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A self-contained introduction to the fundamentals of mathematical analysis Mathematical Analysis: A Concise Introduction presents the foundations of analysis and illustrates its role in mathematics. By focusing on the essentials, reinforcing learning through exercises, and featuring a unique "learn by doing" approach, the book develops the reader's... more... This new-in-paperback edition provides a comprehensive course in Modern Analysis. The first 10 chapters discuss theoretical methods in Measure Theory and Functional Analysis, and contain over 120 end of chapter exercises. The final two chapters apply theory to applications in Probability Theory and Partial Differential Equations. - ;This new-in-paperback... more... An accessible guide to developing intuition and skills for solving mathematical problems in the physical sciences and engineering Equations play a central role in problem solving across various fields of study. Understanding what an equation means is an essential step toward forming an effective strategy to solve it, and it also lays the foundation... more... Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the tactics involved in solving mathematical problems at the Mathematical Olympiad level. With numerous exercises and assuming only basic mathematics, this text is ideal for students of 14 years and above in pure mathematics. more... The chapters in this volume, written by international experts from different fields of mathematics, are devoted to honoring George Isac, a renowned mathematician. These contributions focus on recent developments in complementarity theory, variational principles, stability theory of functional equations, nonsmooth optimization, and several other important... more... Modern Real and Complex Analysis Thorough, well-written, and encyclopedic in its coverage, this text offers a lucid presentation of all the topics essential to graduate study in analysis. While maintaining the strictest standards of rigor, Professor Gelbaum's approach is designed to appeal to intuition whenever possible. Modern Real and Complex Analysis... more...
Find a LincolnwoodConversely, having a weak background in precalculus makes the study of calculus more difficult since the student will feel he has an overwhelming amount of things to learn simultaneously. Precalculus involves memorization. For many students, it is the first math class they have taken where memorization plays such a profound role. ...Many complicated geometries and initial/boundary conditions for differential equations render them impossible to be solved in closed form, and I often (like in my thesis) resort to designing a numerical algorithm for solving them on a computer. I have worked with constant coefficients equations ...
״Created by a high school math teacher with over 25 years of experience in the classroom. Develop your algebraic equation... see more ״Created by a high school math teacher with over 25 years of experience in the classroom. Develop your algebraic equation solving skills through playing a Bingo game. This game will take you step by step through the process of learning how to solve the most basic equation up through multiple-step equations by way of 13 different levels and it will allow you to choose from a variety of Bingo games from straight-line Bingo through multiple patterns of Bingo all the way up to Black-out Bingo if you choose. The game will keep track of your time and points to be able to compete with others as well as yourself.״״This game is great for Pre-Algebra and Algebra I students who are just learning how to solve equations as well as for Algebra II and even Precalculus students needing to review their equation solving skills. It can also be used an Algebra tutorial for anyone preparing for a standardized test such as the S.A.T. or A.C.T.״This app costs $.99 ״Algebra Kaiser (Algebra Emperor) is a game that can vastly enhance both your basic algebraic skills and ability to... see more ״Algebra Kaiser (Algebra Emperor) is a game that can vastly enhance both your basic algebraic skills and ability to concentrate. Practice your ability to handle variables, expressions, terms and simple equations. 17 different types of dynamic exercises ranging from simplifying terms to exponents, linear equations and their graphs.All types of exercises can be combined, including: - Adding and Multiplying of Variables- Common Mistakes- Expansion of Single Brackets- Common Factors- Exponents- Order of Operations- Binomial Theorem- Simple Linear Equations- Graphs of Linear Equations- Geometry: Estimating Angles- Geometry: Sum of the Angles of a TriangleChoose between two game modes: Multiple Choice and Classic. In Classic Mode you are shown an exercise and an answer that might be correct or incorrect. Tap the answer if you think it's correct or strike through, if you believe it's wrong. Time is limited and each mistake you make shortens your life span... With only a few minutes a day your computing power and ability to focus can be vastly enhanced.״This app costs $1.99.'This app costs $2.99 ״Have you forgotten most of your algebra? Algebra Touch will refresh your skills using touch-based techniques built from the... see more ״ Drag Substitution.״This app costs $1.99 'For all the fans of Alice in Wonderland, discover this amazing graphic novel adaptation of the original story by Lewis... see more 'For all the fans of Alice in Wonderland, discover this amazing graphic novel adaptation of the original story by Lewis Carroll. Alice in Wonderland for iPad is a remastered version of the graphic novel by Campfire, specially animated for the iPad, featuring more than 70 pages of gorgeous illustrations. Now you can dive like never before in a comic on your iPad. Try it and be amazed !With Ave!Comics, reading on the go is '100% pleasure' at any time, anywhere.Alice in Wonderlandby Lewis Carroll, Lewis Helfand, Rajesh Nagulakonda & KG Prasanth (C) Campfire74 pages \| 340 panels \| colorsTEXT IN ENGLISH ONLY.Features include: - animated digital version of the complete comic book - A revolutionary reading experience using kinematics that create an animated path through the stories. - Exploration mode to dive into each panel, zoom and pan.- Easy and intuitive navigation between pages and panels. - Full support for multi-touch, swipe, landscape mode. - page mode to display the original layout of the comic book'This app costs $3.99 ״Alice's Adventures in Wonderland (commonly shortened to Alice in Wonderland) is an 1865 novel written by English author... see more ״Alice's Adventures in Wonderland (commonly shortened to Alice in Wonderland) is an 1865 novel written by English author Charles Lutwidge Dodgson under the pseudonym Lewis Carroll. It tells the story of a girl named Alice who falls down a rabbit hole into a fantasy world populated by peculiar and anthropomorphic creatures. The tale plays with logic in ways that have given the story lasting popularity with adults as well as children. It is considered to be one of the best examples of the literary nonsense genre, and its narrative course and structure have been enormously influential, especially in the fantasy genre.״This is a free app 'Holidays are over, it is time for school!With this game, learning math will be fun and effective.Play and learn math with... see more 'Holidays are over, it is time for school!With this game, learning math will be fun and effective.Play and learn math with your cosmic friend! Dress him as you like! Use funny items, clothes, gadgets and send a photo of your cosmic friend by email or share it by FaceBook!You can practice: addition, subtraction, multiplication, division.You will get 1 point for every correct answer. You can exchange points for items like extra clothes, shoes, head items and different gadgets!'This app costs $0.99 'What is All Here!:* Attendance Tracking* Take attendance by photo or name* Reminder Alarms* Export/Email Report CSVs* Import... see more 'What is All Here!:* Attendance Tracking* Take attendance by photo or name* Reminder Alarms* Export/Email Report CSVs* Import students with DropBox* Email Absent Students* Auto & Random Grouping of students* Learn Students with Memory GameApps For Teachers, By Teachers... * We are teachers, we create apps we want to use in our classrooms!Need to take Attendance? Need to email all the absent students with one click? Need to export attendance records? Need to set an alarm to remind you to take attendance? Need to quickly make randomized groups? Maybe you need to learn student names in a hurry? If you said yes to any or all of these, you're in luck. ALL HERE is an app designed for taking attendance and so much more. Import rosters from dropbox, add student photos, and make the attendance work for you, it's that easy. Tap the student's picture once to mark them present, tap twice to mark as absent, or tap three times for tardy. Better yet, if they are all here, simply tap the "All Here" button to mark them all as present. Need to email those absent students? With one click you can email all absent students, or even all students! Use the name game to learn student's names quickly, match the face with the name! Create groups of students with just a tap and email that list if you like! Attendance is something we teachers do every day, with this app you can do so much more!'This app costs $0.99
Foundations of Mathematical & Computational Economics 9780324235838 ISBN: 0324235836 Edition: 1 Pub Date: 2006 Publisher: Thomson Learning Summary: Economics doesn't have to be a mystery anymore. FOUNDATIONS OF MATHEMATICAL AND COMPUTATION ECONOMICS shows you how mathematics impacts economics and econometrics using easy-to-understand language and plenty of examples. Plus, it goes in-depth into computation and computational economics so you'll know how to handle those situations in your first economics job. Get ready for both the test and the workforce with this ...economics textbook. Dadkhah, Kamran is the author of Foundations of Mathematical & Computational Economics, published 2006 under ISBN 9780324235838 and 0324235836. Four hundred ninety nine Foundations of Mathematical & Computational Economics textbooks are available for sale on ValoreBooks.com, one hundred twenty six used from the cheapest price of $0.75, or buy new starting at $80.07 [more0324235836 Brand new book. Ship from multiple locations, including USA, UK, ASIA. 3-5 business days Express Delivery to USA/UK/Europe/Asia/Worldwide. Tracking number will be [more] 0324235836 Brand new book. Ship from multiple locations, including USA, UK, ASIA. 3-5 business days Express Delivery to USA/UK/Europe/Asia/Worldwide. Tracking number will be provided. Satisfaction guaranteed.[less]
The History of Mathematics: An Introduction This text is designed for the junior/senior mathematics major who intends to teach mathematics in high school or college. It concentrates on the ...Show synopsisThis text is designed for the junior/senior mathematics major who intends to teach mathematics in high school or college. It concentrates on the history of those topics typically covered in an undergraduate curriculum or in elementary schools or high schools. At least one year of calculus is a prerequisite for this course. This book contains enough material for a 2 semester course but it is flexible enough to be used in the more common 1 semester edition, Item may show signs of shelf wear....Good. International edition
Beginning and Intermediate Algebra / With 2 CDs - 3rd edition Summary: For college-level courses of combined introductory and intermediate algebra. Elayn Martin-Gay's success as a developmental math author and teacher starts with a strong focus on mastering the basics through well-written explanations, innovative pedagogy and a meaningful, integrated program of learning resources. The revisions provide new pedagogy and resources to build student confidence, help students develop basic skills and understand concepts, and provi...show morede the highest level of instructor and adjunct support. Martin-Gay's series is well known and widely praised for an unparalleled ability to: Relate to students through real-life applications that are interesting, relevant, and practical. Martin-Gay believes that every student can: Test better: The new Chapter Test Prep Video shows Martin-Gay working step-by-step video solutions to every problem in each Chapter Test to enhance mastery of key chapter content. Study better: New, integrated Study Skills Reminders reinforce the skills introduced in section 1.1, "Tips for Success in Mathematics" to promote an increased focus on the development of all-important study skills. Learn better: The enhanced exercise sets and new pedagogy, like the Concept Checks, mean that students have the tools they need to learn successfully. Martin-Gay believes that every student can succeed, and with each successive edition enhances her pedagogy and learning resources to provide evermore relevant and useful tools to help students and instructors achieve success. ...show less Instructor's Edition. With CD! Shows some signs of wear, and may have some markings on the inside. Find out why millions of customers rave about Better World Books. Experience the best customer care a...show morend a 100% satisfaction guarantee
Northeast COLLEGE Department of Mathematics COURSE SYLLABUS MATH 0308: Fundamentals of Math II Fall, 2010 / CRN # 53566 / Tu – Thu 5:00 – 7:00 PM / Northline Campus INSTRUCTOR: Samuel H Dominguez CONFERENCE TIMES: By appointment only CONTACT INFORMATION: sdoming1@houstonisd.org MYMATHLAB COURSE ID: Dominguez84719 Textbook: Prealgebra and Introductory Algebra, 2nd Edition. Bittinger, Marvin L. & Ellenbogen, David J. Pearson /Addison Wesley: Boston, 2008. Catalog Description: Topics include real numbers, basic geometry, polynomials, factoring, linear equations, inequalities, and rational expressions. A departmental final examination must be passed with a score of 60% or higher in order to pass the course. Prerequisites: Math 0306 or MATH 0106: Pass with "C" or better Or Suitable placement test score. Credits: 3 credit hours (3 Lecture). Course Intent: This course is intended for students who have never been exposed to algebra or who have been away from the subject for quite some time. In particular, this course is intended to prepare students for the study of Intermediate Algebra, a course that builds the foundation for the study of College Algebra. Audience: This course is intended for students who require state mandated remediation. 1 Testing policy: Four regular tests worth 100 points each; will drop the lowest test grade and use only three regular tests and a final exam (worth 200 points) to average your semester grade. Make-up policy: There will be no make-up tests in this course. Grading policy: Your final course grade is based on the following standard HCC scale. FINAL AVERAGE FINAL COURSE GRADE 90 ≤ Average ≤ 100% A 80 ≤ Average < 90% B 70 ≤ Average < 80% C 60 ≤ Average < 70% D Average < 60% or F Final Exam Grade < 60 A grade of "IP" (In Progress) will not be given. If your final grade is a "D", then you may be eligible to take the bridge course MATH 0108 instead of repeating the class. To determine eligibility, please contact the math department. A grade of "F" is given if the final average is below 60 or the final exam grade is below 60. Semester Grade = (test #1 + test # 2 + test # 3 + Final Exam + Lab Grade) / 6 Final Examination: The final examination is departmental and consists of 33 multiple-choice problems. The problems cover all the material required in the course. If you score lower than 60% on the final exam, you automatically are given a course grade of "F", as noted under the grading policy. If your score on the final exam is 60% or higher, then your grades are averaged using the formula specified under grading policy. You MUST pass the final exam in order to pass the course. Homework policy: All homework must be completed online using MYMATHLAB. The MyMathLab Course ID to be used for registration purposes is dominguez84719, and the school zip code is 77022. To register for MyMathLab and to access the homework, go to Your homework grade may be used in place of your lowest regular (not final) test. Lab Work: This is a four semester hours course. Therefore, students are expected to log in a minimum of 10 hours computer lab work online. These ten hours will count for 100 points in your semester grade calculation. Students who do not have access to a computer will have the choice to submit paper and pencil work. For this option, students need to obtain a copy of the following workbook: 2 Worksheets for Classroom or Lab Practice for Prealgebra and Introductory Algebra, Math 0308. Custom Edition for Houston Community College Northeast. Judith A. Penna, Laurie Hurley, & Barbara L. Johnson. Pearson Learning Solutions. Boston, MA. Calculators: Only scientific calculators are allowed in this course (no graphing calculators). Attendance policy: Attendance is checked during every class. The instructor may drop you for excessive absences. Tardiness policy: Chronic tardiness will be reported to the college counselors for possible intervention Withdrawal policy: If you decide to drop the class, then IT IS YOUR RESPONSIBILITY TO DROP before the final drop date. If your name is on the roll at the end of the term, you WILL receive a grade. Neither you nor your instructor will be able to perform the drop after the final drop date. Please refer to the following notice before dropping the class. NOTICE: Students who take a course three or more times will face significant tuition or fee increases at HCC and other Texas public colleges and universities. In addition, state law allows students a maximum of 6 course withdrawals during their entire college career. Students with more than 6 drops will be required to pay additional fees. Prior to course withdrawal, you must confer with your professor or counselor about your study habits, homework, test-taking skills, attendance, course participation, and tutoring or other assistance that is available. Student conduct: Students should not engage in disruptive activities while in the classroom. Any conduct that is deemed detrimental to the academic atmosphere, such as cell phone use or consistently talking during instructional delivery, will not be tolerated. Any student found guilty of such conduct will be asked to leave the classroom until further notice. Academic dishonesty: All students are required to exercise academic honesty in completion of all tests and assignments. Cheating involves deception for the purpose of violating testing rules. Students who improperly assist other students are just as guilty as students who receive assistance. A student guilty of a first offense will receive a grade of "F" on the quiz or test involved. For a second offense, the student will receive a grade of "F" for the course. The use of recording devices, including camera phones and tape recorders, is prohibited in all locations where instruction, tutoring, or testing occurs. Students with disabilities who need to use a recording device as a reasonable accommodation should contact the Disability Services Office for information. Resources and supplemental instruction: 3 Any student enrolled in Math 0308 at HCC has access to the math tutoring labs which are staffed with student assistants who can aid students with math problems and offer help with MyMathLab. In addition, free online tutoring is provided. For more information and for tutoring hours and locations, go to the math department web page at and select the tutoring link. One other resource is the student solutions manual that may be obtained from the bookstore. Students with Disabilities: Any student with a documented disability (e.g. physical, learning, psychiatric, vision, hearing, etc.) who needs to arrange reasonable accommodations must contact the Disability Support Services Office at this college at the beginning of the semester. To make an appointment, please call 713-718-7910. Professors are authorized to provide only the accommodations requested by the Disability Support Office. Course Schedule: Chapters and Sections Chapter 8 Geometry 8.1 Basic Geometric Figures 8.2 Perimeter 8.3 Area 8.4 Circles 8.5 Volume and Surface Area 8.6 Relationships Between Angle Measures 8.7 Congruent Triangles and Properties of Parallelograms 8.8 Similar Triangles Chapter 9 Introduction to Real Numbers and Algebraic Expressions 9.1 Introduction to Algebra 9.2 The Real Numbers 9.3 Addition of Real Numbers 9.4 Subtraction of Real Numbers 9.5 Multiplication of Real Numbers 9.6 Division of Real Numbers 9.7 Properties of Real Numbers 9.8 Simplifying Expressions; Order of Operations 4 Chapter 10 Solving Equations and Inequalities 10.1 Solving Equations: The Addition Principle 10.2 Solving Equations: The Multiplication Principle 10.3 Using the Principles Together 10.4 Formulas 10.5 Applications of Percent 10.6 Applications and Problem Solving 10.7 Solving Inequalities 10.8 Applications and Problem Solving with Inequalities Chapter 11 Graphs of Linear Equations 11.1 Graphs and Applications (Omit applications.) Chapter 12 Polynomials: Operations 12.1 Integers as Exponents 12.2 Exponents and Scientific Notation 12.3 Introduction to Polynomials 12.4 Addition and Subtraction of Polynomials 12.5 Multiplication of Polynomials 12.6 Special Products 12.7 Operations with Polynomials in Several Variables 12.8 Division of Polynomials (monomial divisors only) Chapter 13 Polynomials: Factoring 13.1 Introduction to Factoring 13.2 Factoring Trinomials of the Type x2 + bx +c 13.3 Factoring ax2 + bx + c, a  1: The Foil Method 5 13.4 Factoring ax2 + bx + c, a  1: The ac – Method 13.5 Factoring Trinomial Squares and Differences of Squares 13.6 Factoring: A General Strategy Chapter 14 Rational Expressions and Equations 14.1 Multiplication and Simplifying Rational Expressions 14.2 Division and Reciprocals Important Dates: Drop Deadline: TBA Final Review Session: TBA Course Objectives: Upon completion of this course, a student should be able to: 1. find the perimeter and area of rectangles, squares, parallelograms, triangles, trapezoids and circles; volume and surface area, relations between angle measures, congruent and similar triangles, and properties of parallelograms. 2. add, subtract, multiply and divide real numbers, and manipulate certain expressions. 3. solve problems using equations and inequalities. 4. solve problems using scientific notation. 5. factor polynomials using the techniques of the greatest common factor, difference of two squares, special trinomials, and grouping. 6. simplify, multiply, and divide rational expressions. 7. plot ordered pairs and graph linear equations
question book full of test-style practice questions for students studying KS3 Maths. It covers all the topics from the current National Curriculum with a range of exercises that test what you know and how well you can apply your knowledge. The questions are written in a clear, straightforward style aimed at levels 3-6 to give practice on each topic. The separate answer book is included. Matching study notes are available in CGP's Revision Guide (9781841460406).
Course Objectives: 1. To develop the student's Pre-Algebra reasoning powers in preparation for Algebra I and geometry. 2. To develop the student's ability to use Pre-Algebra to solve mathematical problems. 3. To develop student's awareness of the importance of Pre-Algebra in our society. 4. To present applications of Pre-Algebra to everyday situations and to possible future job-related situations. Course Description: A foundation - building course in the fundamentals of arithmetic and algebra. A review of the basic arithmetic functions, fractions, percentages, decimals, positive and negative numbers and simple Pre-Algebra will form the basis for algebra concepts. Pre - Algebra will prepare the student for Algebra I. Non - solid credit is given, but does not count for Math credit for the College Preparatory diploma. PLAY (Planned Learning Activity for You): To be a good basketball player you have to practice a lot. Likewise for Pre-Algebra; to be good at Pre-Algebra you need to practice. Almost everyday a PLAY will be given to you. In class we will take notes as we learn the new lesson. Time will then be given to work on the PLAY. If you use your time efficiently in class you should be able to finish a lot of it in class. Just make sure the PLAY is nearly completed for class the next school day. During class the next school day some time will be given to answer a few questions before we grade it turn it in at the end of class. If it is not completed by the end of class half the points will be subtracted from the total you got correct. All work is to be shown for each question. You will get a zero if PLAY is not turned in by test time, if you are cheating, or if you are helping someone cheat. Materials Needed: 1. Binder to file Pre-Algebra assignments in. 2. 3-holed lined paper & graph paper. 3. Straightedge, protractor, and compass. NO CALCULATORS!! - unless specified to use them for appropriate lessons. Explanation Time: 1. Listen attentively to the explanations and discussion. 2. Be actively reasoning through ideas and try to solve the examples. 3. Be active in our open discussion during explanation time. 4. There should be no talking to others during this time, except when you have the floor when asking a question, or answering a question. Seat Work: After explanation time you should: 1. Work independently - you may ask myself or friends for occasional help. 2. Raise hand & be acknowledged before activating your talking muscles. 3. When you finished all your Pre-Algebra homework - do other homework. Bonus: 1 2. If the "homework update" is signed and returned to myself the next school day after it was issued then 2 bonus points will be added to the chapter homework. If it's returned within a few days of being issued then the student will receive one bonus point. If it is not returned in a timely manner 2 points will be removed from their homework score. 3. See "classroom emergencies" statement 5 for other bonus points available. Tests: During any quiz or test please: 1. Make sure all assignments are turned in before taking the test, or no credit will be given. 2. Make sure you have turned in your notebook at beginning of class for grading. If turned in afterwards, half credit will be given. 3. Close textbook and notebooks and place all materials (except for pencil, pen and eraser) in a pile on the middle of the desk to separate yourself from your partner. 4. When you have finished the test, do not leave the room, move back to a study booth and quietly study there. (If you leave 2 emergencies will be recorded). 5. Remember to pick up any graded assignments from your box. Contents remaining in boxes will be discarded at the end of the day each Wednesday. 6. Tests must be taken during the time that they are scheduled. 20 % per school day will be deducted from a test taken late except: 1) With a parents note excusing you from school for a serious illness or 2) in the case of an official school field trip (you must pre- arrange a test taking time with me). 7. If you are absent on the day before the test you are still expected to write the test as scheduled. Quizzes: 1. Quizzes will be given at the very first part of class. 2. Quizzes can be retaken within one school day. Best score will be recorded. Classroom "Emergencies": If you need to leave the room for any reason, you may simply leave according to these instructions: 1. Only one person at a time may leave the room. 2. You must return in a reasonable length of time (about 5 min.), or an absence will be recorded. 3. Fill out the "Pass or Consultation Slip" then leave quietly without disrupting the class in any way. 4. You have five emergencies per half semester. A "referral" will be issued when this limit is exceeded. 5. If you keep all five, or some emergencies per half semester, then one bonus point per emergency not used will be given to test points. Classroom Integrity and Honor: The best rule of thumb is: "What would Jesus do?": 1. Cheating of any type is totally unacceptable. (Remember a "referral" for academic dishonesty is serious and has considerable consequences. Also a zero will be given for that assignment/test). 2. Copying another person's homework assignment is cheating and will be treated as such. 3. Purposely giving an assignment for someone to copy is also cheating and will be treated as such. Additional Help: Please seek my assistance during class or lab by raising your hand and asking your question. Also: 1. Have a study partner from the class (Your lab partner may be this person). Ask this person for help first before asking me. This way I will be available to help more students. 2. I am available after school from 3:15 till 3.45 pm. On some days I may be available until 4:15 pm. 3. If possible try to come after school but if you are unable to obtain help after school, please call me at home (473 -2606) between 7:30 pm and 8:00 pm, Sunday through Thursday evenings. 4. Because I am available to help you at these times, do not come to me just before the assignment is due and say "I don't understand so this is not complete". I usually do not have time just before class to help with assignments. 5. Seat partners should look out for each other. Please help each other before asking teacher for help. Also relay any announcements to your seat partner, that your seat partner may have missed. Suggestions for success: 1. Believe that Pre-Algebra will be understandable to you this year no mater what others may have told you about mathematics. 2. Ask for God's help in understanding; He made your mind. 3. Listen to the class discussion, even when you didn't ask the question. 4. Write down the examples in a special section of your notebook where you keep only Pre-Algebra notes. 5. Work only on Pre-Algebra in Pre-Algebra class. 6. Memorize any new definitions, postulates, theorems, or corollaries the day they are introduced. 7. If you really don't understand, ask as soon as it is your turn. 8. Start on your assignment as soon as supervised study time begins, and complete as much of it as you can in class. 9. Remember that most of the responsibility for learning something new is yours. 10. Always complete the PLAY before the next Pre-Algebra class. Getting behind in math is deadly. 11. Be seated at your assigned desk with needed materials before the beginning bell. 12. Read the explanations in your book and think about the examples. 13. Learn to do things yourself instead of depending on others. 14. Your instructor is happy to help you during supervised study time in class, after school, or by phone. Phone 473 - 2606 Sunday through Thursday between 7:30 and 8:00 p.m. 15. It is more difficult to do math with too little sleep. Homework Policy Why I Assign Homework: Homework plays an important role in education. It reinforces the material learned in class, prepares students for upcoming classes, teaches them to work independently, promotes students responsibility, and aids in evaluation of performance. It has been demonstrated in scientific studies that consistent homework increases learning. When Homework Will Be Assigned: Homework will be assigned almost every school day. A good portion of it can be completed during class time or study hall. If time is used wisely it should take students no more than one half additional hour. Most homework will reinforce material that has already been taught in class. Student's Homework Responsibilities: - All assignments are to be completed by the next school day and are to be handed in at the end of that class period, (except for long term projects). - Homework must be turned in on time. Half credit will be given for late work. (I want to emphasize personal responsibility. You choose the grade you receive by turning your homework in on time or late). - Students are responsible for making up homework missed due to excused absence. No credit will be given for work turned in after the beginning of the test for that chapter. - Work must be completed neatly and in the proper format. Parent's Homework Responsibilities: Parents are the key to making homework a positive, productive experience for their children. Therefore, I ask that you make his or her homework a top priority. A definite, scheduled time each evening is needed for its completion. This should be a quiet time when television and music are off. (Research has shown that students learn more quickly and retain the information longer when they study in a quiet atmosphere). Mini-breaks are helpful during this time (e.g. walk or jog for a couple of minutes). Assignment books are very helpful to assist him/her in learning to plan and schedule. If you discover your child is spending an unusual amount of time studying for this class please get involved in an effort to make the student's efforts more efficient. One thing to look out for is procrastination. I try to set a reasonable pace throughout the chapter. However, if your child puts off doing assignments till the night before the test he/she will have an unusual amount of homework and will feel overwhelmed. If you have any questions or concerns about homework or any aspect of the class please contact me at 471 -6179 or e-mail satkins@andrews.edu "Homework updates" will be given near the middle of each chapter the day we do the mid-chapter quiz. "Homeletters" will be issued two days after the chapter test. Both these items are to be brought home by the student for you to analyze and sign There are similar procedures for "Homework updates", except bonus points will be given/removed to the homework score. Dates of when homework updates and homeletters will be issued are included in the class schedule so you know when to expect them. Teacher's Homework Responsibilities: My grader or myself will check all homework and keep a record of assignments completed & of those which were not. I will help students who had difficulty with the homework assignment. Because I strongly believe in the value that positive support plays in motivating students to develop good study habits, I will give students praise and other incentives when they complete homework. "Homework updates" will be given near the middle of the chapter the day we do the mid-chapter quiz. "Homeletters" will be issued two days after the chapter test. Both these items are to be brought home by the student for you to analyze and sign. Additional Comments: Pre-Algebra is easy if you learn it step by step. Understanding the first part makes it easier to understand the second part. If you find that a particular problem is troublesome, get help at once because the problem will not go away unless you solve it. ----------------------- We may only do parts of chapter 12 to finish the year -------------------- (Teacher note: See priority of lessons in chapter 12 (listed beside lesson number) to know what lesson(s) to do).
