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Description of Saxon Calculus: Homeschool Kit by Saxpub the Saxon Calculus Homeschool Kit reviews key algebra, trigonometry and analytic geometry topics while introducing limits, functions, and the differentiation and integration of variables. This comprehensive text is ideal for future mathematicians, scientists and engineers! I have used Saxon with all three of my children. I like it because each lesson is clearly defined with instructions and sample problems to reinforce each lesson. There are tests available after every five lessons with clear guidelines on when to administer them. I also like that the child can read through the lesson independently and proceed with it if they have a good understanding. Thus it did not require teacher participation for every lesson.
Boole v1.2 Boole is a program written in Basic that obtains the canonical terms of a boolean function or the result of assigning values to each variable that composes the function. The program is in Spanish as English, and it works with the TI-89, TI-92+ and V200. Accurate Day of the Week Finder Have ya ever wanted to find out what day of the week (Monday, Tuesday, etc) you were born, or when a particular date took place(July 4, 1776)? Well, with this neat little program, all you have to do is enter in the month(12), day(19), and year(1987), and voila, the day of the week appears! I used a formula that I found on the internet somewhere. It is really small, so download it today! This is the ported version of the Day of the Week Finder for the TI-83 Plus/Silver Edition by me. And the day on the screenshot is my birthday for anyone who cares. GPS Math GPS _MATH contains a TI-89 calculator activity which explores the mathematics of Global Positioning in a 2-dimensional world. Graphical and numerical techniques are both used to solve the GPS system of equations. A step-by-step handout is included. Follow instructions in the ReadMe file to set up programs. Human Calculator Trainer This is a program using a revolutionary method to teach you to add and subtrace numerals with speed. You will notice an increase in your addition/subtraction speed withing a few days of use. Irrational Number Trainer Version 1.0 This is an awesome program that helps you memorize common irrational constants. In this version, there are five constants to memorize (pi, e, the square root of two, phi, and the E-M constant). It is relatively small at about four and a half kilobytes. It is extremely effective for fast results. It is aslo fun to recite pi to 200 digits in math class and get extra credit ;) A Must Download! Taverite Taverite is a prog that computes the evaluation of a logical expression. You give the variables names to the program and it gives you the table. Taverite est un programme calculant une table de vérité à partir d'une expression logique.
Students taking this unit will have an opportunity to see how decision-making problems in industry, business, and civic services can be solved using modern mathematical and algorithmic techniques. Students will learn how to make better decisions through mathematical methods in optimization problems such as: production planning, machine scheduling, robotics/vehicle routing, network design, and resource allocation. Techniques covered include linear and integer models as well as game theory models. Four assignments will be offered, out of which students may choose two that suit their own areas of interest.
Calculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series, and which constitutes a major part of modern university education. It has two major branches, differential... Full article >>> Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a real variable x and an interval of the real line, the integral Full article >>> An overview of the background to calculus and a list of some applications. ... Calculus is concerned with comparing quantities which vary in a non-linear way. ... Full article >>> Calculus, Prerequistes and Applications: a Flowchart. Possible ways to Use this Text ... Basic Calculus: From Archimedes to Newton, 2001, and ... Full article >>> Features topic summaries with practice exercises for derivative and integral calculus. Includes solutions. Authored by D. A. Kouba.
History of Mathematics An Introduction 9780073051895 ISBN: 0073051896 Edition: 6 Pub Date: 2005 Publisher: McGraw-Hill College Summary: David Burton covers the history behind the topics typically covered in an undergraduate maths curriculum or in elementary or high schools. He illuminates the people, stories, and social context behind mathematics' greatest historical advances, while maintaining appropriate focus on the mathematical concepts themselves. Burton, David M. is the author of History of Mathematics An Introduction, published 2005 u...nder ISBN 9780073051895 and 0073051896. Nineteen History of Mathematics An Introduction textbooks are available for sale on ValoreBooks.com, fourteen used from the cheapest price of $41.00, or buy new starting at $99.18.[read more]
Develop Algebraic Thinking 3-5 - MAT-925Use manipulatives, real-world problem-solving, and captivating activities to engage your students in developmentally appropriate algebraic thinking. This ONLINE course will provide you with numerous opportunities to reflect on current research and pedagogy related to algebraic thinking in the intermediate grades. Make direct application to your own classroom through the design and development of lessons that explore growth patterns, tables, variables, and coordinate graphs. All of the readings and activities are built upon the Common Core standards. Teachers may complete this course with or without students. Connect With Testimonial "Fresno has a wonderful program. The materials arrived quickly, classes were moderately priced, and I use various aspects of the classes that I've taken in my daily teaching schedule. The classes were practical and I enjoyed them. Well done FPU!"
Mathematics Education and Resource Center The Mathematics Education and Resource Center (MERC) is a free service available to support cadets in math studies. There will be tutors on-hand for help in the following mathematics courses: MA-105/MA-106 - Introduction to Probability and Statistics I & II MA-103 - Fundamentals of Matrix Algebra MA-114 - Pre-Calculus MA-123/MA-124 - Calculus & Analytic Geometry I & II MA-125-MA-126 - Quantitative Methods I & II MA-215 - Multivariable Calculus MA-311 - Differential Equations Location 700 Level of Preston Library, Open Math Lab (OML) No appointments necessary. SUMMER 2014 Open Hours Monday- Thursday: 1300-1600; 1900- 2300 Friday: 1300- 1600 At the Mathematics Education and Resource Center, our mission is to: support our Cadet Corps in their pursuit of mathematical studies, engage our VMI faculty and staff in their goals of maintaining rigor and excellence in their respective disciplines, and work with our community in their desire for productive, inspired, and healthy citizens
1846289718 9781846289712 Sturm-Liouville Theory and Its Applications:Undergraduate textbooks on Fourier series which follow a pointwise approach to convergence miss the rich geometric content which comes with treating the subject within the inner product space L2. This book, developed from a course taught to senior undergraduates, provides a unified introduction to Fourier analysis and special functions based on the Sturm-Liouville theory in L2. The basic results of this theory, namely the orthogonality and completeness of its eigenfunctions, are established in Chapter 2; the remaining chapters present examples and applications. The last two chapters, on Fourier and Laplace transformations, while not part of the Sturm-Liouville theory, extend the Fourier series method for representing functions to integral representations.The treatment relies heavily on the convergence properties of sequences and series of numbers as well as functions, and assumes a solid background in advanced calculus and an acquaintance with ordinary differential equations and linear algebra. Familiarity with the relevant theorems of real analysis, such as the AscoliâArzelà theorem, is also useful for following the proofs.The presentation follows a clear and rigorous mathematical style that is both readable and well motivated, with many examples and applications used to illustrate the theory. Although addressed primarily to undergraduate students of mathematics, the book will also be of interest to students in related disciplines, such as physics and engineering, where Fourier analysis and special functions are used extensively for solving linear differential equations.
Find a Lombard Precalculus TutorDifferential equations are equations involving a function and one or more of its derivatives. Traditionally, they have frequently appeared in the mathematical models constructed by natural scientists and engineers. However, in recent times, their use by social scientists has increased dramaticallyIt has been founded in the 17th century by Leibniz and Newton and was developed primarily to solve problems in physics. Calculus is divided into Differential Calculus, which studies instantaneous rates of change or derivatives, and Integral Calculus, which is concerned with areas of mathematical...
Applied Combinatorics - 2nd edition Summary: For courses in undergraduate Combinatorics for juniors or seniors. This carefully crafted text emphasizes applications and problem solving. It is divided into 4 parts. Part I introduces basic tools of combinatorics, Part II discusses advanced tools, Part III covers the existence problem, and Part IV deals with combinatorial optimization. Book has very light external/internal wear. It may have creases on the cover and some folded pages.This is a USED book. $12.4612
Mathematics for Physicists 9780534379971 ISBN: 0534379974 Pub Date: 2003 Publisher: Thomson Learning Summary: This essential new text by Dr. Susan Lea will help physics undergraduate and graduate student hone their mathematical skills. Ideal for the one-semester course, MATHEMATICS FOR PHYSICISTS has been extensively class-tested at San Francisco State University--and the response has been enthusiastic from students and instructors alike. Because physics students are often uncomfortable using the mathematical tools that they... learned in their undergraduate courses, MATHEMATICS FOR PHYSICISTS provides students with the necessary tools to hone those skills. Lea designed the text specifically for physics students by using physics problems to teach mathematical concepts. Lea, Susan M. is the author of Mathematics for Physicists, published 2003 under ISBN 9780534379971 and 0534379974. Five hundred eighty five Mathematics for Physicists textbooks are available for sale on ValoreBooks.com, one hundred eighteen used from the cheapest price of $32.97, or buy new starting at $69
This module is included inLens:Community College Open Textbook Collaborative By: CC Open Textbook CollaborativeAs a part of collection: "Elementary Algebra"Quadratic Equations: Objectives Summary: This module is from Elementary Algebra</link> by Denny Burzynski and Wade Ellis, Jr. Methods of solving quadratic equations as well as the logic underlying each method are discussed. Factoring, extraction of roots, completing the square, and the quadratic formula are carefully developed. The zero-factor property of real numbers is reintroduced. The chapter also includes graphs of quadratic equations based on the standard parabola, y = x^2, and applied problems from the areas of manufacturing, population, physics, geometry, mathematics (numbers and volumes), and astronomy, which are solved using the five-step method. This module contains the objectives for the chapter "Quadratic Equations".
PrealKEY MESSAGE: Tom Carson'sPrealgebra, Third Editionresponds to individual learning styles with a complete study system. The system begins with a Learning Styles Inventory and then presents targeted learning strategies to guide students to success. Tom speaks to readers in everyday language and walks them through the concepts, explaining not only how to do the math, but also where the concepts come from and why they work. KEY TOPICS: Whole Numbers; Integers; Expressions and Polynomials; Equations; Fractions and Rational Expressions; Decimals; Ratios, Proportions, and Measurement; Percents; More with Geometry and Graphs MARKET: For all readers interested in prealgebra.
major axis the line segment (whose ends are vertices) which passes through the foci of an ellipse. matrix a rectangular collection of numbers, arranged in rows and columns, surrounded by a single set of brackets on either side. minor notated Mij, and corresponding to a square matrix A, it is equal to the determinant of the matrix created by deleting the ith row and jth column of A. minor axis the line segment, perpendicular to the major axis, which passes through the center of an ellipse and has endpoints on the ellipse. modulus the distance from the origin to the point on the coordinate plane representing the graph of the complex number c = a + bi; also called the absolute value of c. natural exponential function the exponential function with Euler's number as its base: f(x) = ex. natural logarithm the logarithmic function of base e, written "ln x" and read either "natural log of x" or "L-N of x." natural numbers the most basic set of numbers, often learned when one is first taught to count: {1, 2, 3, 4, 5, 6, . . . }. They are also called the counting numbers. oblique triangles triangles which do not contain a right angle. odd functions functions such that f (−x) = −f (x). one-to-one a term used to describe a function for which every output has only one corresponding input. Only one-to-one functions have inverses. optimal maximum or minimum values of a function. order describes how many rows and columns are in a matrix. orthagonal describes two vectors which are perpendicular to one another. parabola a set of coplanar points equidistant from a fixed point (the focus) and a fixed line (the directrix). parametric equations two equations (usually "x =" and "y =") defined in terms of a third variable, called the parameter. partial sum sum of the terms of a series whose upper summation limit is finite. Pascal's triangle the triangular arrangement of the coefficients of binomial expansions; the (n + 1)th row of the triangle gives the coefficients for the expression (a + b)n. period the shortest length along the x-axis after which a periodic graph will repeat itself. periodic describes a graph which will repeat itself infinitely after some fixed length of the x-axis, called the period. polar axis the fixed ray in polar coordinates representing the initial side of the angle θ. polar coordinates coordinates in the form (r, θ), where r is the distance from the pole and θ is the angle from the polar axis. pole the fixed point in polar coordinates from which the distance r to the point is measured. principal the initial investment in a compound interest problem. quadrantal an angle in standard position whose terminal side falls upon a coordinate axis. radian measurement of an angle in standard position that, when extended to a circle of radius r centered at the origin, will mark the endpoints of an arc whose length is also r. radius the fixed distance between the center of a circle and any point on that circle. rational numbers any number that can be expressed as a fraction , where a is an integer and b is a non-zero integer. Rational Root Test a method used to determine all possible rational roots for a polynomial. real numbers any number which is either rational or irrational is also a real number, because the real numbers are made up by combining those two, smaller groups. rectangular coordinates coordinates in the form (x,y) in the Cartesian plane. recursive sequence sequence whose terms are defined based on one or more preceding terms of the sequence. reduced row-echelon form the form of a matrix in which the diagonal contains only 1s, all elements above and below the diagonal are 0s, and any rows containing only zeros are placed at the bottom of the matrix. row-echelon form the form of a matrix in which its diagonal contains only 1s, all elements to the left of the diagonal are 0s, and all rows made up entirely of zeros appear at the bottom of the matrix. scalar term used to refer to a numeric, non-vector quantity when dealing with vectors.
make sure each physics problem, math problem, and philosophy argument is analyzed step-by-step. The student will learn (1) how to apply laws of mathematics to solve problems, instead of memorization. They will also learn (2)Proper NOTATION (3)SPECIFIC Problem-Types that are "rare circumstances" in math that require special attention/method.
Math for Health Care Professionals 9781401891794 ISBN: 1401891799 Publisher: Thomson Learning Summary: This workbook provides over 2700 math questions for practice and mastery of the each of the objectives included in Math for Health Care Professionals and Math for Health Care Professionals Quick Review. The use of this workbook to enhance understanding of math concepts also helps the learner become more confident in his or her knowledge and skills. The workbook may be used by anyone wishing to sharpen their math skil...ls. Mike, Kennamer is the author of Math for Health Care Professionals, published under ISBN 9781401891794 and 1401891799. Two hundred fifty two Math for Health Care Professionals textbooks are available for sale on ValoreBooks.com, one hundred twenty one used from the cheapest price of $0.90, or buy new starting at $15
Sentry CA656 56Function Scientific Calculator Quick Overview Sentry CA656 56Function Scientific Calculator Computing is a breeze with the Sentry 56function Scientific Calculator. With its tilting display and 10digit LCD, you'll be able to breeze through highspeed calculations and store them into memory if needed.
Like most math classes it has a few elementary parts that once grasped lend incite to the rest of the topic. Knowing what a derivative means and how an integral is used are just two of the basic elements. Understanding Calculus lead Newton into greater understanding of Physics and which has lead us to many other places as well.
Files in PDF, DVI, and Postscript formats, to help students learn to use calculus in applications and to have confidence in setting up formulas using derivatives and integrals. Contents include: a conceptual approach to applications of integration, max-min problems, definition of the logarithm function and differential, exponential growth and decay, convergence of infinite series, interpolation and numerical integration, discontinuities in one and several variables, curvature, normal and tangential components of acceleration, derivation of Kepler's Second Law, the potential function of a vector field, Green's Theorem, the series solution of a differential equation, and review problems for differential equations.
Synopses & Reviews Publisher Comments: This concisely written book gives an elementary introduction to a classical area of mathematics--approximation theory--in a way that naturally leads to the modern field of wavelets. The exposition, driven by ideas rather than technical details and proofs, demonstrates the dynamic nature of mathematics and the influence of classical disciplines on many areas of modern mathematics and applications. Key features and topics: * Description of wavelets in words rather than mathematical symbols * Elementary introduction to approximation using polynomials (Weierstrass' and Taylor's theorems) * Introduction to infinite series, with emphasis on approximation-theoretic aspects * Introduction to Fourier analysis * Numerous classical, illustrative examples and constructions * Discussion of the role of wavelets in digital signal processing and data compression, such as the FBI's use of wavelets to store fingerprints * Minimal prerequisites: elementary calculus * Exercises that may be used in undergraduate and graduate courses on infinite series and Fourier series"Synopsis" by Springer"Synopsis" by Libri
Study Guides Study Guides: These study guides are great if a) students have been absent, b) students' notes from class are undecipherable, c) students need to study for an upcoming test OR d) you just want to see what kinds of things we're working on in class. Curriculum We are using the College Preparatory Mathematics, Algebra Connections textbook. Click here for the publisher's companion website to this book. Old Calendars 4/1 - 10.2.5 No Homework 4/2 - Study up on 1 past target & 2nd Round OAKs Testing Hmwk: Study 4/3 - Retest on 1 past target & 2nd Round OAKs Testing No Homework 4/4 - Block Day/No Class 4/5 - 10.3.1 Hmwk: 10-105 thru 10-107 4/8 - 10.3.2 Hmwk: 10-114 4/9 - Practice Completing the Square Hmwk: Complete Handout 4/10 - 10.4.1 & 10.4.2 Hmwk: 10-135, 139, & 140 4/11 - Block Day/No Class 4/12 - Teacher Grading Day/No School 4/15 - Review Exponent Rules Hmwk: Finish in-class work 4/16 - Practice Exponent Rules & Simplifying Radicals Hmwk: Finish in-class practice 4/17 - 10.4.3 Hmwk: 10-147 & 10-148 4/18 - Block Day/No Class 4/19 - Chapter 10 Closure Hmwk: Finish Chapter Closure 4/22 - Chapter 10 Test No Homework 4/23 - Proficiency Packets & Test Corrections Homework: Complete Test Corrections 4/24 - 11.1.1 No Homework 4/25 - Block Day/No Class 4/26 - 11.1.2 Hmwk: 11-21 & 11-25
Wednesday, October 05, 2011 In my math classes a typical test is modeled on the character of test questions students will see on their final exams: multiple choice, short answer and long answer. In a grade 10 math class, what we used to call Applied Math 20S, a multiple choice question might be: A factory makes tents. The cost of running the factory is $300 per day plus $50 for each tent made. What is the total cost (C), in dollars, as a function of the number of tents (t) made? (A) C = 350t (B) C = 50t + 300 (C) C = 300t + 50(D) t = 300 + 50C I like this question because it quickly allows a student to show whether or not they understand what a "function" is and it's easy to grade. While they have a 25% chance of getting it correct by guessing, in the context of the entire test, and their classroom experiences with me (read: conversations), I know if a student has grasped the concept. The monthly cost, C, in dollars, of using a cell phone is calculated using the function C(t) = 0.09t + 20 where t is the time in minutes. What is the monthly fee and the cost per minute for this cell phone contract? Another quickie that reveals whether or not the student can decode the information given in a function. Another question might ask them to reverse that; encode a function given the description of a linear relationship. As a matter of fact, there's a fundamental principle there about learning math: Anything you can do you should also be able to undo. i.e. If you can decode the information in a function you should also be able to encode information in a function. Here's a long answer question: The cost of a school graduation dance has a fixed cost of $1500 for the band, security, and so on, and a cost of $22 per plate for every person attending. (a) Write the formula which states how the total cost, C, is related to the number of people attending, n. (b) What is the slope? What does it mean? (c) If the maximum capacity of the hall is 225 people, what is the maximum cost of the dance? (d) State the domain of this function. (e) State the range of this function. The question is not ideal; (d) should be a "gimme" if they understood (c) and (e) depends on the formula they created in (a). Mind you, if they wrote an incorrect formula in (a) but correctly applied it in (e) that's worth full marks in (e). Something these three questions have in common is they require that a student understand the meaning of the marks they're making on the page. While every test has some straight forward calculations, by and large calculations are what computers do best. I want my students to understand what the math means and how it hangs together. Computers don't do that so well; although they're getting better at faking it. That's largely because of the cleverness of people who understand the math behind what computers do. If your assessments largely test mechanical skills that's what your students will focus on learning. If your assessments test for understanding that's what your students will focus on learning. Which would you rather learn?
Math books
This requires sensitivity. Present the concepts in ?chunks? that are too big and they will be incomprehensible. Present them broken down into too much detail and the student will become board and his/her mind will wander. ...In the US, I attended MCC college and majored in mathematics. In the first place, math is a simple logic which needs to be approached methodically. There is no need chewing formulas in your head to solve math problems.
Beginning Algebra Beginning Algebra This is a free online course offered by the Saylor Foundation.'... More This is a free online course offered by the Saylor Foundation. ' exposed to. This course is also intended to provide you with a strong foundation for intermediate algebra and beyond. This course will begin with a review of some math concepts formed in pre-algebra, such as order of operations and simplifying simple algebraic expressions to get your feet wet. You will then build on these concepts by learning more about functions, graphing of functions, evaluation of functions, and factorization. You will spend time on the rules of exponents and their applications in distribution of multiplication over addition/subtraction.'
