BY-SA Boundless is an openly licensed educational resource Logarithms of Quotients The logarithm of the ratio or quotient of two numbers is the difference of the logarithms and can be proven using the first law of exponents. KEY POINTS • The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. • A basic idea in logarithmic math is that the logarithm of a product is the sum of the logarithms of the factors. • A similar idea of the law of products is that the logarithm of the ratio or quotient of two numbers is the difference of the logarithms (Figure 5.26). The logarithm of a product is the sum of the logarithms of the factors: 288 logb(xy) = logbx + logby Similarly, the logarithm of the ratio of two numbers is the difference of the logarithms. logx(a /b) = logxa − logxb To prove this, let m = logxa. Rewrite the above expression as an exponent. xm = a (logxa asks " x to what power is a ?" And the equation answers: "x to the m is a.") Let n = logxb. Thus, xn = b. If we replace a and b based on the previous equations, we get: logx(a /b) = logx(xm /xn ) This can be further simplified to: logx(xm /xn ) = logx(xm−n ) Which, using the first law of exponents, can be written as: logx(xm /xn ) = m − n This is the key step. Therefore, it can be seen that the properties of logarithms come directly from the laws of exponents. Replacing m and n with what they were originally defined as results in this equation: logx(a /b) = logxa − logxb Hence, the previous problem has been proven. Source: logarithms/properties-of-logarithmic-functions/logarithms-of- quotients/ CC-BY-SA Boundless is an openly licensed educational resource 289 The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log2(8) = 3, because 23 = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it. Figure 5.26 Graph of Binary Logarithm Solving General Problems with Logarithms and Exponents Logarithms are useful for solving equations that require an exponential term, like population growth. KEY POINTS • The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. • Both root equations and logarithm equations can be rewritten as exponent equations. • Logarithms are therefore useful in solving equations that require solving for an exponential term, such as those involving population growth then y is the logarithm of x to base b, and is written y = logbx, so log10(1000) = 3. Logarithms have several applications in general math problems. To look at it most simply, both root equations and logarithm equations can be rewritten as exponent equations. √9 = 3 can be rewritten as 32 = 9. These two equations are the same statement about numbers, written in two different ways. √9 asks the question "What number squared is 9?" So the equation √9 = 3 asks this question, and then answers it: "3 squared is 9." We can rewrite logarithm equations in a similar way. Consider this equation: log3 1 3 = − 1 290 This population growth graph shows that it grows exponentially with time. Figure 5.27 Population Growth in Sri Lanka If you are asked to rewrite that logarithm equation as an exponent equation, think about it this way. The left side asks: "3 to what power is (1/3)?" And the right side answers: "3 to the −1 power is (1/3)." 3−1 = 1/3. These two equations, log31/3 = − 1 and 3−1 = 1/3, are two different ways of expressing the same numerical relationship. EXAMPLE A city grows 5% every 2 years. How long will it take for the city to triple its size? Step 1. Use the growth formula: A = P(1 + i)n Assume P = x. Since we want to know when the population will be tripled, A = 3x. For this example n represents a period of 2 years, therefore the n is halved for this purpose. Step 2. Substitute information given into formula: 1. 3 = (1.05)n/2 2. log3 = n /2 * log1.05 3. n = 2log3/log1.05 4. n = 45.034 Step 3. Final answer: It will take approximately 45 years for the population to triple in size. Source: logarithms/properties-of-logarithmic-functions/solving-general- problems-with-logarithms-and-exponents/ CC-BY-SA Boundless is an openly licensed educational resource 291 Simplifying Expressions of the Form logaax and a(logax) The expressions logaax and alogax can be simplified to x, a shortcut in complex equations. KEY POINTS • Because logaa = 1 and logabx = xlogab, the formula for the logarithm of a power says that for any number x, logaax = xlogaa = x. • Because logax and logxa are inverse values, aloga(x) = x. • Simplifying complex-looking equations can greatly facilitate the solving of longer problems. Recall that the logarithm of a number is defined as the exponent by which another fixed value, the base, must be raised to produce that number. When the base is the same as the number being modified, the solution is 1, or: logaa = 1 Here, logaa is asking about what power a must be raised to get a; that power is one. The logarithm of the p-th power of a number is p times the logarithm of the number itself: logabx = xlogab Similarly, the logarithm of a p-th root is the logarithm of the number divided by p: logab1/x = (1/x)logab Because, the formula for the logarithm of a power says that for any number x: logaax = xlogaa = x Once again, is asking about what power a must be raised to get a; that power is x. In prose, we can say that taking the x-th power of a and then the base-a logarithm gives back x. Conversely, if a positive number a is raised to the power of the log base-a of x, the answer again yields x: alogax = x This formula says that first taking the logarithm and then exponentiating gives back x. 292 Population Growth Population size can fluctuate positively or negatively, and growth is capable of being modeled by an exponential function. KEY POINTS • The formula for population growth is of the same form as that of compound interest. That is, P(r, t, f ) = Pi(1 + r) t f, where Pi represents initial population, r is the rate of population growth (expressed as a decimal), t is elapsed time, and f is the period over which time population grows by a rate of r. • Population growth rate is more complex than interest rate; it is dependent on four variables. Population growth rate can be modeled as: ΔP = (B − D) + (I − E), where ΔP represents change in population, B and D are births and deaths, respectively, and I and E are immigrants and emigrants, respectively. • Population growth rate can reveal whether a population size is increasing (positive) or decreasing (negative). It can be calculated for two times with the following equation: PGR = ln(P(t2)) − ln(P(t1)) (t2 − t1) , where t2 and t1 represent the two times. Mathematically, population growth is very similar to the growth of money by compound interest. The equations used to model both are of an exponential form. Population growth can be represented by the following formula: P(r, t, f ) = Pi(1 + r) t f Where Pi represents initial population, r is the rate of population growth (expressed as a decimal), t is elapsed time, and f is the period over which time population grows by a rate of r. The ratio of t to f is often simplified into one value representing the number of compounding cycles. Note that r is more complex in the case of population growth than in interest. In interest, a rate is simply a number specified by a bank. In population growth, it is determined by births (B), deaths (D), immigrants (I), and emigrants (E): ΔP = (B − D) + (I − E) The formula is split into natural growth (B-D) and mechanical growth (I-E). Whereas interest will always increase, population size can fluctuate from growth to decline, and back again. As such, another variable is 295 important when studying population demographics and dynamics, Population Growth Rate (PGR). PGR indicates the rate of change in population over a certain span of time (t2-t1). It can be determined from the formula: PGR = ln(P(t2)) − ln(P(t1)) (t2) Multiplying PGR by 100 affords percentage growth, relative to the population at the beginning of the time period. A positive growth rate indicates an increasing population size, while a negative growth rate is characteristic of a decreasing population. A growth rate of 0 means stagnation in population size. Source: logarithms/growth-and-decay-compound-interest/population- growth/ CC-BY-SA Boundless is an openly licensed educational resource 296 The population of the world has grown at an increasingly staggering rate in recent centuries. Figure 5.29 Population Curve Interest Compounded Continuously Compound interest is accrued when interest is earned not only on principal, but on previously accrued interest: it is interest on interest. KEY POINTS • Interest is, generally, a fee charged for the borrowing of money. The amount of interest accrued depends on the principal (amount borrowed), the interest rate (a percentage of the principal), period (amount of time between interest payments) and time elapsed. • The form of the equation for compound interest is exponential, and thus such interest is accrued much faster than the linear simple interest. • The formula for compound interest is M = p(1 + r) t f where M represents the total value (including principal), p represents principal, r is interest rate (expressed as a decimal), t is time elapsed, and f is the length of time between payments. To calculate interest alone, simply subtract the principal from M. Fundamentally, compound interest is a specific case of exponential functions found very commonly in everyday life. Interest is, generally, a fee charged for the borrowing of money. The amount of interest accrued depends on the principal (amount borrowed), the interest rate (a percentage of the principal), period (amount of time between interest payments) and time elapsed. Simple interest is accrued linearly based on the formula: I = p ⋅ r ⋅ t f Where I represents interest, p is principal, r is interest rate (expressed as a decimal), t is time elapsed, and f is the time elapsed per interest payment. The ratio of t to f is often simplified to the number of interest payments. Simple interest is useful in some applications, such as a home equity line. However, there are many instances in which interest will accrue further interest. Consider a bank savings account opened in the amount of $100, which accrues interest at a rate of 5% per year. If left untouched, after one year, the value of the account will be $105. A year later, it will not be $110, as would be suggested by the simple interest formula. Rather, it would be slightly higher due to interest accrued on not only the initial principal ($100), but the interest previously accrued ($5). This is called compound interest. 297 Compound interest is not linear, but exponential in form. The equation representing investment value as a function of principal, interest rate, period and time is: M = p(1 + r) t f Where M represents the total value (including principal), p represents principal, r is interest rate (expressed as a decimal), t is time elapsed, and f is the length of time between payments. To calculate interest alone, simply subtract the principal from M. EXAMPLE Consider the aforementioned case of the bank account opened in the amount of $100, which accrues interest at 5% per year. If we leave the account untouched for 50 years, we can calculate the compound interest as such: M = 100(1 + 0.05)50 Note that 50 was inserted as the ratio of 50 years and 1 year per interest payment. M = 1146.74 Subtracting the $100 principal, we find that interest accrued is in the amount of $1046.74. Had interest been accrued linearly (in "simple" form), the final value would have been $350, of which $250 would be interest. Thus, it is easy to see how growing interest on interest can make a huge difference in the long term. Source: logarithms/growth-and-decay-compound-interest/interest- compounded-continuously/ CC-BY-SA Boundless is an openly licensed educational resource 298 Starting with a principal of $1000, interest rises exponentially. Notice also that as time passes, a gap forms between the lines as less frequently- compounding methods increase at a lesser rate than more frequently- compounding methods. Figure 5.30 Compound Interest At Varying Frequencies Limited Growth Exponential growth may dampen approaching a certain value, modeled with the logistic growth model: P(t) = c 1 + a ⋅ e−bt . KEY POINTS • Exponential growth may exist within known parameters, but such a functionality may not continue indefinitely. • If a natural maximum is conceivable, the logistic growth model can be used to represent growth. It has the form: P(t) = c 1 + a ⋅ e−bt where P represents population, c is the carrying capacity, b is the population growth rate, t is time, and a is the difference between carrying capacity and initial population. • An example of natural dampening in growth is the population of humans on planet Earth. The population may be growing exponentially at the moment, but eventually, scarcity of resources will curb our growth as we reach our carrying capacity. Exponential models of growth and decay can be used to interpolate and extrapolate. For example, given the population of a country in 1950, 1975, 2000, and at present, one could use the graph of an exponential curve to estimate the rate of population growth in that period, and population at any time between 1950 and the present. This is an example of interpolation. It is also possible to extrapolate, or make predictions beyond the scope of data. For example, using the same exponential trendline as mentioned above, one could estimate population 10, 20, or 50 years from now. There are concerns with the interpolative and extrapolative properties of any trendline, but, in addition, there may be other factors that altogether change the form of population growth. Consider a farm upon which a population of sheep are kept in a constant, comfortable climate in a fully enclosed field. Assume the entire system is closed from gains and losses, but for a flow of a stream of clean drinking water through the field. If left indefinitely, the population of sheep would perfectly fit an exponential model to a certain point. However, eventually, the grass would act as a factor to limit growth. If the amount of grass available to the sheep and its rate of replenishing are constant, eventually the population of sheep will grow to a tipping point at which the grass can no longer feed the sheep. The death rate of sheep will increase as some starve, and thus the model of population growth among sheep will change form. 299 Logistic Growth Model To account for limitations in growth, the logistic growth model can be used: P(t) = c 1 + a ⋅ e−bt where P represents population, c is the carrying capacity (maximum the population approaches as time approaches infinity), b is the population growth rate, t is time, and a is the difference between carrying capacity and initial population. Graphically, the logistic function resembles an exponential function followed by a logarithmic function that approaches a horizontal asymptote. From the left, it grows rapidly, but that growth is dampened as it approaches a maximum (Figure 5.31). Source: logarithms/growth-and-decay-compound-interest/limited-growth/ CC-BY-SA Boundless is an openly licensed educational resource 300 The graph of the logistic function begins with exponential growth from the left, but that growth is dampened as it approaches a horizontal asymptote to the right. Figure 5.31 Standard Logistic Function Exponential Decay Just as a variable can exponentially increase as a function of another, it is possible for a variable to exponentially decrease. KEY POINTS • Exponential decrease can be modeled as: N(t) = N0e−λt where N is the quantity, N0 is the initial quantity, λ is the decay constant, and t is time. • Oftentimes, half-life is used to describe the amount of time required for half of a sample to decay. It can be defined mathematically as: t1/2 = ln(2) λ where t1/2 is half-life. • Half-life can be inserted into the exponential decay model as such: N(t) = N0( 1 2 )t/t1/2 Notice how the exponential changes, but the form of the function will remain. Just as growth of one variable as a function of another can be exponential in form, so can decline. Consider the decrease of a population that occurs at a rate proportional to its value. This rate may be constant, but because the population is continually decreasing, the overall decline becomes less and less steep. Exponential rate of change can be modeled algebraically by the following formula: N(t) = N0e−λt where N is the quantity, N0 is the initial quantity, λ is the decay constant, and t is time. The decay constant is indeed a constant, but the form of the equation (using it as an exponent for e) results in an ever-changing rate of decline. (Figure 5.32) An example of exponential decay is the time-dependent decline in population of a sample of a radioactive isotope. Given a sample of 301 The exponential decline of five different functions is depicted. Note that with higher coefficients of x (λ), the drop is steeper. With lower λ, the drop is more gradual. Figure 5.32 Exponential Decay carbon in an ancient, preserved piece of flesh, the age of the sample can be determined based on the percentage of radioactive carbon-13 remaining (1.1% of carbon is C-13; it decays to carbon-12, which represents approximately all the remaining carbon). C-13 has a half-life of 5700 years—that is, in 5700 years, half of a sample of C-13 will have converted to C-12. Half-life can be mathematically defined as: t1/2 = ln(2) λ It can also be conveniently inserted into the exponential decay formula as follows: N(t) = N0( 1 2 )t/t1/2 Thus, if a sample is found to contain 0.55% of its carbon as C-13 (exactly half of the usual 1.1%), it can be calculated that the sample has undergone exactly one half-life, and is thus 5,700 years old. Source: logarithms/growth-and-decay-compound-interest/exponential-decay/ CC-BY-SA Boundless is an openly licensed educational resource 302 Solving Systems Graphically The graphical method is a simple way to solve a system of equations by looking for the intersecting point or points of the equations. KEY POINTS • The graphical method is a great way to solve a system of equations, and also to check your work if you are solving the system using elimination or substitution. • There are many ways to write any equation, but if you are going to solve the system graphically, it is helpful to first isolate the y term on one side of the equation/s. • You can solve the system by locating the intersections between the different equations in the system. It is possible to have more than one answer that satisfies all equations in a system. A system of equations (also known as simultaneous equations) is a set of equations with multiple variables, often solved with a particular specification of the values of all variables that simultaneously satisfies all of the equations. The most common ways to solve a system of equations are: • The elimination method • The substitution method • The graphical method In this atom we will address the graphical method. Some systems have only one set of correct answers, while others have multiple sets that will satisfy all equations. Shown graphically (Figure 6.1), a set of equations solved with only one set of answers will have only have one point of intersection. A system with two sets that will satisfy both equations has two points of intersection (Figure 6.2). 305 This graph shows a system of equations with two variables with only one set of answers. Figure 6.1 System of linear equations with two variables Before successfully solving a system graphically, one must be comfortable with the system of equations represented in a general format, such as written in the following manner: Ax + By = C. At first, this may look confusing to try to graph, but keep in mind that converting this equation into the slope-intercept form is relatively easy, and will look like this: y = mx + b. Where: m = slope; b = the y intercept. The best way to convert your equation form that is easier to graph is by first isolating the y to yield By = − Ax + C, then dividing the right side by B to yield y = (−Ax + C)/B y = (−A /B)x + (C/B). The (-A/B) is now the slope, m and the (C/B) is the y-intercept, b. If you have a graphing calculator, you can also use that to represent the equations graphically, but it is useful to know how to represent such equations formulaically on your own. An ordered pair is a way of writing the correct values for the system in a manner typically associated with graphs (x,y). Once you have figured out how to represent your system of equations graphically, finding the correct ordered pair to satisfy the system is easy: you simply find the intersections between the graphs. Source: and-matrices/systems-of-equations-in-two-variables/solving-systems- graphically/ CC-BY-SA Boundless is an openly licensed educational resource 306 This is an example of a system of equations shown graphically that has two sets of answers that will satisfy both equations in the system. Figure 6.2 System of Equations with multiple answers The Substitution Method The substitution method is a way of solving a system of equations by expressing the equations in terms of only one variable. KEY POINTS • A system of equations is a set of equations that can be solved using a particular set of values. • The substitution method works by expressing one of the variable in terms of another, then substituting it back into the original equation thus simplifying it. • It is very important to check your work once you have found a set of values for the variables. Do this by substituting the values you found back into the original equations. • The answer to the system of equations can be written as an ordered pair (x,y). A system of equations is a set of equations containing multiple variables. Their solution is a set of variables that satisfy all equations in the system. Although there are many methods to solving a system of equations, the three simplest are: • The substitution method • The elimination method • The graphical method In this atom we will address the substitution method. The substitution method is a way to simplify a system of equations by expressing one variable in terms of another, thus removing one variable from an equation. When all variables in an equation are replaced such that there is only one variable (in different terms), the equation becomes solvable. This method is easier to explain by visually. For this example, let's take the following system of equations: x − y = − 1 x + 2y = − 4 The next step is to chose one variable to express in terms of the other. In this case, we will express x in terms of y: x = y − 1 We can now substitute our new definition of x (in this case y-1) into the second equation: (y − 1) + 2y = − 4 Note that now this equation only has one variable (y). We can then simplify this equation to solve for y: 307 3y = − 3 y = − 1 Now that we know the value of y, we can use it to find the value of the other variable, x: x = − 1 − 1 x = − 2 Thus, the solution to the system is: (-2, -1). On an x-y coordinate plane, this is the point where the two functions intersect. Graphically, one can confirm this to be true, as in Figure 6.3. EXAMPLE Start with this system of equations: x − y = − 1 x + 2y = − 4 Pick a variable to write in terms of the other: x = y − 1 Substitute that back into the original equation: (y − 1) + 2y = − 4 Isolate the variable: y + 2y = − 4 + 1 −3y = − 3 y = − 1 Use this value to solve for the other variable: x = y − 1 x = (−1) − 1 x = − 2 Now that you have your answer (-2,-1), check your answer by plugging these values back into one of the original equations: x + 2y = − 4 (−2) + 2(−1) = − 4 −2 − 2 = − 4 −4 = − 4 It works! Good job! 308 This figure is a graphical representation of the example equations. You can use this to check your answers, and see that the ordered pair you come up with matches where the lines intersect with one another. Figure 6.3 System of Equations with Two Variables Source: and-matrices/systems-of-equations-in-two-variables/the-substitution- method/ CC-BY-SA Boundless is an openly licensed educational resource The Elimination Method The elimination method is used to eliminate a variable in order to more simply solve for the remaining variable(s). KEY POINTS • In order to easily solve the system of equations, it helps to first set the equations up in similar way. For example, x+y=-1 and 2y+x=-4 should be written: x+ y=-1 and x+2y=-4. • Once, the values for the remaining variables have been found successfully, then go back and plug that result into one of the original equation and find the correct value for the other variable. • Always check the work. This is done by plugging both values into one or both of the original equations. A system of equations, also known as simultaneous equations, are a set of at least two equations containing multiple variables. The system can be solved using a specific set of values corresponding to the variables. This section will focus on systems of equations with just two variables. It is important to first learn how to solve simple systems in order to understand the basic concept before moving on to more complex systems of equations with more variables. The values of the variables are often written in the following notation: (x,y). The most basic methods for solving a system of equations are: 309 • The graphical method • The substitution method • and the elimination method In this section, the elimination method will be explored. The elimination method, also known as elimination by addition, is a way to eliminate one of the variables to more simply evaluate the remaining variable. Once, the values for the remaining variables have been found successfully, then go back and plug that result into one of the original equation and find the correct value for the other variable. This elimination method can be demonstrated by using a simple example: Using two equations, both with two variables; 2x+y=8 and y+x=6, the first thing that needs to be done is to write the variables in the same manner, so that the similar variables line up, like so: 2x+y=8 x +y=6.In this example, one variable, y, can easily eliminate using the elimination method, just subtract the bottom equation from the top equation. The remaining variable are: 2x+y=8 minusx+y=6 equals x=2.Then go back to one of the original equations and substitute the value we found for x. It is easiest to pick the simplest equation, but either equation will work: x+y=6 --> 2+y=6 y=4. I has been found, using the method of elimination, that (x,y) is equal to (2,4). It is also easy to go back and check the answer by plugging both of these values in for their respective variables into one of the equations: 2x +y=2(2)+4=4+4=8. Although, this is a very simplified example, make sure to follow these steps, and it will be easy to solve any system of equations with two variables. Now, imagine that there is a more difficult problem, such as: x+2y=4 and 3x-y=5. This system of equations is shown in Figure 6.4. The best way to start every problem is to simplify the equation. Set the equation up so that one of the variables will easily go away. Start by making one of the variables consistent in both equations. This can be done by multiplying the second equation by 310 This image demonstrates that the system of equations has a specific set of values which will solve BOTH equations Figure 6.4 System of equations with two variables 2. Don't forget, the same thing has to be done to both sides of the equation: 2[3x-y]=2 becomes 6x-2y=10.After the equations are placed on top of each other, the equation will be: x+2y=46x-2y=10. Add the equations together the same way as in the first example. Also, it is crucially important to check the work, especially when it is simple to check for a system of equations. EXAMPLE Given the following system of equations: x+2y=4 and 3x-y=5 The answer should be written in the following notation: (x,y) x+2y=4 and 3x-y=5 First, set the system up in a way that makes it easier to manipulate: x+2y=4 3x-y=5 Next, choose a variable to eliminate. In this example, eliminate y, therefore: x+2y=4 x+2y=4 [3x-y]=[5]-----> 6x-2y=10 7x =143. Solve for the remaining variable: 7x=14 x=14/7 x=24 Plug into original equation: x+2y=4(2)+2y=4 2y=2 y=2/2 y=15 We have now found that (x,y) = (2,1). Finally, plug these values into one of the equations to make sure everything checks out! 3x-y=53(2)-(1)=5 6-1=5 5=5. It works! Good job! Source: and-matrices/systems-of-equations-in-two-variables/the-elimination- method/ CC-BY-SA Boundless is an openly licensed educational resource 311 Applications of Systems of Equations Systems of equations can be used to solve many real- life problems in which multiple constraints are used on the same variables. KEY POINTS • If you have a problem that includes multiple variables, you can solve it by creating a system of equations. • The first step to solving a multivariate problem is to identify and label the variables. • Once variables are defined, determine the relationships between them and write them as equations. A system of equations, also known as simultaneous equations, is a set of equations that have multiple variables. The answer to a system of equations is a set of values that satisfies all equations in the system. Systems of equations can have multiple sets of answers that are correct. Solutions to a system of equations are often written as ordered pairs, (x,y). There are many ways of solving a system of equations, including the elimination, substitution, and graphical methods. There are many applications of systems of equations. Whenever you have a problem that has multiple variables, setting up a system of equations is often the best method for solving. The steps you need to take in order to do that are: (1) identify the variables in the problem, (2) name the variables, and (3) set up the equations and solve for each variable. Example 1 Emily is hosting a major after-school party. The principal has imposed two restrictions. First, the total number of people attending (teachers and students combined) must be 56. Second, there must be one teacher for every seven students. How many students and how many teachers are invited to the party? First, we need to identify our variables. In this case, our variables are teachers and students. Now we need to name these variables: number of teachers will be T, and number of students will be S. Now we need to set up our equations. There is a constraint limiting the total number of people in attendance to 56, so: T + S = 56 For every seven students, there must be one teacher, so: 7S = T 312 Now we a system of equations that can be solved by substitution, elimination, or graphically. The solution to the system is S=49 and T=7. Example 2 A group of 75 students and teachers are in a field, picking sweet potatoes for the needy. Kasey picks three times as many sweet potatoes as Davis—and then, on the way back to the car, she picks up five more sweet potatoes than that! Looking at her newly increased pile, Davis remarks "Wow, you've got 29 more potatoes than me!" How many sweet potatoes did Kasey and Davis each pick? To solve, we first define our variables. The number of sweet potatoes that Kasey picks is K, and the number of sweet potatoes that Davis picks is D. Now we can write equations based on the situation: K − 5 = 3D D + 29 = K From here, substitution, elimination or graphing will reveal that K is 41 and D is 12. It is important that you always check your answers. A good way to check solutions to a system of equations is to look at the functions graphically and then see where the graphs intersect (Figure 6.5). Source: and-matrices/systems-of-equations-in-two-variables/applications-of- systems-of-equations/ CC-BY-SA Boundless is an openly licensed educational resource 313 Once you have created your system of equations, one way to solve it is by showing them graphically, and then finding the intersecting point or points. Figure 6.5 Graphical representation of a system of equations Solving Systems of Equations in Three Variables Applications and Mathematical Models Section 2 Systems of Equations in Three Variables 314 Solving Systems of Equations in Three Variables A system of equations in three variables involves two or more equations, each of which involves between one and three variables. KEY POINTS • A system of equations may have no solutions, one unique solution, or infinitely many solutions. • The substitution method involves solving for one of the variables in one of the equations, and plugging that into the rest of the equations to reduce the system. Rinse and repeat until there is a single equation left, and then using this go backwards to solve the previous equations. • The graphical method involves graphing all of the equations and finding points, lines or planes where all of the equations intersect at once, such points, lines or planes are the solutions. • The elimination method involves adding or subtracting multiples of one equation from the other equations, eliminating variables from each of the equations until one variable is left in each equation (if there is a unique solution). In mathematics, simultaneous equations are a set of equations containing multiple variables. This set is often referred to as a system of equations. A solution to a system of equations is a particular specification of the values of all variables that simultaneously satisfies all of the equations. The elementary methods to solve simple systems of equations include graphical method, the matrix method, the substitution method, or the elimination method. Among the systems of equations, the systems of linear equations are especially important. There were the original object of study of linear algebra. Many algorithms have been devised to solve them, which allow to solve huge systems (up to millions of variables). For a system of equations in three variables, you can have one or more equations, each of which may contain one or more of the three variables, usually x, y and z. Finding Solutions Sometimes not all variables can be solved for, and so an answer for at least one variable must be expressed in terms of other variables and so the set of all solutions is infinite. This is typical for the case where the system has fewer equations than variables. If the number of equations is the same as the number of variables, then probably, but not necessarily, the system is exactly solvable in the sense that 315 the set of its solutions is finite. For a system of linear equations in this case, there is exactly one solution, for other systems to have several solutions is also typical. A consistent system is a system of equations with at least one solution. Sometimes a system is inconsistent, or has no solution. Having no solutions is typical for the case where the system has more equations than variables. Graphical Method The graphical method involves plotting the planes that are formed by a system of equations in three variables and finding the intersection point, line, or plane that solves the system. A single point is a unique solution, whereas intersecting lines or planes mean an infinite number of solutions. If all the planes don't intersect at the same spot, there are no solutions. An example of this is given in Figure 6.6. Substitution Method The substitution method involves finding an equation that can be written with a single variable as the subject, in which the left-hand side variable does not occur in the right-hand side of the expression. Next, substitute that expression where that variable appears in the other equations, thereby obtaining a smaller system with fewer variables. After that smaller system has been solved, whether by further application of the substitution method or by other methods, substitute the solutions found for the variables in the above right- hand side expression. Using the simple example above, one can perform the substitution method. Taking the first equation 3x + 2y − z = 6, since the coefficient of z is already 1, let's solve for z to get z = 3x + 2y − 6. Plug this into the other two equations to get the new smaller system: { −2x + 2y + 3x + 2y − 6 = 3 x + y + 3x + 2y − 6 = 4 Which simplifies to: { x + 4y = 9 4x + 3y = 10 . 316 This images shows a system of three equations in three variables. The intersecting point is the unique solution to this system. Figure 6.6 System of Linear Equations Now solving for x in the first equation, one gets x = 9 − 4y. Plug this into last equation in the system to get 4(9 − 4y) + 3y = 10, which simplifies to y = 2. Now having the value of y, work back up the equation. Plug y = 2 into the equation x = 9 − 4y to get x=1. Working up again, plug y = 2 and x = 1 into the first substituted equation, z = 3x + 2y − 6, which simplifies to z = 1. Elimination Method Elimination by judicious multiplication is the other commonly used method to solve simultaneous linear equations. It uses the general principles that each side of an equation still equals the other when both sides are multiplied (or divided) by the same quantity, or when the same quantity is added (or subtracted) from both sides. As the equations grow simpler through the elimination of some variables, a variable will eventually appear in fully solvable form, and this value can then be "back-substituted" into previously derived equations by plugging this value in for the variable. Typically, each "back- substitution" can then allow another variable in the system to be solved. Looking at the example: x + y + z = 2 x − y + 3z = 4 2x + 2y + z = 3 Using the elimination method, first subtract the first equation from the second equation to get x − y + 3z − (x + y + z) = 4 − 2, which simplifies to −2y + 2z = 2, giving us the system of equations: x + y + z = 2 −2y + 2z = 2 2x + 2y + z = 3 Now subtract two times the first equation from the third equation to get 2x + 2y + z − 2(x + y + z) = 3 − 2(2), which simplifies to z = 1. Doing this shows the new system: x + y + z = 2 −2y + 2z = 2 z = 1 Next, subtract two times the third equation from the second equation to get −2y + 2z − 2z = 2 − 2, simplifying to y = 0. Doing this shows the new system: x + y + z = 2 y = 0 z = 1 Finally, subtract the third and second equation from the first equation to get x + y + z − y − z = 2 − 0 − 1, giving x = 1 and the final, solved, system that is: 317 x = 1 y = 0 z = 1 Unsolvable Systems If there is a system of three variables with only two equations, one will end up with either an infinite number of solutions or no solutions. { x + y + z = 2 −x − y + z = 4 Which through any of the methods will simplify to: { x + y = − 1 z = 3 Which has an infinite number of solutions so long as x + y = -1. An example of a system with no solutions is: { x + y + z = 2 2x + 2y + 2z = 6 This can be seen by subtracting two times the first equation from the second equation, which simplifies to 0=2, which is impossible, hence there are no solutions. These situations can arise in any number of equations and variables. Source: and-matrices/systems-of-equations-in-three-variables/solving- systems-of-equations-in-three-variables/ CC-BY-SA Boundless is an openly licensed educational resource 318 Applications and Mathematical Models Systems of equations are problems that have multiple unknowns and multiple observations, and can be used in many practical applications. KEY POINTS • When you have multiple unknown quantities with multiple observations on these quantities and their interactions with each other, then the problem can usually be naturally described with a system of equations. • When a system of equations is laid out, all of the equations need to be satisfied in order for there to be a solution. Sometimes there are no solutions; other times there are infinitely many solutions. • There are numerous applications for systems of equations, such as Physics problems that involve multiple objects with multiple observations, or multiple forces that all need to be balanced. Systems of linear equations are common in science and mathematics, including Physics, Chemistry and maximization/ minimization and constraint problems.A system of equations is a way to evaluate multiple unknown quantities. You will need observations of these quantities in order to properly solve for the unknowns. The simplest and most studied method is a system of linear equations, and we will use this type in our examples. Physics Example Systems of equations are best explained through examples, and for practicality sake, let's start with a physics example. Let's say you have three balls, but you are only told the mass of one of them. Your unknowns are the other two balls, c and h. The one known mass is 2 kg. You take two observations with the three balls balanced on a bar over a fulcrum, Figure 6.7 and Figure 6.8. You therefore know the sum of the moments on the left equal the sum of the moment on the right.The moment of an object is its mass multiplied by its distance from the balance point. The two balances give this system of two 319 Here is the first observation of three balls, two with unknown weight, which are balanced on a bar in a given configuration. Figure 6.7 Physics Example: First Observation equations. In this example, the unknowns are the two masses and the observations are the balances. Using the substitution method on this example, let's use the second equation and solve for c to get c = 2 + 2h. Plug this into the first equation to get 40h + 15(2 + 2h) = 100 which simplifies to h = 1, and plugging this into the previous equation gives us c = 4. Chemistry Example We mix, under controlled conditions, a certain amount of toluene, C7H8, and nitric acid HNO3, to produce trinitrotoluene, C7H5O6N3 and water H2O (conditions have to be controlled; trinitrotoluene is better known as TNT). In what proportion should those components be mixed? The number of atoms of each element present before the reaction must equal the number present afterward. xC7H8 + yHNO3 → zC7H5O6N3 + wH2O Applying that principle to the elements C, H, N, and O results in this system. 7x = 7z 8x + y = 5z + 2w y = 3z 3y = 6z + w Let's use substitution. The first line gives us x = z, giving us the reduced system: 8z + y = 5z + 2w y = 3z 3y = 6z + w The second equation in this system gives us y = 3z, giving us the reduced system: { 8z + 3z = 5z + 2w 9z = 6z + w Which simplifies to: { 6z = 2w 3z = w 320 Here is the second observation of three balls, two with unknown weight, which are balanced on a bar in a given configuration. Figure 6.8 Physics Example: Second Observation Since these two equations are equivalent, we have an infinite number of solutions. This makes sense seeing that, for instance, if you place a certain amount of toluene and nitric acid to produce the TNT and water, then placing a multiple of that amount will give you a multiple of TNT. So we'll simply choose one solution, and know that there are infinitely many multiples of this one solution. Let's choose z=1. Therefore w=3, and propagating these two values up, we find that y=3 and x=1. Therefore one atom of toluene and three atoms of nitric acid produce one atom of trinitrotoluene and three atoms of water. Source: and-matrices/systems-of-equations-in-three-variables/applications- and-mathematical-models/ CC-BY-SA Boundless is an openly licensed educational resource 321 Matrices and Row-Equivalent Operations Two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. KEY POINTS • Because elementary row operations are reversible, row equivalence is an equivalence relation. • An elementary row operation is any one of the following moves: Swap (swap two rows of a matrix), Scale (multiply a row of a matrix by a nonzero constant), or Pivot (add to one row of a matrix some multiple of another row). • If the rows of the matrix represent a system of linear equations, then the row space consists of all linear equations that can be deduced algebraically from those in the system. In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two m × n matrices (Figure 6.9) are row equivalent if and only if they have the same row space. The concept is most commonly applied to matrices that represent systems of linear equations, in which case two matrices of the same size are row equivalent if and only if the corresponding homogeneous systems have the same set of solutions, or equivalently the matrices have the same null space. Because elementary row operations are reversible, row equivalence is an equivalence relation. It is commonly denoted by a tilde (~).There is a similar notion of column equivalence, defined by elementary column operations. Two matrices are column equivalent if and only if their transpose matrices are row equivalent. Two rectangular matrices that can be converted into one another allowing both elementary row and column operations are called simply equivalent. An elementary row operation is any one of the following moves: Swap: Swap two rows of a matrix. Scale: Multiply a row of a matrix by a nonzero constant. 323 Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. Figure 6.9 A Matrix Pivot: Add to one row of a matrix some multiple of another row. For example, the following steps show that these two matrices are row equivalent: A = ( 1 −1 0 2 1 1) B = ( 3 0 1 0 3 1) Start with A, keep the second row, and then add the first to the second: A = ( 3 0 1 2 1 1) Then, keep the first row. Multiply the second row by 3 and then subtract the first row from the second row: A = ( 3 0 1 3 3 2) Keep the first row again and then subtract the first row from the second: A = ( 3 0 1 0 3 1) Now, you can see that A = B, which we achieved through a series of elementary row operations. Row Space The row space of a matrix is the set of all possible linear combinations of its row vectors. If the rows of the matrix represent a system of linear equations, then the row space consists of all linear equations that can be deduced algebraically from those in the system. Two m × n matrices are row equivalent if and only if they have the same row space. For example, the matrices are row equivalent, the row space being all vectors of the form. The corresponding systems of homogeneous equations convey the same information. In particular, both of these systems imply every equation of the form. Source: and-matrices/matrices/matrices-and-row-equivalent-operations/ CC-BY-SA Boundless is an openly licensed educational resource 324 Gaussian Elimination Using elementary operations, Gaussian elimination reduces matrices to row echelon form. KEY POINTS • Since elementary row operations preserve the row space of the matrix, the row space of the row echelon form is the same as that of the original matrix. • There are three types of elementary row operations: swap the positions of two rows, multiply a row by a nonzero scalar, and add to one row a scalar multiple of another. • In practice, one does not usually deal with the systems in terms of equations but instead makes use of the augmented matrix (which is also suitable for computer manipulations). By means of a finite sequence of elementary row operations, called Gaussian elimination, any matrix (Figure 6.9) can be transformed to a row echelon form. Since elementary row operations preserve the row space of the matrix, the row space of the row echelon form is the same as that of the original matrix. The resulting echelon form is not unique. For example, any multiple of a matrix in echelon form is also in echelon form. However, it is the case that every matrix has a unique reduced row echelon form. This means that the nonzero rows of the reduced row echelon form are the unique reduced row echelon generating set for the row space of the original matrix. Before getting into more detail, there are a couple of key terms that should be mentioned: • Augmented matrix: an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. • Upper triangle form: A square matrix is called upper triangular if all the entries below the main diagonal are zero. A triangular matrix is one that is either lower triangular or upper triangular. A matrix that is both upper and lower triangular is a diagonal matrix. • Elementary row operations: Swap rows, add rows or multiply rows. Now, the steps of Gaussian Elimination are as follows: 1. Write the augmented matrix for the linear equations. 2. Use elementary row operations on the augmented matrix [A\| b] to transform A to upper triangle form. If a zero is on the diagonal, switch the rows until a nonzero is in its place. 3. Use back substitution to find the solution. 