Math Study Skills Workbook - 3rd edition Summary: This best-selling workbook helps traditionally unsuccessful students learn to effectively study mathematics. Typically used for a Math Study Skills course Freshman Seminar or as a supplement to class lectures the Nolting workbook helps students identify their strengths weaknesses and personal learning styles in math. Nolting offers proven study tips test-taking strategies a homework system and recommendations for reducing anxiety and improving grades. New features to the Third Editio...show moren include the Learning Modality Inventory and Dan's Take. Since students become more motivated when they understand how math directly relates to their lives a new appendix chart shows how mathematical success can impact their career opportunities and earning potential. Material has also been streamlined and updated to make it more relevant to today's students. New!Learning Modality Inventoryhelps students determine their own learning style. With this knowledge students can identify and implement study habits that are best suited to them. New!Dan's Takeappears in every chapter and presents a student's perspective of different challenges that are encountered in the classroom. These segments are designed to inspire students and decrease their anxiety about math. New!Coverage of cooperative learning has been added to assist instructors who favor group work. New!Another exciting new student support resourceBecoming an Online Learneroffers essential guidance for using technology to study and learn. LikewiseOvercoming Technophobiaa boxed feature from the best-selling Houghton Mifflin series Becoming a Master Student offers helpful suggestions to students intimidated by technology. Updated!The chapter opening sections have been redesigned to include captivating photographs and more clearly defined objectives which help students navigate each chapter more smoothly. ...show less 3. How to Reduce Math Test Anxiety Understanding Math Anxiety How to Recognize Test Anxiety The Causes of Test Anxiety The Different Types of Test Anxiety How to Reduce Test Anxiety 4. How to Improve Your Listening and Note Taking Skills How to Become an Effective Listener How to Become a Good Note Taker The Seven Steps to Math Note Taking How to Rework Your Notes 5. How to Improve Your Reading Homework and Study Techniques How to Read a Math Textbook How to Do Your Homework How to Solve Word Problems How to Work With a Study Buddy The Benefits of Study Breaks Using Online/Computer Resources to Support Learning 6. How to Remember What You Have Learned How You Learn How to Use Learning Styles to Improve Memory How to Use Memory Techniques How to Develop Practice Tests How to Use Number Sense 7. How to Improve Your Math Test-Taking Skills Why Attending Class and Doing Your Homework May Not Be Enough to Pass Pre-Test Check-Off List The Ten Steps to Better Test Taking The Six Types of Test-Taking Errors How to Prepare for the Final Exam Appendix. Math Autobiography Best Jobs Requiring a Bachelor's Degree Bibliography Author Biographies
Download free ebooks at bookboon.com Mathematics for Computer Scientists 5 Introduction Introduction The aim of this book is to present some the basic mathematics that is needed by computer scientists. The reader is not expected to be a mathematician and we hope will find what follows useful. Just a word of warning. Unless you are one of the irritating minority math- ematics is hard. You cannot just read a mathematics book like a novel. The combination of the compression made by the symbols used and the precision of the argument makes this impossible. It takes time and effort to decipher the mathematics and understand the meaning. It is a little like programming, it takes time to understand a lot of code and you never understand how to write code by just reading a manual - you have to do it! Mathematics is exactly the same, you need to do it. 5 Download free ebooks at bookboon.com Mathematics for Computer Scientists 6 Numbers Chapter 1 Numbers Defendit numerus: There is safety in numbers We begin by talking about numbers. This may seen rather elementary but is does set the scene and introduce a lot of notation. In addition much of what follows is important in computing. 1.0.1 Integers We begin by assuming you are familiar with the integers 1,2,3,4,. . .,101,102, . . . , n, . . . , 232582657 − 1, . . ., sometime called the whole numbers. These are just the numbers we use for count- ing. To these integers we add the zero, 0, defined as 0 + any integer n = 0 + n = n + 0 = n Once we have the integers and zero mathematicians create negative integers by defining (−n) as: the number which when added to n gives zero, so n + (−n) = (−n) + n = 0. Eventually we get fed up with writing n+(−n) = 0 and write this as n−n = 0. We have now got the positive and negative integers {. . . , −3, −2, −1, 0, 1, 2, 3, 4, . . .} You are probably used to arithmetic with integers which follows simple rules. To be on the safe side we itemize them, so for integers a and b 1. a + b = b + a 2. a × b = b × a or ab = ba 3. −a × b = −ab 7 Download free ebooks at bookboon.com Mathematics for Computer Scientists 7 Numbers 8 CHAPTER 1. NUMBERS 4. (−a) × (−b) = ab 5. To save space we write ak as a shorthand for a multiplied by itself k times. So 34 = 3 × 3 × 3 × 3 and 210 = 1024. Note an × am = an+m 6. Do note that n0 =1. Factors and Primes Many integers are products of smaller integers, for example 2 × 3 × 7 = 42. Here 2, 3 and 7 are called the factors of 42 and the splitting of 42 into the individual components is known as factorization. This can be a difficult exercise for large integers, indeed it is so difficult that it is the basis of some methods in cryptography. Of course not all integers have factors and those that do not, such as 3, 5, 7, 11, 13, . . . , 2216091 − 1, . . . are known as primes. Primes have long fascinated mathematicians and others see and there is a considerable industry looking for primes and fast ways of factorizing integers. To get much further we need to consider division, which for integers can be tricky since we may have a result which is not an integer. Division may give rise to a remainder, for example 9 = 2 × 4 + 1. and so if we try to divide 9 by 4 we have a remainder of 1 . In general for any integers a and b b = k × a + r where r is the remainder. If r is zero then we say a divides b written a \| b. A single vertical bar is used to denote divisibility. For example 2 \| 128, 7 \| 49 but 3 does not divide 4, symbolically 3 4. Aside To find the factors of an integer we can just attempt division by primes i.e. 2, 3, 5, 7, 11, 19, . . . . If it is divisible by k then k is a factor and we try again. When we cannot divide by k we take the next prime and continue until we are left with a prime. So for example: 1. 2394/2=1197 can't divide by 2 again so try 3 Download free ebooks at bookboon.com Mathematics for Computer Scientists 8 Numbers 9 2. 1197/3=399 3. 399/3 = 133 can't divide by 3 again so try 7 ( not divisible by 5) 4. 133/7 = 19 which is prime so 2394 =2 × 3 × 3 × 7 × 19 Modular arithmetic The mod operator you meet in computer languages simply gives the remainder after division. For example, 1. 25 mod 4 = 1 because 25 ÷ 4 = 6 remainder 1. 2. 19 mod 5 = 4 since 19 = 3 × 5 + 4 . 3. 24 mod 5 = 4. 4. 99 mod 11 = 0. There are some complications when negative numbers are used, but we will ignore them. We also point out that you will often see these results written in a slightly different way i.e. 24 = 4 mod 5 or 21 = 0 mod 7. which just means 24 mod 5 = 4 and 27 mod 7 = 0 Modular arithmetic is sometimes called clock arithmetic. Suppose we take a 24 hour clock so 9 in the morning is 09.00 and 9 in the evening is 21.00. If I start a journey at 07.00 and it takes 25 hours then I will arrive at 08.00. We can think of this as 7+25 = 32 and 32 mod 24 = 8. All we are doing is starting at 7 and going around the (25 hour) clock face until we get to 8. I have always thought this is a complex example so take a simpler version. Four people sit around a table and we label their positions 1 to 4. We have a pointer point to position 1 which we spin. Suppose it spins 11 and three quarters or 47 quarters. The it is pointing at 47 mod 4 or 3. 1 2 3 4 6 K 9 =399 = 133 can't divide by 3 again so try 7 ( not divisible by 5) = 19 which is prime so 2394 =2 × 3 × 3 × 7 × 19 rithmetic erator you meet in computer languages simply gives the remainder n. For example, d 4 = 1 because 25 ÷ 4 = 6 remainder 1. d 5 = 4 since 19 = 3 × 5 + 4 . d 5 = 4. d 11 = 0. me complications when negative numbers are used, but we will ignore so point out that you will often see these results written in a slightly i.e. 24 = 4 mod 5 or 21 = 0 mod 7. which just means 24 mod 5 = od 7 = 0 arithmetic is sometimes called clock arithmetic. Suppose we take a k so 9 in the morning is 09.00 and 9 in the evening is 21.00. If I start 07.00 and it takes 25 hours then I will arrive at 08.00. We can think +25 = 32 and 32 mod 24 = 8. All we are doing is starting at 7 and d the (25 hour) clock face until we get to 8. I have always thought this example so take a simpler version. ple sit around a table and we label their positions 1 to 4. We have a t to position 1 which we spin. Suppose it spins 11 and three quarters rs. The it is pointing at 47 mod 4 or 3. 1 2 3 4 6 K Download free ebooks at bookboon.com Pleaseclicktheadvert Mathematics for Computer Scientists 9 Numbers 10 CHAPTER 1. NUMBERS The Euclidean algorithm Algorithms which are schemes for computing and we cannot resist putting one in at this point. The Euclidean algorithm for finding the gcd is one of the oldest algorithms known, it appeared in Euclid's Elements around 300 BC. It gives a way of finding the greatest common divisor (gcd) of two numbers. That is the largest number which will divide them both. Our aim is to find a a way of finding the greatest common divisor, gcd(a, b) of two integers a and b. Suppose a is an integer smaller than b. 1. Then to find the greatest common factor between a and b, divide b by a. If the remainder is zero, then b is a multiple of a and we are done. 2. If not, divide the divisor a by the remainder. Continue this process, dividing the last divisor by the last remainder, until the remainder is zero. The last non-zero remainder is then the greatest common factor of the integers a and b10 Numbers 11 The algorithm is illustrated by the following example. Consider 72 and 246. We have the following 4 steps: 1. 246 = 3 × 72 + 30 or 246 mod 72 = 30 2. 72 = 2 × 30 + 12 or 72 mod 30 = 12 3. 30 = 2 × 12 + 6 or 30 mod 12 = 6 4. 12 = 2 × 6 + 0 so the gcd is 6. There are several websites that offer Java applications using this algorithm, we give a Python function def gcd(a,b): """ the euclidean algorithm """ if b == 0: return a else: return gcd(b, (a%b)) Those of you who would like to see a direct application of some these ideas to computing should look at the section on random numbers 1.0.2 Rationals and Reals Of course life would be hard if we only had integers and it is a short step to the rationals or fractions. By a rational number we mean a number that can be written as P/Q where P and Q are integers. Examples are 1 2 3 4 7 11 7 6 These numbers arise in an obvious way, you can imagine a ruler divided into 'iths' and then we can measure a length in 'iths'. Mathematicians, of course, have more complicated definitions based on modular arithmetic . They would argue that for every integer n, excluding zero, there is an inverse, written 1/n which has the property that n × 1 n = 1 n × n = 1 Of course multiplying 1/n by m gives a fraction m/n. These are often called rational numbers. We can manage with the simple idea of fractions. Download free ebooks at bookboon.com Mathematics for Computer Scientists 11 Numbers 12 CHAPTER 1. NUMBERS One problem we encounter is that there are numbers which are neither integers or rationals but something else. The Greeks were surprised and confused when it was demonstrated that √ 2 could not be written exactly as a fraction. Technically there are no integer values P and Q such that P/Q = √ 2. From our point of view we will not need to delve much further into the details, especially as we can get good enough approximation using fractions. For example 22/7 is a reasonable approximation for π while 355/113 is better. You will find people refer to the real numbers, sometimes written R, by which they mean all the numbers we have discussed to date. Notation As you will have realized by now there is a good deal of notation and we list some of the symbols and functions you may meet. • If x is less than y then we write x < y. If there is a possibility that they might be equal then x ≤ y. Of course we can write these the other way around. So y > x or y ≥ x. Obviously we can also say y is greater than x or greater than or equal to x • The floor function of a real number x, denoted by x or floor(x), is a function that returns the largest integer less than or equal to x. So 2.7 = 2 and −3.6 = −4. The function floor in Java and Python performs this operation. There is an obvious(?) connection to mod since b mod a can be written b−floor(b÷a)×a. So 25 mod 4 = 25− 25/4 ×4 = 25−6×4 = 1 Download free ebooks at bookboon.com Mathematics for Computer Scientists 12 Numbers 13 • A less used function is the ceiling function, written x or ceil(x) or ceiling(x), is the function that returns the smallest integer not less than x. Hence 2.7 = 3. • The modulus of x written \| x \| is just x when x ≥ 0 and −x when x < 0. So \| 2 \|= 2 and \| −6 \|= 6. The famous result about the modulus is that for any x and y \| x + y \|≤\| x \| + \| y \| • We met ab when we discussed integers and in the same way we can have xy when x and y are not integers. We discuss this in detail when we meet the exponential function. Note however – a0 =1 for all a = 1 – 0b = 0 for all values of b including zero. 1.0.3 Number Systems We are so used to working in a decimal system we forget that it is a recent invention and was a revolutionary idea. It is time we looked carefully at how we represent numbers. We normally use the decimal system so 3459 is shorthand for 3 + 4 × 100+5+9. The position of the digit is vital as it enables us to distinguish between 30 and 3. The decimal system is a positional numeral system; it has positions for units, tens, hundreds and so on. The position of each digit implies the multiplier (a power of ten) to be used with that digit and each position has a value ten times that of the position to its right. Notice we may save space by writing 1000 as 103 the 3 denoting the number of zeros. So 100000 = 105 . If the superscript is negative then we mean a fraction e.g 103 = 1/1000. Perhaps the cleverest part of the positional system was the addition of the decimal point allowing us to include decimal fractions. Thus 123.456 is equivalent to 1×100+2×10+3+ numbers after the point +4×1/10+5×1/100+6×1/1000 Multiplier . . . 102 101 100 . 10−1 10−2 10−3 . . . digits . . . 1 2 3 . 4 5 6 . . . ↑ decimal point However there is no real reason why we should use powers of 10, or base 10. The Babylonians use base 60 and base 12 was very common during the middle ages in Europe. Today the common number systems are 3459 is shorthand for 3 x 1000 + 4 x 100+5x10+9. Download free ebooks at bookboon.com Mathematics for Computer Scientists 20 The statement calculus and logic Chapter 2 The statement calculus and logic "Contrariwise," continued Tweedledee, "if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic. Lewis Carroll You will have encountered several languages - your native language or the one in which we are currently communicating( English) and other natural languages such as Spanish, German etc. You may also have encountered programming languages like Python or C. You have certainly met some mathematics if you have got this far. A language in which we describe another language is called a metalanguage. For almost all of mathematics, the metalanguage is English with some extra notation. In computing we need to define, and use, languages and formal notation so it is essential that we have a clear and precise metalanguage. We begin by looking at some English expressions which we could use in computing. Most sentences in English can be thought of as a series of statements combined using connectives such as "and", "or", "if . . . then . . ." For example the sentence "if it is raining and I go outside then I get wet" is constructed from the three simple statements: 1. "It is raining." 2. "I go outside." 3. "I get wet." Whether the original sentence is true or not depends upon the truth or not of these three simple statements. If a statement is true we shall say that its logical value is true, and if it is false, its logical value is false. As a shorthand we shall use the letter T for true and F for false. 21 Download free ebooks at bookboon.com Mathematics for Computer Scientists 21 The statement calculus and logic 22 CHAPTER 2. THE STATEMENT CALCULUS AND LOGIC We will build compound statements from simple statements like "it is raining", "it is sunny" by connecting them with and and or In order to make things shorter and we hope more readable, we introduce symbolic notation. 1. Negation will be denoted by ¬. 2. "and" by ∧. 3. "or" by ∨. We now look at these connectives in a little more detail. Negation ¬ The negation of a statement is false when the statement is true and is true if the statement is false. So a statement and its negation always have different truth values. For example "It is hot" and "It is not hot." In logic you need to be quite clear about meanings so the negation of, "All computer scientists are men" is "Some computer scientists are men" NOT "No computer scientists are men." The first and third statement are both false! Download free ebooks at bookboon.com Mathematics for Computer Scientists 22 The statement calculus and logic 23 In symbolic terms if p is a statement, say " it is raining" , then ¬p is its negation. That is ¬p is the statement "it is not raining". We summarize the truth or otherwise of the statements in a truth table, see table 2.1. p ¬p T F F T Table 2.1: Truth table for negation (¬) In the truth table 2.1 the first row reads in plain English - "If p is true then ¬p is false" and row two "If p is false then ¬p is true'. Conjunction ∧ Similarly, if p and q are statements, then p ∧ q is read as "p and q" . This (confusingly) is called the conjunction of p and q. So if p is the statement " it is green" while q is the statement " it is an apple" then p ∧ q is the statement "It is green and it is an apple " We often write this in the shorter form: If p=" it is green" and q = " it is an apple" then p ∧ q = "It is green and it is an apple " Clearly this statement is true only when both p and q are true. If either of them is false then the compound statement is false. It will be helpful if we have a precise definition of ∧ and we can get one using a truth table. p q p ∧ q T T T T F F F T F F F F Table 2.2: The truth table for ∧ From table 2 we see that if p and q are both true then p ∧ q is also true. If p is true and q is false then p ∧ q is false. Download free ebooks at bookboon.com Mathematics for Computer Scientists 23 The statement calculus and logic 24 CHAPTER 2. THE STATEMENT CALCULUS AND LOGIC Disjunction ∨ Suppose we now look at "or". In logic we use p∨q as a symbolic way of writing p or q. The truth table in this case is given in table 2.3 This version of "or" , which p q p ∨ q T T T T F T F T T F F F Table 2.3: The truth table for ∨ is the common one used in logic is sometimes known as the "inclusive or" because we can have p ∨ q true if either one of p and q is true or if both are true. You could of course define the exclusive or , say ≡ as having the truth table in 2.4 p q p ≡ q T T F T F T F T T F F F Table 2.4: The truth table for ≡ The Conditional ⇒ A rather more interesting connective is "implies" as in p "implies" q. This can be written many ways, for example • p implies q • If p then q • q if p • p is a sufficient condition for q I am sure you can think of other variants. We shall use the symbolic form p ⇒ q and the truth table for our definition is given in table 2.5. Download free ebooks at bookboon.com Mathematics for Computer Scientists 24 The statement calculus and logic 25 p q p ⇒ q T T T T F F F T T F F T Table 2.5: The truth table for ⇒ We sometimes call p the hypothesis and q the consequence or conclusion. Many people find it confusing when they read that " p only if q" is the same as "If p then q". Notice that " p only if q" says that p cannot be true when q is not true, in other words the statement is false if p is true but q is false. When p is false q may be true or false. You need to be aware that " q only if p" is NOT a way of expressing " p ⇒ q. We see this by checking the truth values. The truth value in line 3 of table 2.5 is the critical difference. You might like to check that " ¬p ∨ q is equivalent to p ⇒ q, see the table below p ¬p q ¬p ∨ q T F T T T F F F F T T T F T F T Table 2.6: The truth table for ⇒ Notice that our definition of implication is rather broader than the usual usage. Download free ebooks at bookboon.com Mathematics for Computer Scientists 25 The statement calculus and logic 26 CHAPTER 2. THE STATEMENT CALCULUS AND LOGIC Typically you might say "if the sun shines today we will have a barbecue" . The hypothesis and the conclusion are linked in some sensible way and the state- ment is true unless it is sunny and we do not have a barbecue. By contrast the statement "If the sun shines today 19 is prime" is true from the definition of an implication because the conclusion is always true no matter if it is sunny or not. If we consider "if the sun shines today 8 is prime" The statement is obviously false if today is sunny because 8 is never prime. How- ever the whole statement is true when the sun does not shine today even though 8 is never prime. Of course we are unlikely to make statements like these in real life. The Biconditional ⇐⇒ Suppose p and q are two statements. Then the statement "p if and only if q" is called the biconditional and denoted by p ⇐⇒ q or iff. Yes there are two f's! It is true only when p and q have the same logical values, i.e., when either both are true or both are false. You may also meet the equivalent • p iff q • p is necessary and sufficient for q The truth table is shown in figure 2.7. For example we might say p q p ⇐⇒ q T T T T F F F T F F F T Table 2.7: The truth table for ⇐⇒ You can go to the match if and only if you buy a ticket. This sort of construction is not very common in ordinary language and it is often hard to decide whether a biconditional is implied in ordinary speech. In math- ematics or computing you need to be clear if you are dealing with implication p ⇒ q or the biconditional p ⇐⇒ q Download free ebooks at bookboon.com Mathematics for Computer Scientists 26 The statement calculus and logic 27 Converse, contrapositive and inverse Propositional logic has lots of terminology. So If p ⇒ q then • q ⇒ p is the converse. • ¬q ⇒ ¬p is the contrapositive. • ¬p ⇒ ¬q is the inverse. Truth tables It is probably obvious that we aim to use logic to help us in checking arguments. We hope to be able to translate from English to symbols. Thus if p is "John learns to cook" and q is " John will find a job" then p ⇒ q represents . "If John learns to cook" and then John will find a job" In problems like these the truth table, while cumbersome can be very helpful in giving a mechanical means of checking the truth values of arguments. To construct tables for compound statements such as p ∨ ¬q ⇒ (p ∧ q) we need to think about the order we work out the truth values of symbols. The table 2.8 gives the order of precedence. Precedence 1(Highest) 2 3 4 5(Lowest) Operator ¬ ∧ ∨ ⇒ ⇐⇒ Table 2.8: Operator precedence So we negate first, then and etc. As in algebra we also use brackets to indicate that we evaluate the terms in brackets first. Thus for (p ∨ q) ∧ r we evaluate the term in brackets (p ∨ q) first. Thus p q (p ∨ q) ¬p (p ∨ q) ∨ ¬p T T T F T T F T F T F T T T T F F F T T precidence - - 1 2 3 The vital point about logical statements and about truth tables is : Two symbolic statements are equivalent if they have the same truth table. and two statements p1 and p2 are equivalent, we will write p1 ⇐⇒ p2. Download free ebooks at bookboon.com Mathematics for Computer Scientists 28 The statement calculus and logic 29 Arguments We now look briefly at logical arguments and begin with some definitions. Defini- tion: • A statement that is always true is called a tautology. • A statement that is always false is called a contradiction. So a statement is 1. A tautology if its truth table has no value F. 2. A contradiction if its truth table has no value T. Notice you may find some writers who say that a formula ( in the statement calculus we have just described ) is valid rather than use the term tautology. The symbol A is often used as a shorthand for "A is a tautology" or " A is valid". Examples 1. The statement p ∨ ¬p is a tautology, while the statement p ∧ ¬p is a con- tradiction. 2. The statement ((p ∨ q) ∧ p) ⇐⇒ p is a tautology. 3. Two statements p1 and p2 are equivalent when p1 ⇐⇒ p2 is a tautology, and so p1 ≡ p2 when p1 ⇐⇒ p2 is a tautology. Definition 1: Given two statements p1 and p2 we say that p1 implies p2 if p1 ⇒ p2 is a tautology. In everyday life we often encounter situations where we make conclusions based on evidence. In a courtroom the fate of the accused may depend the defence prov- ing that the opposing side's arguments are not valid. A typical task in theoretical sciences is to logically come to conclusions given premises. That is to provide principles for reasoning. A scientist might say "if all the premises are true then we have the following conclusion." Thus they would assert that the conditional "if all the premises are true then we have the following conclusion" is a tautology, or that the premises imply his/her conclusion. If his/her reasoning is correct we say that his argument is valid. Download free ebooks at bookboon.com Mathematics for Computer Scientists 29 The statement calculus and logic 30 CHAPTER 2. THE STATEMENT CALCULUS AND LOGIC Definition 2: A conditional of the form ( a conjunction of statements) implies c where c is a statement, is called an argument. Symbolically p1, p2, . . . , pm ⇒ c The statements in the conjunction on the left side of the conditional are called premises, while c is called the conclusion. An argument is valid if it is a tautology, that is, if the premises imply the conclusion ( every line of the truth table is T), otherwise it is invalid. So we might have a sequence of premises p1, p2, p3, . . . , pm for which c is a valid consequence, symbolically p1, p2, p3, . . . , pm c You should note that 1. A conjunction of several statements is true only when all the statements are true. 2. A conditional is false only when the antecedent ( the left hand side) is true and the consequent ( the right hand side) is false. 3. Therefore, an argument is invalid only when there is a situation where all the premises are true, but the conclusion is false. If such a situation cannot occur, the argument is valid. Exercise s: 1. Is the following argument valid? All birds are mammals and the platypus is a bird. Therefore, the platypus is a mammal. Note the premises may be wrong but we are interested in the argument. 2. Sketch how you might show that the statements below below imply that "It rained". Beware this is a big truth table so you are probably best to ensure you understand the method. If it does not rain or if it is not foggy then the regatta will be held and the lifeboat demonstration will go on. If the regatta is held then the trophy will be awarded. and Download free ebooks at bookboon.com Mathematics for Computer Scientists 30 The statement calculus and logic 31 the trophy was not awarded. 3. Show that the following argument is valid. Blodwin works hard. If Blodwin works hard then she is a dull girl. If Blodwin is a dull girl she will not get the job therefore Blodwin will not get the job. So far we have used truth tables only to determine the validity of arguments that are given in symbolic form. However, we can do the same with other arguments by first rewriting them in symbolic form. This is illustrated in the following example. Either I shall go home or stay and have a drink. I shall not go home. Therefore I stay and have a drink. Suppose p= I shall go home and q = I shall stay and have a drink. The argument is ¬p ⇒ q. p ¬p q ¬p ⇒ q T F T T T F F F F T T T F T F F Table 2.10: The truth table for ⇒ From the truth table table 2.10 we have a F and so the argument is not valid is , we do not have a tautology. We summarize the process of determining the validity of arguments as follows. Download free ebooks at bookboon.com Mathematics for Computer Scientists 31 The statement calculus and logic 32 CHAPTER 2. THE STATEMENT CALCULUS AND LOGIC 2.0.4 Analyzing Arguments Using Truth Tables • Step 1: Translate the premises and the conclusion into symbolic form. • Step 2: Write the truth table for the premises and the conclusion. • Step 3: Determine if there is a row in which all the premises are true and the conclusion is false. If yes, the argument is invalid, otherwise it is valid. However truth table can become unwieldy if we have several premises. Consider the following p, r, (p ∧ q) → ¬r ¬q Given we have p, q and r we need 8 rows (23 ) in our table 2.11 as we need all combinations of p, q and r. If we examine line 3 in table 2.11 we can see that when p, r, (p ∧ q) → ¬r are all true ( we can ignore q ) then the result ¬q is true and we have a tautology. p q r p ∧ q ⇒ ¬r ¬q T T T F F T T F T F T F T T T ← T F F T T F T T T T F T F T F F F T T T F F F T T Table 2.11: Truth table with p, q and r Now suppose we have p, q, r, s and t. Our table will have 25 = 32 rows. Take as an example : If I go to my first class tomorrow , then I must get up early, and if I go to the dance tonight, I will stay up late. If I stay up late and get up early, then I will be forced to exist on only five hours sleep. I cannot exist on five hours of sleep. Therefore I must either miss my fist class tomorrow or not go to the dance. • Let p be " I go to my first class tomorrow" • Let q be " I must get up early" • Let r be " I go to the dance " • Let s be " I stay up late ". Download free ebooks at bookboon.com Mathematics for Computer Scientists 32 The statement calculus and logic 33 • Let t be "I can exist on five hours sleep". The premises are (p ⇒ q) ∧ (r ⇒ t), s ∧ q ⇒ t, ¬t and the conclusion is ¬p ∨ ¬r. We will prove that ¬p ∨ ¬r is a valid consequence of the premises. Of course we could write out a truth table, however we can try to be cunning. 1. Take the consequence ¬p ∨ ¬r and assume that it is FALSE. 2. Then both p and r must be TRUE. 3. The first premise (p ⇒ q) ∧ (r ⇒ t) implies that q and t are true. 4. So t is true and the last premise is ¬t is assumed TRUE so we have a contradiction. 5. Thus our premise is valid. I think you might agree that this is a good deal shorter than using truth tables!. Exercises Show that 1. (p ⇒ q) ⇒ ((q ⇒ r)) ⇒ (p ⇒ r)) 2. p ⇒ (¬q ⇒ ¬p) ⇒ q) We add some tables of tautologies which enable us to eliminate conditionals and biconditionals. 