Nature of Mathematics Written for liberal arts students and based on the belief that learning to solve problems is the principal reason for studying mathematics, Karl ...Show synopsisWritten for liberal arts students and based on the belief that learning to solve problems is the principal reason for studying mathematics, Karl Smith introduces students to Polya's problem-solving techniques and shows them how to use these techniques to solve unfamiliar problems that they encounter in their own lives. Through the emphasis on problem solving and estimation, along with numerous in-text study aids, students are assisted in understanding the concepts and mastering the techniques. In addition to the problem-solving emphasis, "The Nature Of Mathematics, 12e, International Edition" is renowned for its clear writing, coverage of historical topics, selection of topics, level, and excellent applications problems. Smith includes material on such practical real-world topics as finances (e.g. amortization, installment buying, annuities) and voting and apportionment. With the help of this text, thousands of students have 'experienced' mathematics rather than just do problems-and benefited from a writing style that boosts their confidence and fosters their ability to use mathematics effectively in their everyday lives.Hide synopsis Description:Good. 083 Item may show signs of shelf wear. Pages may include...Good. 0831133275794Paperback. New Condition. SKU: 9781133275794-1-0-3 Orders ship...Paperback. New Condition. SKU: 9781133275794-138737586Good. Hardcover. May include moderately worn cover, writing,...Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780538737586-4Fine. Hardcover. Almost new condition. SKU: 9780538737586-2-0-3...Fine. Hardcover. Almost new condition. SKU: 9780538737586-2The title of this book does not indicate that it is the teacher's edition, neither is this reflected in the photo. If you read the entire description it is confusing. It does state that the teacher's edition is the same as the student edition, but does not in any way indicate that you are actually purchasing the teacher's
Throughout this course an attempt will be made to develop the new concepts, and to then illustrate them by solving sample problems. Where possible actual demonstrations and computer software to carry out simulations and to provide graphical representation of subjects are used. As in all areas of knowledge a deeper understanding of science involves more than a simple recitation of various facts or laws. It entails the ability to formulate reasoned arguments based on sound principles. Our objective is to be able to apply these definitions and laws to analyze new situations, and to be able to make predictions. This requires not only sound mathematical skills but also a clear understanding of the fundamental concepts. Development of these skills requires a sustained effort. The following is a list of suggestions on how to study physics that I think can help improve understanding and performance. 1. Students should realize there is not sufficient time to discuss every covered section in class, so it is imperative that students read the material before it is discussed in class. This introduces you to the definitions and terms used, some of which are words you use in everyday life, but which have specific, and sometimes different, meanings in physics. 2. As you read (and reread) the text write out the details of the examples in the text. The examples are your first illustrations of how to use the principles presented in the reading. Do not do this with the goal of memorizing the example, but rather understanding which principles or laws are employed, and how each step follows from the preceding ones. If there are points you do not understand note them, and ask your instructor about them. 3. Work on the homework problems on a regular basis, before they are discussed in class. Here you have the chance to apply your understanding in tackling a new problem. This is the ultimate goal, but it requires practice. Although there is no single prescription for solving all problems there are some aids. (i) For most problems it is beneficial to draw a diagram representing the problem and the information given. (ii) In general, the solution will involve relating variables through several relations. I find it helpful to first write out each of these equations, regardless of how simple it may be, and then to combine them to solve for the quantity desired. It is better to work in more short steps rather than a few larger ones. (iii) In addition, I find that it is often advantageous to solve a problem algebraically, and to put in numbers only at the end. Numbers quickly lose their meaning when many appear in a calculation, whereas algebraic variables are more easily identified, especially in conjunction with a good diagram. This makes it easier to check your work. Finally, an algebraic solution is a solution for all choices of numerical values of the variables. It is not necessary to redo the entire problem if values of some of the quantities are changed. Furthermore, an algebraic solution can easily be rearranged to solve for different quantities. GRADING:Grades will be determined on the basis of the number of points earned out of a maximum of 550 points distributed as follows: 3 100point exams to be given during regular class periods, a 150-point comprehensive final, and 100 points for unannounced quizzes/homework. The following table shows the cut-offs for the various grades. A A- B+ B B- C+ C C- D F 93% 90 87 83 80 77 73 70 60 < 60 Exams are during normal class period in the usual classroom. Anyone missing an exam will be given a grade of zero. Students missing an exam must see me as soon as possible. Students are expected to come to my office to see me about an excused absence. Students with an excused absence will be allowed to take a MAKE-UP EXAM to be given near the end of the semester. There will be no make-up for quizzes, but the lowest quiz score will be dropped. There are no extra credit assignments. Cell phones and headphones are to be put away during class.Cell phones and headphonesare not allowedduring exams. There will be an equation sheet provided for the exams and will be re-used during the term. Personal equation sheets are not allowed. Calculators are allowed. Any student having difficulty with the course should come see me for help as soon as possible. Feel free to stop in either during office hours or at other times when I am in my office. Additionally PHY 201 is participating in a Supplementary Instruction (SI) program. A peer facilitator, Stephanie Hemmer, will attend class and hold two one-hour sessions every week in order to assist any student having difficulty. For more information please see SI at UD . Occasionally I may reach you by email. I will communicate with you only through the email address listed in the University Address Book. You may change your email address by going to and following the instructions. I believe this ensures mail sent to your notes address will be forwarded to whatever address you introduce at this web site. I should point out that your Lotus notes address in the form ***@notes.udayton.edu can be accessed from anywhere once you are on the web, but you will need to learn how to do that. If you do not make this change at the above site and employ another email address you are still responsible for checking assignments or announcements which are mailed to you at the address listed in the University Address Book. FINAL EXAM:Wednesday 16 Dec., 12:20-2:10 SC119EVERYONEIS EXPECTED TO TAKE THE FINAL AT THE SCHEDULED TIME. TENTATIVE SCHEDULE - FALL, 2009 - CHAPTERS TO BE DISCUSSED This schedule is subject to change so watch for changes posted here, or for announcements made in class or by email. For each day the date is in italic and the chapter number is in standard font in the center of each box.
Discrete Mathematics for Computing presents the essential mathematics needed for the study of computing and information systems. The subject is covered in a gentle and informal style, but without compromising the need for correct methodology. It is perfect for students with a limited background in mathematics. This new edition includes: An expanded section on encryption Additional examples of the ways in which theory can be applied to problems in computing Many more exercises covering a range of levels, from the basic to the more advanced This book is ideal for students taking a one-semester introductory course in discrete mathematics - particularly for first year undergraduates studying Computing and Information Systems. PETER GROSSMAN has worked in both academic and industrial roles as a mathematician and computing professional. As a lecturer in mathematics, he was responsible for coordinating and developing mathematics courses for Computing students. He has also applied his skills in areas as diverse as calculator design, irrigation systems and underground mine layouts. He lives and works in Melbourne, Australia. Author's Biography PETER GROSSMAN has worked in both academic and industrial roles as a mathematician and computing professional. As a lecturer in mathematics, he was responsible for coordinating and developing of mathematics courses for Computer Science students. He is based in Australia and currently works in industry, in the areas of mathematical modelling and software development.
In this book, Professor Lounesto offers a unique introduction to Clifford algebras and spinors. The initial chapters could be read by undergraduates; vectors, complex numbers and quaternions are introduced with an eye on Clifford algebras. The next chapters will also interest physicists, and include treatments of the quantum mechanics of the electron, electromagnetism and special relativity with a flavour of Clifford algebras. This book also gives the first comprehensive survey of recent research on Clifford algebras. A new classification of spinors is introduced, based on bilinear covariants of physical observables. This reveals a new class of spinors, residing between the Weyl, Majorana and Dirac spinors. Scalar products of spinors are classified by involutory anti-automorphisms of Clifford algebras. This leads to the chessboard of automorphism groups of scalar products of spinors. On the analytic side, Brauer-Wall groups and Witt rings are discussed, and Caucy's integral formula is generalized to higher dimensions. En savoir plus sur l'auteur Première phrase Vectors provide a mathematical formulation for the notion of direction, thus making direction a part of our mathematical language for describing the physical world.&nbspLire la première page Commentaires en ligne Commentaires client les plus utiles sur Amazon.com (beta) Amazon.com: 5 commentaires 24 internautes sur 24 ont trouvé ce commentaire utile For the Physicist - Not the Mathematician13 novembre 2002 Par Un client - Publié sur Amazon.com Format: Broché Lounesto's book is replete with geometric and physical applications. The treatment is informal and non-rigorous and appears to have been designed with developing intuition in the reader. The text starts slowly working through many examples of particular Clifford algebras of interest and their relevence to physical problems. Towards the end Lounesto investigates general Clifford algebras and their associated spin groups as well as some specialized topics. This is a good introductory text but fails to give the reader a firm mathematical basis of the material. Most striking is the almost total lack of proofs of any kind - the author is content merely to state the most important results but seldom leaves the reader with any mathematical justification. As such it is really a primer and the student of Clifford algebras must after working through the material move beyond to a rigorous algebraic text. 19 internautes sur 19 ont trouvé ce commentaire utile The best introduction to Clifford algebras and spinors26 décembre 2006 Par Prof C. R. PAIVA - Publié sur Amazon.com Format: Broché The second edition (2001) of this book (from the late Professor Pertti Lounesto) should be considered, for those interested in Clifford algebras and their applications in physics and in engineering, as a pedagogically brilliant introduction. Chapters 1 and 2 form a pedagogical unit for an undergraduate course on vectors (and the scalar product) and on a geometrical interpretation of complex numbers. Although only the geometric algebra of the plane is addressed in these two chapters, the student will get a firm grasp of the first (real) Clifford algebra (with dimension 4) defined on the 2D linear space RR (with dimension 2). A clear distinction between a common associative algebra (such as any matrix representation of this first geometric algebra) and the Clifford algebra itself is stressed. In fact, by introducing the square of a vector as the square of its length, this important distinction is definitely drawn. Furthermore, with this definition, a clear interpretation of the Clifford product of vectors is made possible as well as the introduction of bivectors. Reflections and rotations in the plane can then be easily handled, although the rotor concept is not explicitly introduced. In Chapter 2 a beautiful and simple introduction to complex numbers clearly explains how the vector plane is the odd part of the first Clifford algebra, whereas the complex plane is the even part of that same Clifford algebra. Therefore, the student will be able to understand the distinction between the structure of C as a real algebra and the structure of C as the field of complex numbers. Moreover, an important distinction between the unit bivector, which anticommutes with every vector, and the number i=sqrt(-1) as the imaginary unit, which commutes with every vector, is then easily understood. Although the author does not include Chapter 3 in the same unit as the one formed by the two previous chapters, I would certainly recommend its inclusion within the same pedagogical unit. Indeed, the second (real) Clifford algebra (with dimension 8), defined on the linear space RRR (with dimension 3), is a natural extension of the concept of a Clifford algebra (defined over the real field) to the ordinary 3D space. In this context it is possible to explain the important distinction between the cross product of vectors (its result being a vector) and the exterior product of vector (its result being a bivector). Through the Hodge dual, it is then possible to understand the connection between a given vector and its corresponding dual bivector as an oriented plane segment. Moreover, it is clear how the cross product does require a metric while the exterior (or outer) product does not. Indeed, the cross product satisfies the Jacobi identity which makes the linear space RR*R a non-associative algebra called a Lie algebra. Finally, it is also transparent what distinguishes this second (real) Clifford algebra from Grassmann's exterior algebra: while the Clifford multiplication of vectors does preserve the norm, the exterior multiplication of vectors does not. In fact, this is what allows rotations to become represented as operations inside the Clifford algebra and to state that this algebra provides us with an invertible product for vectors as a direct consequence of its (graded) multivector structure. Although it is possible to define a cross product of two vectors in seven dimensions (see pages 96-98), the student will readily understand the reason why the cross product of vectors only has a unique direction in three dimensions (indeed, only in this case the dual of a vector is a bivector). Hence, with this undergraduate pedagogical unit formed by Chapters 1-3, we have a real and consistent alternative to the elementary vector algebra solely based on the cross product - as universally promoted since Gibbs misguidedly advocated abandoning quaternions altogether. Of course it is possible to exclaim that, to teach electromagnetism, the cross product is an invaluable tool. But then, it is also possible to defend an alternate viewpoint: electromagnetism can be easily taught with Clifford algebra as shown is Chapter 8. Indeed, if electromagnetism is to be taught in its natural framework (i.e., inside special relativity), then spacetime algebra provides the proper setting for its mathematical formulation. A very useful textbook for classical mechanics, special relativity, classical electrodynamics, quantum mechanics and gravitation using geometric algebra is «Geometric Algebra for Physicists» by Chris Doran and Anthony Lasenby (Cambridge University Press, 2003). A very clear and comprehensive introduction to a somewhat esoteric subject; essential for anyone in theoretical physics (especially field theory). It is possible to teach yourself the subject from this book alone--a rare feature in mathematics texts. Bravo! 21 internautes sur 26 ont trouvé ce commentaire utile Lounesto's Clifford Algebras and Spinors3 avril 2000 Par Osher Doctorow, Ph.D. - Publié sur Amazon.com Format: Broché Clifford algebras make geometry and its applications to advanced physics incredibly simple, and this book is one of the best that I have read on this topic and on spinors. Readers outside physics should also study this book, if necessary with the help of a consultant or tutor to translate into more or less ordinary English, because most fields of science or industry are in need of tremendous simplification. Lounesto's approach is algebraic, as is Okubo's (see my review of his book) and Chisholm's (likewise), and Cambridge University Press as usual is at the head of the field in publishing the deepest and yet most simple topics. Spinors (the word comes from spin plus the ending -ors) describe spin in quantum theory, and Lounesto has the most detailed division of the types of spinors that I have seen in a book together with their physical applications. Weyl and Majorana spinors describe the neutrino (of the weak force, although the former only describe massless neutrinos), flagpole spinors appear to describe the strong nuclear force/interaction and appear to be related to quark confinement, Dirac spinors describe the electron (Weyl and Majorana spinors, unlike Dirac spinors, are singular with a light-like pole/current). Penrose flags (see my reviews of Roger Penrose's books) are related to Weyl and Majorana spinors. Penrose has an interesting theory of twistors which is well reviewed in some of the popular science books. 2 internautes sur 5 ont trouvé ce commentaire utile Review from Russia13 juin 2007 Par Pyotr Ivanshin - Publié sur Amazon.com Format: Broché I would like to see more algebraic theory of spinors in this book (D. Salamon's "Spin geometry and SW invariants" possess better though somewhat one-sided insight in this field); nevertheless, the amount of multifarious information containd here some of which I have been able to get only though Internet (sometimes with controversial results and always with some frustrating notation), is outstanding. Part of the book is dedicated to the application of spinors to physics, and this seems to be useful especially for high-school professors.
More About This Textbook Overview Building on the success of its first five editions, the Sixth Edition of the market-leading text explores the important principles and real-world applications of plane, coordinate, and solid geometry. Strongly influenced by both NCTM and AMATYC standards, the text includes intuitive, inductive, and deductive experiences in its explorations. Goals of the authors for the students include a comprehensive development of the vocabulary of geometry, an intuitive and inductive approach to development of principles, and the strengthening of deductive skills that leads to both verification of geometric theories and the solution of geometry-based real world applications. Updates in this edition include the addition of 150 new problems, new applications, new Discover! activities and examples and additional material on select topics such as parabolas and a Three-Dimensional Coordinate System. Related Subjects Meet the Author Daniel C. Alexander, now retired, taught mathematics at the secondary and college levels for over 40 years. His final 25 years of teaching were at Parkland College in Champaign, Illinois; before retirement, his position at Parkland College was as mathematics professor emeritus. Although Professor Alexander held undergraduate and graduate degrees from Southern Illinois University, he also completed considerable post graduate course work as well. He delivered many talks and participated in various panel discussions at mathematics conferences of IMACC, AMATYC, and ICTM. Further, he had numerous published articles in the ICTM, NCTM,and AMATYC mathematics journals. Geralyn M. Koeberlein, now retired, taught mathematics at Mahomet-Seymour High School in Mahomet, Illinois for 34 years. She taught several levels of math, from Algebra I to AB Calculus. In the last few years of her career, Geralyn was also Chair of the Math and Science Department. After receiving her Master's Degree from the University of Illinois early in her teaching years, Geralyn continued her education by receiving over 90 hours of post graduate credit. She was a member of the the ICTM and the NCT 2007 Challenges the eager mind. This is a great text. I've read some reviews at other sites, claiming that this has to be the worse textbook ever written. Well, I'm here to tell you that those are false statements. This book, is not like your typical high school geometry text. It was clearly written in mind for college student, therefore it is a tad bit more challenging. One has to remember that geometry is all about proofs, not just the figures/shapes. So if you're buying this book for a class or for your own pleasure, be ready to test you reasoning skills. Nevertheless this is a rigorous yet concisely written textbook. And remember, it's all about the logic. 1 out of 1 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Known for a clear and concise exposition, numerous examples, and plentiful problem sets, Jerome E. Kaufmann and Karen L. Schwitters's COLLEGE ALGEBRA is an easy-to-use book that focuses on building technique and helping students hone their problem-solving skills. The eighth edition focuses on solving equations, inequalities, and problems; and on developing graphing techniques and using the concept of a function. Updated with new application problems and examples throughout, the eighth edition is accompanied by a robust collection of teaching and learning resources, including Enhanced WebAssign, an easy-to-use online homework management system for both instructors and students.
Foundation Goals Math and Natural Sciences In 2011, the Math and Natural Sciences faculty began conversations regarding learning goals for the current Math and Natural Science Foundation. Through these conversations, it became apparent that our aspirations for Knox students were greater than the goals that could be encompassed by a single foundation in Math and Natural Science. Through a series of conversations, the decision to request that faculty approve a change in graduation requirements that would: Create a new foundation: Quantitative and Symbolic Reasoning and Change the existing MNS foundation to Natural and Physical Sciences. This foundation would have its core the scientific method. The learning goals for the new foundations, approved by the Knox faculty to take effect in Fall of 2014, are below: Learning Goals Natural and Physical Sciences : Courses in this area lie in the physical or biological sciences that include an experimental component. Natural and Physical Sciences Learning Goals At the completion of a Foundations course in the Natural and Physical Sciences, students will be able to: Students will be able to identify key concepts used in understanding the physical or biological world using a scientific discipline or framework. Students will be able to describe important theories in the physical or biological sciences and the empirical evidence upon which they are based. Students will be able to describe the application of the scientific method to questions using the following concepts: formulate and test a hypothesis, analyze data, draw conclusions. Quantitative and Symbolic Reasoning: Courses in this area focus on methods of abstract or symbolic reasoning including mathematics, logic, algorithmic or statistical reasoning. Quantitative and Symbolic Reasoning Foundations Learning Goals At the completion of a Foundations course in Quantitative and Symbolic Reasoning, students will be able to: Students will be able to translate between real world concepts and quantitative or symbolic abstract structures. Students will be able to perform and interpret quantitative or symbolic manipulations in an abstract structure; Students will be able to construct carefully reasoned logical arguments. Students will be able to use abstract methods to analyze patterns and formulate conjectures with the goal of verifying them rigorously.
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Special attention is given to the powerful matrix-based-multigrid approach, which is particularly useful for problems with variable coefficients and nonsymmetric and indefinite problems. This... more 12 new & used from sellers starting at 11,022 In Stock.Ships Free to India in 5-7 days Written by the recipient of the 1997 MAA Chauvenet Prize for mathematical exposition, this book tells how the theory of Lie groups emerged from a fascinating cross fertilization of many strains of 19th and early 20th century geometry, analysis, mathematical physics, algebra and topology. The reader will meet a host... more 3 new & used from sellers starting at 14,566 In Stock.Ships Free to India in 5-7 days The Structural Theory of Probability addresses the interpretation of probability, often debated in the scientific community. This problem has been examined for centuries; perhaps no other mathematical calculationsuffuses mankind's efforts at survival as amply as probability. In the dawn of the 20th century David Hilbert included the foundations of the... more 10 new & used from sellers starting at 6,356 In Stock.Ships Free to India in 5-7 days This volume aims to provide the fundamental knowledge to appreciate the advantages of the J-matrix method and to encourage its use and further development. The J-matrix method is an algebraic method of quantum scattering with substantial success in atomic and nuclear physics. The accuracy and convergence property of the method... more 3 new & used from sellers starting at 2,038 In Stock.Ships Free to India in 3-5 days Linear algebra permeates mathematics, perhaps more so than any other single subject. It plays an essential role in pure and applied mathematics, statistics, computer science, and many aspects of physics and engineering. This book conveys in a user-friendly way the basic and advanced techniques of linear algebra from the point... more 12 new & used from sellers starting at 6,105 In Stock.Ships Free to India in 5-7 days Examines linear structures, the topology of metric spaces, and continuity in infinite dimensions, with detailed coverage at the graduate level Includes applications to geometry and differential equations, numerous beautiful illustrations, examples, exercises, historical notes, and comprehensive index May be used in graduate seminars and courses or as a reference text... more 1 new & used from sellers starting at 2,535 In Stock.Ships Free to India in 3-5 days Table of Contents Graphs Euler and Hamiltonian Graphs Tree Cut Sets and Network Flow Planar Graphs Vector Spaces of a Graph Matrix Representation of Graphs Colouring of Graphs Enumeration of Graphs Directed Graphs Previous Years Question Papers Index... more
Calculus - A Complete Introduction Teach Yourself Paperback Click on the Google Preview image above to read some pages of this book! This introductory course covers all areas of calculus, including functions, gradients, rates of change, differentiation, exponential and logarithmic functions and intgration. Calculus -, differentiation, exponential and logarithmic functions and integration. Everything you will need to know is here in one book. Each chapter includes not only an explanation of the knowledge and skills you need, but also worked examples and test questions. About the Author Hugh Neill is a maths teacher who has also been an inspector and chief examiner. His books have helped over 100,000 people improve their mathematics.