325 Gauss-Jordan Elimination Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. KEY POINTS • Gaussian elimination places zeros below each pivot in the matrix, starting with the top row and working downwards. • Gauss–Jordan elimination goes a step further by placing zeros above and below each pivot; such matrices are said to be in reduced row echelon form. • Gauss-Jordan elimination, like Gaussian elimination, is used for inverting matrices and solving systems of linear equations. In linear algebra, Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations (Figure 6.10). It is a variation of Gaussian elimination, which places zeros below each pivot in the matrix, starting with the top row and working downwards. Matrices containing zeros below each pivot are said to be in row echelon form. Gauss–Jordan elimination goes a step further by placing zeros above and below each pivot; such matrices are said to be in reduced row echelon form. Gauss-Jordan elimination, like Gaussian elimination, is used for inverting matrices and solving systems of linear equations. Both Gauss–Jordan and Gaussian elimination have time complexity of order O(n3 ) for an n by n full rank matrix (using Big O Notation), but the order of magnitude of the number of arithmetic operations (there are roughly the same number of additions and multiplications/divisions) used in solving an n by n matrix by Gauss- Jordan elimination is n3 , whereas that for Gaussian elimination is 2n 3 . However, the result of Gauss-Jordan elimination (reduced row echelon form) may be retrieved from the result of Gaussian elimination (row echelon form) in arithmetic operations by proceeding from the last pivot to the first one. Thus the needed number of operations has the same order of magnitude for both eliminations. The steps of Gauss-Jordan elimination are very similar to that of Gaussian elimination, the main difference being that we will work in diagonal form instead of putting the augmented matrix into upper triangle form. In diagonal form, we remove any zeros from the diagonal and add them below and above. 327 A matrix is in reduced row echelon form (also called row canonical form) if it is the result of a Gauss–Jordan elimination. Figure 6.10 Matrix in Reduced Row Echelon Addition and Subtraction; Scalar Multiplication There are a number of operations that can be applied to modify matrices, such as matrix addition, subtraction, and scalar multiplication. KEY POINTS • When performing addition, you add each number in the first matrix to the corresponding number in the second matrix. • When performing subtraction, simply subtract a number in one of the matrices from the corresponding number in the other matrix. • Addition and subtraction require that the matrices be the same dimensions. Also, you must begin and end with the same dimensions. • Scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. There are a number of operations that can be applied to modify matrices, such as matrix addition and subtraction and scalar multiplication. These form the basic techniques to deal with matrices. Adding and Subtracting Matrices Adding matrices is very simple. You just add each number in the first matrix to the corresponding number in the second matrix. ( 1 2 3 4 5 6) + ( 10 20 30 40 50 60) = ( 11 22 33 44 55 66) For instance, you can take each number that appears in the upper- right-hand corner to create the calculation 3+30=33. Note that both matrices being added are 2×3, and the resulting matrix is also 2×3. You cannot add two matrices that have different dimensions. As you might guess, subtracting works much the same way, except that you subtract instead of adding. ( 10 20 30 40 50 60) − ( 1 2 3 4 5 6) = ( 9 18 27 36 45 54) Once again, note that the resulting matrix has the same dimensions as the originals, and that you cannot subtract two matrices that have different dimensions. Scalar Multiplication In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. What does it 330 mean to multiply a number by 3? It means you add the number to itself 3 times. Multiplying a matrix by 3 means the same thing; you add the matrix to itself 3 times. 3 ⋅ ( 1 2 3 4 5 6) = ( 3 6 9 12 15 18) Note what has happened: each element in the original matrix has been multiplied by 3. Hence, we arrive at the method for multiplying a matrix by a constant: you multiply each element by that constant. The resulting matrix has the same dimensions as the original. Row Operations Row operations are ways to change matrices. There are three types of row operations: row switching, that is interchanging two rows of a matrix; row multiplication, or multiplying all entries of a row by a non-zero constant; and finally row addition, which means adding a multiple of a row to another row. These row operations are used in a number of ways, including solving linear equations and finding inverses. Source: and-matrices/matrix-operations/addition-and-subtraction-scalar- multiplication/ CC-BY-SA Boundless is an openly licensed educational resource 331 Matrix Multiplication When multiplying matrices, the elements of the rows in the first matrix are multiplied with corresponding columns in the second matrix. KEY POINTS • If A is an n×m matrix and B is an m×p matrix, the result AB of their multiplication is an n×p matrix defined only if the number of columns m in A is equal to the number of rows m in B. • Treating the rows and columns in each matrix as row and column vectors respectively, this entry is also their vector dot product. • The product of a square matrix multiplied by a column matrix arises naturally in linear algebra for solving linear equations and representing linear transformations. Assume two matrices are to be multiplied, the generalization to any number is discussed below. If A is an n×m matrix and B is an m×p matrix, the result AB of their multiplication is an n×p matrix defined only if the number of columns m in A is equal to the number of rows m in B. General Definition The arithmetic process of multiplying numbers in row i in matrix A and column j in matrix B, then adding to obtain entry ij in the final matrix. When multiplying matrices, the elements of the rows in the first matrix are multiplied with corresponding columns in the second matrix. One may compute each entry in the third matrix one at a time. For two matrices, A = ( A11 A12 . . . A1m A21 A22 . . . A2m) B = ( B11 B12 . . . B1p B21 B22 . . . B2p) where necessarily the number of columns in A equals the number of rows in B equals m, the matrix product AB is defined (with no multiplication signs or dots), and where AB has entries defined by the equation: A = ( (AB)11 (AB)12 . . . (AB)1mp (AB)21 (AB)22 . . . (AB)2mp) Treating the rows and columns in each matrix as row and column vectors respectively, this entry is also their vector dot product: 332 Usually the entries are numbers or expressions, but can even be matrices themselves, see block matrix. The matrix product can still be calculated exactly the same way. Illustration Figure 6.11 illustrates diagrammatically the product of two matrices A and B, showing how each intersection in the product matrix corresponds to a row of A and a column of B. a11 a12 ⋅ ⋅ a31 a32 ⋅ ⋅ [ ⋅ b12 b13 ⋅ b22 b23] = ⋅ x12 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ x33 ⋅ ⋅ ⋅ The values at the intersections marked with circles are: x12 = (a11, a12) ⋅ (b12, b22) = (a11b12) + (a12b22) x33 = (a31, a32) ⋅ (b13, b23) = (a31b13) + (a32b23) Source: and-matrices/matrix-operations/matrix-multiplication/ CC-BY-SA Boundless is an openly licensed educational resource 333 This figure illustrates diagrammatically the product of two matrices A and B, showing how each intersection in the product matrix corresponds to a row of A and a column of B. Figure 6.11 Matrix Multiplication Matrix Equations Matrices can be used to compactly write and work with systems of multiple linear equations. KEY POINTS • If A is an m-by-n matrix, x designates a column vector (i.e., n×1-matrix) of n variables x1, x2,..., xn, and b is an m×1- column vector, then the matrix equation: Ax = b Matrices can be used to compactly write and work with systems of equations. A matrix is made of m rows and n columns. It is very common for the dimensions of a matrix to be written like m by n, or mxn. This is an example of a 2x2 matrix. [ a b] This is an example of a 2x1 matrix. As we have learned in previous sections, matrices can be manipulated in any way that a normal equation can be manipulated in. This is very helpful when we start to work with systems of equations. It is helpful to understand how to use matrices to solve these systems before we start to just 'plug them into a calculator'. Lets take the following system of equations: 3x+2y-z = 1 2x+2y+4z = -2 x + 1 2 y − z = 0 Solving a System of Equations Using a Matrix It is possible to solve this system using the elimination or substitution method, but it would be easier to do it with a matrix operation. Before we start setting up the matrices, it is important to do the following: • Make sure that all of the equations are written in a similar manner, meaning the variables need to all be in the same order • Make sure that one side of the equation is only variables and their coefficients, and the other side is just constants. There are 3 matrices that need to be set up. Matrix [A], which is made up of the variable coefficients. This matrix will have as many rows as variables, and as many columns as equations. Matrix [x], which is a matrix made up of just variables. This matrix will have as many rows as variables, but only one column. Matrix [b] is made up of the equation solutions, that is, the right side of the equations. 334 This matrix will have as many rows as variables, but only one column. These matrices will be written as Ax = b. In order to solve the equations, use the following equation: x = A−1 b. We will not cover how to get the inverse of a matrix in this section, because it is outside the scope of this atom. This can easily be plugged into a graphing calculator, or you can refer to that atom. This is how we will write Ax=b for this system: 3 2 −1 2 2 4 −1 1 2 −1 * [ x y z] = [ 1 −2 0 ] When you use the given equation, A-1(b)=x, you will be given a solution in this format: 3 2 −1 2 2 4 −1 1 2 −1 −1 * [ 1 −2 0 ] = [ 1 −2 −2] = [ x y z] Another example, if A is an m-by-n matrix, x designates a column vector (i.e., n×1-matrix) of n variables x1, x2,..., xn, and b is an m×1- column vector, then the matrix equation: Ax = b Figure 6.12 is equivalent to the following system of linear equations: A1,1 x1 + A1,2 x2 + . . . + A1,n xn = b1 ... Am,1 x1 + Am,2 x2 + . . . + Am,n xn = bm Source: and-matrices/matrix-operations/matrix-equations/ CC-BY-SA Boundless is an openly licensed educational resource 335 The vector equation is equivalent to a matrix equation of the form: Ax=b, where A is an m×n matrix, x is a column vector with n entries, and b is a column vector with m entries. Figure 6.12 Matrix equation The Identity Matrix The Inverse of a Matrix Solving Systems of Equations Using Matrices Section 5 Inverses of Matrices 336 The Identity Matrix The identity matrix [I] is defined so that [A][I]=[I][A]=[A], i.e. it is the matrix version of multiplying a number by one. KEY POINTS • For any square matrix, its identity matrix is a diagonal stretch of 1s going from the upper-left-hand corner to the lower- right, with all other elements being 0. • Non-square matrices do not have an identity. That is, for a non-square matrix [A], there is no matrix such that [A][I]=[I] [A]=[A]. • Proving that the identity matrix functions as desired requires the use of matrix multiplication. When multiplying numbers, the number 1 has a special property: when multiplying 1 by any number, the same number back is gotten back. This idea can be expressed with the following property as an algebraic generalization: 1x = x The matrix that has this property is referred to as the identity matrix. Definition of the Identity Matrix The identity matrix, designated as [I ], is defined by the property: [A][I] = [I ][A] = [A]. Note that the definition of [I ] stipulates that the multiplication must commute—that is, it must yield the same answer no matter in which order multiplication is done. This stipulation is important because, for most matrices, multiplication does not commute. What matrix has this property? The first guess might be a matrix full of 1s, but that does not work: ( 1 2 3 4) ( 1 1 1 1) = ( 3 7 3 7) so ( 1 1 1 1) is not an identity matrix. The matrix that does work is a diagonal stretch of 1s, with all other elements being 0. ( 1 3 2 4) ( 1 0 0 1) = ( 1 3 2 4) so ( 1 0 0 1) is the identity matrix for 2x2 matrices. For a 3x3 matrix, the identity matrix is a 3x3 matrix with diagonal 1's and the rest 0's, for example 337 2 π −3 5 −2 1 2 9 8 8.3 ( 1 0 0 0 1 0 0 0 1) = 2 π −3 5 −2 1 2 9 8 8.3 so ( 1 0 0 0 1 0 0 0 1) is the identity matrix for 3x3 matrices. It is important to confirm those multiplications, and also confirm that they work in reverse order (as the definition requires). Remember, when doing matrix multiplication, multiply the first element of the first row of the first matrix by the first element of the first column of the second matrix, add it to the second by the second, etc., to determine the first element of the first row and column of the product matrix. The second element of the first row of the product matrix is the same thing, but done by multiplying the first row of the first matrix by the second column of the second matrix, then summing. In this way, each element of the product matrix can be determined. One mnemonic for remembering how to multiply matrices is that it is similar to taking a row of the first matrix, putting it into a dump truck, and tilting it into a column of the second matrix, then multiplying whatever is next to each other and adding the products (Figure 6.13). The final location of the element will be where the row and column intersect. Why is there no identity for a non-square matrix? There is no identity for a non- square matrix, because of the requirement of commutativity. For a non- square matrix [A] one might be able to find a matrix [I ] such that [A][I ] = [A]. However, if the order is reversed, then an illegal multiplication will be left. The reason for this is because, for two matrices to be multiplied together, the first matrix must have the same number of columns as the second has rows. Source: and-matrices/inverses-of-matrices/the-identity-matrix/ CC-BY-SA Boundless is an openly licensed educational resource 338 One way to remember how to multiply matrices is to imagine a dump truck taking the row of the first matrix and "dumping" it next to the column of the second. Multiply each pair and add the products together to get the element of the product matrix. Figure 6.13 Matrix multiplication The Inverse of a Matrix The inverse of matrix [A] is [A]-1, and is defined by the property: [A][A]-1=[A]-1[A]=[I]. KEY POINTS • Note that, just as in the definition of the identity matrix, this definition requires commutativity—the multiplication must work the same in either order. • To be invertible, a matrix must be square, because the identity matrix must be square as well. • To determine the inverse of the matrix ( 3 4 5 6) , set ( 3 4 5 6) ( a b c d) = ( 1 0 0 1) . Then solve for a, b, c, and d. Having seen that the number 1 plays a special role in multiplication, because 1x = x, the inverse of a number is defined as the number that multiplies by that number to give 1. b is the inverse of a if the inverse of a matrix multiplies by that matrix to give the identity matrix. Definition of Inverse Matrix The inverse of matrix [A], designated as [A]−1 , is defined by the property: [A][A]−1 = [A]−1 [A] = [I ], where [I ] is the identity matrix. Note that, just as in the definition of the identity matrix, this definition requires commutativity—the multiplication must work the same in either order. Note also that only square matrices can have an inverse. The definition of an inverse matrix is based on the identity matrix [I ], and we already said that only square matrices even have an identity. The method for finding an inverse matrix comes directly from the definition, along with a little algebra. Finding an Inverse Matrix Find the inverse of ( 3 4 5 6) . Set ( 3 4 5 6) ( a b c d) = ( 1 0 0 1) This is the key step. It establishes ( a b c d) as the inverse that is being looked for, by asserting that it fills the definition of an inverse matrix: when multiplying this mystery matrix by our original matrix, the result is I. When solving for the four variables a, b, c, and d, then the inverse of the matrix will be found. 339 Next, do the multiplication giving ( 3a + 4c 3b + 4d 5a + 6c 5b + 6d) = ( 1 0 0 1) . Make sure to check this step, it provides great practice. For instance, start by multiplying the first row times the first column, and the result is 3a + 4c. Therefore, one can set: 3a + 4c = 1 3b + 4d = 0 5a + 6c = 0 5b + 6d = 1 For two matrices to be equal: every element in the left must equal its corresponding element on the right. So, for these two matrices to equal each other, all four of these equations must hold. Therefore, the results are: a = − 3 b = 2 c = 2 1 2 d = − 1 1 2 Solve the first two equations for a and c by using either elimination or substitution. Solve the second two equations for b and d by using either elimination or substitution. So, the inverse is ( −3 2 2 1 2 −11 2 ) Solving for the four variables results in the inverse. If an inverse has been found, then when it is multiplied by the original matrix, then the result should be: ( −3 2 2 1 2 −1 1 2 ) ( 3 4 5 6) = ( 1 0 0 1) After multiplying, one gets ( (−3)(3) + (2)(5) (−3)(4) + (2)(6) (2 1 2 )(3)+(−1 1 2 )(5) (2 1 2 )(4) + (−1 1 2 )(6)) = ( 1 0 0 1) Simplifying the problem gives ( −9 + 10 −12 + 12 7 1 2 − 7 1 2 10 − 9 ) = ( 1 0 0 1) Therefore, the inverse works. 340 Note that, to fully test it, one has to try the multiplication in both orders, because, in general, changing the order of a matrix multiplication changes the answer. The definition of an inverse matrix specifies that it must work both ways. Only one order was shown above, so technically, this inverse has only been half-tested. This process does not have to be memorized: it should make logical sense. Everything one knows about matrices should make logical sense, except for the very arbitrary-looking definition of matrix multiplication. If it helps, remember the dump truck mnemonic (Figure 6.14). Source: and-matrices/inverses-of-matrices/the-inverse-of-a-matrix/ CC-BY-SA Boundless is an openly licensed educational resource Solving Systems of Equations Using Matrices A system of equations can be readily solved using the concepts of the inverse matrix and matrix multiplication. KEY POINTS • Using matrices to solve systems of equations can drastically reduce the workload on you. Consider the following three equations: x + 2y − z = 11 2x − y + 3z = 7 7x − 3y − 2z = 2 • To solve these equations using matrices, we first define a 3×3 matrix [A], which is the coefficients of all the variables on the left side of the equal signs: [A] = [ 1 2 −1 2 −1 3 7 −3 −2] Also, define a 3×1 matrix [B], which is the numbers on the right side of the equal signs: [B] = [ 11 7 2 ] . • In order to determine the values of x, y, and z, we simply multiply the inverse of [A] times [B]. This is most readily done using a calculator. The calculator responds with a 3×1 matrix, which is all three answers. In this case, x=3, y=5, and z=2. 341 Remember, when multiplying matrices, that one mnemonic to remember which terms to multiply together is to pretend that you have placed the row from the first matrix onto a dump truck, and are "dumping" the terms onto the column from the second matrix. Figure 6.14 Multiplying matrices At this point, you may be left with a pretty negative feeling about matrices. The initial few ideas—adding matrices, subtracting them, multiplying a matrix by a constant, and matrix equality—seem almost too obvious to be worth talking about. On the other hand, multiplying matrices and taking determinants seem to be strange, arbitrary sequences of steps with little or no purpose. A great deal of it comes together in solving linear equations. We have seen, in the chapter on simultaneous equations, how to solve two equations with two unknowns. But suppose we have three equations with three unknowns? Or four, or five? Such situations are more common than you might suppose in the real world. And even if you are allowed to use a calculator, it is not at all obvious how to solve such a problem in a reasonable amount of time. Surprisingly, the things we have learned about matrix multiplication, about the identity matrix, about inverse matrices, and about matrix equality, all give us a very fast way to solve such problems on a calculator! Consider the following example, three equations with three unknowns: x + 2y − z = 11 2x − y + 3z = 7 7x − 3y − 2z = 2 Define a 3×3 matrix [A], which is the coefficients of all the variables on the left side of the equal signs: [A] = [ 1 2 −1 2 −1 3 7 −3 −2] Define a 3×1 matrix [B], which is the numbers on the right side of the equal signs: [B] = [ 11 7 2 ] Now, in order to determine the values of x, y, and z, we simply multiply the inverse of [A] times [B]. This can be done by hand, finding the inverse matrix of [A], then performing the appropriate matrix multiplication (Figure 6.13). However, if you have a graphing calculator, the situation is much easier. Punch these matrices into your calculator, and then ask the calculator for [A-1][B], that is, the inverse of matrix [A], multiplied by matrix [B]. The calculator responds with a 342 When doing matrix multiplication, it may help to remember the mnemonic device of taking the rows of the first matrix and "dumping" them into the columns of the second matrix. Figure 6.15 Matrix multiplication 3×1 matrix that is all three answers. In this case, x=3, y=5, and z=2. The whole process takes no longer than it does to punch a few matrices into the calculator. And it works just as quickly for 4 equations with 4 unknowns, or 5, etc. Huh? Why the heck did that work? Solving linear equations in this way is fast and easy. The proof, unfortunately, is beyond the scope of this atom. Source: and-matrices/inverses-of-matrices/solving-systems-of-equations- using-matrices/ CC-BY-SA Boundless is an openly licensed educational resource 343 Determinants of Square Matrices The determinant of a square matrix is computed by recursively performing the Laplace expansion to find the determinant of smaller matrices. KEY POINTS • The determinant of a 2-by-2 matrix [ a b c d] is defined to be ad − bc. • The Laplace expansion for a n-by-n square matrix B is n ∑ j′=1 bij′Cij′ for some fixed i or n ∑ i′ bi′jCi′j for some fixed j. • Recursively performing the Laplace expansion on each of the smaller sub-matrices in the Cofactor will eventually produce a small enough sub-matrix for which the determinant is known. In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well. The determinant provides important information when the matrix is, that of the coefficients of a system of linear equations, or when it corresponds to a linear transformation of a vector space: in the first case the system has a unique solution if and only if the determinant is nonzero, while in the second case that same condition means that the transformation has an inverse operation. A geometric interpretation can be given to the value of the determinant of a square matrix with real entries: the absolute value of the determinant gives the scale factor by which area or volume is multiplied under the associated linear transformation, while its sign indicates whether the transformation preserves orientation. Thus a 2 × 2 matrix with determinant −2, when applied to a region of the plane with finite area, will transform that region into one with twice the area, while reversing its orientation. Determinants occur throughout mathematics. The use of determinants in calculus includes the Jacobian determinant in the substitution rule for integrals of functions of several variables. They are used to define the characteristic polynomial of a matrix that is an essential tool in eigenvalue problems in linear algebra. In some cases, they are used just as a compact notation for expressions that would otherwise be unwieldy to write down. The determinant of a matrix A is denoted det(A), det A, or \|A\|. In the case where the matrix entries are written out in full, the 345 aei + bfg + cdh − af h − bdi − ceg. Determinant of a 2-by-2 Matrix For a 2-by-2 matrix, [ a b c d] , the determinant a b c d is defined to be ad-bc. If the matrix entries are real numbers, the matrix A can be used to represent two linear mappings: one that maps the standard basis vectors to the rows of A, and one that maps them to the columns of A. In either case, the images of the basis vectors form a parallelogram that represents the image of the unit square under the mapping. The parallelogram defined by the rows of the above matrix is the one with vertices at (0,0), (a,b), (a + c, b + d), and (c,d), as shown in Figure 6.16. The absolute value of is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by A. Laplace Expansion In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant \|B\| of an n × n square matrix B that is a weighted sum of the determinants of n sub-matrices of B, each of size (n−1) × (n −1). The Laplace expansion is of theoretical interest as one of several ways to view the determinant, as well as of practical use in determinant computation. The i, j cofactor of B is the scalar Cij defined by: Cij = (−1)i+j Mij where Mij is the i, j minor matrix of B, that is, the determinant of the 346 The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides. Figure 6.16 Determinant as Area (n-1)x(n-1) matrix that results from deleting the i-th row and j-th column of B. Then the Laplace expansion is given by the following: Suppose B = (bij) is an n x n matrix and keep one of i, j ∈ {1, 2,..., n} fixed. Then its determinant \|B\| is given by: \|B\| = bi1Ci1 + bi2Ci2 + . . . + binCin = n ∑ j′=1 bij′Cij′ Or: \|B\| = b1jC1j + b2jC2j + . . . + BnjCnj = n ∑ i′=1 bi′jCi′j Determinant of a 3-by-3 Matrix Using the Laplace expansion to find the determinant of a 3-by-3 matrix when given a general 3-by-3 matrix B = a b c d e f g h i , choose to fix a row i or a column j. Choose the first row, and fix i=1.Thus the determinant is: \|B\| = b11C11 + b12C12 + b13C13 = a(−1)1+1 M11 + b(−1)1+2 M12 + c(−1)1+3 M13 = aM11 − bM12 + cM13 = a e f h i − b d f g i + c d e g h = a(ei − hf ) − b(di − fg) + c(dh − eg) = aei + bfg + cdh − ahf − bdi − ceg Laplace Expansion as Recursive Function The Laplace expansion can be used as a recursive function for any arbitrary n-by-n matrix. Start with n x n matrix, and fix either a row or column and perform the Laplace expansion to get n sub-matrices of size (n-1) x (n-1). Then recurse on each of those (n-1) x (n-1) sub- matrices to get (n-1) sub-matrices of size (n-2) x (n-2), for EACH (n-1) sub-matrix. Now there are n(n-1) sub-matrices of size (n-2) x (n-2). Then perform the Laplace expansion on each of those sub- matrices, until eventually ending up with a sub-matrix of a size that the determinant is known, for instance a 3-by-3 matrix. However, to find the determinant of a very large matrix becomes rather difficult quickly. Indeed there will end up being n! sub-matrices of size 1, and the determinant of a matrix of size 1 is just the single element of that matrix. 347 Source: and-matrices/determinants-and-cramer-s-rule/determinants-of- square-matricies/ CC-BY-SA Boundless is an openly licensed educational resource Cofactors The cofactor of an entry (i, j) of a matrix A is the signed minor of that matrix. KEY POINTS • Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n. A k × k minor of A is the determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns. • The first minor of a matrix Mij is formed by removing the ith row and jth column of the matrix, and retrieving the determinant of the smaller matrix. • The cofactor of an element aij of a matrix A, written as Cij is defined as (−1)i+j Mij. In linear algebra, the cofactor (sometimes called adjunct) describes a particular construction that is useful for calculating both the determinant and inverse of square matrices. Specifically the cofactor of the (i, j) entry of a matrix, also known as the (i, j) cofactor of that matrix, is the signed minor of that entry. Minor To know what the signed minor is, we need to know what the minor of a matrix is. In linear algebra, a minor of a matrix A is the 348 determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which in turn are useful for computing both the determinant and inverse of square matrices. Definition Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n. A k × k minor of A is the determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns. More often than not you'll only remove one column and row at a time, and so the (i, j) minor, often denoted Mij, of an n × n square matrix A is defined as the determinant of the (n − 1) × (n − 1) matrix formed by removing from A its ith row and jth column. An (i, j) minor is also referred to as the (i, j)th minor, or simply i, j minor. Mij is also called the minor of the element aij of matrix A. A minor that is formed by removing only one row and column from a square matrix A (such as Mij) is called a first minor. When two rows and columns are removed, this is called a second minor. Example For example, say we are given the matrix: 1 4 7 3 0 5 −1 9 11 The minor M23 is the determinant of the 2 x 2 matrix formed by removing the 2nd row and 3rd column, i.e. 1 4 • • • • −1 9 • = 1 4 −1 9 = (9 − (−4)) = 13 349 Here is a cofactor of an arbitrary 3x3 matrix taking the first row and second column out. Figure 6.17 Cofactor Where a black dot represents an element we are removing. Cofactor The cofactor of aij entry of a matrix is defined as: Cij = (−1)i+j Mij Informal Approach Finding the minors of a matrix A is a multi-step process: 1. Choose an entry aij from the matrix. 2. Cross out the entries that lie in the corresponding row i and column j. 3. Rewrite the matrix without the marked entries. 4. Obtain the determinant of this new matrix. Mij is termed the minor for entry aij. If i + j is an even number, the cofactor of coincides with its minor: Cij = Mij Otherwise, it is equal to the additive inverse of its minor: Cij = − Mij Source: and-matrices/determinants-and-cramer-s-rule/cofactors/ CC-BY-SA Boundless is an openly licensed educational resource 350 Cramer's Rule Cramer's Rule uses determinants to solve for a solution to the equation Ax=b, when A is a square matrix. KEY POINTS • Cramer's Rule only works on square matrices that have a non-zero determinant and a unique solution. • Cramer's Rule is defined as xi = det(Ai) det(A) i = 1,...,n, where Ai is the matrix formed by replacing the ith column of A by the column vector b in the equation. • Cramer's Rule is efficient for solving small systems and can be calculated quite quickly; however, as the system grows, calculating the new determinants can be tedious. Determinants Determinants play a critical role in the application of Cramer's Rule. A determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well. The determinant provides important information when the matrix is that of the coefficients of a system of linear equations, or when it corresponds to a linear transformation of a vector space: in the first case the system has a unique solution if and only if the determinant is nonzero, while in the second case that same condition means that the transformation has an inverse operation. A geometric interpretation can be given to the value of the determinant of a square matrix with real entries: the absolute value of the determinant gives the scale factor by which area or volume is multiplied under the associated linear transformation, while its sign indicates whether the transformation preserves orientation. Thus a 2 × 2 matrix with determinant −2, when applied to a region of the plane with finite area, will transform that region into one with twice the area, while reversing its orientation. The determinant of a matrix A is denoted det(A), det A, or \|A\|. In the case where the matrix entries are written out in full, the : aei + bfg + cdh − bdi − af h. A simple 2x2 matrix: a b c d = ad − bc 351 The images of the basis vectors form a parallelogram that represents the image of the unit square under the mapping (Figure 6.18). The parallelogram defined by the rows of the above matrix is the one with vertices at (0,0), (a,b), (a + c, b + d), and (c,d), as shown in the accompanying diagram. The absolute value of is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by A. The parallelogram formed by the columns of A is in general a different parallelogram. However, since the determinant is symmetric with respect to rows and columns, the area will be the same. Cramer's Rule Cramer's Rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, i.e. a square matrix, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations. Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows: Ax = b where the n by n matrix A has a nonzero determinant, and the vector Ex = (x1, . . . , xn)T is the column vector of the variables. Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns are given by: xi = det(Ai) det(A) i = 1,...,n Where Ai is the matrix formed by replacing the ith column of Aby the column vector b. 352 The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides. Figure 6.18 Determinant as Area Graphs of Linear Inequalities Graphing linear inequalities involves graphing the original line, and then shading in the area connected to the inequality. KEY POINTS • To graph a single linear inequality, first graph the inequality as if it were an equation. If the sign is ≤ or ≥, graph a normal line. If it is > or <, then use a dotted or dashed line. Then, shade either above or below the line, depending on if y is greater or less than mx + b. • If there are multiple linear inequalities, then where all the shaded areas of each inequality overlap is the solutions to the system. • If the shaded areas of all inequalities in a system do not overlap, then the system has no solution. Single Inequality in Two Variables The simplest inequality to graph is a single inequality in two variables, usually of the form y ≤ mx + b where the inequality can be of any type, less than, less than or equal to, greater than, greater than or equal to, or not equal to. Graphing an inequality is easy. First, graph the inequality as if it were an equation. If the sign is ≤ or ≥, graph a normal line. If it is > or <, then use a dotted or dashed line. Then, shade either above or below the line, depending on if y is greater or less than mx + b. As an example, say y ≤ x + 2. Since the equation is less than or equal to, start off by drawing the line y = x + 2. Next, note that y is less than or equal to x+2, which means that y can take on the values along the line, or any values below the line, and so we shade in all the values under the line to get Figure 6.19. Multiple Inequalities in Two Variables Now if there is more than one inequality, start off by graphing them one at a time, just as was done with a single inequality. To find 355 Here is the graph of the single inequality y <= x + 2. Figure 6.19 Graph of Single Inequality solutions for the group of inequalities, observe where the area of all of the inequalities overlap. These overlaps are solutions to the system. This also means that if there are inequalities that don't overlap, then there is no solution to the system. To use a real world example, such as gas mileage: The gas mileage sticker on cars gives two numbers: one for city driving, and one for highway driving. If a sticker says the car gets 25 mpg in the city and 32 mpg on the highway how far can you drive? The abbreviation mpg stands for miles per gallon. The sticker on our car predicts that the car gets between 25 and 32 mpg, but when the car is driven one estimates how far the car is going and how much gas is in the tank. When graphing change the inequality to a function where x is the number of gallons of gas in our tank: 25x < f(x) < 32x. Figure 6.20 has two lines one for y=25x and one for y = 32x. The yellow portion of the picture represents how far the car may be able to drive when there are x gallons of gas. The vertical line at 10 shows that it can drive between 250 and 320 miles on 10 gallons of gas. The difference between these two numbers is 70. If a line was drawn at 1 gallon of gas, it would be seen that the car could drive between 25 and 32 miles, and the difference would be 7. What this graph shows is the predicted range of miles a car can drive on a given amount of gas. The more gas a car has, the more likely the actual mileage is going to be different from what the sticker on the car predicted. If the actual mileage falls outside of this range than the car may need to go to the mechanic to make sure everything is running correctly. If the mileage is too high the odometer on the car may be broken. If the mileage is too low the car might need a tune up. Source: and-matrices/systems-of-inequalities-and-linear-programming/ graphs-of-linear-inequalities/ CC-BY-SA Boundless is an openly licensed educational resource 356 Here is an example showing in what range a car can drive given a city mpg and a highway mpg. Figure 6.20 Gas Mileage Example Solving Systems of Linear Inequalities Solving for a system of linear inequalities requires finding values for each of the variables such that all equations are satisfied. KEY POINTS • To solve a system graphically, draw and shade in each of the inequalities on the graph, and then look for an area in which all of the inequalities overlap, this area is the solution. • If there is no area in which all of the inequalities overlap, then the system has no solution. • To solve a system non-graphically, find the intersection points, and then find out relative to those points which values still hold for the inequality. Narrow down these values until mutually exclusive ranges (no solutions) are found, or not, in which the solution is within your final range. Graphical Method To solve a system of linear inequalities, if possible, the easiest way to do this is by graphing. However, graphing is only possible if there are two or three variables. For two variables, when graphing, first draw all of the lines of the inequalities as if they were an equation, drawing a dotted line if it is either < or >, and a solid line if it is either ≤ or ≥. After the line has been drawn, shade in, or indicate with hash marks, the area that corresponds to the inequality. For instance, if it is < or ≤, shade in the area below the line. If it is > or ≥ shade in the area above the line. Once all of the inequalities have been drawn and shaded in, to find solutions to the system one needs to find areas in which all of the inequalities overlap each other. For example, given the system: y ≥ −2x − 1 y ≥ 2x + 1 y ≤ x + 2 Draw each of the lines and shade in, or indicate, their corresponding inequalities, and then look to see what parts overlap. As can be seen in Figure 6.21, the shaded part in the middle is where all three inequalities overlap. If all of the inequalities of a system fail to overlap over the same area, then there is no solution to that system. For instance, given the following system: y ≥ 1 y ≥ 2x + 2 y ≤ x + 1 357 Again, draw all the inequalities and shade in, or indicate somehow, the area that the inequality covers. Notice that in this graph Figure 6.22, there is no part of the graph where all three inequalities overlap. There are plenty of areas where two of the three overlap at a time, but that is not enough, all three must overlap for those points to be a solution to the system. Non-Graphical Method Sometimes one may not wish to graph the equations, or simply cannot due to the number of variables. In this situation, find intervals in which certain variables satisfy the system by looking at two equations at a time. First, find the intersection point(s) of two of the equations, if there is no intersection, then the two inequalities are either mutually exclusive, or one of the inequalities is a 358 This is a graph showing the solutions to a linear inequality system. Note that it is the overlapping areas of all three linear inequalities. Figure 6.21 Solution to Linear Inequality System This is a graph showing a system of linear inequalities that has no solution as there is no point in which the areas of all three inequalities overlap. Figure 6.22 Linear Inequality System With No Solutions subset of the other. For a simple example, x > 2 and x < 1 are mutually exclusive, whereas x > 2 and x > 1 has x > 2 as a subset of x > 1. If they are mutually exclusive, then there is no solution. Once an intersection point is found, determine on which side(s) of the intersection point the inequalities hold. For example, if there are two equations y ≥ − 2x and y ≤ x + 1, the intersection is found to occur at x = −1 3 . Look at any point greater than that x value (say 0) to see that the first equation gives y ≥ 0, and the second gives y ≤ 1. Since these two equations are not mutually exclusive, these two equations are satisfied for any x ≥ −1 3 . If the entire system had a third equation as well: y ≤ 4, then next find the intersection points between this new equation and the other two. The intersection between y ≤ 4 and y ≥ − 2x occurs at x = − 2, and that it is satisfied when x ≥ − 2. Note that the x ≥ −1 3 found earlier is more restrictive. Therefore, ignore the new inequality for x. Now the last set y ≤ 4 and y ≤ x + 1 intersects at x = 3, further, by looking at x=2 and x=4, it is seen that for these two inequalities, x ≤ 3. Now for the system to hold true, −1 3 ≤ x ≤ 3, and to find valid y values, simply choose any x value in that range, and plug them in to get the range on the y values. The non-graphical method is much more complicated, and is perhaps much harder to visualize all the possible solutions for a system of inequalities. However, having too many equations or too many variables, may be the only feasible method. Source: and-matrices/systems-of-inequalities-and-linear-programming/ solving-systems-of-linear-inequalities/ CC-BY-SA Boundless is an openly licensed educational resource 359 Application of Systems of Inequalities: Linear Programming Linear programming involves finding an optimal solution for a linear equation given a number of constraints. KEY POINTS • The standard form for a linear program is: minimize c ⋅ x, subject to Ax = b, xi ≥ 0. c is the coefficients of the objective function, x is the variables, A is the left-side of the constraints and b is the right side. • The Simplex Method involves choosing an entering variable from the nonbasic variables in the objective function, finding the corresponding leaving variable that maintains feasibility, and pivoting to get a new feasible solution, repeating until you find a solution. • In the Simplex Method, if there are no positive coefficients corresponding to the nonbasic variables in the objective function, then you are at an optimal solution. • In the Simplex Method, if there are no choices for the leaving variable, then the solution is unbounded. Linear programming is a mathematical method for determining a way to achieve the best outcome for some list of requirements represented as linear relationships. Example A factory makes three types of chair, A, B and C. The factory makes a profit of $2 on A, $3 on B and $4 on C. A requires 30 man-hours, B requires 20 and C requires 10. A needs 2m2 of wood; B needs 5m2; C needs 3m2. Given 100 man-hours and 15m2 of wood per week, how many chairs of each type should be made each week to maximize profit? Simplex Method The most common method in linear programing is the simplex method. To use the simplex method, we need to represent the problem by linear equations. Let a be the number of A chairs, b the B chairs, and c the C chairs. 30a + 20b + 10c ≤ 100 ⇒ 3a + 2b + c ≤ 10 2a + 5b + 3c ≤ 15 All three variables must be non-negative as well. All constraints must be satisfied. 360 The function to be maximized (the objective function) is: P = 2a + 3b + 4c Standard Form The standard form for the simplex method is: Minimize c ⋅ x Subject to: Ax = b, xi ≥ 0 Where x = [x1, x2, . . . , xn]T are the variables, c = [c1, c2, . . . , cn] are the coefficients of the objective function, A is the left-side of the constraints, and b = [b1, b2, . . . , bp]T the right. The solution of a linear program is accomplished in two steps. In the first step, Phase I, a starting extreme point is found. Phase I either gives a basic feasible solution or no solution. In the latter, the linear program is infeasible. In the second step, Phase II, the simplex algorithm is applied using the solution found in Phase I as a starting point. The possible results from Phase II are either an optimal solution or an unbounded solution. Achieving Standard Form You may have noticed that we had inequalities, 3a + 2b + c ≤ 10, but standard form calls for equalities. We therefore introduce a slack variable that represents the difference between the two sides of the inequality and is non-negative. This gives us the new equality 3a + 2b + c + s = 10. The other inequality becomes: 2a + 5b + 3c + t = 15. Standard form also requires the objective function be a minimization. If the problem calls for maximization, multiply the objective function by -1. Here are the pieces for standard form: x = [a, b, c, s, t]T c = [−2, − 3, − 4,0,0] A = [ 3 2 1 1 0 2 5 3 0 1] b = [ 10 15] Canonical Tableaux A linear program in standard form can be represented as a tableau of the form [ 1 −c 0 0 A b] 361 The first row defines the objective function and the remaining rows specify the constraints. If the columns of A can be rearranged so that it contains the p-by-p identity matrix (the number of rows in A) then the tableau is said to be in canonical form. The variables corresponding to the columns of the identity matrix are called basic variables, while the remaining variables are called nonbasic or free variables. If the nonbasic variables are assumed to be 0, then the values of the basic variables are easily obtained as entries in b and this solution is a basic feasible solution. Pivots Moving from one basic feasible solution to an adjacent basic feasible solution is called a pivot. First, a nonzero pivot element is selected in a nonbasic column. The row containing this element is multiplied by its reciprocal to change this element to 1, and then multiples of the row are added to the others to change the other entries in the column to 0. The result, is that if the pivot is in row r, then the column becomes the r-th column of the identity matrix. The variable for this column is now basic, replacing the variable which corresponded to the r-th column of the identity matrix. The variable corresponding to the pivot column enters the set of basic variables, and the variable being replaced leaves the set of basic variables. Simplex Method Algorithm The simplex algorithm proceeds by performing successive pivot operations which each improve the basic feasible solution; the choice of pivot element at each step is largely determined by the requirement that this pivot improves the solution. For the entering variable, choose any column in which the entry in the objective row is positive. If all the entries in the objective row are less than or equal to 0 then no choice of entering variable can be made and the solution is optimal. For the choice of pivot row, only positive entries in the pivot column are considered. This guarantees that the value of the entering variable will be non-negative. If there are none in the pivot column, then the entering variable can take any non-negative value with the 362 The blue area is the feasible region for a linear program with the constraints forming the lines. The optimal solution will occur at one of the intersection points of the lines. Figure 6.23 Graph of Linear Inequality solution remaining feasible. Therefore the objective function is unbounded. Next, the pivot row must be selected so that all the other basic variables remain positive. This occurs when the resulting value of the entering variable is at a minimum. If the pivot column is c, then the pivot row r is chosen so that br /acr is at a minimum. Example Using our example, the canonical tableau is 1 2 3 4 0 0 0 0 3 2 1 1 0 10 0 2 5 3 0 1 15 Columns 5 and 6 are the basic variables s and t, and the basic feasible solution is a = b = c = 0,s = 10,t = 15. Columns 2, 3, and 4 can be selected as pivot columns; for this example column 4 is selected. The values of x resulting from the choice of rows 2 and 3 as pivot rows are 10/1 = 10 and 15/3 = 5 respectively. Of these the minimum is 5, so row 3 must be the pivot row. Performing the pivot produces: 1 −2 3 −11 3 0 0 −4 3 −20 0 7 3 1 3 0 1 −1 3 5 0 2 3 5 3 1 0 1 3 5 Now columns 4 and 5 represent the basic variables c and s and the corresponding basic feasible solution is a = b = t = 0,s = 5,c = 5 For the next step, there are no positive entries in the objective row, and in fact: −P = − 20 + 2 3 a + 11 3 b + 4 3 t So we should make 5 chairs of type C to maximize our profits with 20 dollars. Source: and-matrices/systems-of-inequalities-and-linear-programming/ application-of-systems-of-inequalities-linear-programming/ CC-BY-SA Boundless is an openly licensed educational resource 363 Partial Fractions Partial fraction decomposition is a procedure used to reduce the degree of either the numerator or the denominator of a rational function. KEY POINTS • In terms of symbols, partial fraction decomposition turns a function of the form f(x) g(x) , where f and g are both polynomials, into a function of the form ∑ j fj(x) gj(x) , where gj(x) are polynomials that are factors of g(x). • The main motivation to decompose a rational function into a sum of simpler fractions is to make it simpler to perform linear operations on the sum. • If the degree of f(x) is greater than or equal to the degree of g(x), then it is necessary to perform the Euclidean division of f by g, using polynomial long division, giving f(x) = E(X)g(x) + h(x). • If g(x) contains factors which are irreducible, then the numerator N(x) of each partial fraction with such a factor h(x) in the denominator must be sought as a polynomial with degree N < degree h, rather than as a constant. Partial Fraction Decomposition In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function (also known as a rational algebraic fraction). In symbols, one can use partial fraction expansion to change a rational function in the form: f(x) g(x) , where f and g are polynomials, into a function of the form, ∑ j fj(x) gj(x) , where gj(x) are polynomials that are factors of g(x), and in general of lower degree. Thus, the partial fraction decomposition may be seen as the inverse procedure of the more elementary operation of addition of algebraic fractions, which produces a single rational function with a numerator and denominator usually of high degree. The full decomposition pushes the reduction as far as it will go: in other words, the factorization of g is used as much as possible. 