1. p ⇒ q ⇐⇒ ¬p ∨ q 2. p ⇒ q ⇐⇒ ¬(p ∨ ¬q) 3. p ∨ q ⇐⇒ ¬p → q 4. p ∨ q ⇐⇒ ¬(p ⇒ ¬q) 5. p ∨ q ⇐⇒ ¬p → q Download free ebooks at bookboon.com Mathematics for Computer Scientists 33 The statement calculus and logic 34 CHAPTER 2. THE STATEMENT CALCULUS AND LOGIC 6. p ∨ q ⇐⇒ ¬p → q 7. p ∧ q ⇐⇒ ¬(p ⇒ ¬q) 8. p ∧ q ⇐⇒ ¬(¬p ∨ ¬q) 9. (p ⇐⇒ q) ⇐⇒ (p ⇒ q) ∧ (q ⇒ p) Normal forms A statement is in disjunctive normal form (DNF) if it is a disjunction i.e. a sequence of ∨'s consisting of one or more disjuncts. Each disjuncts is a conjunction, ∧, of one or more literals (i.e., statement letters and negations of statement letters. For example 1. p 2. (p ∧ q) ∨ (p ∧ ¬r) 3. (p ∧ q ∧ ¬r) ∨ (p ∧ ¬q) 4. p ∨ (q ∧ r) However ¬(p∨q) is not a disjunctive normal form(¬ is the outermost operator) nor is p∨(q∧(r∨s) as a ∨ is inside a ∧. Converting a formula to DNF involves using logical equivalences, such as the double negative elimination, De Morgan's laws, and the distributive law. All logical formulas can be converted into disjunctive normal form but conversion to DNF can lead to an explosion in the size of of the expression. A formula is in conjunctive normal form (CNF ) if it is a conjunction of clauses, where a clause is a disjunction of literals. Essentially we have the same form as a DNF but we use ∧ rather than ∨. As a normal form, it is useful ( as is the DNF) in theorem proving. We leave with some ideas which are both important and common in mathe- matics. Download free ebooks at bookboon.com Mathematics for Computer Scientists 34 The statement calculus and logic 35 2.0.5 Contradiction and consistency We say a contradiction is a formula that always takes the value F, for example p ∧ ¬p. Then a set of statements p1, p2, . . . , pn is inconsistent if a contradiction can be drawn as a valid consequence of this set. p1, p2, . . . , pn q ∧ ¬q for some formula b if a contradiction can be derived as a valid consequence of p1, p2, . . . , pn q and ¬q Mathematics is full of proofs by contradiction or Reductio ad absurdum (Latin for "reduction to the absurd"). For example There are infinitely many prime numbers. Assume to the contrary that there are only finitely many prime numbers, and all of them are listed as follows: n1, n2 . . . , pm. Consider the number q = n1 × n2 × . . . × pm + 1 Then the number q is either prime or composite. If we divided any of the listed primes ni into q, there would result a remainder of 1 for each i = 1, 2, ..., m Thus, q cannot be composite. We conclude that q is a prime number, not among the primes listed above, contradicting our assumption that all primes are in the list n1, n2 . . . , nm. Thus there are and infinite number of primes. there is no smallest rational number greater than 0 Remember that a ration can be written as the ratio of two integers p/q say. Assume n0 = p/q is the smallest rational bigger that zero. Consider n0/2. It is clear that n0/2 < n0 and n0 is rational. Thus we have a contradiction and can assume that there is no smallest rational number greater than 0. Download free ebooks at bookboon.com Mathematics for Computer Scientists 35 Mathematical Induction Chapter 3 Mathematical Induction I have hardly ever known a mathematician who was capable of rea- soning. Plato (427 BC - 347 BC), The Republic The integers , 1, 2, 3, 4, . . . are also known as the natural numbers and Mathematical induction is a technique for proving a theorem, or a formula, that is asserted about every natural number. Suppose for example we believe 1 + 2 + 3 + ... + n = n(n + 1)/2 that is the sum of consecutive numbers from 1 to n is given by the formula on the right. We want to prove that this will be true for all n. As a start we can test the formula for any given number, say n = 3: 1 + 2 + 3 = 3 × 4/2 = 6 It is also true for n = 4 1 + 2 + 3 + 4 = 4 × 5/2 = 10 But how are we to prove this rule for every value of n? The method of proof we now describe is called the principle of mathematical induction. The idea is simple. Suppose we have some statement that is true for a particular natural number n and we want to prove that it is true for every value of n from 1, 2, 3, . . . If all the following are true 1. When a statement is true for some natural number n, say k. 2. When it is also true for its successor, k + 1. 3. The statement is true for some value n, usually n = 1. 37 Download free ebooks at bookboon.com Mathematics for Computer Scientists 36 Mathematical Induction 38 CHAPTER 3. MATHEMATICAL INDUCTION then the statement is true for every natural number n. This is because, when the statement is true for n = 1, then according to 2, it will also be true for 2. But that implies it will be true for 3; which implies it will be true for 4. And so on. Hence it will be true for every natural number and thus is true for all n. To prove a result by induction, then, we must prove parts 1, 2 and 3 above. The hypothesis of step 1 "The statement is true for n = k" is called the induction assumption, or the induction hypothesis. It is what we assume when we prove a theorem by induction. Example Prove that the sum of the first n natural numbers is given by this formula: Sn = 1 + 2 + 3 + ... + n = n(n + 1)/2 We will call this statement Sn, because it depends on n. Now we do steps 1 and 2 above. 1. First, we will assume that the statement is true for n = k that is, we will assume that Sk is true so Sk = 1 + 2 + 3 + ... + k = k(k + 1)/2 Note this is the induction assumption. 2. Assuming this, we must prove that S(k+1) is also true. That is, we need to show: S(k+1) = 1 + 2 + 3 + ... + (k + 1) = (k + 1)(k + 2)/2 To do that, we will simply add the next term (k + 1) to both sides of the induction assumption, S(k+1) = S(k+1) + (k + 1) = 1 + 2 + 3 + . . . + (k + 1) = k(k + 1)/2 + (k + 1) = (k + 1)(k + 2)/2 This is line 2, which is we wanted to show. 3. Next, we must show that the statement is true for n = 1. We have S(1) = 1 = 1 × 2/2. The formula therefore is true for n = 1. We have now fulfilled both conditions of the principle of mathematical induction. Sn is therefore true for every natural number. Download free ebooks at bookboon.com Mathematics for Computer Scientists 38 Mathematical Induction 40 CHAPTER 3. MATHEMATICAL INDUCTION Another Example We prove this rule of exponents: (ab)n = an bn , for every natural number n. Call this statement S(n) and assume that it is true when n = k; that is, we assume S(k) = (ab)k = ak bk is true. We must now prove that S(k + 1) is true, that is S(k + 1) = (ab)k+1 = ak+1 bk+1 Simply by multiplying both sides of line (3) by ab gives : (ab)k ab = ak bk ab = ak abk b since the order of factors does not matter, (ab)k ab = ak+1 bk+1 . Which is what we wanted to show. So, we have shown that if the theorem is true for any specific natural number k, then it is also true for its successor, k + 1. Next, we must show that the theorem is true for n = 1 which is trivial since (ab)1 = ab = a1 b1 . This theorem is therefore true for every natural number n. Exercises In each of the following 0 ≤ n is an integer 1. Prove that n2 + n is even. 2. Prove that n i=1 n2 = n(n + 1)(2n + 1)/6. 3. Prove that 1 + 4 + 7 + . . . + (3n − 2) = n(3n − 1)/2. 4. Prove that n! ≥ 2n when n > 1 Download free ebooks at bookboon.com Mathematics for Computer Scientists 39 Sets Chapter 4 Sets Philosophers have not found it easy to sort out sets . . . D. M. Armstrong, It is useful to have a way of describing a collection of "things" and the mathe- matical name for such a collection is a set. So the collection of colours {Red,Blue, Green } is a set we might call A and write as A={Red, Blue, Green } . Other examples are 1. {1, 3, 7, 14} 2. {1, 2, 3, 5, 7, 11 . . .} the set of all prime numbers. 3. { Matthew, Mark, Luke, John} 4. {k : k is an integer and k is divisible by 4} here the contents are defined by a rule. 5. { All songs available on iTunes} again the contents are defined by a rule. We do not care about the order of the elements of a set so {1, 2, 3} is the same as {3,2,1}. Of course we may want to do things with sets and there is a whole mathemat- ical language attached as you might expect. For example you will often see the statement a belongs to the set A written as a ∈ A. The symbol /∈ is, of course, the converse i.e. does not belong to. So • Mark ∈ {Matthew, Mark, Luke, John} • Abergail /∈ {Matthew, Mark, Luke, John} . 41 Download free ebooks at bookboon.com Mathematics for Computer Scientists 40 Sets 42 CHAPTER 4. SETS • 7∈ {1,2,3,4,5,6,7} There are some sets that have special symbols because they are used a lot. Examples are 1. The set with nothing in it, called the empty set is written as ∅. 2. N = {1, 2, 3, . . .} the set of natural numbers. 3. Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .} the integers. 4. Q = the set of fractions. 5. R = the set of real numbers. 6. The set that contains everything is called the universal set written S, U or ∅. Finally we will write ¯A when we mean the set of things which are not in A. Subsets It is probably obvious that some set are "bigger" than others, for example {A,B,C,D,E} and {B,C,D}. We formalize this idea by defining subsets. If the set B contains all the elements in the set A together with some others then we write A ⊂ B. We say that A is a subset of B. So {Matthew, Mark, Luke, John} ⊂ {Matthew, Mark, Luke, John, Thomas } We can of course write this the other way around, so A ⊂ B is the same as B ⊃ A. 1. Formally for A ⊂ B we say if a ∈ A then a ∈ B or a ∈ A ⇒ a ∈ B 2. If B is a subset but might possibly be the same as A then we use A⊆B. 3. We will use A = B to mean A contains exactly the same things as B. Note that if A ⊆ B and B ⊆ A then A = B. In our logical symbolism we have (A ⊆ B) ∧ (B ⊆ A) ⇒ A = B. Download free ebooks at bookboon.com Mathematics for Computer Scientists 41 Sets 43 The power set of A, written, P(A), or 2A , is the set of all subsets of A. So if A = { Matthew, Mark, Luke } then P(A) is the set with eight elements { Matthew, Mark, Luke } { Matthew, Mark } { Matthew, Luke } { Mark, Luke } { Matthew } { Mark } { Luke } ∅ The number of elements in a set A is called the cardinality of A and written A . So if A = { Matthew, Mark, Luke, John} then A =4. Venn Diagrams and Manipulating Sets We intend to manipulate sets and it helps to introduce Venn diagrams to illustrate what we are up to. We can think of the universal set S as a rectangle and a set, say A as the interior of the circle drawn in S, see figure 4.1 The speckled area is Figure 4.1: Venn diagram of set A and universal set S A while the remainder of the area of the rectangle is ¯A. We see immediately that A together with ¯A make up S Download free ebooks at bookboon.com Mathematics for Computer Scientists 42 Sets 44 CHAPTER 4. SETS Intersection We can write the set of items that belong to both the set A and the set B as A∩B. Formally (x ∈ A) ∧ (x ∈ B) ⇒ (x ∈ A ∩ B). We call this the intersection of A and B or, less formally, A and B. In terms of the Venn diagram in figure 4.2 the two circles represent A and B while the overlap (in black) is the intersection. As examples Figure 4.2: Venn diagram of A ∩ B 1. {1,2,3,4} ∩ { 3,4,5,6,7} ={ 3,4}. Notice 3 ∈ { 3,4} while 1 /∈ { 3,4}. 2. {1,2,3,4} ∩ { 13,14,15,16,27} =∅. 3. {Abergail, Ann, Blodwin, Bronwin, Clair,}∩ { Abergail, Bronwin, Gareth, Ian} = {Abergail, Bronwin, }. 4. In figure 4.2 we see A ∩ ¯A = ∅ so A and ¯A have nothing in common. 5. A ∩ B ⊂ B and A ∩ B ⊂ A Union: We can write the set of items that belong to the set A or the set B or to both as A ∪ B. Formally (x ∈ A) ∨ (x ∈ B) ⇒ x ∈ (A ∪ B). We call this the union of A and B or, less formally, A or B. The corresponding diagram is 4.3 Here the speckled area represents A ∪ B Download free ebooks at bookboon.com Pleaseclicktheadvert Mathematics for Computer Scientists 43 Sets 45 Figure 4.3: Venn diagram of set A ∪ B (speckled) and universal set S As examples we have 1. {1,2,3,4} { 3,4,5,6,7} ={ 1,2,3,4,5,6,7} 2. { Blue,Green} { Red,Green} ={ Red,Blue , Green} 3. In figure 4.2 we see A ∪ ¯A = S so A and ¯A together make up S. 4. If A ⊂ B then A ∪ B ⊂ B We can now use our basic definitions to get some results.44 Sets 46 CHAPTER 4. SETS 1. A= ¯¯A The set ¯A consists of all the elements of S ( the universal set) which do not belong to A. So ¯¯A is the set of elements that do not belong to ¯A, or the elements of S which do not belong to ¯A . That is the elements that belong to A. Or suppose a ∈ ¯¯A ⇒ a /∈ ¯A ⇒ a ∈ A 2. (A ∩ B) = ¯A ∪ ¯B We have a ∈ (A ∩ B) ⇒ a /∈ (A∩B) ⇒ (a /∈ A)∨(a /∈ B) ⇒ (a ∈ ¯A)∨(a ∈ ¯B) ⇒ a ∈ ¯A ∪ ¯B There is a table of useful results in table 4.1. Notice each rule in the left column has a dual rule in the right. This dual has the ∪ symbol replace by ∩ A ∪ A = A A ∩ A = A (A ∪ B) ∪ C = A ∪ (B ∪ C) (A ∩ B) ∩ C = A ∩ (B ∩ C) A ∪ B = B ∪ A A ∩ B = B ∩ A A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) A ∪ ∅ = A A ∩ S = A A ∪ S = S A ∩ ∅ = ∅ A ∪ ¯A = S A ∩ ¯A = ∅ (A ∪ B) = ¯A ∩ ¯B (A ∩ B) = ¯A ∪ ¯B Table 4.1: Rules for set operations Cartesian Product Suppose we have two sets A and B. We define the Cartesian Product P = A × B to be the set of ordered pairs (a, b) where a ∈ A and b ∈ B. Or P = {(a, b) : (a ∈ A) ∧ (b ∈ B)}. The pair (a, b) is ordered in the sense that the first term (a) comes from the set A in A × B. The obvious example and hence the name comes from the geometry of the plane. We usually write (x, y) to denote the coordinates of a point on the plane. This is an ordered pair! If we take real values x and y with x ∈ R and y ∈ R then the Cartesian product is R × R 1. Suppose A = {a, b} and B = {1, 2} then A × B = {(a, 1), (a, 2), (b, 1), (b, 2)}. 2. We can extend to 3 or more sets so A × B × C is the set of ordered triples (a, b, c). Download free ebooks at bookboon.com Pleaseclicktheadvert Mathematics for Computer Scientists 46 SetsALEXANDER WOLLTE MEHR VERANTWORTUNG. WIR VERTRAUTEN IHM UNSERE WICHTIGSTEN KUNDEN AN. Als Alexander mit der Betreuung seiner neuen Private Banking Kunden startete, war nicht abzusehen, wohin ihn sein Engagement führen würde. Heute berät er 50 unserer grössten Kunden in Nahost. Seine Empfehlungen sind für die Kunden sehr wertvoll – die Erfahrung für ihn unbezahlbar. Lesen Sie Alexanders Geschichte unter credit-suisse.com/careers LOCATION: ZURICH Download free ebooks at bookboon.com Mathematics for Computer Scientists 47 Sets 49 4. asymmetric: for all x and y in X it follows that if xRy then not yRx. "Greater than" is an asymmetric relation, because ifx > y then y > x. 5. transitive: for all x, y and z in X it follows that if xRy and yRz then xRz. "Is an ancestor of" is a transitive relation, because if x is an ancestor of y and y is an ancestor of z, then x is an ancestor of z. 6. Euclidean: for all x, y and z in X it follows that if xRy and xRz, then yRz. 7. A relation which is reflexive, symmetric and transitive is called an equivalence relation. You can now speculate as the name "Relational Database". exercises 1. If A − B is the set of elements x that satisfy x ∈ A and x /∈ B draw a Venn diagram for A − B 2. Prove that for sets A, B and C (a) If A ⊆ B and B ⊆ C then A ⊆ C (b) If A ⊆ B and B ⊂ C then A ⊂ C (c) If A ⊂ B and B ⊆ C then A ⊂ C (d) If A ⊂ B and B ⊂ C then A ⊂ C 3. Recall that Z = {0, 1, 2, 3, 4, . . .} and we define the following sets (a) A = {x ∈ Z : for some integer y > 0, x = 2y} (b) B = {x ∈ Z : for some integer y > 0, x = 2y − 1} (c) A = {x ∈ Z : for some integer x < 10} Describe ¯A, ¯(A ∪ B), ¯C, A − ¯C, andC − (A ∪ B) 4. Show that for all sets A, B and C (A ∩ B) ∪ C = A ∩ (B ∪ C) iff C ⊆ A 5. What is the cardinalty of {{1, 2}, {3}, 1}. 6. Give the domain and the range of each of the following relations. Draw the graph in each case. Download free ebooks at bookboon.com Mathematics for Computer Scientists 49 Counting Chapter 5 Counting There are three types of people in this world: Those who can count, and those who can't. Counting seem quite simple but this is quite deceptive, especially when we have complicated system. If you do not believe me have a look at the probability section. To make like a little simpler we lay down some rules. Sets If we have two sets A and B the number of item in the sets ( the cardinality) is written A and B . Then we can show that A ∪ B = A + B − A ∩ B This is fairly easy to see if you use a Venn diagram. For 3 sets A ∪ B = A + B + C − A ∩ B − B ∩ C − A ∩ C + A ∩ B ∩ C Example Let S be the set of all outcomes when two dice (one blue ; one green) are thrown. Let A be the subset of outcomes in which both dice are odd, and let B be the subset of outcomes in which both dice are even. We write C for the set of outcomes when the two dice have the same number showing. How many elements are there in the following sets? It is useful to have the set S set out as below 51 Download free ebooks at bookboon.com Mathematics for Computer Scientists 51 Counting 53 This extends to several successive actions. Thus 1. If we roll a die 3 times then there are 6 × 6 = 216 possibilities. 2. If we toss a coin 7 times there are 2 × 2 × 2 × 2 × 2 × 2 × 2 = 27 = 128 possibilities. 3. My bicycle lock has 4 rotors each with 10 digits. That gives 10×10×10×10 = 104 combinations. 4. Suppose you have to provide an 8 character password for a credit card com- pany. They say that you can use a to z ( case is ignored) and 0 to 1 but there must be at least one number and at least one letter. there are 26 letters and 10 numbers so you can make 836 possible passwords. Of these there are 810 which are all numbers and 826 which are all letters. This gives 836 − 826 − 810 = 3.245 × 1032 allowable passwords. Permutations Suppose I have n distinct items and I want to arrange them in a line. I can do this in n × (n − 1) × (n − 2) × (n − 3) × · · · × 3 × 2 × 1 We compute this product so often it has a special symbol n!. However to avoid problems we define 1! = 0 and 0! = 1 So 3! = 3 × 2 × 1 = 6 while 5! = 5 × 4 × 3 × 2 × 1 = 120 If we look at the characters in (1D4Y) there are 4! = 24 possible distinct arrangements. Sometimes we do not have all distinct items. We might have n item of which r are identical then there are n!/r! different possible arrangements. So WALLY can be arranged in 5!/2! = 60 ways. It is simpler to just state a rule in the more general case: Suppose we have n objects and • there are n1 of type 1. • there are n2 of type 2. • · · · · · · • there are nk of type k. Download free ebooks at bookboon.com Mathematics for Computer Scientists 54 Counting 56 CHAPTER 5. COUNTING Suppose you were given 3x + 5/x3 8 and you wanted the term in the expansion which did not have an x. From the above the general term is 8 i (3x3 )8−i (5/x3 )i . The x terms cancel when 8 − i = 3i or i = 2. Then the term is 8 2 (3x3 )6 (5/x3 )2 = 8 2 36 52 We can do something similar for non-integral n as follows: (1+x)n = 1+nx+ n(n − 1) 1.2 x2 + n(n − 1)(n − 2) 1.2.3 +· · ·+ n(n − 1)(n − 2) · · · (n − k + 1) 1.2.3 · · · k xk +· · · but this is only true when \|x\| < 1. Thus (1 + x)1/2 = 1 + 1 2 x1/2 + 1 2 −1 2 x−1/2 + 1 2 −1 2 −3 2 x−3/2 + . . . Examples 1. Suppose we look at sports scholarships awarded by American universities. A total of 147,000 scholarships were earned in 2001. Out of the 5,500 schol- arships for athletics, 1500 were earned by women. Women earned 75,000 scholarships in total. How many men earned scholarships in athletics? 2. In clinical trials of the suntan lotion, Delta Sun, 100 test subjects experi- enced third degree burns or nausea (or both). Of these, a total of 35 people experienced third degree burns, and 25 experienced both third degree burns and nausea. How many subjects experienced nausea? 3. A total of 1055 0 MSc degrees were earned in 2002. Out of the 41 MSc degrees in music and music therapy, 5 were earned by men. Men earned 650 MSc degrees. How many women earned MSc degrees in fields other than music and music therapy? 4. A survey of 200 credit card customers revealed that 98 of them have a Visa account, 113 of them have a Master Card, 62 of them have a Visa account and a American Express, 36 of them have a Master Card account and an American Express, 47 of them have only a Master Card account, 32 have a Visa account and a Master Card account and an American Express. Assume that every customer has at least one of the services. The number of customers who have only have a Visa card is? Download free ebooks at bookboon.com Mathematics for Computer Scientists 55 Counting 57 5. So for example from the New York Times According to a New York Times report on the 16 top-performing restaurant chains (a) 11 serve breakfast. (b) 11 serve beer. (c) 10 have full table service i.e. they server alcohol and all meals. All 16 offered at least one of these services. A total of 5 were classified as "family chains," meaning that they serve breakfast, but do not serve alcohol. Further a total of five serve breakfast and have full table service, while none serve breakfast, beer, and also have full table service. We ask (a) ( How many serve beer and breakfast? (b) How many serve beer but not breakfast? (c) How many serve breakfast, but neither have full table service, nor serve beer? (d) How many serve beer and have full table service? 6. When \| x \|< 1 then show that • 1/(1 − x) = 1 + x + x2 + x3 + x4 + · · · + xn + · · · • 1/(1 − x)1/2 = 1 + (1/2)x + (1/2)(−1/2)x2 1.2 + (1/2)(−1/2)(−3/2)x3 1.2.3 + x4 + · · · • 1/(1 − x)2 = 1 + 2x + 3x2 + 4x3 + 5x4 + . . . + nxn−1 + . . . 7. Expand (1 + 2x)7 8. Which is the coefficient of the term without an x in (x + 2/x)11 . 9. Find an approximation for (0.95)11 . 10. Find the first 3 terms of the expansion of (1 + x)1/4 . Download free ebooks at bookboon.com Mathematics for Computer Scientists 56 Functions Chapter 6 Functions Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different. Johann Wolfgang von Goethe One of the most fundamental ( and useful) ideas in mathematics is that of a function. As a preliminary definition suppose we have two sets X and Yand we also have a rule which assigns to every x ∈ X a UNIQUE value y ∈ Y. We will call the rule f and say that for each x there is a y = f(x) in the set Y. This is a very wide definition and one that is very similar to that of a relation , the critical point is that for each a there is a unique value y. A common way of writing functions is f : X → Y which illustrates that we have two sets X and Y together with a rule f giving values in Y for values in X. We can think of the pairs (x, y) or more clearly (x, f(x)). This set of pairs is the graph of the function In what follows we show how functions arise from the idea of relations and come up with some of the main definitions. You need to keep in mind the simple idea a function is a rule that takes in x values and produces y values. It is probably enough to visualize f as a device which when given an x value produces a y. 59 Download free ebooks at bookboon.com Mathematics for Computer Scientists 58 Functions 61 Figure 6.1: Function f Clearly if you think of f as a machine we need to take care about what we are allowed to put in, x, and have a good idea of the range of what comes out, y. It is these technical issues we look at next. The set X is called the domain of the function f and Y is codomain. We are normally more interested in the set of values { f(x) : x ∈ X}. This is the range R sometimes called the image of the function. See figure 6.1 Examples We can have f : X → Y where 1. f(x) = 2x where X = {x : 0 ≤ x < ∞} and Y = {y : 0 ≤ x < ∞} 2. f(x) = √ x where X = {x : 0 ≤ x < ∞} and Y = {y : 0 ≤ y < ∞} 3. f(x) = sin−1 (x) where X = {x : −π/2 ≤ x < pi/2} and Y = {−1 ≤ y ≤ 1} If we think of the possibilities we have • There may be some points in Y (the codomain) which cannot be reached by function f. If we take all the points in X and apply f we get a set Download free ebooks at bookboon.com Mathematics for Computer Scientists 59 Functions 62 CHAPTER 6. FUNCTIONS Range of A Domain A Figure 6.2: An onto function R = {f(x) : x ∈ X} which is the range of the function f. Notice R is a subset of Y i.e.R ⊂ Y. • Surjections (or onto functions) have the property that for every y in the codomain there is an x in the domain such that f(x) = y. If you look at 6.1 you can see that in this case the codomain is bigger than the range of the function. See figure 6.2 If the range and codomain are the same then out function is a surjection. This means every y has a corresponding x for which y = f(x) • Another important kind of function is the injection (or one-to-one function), which have the property that if x1 = x2 then y1 must equal y2. See figure 6.3 • Lastly we call functions bijections, when they are are both one-to-one and onto. A more straightforward example is as follows. Suppose we define f : X → Y where f(x) = 2x and X = {x : 0 ≤ x < ∞} and Y = {y : −∞ ≤ x < ∞}. The range of the function is R = {y : 0 ≤ x < ∞} while the codomain Y has negative values which we cannot reach using our function. Composition of functions The composition of two or more functions uses the output of one function, say f, as the input of another, say g. The functions f : X → Y and g : Y → Z can be Download free ebooks at bookboon.com Mathematics for Computer Scientists 60 Functions 63 Figure 6.3: An 1 to 1 function composed by applying f to an argument x to obtain y = f(x) and then applying g to y to obtain z = g(y). See figure 6.4. The composite function formed in this way from f and g can be written g(f(x)) or g ◦ f. This last form can be a bit dangerous as the order can be different in different subjects. Using composition we can construct complex functions from simple ones, which is the point of the exercise. One interesting function, given f, would be the function g for which x=g(f(x)). In other words g is the inverse function. Not all functions have inverses, in fact there is an inverse g written f−1 if and only if f is bijective. In this case x = f−1 (f(x)) = f(f−1 (x)). The arrows and blob diagrams are not the usual way we draw functions. You will recall that the technical description of f : X → Y is the set of values (x, f(x)) Suppose we take the reals R so our function takes real values and gives us a new set of reals, say f(x) = x3 we take x values , compute y = f(x) for these values and plot them as in figure 6.6. Plotting functions is a vital skill, you know very little about a function until you have drawn the graph. It need not be very accurate, mathematicians often talk about sketching a function. By this they mean a drawing which is not completely accurate but which illustrates the main characteristics of the function, Now we might reasonably does every sensible looking function have an inverse? An example consider f(x) = x2 which is plotted in figure 6.8. There is now problem in the definition of f for all real values of x, that is the domain is R and the codomain R. However if we examine the inverse we have a problem. if we take y=4, this may arise from x=2 or x=-2. So there is not an f−1 = y−1/2 ! If we change the domain we can get around this. Suppose we define R+ = {x : Download free ebooks at bookboon.com Mathematics for Computer Scientists 61 Functions 64 CHAPTER 6. FUNCTIONS Figure 6.4: Composition of two functions f and g Figure 6.5: The inverse f and g = f−1 Examples 1. Suppose f(x) = x2 and g(y) = 1/y then g(f(x)) = 1/x2 . We of course have to take care about the definition if the range and the domain to avoid x = 0 2. When f(x) = x2 and g(x) = x1/2 g is the inverse function when f is defined on the positive reals. Download free ebooks at bookboon.com Mathematics for Computer Scientists 67 Functions 70 CHAPTER 6. FUNCTIONS The power function Suppose we take values x from the reals and consider the function P(x) = xa for some value a. We can suppose that a is also real. So we have R : P → R An example might be P(x) = x2 or P(x) = x1.5 . In the second case we clearly have to redefine the domain. Can you see why? The properties of the power function 1. xa × xb = xa+b 2. x0 = 1 Logarithms We know that we can write powers of numbers, so 100 = 1 101 = 2 102 = 100 102 = 1000 . . . and 100.5 = 3.162278 . . .. Now consider the backwards problem: Given y can we find an x such that y = 10x . In other words if we define the power function y = P(x) = 10x for x ∈ R, as above, then what is the inverse of this P−1 (y)? It may help to look at figure 6.10. We have plotted dotted lines from (1.5,0) to the curve. Going from x vertically to the curve and then to the y axis gives the power value P(x) = y. The reverse path from y to x is the logarithm. Download free ebooks at bookboon.com Mathematics for Computer Scientists 68 Functions 71 −2 −1 0 1 2 020406080100 x y Figure 6.10: Plot of f(x) = 10x The inverse of p(x) is call the logarithm or log and is written log10(x). So log10(1) = 0 log10(10) = 1 log10(100) = 0 log1000(1) = . . . Often we are lazy and drop the 10 and just write log(x) Because we know that log is the inverse of the power function we have some useful rules 1. log(u) + log(v) = log(uv) 2. log(uv ) = v log(uv) 3. log(u − log(v) = log u v 4. −log(u) = log 1 u Of course we did not have to choose 10 in our definitions. We could have choose 2, like many engineers, or any positive number a say. We then write y = loga(x) to indicate the number y which satisfies x = ay . The loga(x) is called the log of x to base a. For reasons which will (we hope) become apparent mathematicians like to use natural logs which have a base e = 2.718282 . . .. because they are used so often rather than write loge(x) you will often see them written as ln(x) or just as log(x). All logs satisfy the rules set out in the list 6.0.8. We shall be lazy and just use logarithms to base e. Download free ebooks at bookboon.com Mathematics for Computer Scientists 69 Functions 72 CHAPTER 6. FUNCTIONS We can of course express logs in one base as logs in another. Suppose x = aloga(x) = blogb(x) then taking logs gives loga(x) = loga(b) logb(x) Sometime it is natural to express powers as base 2 for example y = P(x) = 2x . Mathematicians often use the number e so the power definition is y = ex which you will often see written as y = exp(x) since ex is called the exponential function. 6.1 Functions and angular measure We look briefly at the measurement of angles. Angular measure has been important from the very beginning of human history both in astronomy and navigation. Consider a circle with the angle θ made with the x axis as shown. Unlike maps in mathematics the reference line is not North but along the x axis and if we rotate anti-clockwise we sweep out an angle θ. The angle is traditionally measured in degrees, minutes and seconds. We will stick to degrees for the moment. - ? 3 θ x y - 6 6 If we sweep anti-clockwise through 360 degrees we sweep out a circle. 180 degrees is a half circle and 720 = 3 × 360 two circles. Rotations in a clockwise direction are assumed to be negative degrees, so −90o = 270o To complicate things a little we can also measure the angle in an equivalent way by measuring the length of the arc we make out on the circle as we sweep through the angle θ. Suppose this is s. For a circle of radius 1 s is a measure of the angle, although in different units called radians. So one circle is 2π radians and 90o is π/2 radians. We convert from degrees to radians as follows Download free ebooks at bookboon.com Mathematics for Computer Scientists 70 Functions 6.1. FUNCTIONS AND ANGULAR MEASURE 73 degrees radians θ 2πθ/360 360s/(2π) s If you look at most "scientific calculators" you will see a button for switching from degrees to radians and vice versa. The trigonometric functions Of course we can measure angles in other ways. Suppose we look at the angle θ in the diagram. The ratio of the y and x values is related to the angle. Roman surveyors would often choose and angle by fixing the x value and the y value. As you can. imagine, five steps and then 3 steps vertically gives the same angle no matter where you are x yr θ Thus from the diagram θ is related to y/x. In fact we define y/x to be the tangent of θ written as tan θ = y/x. The inverse function is tan−1 θ = y/x or sometimes arctan θ = y/x The reader might like to examine our triable and see why the tangent of 90o does not exist. We provide a plot of the tangent from 0 to just under 90 degrees in figure 6.11. If we keep the definition on the domain 0 ≤ θ < 90 as is (relatively) simple. While the domain is easily extended we leave this to those of you will interests in this direction. Of course we do not have to use tangents, although they are probably the most practical in applications. Alternative are to use the ratio y/r the height y divided by the radius of the circle r. This is called the sine function and written sin θ = y/x. In a similar we we could use the cosine written cos θ = x/r. Both of these functions are plotted in figure 6.12 There are lots of links between these functions, Download free ebooks at bookboon.com Pleaseclicktheadvert Mathematics for Computer Scientists 71 Functions 74 CHAPTER 6. FUNCTIONS 0 20 40 60 8001020304050 theta tan(theta) Figure 6.11: tan x for example tan θ = sin θ cos θ This can be deduced quite simple from the definitions. Try it yourself! The trigonometric functions are periodic in that if we plot them over a large part of the axis they repeat as in figure 6.13 Out next step is the study of the shapes of functions which brings us to Cal- culus.73 Sequences Chapter 7 Sequences Reason's last step is the recognition that there are an infinite number of things which are beyond it. Pascal We write a sequence a1, a2, a3, · · · , an, · · · as {an} and our interest is normally whether the sequence tends to a limit A written • an → A as n → ∞. • or limn→∞ an = A However there are many interesting sequences where limits are not the main inter- est. For example the Fibonacci sequence. In Fibonacci's Liber Abaci (1202) poses the following problem How Many Pairs of Rabbits Are Created by One Pair in One Year: A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. The resulting sequence is 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . . and each term is the sum of the previous two terms. An interesting aside is that the nth Fibonacci number F(n) can we written as F(n) = [φn − (1 − φ)n ] / √ 5 where φ = (1 + √ 5)/2 1.618 . . . which is a surprise since F(n) is an integer and the formula contains √ 5. For lots more on sequences see njas/sequences/ 77 Download free ebooks at bookboon.com Mathematics for Computer Scientists 74 Sequences 78 CHAPTER 7. SEQUENCES 7.0.1 Limits of sequences We turn our attention to the behaviour of sequences such as {an} as n becomes very large. 1. A sequence may approach a finite value A. We say that it tends to a limit, so for example we write 1, 1 2 , 1 2 2 , 1 2 3 . . . 1 2 n , . . . or 1.0000 0.5000 0.2500 0.1250 0.0625 0.0312 0.0156 0.0078 0.0039 0.0020 . . . as 1 2 n and we shall see that 1 2 n → 0 as n → ∞ 2. If a sequence does not converge it may go to ±∞, that is keep increasing or decreasing. 1 2 4 8 16 32 64 128 256 512 1024 . . . Informally {2n } → ∞ as → ∞. 