Finite Mathematics, 1st Edition Geared toward business and social science majors in a single semester finite mathematics course, this text equips students with the analytical tools and technological skills they will need in the workplace. Plain language and an easy-to-read style help stress conceptual understanding and reinforce key terms and concepts. At the same time, the incorporation of real-life applications, examples, and data help engage students--even those who have never enjoyed mathematics. Pedagogy throughout the text helps students analyze data from a variety of approaches, including numeric, algebraic, graphical, literal, and technological. A robust supplement package and exciting new technology program provide students with extensive learning and support, so that instructors can spend more time teaching266.95 Purchase Options Hardcover $213.49 $213.49 Save $53.46224.46! Rent thru 09/23/14 for $42.49 $42.49 Save $224.46! Rent thru 10/23/14 for $47.99 $47.99 Save $218.96! Rent thru 12/02/14 for $52.99 $52.99 Save $213.96! Rent thru 04/21/15 for $60.99 $60.99 Save $205.96! Rent thru 01/16/16 for $66.49 $66.49 Save $200
Sitemeter Techno Tuesday, March 01, 2011 Calculus is the gateway to math, engineering, and the sciences. It's hard for me to consider anyone educated who doesn't know at least a little. It's a forbiddingly fortified gateway though. Typical calculus texts these days have 1300 pages or so and weigh and cost roughly as much as a Mercedes S600. Could that be overkill? I mean there are only one or two real ideas in calculus - the rest is technology.
Canadian Senior and Intermediate Mathematics Contests The Canadian Senior and Intermediate Mathematics Contests (CSMC and CIMC) are two contests designed to give students the opportunity to have fun and to develop their mathematical problem solving ability. Audience Students in Grades 10 or 9 or below (CIMC) and senior secondary school and CÉGEP students (CSMC); motivated students in lower grades are also encouraged to write these contests. Date Thursday, November 20, 2014 Format 9 questions; 6 are answer only and 3 are full solution marks for full solution questions assigned for form and style of presentation 2 hours 60 total marks non-programmable calculators permitted provided they are without graphical displays Mathematical Content Most of the CIMC problems are based on the mathematical curriculum up to and including Grade 10. Most of the CSMC problems are based on the mathematical curriculum up to and including the final year of secondary school.
More About This Textbook Overview As the title indicates, this book is intended for courses aimed at bridging the gap between lower-level mathematics and advanced mathematics. The text provides a careful introduction to techniques for writing proofs and a logical development of topics based on intuitive understanding of concepts. The authors utilize a clear writing style and a wealth of examples to develop an understanding of discrete mathematics and critical thinking skills. While including many traditional topics, the text offers innovative material throughout. Surprising results are used to motivate the reader. The last three chapters address topics such as continued fractions, infinite arithmetic, and the interplay among Fibonacci numbers, Pascal's triangle, and the golden ratio, and may be used for independent reading assignments. The treatment of sequences may be used to introduce epsilon-delta proofs. The selection of topics provides flexibility for the instructor in a course designed to spark the interest of students through exciting material while preparing them for subsequent proof-based
Permutations, combinations, and basic probability, 12. Matrices. Often coming between Algebra I and Algebra II, Geometry is the study of the properties and uses of geometric figures in two and three dimensions.Organic Chemistry is one of the most difficult subjects for students to understand. It
Catalog Course Descriptions MAT 092: INTRODUCTION TO MATHEMATICS 3 hrs. (BEC) Prerequisite: Placement into MAT 092 is according to placement test scores or on a voluntary basis. This course is designed for students who need to review basic arithmetic skills before taking Beginning Algebra (MAT 094 - formerly MAT 104). Topics include basic operations and applications of whole numbers, fractions, decimals, signed numbers and an introduction to algebra. As calculators are not permitted in MAT 092 except for enrichment purposes, students must be able to add, subtract, multiply and divide without the aid of a calculator. Students who have completed one year of high school algebra should consider enrolling in Beginning Algebra. Repeatable up to a maximum of three times. MAT 094: ELEMENTARY ALGEBRA 5 hrs. (BEC) Prerequisite: Placement into MAT 094 is according to placement test scores or on a voluntary basis. This course is specifically designed for the student with less than one year of credit in high school algebra or for the student who needs a review of elementary algebra. It is considered equivalent to the standard first-year course in algebra. Topics include real numbers, linear equations and inequalities, systems of equations, exponents, polynomials, factoring, quadratic equations, and rational expressions and equations. Repeatable up to three times. MAT 095: ELEMENTARY GEOMETRY 3 hrs. (BEC) Prerequisite: One year of high school algebra or concurrent enrollment in MAT 094 This course is designed for the student with less than one year of credit in high school geometry or for the student who desires a review of elementary geometry. The basic concepts of the standard first-year course in geometry are covered. MAT 097: ELEMENTARY ALGEBRA REVIEW 2 hrs. (BEC) Prerequisite: One year of high school algebra (or equivalent) and department approval or by placement test scores. This course is specifically designed for the student with one or more years of credit in high school algebra who needs a brief review of elementary algebra. Students who need more than a brief review should enroll in MAT 094. Topics include real numbers, linear equations and inequalities, systems of equations, exponents, polynomials, factoring, quadratic equations, and rational expressions and equations. Repeatable up to a maximum of three times. MAT 098: INTERMEDIATE ALGEBRA 3 hrs. (BEC) Prerequisite: MAT 094 or 097 with a grade of "C" or better or an appropriate score on the math placement test This course includes work in linear and quadratic equations, systems of equations, exponents, radicals, functional relationships, and logarithms. It also includes work in graphing linear, quadratic, square root, cubic, exponential, and logarithmic functions. The course is designed for students who have had a minimum of one year of high school algebra or those needing a review of second-year high school algebra. MAT 099: MATHEMATICAL LITERACY FOR COLLEGE STUDENTS 6 hrs. (BEC) Prerequisite: A grade of "B" or higher in MAT 092 (or equivalent) or equivalent placement test score, or department approval This course is a one semester course for non-math and non-science majors integrating numeracy, proportional reasoning, algebraic reasoning, and functions. Students will develop conceptual and procedural tools that support the use of key mathematical concepts in a variety of contexts. Throughout the course, college success content will be integrated with mathematical topics. Credit earned does not count toward any degree, nor does it transfer. Upon successful completion of the course, students will be prepared to take MATH 110 or MATH 111. This course is not a prerequisite course for MATH 115. MAT 106: APPLIED ALGEBRA, GEOMETRY AND TRIGONOMETRY 4 hrs. (OC) Prerequisite: MAT 094 or equivalent This course presents the practical application of arithmetic, algebra, geometry, and trigonometry. Emphasis is placed on calculations, areas, volumes and weights, and special shop applications. Applying problem-solving techniques to industrial applications will be stressed. MATH 110: CONCEPTS OF MATHEMAT with a grade of "C" or better or MAT 099 with a grade of "C" or better or appropriate placement score or department approval. This course introduces the nature of mathematics through a study of elementary logic, set theory, statistics, geometry, and the mathematics of finance. The course will focus on mathematical reasoning and real-life problem solving. This is not intended to be a survey course or a math appreciation course. [IAI: M1 904] MATH 111: GENERAL EDUCATION STATIST or MAT 099 with a grade of "C" or better or appropriate placement score or department approval. Prerequisite: MAT 098 with a grade of "C" or better or an appropriate score on the math placement test This course emphasizes both algebraic and graphical approaches to college algebra. Topics include functions, relations, and inverses with emphasis on polynomial, rational, exponential, and logarithmic functions; systems of equations and inequalities; and theory of equations. MATH 120: COLLEGE TRIGONOMETRY 3 hrs. (TC) Prerequisite: MAT 095 and 098 with a grade of "C" or better or equivalent This course includes a study of the trigonometric functions and their graphs, radian measure, inverse trigonometric functions, solutions of triangles, trigonometric identities and equations, and roots of complex numbers. MATH 122: DISCRETE MATHEMATICS I 3 hrs. (TC) Prerequisite: COMPASS reading score of 81 or higher, or equivalent, or department approval and MATH 115 with a grade of "C" or better or equivalent or department approval This course covers directed and undirected graphs including trees and routing; combinatorics and counting principles; logic, functions, relations and sets; Boolean Algebra and switching theory; and finite state machines. Application problems from the field of computer science will be studied such as speed of sorting, database management, and routing in networks. [IAI: M1 905 ] MATH 130: TECHNICAL ALGEBRA AND TRIGONOMETRY 5 hrs. (OC) Prerequisite: MAT 095 and MAT 098 with a grade of "C" or better, or MAT 106 with a grade of "C" or better, or two years of high school algebra, one year of high school geometry and an appropriate score on the math placement test This course includes the topics: approaches to problem solving, dimensional analysis, the basic use of the calculator and computer, selected topics from college algebra, trigonometry, analytic geometry, and statistics. Included will be systems of equations, basic trigonometric functions, right triangle solutions, two dimensional vectors, common and natural logarithms, and basic conic sections. Scientific calculators and computer software are used. MATH 134: FINITE MATH 4 hrs. (TC) Prerequisite: COMPASS Reading score of 81 or higher, or equivalent, or department approval, and MATH 115 with a grade of "C" or better or equivalent This course covers topics from college algebra with emphasis on systems of linear equations and inequalities, matrix theory, linear programming, probability, statistics, and mathematics of finance. Application problems are chosen from the fields of business and social science. [IAI: M1 906] MATH 135: CALCULUS FOR BUSINESS AND SOCIAL SCIENCES 4 hrs. (TC) Prerequisite: COMPASS Reading score of 81 or higher, or equivalent, or department approval, and MATH 115 with a grade of "C" or better This course covers the basic ideas of calculus including differentiation and integration of polynomial, logarithmic and exponential functions. Application problems are chosen from the fields of business and social science. [IAI: M1 900B] MATH 137: TECHNICAL CALCULUS 3 hrs. (OC) Prerequisite: MATH 130 with a grade of "C" or better or equivalent This course covers topics which include: functions, limits, derivatives, anti-derivatives, integrals, and applications of the definite integral. Emphasis is placed on the physical significance of the derivative and integral to enable the student to relate to the basic underlying mathematical principles. MATH 165: PRECALCULUS 5 hrs. (TC) Prerequisite: MAT 098 with a grade of C or better, or an appropriate score on the math placement test. NOTE: If a student has not previously completed a high school course in trigonometry, enrollment in the separate courses MATH 115 and MATH 120 is recommended. Students may not earn credit for both MATH 115/120 and MATH 165 This course is intended to provide a solid foundation in the skills of algebra and trigonometry that are required for success in elementary calculus. Algebraic topics will include: properties of functions and graphs that are commonly used in calculus, conic sections, solving equations and higher order systems of equations, and sequences and series. Trigonometry topics will include: numerical aspects, including Laws of Sines and Cosines; trigonometry identities and equation solving; powers and roots of complex numbers; and radian measure and conversion. This course will make use of current technology. MATH 190: MATHEMATICAL REASONING FOR THE ELEMENTARY TEACHER I 3 deepen mathematical understanding by providing opportunities to develop problem-solving and reasoning skills. In order to develop depth of understanding, the course concentrates on problems involving fractions, percents, place value and decimals. MATH 200: MATHEMATICS FOR ELEMENTARY TEACHERS I 4 reinforce and strengthen the prospective elementary teacher´s knowledge of the structure of the real number system and the mathematical operations that can be performed within that system. The historical development of the system will be discussed along with many of the applications that an understanding of elementary mathematics permits. Mathematical reasoning and problem solving are consistent themes throughout the course. MATH 201: MATHEMATICS FOR ELEMENTARY TEACHERS II 3 hrs. (TC) Prerequisite: COMPASS reading score of 81 or higher, or equivalent, or department approval, and MATH 190 or MATH 200 with a grade of "C" or better or department approval. This course is designed to survey and to expand the mathematical concepts needed to teach a modern mathematical program in grades K-9 and prepare teachers and prospective teachers for future changes in mathematics curricula. The course includes a study of logic and problem-solving, graphing and analysis of relations, functions and statistical data, non-metric and informal geometry, estimating and measuring, the metric system, and use of calculating devices. [IAI: M1 903] MATH 211: STATISTICAL ANALYSIS 4 hrs. (TC) Prerequisite: COMPASS Reading score of 81 or higher, or equivalent, or department approval and MAT 095 and MATH 115 with a grade of "C" or better or equivalent Prerequisite: COMPASS reading score of 81 or higher, or equivalent, or department approval; MATH 115 and MATH 120 with grades of "C" or better, or MATH 165 with a grade of "C" or better, or an appropriate score on the math placement test or equivalent. This is the first course of a three-semester sequence in Analytic Geometry and Calculus. The course includes the analytic geometry of lines and circles, limits and continuity of functions of one variable and an introduction to the derivative and the definite integral along with applications and the fundamental theorem of calculus. [IAI: M1 9001] MATH 223: CALCULUS AND ANALYTIC GEOMETRY II 4 hrs. (TC) Prerequisite: COMPASS Reading score of 81 or higher, or equivalent, or department approval, and MATH 222 with a grade of "C" or better or equivalent This course is a continuation of MATH 222 and includes the analytic geometry of conic sections, the study of calculus as related to transcendental functions including trigonometric, logarithmic, exponential and hyperbolic functions and their inverses, techniques of integration, indeterminate forms, improper integrals, and infinite series and Taylor's theorem. [IAI: M1 9002] MATH 224: CALCULUS AND ANALYTIC GEOMETRY III 4 hrs. (TC) Prerequisite: COMPASS Reading score of 81 or higher, or equivalent, or department approval, and MATH 223 with a grade of "C" or better or equivalent This course is a continuation of MATH 223 and includes parametric curves, vectors in two and three dimensions, vector valued functions, curves and surfaces in space, curvature, acceleration, quadric surfaces, functions of several variables, partial derivatives and applications, Lagrange multipliers, multiple integrals and integration with polar, cylindrical, and spherical coordinates. [IAI: M1 9003] This course includes first order (e.g., separable, linear, exact) with applications and simple higher order ordinary differential equations; linear independence and the Wronskian; linear differential equations with constant coefficients along with systems and applications; variation of parameters and undetermined coefficients; solution by means of Laplace transforms, solutions of partial differential equations, solution by power series and numerical methods. Prior knowledge of the basic concepts of physics is recommended. [IAI: MTH 912]
subject of partial differential equations has an unchanging core of material but is constantly expanding and evolving. Introduction to Partial Differential Equations with MATLAB is a careful integration of traditional core topics with modern topics, taking full advantage of the computational power of MATLAB to enhance the learning experience. This advanced text/reference is an introduction to partial differential equations covering the traditional topics within a modern context. To provide an up-to-date treatment, techniques of numerical computation have been included with carefully selected nonlinear topics, including nonlinear first order equations. Each equation studied is placed in the appropriate physical context. The analytical aspects of solutions are discussed in an integrated fashion with extensive examples and exercises, both analytical and computational. The book is excellent for classroom use and can be used for self-study purposes. Topic and Features: • Nonlinear equations including nonlinear conservation laws; • Dispersive wave equations and the Schrodinger equation; • Numerical methods for each core equation including finite difference methods, finite element methods, and the fast Fourier transform; • Extensive use of MATLAB programs in exercise sets. MATLAB m files for numerical and graphics programs available by ftp from this web site. This text/reference is an excellent resources designed to introduce advanced students in mathematics, engineering and sciences to partial differential equations. It is also suitable as a self-study resource for professionals and practitioners.
0130850489 9780130850485 Algebra:In an effort to provide a solid foundation in algebra for students who may not have had previous algebraic experience, Martin-Gay (mathematics, U. of New Orleans) incorporates problems and examples that use real-life and real-data applications, data interpretation, conceptual understanding, problem solving, writing, cooperative learning, number sense, critical thinking, and geometric concepts throughout the book. She also includes some optional examples that use scientific and graphing calculators, as well as pretests and review tests. Back to top Rent Algebra 1st edition today, or search our site for K. Elayn textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Prentice Hall.
Advanced Engineering Mathematics : Alt. Edition - 7th edition Summary: Building upon the sequence of topics of the popular 5th Edition, Linear Algebra with Applications, Alternate Seventh Edition provides instructors with an alternative presentation of course material. In this edition earlier chapters cover systems of linear equations, matrices, and determinates. The vector space Rn is introduced in chapter 4, leading directly into general vector spaces and linear transformations. This order of topics is ideal for those preparing to use linear equations...show more and matrices in their own fields. New exercises and modern, real-worldapplications allow students to test themselves on relevant key materialand a MATLAB manual, included as an appendix, provides 29 sections of computational problems. Learn about Gareth Williams'Linear Algebra with Application, Seventh EditionVeryGood Bang-for-Bucks sterling, VA 2009-12-23 Hardcover Very Good We ship daily and provide tracking numbersWe Beat the Lowest Amazon Prices All the time
College Mathematics - 2nd edition Summary: Algebra, plane trigonometry and plane analytic geometryalso aspects of solid analytic geometryget full coverage in this book and lead to an introduction to calculus. The many hundreds of problems with step-by-step solutions and supplementary ones with answers illustrate and amplify the theory and repeat basic principles, permitting the kind of active learning that lets students understand and remember important points00 +$3.99 s/h Good a2zbooks Burgin, KY The text appears clean, the front cover has sticker residue. The pages are yellowing, and there is a light watermark on the lower corner.This is a former free copy. Quantity Available: 1. Category: Ma...show morethematics; ISBN: 0070026645. ISBN/EAN: 9780070026643. Inventory No: 1560760727. 2nd Edition. ...show less $9.00 +$3.99 s/h Good A2ZBooks Ky Burgin, KY Boston, MA 1992 Softcover 2nd Edition Good Condition The text appears clean, the front cover has sticker residue. The pages are yellowing, and there is a light watermark on the lower corner. This is...show more a former free copy. Quantity Available: 1. Category: Mathematics; ISBN: 0070026645. ISBN/EAN: 9780070026643. Inventory No: 1560760727
How Is Algebra Used in Real Life? Answer Algebra is used in real life in a couple of ways. The most obvious way would be if the career you choose requires use of algebra; engineers, mathematicians, teachers, and scientists are examples of people that might need to use algebra. Otherwise, everyone does use algebra even though they may not realize it. For example, you're at the grocery store picking out ingredients for a big recipe. You need 2 pounds of peanuts, but the store only sells quarter pound bags. Well, in a case like that you're going to have to quickly figure out how many bags will equal the 2 pounds you need. It is just like solving an equation!
These modules are intended to provide just-in-time instruction to students currently enrolled in a first year calculus course... see more These modules are intended to provide just-in-time instruction to students currently enrolled in a first year calculus course who need a refresher or remedial instruction in one or all of the following topics:1. Algebraic Operations 2. Factorization 3. Polynomials and Rational Expressions 4. Radical Expressions 5. Linear and Quadratic EquationsStudents can work through the modules in a sequential fashion or they can focus on those learning activities or topics with which they have the most difficulty. Copies of the files that were used to create content for these modules have been provided to allow instructors to develop their own content
97803064614odes on Algebraic Curves This book provides a self-contained introduction to the theory of error-correcting codes and related topics in number theory, Algebraic Geometry and the theory of Sphere Packings. The material is presented in an easily understandable form. This book is devoted to geometric Goppa codes; the recently discovered areas which combines Coding Theory, Algebraic Geometry, Number Theory, and Theory of Sphere Packings. It has an interdisciplinary nature and demonstrates the close interconnection of Coding Theory with various classical areas of mathematics. There are four main themes in the book. The first is a brief exposition of the basic concepts and facts of error-correcting code theory. The second is a complete presentation of the theory of algebraic curves; especially the curves defined over finite fields. The third is a detailed description of the theory of elliptic and modular codes, and their reductions modulo a prime number. The fourth is a construction of geometric Gappa codes producing rather long linear codes with very good parameters coming from algebraic curves, and with a lot of rational points. The aim of the book is to present these themes in a simple, easily understandable manner, and explain their close interconnection. At the same time the book introduces the reader to topics which are at the forefront of current
Prealgebra This book's clear, well-constructed and straightforward writing style makes it accessible to even the most apprehensive math students.The primary ...Show synopsisThis book's clear, well-constructed and straightforward writing style makes it accessible to even the most apprehensive math students.The primary focus of the pedagogy, presentation and other elements is to ease the transition into algebra; for example, emphasis is placed on basic arithmetic operations within algebraic contexts. The Second Edition includes a greater integration of NCTM and AMATYC standards, including more emphasis on visualization, problem solving and data analysis.Hide synopsis Description:Fine. Paperback. Almost new condition. SKU: 9780321955043-2-0-3...Fine. Paperback. Almost new condition. SKU: 978032195504319550432886Description:Good. Paperback. May include moderately worn cover, writing,...Good. Paperback. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780321628862
...Algebra 2 involves difficult concepts, so it is important not to rush and simply memorize formulas. I encourage students to think through a math problem before trying to solve it. This leads to better retention and greater understanding
This website can be used by teachers or students in utilizing dense information with assesments. This exercise is designed... see more This website can be used by teachers or students in utilizing dense information with assesments. This exercise is designed for anyone who wants an introduction or review of the fundamentals of chemistry that will be used in freshman level chemistry classes. The goal of the program is to provide an introduction or a review to incoming freshman chemistry students on the basic mathematical skills that are required to be successful in freshman chemistry. In addition, the materials last year worked to introduce or review basic skills in the use of a calculator
You are here Advertising in MAA Math Alert Your exclusive opportunity to connect with MAA Members monthly! MAA Math Alert is the monthly newsletter of the Mathematical Association of America. MAA Math Alert connects members of MAA to association news, continuing education opportunities, valuable professional resources, highlights of new products and services and exclusive MAA journal content.