365 This is the basic form of partial fraction decomposition. Figure 6.24 Partial Fraction Decomposition The main motivation to decompose a rational function into a sum of simpler fractions is to make it simpler to perform linear operations on the sum. Therefore, the problem of computing derivatives, antiderivatives, integrals, power series expansions, Fourier series, Laplace transforms, residues, and linear functional transformations of rational functions can be reduced, via partial fraction decomposition, to focusing on computing each single element of the decomposition. Basic Principles Assume a rational function R(x) = f(x) g(x) has a denominator that factors as g(x) = P(x) ⋅ Q(x). If P and Q have no common factor, then R may be written as A P + B Q for some polynomials A(x) and B(x). The fraction can then be decomposed by equating R(x) = A P + B Q and solving for each of A and B by substitution, equating the coefficients of terms, or otherwise. There are some important cases to note for partial fraction decomposition. If the degree of f(x) is greater than or equal to the degree of g(x), then it is necessary to perform the Euclidean division of f by g, using polynomial long division, giving f(x) = E(X)g(x) + h(x). Dividing through by g(x) gives f(x) g(x) = E(x) + h(x) g(x) , which you can then perform the decomposition on h(x)/g(x). If g(x) contains factors which are irreducible, then the numerator N(x) of each partial fraction with such a factor h(x) in the denominator must be sought as a polynomial with degree N < degree h, rather than as a constant. For example, take the following decomposition over R: x2 + 1 (x + 2)(x − 1)(x2 + x + 1) = a x − 2 + b x − 1 + cx + d x2 + x + 1 Example 1 f(x) = 1 x2 + 2x − 3 Here, the denominator splits into two distinct linear factors: q(x) = x2 + 2x − 3 = (x + 3)(x − 1) So we have the partial fraction decomposition: f(x) = 1 x2 + 2x − 3 = A x + 3 + B x − 1 Multiplying through by x2 + 2x − 3, we have: 366 Parabolas Parabolas are common in algebra as the graphs of quadratic functions, and they have many important real world applications. KEY POINTS • Parabolas are frequently encountered as graphs of quadratic functions, including the very common equation y=x^2. • All parabolas contain a focus, a directrix, and an axis of symmetry that vary in exact location depending on the equation used to define the parabola. • Parabolas are frequently used in physics and engineering in places such as automobile headlight reflectors and in the design of ballistic missiles. In mathematics, a parabola is a conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. Another way to generate a parabola is to examine a point (the focus) and a line (the directrix), as can be visualized in Figure 7.1. The locus of points in that plane that are equidistant from both the line and point is a parabola. In algebra, parabolas are frequently encountered as graphs of quadratic functions, such as: y = x2 The line perpendicular to the directrix and passing through the focus, that is, the line that splits the parabola through the middle, is called the axis of symmetry. The point on the axis of symmetry that intersects the parabola is called the "vertex", and it is the point where the curvature is greatest. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, all parabolas are similar. 370 Parabolic curve showing directrix (L) and focus (F). The distance from any point on the parabola to the focus (PnF) equals the perpendicular distance from the same point on the parabola to the directrix (PnQn). Figure 7.1 Parabola with Focus and Directrix Parabolas have the property that, if they are made of material that reflects light, then light which enters a parabola traveling parallel to its axis of symmetry is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected, or collimated, into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy. This reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas. Source: parabola/parabolas/ CC-BY-SA Boundless is an openly licensed educational resource Standard Form and Completing the Square The standard form of a quadratic equation is useful for completing the square, which is used to graph the equation. KEY POINTS • Quadratic equations are graphed as parabolas. • The standard form of a parabola is y = a(x − h)2 + k. • Completing the square is a technique used to solve for the x- intercepts, a necessary first step in graphing a quadratic equation as a parabola. In algebra, parabolas are frequently encountered as graphs of quadratic functions, such as: y = x2 In algebraic geometry, the parabola is generalized by the rational normal curves, which have coordinates of x, x2 , x3 , …, xn . The standard parabola is the case n = 2. The case n = 3 is known as the twisted cubic. A further generalization is given by the Veronese variety, when there is more than one input variable. 371 Completing the square is a technique for converting a quadratic polynomial of the form: y = ax2 + bx + c to the form: y = a(: y = a(x − h)2 + k and one must find h and k. This form of quadratic equation is known as the "standard form" for graphing parabolas in algebra; from this equation, it is simple to determine the x-intercepts (y = 0) of the parabola, a process known as "solving" the quadratic equation. Completing the square may be used to solve any quadratic equation. For example: y = x2 + 6x + 5 The first step is to set this equation equal to zero. x2 + 6x + 5 = 0 The second step is to complete the square: (x + 3)2 − 4 = 0 Next, we solve for the squared term: (x + 3)2 = 4 That gives us: x + 3 = − 2 or x + 3 = 2 And therefore: x = − 5 or x = − 1 The result of graphing this equation can be seen in Figure 7.2, where the x-intercepts are indeed at -1 and -5. This can be applied to any quadratic equation. When the x2 has a coefficient other than 1, the first step is to divide out the equation by this coefficient. 372 Source: parabola/standard-form-and-completing-the-square/ CC-BY-SA Boundless is an openly licensed educational resource Applications and Problem Solving Parabolas have important applications in physics, engineering, and nature. KEY POINTS • Projectiles and missiles follow an approximate parabolic path; they are approximate because real-world imperfections affect the movement of an object. • Parabolic reflectors are common in today's microwave and satellite dish receiving and transmitting antennas. • Paraboloids are also observed in the surface of a liquid confined to a container and rotated around the central axis. A parabola is a conic section created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. The parabola has many important applications, from automobile headlight reflectors to the design of ballistic missiles. They are frequently used in areas such as engineering, physics, and nature. The parabolic trajectory of projectiles was discovered experimentally in the 17th century by Galileo, who performed 373 The graph of this quadratic equation is a parabola with x-intercepts at -1 and -5. Figure 7.2 Graph of y = x2 + 6x + 5 experiments with balls rolling on inclined planes. For objects extended in space, such as a diver jumping from a diving board, the object follows a complex motion as it rotates, while its center of mass forms a parabola. As in all cases in the physical world, a projectile's trajectory is an approximation. The presence of air resistance, for instance, distorts parabolic shape, although at low speeds the shape is a good approximation. At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola (Figure 7.3). Paraboloids also arise in several physical situations. Best-known is the parabolic reflector—a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point, or conversely collimates light from a point source at the focus into a parallel beam (Figure 7.4). This principle was applied to telescopes in the 17th century. Today, paraboloid reflectors are common throughout much of the world in microwave and satellite dish receiving and transmitting antennas. Paraboloids are also observed in the surface of a liquid confined to a container then rotated around the central axis. In this case, he centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface (the principle behind the liquid mirror telescope). Aircraft used to create a weightless state for purposes of experimentation, such as NASA's "Vomit Comet," follow a vertically 374 In this image, the water shot from a fountain follows a parabolic trajectory as gravity pulls it back down. Figure 7.3 Parabolic Water Trajectory Note how the source light (Q1, Q2, Q3) reflects off different portions of the parabola, but collects at the same focal point (F). Figure 7.4 Parabolic reflector parabolic trajectory for brief periods in order to trace the course of an object in free fall, which, for most purposes, produces the same effect as zero gravity. Source: parabola/applications-and-problem-solving--3/ CC-BY-SA Boundless is an openly licensed educational resource 375 Circles The set of all points in a plane that are the same distance from a given point forms a circle. KEY POINTS • The mathematical formula for a circle is (x–h)2+(y–k)2=r2, with the circle's center (h, k) and its radius, r. • The length of a circumference of a circle can be found from the radius using the equation: C = 2πr = πd. • The area enclosed by a circle can be found from the radius using the formula: Area = πr2 . You've known all your life what a circle looks like. You probably know how to find the area and the circumference of a circle, given its radius. But what is the exact mathematical definition of a circle? Before you read the answer, you may want to think about the question for a minute. Try to think of a precise, specific definition of exactly what a circle is. Definition of a Circle The set of all points in a plane that are the same distance from a given point forms a circle. The point is known as the center of the circle, and the distance is known as the radius (Figure 7.5). Mathematicians often seem to be deliberately obscuring things by creating complicated definitions for things you already understood, but if you try to find a simpler definition of what a circle is, you will be surprised at how difficult it is to define. Most people start with something like "a shape that is completely round." That does describe a circle, but it also describes many other shapes, such as a pretzel. So, you start adding caveats like "it can't cross itself" and "it can't have any loose ends." And then somebody draws an egg shape that fits all of your criteria, and yet is still not a circle. Hence, the definition for a circle as given above. 377 A circle is defined as all the points that are a certain distance on a place from another point, the center. Figure 7.5 A Circle The Mathematical Formula for a Circle You already know the formula for a line: y=mx+b. You know that m is the slope, and b is the y-intercept. Knowing all of this, you can easily answer questions such as: "Draw the graph of y=2x–3," or "Find the equation of a line that contains the points (3,5) and (4,4)." If you are given the equation 3x+2y=6, you know how to graph it in two steps: first put it in the standard y=mx+b form, and then graph it. All the conic sections are graphed in a similar way. There is a standard form which is very easy to graph, once you understand what all the parts mean. If you are given an equation that is not in standard form, you must put it into the standard form, and then graph it. To understand the formula below, think of it as the y=mx+b of circles. The mathematical formula for a circle is (x − h)2 + (y − k)2 = r2 with the circle's center (h, k) and its radius, r. Length of Circumference The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. The length of the circumference, C, is related to the radius, r, and diameter, d, by: C = 2πr = πd Area Enclosed As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference, and whose height equals the circle's radius, which comes to π multiplied by the radius squared: Area = πr2 Equivalently, denoting diameter by d, Area = πd2 4 ≈ 0.7854d2 , that is, approximately 79 percent of the circumscribing square (whose side is of length d). Terminology of a Circle • Chord: a line segment whose endpoints lie on the circle. 378 • Diameter: the longest chord, a line segment whose endpoints lie on the circle and which passes through the center; or the length of such a segment, which is the largest distance between any two points on the circle. • Radius: a line segment joining the center of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter. • Circumference: the length of one circuit along the circle itself. • Tangent: a straight line that touches the circle at a single point. • Secant: an extended chord, a straight line cutting the circle at two points. • Arc: any connected part of the circle's circumference. • Sector: a region bounded by two radii and an arc lying between the radii. • Segment: a region bounded by a chord and an arc lying between the chord's endpoints. See Figure 7.6 and Figure 7.7 for visual demonstrations. Source: circle-and-the-ellipse/circles--2/ CC-BY-SA Boundless is an openly licensed educational resource 379 Chord, secant, tangent, radius, and diameter. Figure 7.6 Circle terminology displayed Arc, sector, and segment. Figure 7.7 Circle terminology Ellipses An ellipse, which resembles an oval, is defined as all points whose distance from two foci add to a constant. KEY POINTS • An ellipse is like a stretched out circle, sometimes referred to as an oval. In mathematics, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. • An ellipse is also the locus of all points of the plane whose distances to two fixed points (each called a focus) add to the same constant. • The formula for an ellipse is (x − h)2 a2 + (y − k)2 b2 = 1. The area of an ellipse is given by πab, where a and b are one-half of the ellipse's major and minor axes respectively. An ellipse is a sort of stretched out circle, sometimes referred to as an oval. In mathematics, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve (Figure 7.8). Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis. An ellipse has the property that, at any point on its perimeter, the distance from two fixed points (the foci) add to the same constant. Drawing an Ellipse 1. Lay a piece of cardboard on the floor. 2. Thumbtack one end of a string to the cardboard. 3. Thumbtack the other end of the string, elsewhere on the cardboard. The string should not be pulled taut: it should have some slack. 4. With your pen, pull the middle of the string back until it is taut. 5. Pull the pen all the way around the two thumbtacks, keeping the string taut at all times. 6. The pen will touch every point on the cardboard such that the distance to one thumbtack, plus the distance to the other thumbtack, is exactly one string length. And the resulting shape will be an ellipse. The cardboard is the "plane" in our definition, the thumbtacks are the "foci," and the string length is the "constant distance." 380 One definition of an ellipse is the intersection of a cone with an inclined plane. Figure 7.8 An ellipse as part of a cone How often do ellipses come up in real life? You'd be surprised. For a long time, the orbits of the planets were assumed to be circles. However, this is incorrect: the orbit of a planet is actually in the shape of an ellipse. The sun is at one focus of the ellipse (not at the center). Similarly, the moon travels in an ellipse, with the Earth at one focus. Elements of an Ellipse An ellipse is a smooth closed curve which is symmetric about its horizontal and vertical axes (Figure 7.9). The distance between antipodal points on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximum along the major axis or transverse diameter, and a minimum along the perpendicular minor axis or conjugate diameter. The semi-major axis (denoted by a in the figure) and the semi- minor axis (denoted by b in the figure) are one half of the major and minor axes, respectively. These are sometimes called (especially in technical fields) the major and minor semi-axes, or major radius and minor radius. The four points where these axes cross the ellipse are the vertices, points where its curvature is minimized or maximized. The foci of the ellipse are two special points F1 and F2 on the ellipse's major axis and are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis (PF1 + PF2 = 2a). Each of these two points is called a focus of the ellipse. The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to the length of the major axis or e = 2f 2a = f a For an ellipse the eccentricity is between 0 and 1. 381 See text for descriptions of each element of this ellipse. Figure 7.9 Properties of an ellipse The Formula of an Ellipse With ellipses, it is crucial to start by distinguishing horizontal from vertical. The general formula for an ellipse is x2 a2 + y2 b2 = 1. If a>b, the ellipse is horizontal. If a, the the ellipse is vertical. And of course, the usual rules of permutations apply. For instance, if we replace x with x−h, the ellipse moves to the right by h. So we have the more general form: (x − h)2 a2 + (y − k)2 b2 = 1 The area enclosed by an ellipse is πab, where a and b are one-half of the ellipse's major and minor axes respectively. Source: circle-and-the-ellipse/ellipses/ CC-BY-SA Boundless is an openly licensed educational resource Applications and Problem Solving Circles and ellipses are encountered in everyday life, and knowing how to solve their equations is very important. KEY POINTS • The equation for a circle is (x − h)2 + (y − k)2 = r2 . Where r=radius; (h,k)= center point coordinates. • The equation for an ellipse is (x − h)2 a2 + (y − k)2 b2 = 1. Where (h,k) = center point coordinates; a2 = major/minor axis length; b2 = major/minor axis length. Out of a and b, which ever value is greater, that will be the major axis. • In order to determine the parameters of a circle or an ellipse, you must first put the equation into standard form as shown above. Circles are all around you in everyday life, from tires on cars to buttons on coats, as well as on the tops of bowls, glasses and water bottles. Ellipses are less common, mostly encountered as the orbits of planets, but you should be able to find the area of a circle or an ellipse, or the circumference of a circle, based on information given to you in a problem. Some sample problems are shown below. The 382 answers will be given at the end of the atom, try not to look until you have attempted the problem Exercise 1 Let's say you are a gardener, and you have just planted a lot of flowers that you want to water. The flower bed is 15 square feet. You are using a circular sprinkler system, and the water reaches 6 feet out from the center. The sprinkler is located, from the bottom left corner of the bed, 7 feet up, and 6 feet over. 1. If the flower bed was a graph with the bottom left corner being the origin, what would the equation of the circle be? 2. What is the area being watered by the sprinkler? 3. What percentage of the garden that is being watered? Exercise 2 Now, let's take it the other way. (x − 4)2 + (y + 8)2 = 49 is the equation for a circle. a. What are the coordinates of the center of the circle? b. What is the radius? c. Draw the circle. d. Find two points on the circle (by looking at your drawing) and plug them into the equation to make sure they work. (Show your work!) Exercise 3 a. Put 2x2 + 2y2 + 8x + 24y + 60 = 0 into standard canonical circle form, (x − h)2 + (y − k)2 = r2 . b. What are the center and radius of the circle? c. Draw the circle. d. Find two points on the circle (by looking at your drawing) and plug them into the original equation to make sure they work. Exercise 4 4x2 9 + 25y2 = 1 This sort of looks like an ellipse in standard form, doesn't it? It even has a 1 on the right. But it isn't. Because we have no room in our standard form for that 4 and that 25—for numbers multiplied by the x2 and y2 terms. How can we get rid of them, to get into standard form, while retaining the 1 on the right? 383 Answer 3: a. First, divide by the coefficient of x2 and y2 to give x2 + y2 + 4x + 12y + 30 = 0 Next, collect x and y terms together, and bring the number to the other side to give (x2 + 4x) + (y2 + 12y) = − 30 Now, complete the square in both parentheses, subtracting or adding the necessary constant to both sides, (x2 + 4x + 4) + (y2 + 12y + 36) = − 30 + 4 + 36 Notice that each term is a perfect square, which gives (x + 2)2 + (y + 6)2 = 10 b. The center is at (-2,-6) and the radius is 10 c. and d. are left up to the student. Answer 4: a. After dividing the top and bottom by 4, we get x2 9 4 + 25y2 = 1 b. If we divide the top and bottom of the y term (in fraction form) by 25, we get x2 9 4 + y2 1 25 = 1. Now the equation is in our standard form. c. From our standard equation ( (x − h)2 a2 + (y − k)2 b2 = 1), we know that the center is at (h, k). Since these are both zero in our equation, the center is at (0,0). d. The major axis is 2a, or 18 4 . e. The minor axis is 2b, or 2 25 . f. Figure 7.11 385 In order to graph this circle, the program required input in terms of y=, which is not the form for the circle. In order to graph it, we rearrange the equation for a circle to two equations, y=-8-sqrt(33+8 x-x^2) and y=-8+sqrt(33+8 x-x^2) and ignore the imaginary sections. Figure 7.10 A graph of (x-4)^2+(y+8)^2=49 gives a circle Source: circle-and-the-ellipse/applications-and-problem-solving--6/ CC-BY-SA Boundless is an openly licensed educational resource 386 In order to graph this ellipse, we must rearrange the equation in terms of y= (unless you have a calculator that can handle ellipses), so we graph + and - sqrt(9-4x^2)/(15) Figure 7.11 The ellipse given by x^2/(9/4)+y^2/(1/25)=1 Standard Equations of Hyperbolas A standard equation for a hyperbola can be written as x2 /a2 − y2 /b2 = 1. KEY POINTS • Similar to a parabola, a hyperbola is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an ellipse does. A hyperbola consists of two disconnected curves called its arms or branches. • If the transverse axis of any hyperbola is aligned with the x- axis of a Cartesian coordinate system and is centered on the origin, the equation of the hyperbola can be written as x2 /a2 - y2 /b2 = 1. • Likewise, a hyperbola with its transverse axis aligned with the y-axis is called a "North-South opening hyperbola" and has equation y2 /a2 − x2 /b2 = 1. Similar to a parabola, a hyperbola is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an ellipse does. A hyperbola consists of two disconnected curves called its arms or branches. At large distances from the center, the hyperbola approaches two lines, its asymptotes, which intersect at the hyperbola's center. A hyperbola approaches its asymptotes arbitrarily closely as the distance from its center increases, but it never intersects them. Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the x-axis, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±b⁄a, where b=a×tan(θ) and where θ is the angle between the transverse axis and either asymptote. The distance b (not shown in Figure 7.12) is the length of the perpendicular segment from either vertex to the asymptotes. A conjugate axis of length 2b, corresponding to the minor axis of an ellipse, is sometimes drawn on the non-transverse principal axis; its 388 The asymptotes of the hyperbola (red curves) are shown as blue dashed lines and intersect at the center of the hyperbola, C. The two focal points are labeled F1 and F2, and the thin black line joining them is the transverse axis. The perpendicular thin black line through the center is the conjugate axis. The two thick black lines parallel to the conjugate axis (thus, perpendicular to the transverse axis) are the two directrices, D1 and D2. The eccentricity e equals the ratio of the distances from a point P on the hyperbola to one focus and its corresponding directrix line (shown in green). Figure 7.12 Properties of a Hyperbola endpoints ±b lie on the minor axis at the height of the asymptotes over/under the hyperbola's vertices. Because of the minus sign in some of the formulas below, it is also called the imaginary axis of the hyperbola. If b = a, the angle 2θ between the asymptotes equals 90° and the hyperbola is said to be rectangular or equilateral. In this special case, the rectangle joining the four points on the asymptotes directly above and below the vertices is a square, since the lengths of its sides 2a = 2b. If the transverse axis of any hyperbola is aligned with the x-axis of a Cartesian coordinate system and is centered on the origin, the equation of the hyperbola can be written as: x2 /a2 − y2 /b2 = 1 A hyperbola aligned in this way is called an "East-West opening hyperbola." Likewise, a hyperbola with its transverse axis aligned with the y-axis is called a "North-South opening hyperbola" and has equation: y2 /a2 − x2 /b2 = 1 Source: hyperbola/standard-equations-of-hyperbolas/ CC-BY-SA Boundless is an openly licensed educational resource 389 Applications and Problem Solving A hyperbola is an open curve with two branches and a cut through both halves of a double cone, which is not necessarily parallel to the cone's axis. KEY POINTS • Hyperbolas have applications to a number of different systems and problems including sundials and trilateration. • Hyperbolas may be seen in many sundials. On any given day, the sun revolves in a circle on the celestial sphere, and its rays striking the point on a sundial traces out a cone of light. The intersection of this cone with the horizontal plane of the ground forms a conic section. • A As we should know by now, a hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone. The plane may or may not be parallel to the axis of the cone Sundials Hyperbolas may be seen in many sundials. Every day, the sun revolves in a circle on the celestial sphere, and its rays striking the point on a sundial traces out a cone of light. The intersection of this cone with the horizontal plane of the ground forms a conic section. At most populated latitudes and at most times of the year, this conic section is a hyperbola. This conic section can be shown in Figure 7.13. The shadow of the tip of a pole traces out a hyperbola on the ground over the course of a day (this path is called the declination line). The shape of this hyperbola varies with the geographical latitude and with the time of the year, since those factors affect the cone of the sun's rays relative to the horizon (Figure 7.14). Trilateration Trilateration is the a method of pinpointing an exact location, using its distances to a given points. The can also be characterized 390 A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone. The plane may or may not be parallel to the axis of the cone Figure 7.13 Hyperbola as the difference in arrival times of synchronized signals between the desired point and known points.These types of problems arise in navigation, mainly nautical. A ship can locate its position using the arrival times of signals from GPS transmitters. Alternatively, a homing beacon can be located by comparing the arrival times of its signals at two separate receiving stations. This can be used to track people, cell phones, internet signals and many other things.In particular, the set of possible positions of a point that has a distance variation of 2a from two known points is a hyperbola of vertex separation 2a, and whose foci are the two known points. The Kepler Orbit of Particles The Kepler orbit is the path followed by any orbiting body (Figure 7. 15). This can be applied to a particle of any size, a planet or even hydrogen atoms. depending on the particles properties, including size and shape (eccentricity), this orbit can be one of six conic sections. In particular, if the total energy E of the particle is greater than zero (i.e., if the particle is unbound), the path of such a particle is a hyperbola. In the figure, the blue line shows the hyperbolic Kepler orbit. 391 Hyperbolas as declination lines on a sundial. Figure 7.14 Hyperbolas and Sundials A diagram of the various forms of the Kepler Orbit and their eccentricities. Blue is a hyperbolic trajectory (e > 1). Green is a parabolic trajectory (e = 1). Red is an elliptical orbit (e < 1). Grey is a circular orbit (e = 0). Figure 7.15 Kepler Orbits EXAMPLE A Such problems are important in navigation, particularly on water; a ship can locate its position from the difference in arrival times of signals from GPS transmitters. Source: hyperbola/applications-and-problem-solving--4/ CC-BY-SA Boundless is an openly licensed educational resource 392 Nonlinear Systems of Equations and Problem Solving As with linear systems, a nonlinear system of equations (and conics) can be solved graphically and algebraically for all its variables. KEY POINTS • Subtracting one equation from another is an effective means for solving linear systems, but it often is difficult to use in nonlinear systems, in which the terms of two equations may be very different. • Substitution of a variable into another equation is usually the best method for solving nonlinear systems of equations. • Nonlinear systems of equations may have one or multiple solutions. A conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. There are a number of other geometric definitions possible. Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it is sometimes called the fourth type of conic section. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus-directrix definition of a conic, the circle is a limiting case with eccentricity 0. In modern geometry, certain degenerate cases, such as the union of two lines, are included as conics as well. In a system of equations, two or more relationships are stated among variables. A system is solvable so long as there are as many simultaneous equations as variables. If each equation is graphed, the solution for the system can be found at the point where all the functions meet. The solution can be found either by inspection of a graph, typically with the use of software, or algebraically. Nonlinear systems of equations, such as conic sections, include at least one function that is non-linear. Because at least one function has curvature, it is possible for nonlinear systems of equations to contain multiple solutions. As with linear systems of equations, substitution can be used to solve nonlinear systems for one variable and then the other. 394 Solving nonlinear systems of equations algebraically is similar to doing the same for linear systems of equations. However, subtraction of one equation from another can become impractical if the two equations have different terms, which is more commonly the case in nonlinear systems. Consider, for example, the following system of equations (Figure 7. 16): y = x2 (1) y = x + 6 (2) Substituting x2 for y in equation 2: x2 = x + 6 This quadratic equation can be solved by moving all the equation's components to the left before using the quadratic formula: x2 − x − 6 = 0 Using the quadratic formula, with a=1, b=-2 and c=-6, it can be determined that x=3 and x=-2 are solutions. The solutions for x can then be plugged into either of the original systems to find the value of y. In this example, we will use equation 1: y = (−2)2 y = 32 Thus, for x=-2, y=4. And for x=3, y=9. Our final solutions are: (-2, 4) and (3, 9). Source: nonlinear-systems-of-equations-and-inequalities/nonlinear-systems- of-equations-and-problem-solving/ CC-BY-SA Boundless is an openly licensed educational resource 395 The parabola (blue) falls below the line (red) between x=-2 and x=3. For all values of x less than -2 and greater than 3, the parabola is greater than the line. Figure 7.16 Integer values of y=x^2 (blue) and y=x+6 (red) Models Involving Nonlinear Systems of Equations Nonlinear systems of equations can be used to solve complex problems involving multiple known relationships. KEY POINTS • Problems involving simultaneously moving bodies can be solved using systems of equations. If at least one body accelerates or decelerates, the system is nonlinear. • If the relationship between multiple unknown numbers is described in as many ways as there are numbers, all can be found using systems of equations. If at least one of those relationships is nonlinear, the system is nonlinear. • Substitution is the best method for solving for simultaneous equations, although to answer a question, one may not need to solve for every variable. Nonlinear systems of equations are not just for hypothetical discussions; they can be used to solve complex problems involving multiple known relationships. Consider, for example, a car that begins at rest and accelerates at a constant rate of 4 meters per second each second. Its position in meters (y) can be determined as a function of time in seconds (t), by the formula: y = 2t2 (1) Now consider a second car, traveling at a constant speed of 20 meters per second. Its position (y) in meters can be determined as a function of time (t) in seconds, using the following formula: y = 20t (2) 396 Figure 7.17 Nonlinear Simultaneous Equations When the first car begins to accelerate, the second car is 400 meters ahead of it. To express the position of the second car relative to the first as a function of time, we can modify the second equation as such: y = 20t + 400 (3) To determine where the cars are when they are alongside one another and how much time has passed since the first began to accelerate, we can algebraically solve the system of equations using substitution: 2t2 = 20t + 400 (4) Solving for t, we can find that the cars are side-by-side after 20 seconds. Substituting 20 for t in either equation 2 or 3, we can find that the cars meet 800 meters ahead of the first car's starting point. Note that a question on an exam may not prompt solutions for both variables. In addition to practical scenarios like the above, nonlinear systems can be alluded to in word problems. For example, a question could ask: The product of two numbers is 12, and the sum of their squares is 40. What are the numbers? In this case, we could make an equation for each known relationship: x y = 12 (5) x2 + y2 = 40 (6) Substitution can be used to prove that the numbers are 2 and 6. Source: nonlinear-systems-of-equations-and-inequalities/models-involving- nonlinear-systems-of-equations/ CC-BY-SA Boundless is an openly licensed educational resource 397 Nonlinear Systems of Inequalities A nonlinear inequality contains two expressions connected by a "greater than" or "less than" signs and involves a nonlinear function. KEY POINTS • A nonlinear system of inequalities may have at least one solution; if it does, a solution may be bounded or unbounded. • A solution for a nonlinear system of inequalities will be in a region that satisfies every inequality in the system. • The best way to show solutions to nonlinear systems of inequalities is graphically, shading the area that satisfies all the system's constituent inequalities. A system of inequalities consists of two expressions connected by a "greater than" (>) or "less than" (<) sign. A nonlinear inequality is an inequality that involves a nonlinear function - a polynomial function of degree 2 or higher, as shown in Figure 7.18. When operating in terms of real numbers, nonlinear inequalities are written in the forms f(x) < b or f(x) ≤ b where f(x) is a nonlinear function and b is a constant real number. Consider, for example, the system including the parabolic nonlinear inequality: y > x2 and the linear inequality: y < x + 2 All points below the line y=x+2 satisfy the linear equality, and all points above the parabola y=x2 satisfy the parabolic nonlinear inequality. Graphing both inequalities reveals one region of overlap: The area where the parabola dips below the line. This area is the solution to the system. The limits of each inequality intersect at (-1, 1) and (2, 4). Note that the area above y=x2 that is also below y=x+2 is closed between those two points. Whereas a solution for a linear system of equations will contain an infinite, unbounded area (lines can only pass one another a maximum of once), in many instances, a solution for a nonlinear system of equations will consist of a finite, bounded area. 398 Figure 7.18 The graph of a polynomial function of degree 3 This need not be the case with all nonlinear inequalities, however: reversing the direction of both inequalities in the previous example would lead to an infinite solution area. The best way to write solutions to systems of inequalities, both linear and nonlinear, is graphically. Each inequality can be drawn as a solid (if < or > is used) or dashed (if ≤ or ≥ is used) line. The region that satisfies all inequalities can be indicated with shading. Source: nonlinear-systems-of-equations-and-inequalities/nonlinear-systems- of-inequalities/ CC-BY-SA Boundless is an openly licensed educational resource 399 Introduction to Sequences Finding the General Term Sums and Series Notation: Sigma Section 1 Sequences and Series 401 Introduction to Sequences In mathematics, a sequence is an ordered list of objects, often numbers and often defined in terms of the previous member of the set. KEY POINTS • The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. • An arithmetic sequence is arrived at by adding a constant to the previous term of a sequence to arrive at the next term. It can be described by the formula an = am + (n − m)d. • A geometric sequence is one in which the previous member of a sequence is multiplied by a constant to arrive at the next term. It can be described by the formula tn = r ⋅ tn−1. In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function. For example, (M, A, R, Y) is a sequence of letters that differs from (A, R, M, Y), as the ordering matters, and (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,...). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence () is included in most notions of sequence, but may be excluded depending on the context (Figure 8. 1). Examples and Notation: Finite and Infinite A more formal definition of a finite sequence with terms in a set S is a function from {1, 2,..., n} to S for some n > 0. An infinite sequence in S is a function from {1, 2,... } to S. For example, the sequence of prime numbers (2,3,5,7,11, …) is the function 1→2, 2→3, 3→5, 4→7, 5→11, …. 402 An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent. It is, however, bounded. Figure 8.1 Sequence A sequence of a finite length n is also called an n-tuple. Finite sequences include the empty sequence () that has no elements. A function from all integers into a set is sometimes called a bi- infinite sequence or two-way infinite sequence. An example is the bi-infinite sequence of all even integers (…, -4, -2, 0, 2, 4, 6, 8…). Next, we will look at arithmetic, quadratic, and geometric sequences. Arithmetic and Geometric Sequences Many of the sequences you will encounter in a mathematics course are produced by a formula, where some operation(s) is performed on the previous member of the sequence (an−1) to give the next member of the sequence (an). An arithmetic (or linear) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term. An example is 10,13,16,19,22,25. In this example, the first term (which we shall call t1 in this case, though a1 would be equally valid) is 10, and the common difference (d)—that is, the difference between any two adjacent numbers—is 3. Another example is 25,22,19,16,13,10. In this example t1=25, and d=(–3). In both of these examples, n (the number of terms) is 6. A geometric sequence is a list where each number is generated by multiplying a constant by the previous number. An example is 2,6,18,54,162. In this example, t1=2, and the common ratio (r)—that is, the ratio between any two adjacent numbers—is 3. Another example is 162,54,18,6,2. In this example t1=162, and r= 1 3 . In both examples n=5. A recursive definition of a sequence means that you define each term based on the previous. So the recursive definition of an arithmetic sequence is tn=tn−1+d, and the recursive definition of a geometric sequence is tn=rtn−1. An explicit definition of an arithmetic sequence means you define the nth term without making reference to the previous term. This is more useful, because it means you can find (for instance) the 20th term without finding all the other terms in between. To find the explicit definition of an arithmetic sequence, you just start writing out the terms. The first term is always t1. The second term goes up by d so it is t1+d. The third term goes up by d again, so it is (t1+d)+d, or in other words, t1+2d. So we get a chart like this: t1=t1, t2=t1+d, t3=t1+2d, t2=t1+3d,... and so on. From this you can see the generalization that tn=t1+(n−1)d, which is the explicit definition we were looking for. 403 The explicit definition of a geometric sequence is arrived at the same way. The first term is t1; the second term is r times that, or t1r; the third term is r times that, or t1r2; and so on. So the general rule is tn = (t1r)n−1 . Read this as: t1 multiplied by r, (n–1) times. Source: combinatorics/sequences-and-series/introduction-to-sequences/ CC-BY-SA Boundless is an openly licensed educational resource Finding the General Term Given terms in a sequence, there are many ways to find the term based on variables that, when replaced with integers, affords valid terms. KEY POINTS • Plotting points on a Cartesian coordinate plane and using graphing software to solve is the easiest and most effective way to find a general term. Given terms x1, x2 . . . xn, convert these to points (1, x1), (2, x2)... (n, xn) and plot. Then find the best-fit curve. • By hand, one can take the differences between each term, then the differences between the differences in terms, etc, until each difference becomes constant. • Once a constant difference is achieved, one can work backwards to find the relationship among terms in each set of the differences. Eventually, one can work backwards to obtain a sum for the initial series. Given several values in a sequence or series, there are a number of ways to determine the general term. The general term is an expression that consists of variables and constants that, when substituting integers in place of x, produces a valid term in a sequence or series. 404 Find Differences Among Terms In some instances, a sequence or series is simple enough to be solved generally by hand. One quick way is to take the differences between each term, then the differences between the differences in terms, etc, until each difference becomes constant. From there, one can work backwards to find the relationship among terms in each set of the differences. Eventually, one can work backwards to obtain a sum for the initial series. For example, consider the sequence: 4, -7, -26, -53, -88, -131 The difference between 4 and -7 is -11; the difference between -7 and -26 is -19. Finding all these differences, we get a new set: -11, -19, -27, -35, -43 This set is still not yet constant. However, finding the differences between terms once more, we get: -8, -8, -8, -8 The formula for the last row of terms (f(n)) as a function of number of the term (n) is a simple constant: f(n) = − 8 We can use this function in relating the previous row of terms (-11, -19, -27, -35, -43). They all differ from one another by -8, although the terms are not all multiples of -8; they are offset by 3 (or 5, depending on your perspective). Thus: f(n) = − 8n − 3 We started with a constant function and then moved to a first-order function. To solve for the general term of the original sequence (4, -7, -26, -53, -88, -131), we must use a second-order relationship. By this point, it may be clear that a sum is needed. A sum has the form: f(n) = n(n + 1) 2 Modifying the above, we can find: f(n) = − 4n(n − 1) − 3n + 7 fits the terms 4, -7, -26, -53, -88, and -131. This simplifies to: f(n) = − 4n2 + n + 7 Using a Best-Fit Model Sometimes it can be difficult to solve for the general term of a sequence or series by hand. The task can be made much easier if one has access to a graphing calculator or spreadsheet program like Excel or Numbers. 405 Given the terms 4, -7, -26, -53, -88, and -131 again, these can be converted into points for a Cartesian plot: (1, 4), (2, -7), (3, -26), (4, -53), (5, -88), (6, -131). If these points are graphed, a program can calculate the formula relating all the terms (Figure 8.2). Source: combinatorics/sequences-and-series/finding-the-general-term/ CC-BY-SA Boundless is an openly licensed educational resource Sums and Series The summation of all the terms of a sequence is called a series, and many formulae are available for easily calculating large series. KEY POINTS • ::A series is merely the sum of the terms of a series. The notation for this operation is to use the capital greek letter sigma, following the general formula: n ∑ i=m xi = xm + xm+1 + xm+2 + . . . + xn−1 + xn. • The sum of an arithmetic series can be calculated using the equation n 2 (t1 + tn). • The sum of a geometric series can be calculated using the equation: S = t1 (rn − 1) r − 1 . Sums and Series Summation is the operation of adding a sequence of numbers; the result is their406 The program Numbers calculated a trendline of y=-4x^2+x+7 to relate points in a series. Figure 8.2 Series points fit with trendline numbers, real numbers, or complex numbers. Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group (or even monoid). For finite sequences of such elements, summation always produces a well-defined sum (possibly by virtue of the convention for empty sums). A series is merely the sum of the terms of a series. The notation for this operation is to use the capital greek letter sigma, following the general formula: n ∑ i=m xi = xm + xm+1 + xm+2 + . . . + xn−1 + xn where i represents the index of summation; xi is an indexed variable representing each successive term in the series; m is the lower bound of summation, and n is the upper bound of summation. The "i = m" under the summation symbol means that the index i starts out equal to m. The index, i, is incremented by 1 for each successive term, stopping when i = n. Arithmetic Series If you add up all the terms of an arithmetic sequence (a sequence in which every entry is the previous entry plus a constant), you have an arithmetic series. For instance: 10 + 13 + 16 + 19 + 22 + 25 = 105. There is a trick that can be used to add up the terms of any arithmetic series. While this trick may not save much time with a 6- item series like the one above, it can be very useful if adding up longer series. The trick is to work from the outside in. Consider the example given above: 10 + 13 + 16 + 19 + 22 + 25. Looking at the first and last terms: 10 + 25 = 35. Going in, to the second and next-to-last terms: 13 + 22 = 35. Finally, the two inside numbers: 16 + 19 = 35. So we can see that the sum of the whole thing is: 3 ⋅ 35. Pause here and check the following things: • You understand the calculation that was done for this particular example. • You understand that this trick will work for any arithmetic series. • You understand that this trick will not work, in general, for series that are not arithmetic. 