3. A sequence may just oscillate 1 − 1 1 − 1 1 − 1 1 − 1 1 − 1 1 Limit We need a definition of a limit and after 2000 years of trying we use : {an} → A as → ∞ if and only if, given any number there is an N such that for n ≥ N \|an − A\| < . In essence I give you a guarantee that I can get as close as you wish to a limit (if it exists) for all members of the sequence with sufficiently large N, that is after N all the values of the sequence satisfy \| an − A \|< . The idea is that if there is a limit then if you give me some tolerance, here , I can guarantee that for some point in the sequence all the terms beyond that all lie within of the limit. Download free ebooks at bookboon.com Pleaseclicktheadvert Mathematics for Computer Scientists 77 Sequences 7.1. SERIES 81 Bounded If an increasing sequence is bounded above then it must converge to a limit. Simi- larly If an decreasing sequence is bounded below then it must converge to a limit. 7.1 Series A series is the sum of terms of a sequence written u1 + u2 + u3 + · · · + uN = N i=1 ui We use capital sigma ( Σ) for sums and by b i=a ui we mean the sum of terms like ui for i taking the values a to b. Of course there are many series we sum, for example we have met the Binomial series and we have the following useful results. Download free ebooks at bookboon.com Mathematics for Computer Scientists 83 Calculus Chapter 8 Calculus I'm very good at integral and differential calculus, I know the sci- entific names of beings animalculous; In short, in matters vegetable, animal, and mineral, I am the very model of a modern Major-General. The Pirates of Penzance. Act 1. We have looked at limits of sequences, now I want to look at limits of functions. Suppose we have a function f(x) defined on an interval a ≤ x ≤ b. I have a sequence x1, x2, · · · , xn which tends to a limit x0. Can I say that the sequence f(x1), f(x2, . . . , f(xn) tends to and what do I mean? We normally define the limit as follows: We say that f(x) → f(x0) as x → x0 if for any > 0 there is a value δ > 0 such that \| x − x0 \|< δ ⇒\| f(x) − \|< This is in the same spirit as our previous definition for sequences. We can be as close as we wish to the limiting value . For example (x − 2)4 → 0 as x → 2. If you given me an 0 < < 1 then if \| x − 2 \|≤ δ we know \| (x − 2)4 − 0 \|≤ δ4 . So provided δ ≤ we have a limit as x → 0! - 6 1.0 2.0 3.0 4.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 87 Download free ebooks at bookboon.com Pleaseclicktheadvert Mathematics for Computer Scientists 84 Calculus 88 CHAPTER 8. CALCULUS - 6 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.5 -0.5 In the second case we plot sin (1/x). This starts to oscillate faster and faster as it approaches zero and ( it is not quite simple to show) does not have a limit85 Calculus 89 8.0.2 Continuity and Differentiability We did not specify which direction we used to approach the limiting value, from above or from below. This might be important as in the diagram below where the function has a jump at x0. x0 f(x) x → We like continuous functions, these are functions where f(x) → f(x0) as x → x0 from above and below. You can think of these as functions you can draw without lifting your pencil off the page. Continuous functions have lots of nice properties. If we have a continuous function we might reasonably look at the slope of the curve at any point. This may have a real physical meaning. So suppose we have the track of a car. We might plot the distance it travels, East say, against time. If the difference between the distance at times t0 and t1 is D then D/(t1 − t0) gives the approximate speed. This is just the procedure followed by average speed cameras on roads! However what we have observed is an average speed. If we want an estimate of speed at a particular time t we need t0 and t1 to approach t. t0 t1t Download free ebooks at bookboon.com Mathematics for Computer Scientists 86 Calculus 90 CHAPTER 8. CALCULUS x y x + δx f(x + δx) f(x) θ If we take the times to be t and t+δt, where δt means a small extra bit of t, then we want f(t + δt) − f(t) (t + δt − t) as δt becomes small or more explicitly f(t + δt) − f(t) δt as t → 0 This limit gives the derivative which is the slope of the curve f(t) at the point t and is written f (t) or df dx = lim δt→0 f(t + δt) − f(t) δt (8.1) Suppose we take y = f(t) = 3 − 4t, a line with constant negative slope. Using the equation 8.1 we have df dx = lim δt→0 3 − 4(t + δt) − 3 + 4t δt = −4δt δt = −4 If we now have y = x2 − 3 we have, writing x for t df dx = lim δt→0 (x + δx)2 − 3 − x2 + 3 δx = x2 + 2xδx + (δx)2 − 3 − x2 + 3 δx = 2xδx + (δx)2 δx = 2x+δx = 2x So at x=2 the slope is zero while when x is negative the slope is down and then is upwards when x is greater that zero. You might find it useful to consider the plot. Note that if we take a point on a curve and draw a straight line whose slope is f (x) this line is known as the tangent at x. Download free ebooks at bookboon.com Mathematics for Computer Scientists 89 Calculus 93 higher derivatives Since dy dx is a function we might wish to differentiate it again to get d dy dx dx called the second derivative and written d2 y dx2 . If we differentiate 4 times we write d4 y dx4 and in general dn y dxn n = 2, 3, 4, . . . So if y = log(x) we have dy dx = 1 x d2 y dx2 = − 1 x2 d3 y dx3 = 2 x3 d4 y dx4 = − 6 x4 . . . Maxima and minima One common use for the derivative is to find the maximum or minimum of a function. It is easy to see that if we have a maximum or minimum of a function then the derivative is zero. Consider y = 1 3 x3 + 1 2 x2 − 6x + 8 - 6 2 4-2-4 10 20 30 40 -10 -20 -30 f(x) = 1 3 x3 + 1 2 x2 − 6x + 8 We compute df dx = x2 + x − 6 which is zero when x2 + x − 6 = (x + 3)(x − 2) = 0 or x = −3 and x = 2 and from the plot it we see that we have found the turning points of the function. These are the local maxima and minima. However when we step back and look at the whole picture it is possible to we have a stationary point i.e. df dx = 0 which is not a turning point and hence we need a local max or minimum rule: Download free ebooks at bookboon.com Mathematics for Computer Scientists 91 Calculus 95 Example Suppose we make steel cans. If the form of the can is a cylinder of height h and radius r the volume of the can is V = πr2 h and the area of the steel used is A = 2πrh + 2πr2 . We want the volume to be 64cc. and hence V = πr2 h = 64 which gives h = 64/(πr2 ). The area is therefore A = 2πrh + 2πr2 = 128/r2 + 2πr2 To minimize the area we compute dA dr = −128/r2 + 4πr which is zero when 4πr3 = 128 giving r 2.17 and h = 64/(πr2 ) 4.34. To check that this is a minimum d2 A dr2 = 256/r3 + 4π which is positive when r is positive so we have a minimum. The Taylor Expansion We leave you with one useful approximation. If we have a function f(x) then we have f(x + a) = f(x) + a df dx + a2 2! df2 dx2 + . . . + an n! dfn dxn + . . . When a is small and we evaluate the derivatives at x. For example if we take sin x the derivatives are cos x, −sinx, −cosx, sinx, . . . . So at x = 0 since sin 0 = 0 and cos 0 = 1 sin(a) = a − a3 3! + a5 5! − a7 7! − . . . 8.0.3 Newton-Raphson method We now examine a method, known as the Newton-Raphson method, that makes use of the derivative of the function to find a zero of that function. Suppose we have reason to believe that there is a zero of f(x) near the pointx0. The Taylor expansion for f(x) about x0 can be written as: f(x) = f(x0) + (x − x0)f (x0) + 1 2! (x − x2 0f (x0) + . . . If we drop the terms of this expansion beyond the first order term we have f(x) = f(x0) + (x − x0)f (x0) Download free ebooks at bookboon.com Mathematics for Computer Scientists 94 Calculus 98 CHAPTER 8. CALCULUS 8.0.4 Integrals and Integration Many important problems can be reduced to finding the area under a curve between two points a and b a bx → f(x) The obvious idea is to split the area into small rectangles and sum the area of these. So if we take the rectangle between xj and xj+1 this has a height of f(xj) and an area of f(xj)(xj+1 − xj). If we add all such rectangles this gives an gives an approximation to the area. We do better when the width of the rectangles gets small so if we choose all the widths as δ our approximation is f(xj)δx for a = x1, x2, . . . , xn = b When we shrink δx to zero we have the area we need and write b a f(x)dx The sign was originally a capital S, for sum. Download free ebooks at bookboon.com Mathematics for Computer Scientists 95 Calculus 99 a bx → f(x) xj xj+1 f(xj) We avoid technicalities and define the definite integral of a functionf(x) between a and b as b a f(x)dx which is the area under the curve, see figure 8.1 Using the idea of areas we have Figure 8.1: Areas under f(x) some rule for integrals 1. If a ≤ c ≤ b then b a f(x)dx = c a f(x)dx + b c f(x)dx 2. For a constant c b a cf(x)dx = c b a f(x)dx 3. For two functions f(x) and g(x) b a c (f(x) + g(x)) dx = b a f(x)dx+ b a g(x)dx Download free ebooks at bookboon.com Mathematics for Computer Scientists 98 Algebra: Matrices, Vectors etc. Chapter 9 Algebra: Matrices, Vectors etc. The human mind has never invented a labor-saving machine equal to algebra Author Unknown We now meet the ideas of matrices and vectors. While they may seem rather odd at first they are vital for studies in almost all subjects. The easiest way to see the power of the idea is to consider simultaneous equations. Suppose we have the set of equations 3x − 5y = 12 x + 5y = 24 We can find the solution x = 9 y = 3 in several ways . For example if we add the second equation to the first we have equations 4x = 36 x + 5y = 24. Thus x = 9 and substituting 9 in the second equation gives 9 + 5y = 24 or 5y = 15 giving y = 3. Many mathematical models result in sets of simultaneous equations, like these except much more complex which need to be solved, or per- haps just to be examined. To do this more easily the matrix was invented. The essence of the set of equations 3x − 5y = 12 x + 5y = 24 is captured in the array or matrix of coefficients 3 −5 1 5 or the augmented matrix 3 −5 12 1 5 0 These arrays of numbers are called matrices. To save 103 Download free ebooks at bookboon.com Mathematics for Computer Scientists 99 Algebra: Matrices, Vectors etc. 104 CHAPTER 9. ALGEBRA: MATRICES, VECTORS ETC. space we often give matrices names in boldface, for example A =     3 −5 12 1 5 24 6 8 −3 11 0 0     or X = 3 −5 12 0 1 5 24 0 . We define an r × c matrix as a rectangular array of numbers with r rows and c columns, for example A above is a 4 × 3 matrix while X is 2 × 4 is A matrix with just one column is called a column vector while one with just one row is a row vector, for example a column vector a =   3 1 6   and a row vector b = 3 −5 12 −19 We use matrices in ways which keep our links with systems of equations. Before looking at the arithmetic of matrices we see how we can use them to come up with a general method of solving equations. 9.0.5 Equation Solving. If you were to look at ways people use to solve equations you would be able do deduce some simple rules. 1. Equations can be multipled by a non-zero constant 2. Equations can be interchanged 3. Equations can be added or subtracted to other equations If equations are manipulated following these rules they may look different but they have the same solutions as when you started. We can solve equations by writing the coefficients in the augmented matrix form and manipulating as follows 1. rows of the matrix may be interchanged 2. rows of the matrix may be multiplied by a nonzero constant. Download free ebooks at bookboon.com Mathematics for Computer Scientists 103 Algebra: Matrices, Vectors etc. 108 CHAPTER 9. ALGEBRA: MATRICES, VECTORS ETC. We have   1 2 −1 0 1 0 1 3 2 2 0 3   → · · · →   1 2 −1 0 0 1 −2 −3 0 0 0 0   →   1 2 1 −6 0 1 −1 −3/2 0 0 0 0   This corresponds to x + y + z = −69 y − z = −3/2 Now there is a solution for these equations but it is not the explicit unique type we have been dealing with up to now. If z is known, say z0 then it follows x = 3−z0 and y = (2z0−3)/2. We have a solution for every z0 value. Technically there are an infinite number of solutions. It is obvious if you think about it that if you have fewer equations than variables (unknowns) then you will not have a simple solution. If we have 2 rows all zero then we have to give a value to two variables, if 3 then 3 variables and so on. 2. No Solution Of course your equations may not have a solution in that they are contra- dictory, for example: x = 1 y = 3 x = −2 z = 16 We recognize the equations are contradictory ( have no solutions at all ) in the following way. If we have a row of which is all zero except for the very last element then the equations have no solution. For example: Suppose we have the equations x − 2y − 3z = 1 2x + cy + 6z = 6 −x + 3y +(c − 3)z = 0 where c is some constant. We proceed to row echelon   1 −2 3 1 2 c 6 6 −1 3 c − 3 0   →   1 −2 3 2 0 c + 6 0 4 0 1 c 0   Download free ebooks at bookboon.com Mathematics for Computer Scientists 104 Algebra: Matrices, Vectors etc. 109 Before we go further what happens if c = −6? The middle row of our matrix corresponds to 0=4 which is nonsense. Thus the original equation set does not have a solution when c = −6 However we will just carry on   1 −2 3 1 2 c 6 6 −1 3 c − 3 0   →   1 −2 3 1 0 c + 6 2c 6 0 1 c 1   →   1 −2 3 1 0 0 2c − c(c + 6) 4 0 1 c 0   →   1 −2 3 1 0 1 c 0 0 0 2c − c(c + 6) 6   →   1 −2 3 1 0 1 c 0 0 0 −4c − c2 1   Now if −4c − c2 = 0, that is c = 0, or c = −4 our last equation is 0 = 1 which is clearly nonsense! This means that the original equations had no solution. You may feel that this is a bit of a sledge hammer to crack a nut, but there is a real purpose to our exercise. If you move away from the trivial cases then the scheme we have outlined above is the best approach. It is also the technique use in the computer programs available for equation solving. In addition the shape of the reduced row echelon form tell us a lot about matrices. Often we have a system of equations where we have some parameters e.g. using our techniques above we can find the range of values, or perhaps the values themselves when solutions are possible. The row elimination ideas we have outlined are known as Gaussian elimination in numerical circles. The algorithms which bear tis name, while very much slicker are based on these simple ideas. Download free ebooks at bookboon.com Pleaseclicktheadvert Mathematics for Computer Scientists 108 Algebra: Matrices, Vectors etc.109 Algebra: Matrices, Vectors etc. 114 CHAPTER 9. ALGEBRA: MATRICES, VECTORS ETC. As we said the reason for this strange idea is so that it ties in with linear equations, thus if x + 2y = u 4x + 9y = v and v + 4y = 3 2v − y = 0 these can be written in matrix form Ax = 1 2 4 9 x y = u v = u and Bu = 1 4 2 −1 u v = 3 0 So we can write both e can write systems of equations as one matrix equation BAx = 3 0 1 4 2 −1 1 2 4 9 x = 17 38 −2 −5 x = 3 0 This is exactly the same set of equations we would have had if we had eliminated u and v without any matrices. Inverses So we have a whole set of algebraic operations we can use to play with matrices, except we have not defined division since if we can multiply then why not divide? For a ( non-zero) number z we can define the inverse z−1 which satisfies zz−1 = z−1 z = 1. In the same way we say that the matrix A has an inverse A−1 . if there is a matrix A−1 which satisfies A−1 A = AA−1 = I. Beware not all matrices have inverses! Those that do are said to be non-singular otherwise a matrix which does not have an inverse said to be is singular. If you Download free ebooks at bookboon.com Mathematics for Computer Scientists 110 Algebra: Matrices, Vectors etc. 115 think about it you will see that only square matrices can have inverses. Suppose A is an n × n matrix and B is another n × n matrix. If AB = BA = I where I is an n × n unit matrix then B is the inverse of A. Notice A must be square but not all square matrices have inverses. We can of course find the inverse by solving equations. For example a b c d e f g h = 1 0 0 1 So ae + bg af + bh ce + dg cf + dh = 1 0 0 1 we then solve the four equations . ae + bg = 1 af + bh = 0 ce + dg = 0 cf + dh = 1 Not a very promising approach. However we can use the row-echelon ideas to get an inverse. All we do is take a matrix A and paste next to it a unit matrix I . Write this augmented matrix as B = (AI). We row reduce B to reduced row echelon form. The position of the original I is the inverse. For example suppose A = 1 2 4 9 then B = (AI) 1 2 1 0 4 9 0 1 We get using row operations 1 2 1 0 4 9 0 1 → 1 2 1 0 0 1 −4 1 → 1 0 9 2 0 1 −4 1 and the inverse is A−1 = 9 2 −4 1 Of course we check 1 2 4 9 9 2 −4 1 = 9 2 −4 1 1 2 4 9 Download free ebooks at bookboon.com Mathematics for Computer Scientists 114 Algebra: Matrices, Vectors etc. 9.1. DETERMINANTS 119 Geometry We write the point (x, y) in the plane as the vector x = x y . If A is a 2 × 2 matrix Ax transforms x into a new point. Suppose A = 1 1/2 0 1 Then 1. A 0 0 = 0 0 2. A 1 0 = 1 0 3. A 0 1 = 1/2 1 4. A 1 1 = 3/2 1 If we plot the 4 points (0,0),(0,1),(1,1),(0,1) and their transforms we get −2 −1 0 1 2 −2−1012 x y 9.1 Determinants Consider the matrix a b c d . We can show that this has an inverse e f g d when = ad−bc = 0, see 9.0.7. The quantity is called the determinant of the Download free ebooks at bookboon.com Mathematics for Computer Scientists 115 Algebra: Matrices, Vectors etc. 120 CHAPTER 9. ALGEBRA: MATRICES, VECTORS ETC. matrix A = a b c d and is written a b c d or det(A). Similarly   a b c d e f g h i   has an inverse when a b c d e f g h i = a e f h i − b d f g i + c d e g h = 0 The general definition of a determinant of an n × n matrix A is as follows. 1. If n = 1 then det( A) = a11 2. if n > 1 Let Mij be the determinant of the (n−1)×(n−1) matrix obtained from A by deleting row i and column j. Mij is called a minor. Then det(A) = a11M11−a12M12+a13M13−a14M14+. . . (−1)n+1 a1nM1n = n j=1 (−1)j+1 a1jM1j Determinants are pretty nasty but we are fortunate as we really only need them for n = 1, 2 or 3. 9.2 Properties of the Determinant 1. Any matrix A and its transpose AT have the same determinant, i.e. det(A)=det(AT ). Note: This is useful since it implies that whenever we use rows, a similar behavior will result if we use columns. In particular we will see how row elementary operations are helpful in finding the determinant. 2. The determinant of a triangular matrix is the product of the entries on the diagonal, that is   a b c 0 e f 0 0 i   = aei 3. If we interchange two rows, the determinant of the new matrix is the opposite sign of the old one, that is   a b c d e f g h i   = −   d e f a b c g h i   Download free ebooks at bookboon.com Mathematics for Computer Scientists 116 Algebra: Matrices, Vectors etc. 9.2. PROPERTIES OF THE DETERMINANT 121 4. If we multiply one row by a constant, the determinant of the new ma- trix is the determinant of the old one multiplied by the constant, that is  a b c d e f λg λh λi   = λ   a b c d e f g h i   In particular, if all the entries in one row are zero, then the determinant is zero. 5. If we add one row to another one multiplied by a constant, the determinant of the new matrix is the same as the old one, that is   a b c d e f λa + g λb + h λc + i   =   a b c d e f g h i   Note that whenever you want to replace a row by something (through ele- mentary operations), do not multiply the row itself by a constant. Otherwise, it is easy to make errors, see property 4 6. det(AB)=det(A)det(B) 7. A is invertible if and only if det(A) = 0. Note in that case det(A−1 )=1/det(A) While determinants can be useful in geometry and theory they are complex and quite difficult to handle. Our last result is for completeness and links matrix inverses with determinants. Recall that the n×n matrix A does not have an inverse when det(A)=0. How- ever the connection between determinants and matrices is more complex. Suppose we define a new matrix, the adjoint of A say adj(A) as adjA = (−1)i+1 Mij T =     M11 −M12 · · · (−1)n+1 M1,n −M21 M22 · · · (−1)n+2 M2,n · · · · · · · · · · · · (−1)n+1 Mn1 (−1)n+2 Mn2 · · · (−1)2n Mnn     T Here the Mij are just the minors defined above. So if A =   1 2 3 1 3 5 1 5 · · ·   then adj(A)=   11 −7 2 −9 9 −3 1 −2 1   T =   11 −9 1 −7 9 −2 2 −3 1   Why is anyone interested in the adjoint? The main reason is A−1 = adjA det(A) Of course you would have to have a very special reason to compute an inverse this way. Download free ebooks at bookboon.com Mathematics for Computer Scientists 119 Probability Chapter 10 Probability Probability theory is nothing but common sense reduced to calcula- tion. Pierre Simon Laplace In what follows we are going to cover the basics of probability. The ideas are reasonably straightforward, however as it involves counting it is very easy to make mistakes - as we shall see. Suppose we perform an experiment whose outcome is not perfectly predictable e.g. roll a die or toss a coin. Imagine we make a list of all possible outcomes, call this list S the sample space. So • If we toss a coin S consists of {Head, Tail}, we write S = {Head, Tail}, • If we roll a die S={ 1,2,3,4,5,6} • If a princess kisses a frog then we have two possibilities S={ we get a prince, we get an embarrassed frog} • When we roll two dice then S is the set of pairs (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) An event A is a collection of outcomes of interest, for example rolling two dice and getting a double. In this case the event A is defined as 125 Download free ebooks at bookboon.com Mathematics for Computer Scientists 120 Probability 126 CHAPTER 10. PROBABILITY A ={ (1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}. Suppose that the event B is that the sum is less that 4 when we roll two dice, then B={ (1,1),(1,2),(2,1)} . If two events A and B have no elements in common then we say they are mutually exclusive. For example let A be the event {At least one 6} that is A={(1,6),(2,6),(3,6),(4,6),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,)} Since A and B have no elements in common they are mutually exclusive. Define the event C as C={ (2,3),(5,7)} Then A and C are also mutually exclusive. If D={sum exceeds 10} then A and D are not mutually exclusive! Check this yourself. Combining events • It is handy to have a symbol for not A, we use ∼ A but we are not very picky and not A is acceptable. • The event A and B, often written A ∩ B is the set of outcomes which belong both to A and to B. • The event A or B, often written A ∪ B is the set of outcomes which belong either to A or to B or to both. You will recognise the notation from the earlier discussion on sets. Suppose S={0,1,2,3,4,5,6,7,8,9} then if we define A={1,3,5,7,9} and B={4,5,7} we have • A ∩ B = A and B = {5,7} • While A ∪ B = A or B = {1,3,4,5,7,9} • ∼ B=not B = 1,2,3,4,8,9 Download free ebooks at bookboon.com Mathematics for Computer Scientists 121 Probability 127 10.0.2 Probability - the rules Now to each event we are going to assign a measure ( in some way ) called the probability. We will write the probability of an event A as P[A]. We will set out some rules for probabilities, the main ones are as follows: 1. 0 ≤ P[A] ≤ 1. 2. P[S] = 1 3. For mutually exclusive events A and B P[A or B] = P[A] + P[B] We will add a few extra rules (i) For mutually exclusive events A1 and A2 and A3 · · · An then P[A1 ∪ A2 ∪ A3 · · · ∪ An · · · ] = P[A1] + P[A2] + P[A3] + · · · + P[An] + · · · or written differently P[A1 or A2 or A3 · · · or An · · · ] = P[A1] + P[A2] + P[A3] + · · · + P[An] + · · · (ii) For an event A P[ not A] = 1 − P[A] (iii) For events A and B P[A or B] = P[A] + P[B] − P[A and B] All this is a bit fiddley but is not really very hard. If you were not too confused at this point you will have noticed that we do not have a way of getting the probabilities. This is a difficult point except in the case we are going to discuss. 10.0.3 Equally likely events Suppose that every outcome of an experiment is equally likely. Then we can show from the rules above for any event A P[A] = the number of outcomes in A the number of possible outcomes Download free ebooks at bookboon.com Mathematics for Computer Scientists 122 Probability 128 CHAPTER 10. PROBABILITY This means we can do some calculations. examples 1. Suppose that the outcomes • that a baby is a girl • that a baby is a boy are equally likely. Then as there are two possible outcomes we have P[girl]=1/2=P[boy]. 2. Suppose now a family has 3 children, the possibilities are BB BG GB GG and so P[ one boy and one girl]= 2/4=1/2 while P[two girls]=1/4 3. The famous statistician R A Fisher had seven daughters. If you count the possible sequences BBBBBBB to GGGGGGG you will find that there are 27 = 128. Only one sequence is all girl so the probability of this event is 1/128. 4. A pair of dice is thrown. What is the probability of getting totals of 7 and 11. Suppose now we throw the two dice twice. What is the probability of getting a total of 11 and 7 in this case? 5. We draw 2 balls from an urn containing 6 white and 5 black, WHat is the probability that we get one white and one black ball? As you can see we really need some help in counting. ExercisesS 1. A poker hand consists of 5 cards drawn from a pack of 52. What is the probability that a hand is a straight, that is 5 cards in numerical order, but not all of the same suit. 2. What is the probability that a poker hand is a full house, that is a triple and a pair. 3. A and B flip a coin in turn. The first to get a head wins. Find the sample space. What is the probability that A wins? Download free ebooks at bookboon.com Mathematics for Computer Scientists 123 Probability 129 4. The game of craps is played as follows: A player rolls two dice. If the sum is a 2, 3 or 12 he loses. If the sum is a seven or an 11 he wins. Otherwise the player rolls the dice until he gets his initial score, in which case he wins or gets a 7 in which case he loses. What is the probability of winning? 5. A man has n keys, one of which will open his door. He tries keys at ran- dom, discarding those that don't work until he opens the door. What is the probability that he is successful on the kth try. 6. The birthday problem How many people should be in a room to make the probability of two or more having the same birthday more than 0.5? This is quite difficult and a simpler approach is to consider the probability that no two people have the same birthday. It is often a useful dodge in probability to look at P[ not A] when P[A] is hard. So P[ no coincidences] = 365 × 364 × 363 × · · · × (365 − n + 1) 365 × 365 × · · · × 365 = 1×(1−364/365)×(1−364/365)×(1−363/365)×· · ·×(1−(365−n+1)/365) Number Probability 15 0.74709868 16 0.71639599 17 0.68499233 18 0.65308858 19 0.62088147 20 0.58856162 21 0.55631166 22 0.52430469 23 0.49270277 24 0.46165574 25 0.43130030 Download free ebooks at bookboon.com Mathematics for Computer Scientists 124 Probability130 CHAPTER 10. PROBABILITY 10 20 30 40 50 60 70 80 0.20.40.60.81.0 number probability Prob of coincident birthdays 10.0.4 Conditional Probability Sometime it is natural to talk of the probability of an event A given some other event has occurred. We write the probability of A given B as P[A \| B] and define it as P[A \| B] = P[A ∩ B] P[B] Remember this is a fancy way of writing P[A \| B] = P[A and B] P[B] While conditional probabilities can have interesting philosophical implications they also allow one to do calculations. Thus P[A] = P[A \| B]P[B] + P[A \|∼ B]P[∼ B] or more generally if B1, B2, · · · are the only possibilities so n i=1 Bi = 1 then P[A] = n i=1 P[A \| Bi]P[Bi] Download free ebooks at bookboon.com Mathematics for Computer Scientists 127 Probability 133 10.0.5 Bayes We also have Bayes Theorem P[A\|B] = P[B\|A]P[A] P[B] (10.1) or P[A\|B] ∝ P[B\|A]P[A] (10.2) Here ∝ means equal to but multiplied by a constant. You will often find that you can compute P[A \| B] when really you want P[B \| A]. Bayes theorem gives you the means for turning one into the other. Examples 1. Take the data in the example 2 above. We know that P[defective \| A]=0.03 and we found that P[defective]=0.034. Then suppose we pick up a defective component and ask what is the probability that it come from A. Thus we need P[A \| defective]. We can use Bayes to give P[A \| defective] = P[defective \| A]P[A]/P[defective] = 0.03 × 0.6/0.34 = 9/17 = 0.529. 2. Suppose that the probability that a person has a disease P[D] = 0.01. A test is available which is correct 90% of the time. If we use Y to denote that the test is positive and ∼ Y negative we mean P[Y\|D] = P[∼ Y\| ∼ D] = 0.9 Now the probability of a yes is P[Y] = P[Y\|D]P[D] + P[Y\| ∼ D]P[∼ D] = 0.9 × 0.01 + 0.1 × 0.99 = 0.108. The more interesting case is P[D\|Y] = P[Y\|D]P[D] P[Y] = 0.009/0.108 = 0.0833 Download free ebooks at bookboon.com Pleaseclicktheadvert Mathematics for Computer Scientists 128 ProbabilityManche haben Ihren Weg klar vor Augen. Andere lieben das Abenteuer, unbekanntes Terrain zu erkunden. Wie auch immer Sie sich Ihren Karrierepfad vorstellen – bei Oliver Wyman sind Sie genau richtig. GET THERE FASTER Kontakt General Management Consulting: Natalie Bojdo, Telefon +49 89 939 49 409 Kontakt Financial Services Management Consulting: Marlene Nies, Telefon +49 69 955 120 0 Download free ebooks at bookboon.com Mathematics for Computer Scientists 129 Probability 135 Independence If P[A\|B] = P[A] then we say A and B are independent. This is usually written in the equivalent form P[A ∩ B] = P[A]P[B] Independent is very useful and plays a central role in statistics. 10.0.6 Random Variables and distributions If we conduct and experiment and see an outcome we almost always code the outcome in same way, say H,T for head and tail or even 0,1. The coding is known as a random variable, usually written as a capital such as X. If we toss a coin we can say that the outcome is X. The actual values may be head, head, tail, giving the sequence of values of X as H, H, T, . . We use random variables when we have probability distributions, that is lists of possible outcomes and probabilities, such as in the table k 0 1 2 3 P[X = k] 0.1 0.3 0.5 0.1 We point out that the sum of the probabilities must be one, that is 3 k=0 P[X = k] We define the cumulative distribution function (c.d.f.) F(x) as the cumulative sum of the probabilities F(x) = k x=0 P[X = k] So in the example above k 0 1 2 3 P[X = k] 0.1 0.3 0.5 0.1 F(x) 0.1 0.4 0.9 1.0 It is more usual to give a formula for a random variable, for example P[x = k] = 0.3 × 0.7x−1 x = 1, 2, 3, · · · As the formula is commonly shorter you can see why. Download free ebooks at bookboon.com Mathematics for Computer Scientists 130 Probability 136 CHAPTER 10. PROBABILITY 10.1 Expectation We can also view probability from the point of view of what happens in the long run. Given a random variable X define the expected value of X written E[X] as E[X] = allx xP[X = x] The expected value can be regarded as the long run average. So if we roll a fail die and the outcome is X then P[X = i] = 1/6 i = 1, 2, · · · , 6] and so E[X] = 1 × 1 6 + 2 × 1 6 + · · · + 6 × 1 6 = 3.5 You can be sure that if you roll a die you will never get 3.5, however if you rolled a die and kept an average of the score you will find that this will approach 3.5, see the plot below 0 20 40 60 80 100 1.01.52.02.53.03.5 no rolls runningaveragescore For a coin we have Head and Tail. Suppose we count head as 1 and tail as zero, then P[X = 1] = 1/2 and P[X = 0] = 1/2 and so E[X] = 1 × 1 2 + 0 × 1 2 = 1 2 . A similar experiment gives the following Download free ebooks at bookboon.com Mathematics for Computer Scientists 131 Probability 10.1. EXPECTATION 137 0 20 40 60 80 100 0.00.10.20.30.40.50.60.7 no rolls runningaveragescore 10.1.1 Moments Some important expected values in statistics are the moments µr = E[Xr ] r = 1, 2, . . . since we can usually estimate these while probabilities are much more difficult. You will have met the • mean µ = E[X] • The variance σ2 = E[(X − µ)2 ]. • The parameter σ is known as the standard deviation. The central moments are defined as µr = E[(X − µ)r ] r = 1, 2, . . . The third and fourth moments E[(X−µ)3 ],E[(X−µ)4 ] are less commonly used. We can prove an interesting link between the mean µ and the variance σ2 . The result s known as Chebyshev's inequality P[\|X − µ\| > ] ≤ σ 2 (10.3) This tells us that departure from the mean have small probability when σ is small. Download free ebooks at bookboon.com Pleaseclicktheadvert Mathematics for Computer Scientists 133 Probability 10.1. EXPECTATION 139 Applying expectation using the Binomial A more interesting use is: Suppose we wish to test whether N people have a disease. It would seem that the only way to do this is to take a blood test, which will require N blood tests. Suppose we try the following: 1. We pool the blood of k < N people. 2. If the combined sample is negative we have k people without the disease. 