Book Summary of S.Chand's Mathematics For Class Ix Term ... (Paperback) A Complete Textbook of Mathematics for Term -II absolutely based on new pattern of examination CCE for both Formative and Summative Assessments with following key features and worksheets at the end of each chapter for practising the problems based on : True / False \| Fill in the blanks \| Match the Columns \| MCQ's \| Asserstion Reasoning \| Comprehension \| Riddles \| Special Worksheets \| Activities for Lab Manual \| Important Facts \| Ten CBSE Model Papers
Key Message: The" and the New Instructor and Adjunct Support Manual. Geared toward helping students visualize and apply mathematics, Elementary and Intermediate Algebra: Graphs and Models, Second Edition is designed for a two-semester course. The authors make use of illustrations, graphs, and graphing technology to enhance students' mathematical skills. This is accomplished through Interactive Discoveries, Algebraic-Graphical Side-by-Sides, have come to expect with any Bittinger text, we bring you a complete supplements package that now includes an Annotated Instructor's Edition and MyMathLab, Addison-Wesley's on-line course solution. More editions of Elementary and Intermediate Algebra: Graphs and Models: The Sixth Edition of Intermediate Algebra: Concepts and Applications continues to bring your students a best-selling text that incorporates the five-step problem-solving process, real-world applications, proven pedagogy, and an accessible writing style. The Bittinger/Ellenbogen hardback series has consistently provided teachers and students with the tools needed to succeed in developmental mathematics. With this revision, the authors have maintained all the hallmark features that have made this series so successful, including its five-step problem-solving process, student-oriented writing style, real-data applications, and wide variety of exercises. Among the features added or revised are new Aha! exercises that encourage students to think before jumping in to solve a problem, 20% new and added real-data applications, and 50% more new Skill Maintenance Exercises. This series not only provides students with the tools necessary to learn and understand math, but also provides them with insights into how math works in the world around them. Geared toward helping students visualize and apply mathematics, Intermediate Algebra: Graphs and Models, Second Edition uses illustrations, graphs, and graphing technology to enhance students' mathematical skills. This is accomplished throughInteractive Discoveries, Algebraic-Graphical Side-by-Sideshave come to expect with any Bittinger text, we bring you a complete supplements package that now includes an Annotated Instructor's Edition and MyMathLab, Addison-Wesley's on-line course solution. This interactive tutorial CD-ROM provides algorithmically generated practice exercises that are correlated at the objective level to the exercises in the textbook. Every practice exercise is accompanied by an example and a guided solution designed to involve students in the solution process. Selected exercises may also include a video clip to help students visualize concepts. The software provides helpful feedback for incorrect answers and can generate printed summaries of students' progress. More editions of Intermediate Algebra: MathXL Tutorials: Graphs and Models: The Student Solutions Manual contains completely worked-out solutions with step-by-step annotations for all the odd-numbered exercises in the text, with the exception of the thinking and writing exercises. It also includes complete, worked-out answers for all end-of-chapter material. The Student Solutions Manual contains completely
Slope Task 1Smith, S., Charles, R., Dossey, J., & Bittinger, M. (2001). Algebra1 California Edition, Prentice Hall, p321. Slope Task 2 For each pair of points, find a third point that is on the same line. Explain how you used the information given to find the third ... Prentice Hall Algebra1, Geometry, and Algebra 2 help students develop a deep understanding of mathematics through thinking, reasoning, and problem solving. The flexibility of the program components and leveled resources enables teachers to adapt to Smith, S., Charles, R., Dossey, J., & Bittinger, M. (2001). Algebra1, California Edition, Prentice Hall, p. 321. For each pair of points, find a third point that is on the same line. Explain how you used the information given to find the third point. As we approach the end of another successful school year at Charles S. Rushe Middle School, I want to thank all who make our school a great ... Zachary Smith, Bradley Jasper (alternate), finished 1st in the county. ... Our Algebra1 team, Bryan Reeder, Rohil Tuli, Roshan The mission of the Charles Herbert Flowers Counseling ... Pass High School Assessments in English 10, Algebra1, and Biology. AND ... Smith College St. John's College St. Mary's College Swathmore Temple University Towson University Textbook: Algebra 2 with Trigonometry by Smith, Charles, Dossey and Bittinger Course Description: Building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions
Written for the standard calculus sequence. Its purpose, in addition to making it possible to learn calculus, is to teach students to use calculus effectively and to show how knowing calculus can pay off in their professional lives. DLC: Calculus. Written by an experienced author team with expertise in the use of technology and NCTM guidelines, this text provides an emphasis on multiple representations of concepts and an abundance of worked examples. Calculus is explored through the interpretation of graphs and tables as well as through the application of analytical methods. Rich exercises include graphical and data-based problems, and interesting real-life applications in biology, business, chemistry, economics, engineering, finance, physics, the social sciences and statistics. Stepped Explorations throughout the text provide guided investigations of key concepts and help students build problem-solving skills. A grapher is required. Paperback. Pub: 2006 Pages: 486 Publisher: Pearson The main goal of this third edition is to REALIGN with the CHANGES in The Advanced Placement (AP) Calculus Syllabus and the new type of AP * exam questions. We have also more carefully aligned examples and exercises and updated the data used in examples and exercises. Cumulative Quick Quizzes are now provided two or three times in each chapter.Students learn math through practice and the Student Practice Workbook provides practice problems in addition to what students will find in the textbook. Key examples from each section in the book are recreated along with a more detailed solution. Related practice problems follow each example.The main goal of this third edition is to realign with the changes in the Advanced Placement (AP ) calculus syllabus and the new type of AP exam questions. We have also more carefully aligned e... Organized to correspond to the text, the Student Outlines by Joseph Borzellino and Patricia Nelson reinforce important concepts and provide an outline of the important topics, theorems, and definitions, as well as study tips and additional practice problems. Part Two corresponds to chapters 11-16 of Thomas' Calculus, Early Transcendentals, Eleventh Edition. The tenth edition of this clear, precise calculus text with superior applications sets the standard in calculus. The tenth edition of this proven text was carefully revised to give students the solid base they need to succeed in math, science and engineering programs. Through a comprehensive technology package, this edition now includes more opportunity to incorporate optional, but meaningful technology into the course. The updated tenth edition of this clear, precise calculus text with superior applications sets the standard in calculus. This proven text was carefully revised to give students the solid base they need to succeed in math, science and engineering programs. Through a comprehensive technology package, this edition now includes more opportunity to incorporate optional, but meaningful, technology into the course. More editions of Thomas' Calculus, Early Transcendentals (10th Edition):
Chases & Escapes The Mathematics of Pursuit & Evasion 9780691125145 ISBN: 0691125147 Pub Date: 2007 Publisher: Princeton University Press Summary: Ideal both for self-study and as supplemental readings by students and/or professors in any of the mathematical and physical sciences, this text presents the historical development of the differential equations of pursuit theory. Nahin, Paul J. is the author of Chases & Escapes The Mathematics of Pursuit & Evasion, published 2007 under ISBN 9780691125145 and 0691125147. Two hundred seventy six Chases & Escap...es The Mathematics of Pursuit & Evasion textbooks are available for sale on ValoreBooks.com, one hundred six used from the cheapest price of $5.76, or buy new starting at $22Princeton. 2007. Princeton University Press. 1st American Edition. Very Good In Dustjacket. 270 pages. hardcover. 9780691125145. keywords: Mathematics. inventory # 36711. FROM THE PUBLISHER-We all played tag when we were kids. The rules couldn't be easier?one player is designated ?it' and must try to tag out one of the others. What most of us don't realize is that this simple chase game is in fact an application of pursuit theory, and that the same principles of games like tag, dodgeball, and hide-and-seek are at play in military strategy, high-seas chases by the Coast Guard, even romantic pursuits. In Chases and Escapes, Paul Nahin gives us the first complete history of this fascinating area of mathematics. Writing in an accessible style that has been enjoyed by popular-math enthusiasts everywhere, Nahin traces the development of modern pursuit theory from its classical analytical beginnings to the present day. Along the way, he informs his mathematical discussions with fun facts and captivating stories. Nahin invites readers to explore the different approaches to solving various chase-and-escape problems. He draws upon game theory, geometry, linear algebra, target-tracking algorithms?and much more. Nahin offers an array of challenging puzzles for beginners on up, providing historical background for each problem and explaining how each one can be applied more broadly. Chases and Escapes includes solutions to all problems and provides computer programs that readers can use for their own cutting-edge analysis. This informative and entertaining book is the first comprehensive treatment of the subject, one that is sure to appeal to anyone interested in the mathematics that underlie the all-too-human endeavor of pursuit and evasion. Paul J. Nahin is Professor Emeritus of Electrical Engineering at the University of New Hampshire. His books include Dr. Euler's Fabulous Formula, When Least Is Best, and Duelling Idiots and Other Probability Puzzlers (all Princeton).[less]
lab is the Math Works company business mathematics software, mainly applied in engineering design, calculation, the control signal processing and communication, image processing and signal detection, modeling design and analysis, etc. In the above application fields Matlab has powerful functions, in order to cultivate students' scientific spirit and practical ability and innovative consciousness, pay attention to the problems of engineering practice, and modeling ability training in colleges, so the different specialized has opened the Matlab foundation course. The author in teaching practice and experiment, and based on the investigation of different professional open courses and thinking, the idea is: the numerical calculation method and statistics, the optimization module and software mat lab, the typical case study, combine to optimize the teaching effect, enhance students' learning interest, produce good effects on the subsequent courses and the work.
Is math for me? : math math for me? : math math for me? 30 Apr 2012 17:28:56 -0700<!-- SC_OFF --><div class="md"><p>I'm currently an undergraduate math major and taking my first proof based class this quarter (linear algebra). Sadly, I am doing absolutely abysmally in the class and I have no idea why.</p> <p>I enjoy math, in the sense that I like thinking about it and reading about it, and as a result I chose to study it. Additionally a big part of why I chose to declare math as my major was due to the ease with which I got through my computational lower divs. I am at a loss currently as to whether my failure to perform well is a result of me not putting enough work into studying, or whether I simply can't get it, or if I am not studying in the right way, or whatever.</p> <p>If I could get some insight into the nature of undergraduate math studies I would greatly appreciate it. Has anyone else run into this kind of problem before?</p> </div><!-- SC_ON --> submitted by [deleted] <br/> <a href=" <a href=" comments]</a> on Is math for me? you're doing pure mathematics, it's all about understanding concepts well enough to prove theorems about them. Proof-writing itself takes a lot of practice to get good at, but you'll pick that up as you go. However, learning the language and conventions of formal proof is only one part of it. Now that you're not just doing computation, you'll find that it's no longer enough to memorize formulas and theorems; coming up with a proof often requires insight, creativity, and intuition. Surface-level understanding of the statements given in class will generally not be enough. Many times, I've done the reading, known all the relevant theorems, and followed their proofs, only When you get stuck like that, it means you need to do more problems. Reading is good, but it's not enough; the best way to get better in an area of math is to do problems yourself, and by that, I mean writing up the whole proof in a reasonable formal way, not just outlining an argument. If your method of studying just consists of reading or memorizing, you'll have a hard time gaining a deep understanding of the structures and patterns involved. Of course, that's not to say you should do nothing but book problems for studying. Get help when you need it, and assuming the professor's okay with it, consider working with your peers on some problems -- not the whole proof, but enough so you grasp the basic parameters of the problem and the reasoning needed. But if there's one single thing that will improve your proof skills, it's this: pick a problem, think about it for a while, and prove it on your own. Also, here's a bit of advice specific to linear algebra: If you're not already using Axler's [Linear Algebra Done Right]( I highly recommend it. He has a very well-written, straightforward presentation of the basics of linear algebra, and his proofs are generally elegant and informative. It's also a good source of exercises to try and figure out. on Is math for me? As a first year PhD student this is what I have spent a lot of my time doing, filling in gaps in my knowledge. I am however comfortable enough now to identify where the holes in my knowledge are and backtrack until I do understand what I need to, which is really fun. on Is math for me? also currently in Linear Algebra, and I struggled a lot with it at first, but my biggest problem came from not seeing any applications. My professor refuses to do any applications, and I honestly feel as if I've just been manipulating symbols all semester, not really understanding the big picture. Not having something to relate the material to can make things significantly more difficult, especially for someone who is just experiencing abstract algebra for the first time. You aren't always going to understand the material the first time through, so don't be so hard on yourself and keep at it. Here are some things that helped me: - This REALLY helped vector spaces and bases click: [Ignore the physics stuff]( - WATCH KHAN ACADEMY VIDEOS, Sal really does a great job explaining Linear Algebra as a whole. - Go to your campus library and look for a linear algebra book focused on applications and work through problems Hope those help, If i think of/find something else, ill post it here on Is math for me? professor uses some pretty sweet applications in his lecture notes. [Example]( Scroll to the end for the cryptography lesson. on Is math for me? are really cool notes, perfect for a final review, thanks. on Is math for me? want the whole book? on Is math for me? yes, that would be amazing on Is math for me? everything I have from that course.]( It's 11 chapters of Linear Algebra. Chapter Zero - Just an intro. Also, a strange explanation for why he's in the past teaching Math to some king. First Three Chapters are just review of matrices, inverses, transposes. Basic stuff that you should know before he can get into the course. We never did that in the course, it was meant to be "Read if you need it, but I won't lecture on it" material. 4 and 5 are all about vector spaces 6 is about bases. 7 is just a long proof for why the column space has the same dimension as the row space. 8 - Change of basis 9 - Determinants 10 - Eigenvalues, Eigenvectors and solving ODEs using them. I also added in some of our exams and tests if you ever need them. If not, delete! --- Personally, I find the way he introduced vector spaces really great. It gets slightly complicated and extra-rigorous towards the end. Understanding eigenvectors is better done on MIT OCW than through this textbook. But it's pretty rigorous for my standards. I personally liked the fact that he didn't add COMMUTATION as an axiom for vector spaces, instead he proved it using the axioms. Through that, he really drove home the concept of an axiom into everyone's head. --- Disclaimer:* This is meant to be an engineering math course. So, I apologize if it's not what you were looking for. on Is math for me? is sick! I cant thank you enough, this will help tremendously on Is math for me? you are like me, you'll probably prefer [MIT OCW Video Lectures from Gilbert Strang]( on Is math for me? its pretty common for students to struggle when first introduced to proofs after being used to more computational mathematics. The general advice would be: Dont worry much about it if you are just starting. Practice a lot. Ask for help if you need, dont be shy about asking for help, ask your professor when in class if you dont understand, stalk your TA if its necessary ( however, be sure to think about the problem first, dont go just after you hit a wall each time, push your boundaries and think hard, be patient ). Check a lot of sources, from your notes, to different books to the internet, sometimes stuff clicks after you read it different than before. Also, if you need help with some of the stuff you are struggling with you can PM me and I'll be glad to give you my email so we can talk. on Is math for me? had pretty much the same experience. Up through calculus it's mostly memorizing a process and knowing where to apply it. In linear algebra, you're slammed with fundamentally new concepts and have to really understand their implications. I struggled, and I had the feeling that I was missing something, like it was a joke I wasn't in on. The only way to get better at it is to practice, unfortunately. But even if things don't click by the time you finish the class, you will walk away with a new perspective and as you continue to think in this way you will find all sorts of problems that linear algebra makes easy work of. It took me months after the class before I realized why I was learning what I was. Then, when the topics came up again in differential equations, things became even more clear. So I guess what I'm saying is, don't feel bad about hitting a point where math isn't just second nature anymore. It takes more and more study to get comfortable with topics as you go through higher and higher maths. on Is math for me? you explain a theorem to me? Is there something that you understand to such an extent that you could explain it to someone else, meeting their potential objections along the way? Transitioning to proof-based mathematics is hard for everybody, not necessarily because they need to work harder (some do), but because they need to work differently and understand the core ideas. But it's worth it. on Is math for me? Transitioning to proof-based mathematics is hard for everybody Not true. For true mathematician transition is so joyful that any hardness vanishes. on Is math for me? thought of this as I wrote it, but I think these people simply made the transition much earlier, and that it was at one point hard. On the other hand, some people are yearning so much for rigor and real understanding that they welcome this change as a breath of fresh air. on Is math for me? kind of proofs are you doing? At my school my proof based linear algebra class did a great deal of proofs on matrix operations and stuff while barely covering vector spaces, dimensions, eigenvectors, and all the really cool stuff along with linear algebra. If you have a class like that, then push through it and things will get better. on Is math for me? haven't taken linear, so take my advice with a grain of salt, but from what I understand, linear, even proof-based linear, is very different from other (IMO more legit) undergrad level math topics. I took Set Theory last semester; loved it. Real analysis this semester (yeah, without linear...I have no idea what a prereq is); loved it. Both were different from each other and from linear (on my understanding of linear). on Is math for me? what you like about math is the computational stuff, you should consider switching to engineering. on Is math for me? a professor, I see this all the time. Too many earlier courses are too much "plug-and-chug." Sadly, many students get the wrong impression of mathematics. They aren't forced to thinking abstractly, so they don't. Then they hit the upper level courses and the you-know-what hits the fan. For immediate advice, all I can say is talk to your professor and keep trying. Make sure you understand everything. After reading a few proofs, close the book and do the proofs yourself. Alone. Don't rely on getting help from others until you've struggled with it yourself for a long time.
Investigating Prealgebra - 02 edition ISBN13:978-0534453091 ISBN10: 0534453090 This edition has also been released as: ISBN13: 978-0030226243 ISBN10: 0030226244 Summary: Investigating Prealgebra is a flexible worktext that supports a variety of teaching methods and prepares students for their first course in algebra. The book engages students in active learning, allowing them to discover knowledge and describe their understandings. Through a blend of lecture-type Discussions and constructivist Investigations, this book provides a comprehensive foundation in prealgebra concepts. Introduction to the Chapter The Order of Operations Formulas Paired Data and Line Graphs Properties of Equality Equations With More Than One Operation Read, Reflect, and Respond: Learning Styles Glossary and Procedures Review Test 3. INTEGERS Introduction to the Chapter Introduction to Integers Addition of Integers Subtraction of Integers Multiplication and Division of Integers Introduction to Polynomials Read, Reflect and Respond: Negative Numbers Glossary and Procedures Review Test 4. FRACTIONS Introduction to the Chapter Multiples and Factors Introduction to Fractions Multiplication of Fractions Division of Fractions Addition and Subtraction of Fractions Mixed Numbers Arithmetic with Mixed Numbers Solving Equations Read, Reflect, and Respond: The Value of Mathematics in Daily Life Glossary and Procedures Review Test Introduction to the Chapter Introduction to Ratios and Rates Proportions Percents Applications of Percents Circle Graphs Read, Reflect, and Respond: The Value of Mathematics for Citizenship Glossary and Procedures Review TestPAPERBACK Fair 0534453090 Student Edition. Missing many pages. Moderate wear, wrinkling, Curling or creasing on cover and spine. May have used stickers or residue. Fair binding may have a few loose...show more or torn pages. Heavy writing, highlighting and marker. ...show less $25.65 +$3.99 s/h Good Big Planet Books Burbank, CA 2001-08-31 Paperback Good Expedited shipping is available for this item! $250
for Economists, a new text for advanced undergraduate and beginning graduate students in economics, is a thoroughly modern treatment of the mathematics that underlies economic theory. An abundance of applications to current economic analysis, illustrative diagrams, thought-provoking exercises, careful proofs, and a flexible organization-these are the advantages that Mathematics for Economists brings to today's classroom.
Summary: These popular and proven workbooks help students build confidence before attempting end-of-chapter problems. They provide short exercises that focus on developing a particular skill, mostly requiring students to draw or interpret sketches and graphs. New Jersey 2007 Paperback 2nd Revised edition. Revised. Good. Go green, recycle! Book may have wear from reading, may contain some library markings. 480 p. Intended for professional and scholarly au...show moredience0321513576
Find a Humble can almost guarantee that you will have "Aha! So that's how it works!" moments as algebra becomes more familiar and understandable. Algebra 2 builds on the foundation of algebra 1, especially in the ongoing application of the basic concepts of variables, solving equations, and manipulations such as factoring.
Applies the Associative Rule (Resource Book Only) eBook about algebra and functions, students apply the associative rule to algebraic problems and mentally calculate the answers. Includes directions, mat and number cards, checklist, and activity sheet.