407 If we apply this trick to the generic arithmetic series, we get a formula that can be used to sum up any arithmetic series. Every arithmetic series can be written as follows: t1 + (t1 + d) + (t1 + 2d) + . . . + (tn − d) + tn If you add the first and last terms, you get t1 + tn. Ditto for the second and next-to-last terms, and so on. How many such pairs will there be in the whole series? Well, there are n terms, so there are n 2 pairs. So the sum for the whole series is n 2 (t1 + tn). Geometric Series If you add up all the terms of a geometric sequence (one in which each entry is the previous entry multiplied by a constant), you have a geometric series. The common example of a trend that follows a geometric series is the number of people infected with a virus, as each person passes it to several more, as seen in Figure 8.3. The arithmetic series trick will not work on such a series; however, there is a different trick we can use. As an example, let's find the sum 2 + 6 + 18 + 54 + 162. We begin by calling the sum of this series S: S = 2 + 6 + 18 + 54 + 162 Now, if you multiply both sides of this equation by 3 (the same constant that each entry is multiplied by to arrive at the next entry), you get the first equation written below. (The second equation below is just copied from above.) 3S = 6 + 18 + 54 + 162 + 486 (confirm this for yourself!) S = 2 + 6 + 18 + 54 + 162 Here comes the key moment in the trick: subtract the two equations. Notice that the first term of the top equation is the same as the second term of the bottom equation, making much of the subtraction very straightforward. This leaves you with: 2S = 486 − 2, so S = 242. Once again, pause to convince yourself that this will work on all geometric series, but only on geometric series. 408 So the total number of people infected follows a geometric series. Figure 8.3 Each person infects two more people with the flu virus Finally—once again—we can apply this trick to the generic geometric series to find a formula. So we begin with t1 + t1r + t1r2 + t1r3 …t1rn−1 and write… rS = t1r + t1r2 + t1r3 + … + t1rn−1 + t1rn (confirm this!) Again, subtracting and soling, we get... rS − S = t1rn − t1 S(r − 1) = t1(rn − 1) S = t1 (rn − 1) r − 1 So there we have it: a general formula for the sum of any finite geometric series, with the first term t1, the common ratio r, and a total of n terms. Source: combinatorics/sequences-and-series/sums-and-series/ CC-BY-SA Boundless is an openly licensed educational resource Notation: Sigma Sigma notation, which uses the uppercase Greek letter sigma, is used to represent summations—a series of numbers to be added together. KEY POINTS • A summation is performed on a series, or list of numbers. Each term is added to the next, resulting in a sum of all terms. • Sigma notation is used to represent the summation of a series. In this form, the capital Greek letter sigma (Σ) is used. The range of terms in the summation is represented in numbers below and above the Σ; the lowest term is written below and the greatest term is written above. • Less common forms of sigma notation often leave out specific components of the notation, such as the upper and lower boundaries for the summation to occur—if these boundaries are implicit in the context of the mathematical problem. Summation is the operation of adding a sequence of numbers, resulting in anumbers, real numbers, or complex numbers. 409 Aside from numbers, other types of values can be summed: vectors, matrices, polynomials, and generally any elements of an additive group (or even monoid). For finite sequences of such elements, summation always produces a well-defined sum (possibly by virtue of the convention for empty sums). A summation is performed on a series. A series is a list of numbers —like a sequence—but instead of listing them, you add them all up. For instance, 4+9+3+2+17. (This particular series adds up to 35.) Sigma Notation One way to compactly represent a series is with "sigma notation," or "summation notation," which looks like this: 7 ∑ n=3 n2 The main symbol seen is the uppercase Greek letter for sigma. It indicates a series. To "unpack" this notation, n=3 represents the number at which to start counting (3), and the 7 represents the point at which you stop. For each term, plug that value of n into the given formula (n2). This particular formula, which we can read as "the sum as n goes from 3 to 7 of n2" means; 32 + 42 + 52 + 62 + 72 More generally, sigma notation can be defined as: n ∑ i=m xi = xm + xm+1 + xm+2 + . . . + xn−1 + xn Where, i represents the index of summation; xi is an indexed variable representing each successive term in the series; m is the lower bound of summation, and n is the upper bound of summation. The "i = m" under the summation symbol means that the index i starts out equal to m. The index, i, is incremented by 1 for each successive term, stopping when i = n. Other Forms of Sigma Notation Informal writing sometimes omits the definition of the index and bounds of summation when these are clear from context, as in: ∑ xi 2 = n ∑ i=1 xi 2 One often sees generalizations of this notation in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. For example: ∑ 0≤k≤100 f(k) is the sum of f(k) over all (integer) k in the specified range. 410 ∑ xϵS f(x) is the sum of f(x) over all elements x in the set S. ∑ d\|n μ(d) is the sum of µ(d) over all positive integers d dividing n. Source: combinatorics/sequences-and-series/notation-sigma/ CC-BY-SA Boundless is an openly licensed educational resource Recursive Definitions A recursive definition of a function defines its values for some inputs in terms of the values of the same function for other inputs. KEY POINTS • In mathematical logic and computer science, a recursive definition, or inductive definition, is used to define an object in terms of itself. • Most recursive definitions have three foundations: a base case (basis), an inductive clause, and an extremal clause. Often only the inductive clause is discussed, which is the general case, and then as an example one will be asked to demonstrate the inductive clause given a base case or basis. • The recursive definition for an arithmetic sequence is: an = an−1 + d. The recursive definition for a geometric sequence is: an = ran−1. Again, plugging in an initial value (n1 =1 for example} will allow the nth term of sequence to be calculated, if r or d is known. In mathematical logic and computer science, a recursive definition, or inductive definition, is used to define an object in terms of itself. A recursive definition of a function defines values of the functions for some inputs in terms of the values of the same function for other inputs. For example, the factorial function n! is defined by the rules: 411 Formulas relating to summations that you may later encounter in math classes, presented here as a reference. Figure 8.4 Summations Involving Exponential Terms 0! = 1 (n + 1)! = (n + 1)n! This definition is valid because, for all n, the recursion eventually reaches the base case of 0. Thus the definition is well-founded. The definition may also be thought of as giving a procedure describing how to construct the function n!, starting from n = 0 and proceeding onward with n = 1, n = 2, n = 3 etc. An inductive definition of a set describes the elements in that set in terms of other elements in another set. For example, one definition of the set N of natural numbers is: • 1 is in N • If an element n is in N then n+1 is in N. • N is the smallest set satisfying these two rules. There are many sets that satisfy (1) and (2) - for example, the set {1, 1.649, 2, 2.649, 3, 3.649,...} satisfies the definition. However, condition (3) specifies the set of natural numbers by removing the sets with extraneous members. Form of Recursive Definitions Most recursive definition have three foundations: a base case (basis), an inductive clause, and an extremal clause. The difference between a circular definition and a recursive definition is that a recursive definition must always have base cases, cases that satisfy the definition without being defined in terms of the definition itself, and all other cases comprising the definition must be "smaller" (closer to those base cases that terminate the recursion) in some sense. In contrast, a circular definition may have no base case, and define the value of a function in terms of that value itself, rather than on other values of the function. Such a situation would lead to an infinite regress. Examples of Recursive Definitions All of the elementary functions (addition, multiplication, and exponentiation) can be defined recursively. For addition: 0 + a = a (1 + n) + a = 1 + (n + a) Multiplication: 0a = 0 (1 + n)a = a + na Exponentiation can also be defined recursively: a0 = 1 412 a1+n = aan Prime Numbers The set of prime numbers can be defined as the unique set of positive integers satisfying: • 1 is not a prime number. • any other positive integer is a prime number if and only if it is not divisible by any prime number smaller than itself. The primality of the integer 1 is the base case. Checking the primality of any larger integer X by this definition requires knowing the primality of every integer between 1 and X, which is well defined by this definition. That last point can be proved by induction on X, for which it is essential that the second clause says "if and only if". If the clause had said just "if" the primality of, for instance, 4 would not be clear, and the further application of the second clause would be impossible. Recursive Formulae for Sequences When discussing arithmetic and quadratic sequences, one may have noticed that the difference between two consecutive terms in the sequence could be written in a general way: an = an−1 + d The above equation is an example of a recursive equation since the nth-term can only be calculated by considering the previous term in the sequence. Compare this with the equation: an = a1 + d(n − 1) One can directly calculate the nth-term of an arithmetic sequence without knowing previous terms. For quadratic sequences, the difference between consecutive terms is given by equation: an − an−1 = D(n − 2) + d Which can be rewritten as: an = an−1 + D(n − 2) + d This is then a recursive equation for a quadratic sequence with common second difference, D. A geometric sequence follows the formula an an−1 = r. This can also be stated as a sequence in which every number is equal to the previous number times a constant. For example, the spread of the flu virus often follows a geometric sequence (at least in theory), wherein each person infected will infect two more, such that the terms follow a geometric sequence. Visually, it looks like this: (Figure 8.5). 413 Using this equation, the recursive equation for a geometric sequence is: an = r ⋅ an−1 Recursive equations are extremely powerful: one can work out every term in the series just by knowing previous terms. As can be seen from the examples above, working out and using the previous term an−1 can be a much simpler computation than working out an from scratch using a general formula. This means that using a recursive formula when using a computer to work out a sequence would mean the computer would finish its calculations significantly quicker. Source: combinatorics/sequences-and-series/recursive-definitions/ CC-BY-SA Boundless is an openly licensed educational resource 414 Each person infects two more people with the flu virus, making the number of recently-infected people the nth term in a geometric sequence. Figure 8.5 The flue virus is a geometric sequence Arithmetic Sequences An arithmetic sequence is a sequence of numbers wherein the difference between the consecutive terms is constant. KEY POINTS • The behavior of the arithmetic sequence depends on the common difference d. • Arithmetic sequences can be finite or infinite. • Finite arithmetic sequences can be summed to make an arithmetic series. An arithmetic progression, or arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, … is an arithmetic sequence with common difference of 2. • a1 The first term of the sequence • d The common difference of successive terms • an The nth term of the sequence The behavior of the arithmetic sequence depends on the common difference d. If the common difference, d, is: • Positive, the sequence will progress towards infinity. • Negative, the sequence will regress towards negative infinity. The following equation gives an an = a1 + (n − 1) * d 416 Arithmetic sequences increase each number by a constant amount, similar to a straight line. Figure 8.6 Arithmetic Sequence Of course, one can always write out each term until getting the answer sought, but if the 50th term is needed doing so can be cumbersome. An infinite arithmetic series is exactly what it sounds like: an infinite series whose terms are in an arithmetic sequence. Examples are 1 + 1 + 1 + 1 + · · · and 1 + 2 + 3 + 4 + · · ·. The general form for an infinite arithmetic series is: Σ(an + b) where n can range from 0 to infinity. If a = b = 0, then the sum of the series is 0. If either a or b is nonzero, then the series has no sum. Even if one is dealing with an infinite sequence, the sum of that sequence can still be found up to any nth term with the same equation used in a finite arithmetic sequence. Source: combinatorics/arithmetic-sequences-and-series/arithmetic-sequences/ CC-BY-SA Boundless is an openly licensed educational resource Summing Terms in an Arithmetic Sequence An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. KEY POINTS • The sum of the members of a finite arithmetic progression is called an arithmetic series. • The equation Sn = (n /2)[2a1 + (n − 1)d] can be used to find the sum of any arithmetic sequence up to the nth term. • Some arithmetic sequences are infinite, and their general form is: Σ(an + b). While these sequences are infinite, you can still apply the summation equation to find their sum up to a specific nth term. An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, … is an arithmetic progression with common difference of 2. Finite Summation The sum of the members of a finite arithmetic progression is called an arithmetic series. 417 An arithmetic series can be expressed in two different ways: Sn = a1 + (a1 + d) + (a1 + 2d) + . . . + (a1 + (n − 2)d) + (a1 + (n − 1)d) Sn = (an − (n − 1)d) + (an − (n − 2)d) + . . . + (an − 2d) + (an − d) + an Adding both sides of the two equations, all terms involving d cancel: 2Sn = n(a1 + an) Dividing both sides by 2 produces a common form of the equation: Sn = (n /2)(a1 + an) An alternate form results from reinserting the substitution an = a1 + (n − 1)d: Sn = (n /2)[2a1 + (n − 1)d] For example, lets say you have an arithmetic progression of 3, 8, 13, 18, 23,..., and you want to know the sum up to the 50th term. Before you can start, you first need to know what information is relevant. In this case you will need: • The number terms you want to sum (n) • The first term of the sequence (a1) • and the difference in consecutive terms (d) From the question, we know that n = 50. We can easily see what the first term is, a1 = 3. And because we know simple math, we can quickly figure out that d = 5. So now, lets apply the equation: S50 = (50/2) * [2(3) + (49)5] = 6275. That is a lot easier than actually writing out each term and adding them all together. 418 Arithmetic sequences increase in value by a set number. Figure 8.7 Arithmetic Sequence Infinite Summation An infinite arithmetic series is exactly what it sounds like. It is an infinite series whose terms are in an arithmetic progression. Examples are 1 + 1 + 1 + 1 + · · · and 1 + 2 + 3 + 4 + · · ·. The general form for an infinite arithmetic series is ∑ (an + b) where n ranges from 0 to infinity. If a = b = 0, then the sum of the series is 0. If either a or b is nonzero, then the series diverges, which means it has no sum. Source: combinatorics/arithmetic-sequences-and-series/summing-terms-in- an-arithmetic-sequence/ CC-BY-SA Boundless is an openly licensed educational resource Applications and Problem Solving Arithmetic series can simplify otherwise complex addition problems by decreasing the number of terms to be added. KEY POINTS • Arithmetic sequences can sometimes delineate very long lists of numbers that are not practical to add by hand. • Because the sums of first and last terms, the second and the second-to-last terms, the third and the third-to-last-terms, etc., are all identical, summations of arithmetic sequences can be simplified. • By simplifying the math for an arithmetic series, one can calculate the sums of large strings of numbers. Using equations for arithmetic sequence summation can greatly facilitate the speed of problem solving. For example, let's say we wanted to write the series of all the even numbers between 50 and 100. This can be written as: 26 ∑ n=1 48 + 2n 419 This can be written in a few different ways, but they all have one thing in common: adding them up by hand would be difficult. There is a quicker solution. Consider the series: 3+5+7+9+11+13+15+17. Adding the first term to the last term, 3 + 17 = 20. Adding the second term to the second-to-last term also amounts to a sum of 20. Adding to (Figure 8.9), we can see that the third term and third-to-last terms have a similar effect (Figure 8.8). There are eight terms in 3+5+7+9+11+13+15+17, and they add to four 20's, or 80. This trick applies to all arithmetic series. The reason that the sum of the second pair equaled that of the first pair was that we went up by two on the left, and down by two on the right. As long as you go up by the same amount as you go down, the sum will stay the same —and this is just what happens for arithmetic series. Remember, that this can be generalized as: Sn = n 2 ⋅ [2a1 + (n − 1) ⋅ d] To apply this to the first summation of all the even numbers between 50 and 100, we would want to add until the 50th term: S50 = 50 2 ⋅ [2(50) + (50 − 1)(2)] = 4950 Thus, arithmetic series can simplify otherwise complex addition problems! Source: combinatorics/arithmetic-sequences-and-series/applications-and- problem-solving--5/ CC-BY-SA Boundless is an openly licensed educational resource 420 First we should add the first and last terms. Figure 8.9 Adding a Series Adding the first and the last, second term and second to last, etc. all yield the same answer. Figure 8.8 Adding a Series Part 2 Geometric Sequences Summing the First n Terms in an Geometric Sequence Infinite Geometric Series Applications and Problem Solving Section 3 Geometric Sequences and Series 421 Geometric Sequences A geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by the common ratio. KEY POINTS • The sum of the terms of a geometric progression, or of an initial segment of a geometric progression, is known as a geometric series. • The general form of a geometric sequence is: a, ar, ar2 , ar3 , ar4 , … • The n-th term of a geometric sequence with initial value a and common ratio r is given by: an = arn−1 . A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio Figure 8.10. For example, the sequence 2, 6, 18, 54,... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25,... is a geometric sequence with common ratio 1/2. The sum of the terms of a geometric progression, or of an initial segment of a geometric progression, is known as a geometric series. Thus, the general form of a geometric sequence is: a, ar, ar2 , ar3 , ar4 , … and that of a geometric series is: a + ar + ar2 + ar3 + ar4 + … where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start value. The n-th term of a geometric sequence with initial value a and common ratio r is given by an = arn−1 422 Diagram illustrating three basic geometric sequences of the pattern 1(rn-1) up to 6 iterations deep. The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch:,, and, respectively. Figure 8.10 Geometric Sequence Such a geometric sequence also follows the recursive relation an = arn−1 for every integer n ≥ 1 Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio. The common ratio of a geometric series may be negative, resulting in an alternating sequence, with numbers switching from positive to negative and back. For instance: 1, −3, 9, −27, 81, −243,... is a geometric sequence with common ratio −3. The behavior of a geometric sequence depends on the value of the common ratio. If the common ratio is: • Positive, the terms will all be the same sign as the initial term. • Negative, the terms will alternate between positive and negative. • Greater than 1, there will be exponential growth towards positive infinity. • 1, the progression is a constant sequence. • Between −1 and 1 but not zero, there will be exponential decay towards zero. • −1, the progression is an alternating sequence (see alternating series) • Less than −1, for the absolute values there is exponential growth towards positive and negative infinity (due to the alternating sign). Geometric sequences (with common ratio not equal to −1,1 or 0) show exponential growth or exponential decay, as opposed to the Linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, … (with common difference 11). This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population. Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression. An interesting result of the definition of a geometric progression is that for any value of the common ratio, any three consecutive terms a, b and c will satisfy the following equation: b2 = ac A geometric series is the sum of the numbers in a geometric progression: 423 n ∑ k=0 ark = ar0 + ar1 + ar2 + ar3 + ⋯ + arn We can find a simpler formula for this sum by multiplying both sides of the above equation by 1 − r, and we'll see that (1 − r) n ∑ k=0 ark = (1 − r)(ar0 + ar1 + ar2 + ar3 + ⋯ + arn ) = ar0 + ar1 + ar2 + ar3 + ⋯ + arn = − ar1 − ar2 − ar3 − ⋯ − arn − arn+1 = a − arn+1 since all the other terms cancel. If r ≠ 1, we can rearrange the above to get the convenient formula for a geometric series: n ∑ k=0 ark = a(1 − rn+1 ) 1 − r Source: combinatorics/geometric-sequences-and-series/geometric-sequences/ CC-BY-SA Boundless is an openly licensed educational resource Summing the First n Terms in an Geometric Sequence By utilizing the common ratio and the first term of the sequence, we can sum the first n terms: s = a 1 − rn 1 − r . KEY POINTS • The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. • The behavior of the terms depends on the common ratio r. • The sum of a geometric series is finite as long as the terms approach zero; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series being infinite. The sum can be computed using the self-similarity of the series. Geometric series are one of the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role in the early development of calculus, and they continue to be central in the study of convergence of series. Geometric series are used throughout mathematics, and they have important applications in 424 physics, engineering, biology, economics, computer science, queueing theory, and finance. The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. The following table shows several geometric series with different common ratios (Figure 8.11). The behavior of the terms depends on the common ratio r: • If r is between −1 and +1, the terms of the series become smaller and smaller, approaching zero in the limit, and the series converges to a sum. In the case above, where r is one half, the series has a sum of one. • If r is greater than one or less than minus one, the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, and the series has no sum. (The series diverges.) • If r is equal to one, all of the terms of the series are the same. The series diverges. If r is minus one, the terms take two values alternately (e.g., 2, −2, 2, −2, 2,...). The sum of the terms oscillates between two values (e.g. 2, 0, 2, 0, 2,...). This is a different type of divergence and again the series has no sum. The sum of a geometric series is finite as long as the terms approach zero; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series being infinite. The sum can be computed using the self-similarity of the series. For example: s = 1 + 2 3 + 4 9 + 8 27 + . . . This series has common ratio 2/3. If we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on: 2 3 s = 2 3 + 4 9 + 8 27 + 16 81 + . . . This new series is the same as the original, except that the first term is missing. Subtracting the new series (2/3)s from the original series, s cancels every term in the original but the first: 425 Geometric series with different common ratios Figure 8.11 Geometric Series s − 2 3 s = 1, so s = 3 A similar technique can be used to evaluate any self-similar expression. For r ≠ 1, the sum of the first n terms of a geometric series is: a + ar + ar2 + ar3 + ⋯ + arn−1 = n−1 ∑ k=0 ark = a 1 − rn 1 − r , Where a is the first term of the series, and r is the common ratio. We can derive this formula as follows: Let s = a + ar + ar2 + ar3 + ⋯ + arn−1 Then rs = ar + ar2 + ar3 + ar4 ⋯ + arn Then s − rs = a − arn Then s(1 − r) = a(1 − rn ) So s = a 1 − rn 1 − r Therefore, by utilizing the common ratio and the first term of the sequence, we can sum the first n terms of a sequence. Source: combinatorics/geometric-sequences-and-series/summing-the-first-n- terms-in-an-geometric-sequence/ CC-BY-SA Boundless is an openly licensed educational resource 426 Infinite Geometric Series Geometric series are one of the simplest examples of infinite series with finite sums. KEY POINTS • The sum of a geometric series is finite as long as the terms approach zero; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series being infinite. • Historically, geometric series played an important role in the early development of calculus, and they continue to be central in the study of convergence of series. • The general form of an infinite geometric series is: ∞ ∑ n=0 zn . Geometric series, or infinite series whose terms are in a geometric progression, are one of the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important and indispensable role in the early development of calculus, and even today they continue to be central in the study of convergence of series. Geometric series are used throughout mathematics, but they also have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance. A geometric series is a series with a constant ratio between successive terms, as seen in the Figure 8.12. For example, the following series: 1 2 + 1 4 + 1 8 + 1 16 + ⋯ = ∞ ∑ n=0 1 2n is geometric, because each successive term can be obtained by multiplying the previous term by 1 / 2. The general form of a geometric series is ∞ ∑ n=0 zn . The sum of a geometric series is finite as long as the terms approach zero; as the numbers near zero, they become insignificantly small, 427 Each of the purple squares is obtained by multiplying the area of the next larger square by 1/4 (1/2×1/2=1/4, 1/4×1/4=1/16). The sum of the areas of the purple squares is one third of the area of the large square. Figure 8.12 Geometric Series allowing a sum to be calculated despite the series being infinite. The sum can be computed using the self-similarity of the series. An infinite geometric series is an infinite series whose successive terms have a common ratio. Such a series converges if and only if the absolute value of the common ratio is less than one (\| r \| < 1). Source: combinatorics/geometric-sequences-and-series/infinite-geometric- series/ CC-BY-SA Boundless is an openly licensed educational resource Applications and Problem Solving Geometric series have applications in math and science, and are one of the simplest examples of infinite series with finite sums. KEY POINTS • A repeating decimal can be viewed as a geometric series whose common ratio is a power of 1/10. • Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. • The interior of the Koch snowflake is a union of infinitely many triangles. In the study of fractals, geometric series often arise as the perimeter, area or volume of a self-similar figure. Geometric series played an important role in the early development of calculus, and continue as a central part of the study of convergence of series. Geometric series are used throughout mathematics; they have important applications in physics, engineering, biology, economics, computer science, queueing theory and finance. 428 Geometric series are one of the simplest examples of infinite series with finite sums, although not all of them have this property. Repeating Decimal A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example: 0.7777... = 7 10 + 7 100 + 7 1000 + 7 10000 + ⋯ The formula for the sum of a geometric series can be used to convert the decimal to a fraction: 0.7777... = a 1 − r = 7/10 1 − 1/10 = 7 9 The formula works for any repeating term. For example: 0.123412341234... = a 1 − r = 1234/1000 1 − 1/1000 = 1234 9999 0.0909090909 = .09 99 = 1 11 0.143814381438 = .1438 9999 0.9999 = 9 9 = 1 That is, a repeating decimal with repeat length n is equal to the quotient of the repeating part (as an integer) and 10n - 1. Archimedes' Quadrature of the Parabola Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line (Figure 8.13). His method was to dissect the area into an infinite number of triangles. Archimedes' Theorem states that the total area under the parabola is 4/3 of the area of the blue triangle. He determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the area of a green triangle, and so forth. Assuming that the blue triangle has area 1, the total area is an infinite series: 1 + 2 ( 1 8 ) + 4 ( 1 8 ) 2 + 8 ( 1 8 ) 3 + ⋯ 429 Archimedes' dissection of a parabolic segment into infinitely many triangles. Figure 8.13 Archimedes Theorem The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives: 1 + 1 4 + 1 16 + 1 64 + ⋯ This is a geometric series with common ratio 1/4 and the fractional part is equal to 1/3. Fractal Geometry The interior of the Koch snowflake is comprised of an infinite amount of triangles, as shown in Figure 8.14. In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar figure. For example, the area inside the Koch snowflake can be described as the union of many infinitely equilateral triangles. Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and therefore has exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the blue triangle as a unit of area, the total area of the snowflake is: 1 + 3 ( 1 9 ) + 12 ( 1 9 ) 2 + 48 ( 1 9 ) 3 + ⋯ The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this series is geometric with constant ratio r = 4/9. The first term of the geometric series is a = 3(1/9) = 1/3, so the sum is 1 + a 1 − r = 1 + 1 3 1 − 4 9 = 8 5 Thus the Koch snowflake has 8/5 of the area of the base triangle. 430 The interior of a Koch snowflake is comprised of an infinite amount of triangles. Figure 8.14 Koch Snowflake Zeno's Paradoxes Zeno's Paradoxes reveal that the convergence of a geometric series means that a sum involving an infinite number of terms can be finite. Simply stated, Zeno's paradox says: There is a point, A, that wants to move to another point, B. If A only moves half of the distance between it and point B at a time, it will never get there, because you can continue to divide the remaining space in half for ever. However, we know this is not true, otherwise we would never reach our destinations. Economics In economics, geometric series are used to represent the present value of an annuity (a sum of money to be paid in regular intervals). Geometric Power Series Assume this formula for a geometric series: 1 1 − x = 1 + x + x2 + x3 + x4 + ⋯ This can be interpreted as a power series in the Taylor's theorem sense, converging where x < 1. One can thus extrapolate to obtain other power series. Source: combinatorics/geometric-sequences-and-series/applications-and- problem-solving/ CC-BY-SA Boundless is an openly licensed educational resource 431 Sequences of Statements Sequences of statements are logical, ordered groups of statements that are important for mathematical induction. KEY POINTS • A sequence is an ordered list of objects or events. Like a set, it contains members, but unlike a set, the order of the members matters. • A sequence of statements refers to the progression of logical implications of one statement. • Sequences of statements are important for mathematical inductions, which rely on infinite sequences of statements. In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called "elements," or "terms"). The number of ordered elements (possibly infinite) is called the "length of the sequence." Unlike a set, order matters in sequences and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function. For example, (M, A, R, Y) is a sequence of letters that differs from (A, R, M, Y), although the composition is the same; the ordering differs. Also (1, 1, 2, 3, 5, 8) is a valid sequence despite including a repeating term (1). Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,...). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence () is included in most notions of sequence, but may be excluded depending on the context. A sequence of statements refers to the progression of logical implications of one statement. In algebra, a "statement" usually refers to an equation that contains an equal sign. Sequences of statements are necessary for mathematical induction. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers (positive integers). It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one. 433 Part of an infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent. It is, however, bounded within the two dashed lines. Figure 8.15 Sequence Source: combinatorics/mathematical-inductions/sequences-of-statements/ CC-BY-SA Boundless is an openly licensed educational resource Proving Infinite Sequences of Statements Proving an infinite sequence of statements is necessary for proof by induction, a rigorous form of deductive reasoning. KEY POINTS • Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers (positive integers). • It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one. • Proving an infinite sequences of statements can be understood in the context of the domino effect, which by nature mediates a sequential and predictable order of events. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers (positive integers). It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one. 434 Mathematical induction should not be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics (see problem of induction for more information). In fact, mathematical induction is a form of rigorous deductive reasoning. The simplest and most common form of mathematical induction proves that a statement involving a natural number n holds for all values of n. The proof consists of two steps: 1. The basis (base case): showing that the statement holds when n is equal to the lowest value that n is given in the question. Usually, n = 0 or n = 1. 2. The inductive step: showing that if the statement holds for some n, then the statement also holds when n + 1 is substituted for n. The assumption in the inductive step that the statement holds for some n, is called the induction hypothesis (or inductive hypothesis). To perform the inductive step, one assumes the induction hypothesis and then uses this assumption to prove the statement for n + 1. The choice between n = 0 and n = 1 in the base case is specific to the context of the proof: If 0 is considered a natural number, as is common in the fields of combinatorics and mathematical logic, then n = 0. If, on the other hand, 1 is taken as the first natural number, then the base case is given by n = 1. This method works by first proving the statement is true for a starting value, and then proving that the process used to go from one value to the next is valid. If these are both proven, then any value can be obtained by performing the process repeatedly. It may be helpful to think of the domino effect. If one is presented with a long row of dominoes standing on end, one can be sure that: 1. The first domino will fall 2. Whenever a domino falls, its next neighbor will also fall 435 Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. Figure 8.16 Dominoes So it is concluded that all of the dominoes will fall, and that this fact is inevitable. Source: combinatorics/mathematical-inductions/proving-infinite-sequences- of-statements/ CC-BY-SA Boundless is an openly licensed educational resource 436 Permutations A permutation of a set of objects is an arrangement of those objects into a particular order; the numbers of permutations can be counted. KEY POINTS • Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1), or with colored balls. • The number of permutations of n distinct objects is n×(n − 1)×(n − 2)×⋯×2×1, which number is called "n factorial" and written "n!". • When deciding permutations of a subset from a larger set, it is often useful to divide one factorial by another to determine the number of permutations possible. For example, the first six cards from a deck of cards would have 52! 46! permutations possible, or about 14.7 billion. In the game of "Solitaire", also known as "Patience" or "Klondike", seven cards are dealt out at the beginning, as shown in Figure 8.17: one face-up, and the other six face-down. A complete card deck has 52 cards. Assuming that the only card that is seen is the 7 of spades, how many possible "hands" (the other six cards) could be showing underneath? What makes this a "permutations" problem is that the order matters: if an ace is hiding somewhere in those six cards, it makes a difference whether the ace is on the first position, the second, etc. Permutations problems can always be addressed as an example of the multiplication rule, with one small twist. Question: How many cards might be in the first position, directly under the showing 7? Answer: 51. That card could be anything except the 7 of spades. Question: For any given card in first position, how many cards might be in second position? Answer: 50. The seven of spades, and the next card, are both "spoken for." So there are 50 possibilities left in this position. Question: So how many possibilities are there for the first two positions combined? 438 To find out how many possible combinations of cards there are below the seven, we use the concept of permutations to calculate the possible arrangements of cards. Figure 8.17 One stack of cards in a game of solitaire Answer: 51×50, or 2,550. Question: So how many possibilities are there for all six positions? Answer: 51×50×49×48×47×46, or approximately 1.3×1010; about 10 billion possibilities! This result can be expressed, and typed into a calculator, more concisely by using factorials. A "factorial", written with an exclamation mark, means "multiply all the numbers from 1 up to this number." So 5! means 1×2×3×4×5=120. 7! 5! can also be written as, 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 6 ⋅ 7 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 . Most of the terms cancel, leaving only 6×7=42. In another example, 51! 45! , if all of the terms are written out, the first 45 terms cancel, leaving 46×47×48×49×50×51, which is the number of permutations wanted. Therefore, instead of typing into a calculator six numbers to multiply, or sixty numbers or six hundred depending on the problem, the answer to a permutation problem can be found by dividing two factorials. In many calculators, the factorial option is located under the "probability" menu for this reason. General Considerations In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting (rearranging) objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1), or with colored balls, as in Figure 8.18. One might define an anagram of a word as a permutation of its letters. The study of permutations in this sense generally belongs to the field of combinatorics. The number of permutations of n distinct objects is n×(n − 1)×(n − 2)×⋯×2×1, which number is called "n factorial" and written "n!". Permutations occur, in more or less prominent ways, in almost every domain of mathematics. They often arise when different orderings on certain finite sets are considered, possibly only because one wants to ignore such orderings and needs to know how many configurations are thus identified. For similar reasons 439 If one has three different colored balls, there are six distinct ways to organize them into a different order, as shown. Figure 8.18 The 6 permutation of 3 balls permutations arise in the study of sorting algorithms in computer science. Source: combinatorics/combinatorics/permutations/ CC-BY-SA Boundless is an openly licensed educational resource Permutations: Notation; n Objects Taken k at a Time A permutation is an arrangement of objects in a specific order; it is one of many possible ways to permute the set of objects. KEY POINTS • If all objects in consideration are distinct, they can be arranged in n! permutations, where n represents the number of objects. • If not all the objects in a set of n unique elements are chosen, the above formula can be modified to: where k represents the number of selected elements. • When solving for quotients of factorials, the terms of the denominator can cancel with the terms of the numerator, thus eliminating perhaps the majority of terms to be multiplied. To permute objects is to rearrange them. A permutation is an arrangement of objects in a specific order; it is one of many possible ways to permute the set of objects. 440 For example, there are six possible permutations of the numbers 1, 2, and 3. These are: (1, 2, 3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). If all objects in consideration are distinct, they can be arranged in n! permutations, where n represents the number of objects. The quantity n! is to be read "n factorial," meaning it equals the product: n(n − 1) ⋅ (n − 2) ⋅ (n − 3) ⋅ . . .2 ⋅ 1 If not all the objects in a set of n unique elements are chosen, the above formula can be modified for a selection of k elements: n! (n − k)! Knowing the formula for calculating permutations can help solve quandaries that would be otherwise near-impossible to determine. While it's easy enough to count the six possible permutations of three differently-colored balls, Figure 8.18 counting the possible seven-card hands in a game of cards is an altogether different level of difficulty. Plugging in 52 (the number of unique cards in a deck) for n and k (the size of the hand) for k, we can find: 52! (52 − 7)! 52! 48! 6497400 Thus, there are 6497400 possible seven-card hands that can be drawn from a deck of cards. The above can be solved using a calculator, or by hand. Because 52! and 48! both contain terms 4847464544...etc, those terms will cancel. Ultimately, the equation can be simplified to 525150*49. Source: combinatorics/combinatorics/permutations-notation-n-objects-taken- k-at-a-time/ CC-BY-SA Boundless is an openly licensed educational resource 441 Permutations of Nondistinguishable Objects The expression revealing number of permutations of distinct items can be modified if not all items in a set are distinct. KEY POINTS • Some sets include repetitions of certain elements. In these cases, the number of possible permutations of the items cannot be expressed by n!, where n represents the number of elements, because this calculation would include a multiplicity of possible states. • To correct for the "multiplicity" of certain permutations, divide the factorial of the total number of elements by the product of the factorials of the number of each repeated element. • The expression for number of permutations with repeated elements is: n! n1!n2!n3! . . . where n is the total number of terms in a sequence n1, n2, and n3 are the number of repetitions of different elements. The number of possible permutations of a set of n distinct elements n!, or n(n-1)(n-2)...(2)(1). This can be easily tested. The number 1 can be arranged in just one (1!) way. The numbers 1 and 2 can be arranged in two (2!) ways: (1, 2) and (2, 1). The numbers 1, 2, and 3 can be arranged in six (3!) ways: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). This rule holds true for sets of any size, so long as the elements are all distinct. But what if some elements are repeated? Repetition of some elements complicates the calculation of permutations, because it allows for there to be multiple ways in which a specific order of elements can be arranged. For example, given the numbers 1, 3, and 3 in a set, there are two ways to obtain the order (3, 1, 3). To correct for the "multiplicity" of certain permutations, we must divide the factorial of the total number of elements by the product of the factorials of the number of each repeated element. This can generally be represented as: 442 Imagine replacing each red ball with a green ball. In such a case, three of the six permutations would be repeated. Thus, there would only be three true, distinct permutations. Figure 8.19 The 6 permutation of 3 balls n! n1!n2!n3! . . . Where n is the total number of terms in a sequence n1, n2, and n3 are the number of repetitions of different elements. Consider the set of numbers: (15, 17, 24, 24, 28) There are five terms, so n=5. However, two (the number 24) are the same. Thus, the number of possible distinct permutations in the set is: 5! 2! = 60 The same logic can apply to more complicated systems. Consider the set: (0, 0, 0, 2, 4, 4, 7, 7, 7, 7, 7, 8, 8). In total, there are 13 elements. These include many repetitions: 0 is seen three times, 4 and 8 each are observed twice, and there are five instances of the number 7. Thus, the number of possible distinct permutations can be calculated by: 13! 2! ⋅ 2! ⋅ 3! ⋅ 5! = 2162160 Source: combinatorics/combinatorics/permutations-of-nondistinguishable- objects/ CC-BY-SA Boundless is an openly licensed educational resource 443 Combinations A combination is a way of selecting several things out of a larger group, where (unlike permutations) order does not matter. KEY POINTS • A combination is a mathematical concept where one counts the number of ways one can select several elements out of a larger group. • Unlike a permutation, when determining the number of combinations, order does not matter. • Formally, a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k- combinations is equal to the binomial coefficient: ( n k) = n(n − 1) . . . (n − k + 1) k(k − 1) . . .1 , which can be written using factorials as n! k!(n − k)! whenever k ≤ n and which is zero when k>n. Using an example with a deck of 52 cards, deal out a poker hand (5 cards). How many possible poker hands are there? At first glance, this seems like a minor variation on the Solitaire question (how many ways can be make a stack of six cards, a permutation question). The only real difference is that there are five cards instead of six. But in face, there is a more important difference: order does not matter. One does not want to count "Ace- King-Queen-Jack-Ten of spades" and "Ten-Jack-Queen-King-Ace of spades" separately. They are the same poker hand. To approach such question, begin with the permutations question: how many possible poker hands are there, if order does matter? 52×51×50×49×48, or 52! 47! . One has to count every possible hand many different times in this calculation. How many times? The key insight is that this second question—"How many different times is one counting, for instance, Ace-King-Queen-Jack-Ten of spades?"—is itself a permutations question. It is the same as the question "How many different ways can these five cards be rearranged in a hand?" There are five possibilities for the first card, or each of these, four for the second, and so on. The answer is 5, which is 120. So, since every possible hand has been counted 120 times, divide our earlier result by 120 to find that there are 52! (47!)(5!) , or about 2.6 Million possibilities. The question—"how many different 5-card hands can be made from 52 cards?"—turns out to have a surprisingly large number of applications. Consider the following questions: 444 • A school offers 50 classes. Each student must choose 6 of them to fill out a schedule. How many possible schedules can be made? • A basketball team has 12 players, but only 5 will start. How many possible starting teams can they field? • Your computer contains 300 videos, but you can only fit 10 of them on your iPod. How many possible ways can you load your iPod? Each of these is a combinations question, and can be answered exactly like the card scenario. Since this type of question comes up in so many different contexts, it is given a special name and symbol. The last question would be referred to as "300 choose 10" and written ( 300 10 ) . It is calculated as 300! (290!)