3. If the pooled test is positive we then test all k people individually, resulting in k + 1 tests in all. 4. Repeat until everyone is diagnosed What does this save us? MASTER OF SCIENCE PROGRAMMES .bi.edu for more EFMD CLICK HERE for the best investment in your career "My strategy class taught and enabled me to use systems and tools e.g. multvariate decision analysis in my projects. The management lectures at BI provided me with a lot of useful and applicable tools that I can adapt and use in many situations that I face in business life today." Rudiger Braun, Master of Businessand Economics, 2008 BI Norwegian School of Management (BI) offers a wide range of Master of Science (MSc) programmes in: ł Business & Economics ł Strategic Marketing Management ł International Marketing & Management ł Leadership & Organisational Psychology ł Political Economy ł Financial Economics ł Innovation & Entrepreneurship For more information, visit BI also offers programmes at Bachelor, Masters, Executive MBA and PhD levels. Visit for more information. Download free ebooks at bookboon.com Pleaseclicktheadvert Mathematics for Computer Scientists 135 Probability 10.1. EXPECTATION 141 The Poisson distribution Suppose events occur at random P[X = k] = λk e−λ k! k = 1, 2, · · · , n (10.7) • The mean is E[X] = λ • the variance is var(X) = λ The average number of oil tankers arriving per day at a port is 10. The facilities at the port can handle at most 15 arrivals in a day. What is the probability that the port will not be able to handle all the arrivals in a day? The variable X is Poisson λ = 10 so P[X ≥ 16] = ∞ x=16 10x x! exp(−10) = 1 − 15 x=0 10x x! exp(−10) = 1 − 0.9513136 Probability 142 CHAPTER 10. PROBABILITY 10.1.3 Continuous variables All the cases we have considered so far have been where X takes discrete values. This does not have to be true - we can imagine X taking a continuous set of values. SInce we have though of a probability at X=k we might think of the probability of X being in some small interval x, x + δx This probability will be P[x ≤ X ≤ x + δx] = f(x)δx The function f(x) is called the probability density function. - 6 1.0 2.0 3.0 6 ? f(x) - δx x The probability, as can be seen from the sketch is made up of boxes, and if we add these together we get a probability. Personally I find it simpler to think of the cumulative distribution function F(x) which is defined as P[X ≤ x] = F(x). This is just a probability and is what you find in tables. We relate this to the density function by F(x) = x −∞ f(t)dt It is then not difficult to show that P[a ≤ X ≤ b] = b a f(t)dt Typical shapes are Download free ebooks at bookboon.com Mathematics for Computer Scientists 140 Probability 146 CHAPTER 10. PROBABILITY 1. 2 − µ = −σ × 1.645 2. 14 − µ = σ × 1.96 Solving gives (14 − µ) − (2 − µ) = 12 = 0.315σ or σ = 3.32871 and so µ = 7.475728 The Normal approximation to the Binomial A Binomial variable X which is B(n, p) can be approximated by a Normal variable Y, mean np, variance np(1 − p). This can be very useful as the Binomial tables provided are not very extensive. This is known as the Normal approximation to the Binomial. In this case z = (Y − np)/ (np(1 − p)) is standard Normal. Example Suppose X is number of 6's in 40 rolls of a die. Let Y be N(40 6 , 40 6 5 6 ). Then P[X < 5] P[Y < 5] = P[z < 5 − 20/3 50/9 ] = Φ(−0.7071068) = 0.2398 You can refine this approximation but we will settle for this at the moment. Exercises 1. A die is rolled, what is the probability that (a) The outcome is even. (b) The outcome is a prime. (c) The outcome exceeds 2. (d) The outcome is -1. (e) The outcome is less than 12. 2. Two dice are rolled. What is the probability that (a) The sum of the upturned faces is 7? (b) The score on one die is exactly twice the score on the other. Download free ebooks at bookboon.com Pleaseclicktheadvert Mathematics for Computer Scientists 141 Probability 10.1. EXPECTATION 147 (c) You throw a double, that is the dice each have the same score. 3. Suppose we toss a coin 3 times. Find the probability distribution of (a) X=the number of tails. (b) Y = the number of runs. Here a run is a string of heads or tails. So for HTT Y=2. 4. The student population in the Maths department at the University of San Diego was made up as follows • 10% were from California • 6% were of Spanish origin • 2% were from California and of Spanish origin. If a student from the class was to be drawn at random what is the probability that they are (a) From California or of Spanish origin. (b) Neither from California nor of Spanish origin. (c) Of Spanish origin but not from California Download free ebooks at bookboon.com Mathematics for Computer Scientists 142 Probability 148 CHAPTER 10. PROBABILITY 5. For two events A and B the following probabilities are known P[A] = 0.52 P[B] = 0.36 P[A ∪ B] = 0.68 Determine the probabilities (a) P[A ∩ B] (b) P[∼ A] (c) P[∼ B] 6. A hospital trust classifies a group of middle aged men according to body weight and the incidence of hypertension. The results are given in the table. Overweight Normal Weight Underweight Total Hypertensive 0.10 0.08 0.02 0.20 Not Hypertensive 0.15 0.45 0.20 0.80 Total 0.25 0.53 0.22 1.00 (a) What is the probability that a person selected at random from this group will have hypertension? (b) A person selected at random from this group is found to be overweight, what is the probability that this person is also hypertensive? (c) Find P[hypertensive ∪ Underweight] (d) Find P[hypertensive ∪ Not Underweight] 7. Two cards are drawn from an ordinary deck of 52 cards. What is the prob- ability of drawing (a) Two aces. (b) The two black aces. (c) Two cards from the court cards K,Q,J 8. Five cards are drawn from a deck of cards. What is the chance that (a) Four cards are aces (b) Four cards are the same i.e. 4 10's, 4 9'2 etc. (c) All the cards are of the same suit. (d) All the card are of the same suit and are in sequence. Download free ebooks at bookboon.com Mathematics for Computer Scientists 143 Probability 10.1. EXPECTATION 149 9. A student of statistics was told that there was a chance of 1 in a million that there was a bomb on an aircraft. The reasoned that there would be a one in 1012 chance of being two bombs on a plane. He thus decided that he should take a bomb with him ( defused - he was not stupid) to reduce the odds of an explosion. Assuming no security problems is this a sensible strategy? 10. There are four tickets numbered 1,2,3,4. A two digit number is formed by drawing a ticket at random from the four and a second from the remaining three. So if the tickets were 4 and 1 the resulting number would be 41. What is the probability that (a) The resulting number is even. (b) The resulting number exceeds 20 (c) The resulting number is between 22 and 30. 11. Three production lines contribute to the total pool of parts used by a com- pany. • Line 1 contributes 20% and 15% of items are defective. • Line 2 contributes 50% and 5% of items are defective. • Line 3 contributes 30% and 6% of items are defective. (a) What percentage of items in the pool are defective? (b) Suppose an item was selected at random and found to be defective, what is the probability that it came from line 1? (c) Suppose an item was selected at random and found not to be defective, what is the probability that it came from line 1? Download free ebooks at bookboon.com Mathematics for Computer Scientists 146 Looking at Data Chapter 11 Looking at Data It is very much more difficult to handle data rather than to construct nice probability arguments. We begin by considering the problems of handling data. The first questions are the provenance of the data. • Is it reliable? • Who collected it? • Is it what it is said to be? • Is it a sample and from what population? Such questions are important because if the data is wrong no amount of statistical theory will make it better. Collecting your own data is the best as you should know what is going on. Almost all statistical theory is based on the assumption that the observations are independent and in consequence there is a large body of methodology on sampling and data collection. 11.1 Looking at data Once you have the data what is he next step? If it is presented as a table ( do read the description) it may well be worth reordering the table and normalising the entries. Simplifying and rounding can be very effective, especially in reports. After gathering data, it pays to look at the data in as many ways as possible. Any unusual or interesting patterns in the data should be flagged for further investigation. The Histogram Anyone who does not draw a picture of their data deserves all the problems that they will undoubtedly encounter. The basic picture is the histogram. For the histogram we split the range of the data into intervals and count the number of observations in each 153 Download free ebooks at bookboon.com Mathematics for Computer Scientists 148 Looking at Data 11.1. LOOKING AT DATA 155 • A truncated mean is the mean of a data set where some large or small (or both) observations have been deleted. As you might expect the median is much less influenced by outliers - it is a robust estimate. Histogram of oct oct Frequency 0 50 100 150 200 0510152025 Example The Australian Bureau of Meteorology collects data on rainfall across Australia. Given below is the mean monthly rainfall in Broken Hill as well as the median monthly rainfall. Average Monthly Rainfall in Broken Hill (in millimeters) 1900 to 1990 Month Mean Median Jan 23 9 Feb 24 10 Mar 18 9 Apr 19 9 May 22 13 Jun 22 15 Jul 17 15 Aug 19 17 Sep 20 12 Oct 25 15 Nov 19 10 Dec 20 7 Download free ebooks at bookboon.com Mathematics for Computer Scientists 149 Looking at Data 156 CHAPTER 11. LOOKING AT DATA (a) Note that the median monthly rainfall is January is much smaller than the mean monthly rainfall. What does this imply about the shape of the distri- bution of the rainfall data for the month of January? (b) Which measure of central tendency, the mean or the median, is more ap- propriate for describing rainfall in Broken Hill? Justify your answer using knowledge of mean and median. (c) Use the above table to calculate the total yearly rainfall for Broken Hill. (d) In the north of Australia, the wet season occurs from November to April. Broken Hill, in central Australia, is occasionally drenched by a northern storm during these months. These storms tend drop a large amount of rain in a comparatively short time. How does the table reflect this fact? Spread This is the amount of variation in the data. Common measures of spread are the sample variance, standard deviation and the interquartile range. Less common is the range. The traditional measure is the sample variance s2 = 1 n n i=1 (xi − ¯x)2 and the square root of the sample variance known as the standard deviation s2 = 1 n n i=1 (xi − ¯x)2 For the octopods s=36.06159. Alternatives are: The range This is defined as range = largest data value - smallest data value this is obviously not very robust and hence is not often used which is a shame. Interquartile Range The interquartile range Q3-Q1, while simple in concept, has caused much grief to introductory statistics teachers since different respectable sources define it in different respectable ways! First we find the lower quartile Q1, this is the k = (n/4)th of the ordered observations. If k is not an integer we take the integer part of k plus 1 otherwise we take k + 1. The upper quartile Q3 is obtained by counting down from the upper end of the ordered sample. This is a good robust measure of spread. For the Octopods Q3-Q1= 59.25 -19.00 = 40.25. Download free ebooks at bookboon.com Mathematics for Computer Scientists 150 Looking at Data 11.1. LOOKING AT DATA 157 Shape The shape of a dataset is commonly categorized as symmetric, left-skewed, right-skewed or bi-modal. The shape is an important factor informing the decisions on the best measure of location and spread. There are several summary measures. The sample third moment κ3 = 1 ns3 n i=1 (xi − ¯x)3 measures skewness-it is zero for a symmetric distribution. The fourth moment κ4 = 1 ns4 n i=1 (xi − ¯x)3 gives a flat top measure. It is 3 for a normal variable! Outliers Outliers are data values that lie away from the general cluster of other data values. Each outlier needs to be examined to determine if it represents a possible value from the population being studied, in which case it should be retained, or if it is non-representative (or an error) in which case it can be excluded. It may be that an outlier is the most important feature of a dataset. It is said that the ozone hole above the South Pole had been detected by a satellite years before it was detected by ground-based observations, but the values were tossed out by a computer program because they were smaller than were thought possible. Clustering Clustering implies that the data tends to bunch up around certain values. Granularity Granularity implies that only certain discrete values are allowed, e.g. a company may only pay salaries in multiples of £1,000. A dotplot shows granularity as stacks of dots separated by gaps. Data that is discrete often shows granularity because of its discrete- ness. Continuous data can show granularity if the data is rounded. 11.1.2 Diagrams There is much to be said for drawing pictures. It is hard to imagine a data set where a histogram is not useful. If your computer program does not draw pictures then replace it! I rather like to smooth the histogram to get an idea of the shape of the p.d.f. Note however we need to take care even with the humble histogram! Ideally a histogram should show the shape of the distribution of the data. For some datasets but Download free ebooks at bookboon.com Mathematics for Computer Scientists 151 Looking at Data 158 CHAPTER 11. LOOKING AT DATA the choice of bin width can have a profound effect on how the histogram displays the data. Stem and Leaf charts If you are in a computer-free environment a stem-and-leaf plot can be a quick an effective way of drawing up such a chart. Consider the data below 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 stem leaves freq cum freq 2 789 3 3 3 0123456789 10 13 4 0123456789 10 23 5 0123456 7 30 Such a stem and leaf chart is valuable in giving an approximate histogram and giving the basis for some interesting data summaries. As you can see it is fairly easy to find the median, range etc. from the stem and leaf chart. Dotplots A traditional dotplot resembles a stemplot lying on its back, with dots replacing the values on the leaves. It does a good job of displaying the shape, location and spread of the distribution, as well as showing evidence of clusters, granularity and outliers. And for smallish datasets a dotplot is easy to construct, so the dotplot is a particularly valuable tool for the statistics student who is working without technology. Box-Plots Another useful picture is the box plot. Here we mark the quartiles Q1 Q2 on an axis and draw a box whose ends are at these points. The ends of the vertical lines or "whiskers" indicate the minimum and maximum data values, unless outliers are present in which case the whiskers extend to a maximum of 1.5 times the inter-quartile range. The points outside the ends of the whiskers are outliers or suspected outliers. can be very useful, especially when making comparisons. One drawback of boxplots is that they tend to emphasize the tails of a distribution, which are the least certain points in the data set. They also hide many of the details of the distribution. Displaying a histogram in conjunction with the boxplot helps. Both are important tools for exploratory data analysis. Download free ebooks at bookboon.com Mathematics for Computer Scientists 152 Looking at Data11.2. SCATTER DIAGRAM 159 050100150 Octopod Boxplot 11.2 Scatter Diagram A common diagram is the scatter diagram where we plot x values against y values. We illustrate the ideas with two examples. Breast cancer In a 1965 report, Lea discussed the relationship between mean annual temperature and the mortality rate for a type of breast cancer in women. The subjects were residents of certain regions of Great Britain, Norway, and Sweden. A simple regression of mortality index on temperature shows a strong positive relationship between the two variables. Data Data contains the mean annual temperature (in degrees F) and Mortality Index for neoplasms of the female breast. Data were taken from certain regions of Great Britain, Norway, and Sweden. Number of cases: 16 Variable Names 1. Mortality: Mortality index for neoplasms of the female breast 2. Temperature: Mean annual temperature (in degrees F) The Data:
Functions, Statistics and Trigonometry (FST) Course Outcome Summary Information Credits 1 Organization Hartford Union High School: Mission/Description Mission Statement: Our mathematics program is designed to enable all students to function successfully in a number related world, to give students a basis to increase their mathematical knowledge and improve their quality of life, and to develop an understanding and appreciation of mathematics and technology. Course Description: This course is designed as a college preparatory course for the student who is not necessarily majoring in a mathematics or science field. Successful completion of this course should prepare the student for Pre-Calculus. The topics will include functions, logarithms, trigonometry, probability, and statistics. The student will be required to have a graphing calculator (TI83 Plus or TI83 recommended. Cost approximately $100.) Textbooks Advanced Mathematical Concepts Units and Timelines Linear Relations and Functions - 10 days Exploring Graphs of Functions - 15 days Solving Polynomial and Rational Functions - 20 days Exploring Conic Sections - 15 days Exploring Exponential and Logarithmic Functions - 15 days Sequences and Series - 15 days The Trigonometric functions - 15 days Graphs and Inverses of the Trigonometric functions - 15 days Trigonometric Identities and Equations - 15 days Vectors and Parametric Equations - 10 days Combinatorics and Probability - 15 days Statistics and Data Analysis - 20 days Core Abilities Analyze divergent views to resolve conflicts. Assess personal goals, values, and motivations. Assume responsibility for own actions. Balance a lifestyle that is safe and healthy. Demonstrate appropriate communication skills. Devise solutions to complex problems. Initiate positive leadership skills. Integrate self awareness and attitudes appropriate in diverse social situations. Organize data into useable forms. Utilize appropriate behaviors and attitudes in the work environment. Utilize technology to improve efficiency. Competencies, Linked Standards, Objectives and Performance Standards 1. Use statistics to describe data sets. Properties Domain: Cognitive Level: Application Difficulty: Medium Importance: Important Linked External Standards WI.MA.A.12.1 Use reason and logic WI.MA.A.12.2 Communicate logical arguments and clearly show why a result does or does not make sense, why the reasoning is or is not valid, and an understanding of the difference between examples that support a conjecture and a proof of the conjectureA.12.6 Read and understand mathematical texts and other instructional materials, writing about mathematics (e.g., articles in journals), and mathematical ideas as they are used in other contexts2 Organize and display data from statistical investigations WI.MA.E.12.3 Interpret and analyze information from organized and displayed data when given measures of dispersion, including standard deviation and variance; measures of reliability; and measures of correlation WI.MA.E.12.4 Analyze, evaluate, and critique the methods and conclusions of statistical experiments reported in journals, magazines, news media, advertising, etc. Performance Standards o learner calculates statistics from a data set and display the results in a graph with 75% accuracy. o learner determines the probability that a randomly selected member of a sample lies within a specified number of standard deviations from the mean. o learner determines the standard error of the mean to predict the true mean of a population with a 1% and 5% level of confidence. Learning objectives a. Calculate measures of central tendency and spread for data sets. b. Describe relations between measures of central tendency or measures of spread. c. Use sigma notation to represent a sum, mean, mean deviation, standard deviation or variance. d. Interpret data presented in a frequency distribution. e. Read and interpret histograms, stem-and-leaf plots, and box-and-whisker plots. f. Draw a histogram, frequency polygon, stem-and-leaf plot, and box-and-whisker plot, to display data. g. Use the normal distribution curve to represent data. h. Use samples to make inferences about populations. 2. Describe the nature of graphs by examining transformations, symmetry, continuity, extrema, end behavior, and asymptotes. Properties Domain: Cognitive Level: Comprehension Difficulty: Low Importance: Important Linked External Standards WI.MA.A.12.1 Use reason and logic WI.MA.A.12.4 Develop effective oral and written presentations employing correct mathematical terminology, notation, symbols, and conventions for mathematical arguments and display of dataPerformance Standards o learner describes how transformations affect important features of graphs and how those effects are represented in the changes in equations with 75% accuracy. o learner analyzes all aspects of a function using appropriate tests with 75% accuracy. Learning objectives a. Describe the symmetries of graphs. b. Identify transformations of the constant, identity, polynomial, square root, absolute value, greatest integer, and rational functions. c. Graph nonlinear inequalities. d. Solve absolute value inequalities. e. Determine the inverse of a function. f. Graph functions and their inverses. g. Determine whether a function is continuous or discontinuous. h. Identify the end behavior of functions . i. Determine whether a function is increasing or decreasing on an interval. j. Find the critical points of a function. k. Use the first and second derivative tests to determine whether a critical point is a maximum, minimum, or inflection point. l. Graph rational functions. m. Determine vertical, horizontal, and slant asymptotes of a function. 3. Solve exponential and logarithmic equations. Properties Domain: Cognitive Level: Application Difficulty: High Importance: Important Linked External StandardsB.12.3 Perform and explain operations on real numbers WI.MA.B.12.4 Select and use appropriate procedures in problem solving situations involving the application of different number systems (natural, integers, rational, and real) WI.MA.B.12.5 Create and critically evaluate numerical arguments presented in a variety of classroom and real-world situations (e.g., political, economic, scientific, social WI.MA.E.12.1 Work with data in the context of real-world situations WI.MA.F.12.1 Analyze and generalize patterns of change (e.g., direct and inverse variation) and numerical sequences, and then represent them with algebraic expressions and equationsWI.MA.F.12.4 Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations, and inequalities Performance Standards o learner applies the properties of logarithms to solve equations with 75% accuracy. Learning objectives a. Find the value of logarithmic expressions. b. Describe properties of exponential and logarithmic functions. c. Graph exponential and logarithmic functions. d. Use e and natural logarithms to solve applied problems. e. Use logarithmic and exponential functions to model data. 4. Use trigonometry to find sides or angles in triangles. Properties Domain: Cognitive Level: Application Difficulty: Medium Importance: Essential Linked External Standards WI.MA.A.12.1 Use reason and logicWI.MA.D.12.3 Determine measurements indirectly WI.MA.F.12.4 Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations, and inequalities Performance Standards o learner uses properties of trigonometry to solve for missing dimensions in a triangle with 75% accuracy. Learning objectives a. Convert decimal degree measures to degrees, minutes, and seconds and vice versa. b. Find sines, cosines, and tangents of acute angles using the trigonometric ratios. c. Find missing side and/or angle measures of a triangle using the trigonometric ratios. d. Solve problems involving the Law of Sines and the Law of Cosines. e. Determine the area of a triangle given the measure of two sides and the included angle, the measure of two angles and a side, or the measure of the three sides. 5. Solve problems involving circular functions and their graphs. Properties Domain: Cognitive Level: Application Difficulty: Medium Importance: Important4 Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations, and inequalities Performance Standards o learner solves equations involving circular functions with 75% accuracy. o learner graphs the 6 circular functions and be able to graph their image under a linear transformation with 75% accuracy. o learner uses sinusoidal functions to solve problems with 75% accuracy. Learning objectives a. Identify the amplitude, period, frequency, vertical translation, and phase shift of circular functions. b. Evaluate inverse and reciprocal circular functions. c. State properties of inverse and reciprocal circular functions. d. Graph or identify graphs of circular functions. e. Graph or identify graphs of inverse functions. f. Model real-world data using sine and cosine functions. 6. Use the unit circle to find the values of the trigonometric functions for any angle. Linked External Standards WI.MA.B.12.4 Select and use appropriate procedures in problem solving situations involving the application of different number systems (natural, integers, rational, and real)Performance Standards o learner labels the unit circle with 100% accuracy. o learner uses properties of the unit circle to find the value of the six trigonometric ratios with 75% accuracy. Learning objectives a. Identify angles that are conterminal. b. Identify the key points on the unit circle. c. Convert between degrees and radians. d. Find the values of the six trigonometric functions of an angle in standard position given a point on its terminal side. e. Identify the number of degrees or radians in a given number of rotations. 7. Solve problems using combinatorics and probability. Properties Domain: Cognitive Level: Application Difficulty: High Importance: Essential Linked External Standards WI.MA.A.12.1 Use reason and logic5 Determine the likelihood of occurrence of complex events Performance Standards o learner applies the properties and formulas for counting to solve probability problems with 75% accuracy. Learning objectives a. Solve problems related to the Basic Counting Principle. b. Distinguish between dependent and independent events. c. Use combinations to compute the number of ways of selecting objects. d. Solve problems involving permutations and/or combinations. e. Solve problems involving permutations with repetitions. f. Solve problems involving circular permutations. g. Find the probability of an event. h. Find the odds for an event. i. Find the probability of independent and dependent events. j. Identify mutually exclusive events. k. Find the probability of mutually exclusive and inclusive events. l. Find the probability of an event given the occurrence of another event. 8. Calculate or approximate zeros of polynomial functions. Properties Domain: Cognitive Level: Application Difficulty: Medium Importance: Essential uses the techniques of solving polynomials to approximate all the zeroes of a function with 75 % accuracy. Learning objectives a. Determine roots of polynomial equations. b. Apply the Fundamental Theorem of Algebra. c. Solve quadratic equations by factoring and using the Quadratic Formula. d. Describe the roots of a quadratic function using the discriminant. e. Find factors using the Remainder and Factor Theorems. f. Apply the Rational Root Theorem to identify all possible rational roots. g. Determine the number of positive and negative real roots of a function. h. Approximate the real zeros of a polynomial function. i. Solve rational equations and inequalities. j. Decompose a fraction into partial fractions. k. Solve radical equations and inequalities. l. Model real-world data using polynomial functions. m. Interpret real-world data using polynomial functions. 9. Examine the conic sections. Properties Domain: Cognitive Level: Analysis Difficulty: High Importance: Important Linked External Standards WI.MA.A.12.1 Use reason and logic coordinate identifies a conic section from its graph or equation with 75% accuracy. o learner writes an equation to represent a graph of a conic section with 75% accuracy. o learner solves a system of second degree equations with 75% accuracy. Learning objectives a. State the properties of each of the conic sections. b. Use properties of conic sections (circles, ellipses, hyperbolas, and parabolas) to write equations describing them. c. Rewrite the general equations of conic sections in standard form. d. Identify conic sections by their equations in standard form. e. Identify conic sections by their equations in general form. f. Graph the conic sections. g. Graph systems of second degree equations and inequalities. h. Solve systems of second degree equations and inequalities. 10. Study the polar coordinate system. Properties Domain: Cognitive Level: Analysis Difficulty: Low Importance: Essential Performance Standards o learner converts between the rectangular coordinate system and the polar coordinate system by using the conversion formulas with 75% accuracy. o learner graphs in the polar coordinate system with 75% accuracy. Learning objectives a. Graph points in polar coordinates. b. Determine the distance between two points with polar coordinates. c. Graph polar equations. d. Convert between polar and rectangular coordinates. e. Write the polar from of a linear equation. f. Graph the polar form of a linear equation. 11. Calculate powers and roots of complex numbers. Properties Domain: Cognitive Level: Application Difficulty: Medium Importance: Useful Performance Standards o learner finds powers and roots of complex numbers with 75% accuracy. Learning objectives a. Add, subtract, multiply and divide complex numbers. b. Graph complex numbers. c. Represent complex numbers in different forms. d. Explore geometric relationships in the complex plane. e. Find the product and quotient of complex numbers in polar form. f. Use De Moivre's Theorem to find powers and roots of complex numbers in polar form. 12. Perform operations with and solve problems using vectors. Linked External Standards WI.MA.D.12.2 Select and use tools with appropriate degree of precision to determine measurements directly within specified degrees of accuracy and error (tolerance) WI.MA.D.12.3 Determine measurements indirectly Performance Standards o learner performs operations and solves problems using vectors with 75% accuracy. Learning objectives a. Find equal, opposite, and parallel vectors. b. Add and subtract vectors geometrically. c. Find ordered pairs that represent vectors. d. Add, subtract, multiply, and find the magnitude of vectors algebraically. e. Add, subtract,and find the magnitude of the vectors in three-dimensional space. f. Find the inner and cross products of two vectors. g. Determine whether two vectors are perpendicular. h. Solve problems using vectors and trigonometry
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MatricesPresentation Transcript MATRICES María Isabel Cadena Métodos Numéricos TYPES OF MATRICES UPPER TRIANGULAR MATRIX: The matrix A = (aij) a square matrix of order n. We say that A is upper triangular if all elements of A situated below the main diagonal are zero, ieaij = 0 for all i> j, i, j = 1 ,...., nFor example the matrices LOWER TRIANGULAR MATRIX: The matrix A = (aij) a square matrix of order n. We say that A is lower triangular if all elements of A located above the main diagonal are zero, ieaij = 0 for all i <j, i, j = 1 ,...., nFor example, arrays MATRIX TRANSPOSE: Given a matrix A, is called the matrix transpose of the matrix A is obtained by changing sort rows by the columns. OPERATIONS WITH MATRICES SUM OF MATRICES: Given two matrices of the same size, A = (aij) and B = (bij) is defined as the matrix sum: A + B = (aij + bij).The matrix sum is obtained by adding the elements of the two arrays that occupy the same same position. Properties of matrixaddition: Internal:The sum of two matrices of order mxn matrix is another dimension mxn. Associations:A + (B + C) = (A + B) + C Neutral element:A + 0 = AWhere O is the zero matrix of the same dimension as matrix A. Opposite element:A + (-A) = OThe matrix is opposite that in which all elements are changed in sign. Commutative:A + B = B + A Product of a scalar by a matrix: Given a matrix A = (aij) and a real number kR, defines the product of a real number by a matrix: the matrix of the same order as A, in which each element is multiplied by k. kA=(k aij) Product Matrix: Two matrices A and B are multiplied if the number of columns of A matches the number of rows of B.Mm Mn x x n x m x p = M pThe element cij of the matrix product is obtained by multiplying each element in row i of matrix A for each element of column j of the matrix B and adding. Product features matrix: Associations:A ° (B ° C) = (A ° B) ° C Neutral element:A · I = AWhere I is the identity matrix of the same order as the matrix A.