College Algebra and Trigonometry: A Unit Circle Approach With an emphasis on problem solving and critical thinking, Mark Dugopolski's College Algebra and Trigonometry: A Unit Circle Approach, Sixth Edition ...Show synopsisWith an emphasis on problem solving and critical thinking, Mark Dugopolski's College Algebra and Trigonometry: A Unit Circle Approach, Sixth Edition gives students the essential strategies to help them develop the comprehension and confidence they need to be successful in this course. Students will find carefully placed learning aids and review tools to help them do the math1916492 / 9780321916495 College Algebra and Trigonometry: A Unit Approach Plus NEW MyMathLab with Pearson eText -- Access Card Package Package consists of: 0321431308 / 9780321431301 MyMathLab -- Glue-in Access Card 0321654064 / 9780321654069 MyMathLab Inside Star Sticker 0321916522 / 9780321916525 College Algebra and Trigonometry: A Unit Circle ApproachHide synopsis Description:New. 0321644778 New book with very minor shelf wear. STUDENT US...New. 0321644778
Shipping prices may be approximate. Please verify cost before checkout. About the book: Introduction to Applied Math offers a comprehensive introductory treatment of the subject. The authors explanations of Applied Mathematics are clearly stated and easy to understand. The reference includes a wide range of timely topics from symmetric linear systems to optimization as well as illuminating hands-on examples. Hardcover, ISBN 0961408804 Publisher: Wellesley-Cambridge Press, 1986 Hardcover. Renowned applied mathematician Gilbert Strang teaches applied mathematics with the clear explanations, examples and insights of an experienced teacher. This book progresses steadily throug.Shipping may be from multiple locations in the US or from the UK, depending on stock availability. 760 pages. 1.225 Hardcover, ISBN 0961408804 Publisher: Wellesley-Cambridge Press, 1986 Wellesley-Cambridge Press. Hardcover. 0961408804 Some visible wear, and minimal interior marks. Unbeatable customer service, and we usually ship the same or next day. Over one million satisfied customers! . Good. Hardcover, ISBN 0961408804 Publisher: Wellesley-Cambridge Press, 1986961408804 Publisher: Wellesley-Cambridge Press, 1986 Used - Good, Usually ships in 1-2 business days, Some shelf wear. some cover wear Your purchase helps put people to work and learn life skills to reach their full potential, Thank you. Hardcover, ISBN 0961408804 Publisher: Wellesley-Cambridge961408804 Publisher: Wellesley-Cambridge Press, 1986 Acceptable, Usually ships in 1-2 business days, Textbook may contain underlining, highlighting or writing. Infotrac or untested CD may not be included. Hardcover, ISBN 0961408804 Publisher: Wellesley-Cambridge Press, 1986 Used - Good, Usually ships in 1-2 business days, Some visible wear, and minimal interior marks. Unbeatable customer service, and we usually ship the same or next day. Over one million satisfied customers!
College Algebra : Concepts and Models - 5th edition Summary: College Algebra: Concepts and Models provides a solid understanding of algebra, using modeling techniques and real-world data applications. The text is effective for students who will continue on in mathematics, as well as for those who will end their mathematics education with college algebra. Instructors may also take advantage of optional discovery and exploration activities that use technology and are integrated throughout the text. The Fifth Edition en...show morehances problem solving coverage through Make a Decision features. These features are threaded throughout each chapter, beginning with the Chapter Opener application, followed by examples and exercises, and ending with the end-of-chapter project. This edition also features Eduspace, Houghton Mifflin's online learning tool, which allows instructors to teach all or part of a course online, and provides students with additional practice, review, and homework problems. A brief version of this text, College Algebra: A Concise Course, provides a shorter version of the text without the introductory review. New! Make a Decision features thread through each chapter beginning with the Chapter Opener application, followed by examples and exercises, and ending with the end-of-chapter project. Students are asked to choose which answer fits within the context of a problem, to interpret answers in the context of a problem, to choose an appropriate model for a data set, or to decide whether a current model will continue to be accurate in future years. The student must examine all data and decide upon a final answer. Chapter Projects extend applications designed to enhance students understanding of mathematical concepts. Real data is previewed at the beginning of the chapter and then analyzed in detail in the Project at the end of the chapter. Here the student is guided through a set of multi-part exercises using modeling, graphing, and critical thinking skills to analyze the data. A variety of exercise types are included in each exercise set. Questions involving skills, modeling, writing, critical thinking, problem-solving, applications, and real data sets are included throughout the text. Exercises are presented in a variety of question formats, including free response, true/false, and fill-in the blank. New! "In the News" Articles from current media sources (magazines, newspapers, web sites, etc.) have been added to every chapter. Students answer questions that connect the article and the algebra learned in that section. This feature allows students to see the relevancy of what they are learning, and the importance of everyday mathematics. Discussing the Concept activities end most sections and encourage students to think, reason, and write about algebra. These exercises help synthesize the concepts and methods presented in the section. Instructors can use these problems for individual student work, for collaborative work or for class discussion. In many sections, problems in the exercise sets have been marked with a special icon in the instructor's edition as alternative discussion/collaborative problem. Discovery activities provide opportunities for the exploration of selected mathematical concepts. Students are encouraged to use techniques such as visualization and modeling to develop their intuitive understanding of theoretical concepts. These optional activities can be omitted at the instructor's discretion without affecting the flow of the material.49 +$3.99 s/h Good Silver Arch Books St Louis, MO Book has a small amount of wear visible on the binding, cover, pages. Selection as wide as the Mississippi. $12.49 +$3.99 s/h Good AllAmericanTextbooks Ypsilanti, MI 061849281X Multiple available! Some wear to cover. Text is in great shape. ISBN\|061849281X, College Algebra: Concepts and Models (C.)2006 (MAD) WQ Ships today or the next business day. Cover and binding show moderate wear. Text may contain moderate writing/highlightings but is easily readable. 28-12. $2727Good booksmostly Pasadena, CA 2005 Hardcover Good Few if any marks, if so large majority of pages clean, boards/binding with light wear/stickers. $68
Beginning and Intermediate Algebra: An Integrated Approach 9780495831433 ISBN: 0495831433 Edition: 6 Pub Date: 2010 Publisher: Brooks Cole Summary: Brooks Cole published this text book in 2010, and here you have a chance to buy or rent your own copy of this 6th edition version. Buy Beginning and Intermediate Algebra: An Integrated Approach online from us today for an affordable price. Cheap Beginning and Intermediate Algebra: An Integrated Approach couldn't be easier to get. The book is written by David R Gustafson, Rosemary Karr and Marilyn Massey, and you can ...find out everything there is to know about learning algebra at this level. Sell back to us as well if you already own a copy, because our buyback system makes it easy to do so. Gustafson, R. David is the author of Beginning and Intermediate Algebra: An Integrated Approach, published 2010 under ISBN 9780495831433 and 0495831433. Three hundred twenty Beginning and Intermediate Algebra: An Integrated Approach textbooks are available for sale on ValoreBooks.com, one hundred fourteen used from the cheapest price of $49.94, or buy new starting at $245.71 May include moderately worn cover, writing, markings or slight discoloration. SKU:97805387366The most useful thing about this book was the breakdown of examples in each lesson. It really helped me figure out each problem. I also liked that it provided the answers in the back of the book for the odd problems. I liked that each lesson gave plenty of examples for different problems making it easier to follow and go look back if needed. I loved that the book provided definitions for terms even if they were terms I should already know, it was good to have them there just in case. The book also provided a formula key where needed that really helped! I would definatly recommend this book to beginner math students. The way this book is set up and the way it walks you through everything so smoothly makes this the perfect math book to begin the college experience with. It's not overwhelming with math jargon that will confuse you. I needed this book from math 98 and math 100. My teacher wasn't dry good so I was glad I had this book because I was able to go back in the book and figure out what she wasn't teaching me right. My class was a basic math and algebra course.
Description This course fulfills the first half in the fundamentals of plane geometry, covering the vocabulary and concept of geometry through the use of formal proof and algebra. Completion of the geometry sequence prepares students for higher level mathematics courses and for those science courses requiring a working knowledge of geometry.
NY Regents Exam - Integrated Algebra: Test Prep & Practice About the Course If you're a high school student in New York State, you need to earn a Regents Diploma to satisfy graduation requirements. This includes passing Regents Exams in each of the following topics: comprehensive English, mathematics, global history, U.S. history and science. For your math exam, you can choose to focus on integrated algebra, geometry or algebra 2/trigonometry, unless you're seeking a Regents Diploma with Advanced Designation, in which case you'll need to pass all three math exams. Education Portal called on its team of experienced professional math instructors to put together this online study guide for the Integrated Algebra Regents Exam. Our brief video lessons provide a refresher on decimals and fractions, exponents, algebraic distribution and functions. You'll be reminded how to factor by grouping, graph functions and solve problems involving Venn diagrams. You'll also review the basics of trigonometry and well-known equations, such as the quadratic formula and the absolute-value inequality. Other topics covered in this chapter on integrated algebra include the following: Number theory and basic arithmetic Radical expressions and equations Algebraic linear equations and inequalities Matrices and absolute value Quadratics and polynomials Rational expressions Calculations with ratios, percentages and proportions Probability and statistics Working with data Measurement and geometry for algebra students The Integrated Algebra Regents Exam features around 40 questions, which are a mixture of multiple choice and constructed response. You'll be provided with a ruler and a graphing calculator to complete this exam, which has a 3-hour time limit. The Integrated Algebra Regents Exam is offered in January, June and August. Integrated Algebra Regents Exam Preparation Each lesson in Education Portal's integrated algebra study guide features a video presentation, written transcript and self-assessment quiz. The video lessons can be watched in just 5-10 minutes, and their light, easy-to-follow manner makes reviewing for the Regents Exam easy and fun. The transcripts are a great option for students who learn best by reading, and even if you've also watched the video, they can help reinforce what you've learned. You can complete each lesson by taking the short, multiple-choice quiz; the quiz questions are designed to resemble what you'll see on the Regents Exam. Integrated Algebra Regents Exam Scoring Multiple-choice questions on the Integrated Algebra Regents Exam are worth two points each, while constructed-response questions count for 2-4 points, depending on the complexity of the question. Teachers who score the exam use a rubric for the constructed-response questions, and in some cases, you can earn partial credit for a wrong answer as long as you've shown your work. The integrated algebra test is scored on a 100-point scale. You need at least a 65 to qualify for a Regents Diploma. NY Regents is a registered trademark of New York State Education Department, which is not affiliated with Education Portal. Try this course for FREE 98
General Education Numerical Component Style Sheet: The following numerical component style sheet is designed to help all those who are planning to develop a course for the general education curriculum. Many faculty do not realize that they do have a numerical component in their classes. Indicate how (by exam, project & % of exam) this information will be assessed in examinations. Examples of Numerical Components MAPS CHARTS STATISTICAL MATERIAL If the course has a numerical component, include specific information, include specific information as to what skill or knowledge a student is expected to demonstrate. Failure to include this basic information will result in the return of your application. This site is maintained by GENERAL EDUCATION PROGRAM. This page last modified on Monday, June 24, 2013 11:31:01 AM Central US Time. If you find errors please bring them to the attention of Lee Ann Birdwell (leeann.birdwell@wichita.edu).
A+ National Pre-traineeship Maths and Literacy for Retail by Andrew Spencer Book Description Pre-traineeship Maths and Literacy for Retail is a write-in workbook that helps to prepare students seeking to gain a Retail Traineeship. It combines practical, real-world scenarios and terminology specifically relevant to the Retail Industry, and provides students with the mathematical skills they need to confidently pursue a career in the Retail Trade. Mirroring the format of current apprenticeship entry assessments, Pre-traineeship Maths and Literacy for Retail includes hundreds of questions to improve students' potential of gaining a successful assessment outcome of 75-80% and above. This workbook will therefore help to increase students' eligibility to obtain a Retail Traineeship. Pre-traineeship Maths and Literacy for Retail also supports and consolidates concepts that students studying VET (Vocational Educational Training) may use, as a number of VCE VET programs are also approved pre-traineeships. This workbook is also a valuable resource for older students aiming to revisit basic literacy and maths in their preparation to re-enter the workforce at the apprenticeship level. Buy A+ National Pre-traineeship Maths and Literacy for Retail book by Andrew Spencer from Australia's Online Bookstore, Boomerang Books. You might also like... Walmart, the biggest retailer on earth, has a truly global influence that touches consumers and businesses around the world every day. This title offers an insight into how the retailer emerged from its humble roots in rural Arkansas to become a global retailing phenomenon. Describes and analyses the strategies, organization and cultures that have made the supermarkets what they are today. This title includes material on the rise of online retailing, the success of the hard discount chains and the influence of environmental issues on consumer attitudes and behaviour. This resource presents a modern approach to retail and industrial management, suggesting the tools for implementing integrated management based on the open source software, Open ERP. The text describes how to integrate different services and automate tasks including sales and purchases. In eleven chapters by leading scholars, The Market Makers provides a detailed but highly readable analysis of how retailers have become the leading drivers of the new global economy. The analytic core is the "market-making perspective," which refocuses economic analysis away from factories and production to markets and market-making. Books By Author Andrew Spencer Current approaches to morphology, Andrew Spencer argues, are flawed. He uses intermediate types of lexical relatedness in different languages to develop a morphologically-informed model of the lexical entry. He uses this to build a model of lexical relatedness consistent with paradigm-based models. A book for all morphologists and lexicographers Hairdressing context beauty therapy
Find an Edgewater Park, NJ Math ...It is a branch of mathematics that substitutes letters for numbers. An algebraic equation represents a scale, what is done on one side of the scale with a number is also done to the other side of the scale. You can't get to higher math without taking algebra.
Category Archives: Linear Algebra Post navigation Optimization is by far one of the richest ways to apply computer science and mathematics to the real world. Everybody is looking to optimize something: companies want to maximize profits, factories want to maximize efficiency, investors want to minimize risk, the list just goes on and on. The mathematical tools for optimization are also some of the richest mathematical techniques. They form the cornerstone of an entire industry known as operations research, and advances in this field literally change the world. The mathematical field is called combinatorial optimization, and the name comes from the goal of finding optimal solutions more efficiently than an exhaustive search through every possibility. This post will introduce the most central problem in all of combinatorial optimization, known as the linear program. Even better, we know how to efficiently solve linear programs, so in future posts we'll write a program that computes the most affordable diet while meeting the recommended health standard. Generalizing a Specific Linear Program Most optimization problems have two parts: an objective function, the thing we want to maximize or minimize, and constraints, rules we must abide by to ensure we get a valid solution. As a simple example you may want to minimize the amount of time you spend doing your taxes (objective function), but you certainly can't spend a negative amount of time on them (a constraint). The following more complicated example is the centerpiece of this post. Most people want to minimize the amount of money spent on food. At the same time, one needs to maintain a certain level of nutrition. For males ages 19-30, the United States National Institute for Health recommends 3.7 liters of water per day, 1,000 milligrams of calcium per day, 90 milligrams of vitamin C per day, etc. We can set up this nutrition problem mathematically, just using a few toy variables. Say we had the option to buy some combination of oranges, milk, and broccoli. Some rough estimates [1] give the following content/costs of these foods. For 0.272 USD you can get 100 grams of orange, containing a total of 53.2mg of calcium, 40mg of vitamin C, and 87g of water. For 0.100 USD you can get 100 grams of whole milk, containing 276mg of calcium, 0mg of vitamin C, and 87g of water. Finally, for 0.381 USD you can get 100 grams of broccoli containing 47mg of calcium, 89.2mg of vitamin C, and 91g of water. Here's a table summarizing this information: Some observations: broccoli is more expensive but gets the most of all three nutrients, whole milk doesn't have any vitamin C but gets a ton of calcium for really cheap, and oranges are a somewhere in between. So you could probably tinker with the quantities and figure out what the cheapest healthy diet is. The problem is what happens when we incorporate hundreds or thousands of food items and tens of nutrient recommendations. This simple example is just to help us build up a nice formality. So let's continue doing that. If we denote by the number of 100g units of broccoli we decide to buy, and the amount of milk and the amount of oranges, then we can write the daily cost of food as In the interest of being compact (and again, building toward the general linear programming formulation) we can extract the price information into a single cost vector , and likewise write our variables as a vector . We're implicitly fixing an ordering on the variables that is maintained throughout the problem, but the choice of ordering doesn't matter. Now the cost function is just the inner product (dot product) of the cost vector and the variable vector . For some reason lots of people like to write this as , where denotes the transpose of a matrix, and we imagine that and are matrices of size . I'll stick to using the inner product bracket notation. Now for each type of food we get a specific amount of each nutrient, and the sum of those nutrients needs to be bigger than the minimum recommendation. For example, we want at least 1,000 mg of calcium per day, so we require that . Likewise, we can write out a table of the constraints by looking at the columns of our table above. In the same way that we extracted the cost data into a vector to separate it from the variables, we can extract all of the nutrient data into a matrix , and the recommended minimums into a vector . Traditionally the letter is used for the minimums vector, but for now we're using for broccoli. And now the constraint is that , where the means "greater than or equal to in every coordinate." So now we can write down the more general form of the problem for our specific matrices and vectors. That is, our problem is to minimize subject to the constraint that . This is often written in offset form to contrast it with variations we'll see in a bit: In general there's no reason you can't have a "negative" amount of one variable. In this problem you can't buy negative broccoli, so we'll add the constraints to ensure the variables are nonnegative. So our final form is In general, if you have an matrix , a "minimums" vector , and a cost vector , the problem of finding the vector that minimizes the cost function while meeting the constraints is called a linear programming problem or simply a linear program. To satiate the reader's burning curiosity, the solution for our calcium/vitamin C problem is roughly . That is, you should have about 100g of broccoli and 4.2kg of milk (like 4 liters), and skip the oranges entirely. The daily cost is about 4.53 USD. If this seems awkwardly large, it's because there are cheaper ways to get water than milk. Duality Now that we've seen the general form a linear program and a cute example, we can ask the real meaty question: is there an efficient algorithm that solves arbitrary linear programs? Despite how widely applicable these problems seem, the answer is yes! But before we can describe the algorithm we need to know more about linear programs. For example, say you have some vector which satisfies your constraints. How can you tell if it's optimal? Without such a test we'd have no way to know when to terminate our algorithm. Another problem is that we've phrased the problem in terms of minimization, but what about problems where we want to maximize things? Can we use the same algorithm that finds minima to find maxima as well? Both of these problems are neatly answered by the theory of duality. In mathematics in general, the best way to understand what people mean by "duality" is that one mathematical object uniquely determines two different perspectives, each useful in its own way. And typically a duality theorem provides one with an efficient way to transform one perspective into the other, and relate the information you get from both perspectives. A theory of duality is considered beautiful because it gives you truly deep insight into the mathematical object you care about. In linear programming duality is between maximization and minimization. In particular, every maximization problem has a unique "dual" minimization problem, and vice versa. The really interesting thing is that the variables you're trying to optimize in one form correspond to the contraints in the other form! Here's how one might discover such a beautiful correspondence. We'll use a made up example with small numbers to make things easy. So you have this optimization problem Just for giggles let's write out what and are. Say you want to come up with a lower bound on the optimal solution to your problem. That is, you want to know that you can't make smaller than some number . The constraints can help us derive such lower bounds. In particular, every variable has to be nonnegative, so we know that , and so 6 is a lower bound on our optimum. Likewise, and that's an even better lower bound than 6. We could try to write this approach down in general: find some numbers that we'll use for each constraint to form To make it a valid lower bound we need to ensure that the coefficients of each of the are smaller than the coefficients in the objective function (i.e. that the coefficient of ends up less than 4). And to make it the best lower bound possible we want to maximize what the right-hand-size of the inequality would be: . If you write out these equations and the constraints you get our "lower bound" problem written as And wouldn't you know, the matrix providing the constraints is , and the vectors and switched places. This is no coincidence. All linear programs can be transformed in this way, and it would be a useful exercise for the reader to turn the above maximization problem back into a minimization problem by the same technique (computing linear combinations of the constraints to make upper bounds). You'll be surprised to find that you get back to the original minimization problem! This is part of what makes it "duality," because the dual of the dual is the original thing again. Often, when we fix the "original" problem, we call it the primal form to distinguish it from the dual form. Usually the primal problem is the one that is easy to interpret. (Note: because we're done with broccoli for now, we're going to use to denote the constraint vector that used to be .) Now say you're given the data of a linear program for minimization, that is the vectors and matrix for the problem, "minimize subject to ." We can make a general definition: the dual linear program is the maximization problem "maximize subject to ." Here is the new set of variables and the superscript T denotes the transpose of the matrix. The constraint for the dual is often written , again identifying vectors with a single-column matrices, but I find the swamp of transposes pointless and annoying (why do things need to be columns?). Now we can actually prove that the objective function for the dual provides a bound on the objective function for the original problem. It's obvious from the work we've done, which is why it's called the weak duality theorem. Weak Duality Theorem: Let be the data of a linear program in the primal form (the minimization problem) whose objective function is . Recall that the objective function of the dual (maximization) problem is . If are feasible solutions (satisfy the constraints of their respective problems), then In other words, the maximum of the dual is a lower bound on the minimum of the primal problem and vice versa. Moreover, any feasible solution for one provides a bound on the other. Proof. The proof is pleasingly simple. Just inspect the quantity . The constraints from the definitions of the primal and dual give us that The inequalities follow from the linear algebra fact that if the in is nonnegative, then you can only increase the size of the product by increasing the components of . This is why we need the nonnegativity constraints. In fact, the world is much more pleasing. There is a theorem that says the two optimums are equal! Strong Duality Theorem: If there are any solutions to the primal (minimization) problem and the dual (maximization) problem, respectively, then the two problems also have optimal solutions , and two candidate solutions are optimal if and only ifthey produce equal objective values . The proof of this theorem is a bit more convoluted than the weak duality theorem, and the key technique is a lemma of Farkas and its variations. See the second half of these notes for a full proof. The nice thing is that this theorem gives us a way to tell if an algorithm to solve linear programs is done: maintain a pair of feasible solutions to the primal and dual problems, improve them by some rule, and stop when the two solutions give equal objective values. The hard part, then, is finding a principled and guaranteed way to improve a given pair of solutions. On the other hand, you can also prove the strong duality theorem by inventing an algorithm that provably terminates. We'll see such an algorithm, known as the simplex algorithm in the next post. Sneak peek: it's a lot like Gaussian elimination. Then we'll use the algorithm (or an equivalent industry-strength version) to solve a much bigger nutrition problem. In fact, you can do a bit better than the strong duality theorem, in terms of coming up with a stopping condition for a linear programming algorithm. You can observe that an optimal solution implies further constraints on the relationship between the primal and the dual problems. In particular, this is called the complementary slackness conditions, and they essentially say that if an optimal solution to the primal has a positive variable then the corresponding constraint in the dual problem must be tight (is an equality) to get an optimal solution to the dual. The contrapositive says that if some constraint is slack, or a strict inequality, then either the corresponding variable is zero or else the solution is not optimal. More formally, Theorem (Complementary Slackness Conditions): Let be the data of the primal form of a linear program, "minimize subject to ." Then are optimal solutions to the primal and dual problems if any only if all of the following conditions hold. are both feasible for their respective problems. Whenever the corresponding constraint is an equality. Whenever the corresponding constraint is an equality. Here we denote by the -th row of the matrix and to denote the -th entry of a vector. Another way to write the condition using vectors instead of English is The proof follows from the duality theorems, and just involves pushing around some vector algebra. See section 6.2 of these notes. One can interpret complementary slackness in linear programs in a lot of different ways. For us, it will simply be a termination condition for an algorithm: one can efficiently check all of these conditions for the nonzero variables and stop if they're all satisfied or if we find a variable that violates a slackness condition. Indeed, in more mature optimization analyses, the slackness condition that is more egregiously violated can provide evidence for where a candidate solution can best be improved. For a more intricate and detailed story about how to interpret the complementary slackness conditions, see Section 4 of these notes by Joel Sobel. Finally, before we close we should note there are geometric ways to think about linear programming. I have my preferred visualization in my head, but I have yet to find a suitable animation on the web that replicates it. Here's one example in two dimensions. The set of constraints define a convex geometric region in the plane The constraints define a convex area of "feasible solutions." Image source: Wikipedia. Now the optimization function is also a linear function, and if you fix some output value this defines a line in the plane. As changes, the line moves along its normal vector (that is, all these fixed lines are parallel). Now to geometrically optimize the target function, we can imagine starting with the line , and sliding it along its normal vector in the direction that keeps it in the feasible region. We can keep sliding it in this direction, and the maximum of the function is just the last instant that this line intersects the feasible region. If none of the constraints are parallel to the family of lines defined by , then this is guaranteed to occur at a vertex of the feasible region. Otherwise, there will be a family of optima lying anywhere on the line segment of last intersection. In higher dimensions, the only change is that the lines become affine subspaces of dimension . That means in three dimensions you're sliding planes, in four dimensions you're sliding 3-dimensional hyperplanes, etc. The facts about the last intersection being a vertex or a "line segment" still hold. So as we'll see next time, successful algorithms for linear programming in practice take advantage of this observation by efficiently traversing the vertices of this convex region. We'll see this in much more detail in the next post. So far on this blog we've given some introductory notes on a few kinds of algebraic structures in mathematics (most notably groups and rings, but also monoids). Fields are the next natural step in the progression. If