(10!) , for reasons explained above. General considerations In mathematics, a combination is a way of selecting several things out of a larger group, where (unlike permutations) order does not matter. In smaller cases, it is possible to count the number of combinations. For example given three fruit, say an apple, orange and pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient (Figure 8. 20). ( n k) = n(n − 1) . . . (n − k + 1) k(k − 1) . . .1 Which can be written using factorials as n! k!(n − k)! whenever k ≤ n, and which is zero when k > n. The set of all k-combinations of a set S is sometimes denoted by ( S k) . Combinations can refer to the combination of n things taken k at a time without or with repetitions. In the above example, repetitions were not allowed. If, however, it was possible to have two of any one kind of fruit there would be 3 more combinations: one with two apples, one with two oranges, and one with two pears. Number of k-combinations The number of k-combinations from a given set S of n elements is often denoted in elementary combinatorics texts by C(n, k), or by a variation such as Cn k , nCk, n Ck, or even Ck n (the latter form is standard in French, Russian, and Polish texts). The same number, however, occurs in many other mathematical contexts, where it is denoted by 445 ( n k) (often read as "n choose k"). Notably, it occurs as coefficient in the binomial formula, hence its name, binomial coefficient. One can define ( n k) for all natural numbers k at once by the relation (1 + X)n = Σk≥0 ( n k) Xk From which it is clear that ( n 0) = ( n n) = 1 and ( n k) = 0 for k > n. Source: combinatorics/combinatorics/combinations/ CC-BY-SA Boundless is an openly licensed educational resource 446 The binomial coefficient can be arranged to form Pascal's trangle. The first row is the terms for (x+y)^0, the second for (x +y)^1, the third for (x=y)^2, which is 1x^2+2xy+y^2. Figure 8.20 Pascal's triangle Binomial Expansions and Pascal's Triangle Binomial Expansion and Factorial Notation Finding a Specific Term Total Number of Subsets Section 6 The Binomial Theorem 447 Binomial Expansions and Pascal's Triangle The binomial theorem, which uses Pascal's triangles to determine coefficients, describes the algebraic expansion of powers of a binomial. KEY POINTS • According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. • Using summation notation, the binomial theorem can be expressed as: (. • The The binomial theorem describes the algebraic expansion of powers of a binomial. Essentially, it demonstrates what happens when you multiply a binomial by itself (as many times as you want).of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term. For example, (x + y)4 = x4 + 4x3 y + 6x2 y2 + 4xy3 + y4 The coefficient a in the term of xb yc is known as the binomial coefficient or (the two have the same value). These coefficients for varying n and b can be arranged to form Pascal's triangle. These numbers also arise in combinatorics, where gives the number of different combinations of b elements that can be chosen from an n- element set. According to the theorem, it is possible to expand any power of x + y into a sum of the form: (x + y)n = ( n 0) xn y0 + ( n 1) xn−1 y1 + ( n 2) xn−2 y2 + … + ( n n − 1) x1 yn−1 + ( n n) x0 yn where each is a specific positive integer known as binomial coefficient. This formula is also referred to as the Binomial Formula or the Binomial Identity. Using summation notation, it can be written as: (448 Pascal's Triangle Pascal's triangle, Figure 8.21, determines the coefficients which arise in binomial expansions. For an example, consider the expansion: (x + y)2 = x2 + 2xy + y2 = 1x2 y0 + 2x1 y1 + 1x0 y2 Notice the coefficients are the numbers in row two of Pascal's triangle: 1, 2, 1. In general, when a binomial like x + y is raised to a positive integer power we have: (x + y)n = a0xn + a1xn−1 y + a2xn−2 y2 + ⋅ ⋅ ⋅an−1xyn−1 + anyn where the coefficients ai in this expansion are precisely the numbers on row n of Pascal's triangle. In other words, ai = ( n i) Notice that the entire right diagonal of Pascal's triangle corresponds to the coefficient of yn in these binomial expansions, while the next diagonal corresponds to the coefficient of xyn−1 and so on. To see how the binomial theorem relates to the simple construction of Pascal's triangle, consider the problem of calculating the coefficients of the expansion of (x + 1)n+1 in terms of the corresponding coefficients of (x + 1)n (setting y = 1 for simplicity). The A simple construction of the triangle proceeds in the following manner. On row 0, write only the number 1. Then, to construct the elements of following rows, add the number above and to the left with the number above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place. For example, the first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row. 449 Each number in the triangle is the sum of the two directly above it. Figure 8.21 Pascal's Triangle Source: combinatorics/the-binomial-theorem/binomial-expansions-and- pascal-s-triangle/ CC-BY-SA Boundless is an openly licensed educational resource Binomial Expansion and Factorial Notation The binomial theorem describes the algebraic expansion of powers of a binomial. KEY POINTS • According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axb yc , where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. • The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. • Binomial coefficients can be written as nCk and are defined in terms of the factorial function n!. depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term. 450 In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n (Figure 8.22). For example, 5! = 5 × 4 × 3 × 2 × 1 = 120 The value of 0! is 1, according to the convention for an empty product. The factorial operation is encountered in many different areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic occurrence is the fact that there are n! ways to arrange n distinct objects into a sequence (i.e., permutations of the set of objects). The definition of the factorial function can also be extended to non-integer arguments, while retaining its most important properties; this involves more advanced mathematics, notably techniques from mathematical analysis. The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written nCk, and pronounced "n choose k". The coefficient of xn−kyk is given by the formula: ( n k) = n! k!(n − k)! Note that, although this formula involves a fraction, the binomial coefficient ( n k) is actually an integer. 451 The first few and selected larger members of the sequence of factorials (sequence A000142 in OEIS). The values specified in scientific notation are rounded to the displayed precision. Figure 8.22 Factorials Source: combinatorics/the-binomial-theorem/binomial-expansion-and- factorial-notation/ CC-BY-SA Boundless is an openly licensed educational resource Finding a Specific Term The rth term of any expansion can be found with the equation: ( n r − 1)an−(r−1) br−1 . KEY POINTS • The binomial theorem describes the expansion of a binomial: xn−k yk = n ∑ k=0 ( n k)xk yn−k . • Looking at some smaller expansions, for the exponents, one can notice that the number of terms is one more than n (the exponent) and the sum of the exponents in each term adds up to n. • Applying ( n r − 1)an−(r−1) br−1 and nCk = n! (n − k)!k! one can find a particular term of an expansion without going through every single term. depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term. 452 The equation is: xn−k yk = n ∑ k=0 ( n k)xk yn−k Another way of writing this is nCk = n! (n − k)!k! As an example, if we go through a few expansions, (a + b)1 = a + b (a + b)2 = a2 + 2ab + b2 (a + b)3 = a3 + 3a2 b + 3ab2 + b3 (a + b)4 = a4 + 4a3 b + 6a2 b2 + 4ab3 + b4 A few things can be noticed: • The number of terms is one more than n (the exponent). • The power of a starts with n and decreases by 1 each term. • The power of b starts with 0 and increases by 1 each term. • The sum of the exponents in each term adds up to n. • The coefficients of the first and last terms are both 1 and they follow Pascal's triangle Figure 8.23. If there is a short expansion, such as, (x + 2)3 = x3 + 2x2 21 + 2x1 22 + 23 (x + 2)3 = x3 + 4x2 + 8x + 8 then is easy to find a particular term. However, what about longer expansions? Using the binomial theorem a shortcut can be made. If looking for the rth term, then: ( n r − 1)an−(r−1) br−1 Now, find the 5th term of (3x − 4)12 : 453 Each number in the triangle is the sum of the two directly above it. Figure 8.23 Pascal's Triangle ( 12 5 − 1) (3x)12−(5−1) (−4)5−1 ( 12 4 ) (3x)8 (−4)4 Then, if the coefficient equation from above is applied to ( 12 4 ) , and the power is applied to the terms, the result is: = 495 ⋅ 6561x8 ⋅ 256 = 831409920x8 Source: combinatorics/the-binomial-theorem/finding-a-specific-term/ CC-BY-SA Boundless is an openly licensed educational resource Total Number of Subsets The total number of subsets is the number of sets with 0 elements, 1 element, 2 elements, etc. KEY POINTS • These numbers also arise in combinatorics, where nb gives the number of different combinations of b elements that can be chosen from an n-element set. The number of subsets containing k elements is represented by (nk ). • According the binomial theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axb yc , where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. • The total number of subsets of a set with n elements is 2n and can be represented algebraically: ( n 0) + ( n 1) + ( n 2) + . . . + ( n n). The binomial coefficients appear as the entries of Pascal's triangle where each entry is the sum of the two above it. The binomial theorem describes the algebraic expansion of powers of a binomial.454 of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term. The coefficient a in the term of xb yc is known as the binomial coefficient (nb ) or (nc ) (the two have the same value). These coefficients for varying n and b can be arranged to form Pascal's triangle. These numbers also arise in combinatorics, where nb gives the number of different combinations of b elements that can be chosen from an n-element set. The number of subsets containing k elements is represented by (nk). The total number of subsets is the number of sets with 0 elements, 1 element, 2 elements, etc. This can be represented algebraically by: ( n 0) + ( n 1) + ( n 2) + . . . + ( n n) The total number of subsets of a set Figure 8.24 with n elements is 2n . For example, how many subsets are in the set: {P, Q, R, S, T, U}? It has 6 elements, therefore, 2n = 26 = 64 subsets. Source: combinatorics/the-binomial-theorem/total-number-of-subsets/ CC-BY-SA Boundless is an openly licensed educational resource 455 A is a subset of B Figure 8.24 Subsets Experimental Probabilities The experimental probability is the ratio of the number of outcomes in which an event occurs to the total number of trials in an experiment. KEY POINTS • In a general sense, empirical probability estimates probabilities from experience and observation. • In simple cases, where the result of a trial only determines whether or not the specified event has occurred, modeling using a binomial distribution might be appropriate and then the empirical estimate is the maximum likelihood estimate. • If a trial yields more information, the empirical probability can be improved on by adopting further assumptions in the form of a statistical model. If such a model is fitted, it can be used to derive an estimate of the probability of the specified event. The experimental probability (relative frequency or empirical probability) is a probability that pertains to data taken from a number of trials. It is a calculable probability based not on theory, but experience. If a sample of x trials is observed that results in an event, e, occurring n times, the probability of event e is calculated by the ratio of n to x. In statistical terms, the empirical probability is an estimate of a probability. In simple cases, where the result of a trial only determines whether or not the specified event has occurred, modeling using a binomial distribution might be appropriate. In such cases, the empirical estimate is the most likely estimate. If a trial yields more information, the empirical probability can be improved on by adopting further assumptions in the form of a statistical model: if such a model is fitted, it can be used to estimate the probability of the specified event. For example, one can easily assign a probability to each possible value in many discrete cases: when throwing a die, each of the six values 1 to 6 has the probability 1/6. This distribution model is then created (Figure 8.25). 457 Discrete probability distribution for the sum of two dice. Figure 8.25 Dice Probability Advantages An advantage of estimating probabilities using empirical probabilities is that this procedure includes few assumptions. For example, consider estimating the probability among a population of men that they satisfy two conditions: 1. that they are over six feet in height; and they are less than six feet in height. 2. that they prefer strawberry jam to raspberry jam. A direct estimate could be found by counting the number of men who satisfy both conditions to give the empirical probability of the combined condition. An alternative estimate could be found by multiplying the proportion of men who are over six feet in height with the proportion of men who prefer strawberry jam to raspberry jam, but this estimate relies on the assumption that the two conditions are statistically independent. Disadvantages A disadvantage in using empirical probabilities is that without theory to "make sense" of them, it's easy to draw incorrect conclusions. Rolling a six-sided die one hundred times, it's entirely possible that well over 1/6 of the rolls will land on 4. Intuitively we know that the probability of landing on any number should be equal to the probability of landing on the next. Experiments, especially those with lower sampling sizes, can suggest otherwise. This shortcoming becomes particularly problematic when estimating probabilities which are either very close to zero, or very close to one. For example, the probability of drawing a number from 1 to 1000 is 1/1000. If 1000 draws are taken and the first number drawn is 5, there are 999 draws left to draw a 5 again and thus have experimental data that shows double the expected likelihood of drawing a 5. In these cases, very large sample sizes would be needed in order to estimate such probabilities to a good standard of relative accuracy. Here statistical models can help, depending on the context. For example, consider estimating the probability that the lowest of the daily-maximum temperatures at a site in February in any one year is less than zero degrees Celsius. A record of such temperatures in past years could be used to estimate this probability. A model- based alternative would be to select of family of probability distributions and fit it to the data set containing past years′ values. The fitted distribution would provide an alternative estimate of the desired probability. This alternative method can provide an estimate of the probability even if all values in the record are greater than zero. 458 Source: combinatorics/probability/experimental-probabilities/ CC-BY-SA Boundless is an openly licensed educational resource Theoretical Probability Probability theory uses logic and mathematical reasoning, rather than experimental data, to determine probable outcomes. KEY POINTS • Outcomes of an experiment are often equiprobable (as with heads and tails in a coin toss). In such cases, the probability of an event can be calculated logically: it is equal to the number of outcomes comprising this event, divided by the total number of outcomes in the sample space. • If a set of choices or trials, T1, T2, T3, …, Tk, could result, respectively, in n1, n2, n3, …,nk possible outcomes, the entire set of k choices or trials has nk! possible outcomes. This is the Fundamental Rule of Counting. • In the case of simple probabilities (like dice and coin tosses), outcomes are easy to count. In instances in which counting is difficult, permutations and combinations can be used to calculate outcomes. Mathematically, probability theory formulates incomplete knowledge pertaining to the likelihood of an event. For example, a meteorologist might say there is a 60% chance that it will rain tomorrow. This means that in six out of every 10 times when the world is in its current state, it will rain. This probability is 459 determined through measurements and logic, but not through any experimental findings (the future has not yet happened). As such, the meteorologist's 60% verdict is a theoretical probability, and not the result of any proven experiment. Often, in experiments with finite sample spaces, the outcomes are equiprobable (as with heads and tails in a coin toss). In such cases, the probability of an event can be calculated logically: it is equal to the number of outcomes comprising this event, divided by the total number of outcomes in the sample space. For example, the probability of rolling any specific number on a six- sided die is one out of six: there are six, equally probable sides to land on, and each side is distinct from the others. If the six on the die were changed to a one, you could logically conclude that the probability of rolling a one would be two out of six (or one out of three). This is a theoretical probability; testing by rolling the die many times and recording the results would result in an experimental probability. While counting outcomes may appear straightforward, it is in many circumstances a daunting task. For example, consider the number of distinct subsets of the integers {1, 2,..., n} that do not contain two consecutive integers. This number is equal to: ϕn+2 − (1 − ϕ)n+2 5 where ϕ = (1 + 5)/2 is the golden ratio. It can also be obtained recursively through the Fibonacci recurrence relation. Calculating the number of ways that certain patterns can be formed is part of the field of combinatorics. The Fundamental Rule of Counting If a set of choices or trials, T1, T2, T3, …, Tk, could result, respectively, in n1, n2, n3, …,nk possible outcomes, the entire set of k choices or trials has nk! possible outcomes. By the Fundamental Rule of Counting, the total number of possible sequences of choices 460 By the Fundamental Rule of Counting, the total number of possible sequences of choices is a permutation of each of the items. Figure 8.26 The Counting Principle is 5×4×3×2×1 = 120 sequences. Each sequence is called a permutation (or ordering) of the five items (Figure 8.26). Permutations A permutation is an arrangement of unique objects in which order is important. In other words, permutations using all the objects: n objects, arranged into group size of n without repetition, and order being important. The number of possible permutations of a set size of n in which k elements are drawn can be calculated by: n! (n − k)! Combinations A combination is an arrangement of unique objects, in which order is not important. For example, the number of possible combinations of n objects, arranged in groups of size r can be calculated by: n! r!(n − r)! Source: combinatorics/probability/theoretical-probability/ CC-BY-SA Boundless is an openly licensed educational resource 461
Book Description: This accessible text is designed to help readers help themselves to excel. The content is organized into two parts: (1) A Library of Elementary Functions (Chapters 1–2) and (2) Calculus (Chapters 3–9). The book's overall approach, refined by the authors' experience with large sections of college freshmen, addresses the challenges of teaching and learning when readers' prerequisite knowledge varies greatly. Reader-friendly features such as Matched Problems, Explore & Discuss questions, and Conceptual Insights, together with the motivating and ample applications, make this text a popular choice for today's students and instructors
Mathwords A Word Book for Mathematics 9780521455275 ISBN: 0521455278 Publisher: Cambridge University Press Summary: This is a wordbook for mathematics. Series of informal pictures illustrate words and phrases connected with mathematical topics in a relevant context. The index lists the words children in primary schools can be expected to want to find. Thus children can use the book to browse among pictures, acquiring vocabulary in the process, or they can use it as a picture dictionary. Anglesey Language Studies Centre St...aff is the author of Mathwords A Word Book for Mathematics, published under ISBN 9780521455275 and 0521455278. Seven Mathwords A Word Book for Mathematics textbooks are available for sale on ValoreBooks.com, five used from the cheapest price of $6.35
Aleks Upgrades HS Geometry Tool Math Assessment tool provider Aleks has upgraded its High School Geometry offering so that it now leverages the company's AI-2 intelligence engine, which aims to reduce faculty workload and make learning process more rewarding and more targeted to student needs. Specifically, Aleks used instructor and student feedback to develop the course's AI-2 engine so that it could provide more accurate assessments and let teachers more flexibly instruct students. Students navigate learning paths based on their level of readiness. In terms of subject matter, the geometry course covers 217 concepts under the broader categories of algebra and deductive reasoning, lines and angles, triangles, polygons and circles, similarities and transformations, volumes and surface areas, and coordinate geometry. The geometry course includes more than 40 new problem types and a content editor to let teachers remove more than 100 topics and change the sequence of topics to tailor the curriculum to their classes' needs. The course also includes input tools, such as proof and construction tools, to help foster a free response environment. The geometry course is now available and can be used as core class curriculum, or in a supplemental fashion for remediation, intervention, or gifted or
Math and Drug Calculations Online for Nursing and Health Professions is a complete drug calculations online course that provides students an opportunity for application and practice. Since it is not tied to a specific text, this online course is suitable for all drug calculations classes. It incorporates the ratio and proportion, fractional equation, formula, and dimensional analysis methods and presents a step-by-step approach to the calculation and administration of drug dosages. Animations, voice-overs, and interactive self-assessment activities are used to provide an engaging and interactive course platform for students. This online course consists of three comprehensive modules; Module 1: Math Introduction and Overview, Module 2: Medication Administration, and Module 3: Medication Administration in Specialty Areas. Each module includes practice problems to promote active learning and quizzes that instructors can use to evaluate students' understanding of content presented in the course. A comprehensive test bank of approximately 300 questions is also provided for instructor's to build quizzes and test. Print Book Key Features Includes the four drug calculation methods (ratio and proportion, fractional equation, formula, and dimensional analysis) to expose the user to key calculation methods so they can apply the method which works best for them. Modules are organized by topic sections that include an overview, objectives, what you need to know, example problems, practice problems, and one or more quizzes. Follows abbreviation and dose designation recommendations from the Joint Commission on Accreditation of Healthcare Organizations (JCAHO) and the Institute for Safe Medication Practices (ISMP). Many of the math practice problems include a tutorial button for each of the four drug calculation methods that provides a step-by-step tutorial to solving the problem in the chosen method. Animations demonstrate various concepts related to drug calculation and administration, with some animations requiring user participation. Interactive self-assessment activities are incorporated throughout the course to allow users to apply their knowledge in context. Voice-overs enhance the step-by-step explanation of medication administration procedures and the drug calculation methods demonstrated throughout the course. One or more Quizzes are included within each module to evaluate understanding of all the major topics covered in that particular module. Provides the latest drug administration techniques and devices with detailed explanations of the various ways to administering drugs, including oral, intravenous, intramuscular, subcutaneous and other routes. Includes the most up-to-date, commonly used drugs so users have exposure to what is being used in the "real world" of clinical practice. Presents information on infusion pumps (enteral, single, multi-channel, PCA, and insulin) to help users understand their increased use in drug administration.
Summer 2012 Course objective: The objective of this course is to introduce students to the area of algebraic curves and algebraic geometry. Algebraic curves are some of the most classical objects studied in mathematics. Algebraic curves provide the most intuitive approach to introducing students to algebraic geometry. Furthermore, they have many applications in areas of coding theory, cryptography, etc. This course will be a valuable tool of introducing students to these areas. Textbooks The following books are in Albanian and a very good introduction to the algebra needed and the algebraic geometry. The both can be bought through the following links.
Modify Your Results An ideal program for struggling students "Glencoe Algebra: Concepts and Applications" covers all the Algebra 1 concepts. This program is designed for students who are challenged by high school mathematics. Glencoe Algebra: Concepts and Applications includes lessons that will help students prepare for the Texas Essential Knowledge and Skills assessed on the Texas state test. This textbook contains a special section of practice problems specifically for the Texas state test. Algebra: Concepts & Applications, is a comprehensive Algebra 1 program that is available in full and two-volume editions.Algebra: Concepts & Applications uses a clean lesson design with many detailed examples and straightforward narration that make Algebra 1 topics inviting and Algebra 1 content understandable. Volume 1 contains Chapters 1-8 of Algebra: Concepts & Applicationsplus an initial section called Chapter A. Chapter A includes a pretest, lessonson prerequisite concepts, and a posttest. Designed for students who are challenged by high school mathematics, the 2006 edition has many new features and support components. Algebra: Concepts & Applications, is a comprehensive Algebra 1 program that is available in full and two-volume editions. Algebra: Concepts & Applicationsuses a clean lesson design with many detailed examples and straightforward narration that make Algebra 1 topics inviting and Algebra 1 content understandable. Volume 1 contains Chapters 1-8 ofAlgebra: Concepts & Applicationsplus an initial section called Chapter A. Chapter A includes a pretest, lessonson prerequisite concepts, and a posttest. Designed for students who are challenged by high school mathematics, the 2007 edition has many new features and support components. Focus on Physical Science is a practical study of the relationship between matter and energy. Fundamental Physical Science principles are introduced through student involvement rather than by rote memorization. This book comprises seven units, the contents of which include Measurement and Motion, Classification of Matter, Patterns in Matter, Changes in Matter, Light and Sound, Energy, Energy Resources for the Future. Units of Measurement and Laboratory Techniques form part of Appendix. The book's design is up-to-date as it underlines the real-world applications of Maths as well as its connections to other subjects like science, history and music. The Chapter Projects, Hands-On Labs, and Interdisciplinary Investigations makes the book very useful and interesting to the students. To help chart their journeys, wise travelers consult a map before they begin. Just as maps lead travelers to their destinations, the script on the next five pages points out the ways that you use the mathematics in this text in you daily livesGlencoe Pre-Algebra is a key program in our vertically aligned high school mathematics series developed to help all students achieve a better understanding of mathematics and improve their mathematics scores on today's high-stakes assessments. Each chapter of Merrill CHEMISTRY is organized to keep you centered on the topic at hand. The next few pages describe the many features of this book that will help you understand the exciting world
From the author: "The SMILE program is designed to enhance the elementary and high school learning of Science and Mathematics... see more From the author: "The SMILE program is designed to enhance the elementary and high school learning of Science and Mathematics through the use of the phenomenological approach. Since 1986 each summer session participant has been asked to create and publish a single concept lesson plan. These lesson plans include the materials needed, a suggested strategy and expected outcomes. There are currently over 800 lesson plans available. The following is a collection of almost 200 single concept lessons. These lessons may be freely copied and used in a classroom but they remain the copyright property of the author. The Physics lessons are divided into the following categories: Matter, Mechanics, Fluids, Electricity & Magnetism, Waves, Sound and Optics, and Miscellaneous. " The goal of this project was to create a learning object that would allow students to discover and explore the complex mind... see more The goal of this project was to create a learning object that would allow students to discover and explore the complex mind of Isaac Newton through the primary source materials principally available in the Research Collections area of McMaster University's Mills Library. By combining images, audio, animations and interactive segments, the application gives students a sense of Newton's multifaceted mind. It also provides an appreciation of his achievements in mathematics and physics and introduces Newton's diverse interests in theology, religion and philosophy. A Navier-type method for finding the exact three-dimensional solution for isotropic thick and thin rectangular plates is... see more A Navier-type method for finding the exact three-dimensional solution for isotropic thick and thin rectangular plates is presented. The extension of this procedure to the case of multilayered plates (composite structures) is straightforward and can be found in the listed references . The method presented in these lectures uses the Mixed Form of Hooke's Law (MFHL) which leads one to write the boundary conditions on the top and bottom surfaces of the plate directly in terms of transverse stresses. Mixed Form of Hooke's Law is obtained from the Classical Form of Hooke's Law (CFHL). The displacements, stresses and external loads are expressed by using trigonometric functions. It is possible to demonstrate that this particular choice satisfies the simply supported boundary conditions. The elasticity solution must satisfy the equilibrium equations, geometric relations and Hooke's law. After some mathematical derivations it is possible to demonstrate that a set of differential equations in the unknown amplitudes of the displacements and stresses has to be solved. An eigenvalue problem is obtained. Only 2 (over a total of 6) eigenvalues are distinct. Therefore, a basis of eigenvectors is not available and 2 generalized eigenvectors have to be found. The solution is a combination of the eigenvectors and generalized eigenvectors multiplied by functions of the out-of-plane coordinate z. The unknown constants of this combination are determined by imposing that the transverse stresses at the top and bottom surfaces of the plate must match the applied pressures. In a general multilayered structure additional conditions on the interlaminar continuity of the displacements and transverse stresses are required. Once the unknown coefficients are calculated, the elasticity solution is complete. These lectures present an algebraic solution for the particular case of a pressure applied at the top surface. The extension to the case of generic load can be obtained by using Fourier analysis. The lectures also present the FREE software that can be downloaded. The program calculates the amplitudes of the displacements and stresses for a sinusoidal load pressure applied at the top surface of the plate. The amplitude of the pressure distribution, material properties, geometric dimensions of the plate and wave numbers are inputs of the software. In contrast to the common but misleading "entropy is disorganization" line, this tutorial describes entropy as a measure of... see more In contrast to the common but misleading "entropy is disorganization" line, this tutorial describes entropy as a measure of the spreading and sharing of thermal energy. This concept provides rational, non-mathematical explanations of the effects of temperature change on reaction equilibria and on the colligative properties of solutions. This interactive simulation enable user to investigate the effect of temperature of the behavior of an fictional ideal gas.... see more This interactive simulation enable user to investigate the effect of temperature of the behavior of an fictional ideal gas. The audio provided with the simulation discusses the influence of parameters such as pressure and temperature as they relate to gas behavior and its mathematical relationship
Rent Textbook Buy Used Textbook Buy New Textbook Usually Ships in 3-4 Business Days $211.26 eTextbook 180 day subscription $138.679The History of MathematicsAugust 7, 2011 by Jacki Leach The minute I got this textbook I fell in love with it and I was reading it every day. The problems at the end of the chapters are so cool. I would recommend this textbook for a course any day in the history and development of mathematics for those who have had some experience with mathematical proofs. Burton did a wonderful job on this book. The History of Mathematics: An Introduction: 5 out of 5 stars based on 1 user reviews. Summary The History of Mathematics: An Introduction, Seventh Edition, is written for the one- or two-semester math history course taken by juniors or seniors, and covers the history behind the topics typically covered in an undergraduate math curriculum or in elementary schools or high schools. Elegantly written in David Burton's imitable prose, this classic text provides rich historical context to the mathematics that undergrad math and math education majors encounter every day. Burton illuminates the people, stories, and social context behind mathematics' greatest historical advances while maintaining appropriate focus on the mathematical concepts themselves. Its wealth of information, mathematical and historical accuracy, and renowned presentation make The History of Mathematics: An Introduction, Seventh Edition a valuable resource that teachers and students will want as part of a permanent library.
more details The percentage of high school graduates who have taken Algebra II has more than doubled in the last two decades. And now, with a greater emphasis on advanced algebra in standardized tests, the demand for advanced guidance is growing. Algebra II Workbook For Dummies provides students and teachers with fun, engaging, and effective ways to prepare for exams. Building on the concepts taught in basic algebra, this hands-on guide broadens students146; skill sets through exercises that take a closer look at advanced math such as quadratic equations; polynomial, exponential, and logarithmic functions; irrational and complex numbers; and graphing inequalities. Although the workbook functions independently, it can also be cross-referenced with Algebra II For Dummies. Mary Jane Sterling (Peoria, IL) has more than 30 years146; experience as a teacher of higher mathematics. She is the author of Algebra For Dummies (0-7645-5325-9), Algebra Workbook For Dummies (0-7645-8467-7), and Algebra II For Dummies (0-471-77581-9).
Synopses & Reviews Publisher Comments: - Leveled text reviewed by a math curriculum consultant and a reading consultant - Full-color photographs and illustrations support text and aid in comprehension - Examples that show the relevance of math in students' daily lives - Continuation of the successful Math in Our World--Levels 1 and 2 series - Glossary, Index Synopsis: "Synopsis" by Ingram,
Students explore the elemenets of e-learning within a learning management system(LMS). They explore the appropriate use of tools and skills required to successfully implement an e - learning plan and evaluate their readiness to form educational relationships that will cross time and distance. This course sequence is designed to provide students with the mathematical understandings and critical-thinking skills identified for post-secondary studies in programs that do not require the study of calculus. Topics include geometry, measurement, number and logic, logical reasoning, relations and functions, statistics, and probability.
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Offering the most geometric presentation available, Linear Algebra with Applications, Fourth Edition emphasizes linear transformations as a unifying ...Show synopsisOffering the most geometric presentation available, Linear Algebra with Applications, Fourth Edition emphasizes linear transformations as a unifying theme. This elegant textbook combines a user-friendly presentation with straightforward, lucid language to clarify and organize the many techniques and applications of linear algebra. Exercises and examples make up the heart of the text, with abstract exposition kept to a minimum. Extensive problem sets keep students involved in the material, while genuine applications for a broad range of sciences prepares them for the methods and models of contemporary scientists. In addition, the wealth and variety of exercise sets enable instructors to design a course to best suit the goals and needs of their students. This revision reflects careful review and appropriate changes to the wording of each idea, while preserving the content structure of the previous
simple, concise and useful book, explaining MATLAB for freshmen in engineering. MATLAB is presently a globally available standard computational tool for engineers and scientists. The terminology, syntax and the use of the programming language are well defined and the organisation of the material makes it easy to locate information and navigate through the textbook. This new text emphasises that students do not need to write loops to solve many problems. The MATLAB "find" command with its relational and logical operators can be used instead of loops in many cases. This was mentioned in Palm's previous MATLAB texts, but receives more emphasis in this MATLAB 6 edition, starting with Chapter 1, and re-emphasised in Chapter 4.
Rent Textbook Buy Used Textbook eTextbook 180 day subscription $55.19 New Textbook We're Sorry Sold Out More New and Used from Private Sellers Starting at $0 CD edition with a publication date of 5/7 Math Using Excel Summary Business Math Using Excel prepares users for the business world by incorporating math concepts using two approaches. The first approach uses the traditional method of calculating. The second approach teaches those same concepts using the functions of Excel. Today's office workers are often challenged with Excel worksheets that include complex calculations that once were done using desktop calculators. Business Math Using Excel helps users create formulas and use the functions of Excel to make it the powerful application tool it is meant to be and to develop a high level of math skill. Table of Contents Basic Math Functions Numbers Addition Subtraction Multiplication Division Basic Math Operations and Excel Fractions Fractions and Mixed Numbers Add and Subtract Fractions and Mixed Numbers Multiply and Divide Fractions and Mixed Numbers Percent Introduction to Percents Part, Rate, and Base Percent of Increase and Decrease Bank Services The Checking Account Bank Statement Reconciliation Payroll Gross Earnings Gross Pay for Various Compensation Methods Payroll Deductions Employee's Earnings Record and Payroll Register Taxes & Insurance Property Tax and Property Tax Rate State Unemplyment Tax, Federal Unemployment Tax, and Federal Income Tax
Still baffled by the Building Regs? Confused by codes of practice? Mystified by materials and puzzled by planning permission? Then look no further! This is an ideal guide to glance at when you need a quick, precise answer to the requirements of the Building Regulations. 1001 Basic Math & Pre- Algebra Practice Problems For Dummies Practice makes perfect-and helps deepen your understanding of basic math and pre-algebra by solving problems 1001 Basic Math & Pre-Algebra Practice Problems For Dummies, with free access to online practice problems, takes you beyond the instruction and guidance offered in Basic Math & Pre-Algebra For Dummies, giving you 1,001 opportunities to practice solving problems from the major topics in your math course. You begin with some basic arithmetic practice, move on to fractions, decimals, and percents, tackle story problems, and finish up with basic algebra. Every practice question includes not only a solution but a step-by-step explanation. From the book, go online and find:Many Light Water Reactors (LWRs) are in operation worldwide and others are in construction. Analysis techniques and their validation are needed to provide assurances for licensing and for safe reactor operation.
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TI-83 Plus Features Summary The TI-83 Plus is an easy-to-use graphing calculator for math and science that lets students graph and compare functions, as well as perform data plotting and analysis. Its FLASH™ ROM memory allows students to update and add software applications (Apps). Count on TI calculators at exam time. You can use this TI graphing calculator on the PSAT, SAT, and ACT college entrance exams, IB, and AP tests.* Find What You Need TI graphing calculators are learning tools designed to help students visualize concepts and make connections in math and science. Take a look at the TI Calculator Comparison Chart to find which model fits your needs. Key Features: New App Makes Entering Easier You can now see and enter math problems in your TI graphing calculator just like they appear in your textbooks. Download the free Zoom Math Starter Edition™ - App4Math™ App by I.Q. Joe to your TI-83 Plus or TI-84 Plus family calculator. *SAT & AP are registered trademarks of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product. ACT is a registered trademark of ACT, Inc., which does not endorse this product. IB is a registered trademark of the International Baccalaureate Organization.
Mathematics The goal of Hamilton's Mathematics Department is to provide all students with opportunities to hone logical thinking and statistical reasoning skills, while offering concentrators an understanding of fundamental areas of pure and applied mathematics and what constitutes legitimate, rigorous argument through the analysis and construction of written proofs. Overview Ancient thinkers recognized that mathematics was the language of the natural world. Today we recognize that it is also the language of science and social science, of business, commerce and industry, even of art and design. Doing math can be as simple as executing a computer search and as momentous as planning a mass evacuation or tracing a disease epidemic, but it assumes ever-greater importance in our lives. At Hamilton, mathematics is both a popular major and a crucial part of the College's broader liberal arts curriculum. More ... Academic Program Research Opportunities Research is crucial to mathematics and all the sciences. At Hamilton it is also one of the keys to the liberal arts curriculum. It begins with Hamilton's small, interactive classes and continues through the Senior Program. The Mathematics Department also has a number of students return to campus each summer for professional research. Conducting research, either through collaboration with a faculty member or independently, is the means by which students integrate and apply what they have learned in the classroom. Each summer, Hamilton and the Mathematics Department offer exciting opportunities for undergraduates to conduct hands-on collaborative research in math, computer science and other fields. The Summer Science Collaborative Research Program provides grants for dozens of students to work with faculty members on a variety of cutting-edge projects. Many of these projects lead to scholarly papers and presentations co-authored by students. The Senior Program Mathematics majors begin the senior year with a senior seminar. These small seminars focus on different areas of mathematics and are devoted largely to presentations by students and intensive discussion. Then, in the spring semester, students may elect to write a thesis, working independently but under close faculty supervision. The senior thesis is a culmination of each student's undergraduate experience; it synthesizes coursework, research and discussion into a focused statement of intellectual growth and insight. Strong student papers sometimes are published in scholarly journals — a remarkable accomplishment at the undergraduate level. More ... Resources The Quantitative & Symbolic Reasoning Center offers peer tutoring in courses that include a mathematics/quantitative component. Students may drop in to review topics as needed or to use the resources of the computer and video library. Other programs offered by the center include a review for the mathematics portion of Graduate Record Exam and workshops designed to accompany specific courses. Tutors at the center are top students who have been recommended by the departments for which they tutor. They have taken the courses they tutor and have been trained in tutoring techniques and interpersonal relations. Highlights courses
Foundations of Geometry, Second Editionimplements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers ;and encourages students to make connections between their college courses and classes they will later teach. This text's coverage begins with Euclid's Elements, lays out a system of axioms for geometry, and then moves on to neutral geometry, Euclidian and hyperbolic geometries from an axiomatic point of view, and then non-Euclidean geometry. Good proof-writing skills are emphasized, along with a historical development of geometry. The Second Editionstreamlines and reorganizes material in order to reach coverage of neutral geometry as early as possible, adds more exercises throughout, and facilitates use of the open-source software Geogebra.
From: Jim Wysocki <jim.wysocki@chadwick-k12.com> To: Teacher2Teacher Public Discussion Date: 2000020100:17:12 Subject: Re: Purpose of studying algebra Hear me out. I find it interesting that we need to provide justification to the students about why a particular subject is valuable. Apparently the idea that we study different subjects 1) for the sake of knowledge and 2) for the mental discipline we gain are two justifications that nobody accepts, thus we need to come up with artificial applications and "real-world" connections about where we use math. Realistically, if you are in an inner-city school there is no justification for studying it that you can give students because they see to many examples around them that prove you can survive in this world without knowing the solution to "If x squared minus three times x minus four equals zero, what is x?" At the same time, in a well-to-do neighborhood, any "real-world" application will be shallow at best as these students will see algebra as their stepping stone to college... and that is all they need to know about it. Be honest with your students. Why do we study algebra? Because it is a foundational subject in mathematics that trains you to think and reason in a logical and orderly manner. Will we be able to use this stuff later? Sure, but that is not really the point here.