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more
Mathematics REA Real review, Real practice, Real results.An easier path to a college degree get college credits without the classes.CLEP COLLEGE MATHEMATICS with TESTwareIncludes CD with timed practice tests, instant scoring, and more.Based on todays official CLEP examAre you prepared to excel on the CLEP? * Take the first practice test to discover what you know and what you should know* Set up a flexible study schedule by following... Disc... It is fun to figure out the puzzle of how children go about making sense of mathematics and then how to help teachers help kids. John A. Van de Walle, Late of Virginia Commonwealth University This is the philosophy behind Elementary and Middle School Mathematics: Teaching Developmentally. John A. Van de Walle wrote this book to help students understand mathematics and become confident in their ability to teach the subject... Goldsteins Finite Mathematics, Tenth Edition The textbook is supported by a wide array of supplements as well as MyMathLab and... Walles market-leading textbook, Elementary and Middle School Mathematics, this volume offers brand-new... The book has no illustrations or index. Purchasers are entitled to a free trial membership in the General Books Club where they can select from more than a million books without charge. Subjects: Mathematical recreations; Puzzles; Games / Puzzles; Mathematics /... The new edition of this best-selling text includes a new focus on active and collaborative learning, while maintaining its emphasis on developing skills and concepts. With a wealth of pedagogical tools, as well as relevant discussions of standard curricula and assessments, this book will be a valuable textbook and reference for future teachers. With this revision, two new chapters are included to address the needs of future... The Lial series has helped thousands of students succeed in developmental mathematics through its approachable writing style, supportive pedagogy, varied exercise sets, and complete supplements package. With this new edition, the authors continue to provide students and instructors with the best package for learning and teaching supporta book written with student success as its top priority, now with an emphasis on study... ... Meditations on First Philosophy by Rene Descartes is widely considered to be one of the top philosophical books of all time. For many, Meditations on First Philosophy is required reading for various courses and curriculums. And for others who simply enjoy reading timeless pieces of classic philosophical literature, this gem by Rene Descartes is highly recommended. Meditations on First Philosophy would make an ideal gift and...
Word Translations Guide analyzes the GMAT's complex math word problems and provides structured frameworks for attacking each question type. It will enable you to master the art of translating challenging word problems into organized data. Each chapter builds comprehensive content understanding by providing rules, strategies, and in-depth examples of how the GMAT tests a given topic and how you can respond accurately and quickly. The Guide contains a total of 120 "In-Action" problems of increasing difficulty with detailed answer explanations. The content of the book is aligned to the latest Official Guides from GMAC (12th edition). Purchase of this book includes one year of access to Manhattan GMAT's online practice exams and Word Translations question bank.
Preface to the First Edition.- Preface to the Second Edition.- Outline of Contents.- Notation and Symbols.- Introductory Measure Theory.- Random Variables.- Inequalities.- Characteristic Functions.- Convergence.- The Law of Large Numbers.- The Central Limit Theorem.- The Law of the Iterated Logarithm.- Limited Theorems.- Martingales.- Some Useful Mathematics.- References.- Index.
$ 23.29 Shaping the Future with Math, Science, and Technology examines how ingenuity, creativity, and teamwork skills are part of an intellectual toolbox associated with math, science, and technology. The book provides... $ 33.79 In a rapidly evolving local and global economy, skills related to mathematical problem solving, scientific inquiry, and technological innovation are becoming more critical for success in and out of school. Thus,...
More About the Author Product Description About the Author John Bird, the author of over 100 textbooks on engineering and mathematical subjects, is the former Head of Applied Electronics in the Faculty of Technology at Highbury College, Portsmouth, U.K. More recently, he has combined freelance lecturing at Portsmouth University, with technical writing and Chief Examiner responsibilities for City and Guilds Telecommunication Principles and Mathematics, and examining for the International Baccalaureate Organisation. John Bird is currently a Senior Training Provider at the Royal Naval School of Marine Engineering in the Defence College of Marine and Air Engineering at H.M.S. Sultan, Gosport, Hampshire, U.K. The school, which serves the Royal Navy, is one of Europe's largest engineering training establishments. If you are studying Higher GCSE,A Level,BTEC NC or even HNC this is the one to go for. It is a well written using worked examples helping you understand the core material. Infact any John Bird or Bird and May books are great books,and can be found on Amazon at a great price. If you want a GCSE-A Level/HNC Maths Book GET THIS ONE! This book has combined the concept of mathematics in a very nice, clear, easy to understand manner. I am sure this book will be very useful for students doing undergraduate course in mathematics, physics and engineering. I am PhD qualified highly experienced tutor. Most of my students find this book very useful in revising the fundamental mathematics concept. If you need FREE advice, explanation of any topic, solution of a problem, feel free to contact me at info@aonetuition.co.uk. Good luck!! I bought this as I teach maths and was asked to take a maths support class for engineering students at 16+ level. Covers angineering maths up to degree level in a clear way with good examples and worked solutions.
Find a Miami Lakes, FL Algebra 1 Chabad Organization. Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing object
If you've got a budding (or reluctant!) mathematician, you'll want to check out some of these books! Books are an easy and non-threatening way to build interest in and understanding of mathematical concepts!
Undergraduate Commutative Algebra 9780521458894 ISBN: 0521458897 Publisher: Cambridge Univ Pr Summary: Showing the link between commutative ring theory and algebraic geometry, this book contrasts the methods and ideology of modern abstract algebra with concrete applications in algebraic geometry and number theory. It is ideal for anyone seeking a primer on commutative algebra. Reid, Miles is the author of Undergraduate Commutative Algebra, published under ISBN 9780521458894 and 0521458897. Six hundred sixteen... Undergraduate Commutative Algebra textbooks are available for sale on ValoreBooks.com, one hundred eighty nine used from the cheapest price of $25.65, or buy new starting at $420521458897 Good to VERY GOOD physical shape and/or light to moderate markings/highlight! A great value priced book! Same or next day processing! Choose EXPEDITED for super fa [more] 0521458897This item is printed on demand. In this well-written introduction to commutative algebra, the author shows the link between commutative ring theory and algebraic geometry. In [more] This item is printed on demand. In this well-written introduction to commutative algebra, the author shows the link between commutative ring theory and algebraic geometry. In addition to standard material, the book contrasts the methods and ideology of m.[less]
a thorough understanding of the concepts essential to success in mathematics with BASIC MATHEMATICS, 6th Edition. Each chapter-opening application is used as a common thread to introduce new concepts. Diagrams, charts, and graphs make it easier to grasp the material. McKeague's unique and successful EPAS system of Example, Practice, Answer, and Solution prepares you for working the problem sets. Your textbook purchases includes a variety of online and CD-ROM learning aids you can use for test prep, including iLrn Tutorial, Personal Tutor with SMARTHINKING live online tutoring, the Digital Video Companion CD-ROM with MathCue, and a Book Companion Web Site with practice tests for each chapter.
Hodgkins, IL Microsoft ExcelFinite math is an introductory course in discrete math. A
A good grounding in Mathematics is not only intellectually rewarding, but often provides the passport to a wide variety of jobs, as well as further work in scientific research. The Mathematics A-level specifications changed in 2004 and for this reason did not change to the new 4 exam system in September 2008. The specification remains a six module qualification and is unlikely to change until 2014 at the earliest. Each full A Level Course consists of an AS and A2 Sylla… A good grounding in Mathematics is not only intellectually rewarding, but often provides the passport to a wide variety of jobs, as well as further work in scientific research. The Mathematics A-level specifications changed in 2004 and for this reason did not change to the new 4 exam system in September 2008. The specification remains a six module qualification and is unlikely to change until 2014 at the earliest. Each full A Level Course consists of an AS and A2 Syllabus. The distance learning course is split into six modules. The AS and A2 courses can be sat in their own right as an individual course, but you are required to have completed the AS level syllabus before starting on an A2 syllabus. Each lesson begins with a set of clearly stated objectives and an explanation of its place in the overall programme of study.
Students have to write every function in all three forms: standard, vertex, and intercept. Then identify all key features, intercepts, AoS, asymptotes, end behaviors, vertex, translation from parent function, domain/range, and stretch/compression factor. If it does not exist, write that and explain why. No canceling! You are either making one or making zero with equations. Break functions into two clusters. First, linear, quadratic, radical, square root (mix in inequalities). Second, log, exponential, rational, polynomial. Start with solving basic types of the first four. Then cycle back and do intermediate types. Cycle back a third time and teach advanced types. Six weeks of equations and four weeks of graphs, the whole semester. End with conics, sequences, and series. Start January with second cluster. Use graphs to motivate higher algebra. Inverses, transformations, quadratic formula, systems, complex numbers. Complex numbers are what happens when we force the algebra. When two graphs don't intersect, inverting the first graph and finding the intersection gives you the imaginary number solutions. Students use graphs to validate their algebraic answers. Use data from data.worldbank.org for growth and decay for modeling. Compare countries and growth rates to interpret graphs and motivate extension questions. Use credit card bills and real estate websites for exponentially. Resources: infinitesums.com/alg2 Open the calendar in Dropbox for pacing guide. DIY guide for goals and practice problems. Independent practice for just problems. All assessments, including finals and finals review are posted. Financial formulas: log gives number of months to pay off, second gives mortgage payment.
An Introduction to Mathematical Reasoning [NOOK Book]... Customers Who Bought This Also Bought More About This Book wide range of basic methods of proof, and includes some of the classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. Over 250 problems include questions to interest and challenge the most able student as well as plenty of routine exercises to help familiarize the reader with the basic ideas. Editorial Reviews From the Publisher "A student planning to study advanced mathematics would be well served by first mastering the material in this book...a rigorous study of sevearl fundamental topics pervasive in mathematics, including sets, functions, cardinality, combinatorics, and modular arithmetic." D.S. Larson
The Algebra 1: The Complete Course DVD Series will help students build confidence in their ability to understand and solve algebraic problems. In this episode, concrete examples and practical applications show how the mastery of fundamental algebraic concepts is the key to success in today's technologically advanced world. Students will also learn the development of algebraic symbolism as well as the geometric and numeric currents. Grades 5-9. 30 minutes on DVD. Customer Reviews for Algebra 1 - The Complete Course: An Overview DVD This product has not yet been reviewed. Click here to continue to the product details page.
:: 2014 AP* Summer Institutes at UT Austin Pre-AP High School Math: Algebra II and Pre-Cal Focus This Pre-AP Precalculus and Algebra 2 workshop will focus on the level of rigor and skills needed to prepare students for success in AP courses. Participants will learn to: Identify and enhance the level of rigor on assignments, assessments, and EOC style questions Deepen understanding by applying the rule of four Enhance content by previewing calculus ideas of limit, rate, accumulation, and optimization at the Pre-AP level Create AP-style assessments Participants should bring: A favorite lesson to share. Teachers will share lessons with other workshop participants. A graphing calculator (if available). Textbooks using next year. Lead Consultant: Robert Cole Robert Cole has presented at numerous APSIs and College Board One-Day, Two-Day and Mathematics Vertical Teams workshops. A veteran of over 30 years in the public schools, Robert has taught everything from 7th grade math to calculus, including below-level, on-level, Pre-AP, G/T and AP classes. His current assignments at Hill Country Christian School of Austin and Odyssey School in Austin include Geometry, Algebra 2, Precalculus and AP Calculus AB. * Trademark Notice: College Board, AP, Advanced Placement Program, AP Vertical Teams, Pre-AP, and the acorn logo are registered trademarks of the College Board. Used with permission.
More About This Textbook Overview A stimulating view of mathematics that appeals to students as well as teachers, this undergraduate-level text is written in an informal style that does not sacrifice depth or challenge. Based on twenty years of teaching by the leading researcher in graph theory, it offers a solid foundation on the subject. This revised and augmented edition features new exercises, simplifications, and other improvements suggested by classroom users and reviewers. The first three chapters -- on basic graph theory, colorings of graphs, and circuits and cycles -- are basic to the understanding of the rest of the text. Chapters 6, 8, 9, and 10 -- on labeling graphs, drawings of graphs, measurements of closeness to planarity, and graphs on surfaces -- are appropriate for a mathematically intensive course. Chapter 7, on applications and algorithms, is of particular importance to computer science students. Chapters 4 and 5, on extremal problems and counting, are recommended if time is
Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as... $ 71.79 Discrete geometry is a relatively new development in pure mathematics, while computational geometry is an emerging area in applications-driven computer science. Their intermingling has yielded exciting advances... $ 66.49 Well-illustrated, practical approach to creating star-faced spherical forms that can serve as basic structures for geodesic domes. Complete instructions for making models from circular bands of paper with just... $ 13.99 This book presents a concise exposition of modern mathematical concepts, models and methods with applications in computer graphics, vision and machine learning. The compendium is organized in four parts —... $ 79.99 Geometry defines the world around us, helping us make sense of everything from architecture to military science to fashion. And for over two thousand years, geometry has been equated with Euclid's Elements... $ 15.79If you've ever thought that mathematics and art don't mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more... $ 285.79 Algebraic topology (also known as homotopy theory) is a flourishing branch of modern mathematics. It is very much an international subject and this is reflected in the background of the 36 leading experts who... $ 2.49 In this book, the general theory of submanifolds in a multidimensional projective space is constructed. The topics dealt with include osculating spaces and fundamental forms of different orders, asymptotic and... $ 68.49 egghead's Guide to Geometry will help students improve their understanding of the fundamental concepts of geometry. With the help of Peterson's new character, egghead, students can strengthen their math skills... $ 9.99 Requiring no more than a knowledge of high school mathematics and written in clear and accessible language, this book will give all readers a new insight into some of the most enjoyable and fascinating aspects... $ 9.79 The basic goals of the book are: (i) to introduce the subject to those interested in discovering it, (ii) to coherently present a number of basic techniques and results, currently used in the subject, to those... $ 68.49 What exactly is the Golden Ratio? How was it discovered? Where is it found? These questions and more are thoroughly explained in this engaging tour of one of mathematics' most interesting phenomena. The authors...
Symmetry analysis based on Lie group theory is the most important method for solving nonlinear problems aside from numerical computation. The method can be used to find the symmetries of almost any system of differential equations and the knowledge of these symmetries can be used to reduce the complexity of physical problems governed by the equations. This is a broad, self-contained, introduction to the basics of symmetry analysis for first and second year graduate students in science, engineering and applied mathematics. Mathematica-based software for finding the Lie point symmetries and Lie-Bäcklund symmetries of differential equations is included on a CD along with more than forty sample notebooks illustrating applications ranging from simple, low order, ordinary differential equations to complex systems of partial differential equations. MathReader 4.0 is included to let the user read the sample notebooks and follow the procedure used to find symmetries. less
ISBN: 0471615544 / ISBN-13: 9780471615545 Graphs: An Introductory Approach - A First Course in Discrete Mathematics An introduction to discrete mathematics, this new text on graph theory develops a mathematical framework to interrelate and solve different problems. ...Show synopsisAn introduction to discrete mathematics, this new text on graph theory develops a mathematical framework to interrelate and solve different problems. It introduces the concepts of logic, proof and mathematical problem-solving and places an emphasis on algorithms in every chapter.Hide synopsis The only text available on graph theory at the freshman/sophomore level, it covers properties of graphs, presents numerous algorithms, and describes actual applications to chemistry, genetics, music, linguistics, control theory and the social sciences. Illustrated.The only text available on graph theory at the freshman/sophomore level, it covers properties of graphs, presents numerous algorithms, and describes actual applications to chemistry, genetics, music, linguistics, control theory and the social sciences. Illustrated Graphs: An Introductory Approach--A First Course in...Good. Graphs: An Introductory Approach--A First Course in Discrete Mathematics. This book is in Good condition. Buy with confidence. We ship from multiple location. Description:Fair. Book is in good to acceptable condition with minor...Fair. Book is in good to acceptable condition with minor blemishes on the cover. Pages have highlighting and minor writing. Ships next business Minor highlights
Pergjigjet provimi i matematikes matura 2013. science. Students will write programs in the Python programming language. Good program We will use this 9. Second Exam. Working with strings. Matching, coding, cryptography. 10. Application: Two midterm exams, each accounts for 12% of your grade and a final. Matura shteterore 2013 rezultatet kosove. Text: Essential Calculus: Early Transcendentals, Stewart, 2nd edition. Attendance . etc., with plus/minus cut-offs to be determined), though this is not set in stone. (probably) solutions to graded homework and test problems, and important . Matura shteterore 2013 pergjigjet. (1) Hard copy of Volume 1 of "Chemistry – A Molecular Approach" (chapters 1-11 ). (2) Hard copy of the Solutions Manual that shows the worked-out solutions for every. perhaps Wednesdays if there is a more natural break in the content). Mash matura shteterore 2013 rezultatet. The goal of the lecture component of Organic Chemistry I is to introduce students to fundamental Library: Textbook, Solutions Manual & Molecular Models are on Reserve. 5. as soon as possible, and try to get it to be second nature to you. . (d) Write a 'C' program to find the sum of the first n numbers. (e) Write a 'C' program to (b) Write a short note on Data Manupulation. (e) Following is the frequency distribution of number of students. (i) What is Software Engineering ? . Check course homepage for homework Abstract Algebra by Dummit and Foote, third edition. Contact Info with your fellow students but you have to write up your solutions on your own.
products available from sellers starting from: Condition Seller Information Buy Now MathCraft A course in Mathematics Book 8 Description Mathcraft, an Encyclopedia Britannica series of nine Mathematics books, includes a primer for the Kindergarten and eight textbooks for Classes 1 to 8. Conforming to the National Curriculum Framework 2005, the series aims to develop in students an enthusiasm for this craft of numbers, quantity, and space. Discussion : MathCraft A course in Mathematics Book 8
Mathematics Tutorials and Problems (with applets) Mathematics Tutorials and Problems (with applets)Discussion Discussion for Mathematics Tutorials and Problems (with applets) Katie Larkin (Faculty) Great resource. Great organization and scope of material. 4 years ago Seamus Beall (Student) this is a straight forward site which uses graphs to demonstrate different math concepts. I think that students who prefer a direct visual example of what they are learning would appreciate this site and I believe it would stand to benefit a teacher who might want to show graphically what small changes in input numbers can do to graphs. I would have really liked to have known about this particular site when i took my intermediate algebra class because it covers so many topics in a fair ammount of depth. Time spent reviewing site: 1 half hour. 5 years ago Thomas Brown (Faculty) Contains applets which can help students develop an intuition for boxplots and the normal distribution.
Mathematics Why study Maths? Mathematics is one of the most widely used subjects in the world and at DYCA all students study mathematics until the end of Senior 3. The study of mathematics helps students understand how to organise complicated problems into a series of clear, simple and logical steps that can be solved. It promotes deep thinking and an enquiring mind. The mathematics department at DYCA strives to ensure that the subject is both enjoyable and differentiated so that every student can access it and achieve their potential. Course Overview Key Stage 3 - Fresher and Prep During these years, students set the foundations for their mathematical development. We currently follow the Collins Framework, which consolidates and extends students' experiences with Number, Algebra, Handling Data and Shape, Space and Measure. It also includes functional skills topics, which assist students in understanding the applications of their studies to both the world outside the classroom and the new style of Key Stage 4 examinations. Key Stage 4 – Senior 1, 2 and 3 Students begin to study for the Edexcel Linear GCSE examinations during Senior 1and continue it through to Senior 3. The exam is split into Foundation and Higher tiers, with students able to achieve a C grade on either level. The syllabus continues to work on the Number, Algebra, Handling Data and Shape, Space and Measure strands with each block unified under a practical theme that draws together these different braches. It also includes functional skills elements where students have to combine techniques from a variety of topics to solve problems in a real world context. Key Stage 5 All students studying for the International Baccalaureate Diploma must study maths at Higher, Standard or Studies level. Maths Higher is aimed at those students with a strong background and interest in mathematics and is an excellent preparation for university courses with a very high mathematical content. Maths Standard is aimed at those students who require a sound mathematical preparation for application to other subjects such as sciences, business or economics. Maths Studies is designed to build confidence and encourage an appreciation of a variety of mathematical topics. All three tiers are assessed by a combination of examinations and project work. Those students studying for the International Baccalaureate Career-related Certificate can opt to study either Maths Standard or Maths Studies as one of their subject choices. By Year Fresher and Prep Students have two lessons of mathematics each week during which they consolidate and extend their experiences and understanding across a range of core mathematical topics. Senior 1,2 and 3 Students have two or three lessons of mathematics a week during which they work towards their GCSE mathematics qualification assessed through a calculator and a non calculator examination. Sixth Form Students studying for the IB Diploma choose from the Higher, Standard or Studies level syllabus. Those studying for the IB CC can opt for either Standard or Studies if they wish. Extra-curricular There are three maths clubs every Monday that offer either practical applications of maths or extension activities. Students in all years are offered the opportunity to enter national UK Maths Challenge competitions and students are regularly invited to attend events at other institutions. A full programme of support is available to our Key stage 4 and 5 students for revision and exam preparation hosted by subject specialists.
Customer Reviews for Walch Education Power Basics Consumer Mathematics Set Help students add to their real-world knowledge! Power Basics Consumer Math uses a step-by-step approach to strengthen money-management skills and explains how to spend wisely and create a balanced budget. Concepts such as getting a loan and establishing good credit are covered, as well as crucial life skills such as buying insurance and filing a tax return. This Kit Includes: Student Textbook Student Workbook Teacher's Guide Tests Customer Reviews for Power Basics Consumer Mathematics Set This product has not yet been reviewed. Click here to continue to the product details page.
an introduction to the basic ideas, computational techniques, and applications of linear algebra. The most applied of our basic texts in this market, this text has a superb range of problem sets. Also, this book is extremely technology-friendly, integrating optional CAS and a robust website. Topics covered include wavelets; the Leslie Population Model; fractals; dynamical systems; linear equations and matrices; determinants; vectors; eigenvalues and eigenvectors; linear transformations and matrices; linear programming; and more. Ideal as an introduction to Linear Algebra.
Objective:The practices and pitfalls of numerical computation. Topics include floating point representations; precision, accuracy, and error; numerical solution techniques for various types of problems; root finding, fixed point, interpolation, numerical differentiation and integration. We shall cover the first four chapter of the text. Computer labs: Will be announced and will use Maple V10. Tests: There will be three in class tests, 100 points each, February 16th, March 30th and April 27th . Quizzes:A weekly quiz/hw will be given and will mainly cover questions from the homework.Each quiz counts 20 points and the best 10 are counted towards your final grade.
About this course This distance learning course provides the information you will need to prepare for the AQA A-Level in Maths with Statistics. In this home study course, you will focus on four core topics of algebra, geometry, trigonometry and calculus, which make up two-thirds of the A-Level qualification. The remaining third is focused on the study of statistics, including estimation, probability and distributions. The course is optimized for students studying at home and includes full tutor support via email. A-Level Maths with Statistics is a valuable complement to other A-Level courses with a statistical element, such as biology, sociology or psychology, and for those wishing to study these subjects at a higher level. A-Level Maths with Statistics is also applicable to many jobs and careers and is a well-respected qualification that can be used for career progression and further training whilst in employment. Entry requirements English reading and writing skills, and maths to at least GCSE grade C or equivalent are required. You will need to have general skills and knowledge base associated with a GCSE course or equivalent standard. This specification is designed to: develop the student's understanding of mathematics and mathematical processes in a way that promotes confidence and fosters enjoyment develop abilities to reason logically and to recognise incorrect reasoning, to generalise and to construct mathematical proofs extend their range of mathematical skills and techniques and use them in more difficult unstructured problems use mathematics as an effective means of communication acquire the skills needed to use technology such as calculators and computers effectively, to recognise when such use may be inappropriate and to be aware of limitations develop an awareness of the relevance of mathematics to other fields of study, to the world of work and to society in general On this course you will study six units: AS Level Unit 1 MPC1 Core 1 Unit 2 MPC2 Core 2 Unit 3 MS1B Statistics 1B A2 Level Unit 4 MPC3 Core 3 Unit 5 MPC4 Core 4 Unit 6 MS2B Statistics 2 Each unit has 1 written paper of 1 hour 30 minutes. Course Content AS Level Unit 1 MPC1 Core 1 Co-ordinate Geometry Quadratic functions Differentiation Integration Unit 2 MPC2 Core 2 Algebra and Functions Sequences and Series Trigonometry Exponentials and logarithms Differentiation Integration Unit 3 MS1B Statistics 1B Statistical Measures Probability Discrete Random Variables Normal Distribution Estimation A2 Level Unit 4 MPC3 Core 3 Algebra and Functions Trigonometry Exponentials and Logarithms Differentiation Integration Numerical Methods Unit 5 MPC4 Core 4 Algebra and Functions Coordinate Geometry in the (x, y) plane Sequences and Series Trigonometry Exponentials and Logarithms Differentiation and Integration Vectors Unit 6 MS2B Statistics 2 Poisson distribution Continuous random variables The t-distribution Hypothesis Testing Chi-squared tests AS +A2 = A Level in Maths with Statistics. Both AS and A2 level courses and examinations must be successfully completed to gain a full A Level. AQA Specification 6360 The course comes to you as a paper-based packExams are taken at an AQA centre and we can provide an extensive list of centres for you. Please read our FAQs for further information Our A Levels come with tutor support for 24 months. You will have access to a tutor, via email, who will mark your work and guide you through the course to help you be ready for your examinations. In addition you will be supplied with a comprehensive Study Guide which will help you through the study and assessment process. Through UK Distance Learning & Publishing, the fascinating subject of A-Level Biology is now available to distance learning students for study at home using our online learning platform. online learningThis course can be taken at anytime and anywhere using our online learning platform. Through UK Distance Learning & Publishing, the fascinating subject of A-Level Biology is now available to distance learning students for study at home. Closely home study Psychology is an engaging and challenging subject that offers fascinating insights into the workings of the human mind. With this distance learning online course in A-Level Psychology, you can study at home with UK Distance Learning & Publishing, confident that our experienced tutors are always on hand to offer you guidance and support.In this online home study A-Level Psychology course, you will study a broad range of topics. These include important contemporary issues such as memory, attachment, stress, sleep and celebrity. Students will also study psychopathology, specialising in schizophrenia.Through studying psychology, we can learn more about how we think, whilst simultaneously developing our own powers of critical and evaluative thought. This well-respected A-Level subject is ideal for those who aspire to study psychology at university or to work in a related field. It is also suitable for those who simply want to further their own knowledge of this demanding discipline Following the popular AQA English Literature B specification, this home study course offers students the chance to study a variety of texts, including three of their own choice. This distance learning course aims to engage students in the study of literature, encouraging a multi-layered approach to the reading of literary texts. Focusing on the central place of narrative in the construction of texts, encouraging critical debate and fostering a recognition of the role of genre, this comprehensive home study course guides students through the study of A-Level English Literature, including full tutor support via email. An understanding and appreciation of literature can bring an enhanced enjoyment of reading and a lifelong love of language to all students. This distance learning course is suitable for students wishing to increase their general knowledge of English literature or for those seeking to progress to the study English literature or related subjects at a higher level. This online distance learning course in A-Level Religious Studies approaches the subject of religion as an academic discipline, allowing students to develop their own views and values through a critical, reflective and evaluative approach. The course is optimized for study at home and tpotally online, according to the AQA A-Level specification. The units studied include: Introduction to Religious Studies; Studies in Religion; and Religion and Human Experience. Particular attention is given to the topics of World Religions: Buddhism and to the Christian New Testament. This distance learning course is suitable for those who want to learn more about religious studies for their own interest or for those who wish to study for an A-Level qualification as preparation for study at a higher level. A-Level Religious Studies is a particularly good complement for Philosophy or History A-Levels and is suitable for those who intend to study Religious Studies or Theology at university level. The course is suitable for students or all religions or none and no prior knowledge of religious studies is required.
Mathpad Downloads: Views: Rating: Mathpad is an easy to use text editor for mathematics. You can mix together ordinary text and any mathematical expression. Ideal for math teachers to create quizzes, tests and handouts. Also, you can save the formatted text as an image. With most equation editors you choose a template with the mouse, type a few keystrokes and repeat the process. Mathpad doesn't work this way. Mainly, you just type characers at the keyboard. Most people find this to be easier and faster. Mathpad has some unusual features. For example, the implementation of standard italics for math symbols is simplified by the "automatic math" specification. In this mode, the ordinary text and math expressions get italicized correctly. Another example - the macros feature allows macros with arguments and is easier to use than more sophisticated macros as in Tex/ Free download from Shareware Connection - Mathpad is an easy to use text editor for mathematics. You can mix together ordinary text and any mathematical expression. Ideal for math teachers to create quizzes, tests and handouts. Also, you can save the formatted text as an image. MathpadDraw - MathpadDraw is a drawing tool for mathematical objects, both two and three dimensional. A major feature of the program is the ability to annotate a drawing with almost any mathematical text. Mathpad Draw - The Mathpad Draw application was designed to create static drawings of various mathematical objects. Dolphin Text Editor Menu - Dolphin Text Editor Menu expands the functionality of any text editor. It doesn't replace your current text editors - it makes them better. It works with any software where you can type sentences or paragraphs. Boxer Text Editor - ThisShareware Connection periodically updates pricing and software information of 'Mathpad' from company source 'John Gillam' , so some information may be slightly out-of-date. You should confirm all information before relying on it. Software piracy is theft, Using 'Mathpad' crack, password, serial numbers, registration codes, key generators is illegal and prevent future development of Mathpad
... More About This Book instruction. Manhattan GMAT's Quantitative Strategy Guide Set includes our 5 books devoted to this section's major content areas providing detailed and specialized instruction on Number Properties, Fractions, Decimals, and Percents, Word Problems, Algebra, and Geometry. Developed by the best teachers in the business, these 5 guides are the backbone of Manhattan GMAT's quantitative curriculum and the most in-depth, comprehensive materials available. The Manhattan GMAT Quantitative Strategy Guide Set is aligned to the 13th edition GMAC Official Guide and includes: The Number Properties Guide; The Fractions, Decimals, and Percents Guide; The Word Problems Guide; The Algebra Guide; The Geometry Guide. Purchase of this set includes one year of access to Manhattan GMAT's online computer-adaptive practice exams and 5 Quantitative Question Banks. Meet the Author In the last decade, Manhattan GMAT has grown from a single, dedicated tutor to a major test prep company with locations across the US
First Course in Topology : Continuity and Dimension - 06 edition Summary: How many dimensions does our universe require for a comprehensive physical description? In 1905, Poincaré argued philosophically about the necessity of the three familiar dimensions, while recent research is based on 11 dimensions or even 23 dimensions. The notion of dimension itself presented a basic problem to the pioneers of topology. Cantor asked if dimension was a topological feature of Euclidean space. To answer this question, some important topological ...show moreideas were introduced by Brouwer, giving shape to a subject whose development dominated the twentieth century. The basic notions in topology are varied and a comprehensive grounding in point-set topology, the definition and use of the fundamental group, and the beginnings of homology theory requires considerable time. The goal of this book is a focused introduction through these classical topics, aiming throughout at the classical result of the Invariance of Dimension. This text is based on the author's course given at Vassar College and is intended for advanced undergraduate students. It is suitable for a semester-long course on topology for students who have studied real analysis and linear algebra. It is also a good choice for a capstone course, senior seminar, or independent study200698.16 +$3.99 s/h New Stork Group Phoenix, MD Brand new and unread! Join our growing list of satisfied customers!