the reader is comfortable with rings, then a field is extremely simple to describe: they're just commutative rings with 0 and 1, where every nonzero element has a multiplicative inverse. We'll give a list of all of the properties that go into this "simple" definition in a moment, but an even more simple way to describe a field is as a place where "arithmetic makes sense." That is, you get operations for which satisfy the expected properties of addition, subtraction, multiplication, and division. So whatever the objects in your field are (and sometimes they are quite weird objects), they behave like usual numbers in a very concrete sense. So here's the official definition of a field. We call a set a field if it is endowed with two binary operations addition () and multiplication (, or just symbol juxtaposition) that have the following properties: There is an element we call 0 which is the identity for addition. Addition is commutative and associative. Every element has a corresponding additive inverse (which may equal ) for which . These three properties are just the axioms of a (commutative) group, so we continue: There is an element we call 1 (distinct from 0) which is the identity for multiplication. Multiplication is commutative and associative. Every nonzero element has a corresponding multiplicative inverse (which may equal ) for which . Addition and multiplication distribute across each other as we expect. If we exclude the existence of multiplicative inverses, these properties make a commutative ring, and so we have the following chain of inclusions that describes it all The standard examples of fields are the real numbers , the rationals , and the complex numbers . But of course there are many many more. The first natural question to ask about fields is: what can they look like? For example, can there be any finite fields? A field which as a set has only finitely many elements? As we saw in our studies of groups and rings, the answer is yes! The simplest example is the set of integers modulo some prime . We call them or sometimes just for short, and let's rederive what we know about them now. As a set, consists of the integers . The addition and multiplication operations are easy to define, they're just usual addition and multiplication followed by a modulus. That is, we add by and multiply with . This thing is clearly a commutative ring (because the integers form a commutative ring), so to show this is a field we need to show that everything has a multiplicative inverse. There is a nice fact that allows us to do this: an element has an inverse if and only if the only way for it to divide zero is the trivial way . Here's a proof. For one direction, suppose divides zero nontrivially, that is there is some with . Then if had an inverse , then , but that's very embarrassing for because it claimed to be nonzero. Now suppose only divides zero in the trivial way. Then look at all possible ways to multiply by other nonzero elements of . No two can give you the same result because if then (without using multiplicative inverses) , but we know that can only divide zero in the trivial way so . In other words, the map "multiplication by " is injective. Because the set of nonzero elements of is finite you have to hit everything (the map is in fact a bijection), and some will give you . Now let's use this fact on in the obvious way. Since is a prime, there are no two smaller numbers so that . But in the number is equivalent to zero (mod )! So has no nontrivial zero divisors, and so every element has an inverse, and so it's a finite field with elements. The next question is obvious: can we get finite fields of other sizes? The answer turns out to be yes, but you can't get finite fields of any size. Let's see why. Characteristics and Vector Spaces Say you have a finite field (lower-case k is the standard letter for a field, so let's forget about ). Beacuse the field is finite, if you take 1 and keep adding it to itself you'll eventually run out of field elements. That is, at some point. How do I know it's zero and doesn't keep cycling never hitting zero? Well if at two points , then is a time where you hit zero, contradicting the claim. Now we define , the characteristic of , to be the smallest (sums of 1 with itself) for which . If there is no such (this can happen if is infinite, but doesn't always happen for infinite fields), then we say the characteristic is zero. It would probably make more sense to say the characteristic is infinite, but that's just the way it is. Of course, for finite fields the characteristic is always positive. So what can we say about this number? We have seen lots of example where it's prime, but is it always prime? It turns out the answer is yes! For if is composite, then by the minimality of we get , but . This can't happen by our above observation, because being a zero divisor means you have no inverse! Contradiction, sucker. But it might happen that there are elements of that can't be written as for any number of terms. We'll construct examples in a minute (in fact, we'll classify all finite fields), but we already have a lot of information about what those fields might look like. Indeed, since every field has 1 in it, we just showed that every finite field contains a smaller field (a subfield) of all the ways to add 1 to itself. Since the characteristic is prime, the subfield is a copy of for . We call this special subfield the prime subfield of . The relationship between the possible other elements of and the prime subfield is very neat. Because think about it: if is your field and is your prime subfield, then the elements of can interact with just like any other field elements. But if we separate from (make a separate copy of ), and just think of as having addition, then the relationship with is that of a vector space! In fact, whenever you have two fields , the latter has the structure of a vector space over the former. Back to finite fields, is a vector space over its prime subfield, and now we can impose all the power and might of linear algebra against it. What's it's dimension? Finite because is a finite set! Call the dimension , then we get a basis . Then the crucial part: every element of has a unique representation in terms of the basis. So they are expanded in the form where the come from . But now, since these are all just field operations, every possible choice for the has to give you a different field element. And how many choices are there for the ? Each one has exactly . And so by counting we get that has many elements. This is getting exciting quickly, but we have to pace ourselves! This is a constraint on the possible size of a finite field, but can we realize it for all choices of ? The answer is again yes, and in the next section we'll see how. But reader be warned: the formal way to do it requires a little bit of familiarity with ideals in rings to understand the construction. I'll try to avoid too much technical stuff, but if you don't know what an ideal is, you should expect to get lost (it's okay, that's the nature of learning new math!). Constructing All Finite Fields Let's describe a construction. Take a finite field of characteristic , and say you want to make a field of size . What we need to do is construct a field extension, that is, find a bigger field containing so that the vector space dimension of our new field over is exactly . What you can do is first form the ring of polynomials with coefficients in . This ring is usually denoted , and it's easy to check it's a ring (polynomial addition and multiplication are defined in the usual way). Now if I were speaking to a mathematician I would say, "From here you take an irreducible monic polynomial of degree , and quotient your ring by the principal ideal generated by . The result is the field we want!" In less compact terms, the idea is exactly the same as modular arithmetic on integers. Instead of doing arithmetic with integers modulo some prime (an irreducible integer), we're doing arithmetic with polynomials modulo some irreducible polynomial . Now you see the reason I used for a polynomial, to highlight the parallel thought process. What I mean by "modulo a polynomial" is that you divide some element in your ring by as much as you can, until the degree of the remainder is smaller than the degree of , and that's the element of your quotient. The Euclidean algorithm guarantees that we can do this no matter what is (in the formal parlance, is called a Euclidean domainfor this very reason). In still other words, the "quotient structure" tells us that two polynomials are considered to be the same in if and only if is divisible by . This is actually the same definition for , with polynomials replacing numbers, and if you haven't already you can start to imagine why people decided to study rings in general. Let's do a specific example to see what's going on. Say we're working with and we want to compute a field of size . First we need to find a monic irreducible polynomial of degree . For now, I just happen to know one: . In fact, we can check it's irreducible, because to be reducible it would have to have a linear factor and hence a root in . But it's easy to see that if you compute and take (mod 3) you never get zero. So I'm calling this new ring It happens to be a field, and we can argue it with a whole lot of ring theory. First, we know an irreducible element of this ring is also prime (because the ring is a unique factorization domain), and prime elements generate maximal ideals (because it's a principal ideal domain), and if you quotient by a maximal ideal you get a field (true of all rings). But if we want to avoid that kind of argument and just focus on this ring, we can explicitly construct inverses. Say you have a polynomial , and for illustration purposes we'll choose . Now in the quotient ring we could do polynomial long division to find remainders, but another trick is just to notice that the quotient is equivalent to the condition that . So we can reduce by applying this rule to to get Now what's the inverse of ? Well we need a polynomial whose product with gives us something which is equivalent to 1, after you reduce by . A few minutes of algebra later and you'll discover that this is equivalent to the following polynomial being identically 1 In other words, we get a system of linear equations which we need to solve: And from here you can solve with your favorite linear algebra techniques. This is a good exercise for working in fields, because you get to abuse the prime subfield being characteristic 3 to say terrifying things like and . The end result is that the inverse polynomial is , and if you were really determined you could write a program to compute these linear systems for any input polynomial and ensure they're all solvable. We prefer the ring theoretic proof. In any case, it's clear that taking a polynomial ring like this and quotienting by a monic irreducible polynomial gives you a field. We just control the size of that field by choosing the degree of the irreducible polynomial to our satisfaction. And that's how we get all finite fields! One Last Word on Irreducible Polynomials One thing we've avoided is the question of why irreducible monic polynomials exist of all possible degrees over any (and as a consequence we can actually construct finite fields of all possible sizes). The answer requires a bit of group theory to prove this, but it turns out that the polynomial has all degree monic irreducible polynomials as factors. But perhaps a better question (for computer scientists) is how do we work over a finite field in practice? One way is to work with polynomial arithmetic as we described above, but this has some downsides: it requires us to compute these irreducible monic polynomials (which doesn't sound so hard, maybe), to do polynomial long division every time we add, subtract, or multiply, and to compute inverses by solving a linear system. But we can do better for some special finite fields, say where the characteristic is 2 (smells like binary) or we're only looking at . The benefit there is that we aren't forced to use polynomials. We can come up with some other kind of structure (say, matrices of a special form) which happens to have the same field structure and makes computing operations relatively painless. We'll see how this is done in the future, and see it applied to cryptography when we continue with our series on elliptic curve cryptography. A few awesome readers have posted comments in Computing Homology to the effect of, "Your code is not quite correct!" And they're right! Despite the almost year since that post's publication, I haven't bothered to test it for more complicated simplicial complexes, or even the basic edge cases! When I posted it the mathematics just felt so solid to me that it had to be right (the irony is rich, I know). As such I'm apologizing for my lack of rigor and explaining what went wrong, the fix, and giving some test cases. As of the publishing of this post, the Github repository for Computing Homology has been updated with the correct code, and some more examples. The main subroutine was the simultaneousReduce function which I'll post in its incorrectness below It's a beast of a function, and the persnickety detail was just as beastly: this snippet should have an instead of a . if nonzeroCol == numCols: j += 1 continue This is simply what happens when we're looking for a nonzero entry in a row to use as a pivot for the corresponding column, but we can't find one and have to move to the next row. A stupid error on my part that would be easily caught by proper test cases. The next mistake is a mathematical misunderstanding. In short, the simultaneous column/row reduction process is not enough to get the matrix into the right form! Let's see this with a nice example, a triangulation of the Mobius band. There are a number of triangulations we could use, many of which are seen in these slides. The one we'll use is the following. It's first and second boundary maps are as follows (in code, because latex takes too much time to type out) The first reduced matrix looks fine; there's nothing we can do to improve it. But the second one is not quite fully reduced! Notice that rows 5, 8 and 10 are not linearly independent. So we need to further row-reduce the nonzero part of this matrix before we can read off the true rank in the way we described last time. This isn't so hard (we just need to reuse the old row-reduce function we've been using), but why is this allowed? It's just because the corresponding column operations for those row operations are operating on columns of all zeros! So we need not worry about screwing up the work we did in column reducing the first matrix, as long as we only work with the nonzero rows of the second. Of course, nothing is stopping us from ignoring the "corresponding" column operations, since we know we're already done there. So we just have to finish row reducing this matrix. This changes our bettiNumber function by adding a single call to a row-reduce function which we name so as to be clear what's happening. The resulting function is Indeed, finishRowReducing finishes row reducing the second boundary matrix. Note that it doesn't preserve how the rows of zeros lined up with the pivot columns of the reduced version of as it did in the previous post, but since in the end we're only counting pivots it doesn't matter how we switch rows. The "zeros lining up" part is just for a conceptual understanding of how the image lines up with the kernel for a valid simplicial complex. In fixing this issue we've also fixed an issue another commenter mentioned, that you couldn't blindly plug in the zero matrix for and get zeroth homology (which is the same thing as connected components). After our fix you can. Of course there still might be bugs, but I have so many drafts lined up on this blog (and research papers to write, experiments to run, theorems to prove), that I'm going to put off writing a full test suite. I'll just have to update this post with new bug fixes as they come. There's just so much math and so little time :) But extra kudos to my amazing readers who were diligent enough to run examples and spot my error. I'm truly blessed to have you on my side. Also note that this isn't the most efficient way to represent the simplicial complex data, or the most efficient row reduction algorithm. If you're going to run the code on big inputs, I suggest you take advantage of sparse matrix algorithms for doing this sort of stuff. You can represent the simplices as entries in a dictionary and do all sorts of clever optimizations to make the algorithm effectively linear time in the number of simplices. If you really want to impress your friends and confound your enemies, you can invoke tensor products… People run in terror from the symbol. He was explaining some aspects of multidimensional Fourier transforms, but this comment is only half in jest; people get confused by tensor products. It's often for good reason. People who really understand tensors feel obligated to explain it using abstract language (specifically, universal properties). And the people who explain it in elementary terms don't really understand tensors. This post is an attempt to bridge the gap between the elementary and advanced understandings of tensors. We'll start with the elementary (axiomatic) approach, just to get a good feel for the objects we're working with and their essential properties. Then we'll transition to the "universal" mode of thought, with the express purpose of enlightening us as to why the properties are both necessary and natural. But above all, we intend to be sufficiently friendly so as to not make anybody run in fear. This means lots of examples and preferring words over symbols. Unfortunately, we simply can't get by without the reader knowing the very basics of linear algebra (the content of our first two primers on linear algebra (1)(2), though the only important part of the second is the definition of an inner product). So let's begin. Tensors as a Bunch of Axioms Before we get into the thick of things I should clarify some basic terminology. Tensors are just vectors in a special vector space. We'll see that such a vector space comes about by combining two smaller vector spaces via a tensor product. So the tensor product is an operation combining vector spaces, and tensors are the elements of the resulting vector space. Now the use of the word product is quite suggestive, and it may lead one to think that a tensor product is similar or related to the usual direct product of vector spaces. In fact they are related (in very precise sense), but they are far from similar. If you were pressed, however, you could start with the direct product of two vector spaces and take a mathematical machete to it until it's so disfigured that you have to give it a new name (the tensor product). With that image in mind let's see how that is done. For the sake of generality we'll talk about two arbitrary finite-dimensional vector spaces of dimensions . Recall that the direct product is the vector space of pairs where comes from and from . Recall that addition in this vector space is defined componentwise ()) and scalar multiplication scales both components . To get the tensor product space , we make the following modifications. First, we redefine what it means to do scalar multiplication. In this brave new tensor world, scalar multiplication of the whole vector-pair is declared to be the same as scalar multiplication of any component you want. In symbols, for all choices of scalars and vectors . Second, we change the addition operation so that it only works if one of the two components are the same. In symbols, we declare that only works because is the same in both pieces, and with the same rule applying if we switch the positions of above. All other additions are simply declared to be new vectors. I.e. is simply itself. It's a valid addition — we need to be able to add stuff to be a vector space — but you just can't combine it any further unless you can use the scalar multiplication to factor out some things so that or . To say it still one more time, a general element of the tensor is a sum of these pairs that can or can't be combined by addition (in general things can't always be combined). Finally, we rename the pair to , to distinguish it from the old vector space that we've totally butchered and reanimated, and we call the tensor product space as a whole . Those familiar with this kind of abstract algebra will recognize quotient spaces at work here, but we won't use that language except to note that we cover quotients and free spaces elsewhere on this blog, and that's the formality we're ignoring. As an example, say we're taking the tensor product of two copies of . This means that our space is comprised of vectors like , and moreover that the following operations are completely legitimate. Cool. This seemingly innocuous change clearly has huge implications on the structure of the space. We'll get to specifics about how different tensors are from regular products later in this post, but for now we haven't even proved this thing is a vector space. It might not be obvious, but if you go and do the formalities and write the thing as a quotient of a free vector space (as we mentioned we wouldn't do) then you know that quotients of vector spaces are again vector spaces. So we get that one for free. But even without that it should be pretty obvious: we're essentially just declaring that all the axioms of a vector space hold when we want them to. So if you were wondering whether The answer is yes, by force of will. So just to recall, the axioms of a tensor space are The "basic" vectors are for , and they're used to build up all other vectors. Addition is symbolic, unless one of the components is the same in both addends, in which case and . You can freely move scalar multiples around the components of . The rest of the vector space axioms (distributivity, additive inverses, etc) are assumed with extreme prejudice. Naturally, one can extend this definition to -fold tensor products, like . Here we write the vectors as sums of things like , and we enforce that addition can only be combined if all but one coordinates are the same in the addends, and scalar multiples move around to all coordinates equally freely. So where does it come from?! By now we have this definition and we can play with tensors, but any sane mathematically minded person would protest, "What the hell would cause anyone to come up with such a definition? I thought mathematics was supposed to be elegant!" It's an understandable position, but let me now try to convince you that tensor products are very natural. The main intrinsic motivation for the rest of this section will be this: We have all these interesting mathematical objects, but over the years we have discovered that the maps between objects are the truly interesting things. A fair warning: although we'll maintain a gradual pace and informal language in what follows, by the end of this section you'll be reading more or less mature 20th-century mathematics. It's quite alright to stop with the elementary understanding (and skip to the last section for some cool notes about computing), but we trust that the intrepid readers will push on. So with that understanding we turn to multilinear maps. Of course, the first substantive thing we study in linear algebra is the notion of a linear map between vector spaces. That is, a map that factors through addition and scalar multiplication (i.e. and ). But it turns out that lots of maps we work with have much stronger properties worth studying. For example, if we think of matrix multiplication as an operation, call it , then takes in two matrices and spits out their product Now what would be an appropriate notion of linearity for this map? Certainly it is linear in the first coordinate, because if we fix then And for the same reason it's linear in the second coordinate. But it is most definitely not linear in both coordinates simultaneously. In other words, In fact, if the only operation satisfying linearity in its two coordinates separately and also this kind of linearity is the zero map! (Try to prove this as an exercise.) So the strongest kind of linearity we could reasonably impose is that is linear in each coordinate when all else is fixed. Note that this property allows us to shift around scalar multiples, too. For example, Starting to see the wispy strands of a connection to tensors? Good, but hold it in for a bit longer. This single-coordinate-wise-linear property is called bilinearity when we only have two coordinates, and multilinearity when we have more. Here are some examples of nice multilinear maps that show up everywhere: If is an inner product space over , then the inner product is bilinear. The determinant of a matrix is a multilinear map if we view the columns of the matrix as vector arguments. The cross product of vectors in is bilinear. There are many other examples, but you should have at least passing familiarity with these notions, and it's enough to convince us that multilinearity is worth studying abstractly. And so what tensors do is give a sort of classification of multilinear maps. The idea is that every multilinear map from a product vector space to any vector space can be written first as a multilinear map to the tensor space Followed by a linear map to , And the important part is that doesn't depend on the original (but does). One usually draws this as a single diagram: And to say this diagram commutes is to say that all possible ways to get from one point to another are equivalent (the compositions of the corresponding maps you follow are equal, i.e. ). In fuzzy words, the tensor product is like the gatekeeper of all multilinear maps, and is the gate. Yet another way to say this is that is the most general possible multilinear map that can be constructed from . Moreover, the tensor product itself is uniquely defined by having a "most-general" (up to isomorphism). This notion is often referred to by mathematicians as the "universal property" of the tensor product. And they might say something like "the tensor product is initial with respect to multilinear mappings from the standard product." We discuss language like this in detail in this blog's series on category theory, but it's essentially a super-compact (and almost too vague) way of saying what the diagram says. Let's explore this definition when we specialize to a tensor of two vector spaces, and it will give us a good understanding of (which is really incredibly simple, but people like to muck it up with choices of coordinates and summations). So fix as vector spaces and look at the diagram What is in this case? Well it just sends . Is this map multilinear? Well if we fix then and And our familiarity with tensors now tells us that the other side holds too. Actually, rather than say this is a result of our "familiarity with tensors," the truth is that this is how we know that we need to define the properties of tensors as we did. It's all because we designed tensors to be the gatekeepers of multilinear maps! So now let's prove that all maps can be decomposed into an part and a part. To do this we need to know what data uniquely defines a multilinear map. For usual linear maps, all we had to do was define the effect of the map on each element of a basis (the rest was uniquely determined by the linearity property). We know what a basis of is, it's just the union of the bases of the pieces. Say that has a basis and has , then a basis for the product is just . But multilinear maps are more nuanced, because they have two arguments. In order to say "what they do on a basis" we really need to know how they act on all possible pairs of basis elements. For how else could we determine ? If there are of the 's and of the 's, then there are such pairs . Uncoincidentally, as is a vector space, its basis can also be constructed in terms of the bases of and . You simply take all possible tensors . Since every can be written in terms of their bases, it's clear than any tensor can also be written in terms of the basis tensors (by simply expanding each in terms of their respective bases, and getting a larger sum of more basic tensors). Just to drive this point home, if is a basis for , and a basis for , then the tensor space has basis It's a theorem that finite-dimensional vector spaces of equal dimension are isomorphic, so the length of this basis (6) tells us that . So fine, back to decomposing . All we have left to do is use the data given by (the effect on pairs of basis elements) to define . The definition is rather straightforward, as we have already made the suggestive move of showing that the basis for the tensor space () and the definition of are essentially the same. That is, just take . Note that this is just defined on the basis elements, and so we extend to all other vectors in the tensor space by imposing linearity (defining to split across sums of tensors as needed). Is this well defined? Well, multilinearity of forces it to be so. For if we had two equal tensors, say, , then we know that has to respect their equality, because , so will take the same value on equal tensors regardless of which representative we pick (where we decide to put the ). The same idea works for sums, so everything checks out, and is equal to , as desired. Moreover, we didn't make any choices in constructing . If you retrace our steps in the argument, you'll see that everything was essentially decided for us once we fixed a choice of a basis (by our wise decisions in defining ). Since the construction would be isomorphic if we changed the basis, our choice of is unique. There is a lot more to say about tensors, and indeed there are some other useful ways to think about tensors that we've completely ignored. But this discussion should make it clear why we define tensors the way we do. Hopefully it eliminates most of the mystery in tensors, although there is still a lot of mystery in trying to compute stuff using tensors. So we'll wrap up this post with a short discussion about that. Computability and Stuff It should be clear by now that plain product spaces and tensor product spaces are extremely different. In fact, they're only related in that their underlying sets of vectors are built from pairs of vectors in and . Avid readers of this blog will also know that operations involving matrices (like row reduction, eigenvalue computations, etc.) are generally efficient, or at least they run in polynomial time so they're not crazy impractically slow for modest inputs. On the other hand, it turns out that almost every question you might want to ask about tensors is difficult to answer computationally. As with the definition of the tensor product, this is no mere coincidence. There is something deep going on with tensors, and it has serious implications regarding quantum computing. More on that in a future post, but for now let's just focus on one hard problem to answer for tensors. As you know, the most general way to write an element of a tensor space is as a sum of the basic-looking tensors. where the may be sums of vectors from themselves. But as we saw with our examples over , there can be lots of different ways to write a tensor. If you're lucky, you can write the entire tensor as a one-term sum, that is just a tensor . If you can do this we call the tensor a pure tensor, or a rank 1 tensor. We then have the following natural definition and problem: Definition: The rank of a tensor is the minimum number of terms in any representation of as a sum of pure tensors. The one exception is the zero element, which has rank zero by convention. Problem: Given a tensor where is a field, compute its rank. Of course this isn't possible in standard computing models unless you can represent the elements of the field (and hence the elements of the vector space in question) in a computer program. So we restrict to be either the rational numbers or a finite field . Even though the problem is simple to state, it was proved in 1990 (a result of Johan Håstad) that tensor rank is hard to compute. Specifically, the theorem is that Theorem: Computing tensor rank is NP-hard when and NP-complete when is a finite field. The details are given in Håstad's paper, but the important work that followed essentially showed that most problems involving tensors are hard to compute (many of them by reduction from computing rank). This is unfortunate, but also displays the power of tensors. In fact, tensors are so powerful that many believe understanding them better will lead to insight in some very important problems, like finding faster matrix multiplication algorithms or proving circuit lower bounds (which is closely related to P vs NP). Finding low-rank tensor approximations is also a key technique in a lot of recent machine learning and data mining algorithms. With this in mind, the enterprising reader will probably agree that understanding tensors is both valuable and useful. In the future of this blog we'll hope to see some of these techniques, but at the very least we'll see the return of tensors when we delve into quantum computing.