The course is designed to give students experience solving novel and often open-ended problems in multiple ways. Posing and solving problems drawn from rational numbers, number theory, algebra, measurement and geometry will help students to reinforc,e and integrate various strands of their high school mathematics curriculum and to reason and communicate more effectively. Prerequisite: Certification in Secondary Mathematics Education or ED 339: Meth of Teach & Eval Math and permission of instru,ctor. Offered: Summer\| MAED 585 App & Math Model For Teacher A course to use mathematics in representing and solving real world problems. Mathematical concepts from middle school and high school mathematics will be applied to problems in social, physical, and biological sciences. Pedagogical and assessment i,ssues will be studied. Prerequisite: Certification in teaching mathematics, or mathematical science teachers in grades 7 through 12. Highly recommended: calculus, linear algebra, differential equations, or statistics. Offered: Summer.\| MAED 590 Historical Dev Of Mathematics Students will explore the development of mathematics, from the early development of numeration systems to modern mathematics. They will learn how the technology and culture of different places and times affected the topics developed as well as the m,ethods used. Students will also discuss and develop ways to incorporate this information into their classroom teaching and to encourage cross disciplinary connections. Prerequisite: Degree in mathematical sciences or permission.\| Deals briefly with the history of marine biology, and with the interrelationship of chemical and physical oceanography and marine biology. Marine habitats will be studied in more detail with reference to the adaptations of plant and animal inhabitan,ts. Course taught at the Marine Science Consortium (MSC), Wallops Island, Virginia.\| MARS 345 Marine Ornithology An introductory course on the bird fauna of the sea coast, including avian diversity and ecology. Students will learn to identify coastal birds using plumage, behavior and songs. Lecture material will include information on distribution, behavior,,physiology and anatomy of birds. Course taught at the Marine Science Consortium (MSC), Wallops Island, Virginia.\| MARS 360 Marine Ecology Interrelationships among animals, plants, and physical and chemical aspects of the environment will be studied, with an emphasis on adaptations for survival which are unique to the marine environment. Zoogeography and food chain interactions will al,so be studied with some emphasis placed on the impacts of human activities. Course taught at the Marine Science Consortium (MSC), Wallops Island, Virginia.\| MARS 403 Research Diver Methods Study and practice of aquatic research methods using SCUBA as a tool. Advanced research diving topics include areas such as navigation, search and recovery, underwater photography, survey methods, estimating population parameters, and data acquisiti,on while under water. Specific research techniques will be presented in the context of specific aquatic research project conducted by students under the director of the instructor. Each summer.\| MARS 431 Ecology Of Marine Plankton Study of the phytoplankton and zooplankton in marine and brackish environments. Qualitative and quantitative comparisons will be made between the planktonic populations of various types of habitats in relation to primary and secondary productivity., Course taught at the Marine Science Consortium (MSC), Wallops Island, Virginia.\| MARS 443 Marine Ichthyology Study of the internal and external structure of fishes, their classification and evolutionary relationships, their ecology and behavior, their physiology and environmental requirements, and their distribution patterns in marine and brackish environme,nts. Course taught at the Marine Science Consortium (MSC), Wallops Island, Virginia.\| MARS 471 Scan Electron Microscop Marine Trains students in the use of a portable scanning electron microscope (PSEMII), including principles of operation and use the SEM, preparation of geological and biological samples for imaging, size and shape analysis, and elemental and mineralogical,analysis with the energy dispersive X-ray spectrometer (EDX). The latter portion of the course is dedicated to the design and execution of independent research projects in investigating a problem in marine science using the SEM and/or EDX. Course ta,ught at the Marine Science Consortium (MSC), Wallops Island, Virginia.\| Introduces basic arithmetic and algebraic concepts, including an introduction to real numbers and algebraic expressions, solving equations and inequalities, polynomials, factoring, graphing, and systems of equations. Credits in this course do not cou,nt toward general education or graduation. Prerequisite: Appropriate placement determined by the Mathematics Department. Each semester.\| MATH 110 Intermediate Algebra Covers topics in algebra beyond the introductory level, yet less than the precalculus level. No student who has satisfactorily completed MATH 131 or a higher-numbered mathematics course may subsequently receive credit for MATH 110. Prerequisite: C or, better in MATH 050 or satisfactory score on the department's placement examination. Each semester.\| MATH 111 Math Conc Grades K-8 Examines operations and properties of integers, fractions, and decimals. Includes elementary set theory, number theory, and functions. Covers conceptual foundations of the numerical content of the mathematics curriculum in the elementary and middle g,rades, emphasizing problem solving. MATH 111 is the first in a two-part sequence; the study of measurement, geometry, data gathering, and other topics are included in MATH 211. Prerequisite: C or better in MATH 050 or satisfactory score on the depart,mental placement examination. Each semester.\| MATH 112 Excursions Math Acquaints students with the nature and scope of modern mathematics and its applications. Emphasizes concepts and understanding rather than acquisition of techniques. Prerequisite: C or better in MATH 050 or satisfactory score on the departmental plac,ement examination. Each semester.\| MATH 113 Quantitative Reasoning Helps students develop quantitative reasoning skills. Core content, common to all sections, covers elements of descriptive statistics with particular emphasis on the representation and interpretation of data. Additional topics, covered at the instr,uctor's discretion, may include basic finance, probability, and logic. Prerequisite: MATH 112. Each semester.\| MATH 117 Mathematics for Health Science Mathematics for Health Sciences (MATH 117), a three (3) credit course, will prepare students in health science related majors for the mathematical applications within the healthcare field. Topics that will be included are algebra; fractions and frac,tional equations; ratios and proportions; percentages, variation; relationships among systems of measure; scientific notation; logarithms; calculation of dosages; and basic geometry, graphing, and statistics. The course will incorporate real world ex,amples in an effort to foster the development of critical thinking and problem solving skills necessary for success in today's healthcare space. Concepts included will be relevant for the various discipline specific certification exams. Prerequisite\| MATH 131 Applied Finite Math Covers mathematical techniques with special applications in business and related areas. Includes matrices, linear programming, and mathematics of finance. Prerequisite: MATH 110 or satisfactory score on the departmental placement examination. Each se,mester.\| MATH 170 Algebra & Trig Mid Sch Tchrs Investigates topics in algebra and trigonometry including rational expressions, exponential equations, linear equations and systems of linear equations, and right triangle trigonometry. This course is specifically designed to prepare students for to,pics in algebra and trigonometry that are encountered in the middle school classroom. Prerequisite: MATH 110 or equivalent Mathematics Department Placement and enrollment in the College of Education and Human Services.\| Students will explore mathematical reasoning and learn the basic logic behind mathematical proofs. Prerequisite: MATH 111 and 170 or equivalent. Restricted to students in the College of Education and Human Services.\| MATH 201 Technology in Teaching Math Investigates the use of technology in the practice of teaching mathematics. Topics will include the appropriate use of technology to both present and study mathematics. Prerequisite: MATH 110 or equivalent Mathematics Department Placement and enro,llment in the College of Education and Human Services.\| MATH 211 Fund Topics In K-8 Math Investigates selected topics considered essential to the basic mathematics curriculum in the elementary and middle grades. Includes introductions to mathematical reasoning, additional problem-solving techniques, probability and data analysis, geometr,y and measurement. Math 211 is the second in a two-part sequence beginning with MATH 111. Prerequisite: MATH 111. Each semester.\| MATH 212 Intuitive Geometry Intuitive overview of geometry; Euclid's Axioms, exploration of relationships, measurement and coordinate geometries, geometrics on other surfaces, and geometry in nature and art. Prerequisite: Any 100-level mathematics course. On demand.\| MATH 213 Intuitive Calculus Explores development of the basic properties of the real number system and the calculus, including functions, sequences, limits, continuity, integrals, and derivatives. Examines topics graphically, symbolically, and numerically. Prerequisite: Any 100,-level mathematics course. On demand.\| MATH 214 Finite Math Introduces basic properties of finite mathematics, including logic, counting techniques, elementary probability, and application to social and computer science. Prerequisite: Any 100-level mathematics course. On demand.\| MATH 215 Math Concept Lab Develops certain concepts of mathematics using an activity-oriented approach. Conducted in a laboratory atmosphere. Includes the rational number system, number theory, induction, measurement, geometric shapes. On demand. Prerequisite: Any 100-level, mathematics course.\| MATH 216 Prob Solv Strat In Math An introduction to the problem solving nature of mathematics. Focus is on using quantitative reasoning and intuitive logical thought techniques to solve problems (rather than formal rigid processes). Selected topics may include, but are not limited t,o, set theory, algebra, geometry, number theory, mathematical puzzles and games, estimation, and infinity. Prerequisite: 'C' or better in MATH 110 or equivalent mathematics placement and enrollment in the College of Education and Human Services.\| MATH 217 Historical Dev of Math Teacher Investigates topics in the history of mathematics, including development of number systems, development of important fields of mathematics. Some important mathematical ideas and developments will be seen as products of their times. Prerequisite: M,ATH 211. Restricted to students in the College of Education and Human Services.\|
Teaching Algebra to Middle School Students™ Online Course description: Evaluate the major concepts fundamental to teaching algebra to middle school students. Topics include algebraic number sense, order of operations, linear functions, properties of slope, and an introduction to quadratic equations. Explore methods of teaching mathematical concepts that employ the use of narrative and storytelling to describe, analyze, and solve contextual mathematical problems in real-life applications. (Writing to Learn Mathematics: Strategies That Work by J. Countryman, published by Heinemann, 1992, and Navigating Through Algebra in Grades 6–8 by S. Friel, published by National Council of Teachers of Mathematics, Inc., 2001, are required for this course.)
MTH60 Introductory Algebra- 1st Term Introduction to algebraic concepts and processes with a focus on linear equations and inequalities in one and two variables. Applications, graphs, functions, formulas, and proper mathematical notation are emphasized throughout the course. A scientific calculator is required. The TI-30X II is recommended. Prerequisites: MTH 20 and RD 80 (or ESOL 250). Audit available. (For detailed information, see the Course Content and Outcome Guide ). MTH 60 is a first term beginning algebra course that focuses on the symbolic algebra skills needed for further course work in mathematics and science. In this telecourse, you will watch lessons on Comcast Cable channel 27 (Portland area only), on your computer via youtube, or on VHS tapes that can be viewed at the library. Each week you will be expected to watch two lessons. Hopefully, you'll find the lessons thorough and comprehensive. Nonetheless, it will be important that you also read the corresponding sections of your textbook and do the suggested practice problems. You will also need to log in to Desire2Learn several times a week as I send announcements via Desire2Learn. The youtube links will also be available in Desire2Learn. You need to purchase the text at the Sylvania bookstore. THOUGHTS ABOUT DISTANCE LEARNING AND MATHEMATICS: It is not easy to learn mathematics via pre-recorded lessons. The time you would spend in class will instead be spent watching the lessons, reading and studying mathematics in your textbook. To be successful in this class, you must be an independent learner. This course, like other math courses, is time-intensive. Classes like this one typically require about five hours each week watching, pausing and reviewing the lessons (just like attending class) and an additional eight to twelve hours reading the textbook, doing homework, and studying. Please assess your situation, and determine if you will be able to commit this kind of time to the class. Also think about the type of learner you are. Teleweb courses are a terrific option, especially for independent, self-motivated learners. If this does not describe you, consider why it is you are thinking about taking this type of class, and if it really is a medium that will give you the best chance to succeed. DO NOT TAKE MTH 60 via TELEWEB BECAUSE YOU THINK IT WILL BE EASIER THAN AN ON-CAMPUS CLASS. For many students, it is HARDER via teleweb than on-campus. This is not a self-paced course. There are specific due dates and I do not accept late work. Course Specific Requirements: IMPORTANT DETAILS: There are three proctored paper-and-pencil, no-notes, no-books, no calculator exams (two midterms and a final). There will be scheduled times to take those exams at the Sylvania Campus, but if you live out of town, you can make arrangements with me to take the exam at an approved college testing center. If you live in the Portland area, you must take the exam at the Sylvania Campus. I do not offer testing at the other PCC campuses. Otherwise, the entire course can be completed from home. The five graded worksheets can be mailed. Students with disabilities should notify their instructor if accommodations are needed to take this class. For information about technologies that help people with disabilities in taking Web based distance learning classes please visit the Office for Students with Disabilities website.
MATH-329 Introduction to Teaching Sec School Math This course introduces prospective teachers to important curricular and pedagogical issues related to teaching secondary school mathematics. These issues are addressed in the context of mathematical topics selected from the secondary school curriculum. The course should be taken during the spring preceding student teaching. Prerequisite: A 300-level course in mathematics.
Description: "Do not worry about your difficulties in mathematics, I assure you that mine are greater." Einstein, Albert (1879-1955). If you are not Einstein, or are actually having difficulties with mathematics, let this website help. The page contains a wealth of online math resources. Some of these include: basic math, everyday math, pre-algebra, algebra, geometry, trigonometry, statistics, calculus, advanced topics, and math tutoring.
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
Have you ever been frustrated at calculation problems? Have you ever thought that algebra might be impossible? If you can read, you can learn algebra. Believe it or not, math is just a language--a language of patterns and numbers. Here's where you start. Ad Steps 1 Review every time your teacher teaches you. If you do not understand something, ask your parents, your friends, or even your older siblings. They would definitely be glad to help you. You should review the material very day that the teacher teaches you because who knows? Maybe the next day when you try to review, you realize that you have forgotten an important part. Review while it is still fresh in your mind. Ad 2 Do a prep study. Preparation studies help you. If you prepare yourself to "think in math language", you will know what the teacher is talking about the next time she teaches you a lesson. 3 Focus and ask questions during class. If you don't focus, you won't get much information out of what the teacher says. If you have questions about the lesson, you can ask the teacher, or, if you are a shy person, ask the person sitting next to you without distracting the class or the teacher. Or better yet, maybe you can ask your friends after class. It has to be as soon as the class is over so you don't forget. You can also ask your teacher after school, if he/she is free. 4 Study, study, study! But never cram. Some people think that cramming is good because it makes them study more. However, cramming will make you forget the material the next day when you have a test. Too many new things at once just won't stay in your mind. Two weeks before the test, study somewhere between 30 minutes to 1 hour everyday. 5 The more problems you solve, the better. Practicing is always the best way to study algebra. The point in solving problems is to get used to using the formulas! That way, you can solve the problems faster and you can think faster. Practice, practice, practice. This applies to anything, really. If you practice you will get better at it and you will succeed! 6 Never cheat! Never cheat on a test. Cheating can get you a big old goose egg, "0". You don't want that to happen do you? Cheating can lead to suspension and even expulsion, so if you ever feel the need to glance over and compare answers control yourself, you never know what the consequence can be..but the worst part is that when you cheat, you don't learn. If you go to school and you don't learn, the whole thing seems like a waste of time. Cheating cheats YOU out of an education. 7 Do not panic during a test. Your mind won't be able to think clearly and you will forget the formulas. If you have studied and can solve the problems you do for homework, you will probably do fine on the test. Do not panic. 8 Accept your grades. That's right, accept it. When you get your test back and find out that you have gotten a bad grade, accept it as your own responsibility. That will make you want to figure out how to learn the material. Most importantly, make sure to put in extra effort the next time you study for a test Did you get a bad grade because you didn't study enough? Did you make careless errors? Whatever the reason, locate it and fix it as soon as possible without getting too down on yourself. Look at your score. Do you want to make it better? Never think, "Hurray! I got 100! Now I don't have to study anymore!" If you think that, then you will get a bad grade next time because you will feel that there is no need to study. If you get a 100, just think, "I'll have to keep it up!" Make corrections. Look at the problems you got wrong and find out where you went wrong. Practice them again and again until you get it right. Review each day. The whole point of the tests you take in school is to figure out what things you still need to learn, then to work on those things. Reflect on your scores. 9 You will be better in algebra if you develop and keep up good efforts so that you can build your confidence by establishing good study habits! Ad Know another method for How to Become Better at Algebra? Add it here... Tips Practice daily. Not only it will help to memorize methods and steps, but also helps to increase your speed at doing math! If you are like most people, the best time to study is when you are fresh. If you are too tired, your mind can go blurry. Try a quick review of your studies in the morning as soon as you wake up, or maybe after breakfast. Go through simple calculations, review steps, techniques, and rules that you have already begun to understand before moving on to the parts that you don't yet understand. That will keep you from beating yourself up about something you don't understand. Studying right can clear your mind and might even help you have greater confidence toward other subjects. Try to understand, rather than plainly memorizing; the former leads you to appreciate algebra (and mathematics as a whole), and will help create an enjoyable environment for you to continue your journey into the world of mathematics. Ad Warnings Don't over study (called cramming for a test) so much that you get no sleep, get so tired that you can't see straight, or get too nervous to think. Take breaks and don't overdo it. Over-studying won't make you learn more, but it could give you a bad headache
Category: Format: Video, Level: Basic, Topic: GUI or GUIDE These postings are the author's and don't necessarily represent the opinions of MathWorks.
Basic College Mathematics The Tobey/Slater series builds essential skills one at a time by breaking the mathematics down into manageable pieces. This practical "building block ...Show synopsisThe Tobey/Slater series builds essential skills one at a time by breaking the mathematics down into manageable pieces. This practical "building block" organization makes it easy for readers to understand each topic and gain confidence as they move through each section. The authors provide a "How am I Doing?" guide to give readers constant reinforcement and to ensure that they understand each concept before moving on to the next. With Tobey/Slater, readers have a tutor and study companion with them every step of the way. Whole Numbers, Fractions, Decimals, Ratio and Proportion, Percent. For all readers interested in basic college mathematics
Every year, hundreds of the most beautiful people in the world come to New York to become models. At age fourteen, Cheryl Diamond was one of them. Living on her own in a run-down apartment, Cheryl spent her days on go-sees, runways, and shoots, surviving hand-to-mouth, while taking in everything she could about the tough and sleazy modeling industry. She watched other girls make mistakes, and swore she wouldn't be a victim. . . until a career-altering event changed her life and nearly ruined her shot at her dream. This is the riveting, true account of Cheryl's triumphant rise, disastrous fall, and phoenix-like comeback in one of the hottest and most demanding industries in the world. An innovative course that offers students an exciting new perspective on mathematics,Modeling With Mathematicsexplores how mathematics can help explore problems real people encounter in their jobs and lives. Mathematical modeling and a data-driven approach to exploring functions helps students deepen their mathematical skills and maturity. Modeling With Mathematics: A Bridge To Algebra IIhas been designed for students who have completed Algebra I or Algebra I and Geometry but need review practice and motivation to succeed in Algebra II. In addition the course gives students a look ahead to many Algebra II topics. Modeling With Mathematics: A Bridge To Algebra II list serv whfreeman. com/bridgelistserv. pdf As a service to instructors usingModeling With Mathematics: A Bridge To Algebra II, a listserv has been designed as a forum to share ideas, ask questions and learn new ways to enhance the learning experience for their students.The world we live in is complex and diverse. It contains the hardness of steel and the softness of cotton, the brilliance of diamonds and the blackness of coal, the heat of flames and the chill of ice, the tranquility of a slow moving river and the din of a busy city street. All the colors, textures, objects, animals, and plants that exist are too numerous to count. Yet scientists have learned that there are approximately 90 different ingredients, or elements, that make up all the material things we see and experience. Explore the world around you with pages of colorful photos, helpful illustrations, detailed Sample Problems, and hands-on activities using everyday materials. Learn how chemistry concepts are connected to your everyday life.Modern Livestock and Poultry Production, seventh edition is the textbook students and instructors turn to for thorough coverage of the animal agriculture industry. For each of the many species discussed in the text, students will find essential information about the breeds, selection of breeding stock, feeding and species management, diseases and parasites common to the species and marketing. Material is derived from the latest research and most current information about animal genetics, breeding, nutrition, biotechnology and product promotion. The text provides complete coverage of such major farm animal enterprises as beef cattle, swine, sheep, goats, horses, poultry and dairy cattle. Minor animal enterprises addressed include rabbits, bison, ratites, llamas and alpacas. Students will gain a comprehensive understanding of the livestock industry, including career opportunities. Modern Residential Wiring provides essential information about the tools, materials, equipment, and processes encountered in the electrical trade. The 2008 edition of this comprehensive textbook includes the latest information on installation and repair techniques, as well as recentdevelopments in wiring systems, personal protection equipment, and computer wiring. References to the 2008 National Electrical CodeRG are made throughout this text to reinforce the importance of installing residential wiring in a safe and professional manner. Apprentices, vocational students, andanyone interested in electrical wiring will find Modern Residential Wiring a valuable aid in learning how electrical systems are designed, installed, and maintained. Experienced electricians who want to review basic wiring techniques or study the recent developments in the electrical field will alsofind this book helpful. Based on the 2008 National Electrical CodeRG. Modern Welding is a comprehensive text that has long been the standard for teaching the theory, fundamentals, equipment, and techniques of welding technology. In addition to covering a very wide range of welding and cutting processes, the text includes thorough coverage of welding symbols, testing and inspection, and getting a job in the welding industryIn to better themselves and
Students may rely on calculators to bypass a more holistic understanding of mathematics, researcher says Nov 12, 2012 (Phys.org)——a mathematical function that describes a smooth repetitive oscillation—and—especiallyRelated Stories Calculators are useful tools in elementary mathematics classes, if students already have some basic skills, new research has found. The findings shed light on the debate about whether and when calculators should be used in ... Choosing a college is one of the most important financial decisions that a teenager - and his or her parents - will ever make. But families that are going through the college-application process have usually had little in ... Taken very literally, not all students are created equal — especially in their math learning skills, say Texas A&M University researchers who have found that not fully understanding the "equal sign" in a math problem could ... A new study co-written by a University of Illinois expert in math education suggests that incorporating technology in high school-level geometry classes not only makes the teaching of concepts such as congruency ... Discipline-based education research (DBER) has generated insights that could help improve undergraduate education in science and engineering, but these findings have not yet prompted widespread changes in teaching practice 18 As an engineering tutor, I'm seeing this more and more often. Students will come wanting an equation that fits every situation instead of recognizing that each problem is unique and often can be approached from different directions. They just want to get a number to write down and don't care where it comes from. I used to tutor Math in college to students of algebra and calculus. A calculator was useless in trying to get them to understand the key points. Like one of my professors said, you don't have to come up with the right number on a problem to get most of the credit. If your analysis and process were correct, you could get 90% of the credit for that question. I think that is what math is all about. It's a sign that these students are illiterate with the applied mathematics and science subject matters at hand. During my Mechanical Engineering undergraduate days people has HP 28S and I had my Casio fx 7000 which didn't do all the fancy stuff. Who finished their Machine Design exam first with the top score? I did. When you know the Calculus, DiffEq, Vector Equations, etc., all you do is reduce down to the correct units and then plug and play into the calculator. Calculators are the last step. It appears these kids don't see patterns in unit conversions and more. The sooner they do the sooner they can pay attention and understand the subject(s). I had a great teacher who always approach class the other way around; he would NEVER show any formula, but instead describe a real world problem where the class essentially would need to come up with a formula to solve it! This does require fundamental math understandings on the teachers part - we would typically end up with far more complicated formulas than those presented in the book, before we then looked at ways to simplify them and thereby find the formulas presented in the book! The really surprising result of the class with this this teacher was that though we initially we had quite a few students who hated math, at the end of the course all of them overcame that and several even stated they now liked math! Disclaimer - we had no calculator in our classes so I cant say if this would case a problem, however, I do kind of believe that teachers approach would be effective even when a calculator is present. Absolutely nothing new here. I noticed this problem in graduate school in the late '70s. I was a MBA student and took some engineering classes for fun. Most of the students had zero feel for correct solutions. They'd plug numbers into their calculators and even if they made a large error in data entry, just assume the answer was correct. I had received a bachelors in Engineering Physics just before calculators arrived on the scene and estimating the correct answer was a necessary skill.If so-why to bother students with some math at all? Why not to teach them MathLab or similar simulators directly?Anyone remember the slide rule? I still have several and although it may be faster and more accurate to use a calculator, the slide rule certainly gave one a better feel for the numbers and of course you had to take care of your powers of 10 yourself. With regular use you could almost visualise the result in your head and that certainly helps me to see when I have pressed the wrong key on my calculator even now. I used to tutor Math in college to students of algebra and calculus. A calculator was useless in trying to get them to understand the key points. That mirrors my experiences as a tutor. While calculators free them from the nitty-gritty stuff it is just that nitty-gritty stuff that you need to have a solid foundation in. Understanding builds on solid foundations. If you're unsure about the basics without your electronic crutch then you will not trust your own understanding of anything further up. And that insecurity translates into an inability to efficiently and effectively employ that understanding to solve new problems (or combine knowledge to find new ways of solving problems - e.g. in physics) If your analysis and process were correct, you could get 90% of the credit Yes. The numbers don't really matter. Because once you hit real life you WILL use computers to solve your math problems. (Almost no real problem can be solved analytically).Do you really want to carry the math simulator when you go shopping? Imagine the extra time required to determine whether the 8, 16, or 24 jar of peanut butter is cheaper and more this to everything else you want to buy.(FYI the biggest is not always cheapest.) Of course if you are wealthy it does not matter. Clearly you view on programming is to always buy more bigger and more hardware. Sorry, this does not work in the real world. Programmers need to optimize their code and optimizing compilers do not alway work. If so-why to bother students with some math at all? Why not to teach them MathLab or similar simulators directly? I have this silly little quirk. I just feel more comfortable with the idea that (for instance) the engineer who designed the bridge I use when driving to work has a good understanding of how numbers and equations work, which ones to use to solve which problems, and is not just blindly plugging data into a computer program. Me too. Especially when it gets down to doing any kind of serious analysis you will get into the area of statistics and simulations. If you have no real grasp of the intricacies of statistics (or the underlying principles and limitations of the simulation) then the likelyhood that you will get misleading results is enormous (as can be witnessed by many semi-informed statements in the comment sections on physorg, BTW). 'Blindly plugging in data' will just lead to GIGO (garbage in, garbage out) The problem is, that when the students use only simulation tools, they do not acquire an understanding of the underlying math and processes. You may beleive that the students - freed of the tedious calculating - would have more time to grasp the core concepts. All empirical data, however, indicates the opposite. When someone encounters a problem in the real world, knowing if the basic assumptions and if the results are reasonable is really valuable. Even routine simulations can have incorrect data at the input, and this is why judging if the result is reasonable or not is necessary for any true engineer. The problem is, that when the students use only simulation tools, they do not acquire an understanding of the underlying math and processes Indeed. And they can solve the real-life problems in addition! Whereas in contemporary version of educational system they cannot do both math, both real-life simulations well. What such learning is about, after then? In the same way, the programmers aren't required to understand the assembler and machine code programming. Why they should understand it? Will it help them in solving of real life problems better? Not to say, I've persistent problem even with professional physicists: they do understand math (sometimes) - but they (usually) don't understand the physics. They cannot imagine, what it is possible in it and what not. I know, where their problem actually is - they cannot imagine even the Schrodinger wave solution. They did never see, what the quantum mechanics or relativity really predict. Not to say about experimental physics. The problem is, that when the students use only simulation tools, they do not acquire an understanding of the underlying math and processes. This is correct - but who of them actually did see, how the differential or integral or another functional is working? They should understand the principle of the calculus - but should they spend their most intellectually productive time with memorizing of algorithms, which most of programs already handle a way better? That is to say, even the courses of math should be more illustrative and interactive. It will support the creativity of thinking and multidimensional imagination. I've no problem if the students at the secondary school level become familiar with quantum mechanics. But they should learn about it in interactive way. Unfortunately, we have no good simulators for it developed
: Maths, Science & Technical New Maths Frameworking From 13 To 14 English UK School Key Stage 3 0007268076 Detailed item information Description Year 9 Practice Book 3 is aimed at students working at levels 6-8, helping them to progress with confidence, succeed in the National Tests and take the next step towards GCSE. With hundreds of levelled practice questions corresponding to topics covered in Year 9 Pupil Book 3, it is ideal for extra class work, homework and catch-up classes. Key Features Author(s) Brian Speed, Keith Gordon, Kevin Evans, Trevor Senior Publisher HarperCollins Publishers Date of Publication 09/06/2008 Language(s) English Format Paperback ISBN-10 0007268076 ISBN-13 9780007268078 Genre School Textbooks & Study Guides: Maths, Science & Technical Series Title New Maths Frameworking Series Part/Volume Number No. 41 Publication Data Place of Publication London Country of Publication United Kingdom Imprint Collins Educational Dimensions Weight 280 g Width 192 mm Height 265 mm Pagination 64 Editorial Details Edition Statement 2nd Revised edition Age Details Educational Level UK School Key Stage 3 Interest Age From 13 To 14 Description Author Biography Collectively, Evans, Gordon, Speed and Senior have over 100 years of teaching experience, both in the classroom and leading maths departments. They all currently hold senior positions within examining bodies and have been extensively involved in the development and piloting of new specifications7.96
Discrete Mathematics 9780130890085 ISBN: 0130890081 Edition: 5 Pub Date: 2000 Publisher: Prentice Hall PTR Summary: For one or two term introductory courses in discrete mathematics. This best-selling book provides an accessible introduction to discrete mathematics through an algorithmic approach that focuses on problem- solving techniques. This edition has woven techniques of proofs into the text as a running theme. Each chapter has a problem-solving corner that shows students how to attack and solve problems. Johnsonbaug...h, Richard is the author of Discrete Mathematics, published 2000 under ISBN 9780130890085 and 0130890081. Fifty seven Discrete Mathematics textbooks are available for sale on ValoreBooks.com, fifty four used from the cheapest price of $0.76, or buy new starting at $40
More About This Textbook Overview Very little prior mathematical knowledge is assumed, other than the rudiments of algebra and geometry, so the book may be used as a source of enrichment material and project work for college students. A chapter on games using goldpoint tiles is included at the end, and it can provide much material for stimulating mathematical activities involving geometric puzzles of a combinatoric
This is the first semester in a four semester calculus sequence (175, 176, 177, 178, 277 and 279) for engineering, mathematics, computer science and science majors. You should have taken a precalculus class and a trigonometry class and passed with a grade of C or better in each. If this is your first college level math course, you should have taken 4 years of high school math, including Algebra I and II, Geometry, Trigonometry, and a course on functions. MTH 163 does not satisfy this requirement since Trigonometry is not a topic in MTH 163 and is a requirement for MTH 175. MTH 166 does satisfy this requirement MTH 271 is a one-semester introduction to topics in calculus, emphasizing applications, and primarily oriented for students who are business or social science majors. Concepts are developed more intuitively and less formally than in MTH 173. The prerequisite is a course that may have a title such as college algebra, precalculus or functions. All students enrolled in math classes can take advantage of the free tutoring service available at Academic Assistance on the NRCC campus in Dublin. Go to Martin Hall room 109 to sign up. The hours of operation are If you have never taken a college mathematics course you must take a math placement test before enrolling in any math courses. You must submit your application for admission before taking the placement test. The computerized placement test is divided into five levels of material: Prealgebra, Algebra, College Algebra, Geometry, and Trigonometry. Your answer in any particular section of the test will determine how many problems and which sections of the test you will see. As long as your answers are correct, the computer will continue to give you harder questions. Your answers will determine how many questions you will receive and the areas in which you will be tested. The computer will ask you to select a starting level for the test. Your choices are "easy" (primarily Arithmetic), "medium" (Algebra,) or "difficult" (College Algebra.) For placement purposes, you must begin the test in either "easy" or "medium" level. The test is untimed, but you should allow at least 2 hours to complete the test. The computer decides when you are finished. You must complete the test to receive a score. The placement test may be taken at most once per semester and at most three times per year. Hand-held calculators are not allowed. There is an online calculator for use during the test. Reviewing material that you have previously studied is recommended before you take the placement test. Sample questions for each area are available online and in the Counseling Center. Students can take the test at the Learning Center in Rooker Hall.
Office: MP 314 LEARNING OUTCOMES: At the end of this course, a student should be able to demonstrate: An understanding of the properties of the integers, rational numbers, real numbers and complex numbers. An understanding of the formula for solving a quadratic of cubic equation An understanding of the Euclidean algorithm for integers and polynomials. A basic understanding of the concepts of rings, fields and groups. TEXT: Course notes will be provided. A reference book is Abstract Algebra: A Geometric Approach by Theodore Sifrin (Prentice Hall) TESTS: There will be two exams on Friday, October 3 and Friday, November 14. Each will count 100 points towards your final grade. FINAL: The final is Friday, December 12 at 11:00-1:00. It counts 150 points towards your final grade and will be comprehensive. HOMEWORK: Homework will be assigned weekly and collected. You should try your best at all the homework problems. Each assignment will be assigned a score of 1-8 based on effort and results. You can obtain a maximum of 100 homework points. BOARD WORK: You will "volunteer" to do problems and proofs on the board during class. Every time you go to the board, you will get a score of 0-5 for the problem. 3 points will be based on "mathematics" and 2 points will be based on "presentation". You do not have to do a fancy presentation to get 2 points, but your presentation should be organized, everything you write should be legible, your speaking should be comprehensible and you should be able to talk and write together. Your score will be scaled to 50 points when compared to other scores in the class. GRADES: Your grade will be determined based on the following formula TESTS 200 points +FINAL 150 points +HOMEWORK 100 points +BOARDWORK 50 points TOTAL 500 points POINTSGRADE 450-500 A 400-449 B 350-399 C 300-349 D 0-299 F There will be no curve or extra credit assignments, so don't ask! If you ever have any questions or suggestions, feel free to come by my office at any time. I will definitely be there during my office hours, you can just stop by. You can also stop by or call to see if I am there at other times.
MIST Academy's Advanced Problem Solving program is aimed at students with very strong problem solving skills. It is suggested that students taking this class be confident in their ability to qualify for the American Invitational Mathematics Examination (AIME). Little time will be spent on traditional curriculum, except to extend and generalize important concepts. Topics covered will include synthetic geometry, the pigeonhole principle, the principle of inclusion-exclusion, mathematical induction, and many others. Some class time will be spent developing students' abilities to write rigorous proofs. The Advanced Problem Solving course is listed for grades 9 through 12, but MIST Academy actually places students in classes by ability, not age or grade. The grade range has more to do with the competitions that make up some of the focus of this course. Problems and lessons are taken from many sources including the AMC 12, the Mandelbrot competition, Mu Alpha Theta, the AIME, ARML, and even sometimes from Olympiad competitions such as the USA Mathematical Olympiad. The theme of the Spring Advanced Problem Solving class is Inequalities and Optimization. Topics include tactics and strategies for solving inequalities, the Power Mean inequality chain, the Cauchy Schwarz inequality, geometric inequalities, and much more. Much of the class will be aimed at the levels of problems seen at ARML, and on the AIME and USA Mathematical Olympiad (USAJMO/USAMO) exams. This material should also be helpful for students competing in Alabama math team events, or the Alabama ARML team. Additional course information, instructor information, and a complete schedule can be found here. Cost of enrollment in this course is $420. To enroll or ask questions about class, please contact us. The theme of the Summer Advanced Problem Solving class is Number Theory. Topics include Diophantine equations, modular arithmetic, Fermat's Little Theorem, Euler's Theorem, Pell equations, and much more. Part of the goal of this class will be to teach a rigorous development of these topics, including proofs of many famous and useful results. Another goal will be to prepare students for the kinds of problem solving techniques helpful to a wide variety of analytical disciplines such as those seen on the AIME and USA Mathematical Olympiad (USAMO) exams. This material should also be helpful for students competing in Alabama math team events, or the Alabama ARML team. More information can be found here. Cost of enrollment in the two-week course is $640, which includes all fees, materials, and registration. To enroll, please contact us. The theme of the Fall Advanced Problem Solving class is Geometry. Topics include advanced problems involving triangles and other polygons, 3D geometry, symmetries, centers of triangles, locuses, transformations, and much more. Some of the class focuses on learning to develop better proof writing skills. Much of the class will be aimed at the levels of problems seen on the AIME and USA Mathematical Olympiad (USAMO) exams. This material should also be helpful for students competing in Alabama math team events, or the Alabama ARML team. Click here for the Fall schedule. Cost of enrollment in the ten-week course is $420, which includes all fees, materials, and registration. To enroll, please contact us. The theme of the Winter Advanced Problem Solving class is Combinatorics. Topics include advanced problems involving generating functions, discrete and continuous probability, recursion, set theory, graph theory, and much more. Much of the class will be aimed at the levels of problems seen on the AIME and USA Mathematical Olympiad (USAJMO/USAMO) exams. This material should also be helpful for students competing in Alabama math team events, or the Alabama ARML team. A complete schedule can be found here. Cost of enrollment in the ten-week course is $420, which includes all fees, materials, and registration. To enroll, please contact us.
Peter Kattan Books This book includes the prayers of an eight-year old girl called Lara. She started praying and writing her prayers after her family went through a severe financial crisis. During their time of suffering, Lara turned to God where she found safety and protection. She also started scribbling and drawing. This book has her prayers and her drawings. The book has an elegant and devotional message. This is a concise article detailing mathematical equations showing when and where chaos occurs in linear algebraic equations and systems of linear simultaneous algebraic equations. The approach used is very simple and easy to understand by students. This concise article of twenty pages takes you on a short tour on how to solve algebraic equations using MATLAB. The presentation covers both numerical and symbolic (analytical) solution of equations. This article is taken form the bestselling book "MATLAB for Beginners: A Gentle Approach." This concise article of forty pages takes you on a short tour on how to handle matrices using MATLAB. Topics covered include how to generate matrices in MATLAB, different operations on matrices, and how to handle matrix, vector, and scalar quantities. Various MATLAB functions associated with matrices are also explored. This concise article takes you on a short tour on how to plot graphs using MATLAB. The presentation covers both two-dimensional and three-dimensional graphs. This article is taken form the bestselling book "MATLAB for Beginners: A Gentle Approach." This concise article of thirty pages takes you on a short tour on how to write small programs using MATLAB. The presentation covers both script files and function files. Several programming constructs are illustrated with simple examples. Loops are discussed using the For and While loops, while decisions are implemented using the If Else and Switch Case constructs. This concise article of twenty pages takes you on a short tour of MATLAB. The article covers briefly some selected MATLAB commands. The presentation is made in the form of solved examples showing the exact MATLAB commands used and the output. The solutions to all the exercises are also provided. This article is taken form the bestselling book "MATLAB for Beginners: A Gentle Approach." This book is written for beginners and students who wish to learn MATLAB. The material presented is very easy and simple to understand - written in a gentle manner. The topics covered in the book include arithmetic operations, variables, mathematical functions, complex numbers, vectors, matrices, programming, graphs, solving equations, and an introduction to calculus. This is a book containing simple mazes for children in kindergarten - ages 4 - 8 years and up. All the mazes in the book are designed specially for these children in mind. There are 78 mazes in the book with seven levels of difficulty categorized as: Super Easy, Easy, Easy-Medium, Medium, Medium-Difficult, Difficult, and Super Difficult. The mazes increase gradually in difficulty from the beginnin
This web page, authored and curated by David P. Stern, introduces vectors as an extension of numbers having both magnitude and direction. The initial motivation is to describe velocity but the material includes a... This online exercise lets students practice vector addition. They choose the precision of the test by selecting a target size, then estimate the sum of the two vectors by dragging and dropping a third arrow. Points are... This lesson was created by Larry Friesen and Anne Gillis for Butler Community College. It will help physics and calculus students differentiate between the uses of vectors in mathematics vs. physics. This website... Linked essay covering the beginning of the vector concept and the move away from coordinate methods through the beginning of the 20th century with Peano, Hilbert, Schmidt and Banach, with 13 references (books/articles). An experiment in the use of the World Wide Web as a teaching aid for a course in multivariable calculus, using Maple as a symbolic calculator. Topics include Review of Calculus 1; Vector Geometry; Geometric Algebra;...
LOCATION Statisticians, Mathematicians and Programmers, who are masters or Phd students studying in Mathematics and Computer Science. We can help you solve many advanced problems in Mathematics, Statistics, and Programming with C++, C#, Java, Visual Basic, Fortran, Matlab, Mathematica, Maple, R, SPSS, STATA, STATISTICA, Eviews, Minitab, etc.
Free Printable Algebra 2 Worksheets - Also Available Online There are a number of free algebra 2 worksheets for you to download, print, or solve online. The worksheets cover matrix operations, equations, and factoring quadratic expressions. Begin by selecting the free algebra worksheet you would like to have. This will take you to the web page of the algebra 2 worksheet. You then have several options. You can print the worksheet, download the corresponding PDF file, or complete the free algebra worksheet online. The online feature works as long as you are using a modern web browser, your iPad or other tablet device. Now you are all ready to start solving algebra equations.
ExploreLearning offers a catalog of modular, interactive simulations in math and science for teachers and students. They call these simulations Gizmos and they are meant to be fun, easy to use, and flexible enough to... A non-traditional Algebra text (high school and early college levels) placed on the Web by the Science Education Team at Los Alamos National Laboratory. Browse it on the Web or download a PDF version. Chapter headings... An idiosyncratic and personal selection of particularly important or particularly intriguing mathematical equations, not all of them complicated. Mathematical Constants; The definition of Pi; The definition of e; A... The algebra index of the extensive S.O.S.Math site has lessons and reference material on units of conversion, complex numbers, equations, and much more. Each section features a concise review, notation, examples, and...
Edgewater, CO Precalculus ...Building a differential equation is a major step in creating a mathematical model for a physical system, be it mechanical, electrical, biological, ecological, economic or anything else; it is a step that requires insight into the nature of the system. Solving a differential equation, which is wh...