Trigonometry - 5th edition Summary: As the best-seller in its field, Trigonometry, 5/e, offers both instructors and students a more solid, comprehensive, and flexible program than ever before. Designed for the one- or two-term precalculus course, the text introduces trigonometry first with a unit circle approach and then with the right triangle. The text's unparalleled exercises, motivating real-life applications, cutting-edge design, and innovative ancillaries and technology resources make it the most...show more complete program available. New! Interactive Trigonometry 2.0 CD-ROM provides detailed solutions to every odd exercise in the text, self-assessment testing, a built-in Meridian Graphing Calculator Emulator, animations, video clips, simulations and editable graphs, and the entire contents of the text. For those who prefer a web-based program, Internet Trigonometry 1.0 is an interactive text-specific subscription web site offering all of the above features plus opportunities for interaction online with peers and instructors. New! Four new student success tools designed to help students approach the material with confidence include a chapter-opening guide, How to study this chapter; section-opening objectives and real-life applications, What you should learn and Why you should learn it; and chapter summaries, What did you learn? New! Each exercise set now concludes with Synthesis Exercises, promoting further exploration, critical thinking, and writing; and ReviewExercises. Abundant current, real-life applications are integrated throughout the examples and exercises and indicated by a globe icon. The wide variety of computational, conceptual, and applied problems are graded in difficulty. Special Algebra of Calculus exercises help prepare students for the future study of calculus. Exploration sections introduce select topics to engage students in active discovery of mathematical concepts and relationships, often using the power of technology. Carefully chosen examples contain step-by-step solutions and side-by-side explanations that lead students through mathematical concepts
ISBN: 0321756452 / ISBN-13: 9780321756459 Prealgebra This clear, accessible treatment of mathematics features a building-block approach toward problem solving and realistic, diverse applications. The ...Show synopsisThis clear, accessible treatment of mathematics features a building-block approach toward problem solving and realistic, diverse applications. The "Putting Your Skills to Work" and new chapter-end feature, "Math in the Media," present readers with opportunities to utilize critical thinking skills, analyze and interpret data, and problem solve using applied situations encountered in daily life. The goal of the changes in the 2nd edition is to upgrade the level of algebra in the book--This is accomplished by introducing equations, evaluating expressions, and properties of exponents earlier and revisiting the topics more often. Readers now learn how to solve equations using one principle first (Chapters 1, 3, 4, and 5)--Using both principles together is covered (Ch. 6) after readers have had substantial practice using one principle of equality. Contains 2 chapters dedicated to algebra skills (Ch. 3 and 6). A substantial increase in coverage of evaluating expressions (nearly double) from the first edition. Signed numbers are now covered earlier in Chapter 2 and Whole number operations are covered in one chapter vs. two in the previous edition.Hide synopsis ...Show more56452 Used book, good condition with normal wear...Good. 0321756452 Used book, good condition with normal wear/tears/ markings/writings. US edition as shows. Will ship with tracking. Standard/ Expedited/ Second day Shipping methods available
ISBN: 0321448049 / ISBN-13: 9780321448040 Mathematics for Elementary School Teachers Future elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be ...Show synopsisFuture elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be called upon to teach. This text uniquely balances "what" they will teach (concepts and content) with "how" to teach (processes and communication). As a result, students using "Mathematics for Elementary School Teachers"Hide synopsis ...Show more To see available supplements that will enliven your course with activities, classroom videos, and professional development for future teachers, visit ... Please read description before purchase >>> instructor...Good. Please read description before purchase >>> instructor annotated version printed on cover with all identical Students content with teaching tips, and all solutions text only no access code. Used has highlighting throughout book good binding Quicker shipper with tracking # Description:Like New. U.S. Students hardcover edition in plastic wrap text...Like New. U.S. Students hardcover edition in plastic wrap text only no supplements >light scratches on cover otherwise New
Description of Saxon Advanced Math: Solutions Manual by SaxonBased on Saxon's proven methods of incremental development and continual review strategies, the Saxon Advanced Math program builds on intermediate algebraic concepts and trigonometry concepts introduced in Algebra 2 and prepares students for future success in calculus, chemistry, physics and social sciences. This Solutions Manual contains solutions to all student textbook practices and problem sets. Product: Saxon Advanced Math: Solutions Manual Author: Saxon Other: Jason Roucloux Edition Number: 2 Edition Description: Teacher Series: Saxon Advanced Math Binding Type: Paperback Media Type: Book Number of Pages: 440 Weight: 1.95 pounds Length: 10.87 inches Width: 8.48 inches Height: 0.84 inches Publisher: Saxon Publishers Publication Date: January 2005 Subject: Algebra, Calculus & Trig, Math Curriculum Name: Saxon Learning Style: Auditory, Kinesthetic, Visual Teaching Method: Traditional There are currently no reviews for Saxon Advanced Math: Solutions Manual. This helped my first two children to be ready for calculus in college--they were able to test out of college algebra. My youngest child needed to go at a slower rate and had to take college Math in college. So it is great for those who have a bent for math and science. 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.
"Two stone cutters were asked what they were doing. One said that he was cutting stones into blocks. The other said he was a member of a team building a cathedral." The Need to Refocus College Algebra The pragmatic reason students take College Algebra is to fulfill a college or state requirement. Most of these students will confront a traditional College Algebra course based on a 1950's curriculum developed to prepare students to take Calculus, a course that over 90% of them will never enter. Typically students view the content of traditional College Algebra as irrelevant. and the course itself as a tedious and incomprehensible barrier to obtaining a college education. Currently this College Algebra barrier is terminating the college graduation ambition of approximately half a million students per semester. The use of technology, in the form of graphing calculators, is slowly penetrating traditional College Algebra courses. However, it has not materially changed the courses even though the use of technology can trivialize the majority of the exercises found in these courses. The outdated curriculum, lack of goals to address student needs, student and faculty dissatisfaction, and unacceptably high "F or D or withdraw" (FDW) rates, support the claim that traditional College Algebra is not working. In the words of a Dean of Science and Mathematics, at the Conference to Improve College Algebra held at the U.S. Military Academy, February, 2002, "Traditional College Algebra is a boring, archaic, torturous course that does not help students solve problems or become better citizens. It turns off students and discourages them from seeking more mathematics learning." A new approach to College Algebra Rather than preparing students for Calculus, today's reality is that College Algebra is, currently, the terminal mathematics course for most students. If, rather than fixate on pre-calculus, College Algebra could refocus on the practical, quantitative needs of today's students ... in society ... in the workplace ... and in other disciplines .... then College Algebra could not only help reverse today's unacceptably high rate of college failure but it could also give students a more positive attitude toward mathematics and a greater ability to use mathematics in their personal and professional lives. While many factors contribute to the current high FDW rate (high school preparation, placement, content, attitude, pace, pedagogy, etc.), several improved College Algebra programs have already succeeded in lowering FDW rates by 15 to 25 percentage points while reversing the negative student attitudes. Students in these programs were drawn from the same pool and were subject to the same placement as students in the traditional sections. The difference was in the content and the pedagogical focus. Improved college algebra courses better serve the approximately 90% of students who do not enter into math-intensive programs. They also provide a more effective preparation for those who do go on to Calculus I. More importantly, improved College Algebra courses have demonstrated in a variety of schools ways in which College Algebra can be transformed from extracting an unacceptable cost to providing a valuable asset. Contemporary College Algebra To meet the needs of today's students, College Algebra needs to be transformed by refocusing both the content and pedagogy in order to develop competent and confident problem-solvers. Content needs to be real-world problem-based, to emphasize problem solving in the modeling sense and to include elementary data analysis. Pedagogy needs to be student-centered, involving development of communication skills, appropriate use of technology and small group activities. Projects should be designed to create student confidence and positive experiences. The gateway function of college algebra means that transformed or improved courses can provide a basis on which to develop a college-wide quantitative literacy program. The interdisciplinary collaboration that is important to the development and ongoing assessment of an improved college algebra course provides opportunities to link problem solving to the quantitative needs of other disciplines. In this sense, these disciplines provide laboratory experiences for college algebra students. The Contemporary College Algebra website exists to share resources that can assist educators in bringing about this transformation in the College Algebra curriculum. Resources at this website A textbook based on work supported by the Exxon Education Foundation and by the National Science Foundation, developed as a practical tool for mathematics educators wishing to offer a refocused College Algebra course — a Contemporary College Algebra course in which students are empowered to use mathematics to quantify real-life situations and evaluate potential solutions to these problems through modeling. Students empowered with the necessary mathematical knowledge, confidence, and skills, enabling them to continue in more advance mathematics or quantitatively based courses, to get degrees in these areas, and to be successful in mathematics - dependent careers; A curriculum that changes from one of symbolic manipulation, skill building and emphasis on mechanics, and memorization of algebraic techniques to one that emphasizes variables and functions, mathematical models and representations, data based interdisciplinary applications that are relevant and meaningful, and more and better use of technology. Energized and enthusiastic teachers with high expectations, who are using multiple approaches, teaching and learning with technology, accomplishing learning objectives using exercises, small group activities, and projects connecting mathematical ideas within the discipline and across disciplines, using a variety of assessment methods, and discussing issues, questions, and ideas with colleagues; Students who are actively involved in learning algebra through individual and group activities which involve mathematical modeling, who are solving problems arising from a variety of disciplines, and using computers and calculators to generate numerical examples, graph data points, and, conjecture and reason about mathematics; The link above will take you to a library of articles in support of the National Movement to Refocus College Algebra. These articles provide useful arguments, explanations and examples for educators seeking to develop a more relevant, contemporary college algebra program in their own institutions. We welcome your contributions to our interactive Discussion Forum. You may use this Forum to ask questions, answer questions by others or simply contribute your thoughts, insights and experiences relating to College Algebra and its related topics. We thank you in advance for your contributions. We welcome feedback from both educators and students that will help improve the Contemporary College Algebra website. Any contribution you can make that will help our visitors better understand the "algebra failure" problem — and ways in which we can develop workable solutions to it — will be greatly appreciated. To send your feedback, click on the link above.
Matrix groups are a beautiful subject and are central to many fields in mathematics and physics. They touch upon an enormous spectrum within the mathematical arena. This textbook brings them into the undergraduate curriculum. It is excellent for a one-semester course for students familiar with linear and abstract algebra and prepares them for a graduate course on Lie groups. Matrix Groups for Undergraduates is concrete and example-driven, with geometric motivation and rigorous proofs. The story begins and ends with the rotations of a globe. In between, the author combines rigor and intuition to describe basic objects of Lie theory: Lie algebras, matrix exponentiation, Lie brackets, and maximal tori. The volume is suitable for graduate students and researchers interested in group theory. Editorial Reviews Review "this is an excellent, well-written textbook which is strongly recommended to a wide audience of readers interested in mathematics and its applications. The book is suitable for a one-semester undergraduate lecture course in matrix groups, and would also be useful supplementary reading for more general group theory courses." ---- Mathematical Reviews Most Helpful Customer Reviews This book is a great introduction to matrix groups and related ideas. The author explains the basic ideas in a clear, concise, and precise way. Although there are many excellent texts on matrix groups and more abstract properties of groups, this book provides the most accessible introduction to the subject that I have found. The book is short and easy to read through, compact, and economically priced. I strongly recommend reading this book before attempting to delve into more advanced texts. The clear and unified treatment of the real, complex, and quaternion groups is very nice. Overall, the writing style is so lucid, it is the kind of book where you feel that the writer is teaching you personally, rather than lecturing to an empty hall. Because the book provides such an excellent introduction to the subject, I give it the full 5 stars. The book has a few typos and gaps, but most are pretty obvious. I hope that the author will expand on this book in a future edition, perhaps including a chapter on basic group theory. When you finish reading the book, your only complaint will be that it isn't longer! Given the excellent exposition, I will be on the lookout for any future texts from this author. It's true as the other reviewers have said: this is an exceptionally good introduction to Lie groups and Lie algebras via matrix groups. It's also suitable for self-study provided you have the required math background. Although it's very much in the definition-theorem-proof mode, there's plenty of insightful and well-written exposition motivating the formal development. My only complaint is that there are no solutions to any exercises and even worse, the proofs of a number of propositions are left to the reader. This detracts from the value of the book for self-study, but not too much because on balance, in my view, the sheer excellence of the presentation pretty much makes up for that all too common "sin of omission". This book gathers the important properties of matrix groups and shows where the studies of these groups tend to. Indeed, the great thing about such groups is that they are : a.) groups b.) topological groups c.) smooth manifolds (for most of them) And usually in literature, you don't find these properties treated in the same book. That would be the original thing about this one. It's clearly undergraduate level, but well explained. The only regret if I may say, is that many proofs are left as exercise, and I know it must be a size problem with the editor, but it's always better to have an "adult" view on the proofs instead of doing them on our own (that's why we have exercises, to use the theorems and properties).
Maths Quest Manual for the TI-Nspire CAS Calculator 4E & eBookPLUS This third edition of Maths Quest 11 Mathematical Methods CAS is a comprehensive text designed to meet the requirements of the VCE Units 1 and 2 Mathematical Methods CAS course. The textbook also benefits from new supporting calculator companions containing comprehensive step-by-step CAS calculator instructions, fully integrated into worked examples, for the TI-Nspire CAS calculators. The student textbook contains the following new features: • cumulative exam practice questions to assist with exam preparation The textbook continues to offer the following award-winning features: • exam practice sections with allocated time and marks. Fully worked solutions to these sections are available on eBookPLUS. • technology-free questions • electronic tutorials for key worked examples in each chapter • interactivities • eLessons • full colour with stimulating photographs and graphics • carefully graded exercises with many skill and application problems, including multiple-choice questions • easy to follow worked examples in the Think–Write format • cross references throughout exercises to relevant worked examples • comprehensive chapter reviews with exam-style questions • eBookPLUS references throughout to guide students and teachers to relevant online material. What is eBookPLUS? Maths Quest 11 Mathematical Methods CAS 3e eBookPLUS is an electronic version of the textbook and a complementary set of targeted digital resources. These flexible and engaging ICT activities are available to you online at the JacarandaPLUS website ( Students and teachers are given step by step instructions as well as clear and colourful calculator screen shots to enable them to learn to use the TI-Nspire CAS calculators loaded with the latest Operating System. Maths Quest Manual for the TI-Nspire CAS calculator (Operating System v3) is a comprehensive step-by-step guide to using the TI-Nspire CAS calculator. It is designed to help students and teachers to integrate Computer Algebra Systems (CAS) into their learning and teaching of Mathematics. Maths Quest Manual for the TI-Nspire CAS calculator is suitable for use by students from Years 9 to Year 12. Features: • Calculator screen shots are now in full colour • New calculator functions, such as Vernier Dataquest, are explained • Each chapter is divided into 'How to' sections that provide clear, step-by-step instructions to the user • Easy-to-follow keystrokes and screen shots are accompanied by explicit explanations • Worksheets for almost every section are provided for further practice • A chapter of problem-solving questions is included with fully worked solutions on the eBookPLUS
Provides an examination of the theory of super-recursive algorithms and its ramifications for the computer industry, networks, artificial intelligence, embedded systems, and the Internet. This book demonstrates how these algorithms are more appropriate as mathematical models for modern computers. more... Pyramid Algorithms presents a unique approach to understanding, analyzing, and computing the most common polynomial and spline curve and surface schemes used in computer-aided geometric design, employing a dynamic programming method based on recursive pyramids. The recursive pyramid approach offers the distinct advantage of revealing the entire... more... This work is a self-contained elementary exposition of the theory of recursive functionals, that also includes a number of advanced results. Although aiming basically at a theory of higher order computability, attention is restricted to second order functionals, where the arguments are numerical functions and the values, when defined, are natural numbers.... more...
If you read the chapters in the book, pay attention in class, and actually do the homework, you're already in the top third of the class or so. It's fairly straightforward, but the homework will take a while. Don't just Google the problems unless you're stuck--no pain, no gain.
Elements of Problem Solving Become an expert problem solver with this imperishable volume. Capable of being enjoyed in one or two sittings, this concise volume offers valuable tips, strategies, and habits for anyone learning calculus, physics, or engineering. Product Description This concise volume seeks to address the silent void that university students embarking on mathematical coursework inevitably face: How does one apply ideas, concepts, and theory to story problems that at first seem intractable? We pay equal attention to the abstract aspects of problem solving as well as the practical. We discuss common misperceptions of ability and pay close attention to the attitudes and habits employed by successful problem solvers. We reveal problem solving as both achievable and fun, and illustrate these themes with specific examples and stories throughout the text. We also enumerate simple, executable steps that can be implemented when faced with virtually any new problem: understand the problem; draw a picture; define variables; be consistent; utilize principles. Far from trivial, these steps often lead novice problem solvers to error. In sum, we hope you find this charming work to be an indispensable guide on your journey and an effective means to improving and maturing your problem-solving capabilities.
Beginning Topology - 05 edition Summary: With a nice balance of mathematical precision and accessibility, this text provides a broad introduction to the field of topology. Author Sue Goodman piques student curiosity and interest without losing necessary rigor so that they can appreciate the beauty and fun of mathematics. The text demonstrates that mathematics is an active and ever-changing field with many problems still unsolved, and students will see how the various areas of mathematics-algebra, combinator...show moreics, geometry, calculus, and differential equations-interact with topology. Students learn some of the major ideas and results in the field, do explorations and fairly elementary proofs, and become aware of some recent questions. By presenting a wide range of topics, exercises, and examples, Goodman creates an interactive and enjoyable atmosphere in which to learn topology. Benefits: The first three chapters provide the foundation for the course and then Chapters 4-7 branch out into some of the various fields of topology: combinatorial topology and map coloring, differential and algebraic topology, and knot theory. The book introduces the flavor of topology as a whole while introducing the essentials in the context in which the students work (primarily subsets of Euclidean space). A wealth of examples and exercises appear throughout each chapter (as opposed to the end of the chapter) to keep students actively involved in the process of learning and discovery. The projects are designed to be less routine and are ideal for group work. They may involve some exploratory activities where there is not a simple answer, and may lead students through more complicated arguments. Maps and Map Coloring. The 5-color Theorem for S². Introduction to Graphs. Graphs in Surfaces. Imbedding the Complete Graphs, and Graph Coloring. 5. VECTOR FIELDS ON SURFACES. Vector Fields in the Plane. Index of a Critical Point. Limit Sets in the Plane. A Local Description of a Critical Point. Vector Fields on Surfaces. Proof of the Poincaré-Hopf Index Theorem. 6. THE FUNDAMENTAL GROUP. Path Homotopy and the Fundamental Group. The Fundamental Group of the Circle. Deformation Retracts. Further Calculations. Presentations of Groups. The Seifert-van Kampen Theorem and the Fundamental Groups of Surfaces. Proof of the Seifert-van Kampen Theorem. 7. INTRODUCTION TO KNOTS. Knots--What they are and How to Draw Them. Prime Knots. Alternating Knots. Reidemeister Moves. Some Simple Knot Invariants. Surfaces with Boundary. Knots and Surfaces. Knot Polynomials. A Short Knot Table.59.19 +$3.99 s/h New mermaidbookstore Cupertino, CA 0534424260 HARDCOVER. NEW. Never used. We do not ship to PO Box/APO/FPO addresses. $75
polynomial polynomial In algebra, an expression consisting of numbers and variables grouped according to certain patterns. Specifically, polynomials are sums of monomials of the form axn, where a (the coefficient) can be any real number and n (the degree) must be a whole number. A polynomial's degree is that of its monomial of highest degree. Like whole numbers, polynomials may be prime or factorable into products of primes. They may contain any number of variables, provided that the power of each variable is a nonnegative integer. They are the basis of algebraic equation solving. Setting a polynomial equal to zero results in a polynomial equation; equating it to a variable results in a polynomial function, a particularly useful tool in modeling physical situations. Polynomial equations and functions can be analyzed completely by methods of algebra and calculus. See alsoorthoganal polynomial. This entry comes from Encyclopædia Britannica Concise. For the full entry on polynomial, visit Britannica.com.
Concise Introduction to MATLAB 9780073385839 ISBN: 0073385832 Pub Date: 2007 Publisher: McGraw-Hill Companies, The Summary: A Concise Introduction to Matlab is a simple, concise book designed to cover all the major capabilities of MATLAB that are useful for beginning students. Thorough coverage of Function handles, Anonymous functions, and Subfunctions. In addition, key applications including plotting, programming, statistics and model building are also all covered. 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 organization of the material makes it easy to locate information and navigate through the textbook. Palm, William J., III is the author of Concise Introduction to MATLAB, published 2007 under ISBN 9780073385839 and 0073385832. Five hundred fifty four Concise Introduction to MATLAB textbooks are available for sale on ValoreBooks.com, one hundred seventeen used from the cheapest price of $21.93, or buy new starting at $63.96.[read more]Writing programs to do tedious computations quickly and making a graphical representation of that data. One example is the differential equation solver. Very versatile built in functions to do that task.
{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":57.47,"ASIN":"0521575400","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":54.28,"ASIN":"0521347939","isPreorder":0}],"shippingId":"0521575400::UOAb4jeOgm62AS5HZ1Tl08R32OvSi74Ft%2B6x3UCbYSnMChW0sisajC9%2BWg%2B8GwievH%2B6bks9FjD33SYUseWDwohKrRXUWIFpv%2FAaKiNMgFw%3D,0521347939::Knapk7ufFWs7bFAYhnhSJuFWOMTlzZvKB%2FxJMSGjxzcS%2FCh8zgH3Z0yBkYJOtgsTEfUmWVJU8hG12UmOZaUoMBpYJf5XY3KehqkTcehB'm pleased to report there is a new edition of R. Burn's A Pathway into Number Theory, a book that takes readers quickly and painlessly from simple facts about whole numbers to the wonders of the quadratic forms, Pell's equation and Minkowski's theorem.' Ian Stewart, New Scientist '... admirably suitable for those meeting number theory for the first time and for unsupported individual study.' Nick Lord, The Mathematical Gazette Book Description Now in its second edition, this book consists of a sequence of exercises that will lead readers from simple number work to the point where they can prove algebraically the classical results of elementary number theory for themselves. A high-school course in mathematics is the only requirement. This book is a carefully sequenced set of problems along with answers and a few comments. Burn uses those problems to introduce important number theory ideas. I enjoyed working through the problems to learn more about number theory. Most problems are accessible to those with a good high school mathematics background.
Algebra The Boundless Algebra textbook is a college-level, introductory textbook that covers the important subject of Algebra -- one of the basic building blocks of studies in higher mathematics Building Blocks of Algebra -- Real Numbers, Exponents, Scientific Notation, Order of Operations, Working with Polynomials, Factoring, Rational Expressions, Radical Notation and Exponents, Basics of Equation Solving Systems of Equations and Matrices -- Systems of Equations in Two Variables, Systems of Equations in Three Variables, Matrices, Matrix Operations, Inverses of Matrices, Determinants and Cramer's Rule, Systems of Inequalities and Linear Programming, Partial Fractions Conic Sections -- The Parabola, The Circle and the Ellipse, The Hyperbola, Nonlinear Systems of Equations and Inequalities
LoviOtvet Calculator Education Screenshots Developer Notes Universal Calculator or solver - as any name, and unique to the application is not found. Teaching tool for solving mathematical examples and algebraic expressions. The application provides a solution to the broken steps, and if necessary - all math will be solved graphically - in a...
Intermediate Algebra The Student Study Guide contains additional step-by-step worked out examples, exercises, practice tests, and practice finals. Solutions to all ...Show synopsisThe Student Study Guide contains additional step-by-step worked out examples, exercises, practice tests, and practice finals. Solutions to all exercises, tests, and final examinations are found in the Student Study Guide. It also contains Study Skills and note-taking suggestions
Algebra : Combined Approach - 4th edition Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief.Algebra: A Combined Approach, Fourth Editionwas written to provide students with a solid foundation in algebra and help them effectively transition to their next mathematics course. The new edition offers new resources like theStudent Organizerand now includesStudent Resourcesin the ...show moreback of the book to help students on their quest5227.3803 +$3.99 s/h Good Textbook Bookie Little Rock, AR 4th Edition. With used stickers on front and back cover. Ships fast! Expedited shipping 2-4 business days; Standard shipping 7
Ozone Park Cal idea of a function and its inverse is introduced. Extensive use is made of exponential and logarithmic functions, including graphing and solving equations. Applications include compound interest problems and radioactive decay
Algebra and Trigonometry Mike Sullivann++ s time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing ...Show synopsisMike Sullivann++ s time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing with homework, and reviewing the concepts. In the Ninth Edition , Algebra and Trigonometry has evolved to meet todayn++ s course needs, building on these hallmarks by integrating projects and other interactive learning tools for use in the classroom or online16569....New in new dust jacket. Brand New as listed. ISBN 9780321716569. Clean! Out of sight Shipping & Customer Service! We process all orders same day! ! The previous version of this textbook is better than this version. The examples are not as defined as the previous version, I think that is important in a book. In buying this textbook, I recommend one transfers to another classroom that uses anohter
working through this Study Guide you will definitely improve your results - whether you are working towards being the top performer in your class or whether you regularly break out in a sweat when you have to present your test scores or school report at home! This marvellous resource provides you with: • Introductions to and discussions of the various themes and topics relevant to Grade 10 Mathematics • Fully worked-out examples with their answers • Loads of exercises and questions to practise your newly gained skills • Answers to these exercises at the back of the book • Exemplar examination papers for you to work through and their answers This Study & Master Guide is written according to the NCS for Mathematics.
Description this Table of Contents 1. Critical Thinking Skills 1.1 Inductive Reasoning 1.2 Estimation 1.3 Problem Solving 2. Sets 2.1 Set Concepts 2.2 Subsets 2.3 Venn Diagrams and Set Operations 2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets 2.5 Applications of Sets 2.6 Infinite Sets 3. Logic 3.1 Statements and Logical Connectives 3.2 Truth Tables for Negation, Conjunction, and Disjunction 3.3 Truth Tables for the Conditional and Biconditional 3.4 Equivalent Statements 3.5 Symbolic Arguments 3.6 Euler Diagrams and Syllogistic Arguments 3.7 Switching Circuits 4. Systems of Numeration 4.1 Additive, Multiplicative, and Ciphered Systems of Numeration 4.2 Place-Value or Positional-Value Numeration Systems 4.3 Other Bases 4.4 Computation In Other Bases 4.5 Early Computational Methods 5. Number Theory and the Real Number System 5.1 Number Theory 5.2 The Integers 5.3 The Rational Numbers 5.4 The Irrational Numbers and the Real Number System 5.5 Real Numbers and Their Properties 5.6 Rules of Exponents and Scientific Notation 5.7 Arithmetic and Geometric Sequences 5.8 Fibonacci Sequence 6. Algebra, Graphs, and Functions 6.1 Order of Operations 6.2 Linear Equations in One Variable 6.3 Formulas 6.4 Applications of Linear Equations In One Variable 6.5 Variation 6.6 Linear Inequalities 6.7 Graphing Linear Equations 6.8 Linear Inequalities In Two Variables 6.9 Solving Quadratic Equations By Using Factoring and By Using the Quadratic Formula 6.10 Functions and Their Graphs 7. Systems of Linear Equations and Inequalities 7.1 Systems of Linear Equations 7.2 Solving Systems of Linear Equations by the Substitution and Addition Methods 7.3 Matrices 7.4 Solving Systems of Linear Equations by Using Matrices 7.5 Systems of Linear Inequalities 7.6 Linear Programming 8. The Metric System 8.1 Basic Terms and Conversions Within the Metric System 8.2 Length, Area, and Volume 8.3 Mass and Temperature 8.4 Dimensional Analysis and Conversions To and From the Metric System
Overview Editorial Reviews Booknews Bittinger (mathematics, Indiana U. and Purdue U.) uses a five step problem solving approach with real data applications to make algebra both straightforward and connected to everyday life. Detailed graphs and color drawings and photographs also help students to visualize mathematical concepts. The book is designed to assist in every step of curriculum, from review exercises with answers, to pre and post-tests. There are also a number of supplemental materials available for the instructor to use in conjunction with this text. Annotation c. Book News, Inc., Portland, OR (booknews.com) Systems of Equations in Two Variables. Solving by Substitution or Elimination. Solving by Applications: Systems of Two Equations. Systems of Equations inThree Variables. Solving Applications: Systems of Three Equations. Systems of Linear Inequalities in Two Variables. Business and Economic
More About This Textbook Overview Although extensively revised, this new edition continues in the fine tradition of its predecessor. Major changes include: a notation that formalizes the distinction between equality and congruence and between line, ray and line segment; a completely rewritten chapter on mathematical logic with inclusion of truth tables and the logical basis for the discovery of non-Euclidean geometries; expanded coverage of analytic geometry with more theorems discussed and proved with coordinate geometry; two distinct chapters on parallel lines and parallelograms; a condensed chapter on numerical trigonometry; more problems; expansion of the section on surface areas and volume; and additional review exercises at the end of each chapter. Concise and logical, it will serve as an excellent review of high school geometry
Questions About This Book? The Used copy of this book is not guaranteed to inclue any supplemental materials. Typically, only the book itself is included. 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.