... More About This Book comprehensive arithmetic book that explains the subject matter in a way that makes sense to the reader. It begins with the most basic principles and progresses through more advanced topics to prepare a student for algebra. Basic Math and Pre-Algebra explains the principles and operations of arithmetic, provides step-by-step procedures and solutions, and presents examples and applications. Basic Math and Pre-Algebra is a reference book for grade school and middle school students that explains and clarifies the arithmetic principles they are learning in school. It is also a comprehensive reference source for more advanced students currently learning algebra, pre-calculus and calculus. Basic Math and Pre-Algebra is invaluable for students, parents, tutors and anyone needing a basic arithmetic reference source. The information provided in each book and in the series as a whole is progressive in difficulty and builds on itself, which allows the reader to gain perspective on the connected nature of mathematics. The skills required to understand every topic presented are explained in an earlier chapter or book within the series. Each of the books contains a complete table of contents, a comprehensive index, and the tables of contents of the other books in the series that specific subjects, principles and formulas can be easily found. The books are written in a simple style that facilitates understanding and easy referencing of sought-after principles, definitions and explanations. Basic Math and Pre-Algebra and the Master Math series are not replacements for textbooks but rather reference books providing explanations and perspective. The Master Math series would have been invaluable to me during my entire education from grade school through graduate school. There is no other source that provides the breadth and depth of the Master Math series in a single book or series. Finally, mathematics is a language-the universal language. A person struggling with mathematics should approach it in the same fashion he or she would approach learning any other language. If someone moves to a foreign country, he or she does not expect to know the language automatically. It takes practice and contact with a language in order to master it. After a short time in the foreign country he or she would not say, 'I do not know this language well yet. I must not have an aptitude for it.' Yet many people have this attitude toward mathematics. If time is spent learning and practicing the principles, mathematics will become familiar and understandable. Don't give up. Debra Ross, the author, February 16, 1999 Note to the reader: There is a misprint in books from the first printing on page138: ln (infinity) = infinity, not zero. Related Subjects Meet the Author Debra Anne Ross has a double BA in Chemistry and Biology from the University of California, Santa Cruz, and and MS in Chemical Engineering from Stanford University. Debra's career encompasses biology, chemistry, biochemistry, engineering, biosensors, pharmaceutical drug discovery, and intellectual property. She is the author of the popular Master Math books, The 3:00 PM Secret: Live Slim and Strong Live Your Dreams, The 3:00 PM Secret: Ten Day Dream Diet (2009), and Arrows Through Time: A Time Travel Tale of Adventure, Courage, and Faith Algebra Book Available! This is the best book on learning basic algebra. It is thorough yet concise. The information is presented very clearly. The author has obviously tried to explain the concepts so that they `make sense¿ to students - and their parents. I use the book to explain algebra to my students. Like the other Master Math books by Ross, the topics flow logically and build in difficulty. What a breath of fresh air after the often confusing text books students are given in school. This book is helpful for students struggling with algebra and the parents who are tutoring them. This book is also extremely useful for older students who did not adequately learn algebra, yet find they need to know it later. Topics can easily be looked up and reviewed or learned. I highy recommend this book! 5 out of 7 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Log-IC Posted October 10, 2009 ALG 101 Launchpad Splendid clarity and progression. A cherry on top of this fine sundae, maybe a single page of exponential, radical,and complex properties for a quick reference,( but that's nitpicking). 3 out of 5 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted November 20, 2007 Outstanding Trigonometry Presentation! This is the best book out there on learning trigonometry. I especially appreciate the visually-oriented focus. Each concept is described in all its forms, such as sine. Do you know each of the different ways sine can be described? Like the other Master Math books by Ross, the topics flow logically and are in context with what precedes and follows. It is thorough yet concise, and packed full of everything you, as tutor, or your kids need to know. The real world and fun applications are wonderful! The information is explained clearly and in a way that makes sense, so that a given concept is explained in such a way you understand what is being discussed rather than just memorizing formulas. What a breath of fresh air after the often confusing text books I was and my children are given in school. I really feel I can explain trigonometry to young people using this book! if I were going back to school, and taking math or science, this book would be in my backpack.page 14 e reader i am not sure if i am missing something but on page 14 the book says x times $1.00 per glass will equal 20. then shows this equation X + ($1.00 per glass)=$20.00the internet is more helpful This book is terrible. There are many books that are better. I found websites more helpful. 1 out of 3 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Secondary Navigation Utility You are here Hands-On Learning In May 2013, eight students taking Numb3rs (CIVT 202, crosslisted MATH 140) spent a windy afternoon at Makoce Waste, a grassland near Sioux Falls. Students gathered and analyzed data, looking for differences in the proportions of native and invasive grass species on the eastern and western halves of the grassland. Dr. Amy Lewis points out a Pasque Flower, the state flower of South Dakota. Students Alaina Heiskary (2013) and Sarah Kocer (2016) estimate the percentage of native and invasive grasses in a sample of the grassland, using a Daubenmire quadrat. Some students recently involved are: Alexis Preheim (Math/Secondary Education, 2013): Alexis read a published journal article on creating inquiry based activities from standard textbook problems, and created her own example. In the process she explored the area of Inquiry Based Learning and learned how to use interactive geometry software (Geometer's Sketchpad). Michael Arhndt (Math): Michael read a published research article on an algorithm for computing a matrix with a given sign pattern. He prepared an expository talk with an example of the algorithm and an application to a practical problem. Trent Anderson (Math/Physics/Chemistry, 2011): Trent used modeling software (STELLA) to construct a model of a disease spreading on a college campus. He was able to use the model to quite accurately represent the fall 2009 H1N1 outbreak on the Augustana campus. Trent presented his work at the Augustana Symposium in April 2011. He is currently pursuing graduate studies in chemistry at NDSU. Nicole Winkler (Math, 2011): Nicole produced mathematical models of swarming in one and two dimensions. Her model is based on behavioral characteristics of the swarming organisms which explain swarming behavior. Nicole traveled to California to present her results in 2010. She is currently employed with a national actuarial science firm. Peder Thompson (Math/English, 2010): Peder read a published journal article on the use of interactive algebra software as a tool to aid in two-dimensional visualization of binomial inequalities. He wrote a paper expanding the work to trinomial inequalities. Peder presented his work at Math on the Northern Plains, an undergraduate research conference, in the spring of 2010. He is now a graduate student in mathematics at UNL. Facilities: The math department has a number of facilities available to students. The mathematics hallway serves as a hub of activity in our department. When it's not being used for a class, you'll find the seminar room an invaluable resource in your study—whether for bound copies of Mathematical Reviews going back to the original issues of 1940 — or simply to get help from faculty and fellow students who study there. In addition, the Mikkelsen Library holds more than 17,000 volumes of books and journals on our subject, and we have access to much more material electronically and through interlibrary loan. The dedicated PC lab in the Gilbert Science Center is open many hours every day. The department's central office also houses three computers specifically for use in student research projects.
Elementary Differential Geometry 9781852331528 ISBN: 1852331526 Pub Date: 2000 Publisher: Springer Verlag Summary: Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions, and with trying to answer them using calculus techniques. It is a subject that contains some of the most beautiful and profound results in mathematics, yet many of them a...re accessible to higher level undergraduates.Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to an absolute minimum. Nothing more than first courses in linear algebra and multivariate calculus are required, and the most direct and straightforward approach is used at all times. Numerous diagrams illustrate both the ideas in the text and the examples of curves and surfaces discussed there. Pressley, Andrew is the author of Elementary Differential Geometry, published 2000 under ISBN 9781852331528 and 1852331526. Sixty three Elementary Differential Geometry textbooks are available for sale on ValoreBooks.com, fifteen used from the cheapest price of $7.32, or buy new starting at $56
Hi, I am considering using Abeka Precaculus this year. I used Abeka Geometry this past year and hated it, but I loved their math for all the grades before that. If you've used Abeka Precalculus, please let me know what you thought of it! Thank you! Thinking outside the box - pre-calc is basically just an amalgam of trig functions, logarithms, exponents, etc. Assuming you covered everything in algebra II properly, you should be able to skip straight to calculus I, since calculus textbooks and calculus courses always cover the pre-calc ground at the start. No point doing it twice. Trig and pre-calc by themselves are pretty boring, because you don't get to see -why- you're studying all the material. Same reason why language arts seems pointless until you're studying a foreign language. I went straight from algebra II to calculus I. I skipped over advanced math, trig, and pre-calc. Never did a formal geometry course either, for that matter. Given, I'm not personally familiar with how Abeka's textbooks are set up, and I'm fairly good with math, so take my advice for what it is
18 Lecture Series for Introductory &Intermediate Algebra for College Students DVD Lecture Series for for Introductory &Intermediate Algebra for College Students MathXL Tutorials on CD for Introductory &Intermediate Algebra for College Students MyMathLab Edition PACKAGE: INTRO& INTERMED ALG COLL Student Solutions Manual for for Introductory &Intermediate Algebra for College Students Student Solutions Manual for Introductory and Intermediate Algebra for College Students Worksheets for for Introductory &Intermediate Algebra for College Students Summary KEY MESSAGE: TheBlitzer Algebra Seriescombines mathematical accuracy with an engaging, friendly, and often fun presentation for maximum student appeal. Blitzerrs"s Intermediate Algebra for College Students, Bob has written textbooks covering introductory algebra, college algebra, algebra and trigonometry, precalculus, and liberal arts mathematicsA Brief Guide to Getting the Most from This Book Preface To The Student About the Author Variables, Real Numbers, and Mathematical Models Introduction to Algebra: Variables and Mathematical Models Fractions in Algebra The Real Numbers Basic Rules of Algebra Mid-Chapter Check Point Section 1.1-Section 1.4 Addition of Real Numbers Subtraction of Real Numbers Multiplication and Division of Real Numbers Exponents and Order of Operations Chapter 1 Group Project Chapter 1 Summary Chapter 1 Review Exercises Chapter 1 Chapter Test Linear Equations and Inequalities in One Variable The Addition Property of Equality The Multiplication Property of Equality Solving Linear Equations Formulas and Percents Mid-Chapter Check Point Section 2.1-Section 2.4 An Introduction to Problem Solving Problem Solving in Geometry Solving Linear Inequalities Chapter 2 Group Project Chapter 2 Summary Chapter 2 Review Exercises Chapter 2 Test Cumulative Review Exercises (Chapters 1-2) Linear Equations in Two Variables Graphing Linear Equations in Two Variables Graphing Linear Equations Using Intercepts Slope The Slope-Intercept Form of the Equation of a Line Mid-Chapter Check Point Section 3.1-Section 3.4 The Point-Slope Form of the Equation of a Line Chapter 3 Group Project Chapter 3 Summary Chapter 3 Review Exercises Chapter 3 Test Cumulative Review Exercises (Chapters 1-3) Systems of Linear Equations Solving Systems of Linear Equations by Graphing Solving Systems of Linear Equations by the Substitution Method Solving Systems of Linear Equations by the Addition Method Mid-Chapter Check Point Section 4.1-Section 4.3 Problem Solving Using Systems of Equations Systems of Linear Equations in Three Variables Chapter 4 Group Project Chapter 4 Summary Chapter 4 Review Exercises Chapter 4 Test Cumulative Review Exercises (Chapters 1-4) Exponents and Polynomials Adding and Subtracting Polynomials Multiplying Polynomials Special Products Polynomials in Several Variables Mid-Chapter Check Point Section 5.1-Section 5.4 Dividing Polynomials Long Division of Polynomials Synthetic Division Negative Exponents and Scientific Notation Chapter 5 Group Project Chapter 5 Summary Chapter 5 Review Exercises Chapter 5 Test Cumulative Review Exercises (Chapters 1-5) Factoring Polynomials The Greatest Common Factor and Factoring By Grouping Factoring Trinomials Whose Leading Coefficient Is 1 Factoring Trinomials Whose Leading Coefficient Is Not 1 Mid-Chapter Check Point Section 6.1-Section 6.3 Factoring Special Forms A General Factoring Strategy Solving Quadratic Equations By Factoring Chapter 6 Group Project Chapter 6 Summary Chapter 6 Review Exercises Chapter 6 Test Cumulative Review Exercises (Chapters 1-6) Rational Expressions Rational Expressions and Their Simplification Multiplying and Dividing Rational Expressions Adding and Subtracting Rational Expressions with the Same Denominator Adding and Subtracting Rational Expressions with Different Denominators
Real Analysis 9781852333140 ISBN: 1852333146 Publisher: Springer Verlag Summary: Understanding the concepts and methods of real analysis is an essential skill for every undergraduate mathematics student. Written in an easy-to-read style, Real Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. Combining an informal style with precision mathematics, Real Analysis covers all the key topics with f...ully worked examples and exercises with solutions. Featuring: * Sequences and series - considering the central notion of a limit * Continuous functions * Differentiation * Integration * Logarithmic and exponential functions * Uniform convergence * Circular functions All these concepts and techniques are deployed in examples in the final chapter to provide the student with a thorough understanding of this challenging subject. Howie, John M. is the author of Real Analysis, published under ISBN 9781852333140 and 1852333146. Four hundred seventy three Real Analysis textbooks are available for sale on ValoreBooks.com, one hundred one used from the cheapest price of $31.72, or buy new starting at $31
Key To Algebra offers a unique, proven way to introduce algebra to your students. New concepts are explained in simple language and examples are easy to follow. Word problems relate algebra to familiar situations, helping students understand abstract concepts. Students develop understanding by solving equations and inequalities intuitively before formal solutions are introduced. Students begin their study of algebra in Books 1-4 using only integers. Books 5-7 introduce rational numbers and expressions. Books 8-10 extend coverage to the real number system. This kit contains only Books 1-10. Answers Notes for Books 1-4 Books 5-7 and Books 8-10 are available separately, as well as the Key to Algebra Reproducible Tests. book, is what makes this book the market leader.sta nding... Less Powered by Frooition Pro Shop Search Horizons Pre Algebra Complete Set Click here to view full size. Full Size Image Click to close full size. Item Description Horizons Pre Algebra Complete Set Compare at $84.95 An understanding of the principle elements of algebra is essential to upper-level math and good standardized test scores. Introduce your junior high students to advanced math with this kit's 160 colorful lessons. The colorful student workbook reviews basic math skills before introducing Kelley Wingate's Algebra helps students in grades 5 and up master the skills necessary to succeed in algebra. Aligned to the Common Core State Standards, practice pages will be leveled in order to target each student's individual needs for support. The activities cover skills such as operations with real numbers, variables and equations, factoring, rational expressions, ratios and proportions, graphing, and radicals. This well-known series, Kelley Wingate, has been updated to align content to the Common Core State Standards. The 128-page books will provide a strong foundation of basic skills and will offer differentiated practice pages to make sure all students are well prepared to succeed in today's Common Core classroom. The books will include Common Core standards matrices, cut-apart... LessALGEBRA 2 STUDENT WORKBOOK [Paperback]AGS Secondary (Author)Take students a step further in learning algebra Specially written for low-level learners,Algebra 2 covers several methods for solving quadratic equations, such as factoring, completing the square, and graphing. The text also introduces trigonometry and exponential functions—vital concepts for real world applications. Filled with full-color illustrations and examples throughout, Algebra 2 motivates students to learn.Overall, this high Make math matter to students in grades 6 and up using Algebra: Daily Skill Builders! This 96-page book features two short, reproducible activities per page and includes enough lessons for an entire school year. It covers topics such as number patterns, word problems, equations, tables, graphs, linear relationships, variables, contextualized problems, properties, order of operations, and exponents. Activities become more challenging as students build upon what they have learned. The book is perfect for review and practice and supports NCTM and Common Core State Standards Elementary ElementaryAlgebra 2 covers all topics that are traditionally covered in second-year algebra as well as a considerable amount of geometry. In fact, students completing Algebra 2 will have studied the equivalent of one semester of informal geometry. Time is spent developing geometric concepts and writing proof outlines. Real-world problems are included along with applications to other subjects such as physics and chemistry. Includes the student textbook and the homeschool packet with test forms.Student Text Normal 0 false false false MicrosoftInternet Explorer4 Algebra: A Combined Approach is intended for a 2-semester sequence of Introductory and Intermediate Algebra where students get a solid foundation in algebra, including exposure to functions, which prepares them for success in College Algebra or their next math course. Prealgebra Review; Real Numbers and Introduction to Algebra; Equations, Inequalities, and Problem Solving; Graphing Equations and Inequalities; Systems of Equations; Exponents and Polynomials; Factoring Polynomials; Rational Expressions; Graphs and Functions; Systems of Equations and Inequalities, and Variation; Rational Exponents, Radicals and Complex Numbers; Quadratic Equations and Functions; Exponential and Logarithmic Functions; Con
A Survey of Mathematics with Applications In a Liberal Arts Math course, a common question students ask is, "Why do I have to know this?" A Survey of Mathematics with Applications continues ...Show synopsisIn a Liberal Arts Math course, a common question students ask is, "Why do I have to know this?" A Survey of Mathematics with Applications continues to be a best-seller because it shows students how we use mathematics in our daily lives and why this is important. The Ninth Edition further emphasizes this with the addition of new "Why This Is Important" sections throughout the text. Real-life and up-to-date examples motivate the topics throughout, and a wide range of exercises help students to develop their problem-solving and critical thinking skills. Angel Note: This is a standalone book, if you want the book/access card please order the ISBN listed below: 0321837533 / 9780321837530 A Survey of Mathematics with Applications plus MyMathLab Student Access Kit59664 / 9780321759665 Survey of Mathematics with Applications, AHide3928839288 Missing components. May include moderately...Good. Hardcover. Missing components. May include moderately worn cover, writing, markings or slight discoloration. SKU: 97803217596659th edition). Text Only! ! ! Description:Fine. Hardcover. Instructor Edition: Same as student edition...Fine. Hardcover. Instructor Edition: Same as student edition with additional notes or answers. Almost new condition. SKU: 9780321639288Hardcover. Instructor Edition: Same as student edition with...Hardcover. Instructor Edition: Same as student edition with additional notes or answers. New Condition. SKU: 9780321639288Fine. Hardcover. Almost new condition. SKU: 9780321759665-2-0-3...Fine. Hardcover. Almost new condition. SKU: 9780321759665
Algebra What is Algebra ? As per Wikipedia - Algebra is a branch of mathematics that deals with relations, operations and their constructions. It is one of building blocks of mathematics and it finds a huge variety of applications in our day-to-day life. Apart from its significance as a core subject of mathematics, Algebra helps students and kids a lot in developing an overall understanding of other advanced branches of mathematics such as Calculus, Geometry, Arithmetic etc. Our objective with this course content is to present Algebra and its associated concepts in an interesting and easy manner so that the school going kids can learn the subject with ease and interest. Advanced Definition of Algebra: Algebra (from Arabic al-jebr meaning "reunion of broken parts") is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Concepts Associated with Algebra: There is no limit on the complexity of all these concepts as far as "Algebra" is concerned. Newer concepts and ideas may be added as the level of education rises. All these concepts are very useful and easy-to-understand if taught properly.