Applied Mathematics e...more each chapter. Sample Problems accompany new concepts to show each step in the process involved. Practice Problems appear at the end of each major concept. Test Your Skills Problems cover all concepts taught throughout the chapter. Problem-Solving Activities allow students to relate concepts learned in the chapter to real-world problem solving with tools and materials. (less) Hardcover, 397 pages Published January 25th 2010 by Goodheart-Willcox (first published January 1st 1997)
MATH 308 – Numerical Analysis Carl Toews Spring 2008 Welcome to Numerical Analysis! The techniques you will learn in this course form the bedrock of what is generically called scientific computing. At the heart of most computational problems in industry, engineering, or science lies a basic question: what is the relation between the computer's answer and the true answer? If computers had infinite precision, and if programers never made errors, and if mathematical models were perfect, one might reasonably hope the two answers to be identical. The world being as it is, however, we are reduced to seeking bounds on the extent to which the two answers can diverge. The purpose of this course is thus twofold: on the one hand, to develop a body of techniques for solving applied problems, and on the other, to formally estimate the accuracy of the solutions. This is a math course. As such, it will focus on proofs and logical arguments. But the purpose of the mathematical results is to provide rigorous performance guarantees for computer implementations, and in this sense the course is also a computer course. Much of the beauty of this subject lies in the way the formalism of the mathematics captures the awesome computational power of the modern desktop. Expect to both prove and program in this course. Text: Numerical Analysis, 8th Edition, by Richard Burden and Douglas Faires. Class Hours: MWF, 10:00-10:50, ColH 446. Course/Topic Agenda Week Topic(s) Week 1-2 Error Analysis Weeks 3-4 Solutions to Equations in One Variable Weeks 5-6 Solutions to Linear Systems Weeks 7-8 Solutions to Nonlinear Systems Weeks 9-10 Interpolation Weeks 11-12 Integration and Differentiation Weeks 13-14 Solving Differential Equations The prerequisites for this course are solid coursework in calculus, linear algebra, and ordinary differential equations. (Though as the mathematician Paul Halmos says, "The beginner should not be discouraged if he finds he does not have the prerequisites for the prerequisites"–be bold, take the plunge.) Homework, Quizzes, and Participation (20%) You learn mathematics by solving problems: the importance of doing homework cannot be overstated. In general, I will assign homework on Monday, and it will be due the following Monday. You are encouraged to work with other students, but must write up your own solutions. Grading will be purely cursory, so it will be easy to get all the homework points. However, tests and quizzes will largely follow the homework, so it behooves you to have a thorough understanding of the problems. Project (10%) The project is a paper of about five pages in which you analyze a problem of your choice. It will be due the last day of class–more information will be distributed later in the semester. Exams (40%) There will be two cumulative in-class exams, spaced at roughly equal intervals throughout the semester. As with quizzes, exams will be based on homework problems, so the best preparation is to thoroughly understand the homework. Tentatively, the exams will be on February 14 and March 27. Final (30%) A comprehensive closed-book exam to be administered Thursday, April 24th, from 8:45 a.m. to 10:45 a.m. in 446 College Hall. Grading Final grades will be weighted as per above, with letter grades roughly as follows: A 90-100 B 80-89 C 70-79 D 60-69 F < 60 Pluses and minuses will be assigned to scores on the high or low ends of the scale, respectively. Getting Help Since this is a rather high level class, finding tutors is not easy. But your peers are a good substitute: a group of minds is generally a better problem solver than a mind working alone. You are encouraged to discuss the material with your classmates, and to work on the homework in groups. (Caveat: homework can be solved in groups, but should be written up independently.) You should also feel free to drop by my office, even if it's just to say hello or ask a quick question. If my office hours don't work for you, drop in some other time, shoot me an email, or give me a call. (Contact information below.) Statement on Disabilities Students with documented disabilities are entitled to reasonable accommodations if needed. If you need accommodations, please contact the Office of Freshman Development and Special Student Services in 309 Duquesne Union (412-396-6657) as soon as possible. Contact Information Carl Toews College Hall 418 Tel: (412) 396-4851 toewsc@duq.edu Office Hours : MW 11-12, T 10-10:50, or by appointment "A scientist worthy of his name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature." –Henri Poincar´
Curriculum and Assessment (5 ECTS): In this module, students examine principles, theories and international trends in curriculum and assessment and engage in the curriculum planning process with a particular focus on contemporary issues in Irish curriculum policy and practice. Practical Teaching Programme: Special Educational Needs (5 ECTS): Student teachers will complete 20hrs in the classroom with a focus on issues of catering for diverse learners and accommodating students with special educational needs. Tasks will engage student teachers in designing learning activities and creating resources and materials to support a variety of learning needs. History of Mathematics (5 ECTS): A selection of topics concerned with the ancient and recent history of mathematical thinking, and the development of modern mathematics. Topics will include the history of number theory, the history of geometry, the history of calculus, and major developments in the last century. Discrete Mathematics (5 ECTS): This module offers an introduction to combinatorial mathematics, including the following topics: enumeration techniques, permutations and combinations, graph theory, algorithms and applications. Semester 2: Professional Studies: Integrated Project (5 ECTS): In this module, students draw from learning throughout the programme to design their own summer programme. In a collaborative effort, they consider important issues of school organisation, administration, curriculum design and teaching and learning to develop a two-week mathematics enrichment programme for local second-level pupils. On an administrative level, students create a programme handbook, outlining its aims, structure and policies. They design the curriculum based on the potential pupils, advertise their programme and offer it during the summer as part of their Practical Teaching Programme. Mathematical Software (5 ECTS): The aim of this module will be to introduce students to the many software packages that are available to mathematicians for various purposes including research, learning, teaching, preparation of presentations and mathematical typesetting. Complex Variables (5 ECTS): This module entails a detailed study of the theory of complex numbers including their arithmetic properties and an introduction to the analysis of functions of one complex variable. Topics to be covered include : De Moivre's theorem, roots of unity, analytic functions and the Cauchy-Riemann equations, Cauchy's Integral Theorem, the Residue Theorem and applications. Full-Year Modules: Psychology, Sociology and Catering for Diversity (10 ECTS): This module presents a range of theoretical perspectives in Psychology and Sociology and offers practical support for both meeting diverse educational needs and managing important variables such as social class, race and ethnicity in the context of the post-primary school. The focus is on preparation for promoting educational inclusion in diverse settings and developing strategies for supporting the needs of diverse learners in the mainstream classroom. Applied Mathematics: Mathematical Modeling (10 ECTS): In these modules students will investigate applications of mathematical and statistical theory and uses of mathematical models in diverse fields. Summer Session: Practical Teaching Programme: Mathematics Enrichment Programme (5 ECTS): This programme will run during the summer following Year 3. Student teachers are notified of this component from Year 1 so that they can plan for it in advance; however, in the case that someone is unable to participate outside of the University's regular academic timetable, s/he may petition for completion of an alternative placement during the semester but must still partake in all planning and preparation efforts throughout the semester Thursday, April 22, 2010
, it looks like Mathematics HL includes everything up through calc (including diff eq and series), and some linear algebra. You can almost do actual physics! The books that you have been reading look more pop-sci that it sounds like you are interested in, so here are some options that are a little more rigorous. Having said that, you need to know the basics before you can learn the upper level stuff: IB HL Math may or may not include some diff eq and series topics. IB gives each school a choise of option topics to cover, happens with all the sciences and math. The other topics offered are stats, discrete, and some group topics if I remember correctly. So there's a possibility that the diff eq and series topics may not be covered. If diff eq hasn't been seen, griffiths QM isn't going to make much sense to start. In that case, I would go through your calc book and look at a few differential equation sections (hopefully there are some). It should at least make things make more sense. Here is the IB HL Math curriculum, so the only diff eq and series presented are taylor series, convergence tests, Euler's method, homogeneous diff eq's, and using integrating factors, if you're interested. The topic outline begins on p. 44, with the core calc curriculum p. 30-34.
An accessible introduction to real analysis and its connection to elementary calculus Bridging the gap between the development and history of real analysis, Introduction to Real Analysis: An Educational Approach presents a comprehensive introduction to real analysis while also offering a survey of the field. With its balance of historical background, key calculus methods, and hands-on applications, this book provides readers with a solid foundation and fundamental understanding of real analysis. What better way to teach your children the English language than with Walt Disney characters they all love! A perfect combination between education and diversion: stories, songs and games have been developed and made by a team of qualified experts in the education of English. Disney Magic English is a fun way for children to learn the English language along with their favourite Disney characters! The interactive DVD and activity books bring educational adventures to life with stories, songs, and games! Disney?s Magic English makes learning fun and enjoyable ? just like playing a game! Each part of the program has been developed in close collaboration with highly qualified education specialists in the English Language
Contact Mathematics - Grade 5 Curricular Materials Submitted by Anonymous (not verified) on Tue, 09/27/2011 - 12:54pm Posted: Tue, 09/27/2011 - 12:54pmUpdated: Fri, 02/22/2013 - 2:55pm The reviews below represent an analysis of curricular materials aligned to the Common Core State Standards. These reviews are intended only to provide support for corporations and should not be viewed as the only options for textbook adoption at the local level. Textbooks were reviewed for alignment to two parts of the Common Core State Standards: the Standards for Mathematical Content and the Standards for Mathematical Practice. The review of content standards was conducted by Indiana teachers and educators, and the review of practice standards was conducted by the Charles A. Dana Center. To review comments made by either group, please download the PDF document in the appropriate column below. Minimal: This is the lowest rating a book can receive. In general, a book that was rated as "minimal" scored mostly 1s and 2s on a 4-point scale. Limited: This is the middle rating a book can receive. In general, a book that was rated as "limited" scored mostly 2s and 3s on a 4-point scale. Moderate: This is the highest rating a book can receive. In general, a book that was rated as "moderate" scored mostly 3s and 4s on a 4-point scale.
books.google.com - This is a graduate-level text on algebraic geometry that provides a quick and fully self-contained development of the fundamentals, including all commutative algebra which is used. A taste of the deeper theory is given: some topics, such as local algebra and ramification theory, are treated in depth.... Geometry
Mathematics Welcome to Mathematics at Springwood High School. This page aims to provide students and parents with information regarding Mathematics at our school. Subject Description Mathematics is used to identify, describe and apply patterns and relationships. It provides a precise means of communication and is a powerful tool for solving problems both within and beyond Mathematics. In addition to its practical applications, the study of Mathematics is a valuable pursuit in its own right, providing opportunities for originality, challenge and leisure. Mathematics education provides opportunities for students to develop their numeracy (basic skills), be able to solve mathematical problems, to communicate and make informed decisions after analysing data. The syllabus develops students' mathematical thinking, understanding, competence and confidence in the application of Mathematics. Subjects Year 7 and 8 Mathematics Stage 4 Year 9 and 10 Mathematics Stage 5.1, 5.2, 5.3 and Life Skills CSC Life Skills, General Mathematics, Mathematics, Mathematics Extension I Mathematics Extension II is available for Senior Year 2 students who have completed Mathematics Extension I in Senior Year 1 Curriculum The NSW Board of Studies syllabi target the strands of number, algebra, measurement, trigonometry, functions and relations, chance and data, and calculus. RoSA Satisfactory completion of the mandatory study of Mathematics during Stage 5 (year 9 and 10) will be recorded with a school assessment of either A10, A9, B8, B7, C6, C5, D4, D3 or E2 Equipment Students need to be prepared for every lesson. In all years students will need a grid book, a geometry set, pens, pencils and an eraser. A scientific calculator is imperative for every lesson. Most scientific calculators are acceptable. The school office sells SHARP calculators at a reasonable price. Senior General Mathematics students would also benefit from having a graphics calculator. Homework Homework is a means of consolidation, extension or enrichment of the work that is studied in Mathematics lessons. Homework is set regularly via a homework sheet written specifically for each class and is a compulsory part of a student's workload. Homework is followed up regularly, and students found to be neglecting to complete their homework will have a note sent home, informing their parents/carers of the situation, and encouraging completion of the work in question. Maths on Line All students at Springwood High School have been encouraged to registered for Maths on Line which is a fabulous learning/teaching resource. Students can purchase a login for $20 at the front office then the receipt should be taken to Mrs Hume so each student can be issued with an individual login and password. The site can be accessed from Some of the features of this program include: Full lessons for each concept with a voice over and numerous examples of varying difficulty. Every lesson has a pause and rewind facility to help students grasp concepts at their own pace. Every lesson has an accompanying printable worksheet to consolidate the concepts, with full worked solutions provided also. A student's ongoing use and progress in the program is stored for them and their teacher's perusal at any time. The program can be accessed from any internet ready site, not just at school. The entire Year 7 to senior programs is available to every student, not just their particular year group. Australian Mathematics Competition The competition is held in August each year. All students should look out for the announcements at the beginning of each year for entry information.
Discovering Geometry Overview Build your students' reasoning and proof abilities with Discovering Geometry, a CCSS-aligned high school curriculum that helps students internalize geometric properties as they test their reasoning with physical models. Click here for an overview of the Discovering Mathematics series. To sign up for a free 30-day online trial, click here. The developmental focus of Discovering Geometry will help you meet your students where they are and guide them all to a high level of understanding. You'll find many opportunities both to support students whose progress is slower and to challenge more advanced students. Discovering Geometry will help you bolster your students' understanding and retention of theorems, definitions, and properties as they perform constructions, measure figures, relate patterns and properties, and discuss their findings. Algebra Review exercises—integrated when applicable to the geometry being taught—will make it easy to reinforce your students' skills and keep them on track for continued learning. You will build your students' reasoning and proof skills by using the exercises and group activities in the new Developing Proof strand. Investigations, visual representations, and opportunities for discussion will enable you to integrate multiple teaching modes into your classroom so that visual, auditory, and kinesthetic learners all benefit. The real-world applications, puzzles, and extensions in Discovering Geometry will help you motivate students and keep them thinking. The book's focus on critical problem-solving skills will help you cultivate a classroom of self-motivated, independent thinkers. Author Michael Serra's extensive classroom experience helped him shape a new approach to teaching geometry. Now in its fourth edition, Discovering Geometry has proven effective in countless classrooms across the country.
Synopses & Reviews Publisher Comments: This contemporary first course focuses on concepts and ideas of Measure Theory, highlighting the theoretical side of the subject. Its primary intention is to introduce Measure Theory to a new generation of students, whether in mathematics or in one of the sciences, by offering them on the one hand a text with complete, rigorous and detailed proofs--sketchy proofs have been a perpetual complaint, as demonstrated in the many Amazon reader reviews critical of authors who "omit 'trivial' steps" and "make not-so-obvious 'it is obvious' remarks." On the other hand, Kubrusly offers a unique collection of fully hinted problems. On the other hand, Kubrusly offers a unique collection of fully hinted problems. The author invites the readers to take an active part in the theory construction, thereby offering them a real chance to acquire a firmer grasp on the theory they helped to build. These problems, at the end of each chapter, comprise complements and extensions of the theory, further examples and counterexamples, or auxiliary results. They are an integral part of the main text, which sets them apart from the traditional classroom or homework exercises. JARGON BUSTER: measure theory Measure theory investigates the conditions under which integration can take place. It considers various ways in which the "size" of a set can be estimated. This topic is studied in pure mathematics programs but the theory is also foundational for students of statistics and probability, engineering, and financial engineering. Key Features * Designed with a minimum of prerequisites (intro analysis, and for Ch 5, linear algebra) * Includes 140 classical measure-theory problems * Carefully crafted to present essential elements of the theory in compact form
Product Description The Algebra 2 Tutor DVD Series teaches students the core topics of Algebra 2 and bridges the gap between Algebra 1 and Trigonometry, providing students with essential skills for understanding advanced mathematics. This lesson teaches students the different forms of the equation of a line. The y-intercept is presented along with the point-slope form with numerous examples that show how to write the equation of the line when given the slope and a point on the line. In addition, parallel and perpendicular lines are also covered. Grades 8-12. 31
Trigonometry (2) 2000 Our web site, accessible in both Polish and English, introduces the user to four basic trigonometric functions: sine, cosine, tangent, and cotangent. Each function is defined in the context of a right triangle and a coordinate system, with diagrams for each. Graphs of each function are also shown, and reduction formulas are given. Information pertaining to the functions as a whole include: mathematical relationships between the functions, an overview of reduction formulas, and basic trigonometric formulas. Finally two sample problems are provided to test what the user has learned. About this site 1998 This is a great trigonometry tutorial, put together by a team of young mathematicians. A nice help section explains the site's navigation controls. Much of the text can also be heard since an audio player option is provided. All the necessary formulas and equations that you should know in order to master trigonometry are explained. The extensively used bulletin board allows visitors to pose trig questions and get help when they're stuck. About this site
Barron's E-Z Precalculus - 10 edition Summary: An experienced math teacher breaks down precalculus into a series of easy-to-follow lessons designed for self-teaching and rapid learning. The book features a generous number of step-by-step demonstration examples as well as numerous tables, graphs, and graphing-calculator-based approached. Major topics covered include: algebraic methods; functions and their graphs; complex numbers; polynomial and rational functions; exponential and logarithmic functions; trigonometry...show more and polar coordinates; counting and probability; binomial theorem; calculus preview; and much more. Exercises at the end of each chapter reinforce key concepts while helping students monitor their progress. Barron's continues its ongoing project of improving, updating, and giving contemporary new designs to its popular Easy Way books, now re-named Barron's E-Z Series. The new cover designs reflect the books' brand-new page layouts, which feature extensive two-color treatment, a fresh, modern typeface, and many more graphics. In addition to charts, graphs, and diagrams, the graphic features include instructive line illustrations, and where appropriate, amusing cartoons. Barron's E-Z books are self-teaching manuals designed to improve students' grades in many academic and practical subjects. In most cases, the skill level ranges between senior high school and college-101 standards. In addition to their self-teaching value, these books are also widely used as textbooks or textbook supplements in classroom settings. E-Z books review their subjects in detail and feature short quizzes and longer tests to help students gauge their learning progress. All exercises and tests come with answers. Subject heads and key phrases are set in a second color as an easy reference aidVeryGood TGChavez La Mesa, CA Paperback Very good Clean copy w/ very light signs of use6.50 +$3.99 s/h LikeNew Pre-owned Books, etc. Vienna, VA The lower right corner of the front cover is bent. The inside pages are unused, unmarked and in perfect condition. $6.95 +$3.99 s/h VeryGood arcfoundationthriftstore Ventura, CA 0764144650
Educational Use Keyword: Quadratic Equations (46) Number systems and the rules for combining numbers can be daunting. This ... (more) Number (less) The students will develop an algebraic expression from geometric representations and ultimately ... (more) The students will develop an algebraic expression from geometric representations and ultimately graph quadratic equations with understanding. The students will also develop a better understanding of algebraic expressions by comparing with geometric, tabular, and graphical representations. (less) This applet is designed to allow students to explore how the coefficients ... (more) This applet is designed to allow students to explore how the coefficients of a quadratic equation affect the shape and location of its graph. The applet can be used to enable students to discover a formula for the Axes of symmetry equation, or to use the (less) This applet allows students to change the Vertices of a parabola and ... (more) This applet allows students to change the Vertices of a parabola and how it affects the equation of the parabola. This applet uses a = 1 for all of the parabolas. Integer values for Vertices location. (less) This lesson unit is intended to help teachers assess how well students ... (more) This lesson unit is intended to help teachers assess how well students are able to solve quadratics in one variable. In particular, the lesson will help teachers identify and help students who have the following difficulties: making sense of a real life situation and deciding on the math to apply to the problem; solving quadratic equations by taking square roots, completing the square, using the quadratic formula, and factoring; and interpreting results in the context of a real life situation. (less)
Finite Mathematics for the Managerial, Life, and Social Sciences - With CD - 8th edition ISBN13:978-0534492144 ISBN10: 0534492142 This edition has also been released as: ISBN13: 978-0495010289 ISBN10: 0495010286 Summary: In® Excel exercises to help stimulate student motivation. An exciting new array of supplements, in...show morecluding iLrn Tutorial and the Interactive Video Skillbuilder CD-ROM, provides students with extensive learning support so instructors will have more time to focus on teaching the core concepts. ...show less The Cartesian Coordinate System. Straight Lines. Using Technology: Graphing a Straight Line. Linear Functions and Mathematical Models. Portfolio. Using Technology: Evaluating a Function. Intersection of Straight Lines. Using Technology: Finding the Points of Intersection of Two Graphs. The Method of Least Squares (Optional). Using Technology: Finding an Equation of a Least-Squares Line. Summary of Principal Formulas and Terms. Review Exercises. Compound Interest. Using Technology: Finding the Accumulated Amount of an Investment, the Effective Rate of Return, and the Present Value of an Investment. Annuities. Using Technology: Finding the Amount of an Annuity. Amortization and Sinking Funds. Portfolio. Using Technology: Amortizing a Loan. Arithmetic and Geometric Progressions (Optional). Summary of Principal Formulas and Terms. Review Exercises. 6. SETS AND COUNTING. Sets and Set Operations. The Number of Elements in a Finite Set. The Multiplication Principle. Portfolio. Permutations and Combinations. Using Technology: Evaluating n!, P(n, r), and C(n, r). Summary of Principal Formulas and Terms. Review Exercises.
Math Inductive And Deductive Reasoning Pdf Inductive Versus Deductive Reasoning - University of Houston although inductive reasoning can sometimes lead to false conclusions it can often be a useful first step in the process of applying deductive reasoning to determine Inductive Versus Deductive Reasoning - University of Houston House and Holmes A Guide to Deductive and Inductive Reasoning house and holmes a guide to deductive and inductive reasoning summary maybe youve seen dr house in action figuring things out from what seem like totally House and Holmes A Guide to Deductive and Inductive Reasoning 1 2 An Application of Inductive Reasoning Number Patterns 1 2 an application of inductive reasoning number patterns 15 figure 5 example 3 use the formulas to find each of the following a the seventh triangular 1 2 An Application of Inductive Reasoning Number Patterns 5TH GRADE MATH SCOPE AND SEQUENCE - Cleveland 5th grade math scope and sequence july 2008 early third quarter number number sense and operations e use order of operations including use of 5TH GRADE MATH SCOPE AND SEQUENCE - Cleveland
A portion of the Partnership for Plant Genomics Education, hosted by the University of California-Davis, this PDF presents a student activity where students will use agarose gel electrophoresis to separate several... Created by Work-Ready Electronics, a project of the Advanced Technological Education program, this module walks visitors through the basics of micro embedded controllers. The content of the site is divided into four... GeoGebra is a free and multi-platform dynamic mathematics software for schools that joins geometry, algebra and calculus. As an interactive geometry system, GeoGebra can help you do constructions with points, vectors,... The National Security Agency (NSA) has worked to craft these educational materials they are calling "concept development units" (CDUs). The units are divided into 11 sections, including Algebra, Calculus, and Data... The documents presented here as URLs are a selection from a larger course, offered at Madison Area Technical College, designed to improve the ability of students to complete the various types of math problems commonly...
TPP7180 Preparatory Mathematics Units : 1 Faculty or Section : Open Access College School or Department : Open Access College Version produced : 10 March 2014 Synopsis Using concepts of self-paced instruction and adult learning the course guides students through a carefully sequenced series of topics which will provide the foundation for mathematics that will be encountered in tertiary studies and further pre-tertiary studies in mathematics. The self-paced structure allows students to work at their own pace within a specified framework developing confidence with mathematics, mathematical communication and general problem solving.
The Math You Need When You Need It (TMYN) tutorial modules were introduced as a pilot project into introductory geoscience courses at Los Angeles Valley College in summer 2010 at a time when the Oceanography and Geology laboratory courses were experiencing shrinking enrollments and completion rates. Students expressed frustration and surprise that introductory laboratory courses required math skills, and the fear of, and deficiency in basic math skills appeared to be major obstacles to student success. Based on the success of the pilot project, TMYN tutorial modules were incorporated into the Geology and Oceanography laboratory courses which meet once a week for three hours. Although the modules remain open to the students 24 hours a day for the entire semester, the first 45 minutes of the three-hour lab is dedicated to reviewing and practicing the module/s needed to complete the laboratory exercise for that class session. Students complete the required module assessment during the laboratory class if time permits, or on their own. This session will include discussions on the development and selection of module content and the implementation process. In addition, assessment results from the laboratory classes will be presented and compared, as well as the results of a student attitudinal survey toward math which has been administered since the pilot program in 2010.
Intermediate Algebra - 9th edition Summary: The new edition of INTERMEDIATE ALGEBRA is an exciting and innovative revision that takes an already successful text and makes it more compelling for today's instructor and student. The new edition has been thoroughly updated with a new interior design and other pedagogical features that make the user both easier to read and easier to use. Known for its clear writing and an engaging, accessible approach that makes algebra relevant, INTERMEDIATE ALGEBRA helps users to develop problem-...show moresolving skills and strategies that they can use in their everyday lives. The new edition welcomes two new co-authors Rosemary Karr and Marilyn Massey who along with David Gustafson have developed a learning plan to help users succeed in Intermediate Algebra and transition to the next level in their coursework46 +$3.99 s/h Acceptable Borgasorus Books, Inc. MO Wentzville, MO Hardcover Fair 049583142520.28 +$3.99 s/h Good SellBackYourBook Aurora, IL 0495831425
Algebra 1/2 covers traditional algebra 1/2 topics, as taught in the 2nd Edition Saxon textbook, as well as topics from geometry and discrete mathematics. For use with 2nd Edition texts. Please Note! The current edition of Saxon Math Algebra 1/2 is the 3rd Edition. This 2nd edition is offered for families using older versions of Saxon. Saxon DIVE CD Algebra .5 You sent me a link to view demonstrations of the DIVE CD products for all the Saxon math but I needed a sample of the second edition and it only had the third edition. Please could you send me a sample of the second edition DIVE CD saxon algebra.5 and not the third edition. Thank you. asked 2 years, 8 months ago by Sarah Zimbabwe on Saxon Math Algebra 1/2, 2nd Edition DIVE CD-Rom 0points 0out of0found this question helpful. 1 answer Answers answer 1 No sample clips are available for the 2nd edition. The format and lesson presentation would be same as the 3rd edition, but would correspond to the lessons in the 2nd edition book.
Course Materials (What You'll Need) from the bookstore: • A copy of Flatland, by E. A. Abbott. • Rules for Writers, by Diana Hacker • Viewpoints: Mathematical Perspective and Fractal Geometry in Art miscellaneous • Access to a word-processor and to an e-mail account. • A stapler. • A brain. from the art store (see the attached list) Attendance You will be learning a lot from your classmates and they will be learning a lot from you, so I expect that you come prepared, with proper materials, and that you participate fully. Please be advised that Math Department and F&M policy state that penalties (including grade reduction and/or dismissal from the course) may be assessed for excessive, unexcused absences. Weekly Writing Assignments With a few exceptions, you will have a 1-page paper due every Wednesday at 4:45 p.m., and I will count the best eight of these papers toward your final grade. There are 14 weeks in the class, and you will have two 5-page papers due (see below); there will be one Wednesday with no paper because of Thanksgiving break. This means you may choose to skip writing a paper twice during the semester without any serious danger to your grade. I will usually announce the topic of these papers one week in advance, so that you may begin writing during the week and revise your paper over the course of the following weekend. You should write these papers in a nice 12-point font; the margins should be no narrower than 0.75 inches; the lines should be 1.5 spaced, and there should be a title, the due-date, and your name at the top of the page. You may NOT hand in more than one page, despite various temptations to do so. I want you to think hard about making your writing as concise and forceful as possible. Weekly Math /Art Assignments Every week, you will have assignments that involve drawing. You will be learning to draw by using math (or perhaps you will be learning to solve mathematics problems by drawing). A good drawing takes time; I expect that you will spend at least 2 hours on your sketches, so make sure you carve out time for this. Most students tell me that these sketches (a) take longer than they would have expected and (b) give them a real sense of accomplishment. The long papers Over the course of the semester, you will "adopt" a piece of art that uses (or perhaps deliberately misuses) perspective to create an illusion—I will say more about this assignment once class begins. During the first few weeks of class, you will choose the piece of art and make sure that I approve of your choice. In October, you will write a paper describing the piece and placing it in an artistic context. In November, you will write a paper describing the mathematics of the perspective in this piece. Each of these two papers will become part of the reading assignments for the entire class. For this reason, each of these papers will have 3 different deadlines: one for one-third of the class; another for another third of the class; and a third for the last five students. We're going to learn a lot about library research: this link (developed by our class librarian, Louise Kulp) might help you get started. The class times on the Fridays after these assignments are due will be devoted to discussing and critiquing the papers that your peers have written. All students will be expected to participate in these discussions. On December 16, the final ten-page paper will be due (your final art project will be due the same day). This paper will describe both the artistic and mathematical aspects of your work . . . that is, it will combine and improve upon the ideas and prose of the previous two papers. Reading Assignments I expect you to read your text, your classmates' papers, and (eventually) Flatland and to be prepared to discuss these assignments. In addition, you will be reading scholarly work related to your "adopted" piece of art. Help I love to talk about math, art, or life in general. Please feel free to ask me questions both in and out of class; I've got a lot of office hours and am around a good deal of the time. I'm even more congenial if you bring me chocolate. Your preceptor, Andrew Masterleone, is brilliant and creative. In addition, he writes prose that is specific and evocative. May you do the same. The Writing Center (717-291-3866) is available for those who want help with their written projects, and there are also several undergraduate mathematics teaching assistants who hold regular hours (their schedules will be announced shortly).
ALEX Lesson Plans Title: Now, where did THAT come from? Deriving the Quadratic Formula DescriptionStandard(s): [MA2013] AL1 (9-12) 18: Solve quadratic equations in one variable. [A-REI2 (9-12) 4: Solve quadratic equations with real coefficients that have complex solutions. [N-CN7] [MA2013] ALC (9-12) 6: Use the extreme value of a given quadratic function to solve applied problems. (Alabama) Subject: Mathematics (9 - 12) Title: Now, where did THAT come from? Deriving the Quadratic Formula Description Subject: Business, Management, and Administration (9 - 12), or Mathematics (9 - 12) Title: What's The Real Cost of That Car? Description: This is a Commerce and Information Technology lesson plan. A project requiring research, critical thinking and complex decision-making about factoring all the costs of purchasing a large ticket item... a car. Thinkfinity Lesson Plans Title: Modeling Orbital Debris Problems Description: In Standard(s): [S1] PHS (9-12) 7: Relate velocity, acceleration, and kinetic energy to mass, distance, force, and time. [S1] PHY (9-12) 1: Explain linear, uniform circular, and projectile motions using one- and two-dimensional vectors. [S1] PHY (9-12) 3: Explain planetary motion and navigation in space in terms of Kepler's and Newton's laws. ALC (9-12) 6: Use the extreme value of a given quadratic function to solve applied problems. (Alabama) [MA2013] AL2 (9-12) 32: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). [F-IF9] Subject: Mathematics,Science Title: Modeling Orbital Debris Problems Description: In Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Getting from Arithmetic to Algebra Balanced Assessments for the Transition Getting from Arithmetic to Algebra by Judah L. Schwartz Book Description The title of this book is Getting from Arithmetic to Algebra and is written by author Judah L. Schwartz. The book Getting from Arithmetic to Algebra is published by Teachers' College Press. The ISBN of this book is 9780807753200 and the format is Paperback. The publisher has not provided a book description for Getting from Arithmetic to Algebra by Judah L. Schwartz. You might also like... Reinforces the following common thread: learn a skill; practice the skill to help solve equations; and then apply what you have learned to solve application problems. This book includes algebraic ideas that are developed in a logical sequence, and in an easy-to-read manner, without excessive vocabulary and formalism. Helps you to succeed on your college algebra and trigonometry midterm and final exams. This title includes questions and answers that are similar to the ones you'll find on a typical college exam, so you will know what to expect on test day. It helps you achieve the grade you desire. Books By Author Judah L. Schwartz This comprehensive, research-based book helps teachers use standards-linked assessments to collect accurate formative data about students' strengths and weaknesses and increase mathematical understandings for all learners. Brings together leading experts to offer an in-depth examination of how computer technology can play an invaluable part in educational efforts through its unique capacities to support the development of students; understanding of difficult concepts
Michael Schwartz Library The Math Emporium Come to the Math Emporium Grand Opening Celebration, November 19th! Find out more. The Math Emporium, located on the second floor of the Michael Schwartz Library (RT 230), is a state-of-the-art learning center where students can interact with adaptive MyMathLab Plus software and a highly skilled instructional team. Students learn through an innovative, engaging and easy-to-use program designed to help them become comfortable and proficient in basic mathematics. The main use for this 120 computer facility will be for students enrolled in MTH 87 (Basic Algebra) and MTH 115 (Applied Algebra), who will meet in the Math Emporium for all of their scheduled classes. When seats are available, students enrolled in MTH 116, 117, 148 and 149 can use the Emporium to work on the online homework. Hours: The Math Emporium will be open when the library is open (see Library Hours) and student assistants will be available Monday-Thursday until 6pm, Fridays until 3pm, and Saturdays 10am-2pm. Also know that Mathematics Learning Center (MC 230) is open Mondays through Saturdays. Please check the Math Learning Center website for current hours. Students enrolled in MTH 87 and MTH 115 (except for one section which will meet in RT 403) will meet in the Emporium for all of their scheduled class time. The Emporium is not a tutoring center, it is where students will meet for class. There will be a student to instructor ratio of 35:1. The Emporium will also have several graduate assistants and undergraduate tutors assisting during class time. In MTH 87, there are 14 modules of material that have 4 components: a pre-test, instructional videos, homework and a post-test. Students earning a score of 80% or higher on the pretest can automatically move on to the next module. If a student scores below 80% on the pretest, he or she will watch the instructional videos. The student will work on the homework until 80% of the homework is correct. (Multiple tries available.) A student will need a score of 80% or higher on the post test to move on (multiple tries are available). Instructors will monitor the progress of each of his or her students to determine what might be preventing a student from moving on. The great thing about this approach is that students can work through modules for which they know the material very quickly and spend more time on material they are less familiar with. Additionally, if a student does not complete all 14 modules in one semester, when they enroll in 87 once again, they can simply pick up where they left off previously. Cleveland State University \| 2121 Euclid Avenue, Cleveland, OH 44115-2214 \| 216.687.2000 . The Cleveland State University is an equal opportunity educator and employer.
Recommended buying guides Almost everyone in high school and college needs a scientific calculator at some point of their student life, whether it is to solve simple numerical questions or to calculate various scientific equations. No..... Read more Scientific calculators simplify the process of solving complex mathematical equations. Along with basic operations, such as addition and subtraction, scientific calculator models also offer a variety of..... Read more When preparing to buy a used calculator, buyers research to determine a few key factors. First, buyers should decide which type of calculator they need. Calculators come in several different types, including..... Read more Calculators are used by students and professionals as a tool that offers a range of features and functions to make mathematics easier. Although there are only about four categories of calculators, the purposes..... Read more About Scientific Calculator If you thought scientific calculators were for nerdy geeks in glasses, for guys with bad acne, and for guys in white laboratory coats with bad hairstyles, then you would be just plain wrong. Yes, we can all use a basic calculator to add and subtract numbers, but what about when it gets a little more complicated? What if you wanted to measure the volume of, say, a pizza you'd just baked? Using a scientific calculator with the Greek letter 'Pi' on it, you can very easily. The formula is: PiZZA. This is where Pi is 3.1415927, Z is the radius of the pizza (half its diameter), and A is its thickness. You can work out the area of your pizza by the formula PiZ*Z, but this doesn't look so good on paper. Gone are the days of having to use log tables and slide rules, when working out the square root of a number was a matter of painful calculation. Nowadays, input your number, press the button, and you're away. Functions like sin, cosine, and tangent are a breeze, while working in engineering math, all you need do is press the button to set it all up, and you're away. Scientific calculators aren't just for geeks: they're for cool people too. Nowadays, you can even programme your scientific calculator for more repetitive tasks, and as many have cool LCD graph function displays, you can even type in a formula and see its output graphically. Gone are the days when a calculator had only a few big buttons on. Have fun learning about just what all the different functions will do! As many scientific calculators are now also solar-powered, this means expensive battery replacements are also a thing of the past. A reference to a specific individual, commercial product or brand does not constitute or imply endorsement or sponsorship of eBay by that individual or brand or their affiliation with eBay.
To ensure that mathematics is accessible within DAISY books, the DAISY Consortium [4] formed a working group, which developed a mechanism for extending DAISY to include MathML [5] support. MathML is a standard language for marking up mathematical formulas so that their content and structure can be displayed visually on the web and can be communicated to screen reader users or converted to Braille. The result of the working group's work is the Digital Talking Book Modular Extension for Mathematics [6], informally known as the DAISY MathML Modular Extension. This extension allows all of the features of a digital talking book (e.g., support for large print, customizable speech, Braille, navigation, and synchronized highlighting) to work for math just as they do for literary text.
In this course students integrate the concepts of higher level algebra and trigonometry. Major emphasis is given to polynomial, rational, exponential, logarithmic, trigonometric and inverse functions, sequences and... In this course students integrate the concepts of higher level algebra and trigonometry. Major emphasis is given to polynomial, rational, exponential, logarithmic, trigonometric and inverse functions, sequences and... This course is designed to be a college preparation course that continues to develop a student's mastery of algebra and trigonometry. Students will learn the study of operations, inequalities, absolute value, powers,... Edgenuity Geometry is a two-semester, hands-on and lecture-based course featuring an introduction to geometry, including reasoning and proof and basic constructions. Triangle relationships (similarity and congruency) and... Edgenuity Algebra is a two-semester course that provides in-depth coverage of writing, solving and graphing a variety of equations and inequalities, as well as linear systems. Functions are a central theme of the course... The purpose of this Algebra I course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics in a year long course. Topics included are real... In this course students will use their prior knowledge to learn and apply Algebra II skills. This course includes topics such as functions, radical functions, rational functions, exponential and logarithmic functions,... Trigonometry teaches students to use the trigonometric ratios and the unit circle to find and explore the relationship between the measure of angles and lengths of sides of triangles. Students learn how to solve directly... Algebra II begins with a review of algebraic properties and equation and inequality solving. Students will study relations and functions, including linear, quadratic, and radical functions, and be able to graph these...
978088385 Does Math!: Real-Life Problems from Women on the Job (Classroom Resource Materials) Professor H. S. Wall wrote Creative Mathematics with the intention of leading students to develop their mathematical abilities, to help them learn the art of mathematics, and to teach them to create mathematical ideas. Creative Mathematics, according to Wall, 'is not a compendium of mathematical facts and inventions to be read over as a connoisseur of art looks over paintings. It is, instead, a sketchbook in which readers try their hands at mathematical discovery.' The book is self contained, and assumes little formal mathematical background on the part of the reader. Wall is earnest about developing mathematical creativity and independence in students. Wall developed Creative Mathematics over a period of many years of working with students at the University of Texas -Austin. In less than two hundred pages, he takes the reader on a stimulating tour starting with numbers, and then moving on to simple graphs, the integral, simple surfaces, successive approximations, linear spaces of simple graphs, and concluding with mechanical systems. The student who has worked through Creative Mathematics will come away with heightened mathematical maturity
About this product Book Information Spectrum Enrichment Math helps students develop real-world applications of math curriculum The lessons strengthen math skills by focusing on fractions, decimals, percents, measurements, preparing for algebra, graphs, probability, and moreBook description Spectrum Enrichment Math helps students develop real-world applications of math curriculum! The lessons strengthen math skills by focusing on fractions, decimals, percents, measurements, preparing for algebra, graphs, probability, and more! The
Synopses & Reviews Publisher Comments: Mathematics: Applications and Concepts is a three-text Middle School series intended to bridge the gap from Elementary Mathematics to High School Mathematics. The program is designed to motivate middle school students, enable them to see the usefulness of mathematics in the world around them, enhance their fluency in the language of mathematics, and prepare them for success in Algebra and Geometry
This course emphasizes real world applications of beginning and intermediate algebra. Topics include a review of fractions, decimals and percents, and the development of linear, quadratic, rational, radical, exponential and logarithmic functions. This course is designed for those students whose major and transfer institution requires only statistics or math for liberal arts as the transfer level math course for the degree. This course does not meet the prerequisite for trigonometry, college algebra or any other transfer level math course. Student Learning Outcomes Upon successful completion of this course the student will be able to demonstrate knowledge related to the interrelatedness of the equation of a quadratic function with its graph, including the vertex and x and y intercepts. Additionally, specific course objectives include: 1. Perform the basic arithmetic operations with real numbers, exponents and the order of operations. 2. Apply properties of equality to solve linear equations and related application problems. 3. Determine the equation for a linear function and graph it. 4. Perform the basic arithmetic operations with polynomials. 5. Factor polynomial expressions using a variety of methods. 6. Solve quadratic equations by factoring and use of the quadratic formula. 7. Graph quadratic functions. 8. Identify functions from their equations and graphs and use appropriate functional notation. 9. Perform the basic arithmetic operations with rational expressions. 10. Solve systems of linear equations in two variables graphically and algebraically. 11. Solve exponential and logarithmic equations and applications. 12. Use correct notation to identify, simplify and use arithmetic and geometric sequence and series.