Mat117 Week9 DQ1 From the concepts you have learned in this course, provide a real-world application of something that y Mat117 Week9 DQ1 From the concepts you have learned in this course, provide a real-world application of something that you think has been the most valuable to you? Why has it been valuable?
Applied Mathematics for Business, Economics and the Social Sciences 4th Revised edition Applied Mathematics for Business, Economics and the Social Sciences by Frank S. Budnick Book Description Offering comprehensive treatment of selected topics in finite mathematics and calculus, the fourth edition of this text provides an informal, non-intimidating presentation of the mathematical principles, techniques, and applications most useful to students in business, economics, and the life and social sciences. Oriented towards the needs of the student, the text retains such pedagogical features as "Algebra Flashbacks", "Notes to the Student", "Points for Thought or Discussion", and an extensive array of problems and applications to support the learning process. In the new edition a new chapter entitled "Some Preliminaries" provides a smooth transition from algebra to applied mathematics, and a new chapter on exponentials and logarithms has been added. Actual data from real applications has been included to allow students to see the connection between mathematics and the world around them. There is a new design to provide visual reinforcement, more preliminary material, new motivational scenarios, an increased orientation towards using the computer as a tool for performing mathematical analysis and increased emphasis on the nature of estimation in mathematics. Also available to accompany this text are an "Instructor's Manual" (ISBN 0-07-008903-5), a "Student Solutions Manual" (ISBN 0-07-008904-3) and a "Printed Test Bank" (ISBN 0-07-008905-1). Buy Applied Mathematics for Business, Economics and the Social Sciences book by Frank S. Budnick from Australia's Online Bookstore, Boomerang Books. You might also like... Maths for Economics provides a solid foundation in mathematical principles and methods used in economics, beginning by revisiting basic skills in arithmetic, algebra and equation solving and slowly building to more advanced topics, using a carefully calculated learning gradient. "The mathematical concepts of distance, rate, and time are explored as students partake in their school's annual field day. Readers also learn about proportions, ratios, and cross products. This book includes a discovery activity, connections to science and engineering, and mathematical vocabulary introduction"-- Suitable for students and researchers in political science and sociology, this book begins with the fundamental building blocks of mathematics and basic algebra, then goes on to cover essential subjects such as calculus in one and more than one variable, including optimization, constrained optimization, and implicit functions. Presents a study of the theory of totally nonnegative matrices, defined by the nonnegativity of subdeterminants. This title explores methodological background, historical highlights of key ideas, and specialized topics
Often, trigonometry students leave class believing that they understand a concept but are unable to apply that understanding when they get home and attempt their homework problems. This mainstream yet innovative text is written by an experienced professor who has identified this gap as one of the biggest challenges that trigonometry professors face. She uses a clear voice that speaks directly to students- similar to how instructors communicate to them in class. Students learning from this text will overcome common barriers to learning trigonometry and will build confidence in their ability to do mathematics.
Many students acquire knowledge of a large number of theorems and methods of calculus without being able to say how they work together. This book provides those students with the coherent account that they need. A Companion to Analysis explains the problems that must be resolved in order to procure a rigorous development of the calculus and shows the student how to deal with those problems. Starting with the real line, the book moves on to finite-dimensional spaces and then to metric spaces. Readers who work through this text will be ready for courses such as measure theory, functional analysis, complex analysis, and differential geometry. Moreover, they will be well on the road that leads from mathematics student to mathematician. With this book, well-known author Thomas Körner provides able and hard-working students a great text for independent study or for an advanced undergraduate or first-level graduate course. It includes many stimulating exercises. An appendix contains a large number of accessible but non-routine problems that will help students advance their knowledge and improve their technique. Readership Advanced undergraduates, graduate students and research mathematicians interested in analysis. Reviews "This book not only provides a lot of solid information about real analysis, it also answers those questions which students want to ask but cannot figure how to formulate. To read this book is to spend time with one of the modern masters in the subject." -- Steven G. Krantz, Washington University, St. Louis "T. W. Körner's A Companion to Analysis is a welcome addition to the literature on undergraduate-level rigorous analysis. It is written with great care with regard to both mathematical correctness and pedagogical soundness. Körner shows good taste in deciding what to explain in detail and what to leave to the reader in the exercises scattered throughout the text. And the enormous collection of supplementary exercises in Appendix K, which comprises almost one-third of the whole book, is a valuable resource for both teachers and students. "One of the major assets of the book is Körner's very personal writing style. By keeping his own engagement with the material continually in view, he invites the reader to a similarly high level of involvement. And the witty and erudite asides that are sprinkled throughout the book are a real pleasure." -- Gerald Folland, University of Washington, Seattle "This is a remarkable book. It provides deep and invaluble insight into many parts of analysis, presented by an accomplished analysist. Korner covers all of the important aspects of an advanced calculus course along with a discussion of other interesting topics." -- Paul Sally, University of Chicago "The book is a very useful companion to standard analysis textbooks. It stands out in virtue of the author's style of writing, characterized by a pleasant mixture of various erudite reflections."
OxtonHouse Publishers,LLC Pathways from the Past II Pathways from the Pastis a pair of teacher's manuals and two sets of activity sheets for helping students master various topics in arithmetic, pre-algebra, and algebra. Building on ideas from their award-winning book, Math through the Ages, Drs. Berlinghoff and Gouvêa have crafted teacher-friendly tools for helping students understand basic topics that often cause trouble. Their approach implements the spirit of the NCTM document, Reasoning and Sense Making, as well as the "Standards for Mathematical Practice" of the Common Core State Standards for Mathematics (CCSSM). The second set is Pathways from the Past II: Using History to Teach Algebra This is a 72-page manual and a set of reproducible masters for 18 two-page activity sheets that you can use with your students to strengthen their ability to think clearly and reason effectively as they learn algebra. Contents First Thoughts for Teachers 1. Writing Algebra: Using Algebraic Symbols Sheet 1-1: Symbols of Arithmetic Sheet 1-2: Algebra in Italy, 1200 - 1550 Sheet 1-3: Germany and France, 1450 - 1600 Sheet 1-4: Letters for Numbers 2. Linear Thinking: Ratio, Proportion, and Slope Sheet 2-1: The Rule of Three Direct Sheet 2-2: The Rule of Three Inverse A section for each Activity Sheet starts with its main mathematical ideas and its pedagogical purpose, followed by a detailed solution for each question. The Content The four chapters in this set progress from deciphering basic algebraic symbols to tracing how the pursuit of solutions to cubic equations led to major changes in the way we think about numbers. There's something for every level of algebra instruction, from the basic ideas of proportion and linearity to the equation-solving that led to the complex number system. Connections to European and early American history provide a fertile ground for interdisciplinary student activities.
Follow Us CK-12 Foundation Blog Algebra Help: 5 Websites to Check Out Algebra is not an easy subject for everyone. It's important to know where you can go for help when you need a reminder of an algebraic equation or want to see a step-by-step solution to a problem. Below you will find a list of websites that will provide algebra help when you get stuck and provide additional resources to learn and get ahead in your algebra class. Algebra Help The Algebra Help website provides lessons, calculators and worksheets to help both teachers and students with algebra homework and classwork. Try the algebra calculator which will show you step by step instructions on how to solve a problem. CK-12 Foundation This is a great all-in-one location for all algebra help. Find entire textbooks free for download, practice problems, quizzes, simulations, real world applications and more for algebra. You can even form a study group with other algebra students to work through homework problems and get answers about lessons in real time. Purplemath If you're looking for detailed lessons on algebra topics like graphing absolute value or inverse functions, Purplemath has them. Browse around to find just the right algebra help topic you're looking for and start a lesson with the click of a button. For extra help, visit their algebra help forum. Tutor Chat LiveIf you're in need of quick algebra help or just need clarification on a topic, you might consider an online algebra tutor. If a tutor is online, the website will match you with an algebra tutor immediately. Algebra Cheat Sheet While this link might not be a full website, it does have a listing of just about any algebra formula you might need. Scroll to the bottom of the page for common algebraic errors to avoid. Do you have an algebra help site that you love, but it wasn't listed? Share it below!
Shipping prices may be approximate. Please verify cost before checkout. About the book: Using high-quality, real-world case studies and examples, this introduction to mathematical statistics shows how to use statistical methods and when to use them. This book can be used as a brief introduction to design of experiments. This successful, calculus-based book of probability and statistics, was one of the first to make real-world applications an integral part of motivating discussion. The number of problem sets has increased in all sections. Some sections include almost 50% new problems, while the most popular case studies remain. For anyone needing to develop proficiency with Mathematical Statistics Gibson via United States Hardcover, ISBN 0139223037 Publisher: Prentice Hall, 2000 Usually ships in 1-2 business days Used books: 1 - 25 of 94 # Bookseller Notes Price 1. A&B Media Services via United States Softcover, ISBN 0139223037 Publisher: Pearson Education (US), 2000 Used - Acceptable. A used book that may have some cosmetic wear (i.e. shelf-wear, slightly torn or missing dust jacket, dented corner.) All text in great shape! Hardcover, ISBN 0139223037 Publisher: Prentice Hall, 2000 Used - Acceptable, Usually ships in 1-2 business days, Textbook may contain underlining, highlighting or writing. Infotrac or untested CD may not be included.
Mathematical skills and concepts lie at the heart of chemistry, yet they are an aspect of the subject that students fear the most. Maths for Chemistry recognizes the reality of chemical education today, and the challenges faced by many students in equipping themselves with the maths skills necessary to gain a full understanding of chemistry. Working from basic yet essential principles, the book builds the student's confidence by leading them through the subject in a steady, progressive way from basic algebra to quantum mathematics. Opening with an introduction to the 'language' of maths and fundamental rules of algebra, the book goes on to cover powers, indices, logs and exponential functions, graphical functions, and trigonometry, before leading the student through both differentiation and integration and on to quantum mathematics. With its modular structure, the book presents material in short, manageable sections to keep the content as accessible and readily digestible as possible. Maths for Chemistry is the perfect introduction to the essential mathematical concepts which all chemistry students should master. Paul Monk, Team Vicar, Medlock Head Parish, Oldham, and formerly Senior Lecturer in Physical Chemistry: School of Biology, Chemistry and Health Science, Manchester Metropolitan University, and Lindsey J. Munro, School of Biology, Chemistry and Health Science, Manchester Metropolitan University "The new edition of Maths for Chemistry extends, expands and improves on the excellent first edition. Changes include new chapters covering more introductory material as well as more advanced material, making this the most comprehensive mathematics text book specifically written for chemists that I have seen. REviews, The Higher Education Academy UK Physical Sciences Centre" "The book is well written throughout and has an admirable step-by-step approach to teaching" - Chemistry World "Maths for Chemistry is the best mathematical toolkit currently available to chemists." - Matthew Ryder, student, Heriot-Watt University
Do the Math: Secrets, Lies, and Algebra About the Book 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.
Algebra and Trigonometry: A Graphing Approach Part of the market-leading Graphing Approach series by Larson, Hostetler, and Edwards, Algebra and Trigonometry: A Graphing Approach, 5/e, is an ...Show synopsisPart of the market-leading Graphing Approach series by Larson, Hostetler, and Edwards, Algebra and Trigonometry: A Graphing Approach, 5/e, is an ideal student and instructor resource for courses that require the use of a graphing calculator. The quality and quantity of the exercises, combined with interesting applications and innovative resources, make teaching easier and help students succeed. Continuing the series' emphasis on student support, the Fifth Edition introduces Prerequisite Skills Review. For selected examples throughout the text, the Prerequisite Skills Review directs students to previous sections in the text to review concepts and skills needed to master the material at hand. In addition, prerequisite skills review exercises in Eduspace (see below for description) are referenced in every exercise set. The Larson team achieves accessibility through careful writing and design, including examples with detailed solutions that begin and end on the same page, which maximizes the readability of the text. Similarly, side-by-side solutions show algebraic, graphical, and numerical representations of the mathematics and support a variety of learning styles
Shipping prices may be approximate. Please verify cost before checkout. About the book: This is a 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 4Exotiquewon via United States Softcover, ISBN 0072349832 Publisher: Mcgraw-Hill College, 2000 Usually ships in 1-2 business days, front cover has a crease shows shelf wear ships same or next day Softcover, ISBN 0072349832 Publisher: Mcgraw-Hill College, 2000 Mcgraw-Hill College. Used - Good. GOOD with average wear to cover and pages. We offer a no-hassle guarantee on all our items. Orders generally ship by the next business day. Default Text Softcover, ISBN 0072349832 Publisher: Mcgraw-Hill College2349832 Publisher: Mcgraw-Hill College, 2000 Used - Acceptable, Usually ships in 1-2 business days, Used may contain ex-library marking, notes or highlighting, may no longer have it a dust jacket Softcover, ISBN 0072349832 Publisher: Mcgraw-Hill College72349832 Publisher: Mcgraw-Hill College, 2000 Used - Very Good. Appearance of only slight previous use. Cover and binding show a little wear. All pages are undamaged with potentially only a few, small markings.
Elementary Algebra : Concepts and Application - 7th edition Summary: The goal of Elementary Algebra: Concepts and Applications, 7e is to help today's students learn and retain mathematical concepts by preparing them for the transition from ''skills-oriented'' elementary algebra courses to more ''concept-oriented'' college-level mathematics courses, as well as to make the transition from ''skill'' to ''application.'' This edition continues to bring your students a best-selling text that incorporates the five-step problem-solving process...show more ...show less Exponents and Their Properties Polynomials Addition and Subtraction of Polynomials Multiplication of Polynomials Special Products Polynomials in Several Variables Division of Polynomials Negative Exponents and Scientific Notation Systems of Equations and Graphing Systems of Equations and Substitution Systems of Equations and Elimination More Applications Using Systems Linear Inequalities in Two Variables Systems of Linear Inequalities Direct and Inverse Variation
048622256X 9780486222561 Puzzles in Math and Logic:100 original problems in math and logic, featuring permutations, combinations, properties of numbers, algebra, solid and plane geometry, logic, and probability. Although only high school math is needed to solve these problems, even accomplished mathematicians are likely to find some surprises here. Complete and ingenious solutions provided for all problems. 31 drawings. Back to top Rent Puzzles in Math and Logic 1st edition today, or search our site for Aaron J. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Dover Publications, Incorporated.
Introductory Algebra (Looseleaf) - 11th edition Summary: KEY BENEFIT: The supporting students with qualit...show morey applications, exercises, and new review and study materials to help students apply and retain their knowledge. KEY TOPICS: Prealgebra Review, Introduction to Real Numbers and Algebraic Expressions, Solving Equations and Inequalities, Graphs of Linear Equations, Polynomials: Operations, Polynomials: Factoring, Rational Expressions and Equations, Systems of Equations, Radical Expressions and Equations, Quadratic Equations, MARKET:4441-5-0 $21.95 +$3.99 s/h VeryGood Bookbyte-OR Salem, OR Has minor wear and/or markings. SKU:9780321654441-3-0 $25.92 +$3.99 s/h New BookCellar-NH Nashua, NH 0321654447
More About This Textbook Overview Students and professors of an undergraduate course in differential geometry will appreciate the clear exposition and comprehensive exercises in this book that focuses on the geometric properties of curves and surfaces, one- and two-dimensional objects in Euclidean space. The problems generally relate to questions of local properties (the properties observed at a point on the curve or surface) or global properties (the properties of the object as a whole). Some of the more interesting theorems explore relationships between local and global properties. A special feature is the availability of accompanying online interactive java applets coordinated with each section. The applets allow students to investigate and manipulate curves and surfaces to develop intuition and to help analyze geometric phenomena. What People Are SayingEditorial ReviewsProduct Details Related Subjects Meet the Author Thomas F. Banchoff is a geometer and has been a professor at Brown University since 1967. Banchoff was president of the MAA from 1999-2000. He is published widely and known to a broad audience as editor and commentator on Abbotts Flatland. He has been the recipient of such awards as the MAA National Award for Distinguished College or University Teaching of Mathematics and most recently the 2007 Teaching with Technology Award. Stephen Lovett is an associate professor of mathematics at Wheaton College in Illinois. Lovett has also taught at Eastern Nazarene College and has taught introductory courses on differential geometry for many years. Lovett has traveled extensively and has given many talks over the past several years on differential and algebraic geometry, as well as cryptography
popular textbook/workbook meets the needs of students who will be directly or indirectly involved in the activities of merchandising and buying at the retail level. Tepper continues to explain the essential concepts, practices, procedures, calculations, and interpretations of figures that relate to the many factors that produce profitBette Tepper is a founding faculty member of the Fashion Management Department at the Fashion Institute of Technology (FIT) where she taught for more than 30 years, and served as Assistant Chairperson of Marketing Fashion and Related Industries Department. Prior to FIT, Tepper taught retail buying at the Laboratory Institute of Merchandising (now LIM College). She is a former fashion apparel buyer for Bloomingdale's and continually conducted seminars for major fashion apparel manufacturers showing their relevancy to the retailer's basic math fundamentals. Tepper was the primary contributor to a U.S. government grant to FIT and coauthored a specialized instruction pamphlet issued by the U.S. Office of Education designed to develop or enhance for Distributive Education competencies associated with the marketing of apparel, with special emphasis given to fashion buying. In retirement as an Associate Professor, she continued to teach math courses. Most Helpful Customer Reviews This covers all the basic concepts of mathematics for retailing, with exercises/problems after each section to reinforce your learning. I am a junior Merchandise Planner and have found this a great additional resource to on-the-job training. Fantastic textbook for anyone new to sales analysis and forecasting.
Lowell, MA ACTAt the beginning of my calculus courses, I told my students that the new calculus concepts they would be learning were not the hard part of the course; the hard part was that calculus uses every math concept that they were supposed to have learned before calculus. So I strongly suggested that th
self-contained text, suitable for advanced undergraduates, provides an extensive introduction to mathematical analysis, from the fundamentals to more advanced material. It begins with the properties of the real numbers and continues with a rigorous treatment of sequences, series, metric spaces, and calculus in one variable. Further subjects include Lebesgue measure and integration on the line, Fourier analysis, and differential equations. In addition to this core material, the book includes a number of interesting applications of the subject matter to areas both within and outside the field of mathematics. The aim throughout is to strike a balance between being too austere or too sketchy, and being so detailed as to obscure the essential ideas. A large number of examples and 500 exercises allow the reader to test understanding, practise mathematical exposition and provide a window into further topics. less

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