5 Answers I came from an engineering background and moved into comp science and I found Information Theory quite easy and interesting. It was not so much that I was more talented than everyone else, but because it used mathematics that I was already familiar with at the time. I had a natural 'feel' for information theory because I knew how it functioned. For mathematics, I find the easiest way is to drill it. Just keep doing math problems. That's why they teach it separately and need it as a requirement. Think of it like your ABCs in school... you had to memorize letters before learning to read and write. Much like learning to read, it will click intuitively after a while. Memorization of formulas is a poor method, because you will forget it, especially when you are stressed and have little sleep. Drilling it, by repeating and reusing the formulas dozens of times works. If you have more time and willpower, derive equations. It is a technique used by and taught to elite students because it requires a lot more hard work, but it will quickly deepen your understanding. You will need to grasp a solid foundation of these principles first before using them in more complex applications. If you don't know half the letters of the alphabet, reading will be a struggle to say the least. Mathematical formulas are much like words, except with one symbol representing a sentence. If you don't have time to master the basics, print out the notes, and write the formulas as sentences. For example, p is probability of something happening 1-p is probability of something not happening p(x) is probability of x happening Σ with the x ∈ X under it, means the sum of everything in X and so on. Have this cheat sheet/glossary in your notes somewhere and use it to 'translate' weird mathematical things. That way, you only have to look it up once and not flip through a thick math book every line. Also, learn how something works as you learn what it does. Almost every good textbook teaches this for good reason. The ones that don't assume you already know how all the components go together. As with math, you can learn how it works a few years after you learn what it is, and there's no real hard guideline to this. Well, that would change from person to person. Different people learn differently. For me, I need to have a clear understanding of the topic and it's basics or fundamentals to learn it. Good analogues, or real world examples help. Also, applying the knowledge helps. In 11th grade we had programming, and I had coded a bunch of equations from the Physics book - I didn't forget those equations for a long time, without ever having to revise them or memorize them again. Usually I have a very hard time remembering equations, unless I understand the relationship between the variables. Then I can derive the equation on my own without having to memorize it. In fact, one of the most brilliant students I've known, used to derive basic equations on the back of his answer sheet at the beginning of an exam, instead of trying memorizing them. Understand the basics of how things work, and how they are related, things will fall into place themselves. If you have a feeling that you are studying, revising, trying to memorize and got no effect it can mean you need to change the way. Something what you do, become too much routinized. There can be a flow somewhere and you don't see it. You can find a friend who does worst than you and offer him/her your help. Set up some meeting with clear goals. For example, today we going through "integration by parts" or some other chunk of the material you study. Try to explain to your friend what you already know and how the expressions work. Very soon you will find it as a very powerful technique. Your friend will start to ask you questions which point material not well explained. The reason of course is that you not always entirely confident with it. That way you achieve a few things: reformulate your knowledge find weak points in your knowledge learn to express what you know (very helpful on exams) say mathematical expression in words (good for better memorization) find analogies find "hidden" connection between theories and last but not least you can make a better friendship After you start working with someone, it will be much easier to work throughout big sections of material and also easier to spot mistakes in each other works. +1 for teaching someone else. Though don't ever do it right before an exam, whenever I did that, I ended up forgetting the topic I taught and it would always be on the paper. – elssarFeb 11 '13 at 10:32 It is important not to over do things, try not to spend more than 45 minutes studying something. After that take a break and do something else, best if you can do something totally unrelated. This gives your brain a chance to focus on something else and process the information you've given it. If you can get up and go for a walk, or do some push ups or make some tea and have a quiet 10 minutes. When you come back to your subject start with some revision of the previous 45 minutes. If you like Tony Buzan's mind maps then that could be a good way to jot down what you remember, making the associations will also help you lock the information into your mind. The more 'hooks' you can hang ideas on the easier remembering becomes. When the subject is abstract, as in the case of formulas and equations, then it is harder to find mnemonics and the approach of 'making up a funny story' wherein you place your items to remember is less applicable. But if you can make parts of your equation tell a story or relate them to other equations then you begin to build up a mental map of related functions. Derren Brown has good advice on memorising things as well; a nice summary can be found at How to remember things, how he memorises decks of cards translates nicely to remembering formulas and equations because the subject (playing cards) is also abstract enough that there aren't many places in normal life that we encounter H3, S9 etc. Consider the study by Dale (1969). You will learn more if you write a summary of the stuff yourself, rather than just reading it in a textbook. For best results, you should explain the stuff to others, possibly in a presentation. This also applies to learning stuff with a lot of mathematical notation. What I hear, I forget; What I see, I remember; What I do, I understand.
More About This Textbook Overview Intended for precalculus courses requiring a graphing calculator, Functions and Change emphasizes the application of mathematics to real problems students encounter each day. Applications from a variety of disciplines, including Astronomy, Biology, and the Social Sciences, make concepts interesting for students who have difficulty with more theoretical coverage of mathematics. In addition to these meaningful applications, the authors' easy-to-read writing style allows students to see mathematics as a descriptive problem-solving tool. An extended version of the successful Functions and Change: A Modeling Approach to College Algebra, this text includes three chapters of trigonometry. Related Subjects Meet the Author Bruce Crauder received his Ph.D. from Columbia University in 1981, after which he spent a post-doctoral year at the Institute for Advanced Studies. Since then, he has held positions at the University of Utah, University of Pennsylvania, University of North Carolina, Colorado State University, and is now at Oklahoma State University as a Professor of Mathematics. Crauder has, in the past, held NSF research grants in algebraic geometry as well as several NSF grants for mathematics curricular reform and development with Benny Evans and Alan Noell. He is currently the Associate Dean for Instruction of the College of Arts & Sciences at Oklahoma State. Benny Evans received his Ph.D. in Mathematics from the University of Michigan in 1971. After a year at the Institute for Advanced Study he came to Oklahoma State University where he has served as Undergraduate Director, Associate Head, and Head, and is currently Professor of Mathematics. His academic interests are in topology and mathematics education. He is the author of 28 publications in refereed journals and a number of books and expository articles. Alan Noell earned his Ph.D. degree from Princeton University in 1983 and had a two-year postdoctoral position at Caltech. His scholarly activities include complex analysis and curriculum development. He has been a Co-Principal Investigator on grants funded by the National Science Foundation. He has served as Undergraduate Director and Graduate Director in the Department of Mathematics at Oklahoma State, and he is currently Professor of Mathematics
This book is a unique reference for anyone with a serious interest in mathematics. It is edited by Timothy Gowers, a recipient of the Fields Medal, and it includes entries written by several of the world's leading mathematicians. The entries have many goals, such as to introduce basic mathematical tools and vocabulary, to trace the development of modern mathematics, to explain essential terms and concepts, to describe the achievements of scores of famous mathematicians and to explore the impact of mathematics on other disciplines such as biology, finance and music. The book is divided into eight chapters. The introduction presents basic mathematical language and some fundamental mathematical definitions. The origins of modern mathematics are described in the second chapter. It starts with numbers, geometry and the development of abstract algebra and through algorithms and proofs in mathematics gets to the crisis in the foundations of mathematics. The next chapter is devoted to important mathematical concepts presented in alphabetical order. Thus, one can begin reading with an axiom of choice and end with the Zermelo-Fraenkel axioms. Then the book turns its attention to mathematical branches. From algebraic geometry to stochastic processes, brief descriptions of the main areas of mathematics are provided. Crucial theorems and fundamental open problems occupy the fifth chapter. Then the book presents some achievements of famous mathematicians, starting with Pythagoras and ending with Nicolas Bourbaki. After this, several examples of the influence of mathematics on other branches of science are presented. The companion ends with general thoughts on mathematics and mathematicians. The book contains some valuable surveys of the main branches of mathematics that are written in an accessible style. Hence, it is recommended both to students of mathematics and researchers seeking to understand areas outside their specialties.
This book gives an introduction to algebraic number theory. The authors concentrate on the algorithmic aspects of the theory. Many algorithms are given to compute properties of algebraic number fields and their subrings. The book deals with the following subjects: Galois theory, resolvents and discriminants, normal bases, geometry of numbers (lattice reduction), valuation theory, Newton polygon, units and computation of the class group. The book finishes with a collection of tables. These tables involve permutation groups of degree ≤12, fundamental units and class groups of fields with degrees up to 7. The last table contains two computations of integral bases. The first one is for an 11th degree field, the second one for a 55th degree field. Both are given by a polynomial over ℤ.
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Tag Info A few suggestions for things that might improve the situation: (1) Buy the books that are well-illustrated. This will indicate to authors and publishers that illustrations are important. Another very well illustrated one is Trefethen's book on approximation. See also this question. (2) Learn to use the graphical tools in packages like Matlab and ... Set theory is absolutely necessary to learn more advanced mathematics. It is needed for just about every branch of mathematics, if not every branch. In my opinion, it would be a good idea to start learning some basic set theory notions at least. It will definitely show up in classes like real analysis, complex analysis, and probability just to name a few.It could also be a more "well-known" result, which is normally found in big textbooks but is sufficiently independent that you can treat it thoroughly in a dozen of pages. This happens often for example in computational number theory, where many algorithms use elementary notions (so not much background material needed) but are based on sufficiently ... Completely understand a single "thing" (the solution to a specific problem in a paper for example) and explain it in detail in your -own- words would be my suggestion. This may include filling out the details of the paper that have been left to the reader. Alert .. highly iconoclastic answer follows :) . I believe you have to just have the "gift" for it. You either do or don't. It's like music. Unfortunately I think that's the truth. (Consider too Douglas Hofstadter's writings on how he had a gift for math -- BUT -- only to a certain level. Very astute!) In terms of just getting through, it's ... If you have already learnt group theory, I may suggest you to go through the book 'theory of algebraic numbers' by Pollard & Diamond. It's a really good treatise to start off. You don't even need to know the definition of ring to read this book. Everything is given there in a very well setup. After having finished that book, you may pay a look at 'A ... Don't go it alone. Don't be afraid to talk to your professor/TAs when you're having trouble understanding something. (Actually, this is a good rule for any class.) Don't be afraid to collaborate with your classmates, either -- doing so is silently discouraged in most high school classes, where homework is a big part of your grade, but it should be ... I can't comment or even upvote, so I'll put this as an answer. Several people already told you this, but it can't be stressed enough: Solve a lot of problems! After that, solve even more. Solve problems from the textbooks, try to solve problems which you've heard about from your professors. Try to solve the famous ( P = NP ? ) problem - in this case you will ... The best advice I have ever gotten: Do the exercises without looking at the suggested solutions. This will force you to think critically. In the beginning this will probably be very time consuming, but do not give up, because after a while the payoff will be huge. If you really get stuck, discuss the problem with someone else. In this way, you will have to ... The best advice I got while an undergrad Math major was to take programming/software courses. After picking up a minor in Computer Science (and then a graduate degree in the same), I'm now a full-time software developer and loving it! Once you get calculus out of the way and have time for a few electives, try your hand at discrete mathematics: ... Get some old test questions from previous years for the same professor. Hate to say it, but after wondering why the dumb frat guys were scoring so well in tests, I came upon this 'secret'. Turns out professors are lazy and recycle old questions. You can beat yourself over the head doing hundreds of example questions from books, but there's nothing that works ... Apart from all the good answers that the other guys provided, I have one suggestion: Use pen and paper! In other words, do the exercises (or new concepts) instead of studying them. Imagine a day in the future when you are reading a question or studying a new concept. Then you start a conversation like this with yourself while looking at the textbook: "Nah! ... I completed two bachelor degrees in mathematics and physics in Vienna. I don't know how it compares to an non-European bachelor degree, but I think my experience may be of help. I can't give advice on (calculus) textbooks but maybe I can give you some general advice about studying mathematics in university: Be precise: precision and adherence to the ... Okay I'm not all that good with the advice. But I will tell you this. I am not a fan of Stewart's book. Trust me if you really want to grasp the roots such as Mitch Knight suggests above I strongly urge you to consider a few other texts as supplements. These are my recommendations. You won't go through them all, obviously. But if you can properly grasp a ... I recently completed second year undergrad down at Melbourne Uni, and although we had prescribed texts, I used Stewarts calc. It's important to realise that although people stress that repetition (eg doing question after question) results in success, I disagree. Understanding is the key, persevere to understand the thinking behind the mathematics and you'll ... A wonderful book is Proofs from THE BOOK by Ziegler and Aigner (the capitals are in the title - I am not actually shouting). This book contains lots of wonderfully elegant proofs from number theory, geometry, analysis and so on. The proofs are almost always at a low-enough level for a good undergraduate student to understand (the idea is that they are simple ... You may try to understand the key points in the AKS primality test. The book Primality Testing in Polynomial Time by Dietzfelbinger is very nice. There is also the more recent, and probably simpler, Primality Testing for Beginners by Rempe-Gillen and Waldecker. See also the survey It is easy to determine whether a given integer is prime by Granville. A ... It's a bit of a challenge but perhaps you could summarize the activity following Yitang Zhang's recent paper on the gaps between primes. A friend of mine is doing so as his Master's dissertation; it's doable and it'd get you up-to-date. You'd have to black-box a few results as an undergraduate though. One possibility is to study the so called "Euclidean proofs" for Dirichlet's theorem on arithmetic progressions, which are sometimes possible and are much easier than Dirichlet's methods form analytic number theory. However, it is an interesting question to say what an "Euclidean proof" should be, and for which arithmetic progressions $an+b$ it works. For ... My 2 cents (Tips): When you read a solution, try to figure out why the person decided to do that. Getting to a solution is never as simple as it is stated. The writer didn't just read the problem and "aha! this is how you do it!" it took some inspection and trial and error. Recognize patterns. No problem is unique -- you can always apply past knowledge to ... I think the most important thing is to find right problem set to work on. Too easy or to hard will not help much. Your strategy 1) is not recommended, you will not improve your problem solving skill by just reading the solution without much tries and fails. So my suggestion is to use strategy 2), but start from easier problems, the ones that you can solve in ... One of the best and biggest site for math contests is Art of Problem Solving. On its forum there are thousands of problem and they are nicely divided in category, so you can practise problems of specific area. Another quite good site is Imo Math. All previous IMO math problems are there. Also if you spend some more time you can find other problem. Also ... Here are some things to keep in mind: Not everybody solves every problem, nor solves every problem quickly. It is likely that you'll learn the most from trying to solve problems. You can probably learn helpful strategies about solving problems from "coaching" texts, but you are likely to learn (and retain) a lot more from practical experience. Don't "try ... I realize I'm late to this party, but I'd note that the other answers try to describe a protocol best for the recipient. What's best for you? I fly on planes a lot (usually within the United States). I study both Computer Science and Abstract Mathematics (primarily separately). Often, when I sit down, the person sitting next to me will ask me what I do. ... I think it was Einstein who said, "If you can't explain it to a six year old you don't understand it yourself." I find that it is always best to stick to the very basics, omitting superfluous details that matter only to someone working in the field. A colleague of mine once said: "I do stuff with numbers that you put in a short and wide box to make the world a better place." (Frame Theory: generalization of Linear Algebra with overcomplete bases) See this blog entry for more: My research explained, using only the 1000 most common English words (Duston G. Mixon) I work with two algebraic objects that are closely related called algebraic groups and Lie algebras. These objects can act on spaces (like three-dimensional space) by transforming them in a nice way, and I study these actions. Can you make a nice analogy that illustrates what a "space" and what "transforming in a nice way" looks like in everyday ... Layman's terms first; you can try to guess the mathematical problem. There are two tennis players. They decide to play each other with a new set of rules: They play point by point. The referee counts the points. The first time that one of them is three points ahead he's the winner. Now you'll notice that this game could go on forever, if they play ... Look at some Richard Feynman video. He is great at explaining and visualizing concept, he makes imagination works for the layman, not just pure logic. However, when asked some difficult question he can't respond in layman terms, he says "I can't explain it, because it is not familiar to anything you have seen and don't have the necessary vocabulary to ... Despite not being a mathematician myself, I would like to contribute a bit as in my field I share the same problem often (depending on the project I am working on). What I have learned is that what works best is to bypass the entire description of what you're actually doing and just jump straight ahead to applications even if you will never get anywhere ... "There are only two kinds of math books: Those you cannot read beyond the first sentence, and those you cannot read beyond the first page." -- C.N. Yang One of my favourite sayings from a Nobel laureate mathematician. In my view, the ability to communicate complex mathematical ideas in a simple way is a very special gift that some people have and others ... As a laymen myself, let me shed some light on this, and perhaps prevent this potentially awkward conversation from happening. The problem is you're trying to explain your research in a few sentences and induce wonder and so on. There is no solution. You'd have a better response if, instead of explaining what you're doing or how you're doing it, you told ... I work in a branch of mathematical logic. Although I'm a mathematician, logic is multidisciplinary, and is also studied by philosophers, computer scientists, and linguists. The essential idea in logic is the relationship between syntax and semantics. Syntax refers to symbols and language. Most of mathematics is a linguistic exercise: we manipulate ... On my field: I study curvature. I study higher-dimensional shapes. I study a mix of calculus and geometry. But not the geometry of flat things like triangles and squares, but smooth curvy wavy things. On (one aspect of) my subfield: I study shapes that are like soap films. Like, if you dip a wire in a soapy water, then the bubble will ... My problem is in some sense the opposite of the one described by the OP. Although my background is in pure mathematics, my current work is in statistics. The problem I have is that when I tell people what I do, they think they know about the subject when they really don't. In a way, that's a much harder issue to deal with--to dispel preconceived notions ... Thanks for asking this question. As a layman (but fan of math) -- and one with recent experience asking mathematicians what they do -- I think I have a pretty relevant perspective to offer on this. Let me first suggest what not to do (not that I assume you do this, but just in case, and for anyone else). Almost every mathematician or scientist I've asked ... First, I think it's important to consider the person's motivation for asking this question. If they're just asking to be polite, then I think your answer, or basically any answer, would be fine. They're not going to take away much from whatever few words you say, so those words aren't really important. If they truly are interested in your work, then it's ... I'm a little disappointed by the comments. Granted, it's hard to explain mathematics, but having the attitude that you're not even going to try is not doing mathematics PR any favors. We can't in good faith expect the public to fund our research if we're not even going to try to tell them what we're doing with their money. First, I hope you won't take this ... Last time I tried to explain it went like this: "So, mathematicians sometimes like to have a theory on hand from which you can derive as much of mathematics as possible, something to check your intuitions against when the stuff you're studying gets really flighty. There are a few alternate formulations of such a theory, and I study one of the weirder ones." ... Sam sat with his eyes closed for several minutes, then said softly: "I have many names, and none of them matter." He opened his eyes slightly then, but he did not move his head. He looked upon nothing in particular. "Names are not important," he said. "To speak is to name names, but to speak is not important. A thing happens once that has never happened ... Get Friedberg, Insel and Spence's "Linear Algebra" 4th edition, sit down, and do every non-computational problem in the first three chapters. It's my favorite math book. It introduces linear algebra via an axiomatic approach that you'll probably see often as you go on in math. It's rigorous, it provides intuition and a meaningful framework to linear ... The tension between following abstract rules as against intuition has been present in mathematics for centuries if not much longer. From the time of Newton and Leibniz onwards mathematics became more algebraic due to the calculus. For example, the eighteenth century mathematician Lagrange played a critical role in moving away away from diagrams towards ... No, don't abandon your love of analogies and your search for connections to the "real world". But a caveat: be guided by it, not shackled to it. A few more remarks. (1) Linear algebra can be presented sevaral different ways: computationally, conceptually, geometrically, physically, etc. It sounds like you've encountered a mismatch between your course and ... Linear algebra (and also functional analysis to some extend) are fields where it's still possible to have geometric interpretations. Linear transformations (in particular, matrices) can represent reflections, rotations and scalings which transform vectors. You lose some exact graphical interpretation when you move from 2- and 3-dimensional vector spaces to ... Linear algebra is by no means about computations over concepts. There's actually a rather precise dichotomy that approximates that between computations and concepts in linear algebra, namely that between matrix algebra and the theory of abstract linear transformations. It's from the latter perspective linear algebra most naturally displays its ability to ... You've written this post as a leading question, you'll get much too bloomy answers. My advice is to search for the motivations behind the introduction of this and that concept - knowing that these exist will let you concentrate on the plug and chuck you need to get your answers. I want to add that not all mathematical object are physical things - e.g. the ... If you want single variable stuff, Abbott's Understanding Analysis is a good start. It is sort of like a baby Baby Rudin. If you want both single and multivariable analysis, my personal favorite is Fitzpatrick's Advanced Calculus. The texts by Rudin are also of course "the standard" in analysis. If you want to look at complex analysis, I like a book by ...
Mathematics Page Content The Georgia Performance Standards Mathematics curriculum is designed to achieve a balance among concepts, skills, and problem solving. The curriculum stresses rigorous concept development, presents realistic and relevant tasks, and maintains a strong emphasis on computational and procedural skills. At all grades, the curriculum encourages students to reason mathematically, to evaluate mathematical arguments both formally and informally, to use the language of mathematics to communicate ideas and information precisely, and to make connections among mathematical topics and to other disciplines. Mathematics CCGPS In 2010, the Georgia State School Board of Education adopted a new set of standards known as CCGPS for Mathematics to be implemented in kindergarten through grade nine in school year 2012-2013. The CCGPS for Mathematics was finalized after multiple rounds of development and feedback from states and national organizations. Representatives from Georgia provided feedback at each round. The Georgia Performance Standards (GPS) Mathematics curriculum was one of the state curricula used to inform the creation of the CCGPS for Mathematics. The rigor and relevance, as well as the balance of skills, concepts, and problem solving, found in GPS mathematics is mirrored in the CCGPS. The CCGPS, like the GPS, is evidence and/or research based, vertically aligned, and internationally benchmarked so that all students are prepared to succeed in our global economy and society. Although some content may be in different grade levels, all of the standards addressed in the CCGPS are also addressed in GPS mathematics. Most importantly, our work with GPS has prepared Georgia for the implementation of the CCGPS. It is appropriate that the name given to Georgia's CCGPS Mathematics reflect the influence of the GPS in the teaching and learning of mathematics. Therefore, Georgia's Mathematics curriculum is called CCGPS. Information for Parents and Students CCGPS Mathematics information is available at the GaDOE Parent Information Webpage.
Do the Math: Secrets, Lies, and Algebra In the eighth grade, 1 math whiz < 1 popular boy, according to Tess's calculations. That is, until she has to factor in a few more variables, like: 1 stolen test (x), 3 cheaters (y), and 2 best friends (z) who can't keep a secret. Oh, and she can't forget the winter dance (d)! Then there's the suspicious guy Tess's parents know, but that's a whole different problem— Sydney (Fair Oaks Ranch, TX) Do the Math: Secrets, Lies, and Algebra by Wendy Lichtman was awesome! I really enjoyed the various chacters and the fast paced plot! I might have learned some math along the way too! :) I rate this book an 8/10! — Allie (Forest Hill, MD) This was a very interesting book. It had a new way of looking at life: through math. As the main character discovers, math is so logical that it can often help to solve problems in real life--and she has some big ones. Any math lover would instantly love this book, and anyone else would love it also for its unique perspective on life. I would highly recommend it to anyone, even those who think math is useless (maybe this will change their minds). — Molly (Agua Dulce, CA) This wonderful, witty book puts things in a refreshingly new perspective, relating everyday things to math in a way that will have you thinking. This book evokes an interest in math without being a textbook and also allows us to enter the world of a typical teenage girl. This book combines typical teenage life and math in a way that will make you excited for math class
Product Description The Matrix Algebra Tutor: Learning by Example DVD Series teaches students about matrices and explains why they're useful in mathematics. This episode teaches students about inconsistent and dependent systems in matrix math. Sometimes, a system of equations does not have a well-defined solution. For example, if we have two equations that represent two lines and these lines are exactly parallel and never intersect, then these equations do not have a solution. In this program, these ideas are explored. Grades 9-College. 54 minutes on DVD
Synopses & Reviews Publisher Comments: Do you remember being hopelessly confused in calculus class? Afterwards, you asked your brainy friend over a cup of coffee. "What was going on in that class?" Your friend then explained it all to you in five minutes flat, making it crystal clear. "Oh", you said, "is that all there is to it?" Later, you wished that friend was around to explain all the lectures to you. How to Ace Calculus will play the role of that friend. Written by three gifted teachers, it provides brief and highly readable explanations of the key topics of calculus without the technical details and fine print that would be found in a formal text. Capturing the tone of students exchaging ideas among themselves, this unique guide also explains how calculus is taught, how to get the best teachers, what to study, and what is likely to be on exams?all the tricks of the trade that will make learning the material of first-year calculus a piece of cake Funny, irreverent, and flexible enough to use with any traditional or reform-based calculus text. How to Ace Calculus shows why learning calculus can be not only a mind-expanding experience but also fantastic fun. Synopsis:Synopsis: Written by three gifted teachers, this book provides brief and highly readable explanations of the key topics of calculus without the technical details and fine print found in a formal text. About the Author Colin Adams is Professor of Mathematics at Williams College. He is the author of The Knot Book and winner of the Mathematical Association of America Distinguished Teaching Award for 1998. Joel Hass is Professor of Mathematics at the University of California at Davis, and Abigail Thompson is also Professor of Mathematics at the University of California at Davis. Both have held fellowships from the Sloan Foundation and the National Science Foundation."Synopsis" by Libri, Written by three gifted teachers, this book provides brief and highly readable explanations of the key topics of calculus without the technical details and fine print found in a formal
Introduction to Teaching Mathematics at the College Level college mathematics can be a daunting task, indeed. It's nice for seasoned professionals and others to have a solid primer on the subject and this guide from Professor Suzanne Kelton is quite useful. The 54-page guide is divided into four chapters that cover documenting teaching practices, classroom basics, course policies, and discussion sections. Each of these chapters has additional subsections, such as syllabi, pace, and proofs. As a bonus, the document has an appendix that covers graphing techniques. The language of the document is jargon-free and accessible, which is a welcome departure from other guides. Visitors will want to share this item with friends in the field and even those who are teaching in other disciplines.Fri, 8 Nov 2013 14:43:31 -0600Lessons in Electric Circuits Vol V Reference USEFUL EQUATIONS AND CONVERSION FACTORS Chapter 2: COLOR CODES Chapter 3: CONDUCTOR AND INSULATOR TABLES Chapter 4: ALGEBRA REFERENCE Chapter 5: TRIGONOMETRY REFERENCE Chapter 6: CALCULUS REFERENCE Chapter 7: USING THE SPICE CIRCUIT SIMULATION PROGRAM Chapter 8: TROUBLESHOOTING -- THEORY AND PRACTICE Chapter 9: CIRCUIT SCHEMATIC SYMBOLS Chapter 10: PERIODIC TABLE OF THE ELEMENTSFri, 14 Jun 2013 13:21:33 -0500Webmath from Discovery Education, provides help for mathematics students. Categories include general mathematics, K-8 math, algebra, geometry, trigonometry and calculus. The site covers everything you need to know, whether you need help with a specific topic or are looking to brush up on some math skills.Mon, 10 Jun 2013 13:23:38 -0500National Security Agency: High School Concept Development Units National Security Agency (NSA) has worked to craft these educational materials they are calling "concept development units" (CDUs). The units are divided into 11 sections, including Algebra, Calculus, and Data Analysis. Clicking on each of these sections will bring up a complete list of all the CDUs currently available. Each list offers a paragraph-long description of each activity, along with an indication of the appropriate grade level for each activity. Some of the activities include "Understanding Proportions and Scale Drawings," "Scatter Brained," "Fashion Sense and Dollar Wise" and "Squares in the Light." These are all terrific resources for educators, and the site also contains links to information about the Math and Related Sciences Camp (MARS) sponsored by the National Security Agency and links to other educational centers.Fri, 15 Feb 2013 10:50:45 -0600Bates College Online Resources for Calculus and Linear Algebra College in Maine has worked diligently to bring together this set of mathematical resources to the public, and it's a nice find. The materials here are drawn from four courses at the school: Math 105, Math 106, Math 205, and Math 206. The first couple of resources in each section contain past quizzes and exams from each course, complete with information on each topic. Additionally, each area contains drill problems, tutorials, and a fun "Find the Error!" feature. The topics covered here include linear algebra, quadric surfaces, functions, and abstract vector spaces. Moving on, the site also includes links to external sites from Harvey Mudd College and the University of California-Davis that address advanced math topics. For those persons interested in learning more about the mathematics department at Bates College, there's a link to its official website at the bottom of the page.Mon, 25 Jun 2012 10:53:18Do the Math by staff members at the University of Arizona's Center for Recruitment & Retention of Mathematics Teachers (CRR), Do the Math is a weekly cable television show that features mathematics teachers explaining key mathematical concepts. Recently, the folks at CRR decided to create a "best of" playlist that offers segments from this popular program. Here visitors will find 18 segments that last between 26 and 38 minutes. Some of the subjects covered include geometry, advanced algebra, and calculus. Visitors may be interested in the materials on the left-hand side of the page, such as an AP Calculus practice exam, information about the related academic programs offered at the University of Arizona, and more. Also, the site contains a listserv for mathematics teachers and information on upcoming conferences that may be of interest.Thu, 15 Mar 2012 03:00:05 -0500AP Central: AP Calculus AB Course Home Page Educational Testing Service (ETS) and the AP College Board have a number of excellent resources at their disposal, and this site provides a cornucopia of materials about teaching and learning calculus. First up is the information about the actual AP Calculus AB course, which may be most useful to those teaching the course at the high school level. Most visitors will want to take their time looking over the "Classroom Instruction and Resources" area. Here visitors will find special focus materials on approximation and differential equations, along with sample lesson plans and curriculum modules. The modules cover extrema, motion, and reasoning from tabular data. History of mathematics-types shouldn't miss the "History of Calculus" area, as it is quite a pip.Fri, 17 Jun 2011 22:36:41Precalculus Wed, 22 Dec 2010 03:00:02Vector Calculus is a series of lectures, authored by Chris Tisdell of the University of New South Wales, for MATH2111 "Higher Several Variable Calculus" and "Vector Calculus", which is a 2nd-year mathematics subject taught at UNSW, Sydney. This playlist provides a shapshot of some lectures presented in Session 1, 2009. These lectures focus on presenting vector calculus in an applied and engineering context, while maintaining mathematical rigour. Thus, this playlist may be useful to students of mathematics, but also to those of engineering, physics and the applied sciences. There is an emphasis on examples and also on proofs.Thu, 16 Dec 2010 03:00:01 presented by MIT and taught by Professor Denis Auroux, presents multivariable calculus. It is intended for use in a freshman calculus course. It includes material relating to vectors and matrices, partial derivatives, double and triple integrals and vector calculus in 2 and 3-space. The material includes video lectures, lecture notes, exams (with solutions) and student assignments (without solutions). MIT presents OpenCourseWare as free educational material online. No registration or enrollment is required to use the materials.Fri, 10 Dec 2010 03:00:02 -0600
Rent Book Buy Used Book Buy New Book In Stock Usually Ships in 24 Hours $12.68 eBook We're Sorry Not Available More New and Used from Private Sellers Starting at $1 general review covers equations, functions, and graphs; limits, derivatives; integrals and antiderivatives; word problems; applications of integrals to geometry; and much more. Additional features make this volume especially helpful to students working on their own. They include worked-out examples, a summary of the main points of each chapter, exercises, and where needed, background material on algebra, geometry, and reading comprehension. Table of Contents Introduction vii 1 Equations, Functions, and Graphs 1 (16) 1.1 Equations 1 (1) 1.2 Functions 2 (6) 1.3 Graphs 8 (5) 1.4 Summary of Main Points 13 (1) 1.5 Exercises 14 (3) 2 Change, and the Idea of the Derivative 17 (30) 2.1 Average Rates of Change 17 (6) 2.2 Instantaneous Rates of Change 23 (10) 2.3 The Derivative 33 (4) 2.4 The Geometric Meaning of Derivatives 37 (6) 2.5 Summary of Main Points 43 (1) 2.6 Exercises 44 (3) 3 The Idea of Limits 47 (22) 3.1 The Basic Definition 47 (3) 3.2 Some Complications with the Definition of Limits 50 (2) 3.3 The Problem of Division By Zero 52 (4) 3.4 The Case of (0/0) 56 (3) 3.5 Continuity and Differentiability 59 (4) 3.6 Summary of Main Points 63 (2) 3.7 Exercises 65 (4) 4 Computing Some Derivatives 69 (14) 4.1 Powers of x 70 (3) 4.2 Fractional Expressions with the Variable in the Denominator 73 (6) 4.3 Square Roots 79 (1) 4.4 Some Harder Examples 80 (2) 4.5 Summary of Main Points 82 (1) 4.6 Exercises 82 (1) 5 Formulas for Derivatives 83 (20) 5.1 Derivatives of Some Particular Functions 84 (2) 5.2 Derivatives of Combinations of Functions 86 (7) 5.3 The Chain Rule 93 (8) 5.4 Summary of Main Points 101 (1) 5.5 Exercises 101 (2) 6 Extreme Values, the Mean Value Theorem, and Curve Sketching 103 (22) 6.1 Extreme Values 104 (6) 6.2 The Mean Value Theorem 110 (4) 6.3 Curve Sketching 114 (7) 6.4 Summary of Main Points 121 (1) 6.5 Exercises 122 (3) 7 Word Problems 125 (30) 7.1 A Review of Geometry 125 (6) 7.2 Max-Min Word Problems 131 (9) 7.3 Related Rate Word Problems 140 (9) 7.4 Summary of Main Points 149 (1) 7.5 Exercises 150 (5) 8 The Idea of the Integral 155 (34) 8.1 The Basic Idea 155 (9) 8.2 Terminology and Notation 164 (16) 8.3 The Definite Integral: Definition and Notation 180 (3) 8.4 Summary of Main Points 183 (2) 8.5 Exercises 185 (4) 9 Computing Some Integrals 189 (18) 9.1 Summation Rules and Formulas 189 (10) 9.2 Computing Limits of Approximate Sums 199 (3) 9.3 Summary of Main Points 202 (2) 9.4 Exercises 204 (3) 10 Formulas for Integrals: Integrals, Antiderivatives and the Fundamental Theorem of Calculus 207 (28) 10.1 Introduction 207 (1) 10.2 The Fundamental Theorem of Calculus--The Main Idea 207 (6) 10.3 The Fundamental Theorem of Calculus--An Idea of the Proof 213 (3) 10.4 Computing Some Antiderivatives 216 (8) 10.5 Antiderivatives Involving the Chain Rule 224 (6) 10.6 Summary of Main Points 230 (2) 10.7 Exercises 232 (3) 11 Geometric Applications of the Integral 235 (58) 11.1 Horizontal vs. Vertical, x vs. y 235 (8) 11.2 Area 243 (17) 11.3 Volumes of Solids of Revolution--The Method of Cross-sectional Areas 260 (15) 11.4 Volumes of Solids of Revolution--The Method of Cylindrical Shells
More About This Textbook Overview Users discover the many ways in which mathematics is relevant to their lives with MATHEMATICS: A PRACTICAL ODYSSEY, 7E and its accompanying online resources. They master problem-solving skills in such areas as calculating interest and understanding voting systems and come to recognize the relevance of mathematics and to appreciate its human aspect. Editorial Reviews Booknews Johnson and Mowry (Diablo Valley College) introduce topics in mathematics that are usable and relevant to any educated person, but particularly to liberal arts students who have just completed a course in intermediate algebra
more details The Multivariable portion of the Soo Tan Calculus textbook tackles complex concepts with a strong visual approach. Utilizing a clear, concise writing style, and use of relevant, real world examples, Soo Tan introduces abstract mathematical concepts with his intuitive style that brings abstract multivariable concepts to life. The Multivariable text provides a great deal of visual help by introducing unique videos that assist students in drawing complex calculus artwork by hand. In keeping with this emphasis on conceptual understanding, each exercise set begins with concept questions and each end-of-chapter review section includes fill-in-the-blank questions which are useful for mastering the definitions and theorems in each chapter. Additionally, many questions asking for the interpretation of graphical, numerical, and algebraic results are included among both the examples and the exercise sets5 inches width x 0.75
Math Trek 04/01/05 The NECTAR Foundation's new Math Trek suite of products ( uses curriculum-based programs that cover the foundations of math for grades 1-12. These engaging programs feature sound, graphics, animation and music clips through interactive tutorials, problem-solving activities, assessment components and student tracking. The Macintosh- and Windows-compatible learning aids include a comprehensive teacher resource document with print support materials, as well as individual, group and culminating performance tasks that incorporate many skills into a meaningful context. The NECTAR Foundation also offers specialized programs for algebra I, calculus and trigonometry
A+Click Math Self-study Tests and Skill Assessment for Grade 1 to Grade 12. It includes more than 1000 challenging problems... see more A+Click Math Self-study Tests and Skill Assessment for Grade 1 to Grade 12. It includes more than 1000 challenging problems and answers and tons of illustrations. The practice tests adapt to student ability. This website has a graduated set of problems, starting from very simple, to quite difficult. To progress to a new level, you have to answer five consecutive questions correctly. The questions are appropriate to elementary students; almost any second grader could answer the easiest. This is a good assessment test without being insulting or frustrating An author's Snapshot for Foundations of Computer Science for the material found in MERLOT at... see more An author's Snapshot for Foundations of Computer Science for the material found in MERLOT at This snapshot shows an overview of the material. This was created in the MERLOT Content Builder. PC-based software and interactive tutorial to learn how to write mathematical proofs. Makes use of innovative proof-checking... see more PC-based software and interactive tutorial to learn how to write mathematical proofs. Makes use of innovative proof-checking technology to provide instant feedback to students/users. Free, full-version download available at website.
A seminar course required as a culminating experience for mathematics majors. Students prepare and present a portfolio following departmental guidelines to document achievement of the learning goals for the mathematics major. Additionally, students present a lecture on a topic not covered in core courses in consultation with the instructor and take the Major Field Test for Mathematics. Grade of C or higher required. Prerequisite: Senior standing. Prerequisite(s) / Corequisite(s): Senior standing. Course Rotation for Day Program: Offered Fall and Spring. Text(s): Course Objectives To understand the theory and application of algebra and analysis. To write a computer program in a high-level language to solve a mathematical problem. To identify and apply appropriate technologies to solve mathematical problems. To write rigorous mathematical proofs. To solve real-world problems from a variety of disciplines using mathematical techniques. To approach a topic in calculus from the four perspectives: numerical, graphical, analytical, and verbal. Measurable Learning Outcomes: Demonstrate a fluency in the language of mathematics and with the fundamental results of analysis and algebra. Topical Outline: The Major Field Test in Mathematics will be administered and students will make the following presentations (all materials will be archived in the students' portfolios): Write a computer program to implement a mathematical algorithm Demonstrate an example of solving a mathematical problem using technology Present a summary and critique of a selection of proofs written for the core courses of Linear Algebra, Abstract Algebra, Probability Theory, and Advanced Calculus Present a lecture on a topic not covered in core courses in consultation with the instructor Present three examples of solving problems in other disciplines using mathematics Present a topic from analysis demonstrated from the four viewpoints Culminating Experience Statement: Material from this course may be tested on the History Assessment Test (HAT) administered during the Culminating Experience course for the degree. During this course the ETS Proficiency Profile may be administered. This 40-minute standardized test measures learning in general education courses. The results of the tests are used by faculty to improve the general education curriculum at the College
Find a Marana Algebra 2 ...In Algebra students will learn to solve equations (1st and 2nd degree) by connecting and disconnecting numbers and letters that represent numbers. I will ask them to "al-jabr". It was once known as the Cossic Art. "Coss" is Latin for "the thing".

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