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IGCSEMathematics (4400) London Examinations November 2004 delivered locally, recognised globally ... Mathematics 4400, November 2004 PAPER 4H General Comments There were few errors which occurred regularly and hardly any at all on the first half of the Curriculum Description for London IGCSEMathematics (4400) IGCSEMathematics will • meet the needs of students of all abilities • provide a solid basis for AS and Advanced GCE or equivalent qualifications IGCSE in Mathematics 27 IGCSE in Music 28 IGCSE in Physics 30 IGCSE in PE 34 Options : Advice to Students 35 ... A full set of drawing pencils from 6B to 4H, a 30cm ruler, eraser, protractor, compass, scissors, sketchbook (A3), ... IGCSEMathematics Overview IGCSE in Mathematics aims to give students a foundation in mathematical skills and develop their ... Papers 3H and 4H (Higher Tier) Higher Tier students will take Papers 3H and 4H. Questions in the Higher Tier paper are targeted at
.... Show More. Along the way, readers will augment their understanding of both mathematics and music. The text explores the common foundations of the two subjects, which are developed side by side. Musical and mathematical notions are brought together, such as scales and modular arithmetic, intervals and logarithms, tone and trigonometry, and timbre and harmonic analysis. When possible, discussions of musical and mathematical notions are directly interwoven. Occasionally the discourse dwells for a while on one subject and not the other, but eventually the connection is established, making this an integrative treatment of the two subjects. The book is a text for a freshman level college course suitable for musically inclined or mathematically inclined students, with the intent of breaking down any apprehension that either group might have for the other subject. Exercises are given at the end of each chapter. The mathematical prerequisites are a high-school level familiarity with algebra, trigonometry, functions, and graphs. Musically, the student should have had some exposure to musical staffs, standard clefs, and key signatures, though all of these are explained
Share this Page MATH TREK Algebra 1 04/01/04 For curriculum-based algebra instruction, teachers and students can use MATH TREK Algebra 1. The multimedia program includes tutorials, assessments and student tracking. Students can use the program's scientific calculator, glossary and journal to help them complete the various exercises and activities. The assessment and student-tracking features provide immediate feedback to students so that they can stay on top of their progress. This engaging program, complete with sound, animation and graphics, can be used on stand-alone computers or a network. NECTAR Foundation, (613) 224-3031
There are large chunks of mathematical knowledge that we often assume students have down. In fact, they often lack that knowledge. A good example is this: most students I work with have virtually no sense as to the value of the square root of 2 or the square root of 3. This post offers a few suggestions on how to help students in this regard. One of the legends of math relates to the invention of the coordinate plane system. The legend states that Descartes was inspired to invent this system after watching a fly move around on the ceiling over his bed. This post tells the story and points out its relevance for modern math teachers. Welcome! Here at "I'm a returning college adult now in the fourth week of my College Algebra course. The Algebra Survival Guide has FINALLY filled in the gaps in my earlier education. Thank you to the third power. Thirty years of math phobia gone in three hours of reading." — Mary Ellen Kirian, Lake Oswego, OR
Introduction to Real Analysis This introductory textbook details the rigorous treatment of the basic theory of functions of one real variable. Topological concepts such as open ...Show synopsisThis introductory textbook details the rigorous treatment of the basic theory of functions of one real variable. Topological concepts such as open set and closed set have been gathered together to provide a more unified discussion.Hide synopsis 1. Softcover, JOHN WILEY & SONS, 1982 Description:Fair. Corners/edges are worn/bent, covers slightly scuffed 008...Fair. Corners/edges are worn/bent, covers slightly scuffed2. Softcover, Wiley, 1999 Description:Good. INDIA EDITION with white labels covering up such...Good. INDIA EDITION with white labels covering up such information on red cover, wear to cover and edges with slight scratch to back cover, clean text in black and white print471572565 Introduction to Real Analysis Introduction to Real Analysis by Bartle and Sherbert is very appropriate for an Introductory Course in Real Analysis because it is standard, formal, and presents clear demonstrations. This third edition is more comprehensible than previous editions because it develops its topics with more details, presents more examples and even some applications. Many students will find this book accesible because It is not required to know concepts about Topology, and the students only have to know integral and differential Calculus. It is highly reccomended for Real Analysis professors
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Students will receive in-depth coverage of trigonometry, logarithms, analytic geometry, and upper-level algebraic concepts, as well as continued intermediate algebra and trigonometry concepts first introduced in Algebra 2; new lessons on functions, matrices, statistics, and the graphing calculator are also included. Saxon Teacher provides comprehensive lesson instructions that feature complete solutions to every practice problem, problem set, and test problem, with step-by-step explanations and helpful hints. These Advanced Math CD-ROMs contain hundreds of hours of instruction, allowing students to see and hear actual textbook problems being worked on a digital whiteboard. A slider button allows students to skip problems they don't need help on, or rewind, pause, or fast-forward to get to the sections they'd like to access. Problem set questions can be watched individually after the being worked by the student; the practice set is one continuous video that allows for easy solution review. For use with Advanced Math 2nd Edition. Four Lesson CDs and 1 Test Solutions CD included. Although both teach the lesson concepts using a digital whiteboard format, there are significant differences between "DIVE" and "Saxon Teacher". "DIVE" (Digital Interactive Video Educator) is an excellent tutorial, and uses different illustrations than those presented in the Saxon Textbook. It does not have solutions to the problem sets. "Saxon Teacher" contains the lesson instructions, practice problems and problem sets as they are presented in the Saxon Textbook, with some additional commentary. "Saxon Teacher" can also serve as a solutions manual for the practices, lessons and tests. This is a complete set which includes the "Saxon Teacher CD-ROMs". These CD-ROMs include step-by-step solutions to all of the questions in the problem sets and tests, working much like an audio-visual solutions manual. There is no book version of the Solutions Manual with this set, but it can be ordered separately as CBD stock number WW57311. Saxon Teacher is a CD-Rom that contains comprehensive lesson instructions with solutions to the practice problems as they are presented in the Saxon Textbook. It also serves as a solutions manual with step-by-step audio and visual solutions to the problem sets and tests. Teaching Tapes are DVD's that feature instructions by a state-certified teacher who explains and demonstrates each concept, example, practice problem and investigation. Problem set and test solutions are not included.
Description This revised and updated book is intended for undergraduates, graduate students, and practicing physicists who want to learn new Mathematica techniques for solving a general class of physics problems. The experimental activities included are designed to deepen and broaden the reader's understanding of physics. The book assumes some familiarity with Mathematica, but the focus is on physics rather than on rudimentary Mathematica techniques. Contents Getting Started \| General Physics \| Oscillating Systems \| Nonlinear Oscillating Systems \| Discrete Dynamical Systems \| Langrangians and Hamiltonians \| Orbiting Bodies \| Electrostatics \| Quantum Mechanics \| Relativity and Cosmology \| Index Related TopicsPhysics, Tutorial and Reference
Product Details See What's Inside Product Description By James King, Yuria Orihuela, Eric Robinson Classically, geometry has been the subject in which students encounter mathematical proof based on formal deduction. Attention to proof in the geometry curriculum is strengthened by a focus on reasoning and sense making. The authors examine the four key elements (conjecturing about geometric objects, construction and evaluation of geometric arguments, multiple geometric approaches, and geometric connections and modeling) identified in Focus in High School Mathematics: Reasoning and Sense Making in more detail and elaborates on the associated reasoning habitsRelated ProductsFocus in High School Mathematics: Geometry (eBook) -- A downloadable version of the book development of statistical reasoning must be a high priority for school mathematics. This book offers a blueprint for emphasizing statistical reasoning and sense making in the high school curriculum$27.95 Customers Who Bought This Also Bought...The development of statistical reasoning must be a high priority for school mathematics. This book offers a blueprint for emphasizing statistical reasoning and sense making in the high school curriculum. Learn how to use current research to improve the teaching and learning of mathematics. The Teaching and Learning Mathematics series presents ideas from research to improve mathematics education in schools. Each book presents findings from research to enhance the quality of classroom mathematics teaching and learning. Explore successful models of intervention. No Child Left Behind has set the high expectation that every child meet grade level expectations. This publication synthesizes the research on intervention programs and best practices related to mathematical instructional pedagogy and differentiation to assist teachers, schools, and school districts in improving the manner in which they serve children with challenges in mathematics. The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
See also: Courses MA-100 Precalculus (Fall & Spring; Yearly; 3.00 Credits; N,QM) This course is designed for students who need a structured review of precalculus mathematics. Topics covered include solving equations and inequalities, graphing, and analysis of functions, including polynomial and rational functions, exponential and logarithmic functions and trigonometric functions. Integrates the use of the software package Maple in classroom demonstrations and homework assignments. This course cannot be included in a mathematics POE. Prerequisites: High school algebra and trigonometry. MA-103 Quantitative Methods (Fall & Spring; Yearly; 3.00 Credits; Q) This course prepares students to be quantitatively literate citizens in today's world. By learning to think critically about quantitative issues, students will be able to make responsible decisions in their daily lives. Problems are analyzed and solved using numerical, graphical, statistical, and algebraic reasoning. Technology is used to help visualize data and facilitate calculations, as well as to present quantitative output and verbal arguments. MA-130 Calculus I (Fall & Spring; Yearly; 4.00 Credits; N,QM) An introduction to calculus including differentiation and integration of elementary functions of a single variable, limits, tangents, rates of change, maxima and minima, area, volume, and other applications. Integrates the use of computer algebra systems, and graphical, algebraic and numerical thinking. MA-155 The Heart of Mathematics (Spring; Yearly; 3.00 Credits; N) The goal of this course is to give non-mathematics students the hands-on experience of doing mathematics. Topics include infinity, higher dimensions, chaos, and probability. The emphasis will be on the process of doing mathematics: generating examples, looking for patterns, making conjectures, and proving these conjectures. May not be taken if you have completed MA130. MA-210 Foundations of Mathematics (Fall; Yearly; 3.00 Credits; CW) An introduction to the logical and set-theoretic basis of modern mathematics. Topics covered include propositional and predicate logic; induction; naive and axiomatic set theory, binary relations, mappings, infinite sets and cardinality; finite sets and combinatorics; and an introduction to the theory of computability. Students will learn to read and to express mathematical ideas in the set-theoretic idiom. Prerequisites: MA160 or MA116 or PL208 or MA208 or permission of the instructor. MA-233 Integrals Series & Differential Equations (Fall; Yearly; 2.00 Credits; N) Integration, Taylor and Fourier series, and an introduction to differential equations, with applications and the use of the software package Maple. (Course meets four times per week and concludes at midterm.) Note: A student may receive credit for MA233 or MA235, but not for both. Prerequisite: MA130. MA-299 Special Topics (Variable; Variable; 1.00-4.00 Credits) An introduction to one of the branches of mathematics not currently in the regular course offerings. Prerequisites: Vary depending on course offering. MA-303 Mathematical Modeling (Fall; Yearly; 3.00 Credits; N,QM,CW) How to use mathematics to model " real-world " problems. Modeling topics range from population dynamics to economics to the nuclear arms race. Mathematical tools range from calculus to curve fitting to computer simulation. How to make a little bit of mathematics go a long way. Note: MA160 is recommended. Prerequisite: MA130 and experience with programming and Minitab. MA-316 Combinatorics (Fall; Even Years; 3.00 Credits; QM,N) Advanced counting: what they didn't teach you on Sesame Street. An introduction to graphs, trees, and enumeration techniques with applications to computer science and biology. Prerequisites: MA116 or MA210 or MA220 or permission of the instructor. MA-325 Statistical Consulting (Variable; Variable; 2.00 Credits; N,QS,CW) The participating students will receive some training during the semester in consulting on statistical problems and will be expected to staff the Statistics Center and to assist in collaborative efforts with faculty and/or staff on projects. This provides the student with both real work experience in the field of statistics and a project based learning experience. May be taken multiple times for credit. Prerequisites: MA220 or BI305 or PY214. MA-350 Topics in Geometry (Spring; Odd Years; 3.00 Credits; N) Examines the history and development of geometry with an axiomatic development of Euclidean geometry leading to an investigation of hyperbolic and elliptical non-Euclidean geometries. The roles of these discoveries in the history of mathematics are emphasized. Prerequisites: MA210 or PL208 or MA208. MA-355 Nature of Mathematics (Spring; Odd Years; 1.00 Credit) An introduction to the history and philosophy of mathematics. Briefly traces the historical development of mathematics from its Oriental and Greek origins to modern times. Surveys the different philosophies of mathematics and provides some insight into the current crisis in the foundations ofmathematics. Corequisite: MA350. Prerequisite: MA210 or PL208 or MA208. MA-360 Abstract Algebra (Spring; Even Years; 3.00 Credits) Investigates the algebraic properties of the real numbers and their generalizations. Emphasis on group theory, with introductions to integral domains, rings, fields and vector spaces. Prerequisites: MA160 and MA210. MA-365 Number Theory (Fall; Odd Years; 3.00 Credits; N) An investigation of topics in Elementary Number Theory including divisibility, primes, congruence, congruence equations, quadratic residues and quadratic reciprocity, arithmetic and multiplicative functions, Diophantine equations, and other topics selected according to interest. Prerequisites: MA210 or permission of the instructor. MA-399 Special Topics (Variable; Variable; 1.00-4.00 Credits) An introduction to one of the branches of mathematics not currently included in the regular course offerings, such as number theory, history of mathematics, chaos and fractals, topology, graph theory, mathematical logic. Prerequisites: Vary depending on course offering. Note: abbreviated ST: Students may take each ST: course for credit. MA-480 Mathematics Seminar I (Fall; Yearly; 1.00 Credit; N) A discussion-based study of current mathematical literature and modeling problems. Students will both apply their previous mathematical knowledge and explore new topics. In addition, students may use this course as preparation for an individual research project to be completed in MA485. A junior taking this course may repeat it as a senior as MA481. Prerequisites: junior or senior standing, MA160, MA210 and MA235 or permission of the instructor. MA-485 Mathematics Research (Fall & Spring; Yearly; 3.00-5.00 Credits; N) Allows students to pursue a program of directed original research in pure or applied mathematics. Required of candidates for distinction in mathematics POE. Prerequisites: MA480.
Discrete math was a part of my Bachelors in Mathematics. I also have significant amounts of coursework in computer science including optimization of algorithms, and in mathematics including logic, set theory, combinatorics, number theory, and graph theory. The focus of my undergraduate work was geometry and topology.
Mathematics for Elementary Teachers: A Conceptual Approach 9780073519579 ISBN: 007351957X Publisher: McGraw-Hill Summary: Would you like to rent Mathematics for Elementary Teachers: A Conceptual Approach online from Valore Books now? If you would like to take advantage of discounted prices on pre-owned copies of this book published by McGraw-Hill, look at our selection now. Written by Albert B Bennett, Laurie J Burton and Leonard T Nelson, you can find the cheapest copies of this text book by using our site now. Buy Mathematics for Elem...entary Teachers: A Conceptual Approach online from us today and find out why so many people rent and buy books for college from us. Try our website now for the cheapest deals. Bennett, Albert B. is the author of Mathematics for Elementary Teachers: A Conceptual Approach, published under ISBN 9780073519579 and 007351957X. Nine hundred twenty three Mathematics for Elementary Teachers: A Conceptual Approach textbooks are available for sale on ValoreBooks.com, four hundred twenty one used from the cheapest price of $61.97, or buy new starting at $117.019th edition. Does Not include CD's, DVD's, access codes or other supplements. PLEASE NOTE: Minor ... [more] [less] Ships From:Salem, ORShipping:Standard, ExpeditedComments:ALTERNATE EDITION: International Edition: New Condition. SKU:9780071310024-1-0-15 Orders ship the same or next busin... [more] Minor water damage is present along the side edge in some areas (minor). M [more] .[less] This is my first college math book I've ever even opened. The examples are great! A lot of the odd numbered problems show the answers at the back of the book and that was a big help to know if we were doing the problem correct or not. I was taking Math 155 & 156. It used the book all the time and it helps to get the manipulative kit.
College Algebra Focusing on helping students to develop the conceptual understanding and analytical skills necessary to experience success in mathematics, this book ...Show synopsisFocusing on helping students to develop the conceptual understanding and analytical skills necessary to experience success in mathematics, this book presents mathematical topics using a learning system to actively engage students in the learning process. It addresses the diverse needs of students through an open design and other helpful features
A revision of McGraw-Hill's leading calculus text for the 3-semester sequence taken primarily by math, engineering, and science majors. The revision is substantial and has been influenced by students, instructors in physics, engineering, and mathematics, and participants in the national debate on the future of calculus. Revision focused on these key areas: Upgrading graphics and design, expanding range of problem sets, increasing motivation, strengthening multi-variable chapters, and building a stronger support package. Geometry: A Guided Inquiry by G.D. Chakerian, Calvin D. Crabill and Sherman K. Stein is designed to involve students in the teaching-learning process. It is through cooperative work, encouraged throughout the book, that students of varying abilities can learn together in the same classroom. Maths enthusiasts aren't necessarily the number-crunching geeks we tend to assume. In fact, they know that maths is about much more than numbers; it is a profoundly philosophical endeavour, as well as a stimulating mental exercise. In "How the Other Half Thinks", Sherman Stein emphasizes the creative elements of mathematics by exploring some significant mathematical discoveries through simple, intuitive manipulations. With an ingenious technique that uses no algebra or trigonometry, and only a minimum of arithmetic, Stein takes us through the thought process behind some of math's great discoveries and applications. Each chapter begins with a simple question about strings made up of the letters "a" and "b", which leads to other, more profound questions. Along the way, we become familiar with concepts from such fields as topology and probability, and learn how they have led to applications such as codes and radar, computing and even baseball statistics. Recreational and instructive, "How the Other Half Thinks" should appeal to die-hard maths enthusiasts (of which there are many) as well as those "right-brainers" who are looking for a way to understand and enjoy maths. More editions of How the Other Half Thinks: Adventures in Mathematical Reasoning: This highly readable volume covers a vast array of subjects, including number theory, topology, set theory, geometry, algebra, and analysis. Additional topics include primes, fundamental theory of arithmetic, rationals and irrationals, representation of numbers, congruence, probability, and more. A solutions manual is available upon request. 1994 edition. An Easygoing, Highly Entertaining Refresher on all the Math You'll Ever Need. What do two goats and a car have to do with making good decisions? Was the golden ratio used to build the Great Pyramid of Khufu? Can it be that some numbers are unmistakably "hot," while others are inherently "cool"? With his infectiously enthusiastic and engaging style, award-winning teacher and author Sherman K. Stein offers a new appreciation for mathematics, from the beauty of its logic ("as inevitable and memorable as a Mozart symphony") to its amazing power and pervasiveness in our lives. Requiring no math knowledge beyond basic arithmetic and high school geometry, Strength in Numbers is an enlightening introduction to all the math you'll ever need. More editions of Strength In Numbers: Discovering the Joy and Power of Mathematics in Everyday Life: Insiders, such as officials, experts, and pundits, are in the position to manipulate us outsiders, who are known as John or Jane Doe. Survival Guide for Outsiders discusses why outsiders are vulnerable and reveals some of the insiders techniques. At the same time it suggests ways to protect ourselves from falling victims to the insiders tricks. Throughout it utilizes real-world examples. Survival Guide for Outsiders shows there are two types of experts, and urges us to distinguish between them. Besides exploring the limits of prediction accuracy, it describes the inevitable gaps in reason and control when we make choices and commit ourselves to action. It warns about the abuse of numbers and computers, and demonstrates why they are often useless when making a choice in a controversial issue. Along the way, it examines the significance of the two components of a job; introduces the GRIMP, the model, the choice between incomparables, the dead-fish principle, distortions by commission and by omission, and the action syndrome. Awareness of these concepts will help you defend yourself and, incidentally, help preserve a functioning democracy. More editions of Survival Guide for Outsiders: How to protect yourself from politicians, experts, and other insiders:
Professor Pearson's book starts with an introduction to the area and an explanation of the most commonly used functions. It then moves on through differentiation, special functions, derivatives, integrals and onto full differential equations. As with other books in the series the emphasis is on using worked examples and tutorial-based problem solving... more... The book shows how the abstract methods of analytic semigroups and evolution equations in Banach spaces can be fruitfully applied to the study of parabolic problems. Particular attention is paid to optimal regularity results in linear equations. Furthermore, these results are used to study several other problems, especially fully nonlinear ones. Owing... more... This book deals with the constructive Weierstrassian approach to the theory of function spaces and various applications. The first chapter is devoted to a detailed study of quarkonial (subatomic) decompositions of functions and distributions on euclidean spaces, domains, manifolds and fractals. This approach combines the advantages of atomic and wavelet... more... This book gives an introduction to the basic theory of stochastic calculus and its applications. Examples are given throughout the text, in order to motivate and illustrate the theory and show its importance for many applications in e.g. economics, biology and physics. The basic idea of the presentation is to start from some basic results (without... more... Differential Equations for Scientists and Engineers is a book designed with students in mind. It attempts to take a concise, simple, and no-frills approach to differential equations. The approach used in this text is to give students extensive experience in main solution techniques with a lighter emphasis on the physical interpretation of... more...
ISBN: 0132999188 / ISBN-13: 9780132999182 Basic College Mathematics This clear, accessible treatment of mathematics features a building-block approach toward problem solving, realistic and diverse applications, and ...Show synopsisThis clear, accessible treatment of mathematics features a building-block approach toward problem solving, realistic and diverse applications, and chapter organizer to help users focus their study and become effective and confident problem solvers. The "Putting Your Skills to Work" and new chapter-end feature, "Math in the Media," present readers with opportunities to utilize critical thinking skills, analyze and interpret data, and problem solve using applied situations encountered in daily life. Chapter 7, Geometry, has been extensively revised and re-organized to include a new section 7.1 on angles and new section 7.4 devoted to triangles. Increased coverage of estimating with fractions and decimals with new " To Think About" exercises in Sections 2.5, 2.8, and 3.3 and a new lesson in Section 3.7. Coverage of fractions in Chapter 2 has been expanded as follows: " Section 2.6 now begins with a discussion of least common multiples so that the subsequent coverage of least common denominators is more complete; a new lesson on order of operations in Section 2.8 offers readers additional review of these rules and practice applying them to fractions; and a new mid-chapter test on fractions appears after Section 2.5. Percent applications are now covered in two sections (Sections 5.4 and 5.5) to allow for a more patient presentation of this important topic.Hide synopsis Basic College Mathematics The book was in good condition, as stated by the seller. I looked through it and it seems to be just what I need to help me brush up on the "basics" before I go ahead with an advanced Math class later this year. Thank you! 61 and taking a Career Readiness class that includes basic math. This book helped me to score in the highest level. Easy to understand, even for a silver hair who embraced her Texas Instruments Calculator decades ago and never let go
Ruskeepaa gives a general introduction to the most recent versions of Mathematica, the symbolic computation software from Wolfram. The book emphasizes graphics, methods of applied mathematics and statistics, and programming. Mathematica Navigator can be used both as a tutorial and as a handbook. While no previous experience with Mathematica is required, most chapters also include advanced material, so that the book will be a valuable resource for both beginners and experienced users. - Covers both Mathematica 6 and Mathematica 7 - The book, fully revised and updated, is based on Mathematica 6 - Comprehensive coverage from basic, introductory information through to more advanced topics - Studies several real data sets and many classical mathematical modelsFrom Book News, Inc. A general introduction to the popular mathematics software product, emphasizing graphics, methods of applied mathematics, and programming. Assumes no previous experience with the software, but at least some background in differential and integral calculus, equations, optimization, interpolation, approximation, differential and partly differential equations, probability, statistics, and the other mathematical functions described. The CD-ROM allows the entire text to be entered into the help system of ; it can be read with Windows, Macintosh, or Unix. Book News, Inc.r, Portland, OR Praise for the 2nd Edition: "Each [chapter] is a gem of clarity and concise application, but space limits the praise. If the book has any failings, it is in leaving the reader begging for more." - John A. Wass, Scientific Computing "Novices or beginners who may be students of engineering, science or mathematics ... definitely needs Mathematica Navigator. ... Those who do not use Mathematica everyday, and may work with it for a period of time and then not again for several weeks or more ... finds that Mathematica Navigator is totally indispensable. ... Finally, some users are constantly working with Mathematica, probably teaching courses or writing programs ... should definitely have Mathematica Navigator. ... Run, do not walk, to get your copy." - Robert M. Lurie, Mathematica in Education and Research "The present book is one of the best sources in many respects: its perfect layout, carefully thought-of exposition of the fine points of the software, sophisticated and illuminating examples that are all available on the accompanying CD-ROM. The topics discussed cover many topics which occur in the first two or three years of universioty curriculum in Mathematics such as calculus in single and several variables, linear algebra, differential equations, numerical analysis, partial differential equations." - Matti Vuorinen, Zentralblatt MATH "The book is a must for all beginners in Mathematica, and a great help as a reference for those who already know Mathematica." - K. Waldhör, Computing Reviews "... does a fantastic job at introducing Mathematica for the applied scientist. The book's use of Mathematica is slick, intelligent and comprehensive. It emphasizes Mathematica's strengths, and does it in the best possible way." - Joaquin Carbonara, Buffalo State University "This is an excellent reference book that I would recommend to any one who is thinking about becoming (or already is) a serious user of Mathematica ... I am particularly impressed by the organization which allows a mathematician to approach Mathematica by specific mathematical topics rather than using the organization of Wolfram's Mathematica Book. In addition, I find that the text is clearly written and the examples are well-chosen." - Bill Emerson, Metropolitan State University "There is a great need for this book. The outstanding feature of Mathematica Navigator is the great variety of Mathematica programs." - Mike Mesterton-Gibbons, Florida State University Most Helpful Customer Reviews This comprehensive volume covers many topics. Indeed, I would probably call it the most COMPREHENSIVE yet GENERAL text on the inner workings of the Mathematica program. For instance, Dr. Ruskeepaa treats the topic of 4 dimensional graphics, and I have not found that topis highlighted in any text. Furthermore, when it comes to specific mathematical models, such as difference equations, more information is located in this volume that anywhere else [except perhaps the on-line MathSource Mathematica Library]. Dr. Ruskeepaa's book not only demonstrates the basics in each category, but goes beyond what other resources have taught me. I highly recommend this book! Additionally, the CD-ROM which accompanies the book is quite handy. Lastly, I have had occaision to ask Dr. Ruskeepaa specific questions, and he has been MORE THAN HELPFUL and PUNCTUAL in providing me with solutions from his vast Mathematica knowledge. FIVE STARS -- better than any Mathematica book [I've seen most Mathematica books about graphics, physics and science; and programming]. The positive reviews were right on the money, this book is the best! It allowed me to start using Mathematica with ease, compared to the many hours of past frustration while working on optimizations. I fought endlessly, struggling to decipher the software's baroque navigation, cryptic errors, hostility to the user which borders on abuse - but now Mathematica and I are becoming great friends thanks to Heikki Ruskeepaa's wonderful tome. THANK YOU FOR WRITING THIS BOOK!!! Of particular benefit is the guidance s/he provides for best practices in formatting cells to avoid common, productivity-killing pitfalls. I have many books on Mathematica and use Mathematica a lot to do both symbolic and numerical calcualtions. This book is the best I have seen. I would recommend it to anyone using Mathematica for serious symbolic or computational work. If you are looking for a book about applied mathematics and numerical methods with Mathematica this is the one. The treatment of the graphical capabilities of Mathematica is complete and very useful. The only minor criticism I would have for this book is that it has no unsolved exercises and problems. However the examples presented are excellent. I have the highest regard for the author of this book. He has produced a superb piece of work! Mathematica is an excellent program, with good built-in and online documentation - although many feel the Documentation Centre in version 6 is not as good as the Help Browser in version 5.2. But sooner or later you will find a problem for which the official Mathematica documentation does not help. The program can then be very frustrating, as the syntax is complicated and error messages can be very cryptic. Help online is painfully slow, as the official resource, the news group comp.soft-sys.math.mathematica (Mathgroup), is moderated, so help requests have to be approved before they will appear. (The newsgroup sci.math.symbolic is sometimes helpful and since its not controlled by Wolfram Research, posts appear immediately). Hence Mathematica users will need one or more good books - more so than with other similar programs such as Maple or Matlab. I started learning Mathematica with this book, and it got me up and running quickly. Concepts are presented in order (if a technique is used that hasn't been introduced yet, the reference to later in the book is always included). Explanations are clear. Multiple examples are included for more difficult concepts. What's more, it's a great reference... A good index and appropriate references are provided. I really couldn't fault this book in any way. While it is unlikely that there is a pure Mathematica course anywhere, the textbook for it already exists! Mathematica Navigator by Heikki Ruskeepaa is an absolutely wonderful book that teaches you everything you ever wanted to know about Mathematica. Out of about 10 books I've used, this book is THE BEST on Mathematica out there. As another reviewer has already pointed out, much of the material in this book is just a rehashed version of the Mathematica v6.0 documentation -- written by someone with a rather quirky grasp of English as a second language. However, Mathematica is now up to version 8, so the book is already out of date. Given that "Mathematica Navigator" is nothing but a feature-by-feature overview of Mathematica, it's not clear what value it offers. In comparing the third edition of "Mathematica Navigator" to a friend's copy of the second edition, I also noticed that the publisher has significantly decreased the size of the typeface used in the body of the text. This appears to be the publisher's strategy for including extra topics without adding many extra pages. Unfortunately, it makes the book very painful to read. At first glance, "Mathematica Navigator" gives the impression of being useful, but after reading it, the truth becomes clear: It's just the Mathematica documenation repackaged, printed in a typeface guaranteed to induce eye strain.
This lesson received an honorable mention in the 2014 SoftChalk Lesson Challenge.'We have seen quadratic functions which... see more This lesson received an honorable mention in the 2014 SoftChalk Lesson Challenge.'We have seen quadratic functions which are either concave up or concave down, but we may need more flexibility than that. In this section we are going to discuss higher orderpolynomial functions and models. Linear functions are actually a first order polynomial and quadratics are a second order polynomial.' This lesson won an honorable mention in the 2014 SoftChalk Lesson Challenge.This lesson is an introduction to linear... see more This lesson won an honorable mention in the 2014 SoftChalk Lesson Challenge.This lesson is an introduction to linear functions. Topics included are: Topic 1: Definitions of Relations and FunctionsTopic 2: Domain and Range of FunctionsTopic 3: Vertical Line TestTopic 4: Function Notation For my technology class I was asked to complete a StaIR. This stands for Stand-Alone Instructional Resource. This is a... see more For my technology class I was asked to complete a StaIR. This stands for Stand-Alone Instructional Resource. This is a resource that a student could use to learn a topic completely independently. It teaches the topic, includes independent practice, and feedback. When asked to do this I wanted to make this for something that could be used within my classroom. I chose to complete a resource for the Pythagorean Theorem. This is a standard that is necessary for all seventh grade students. I will be able to use this to differentiate this in both my Math 7 and Pre-Algebra class. A StAIR or Stand-Alone Instructional Resource is a resource designed to help teach certain objectives without an instructor... see more A StAIR or Stand-Alone Instructional Resource is a resource designed to help teach certain objectives without an instructor being present. This means that students can learn by accessing a StAIR at home or some other place where a teacher is not present. This StAIR focuses on teaching students the basic details about equivalent fractions. If your school uses the Common Core State Standards, this StAIR focuses on the following objective:4.NF.1 – Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visualfraction models, with attention to how the number and size of the parts differ even thoughthe two fractions themselves are the same size. Use this principle to recognize and generateequivalent fractions.
Introduction to Geometric Computing Computing is quickly making much of geometry intriguing. What is the core set of topics that a practitioner needs to study before embarking on the design and implementation of a geometric system in a specialized discipline? This book attempts to find the answer. Although geometry has been a flourishing discipline for millennia, most of it has seen either no practical applications or only esoteric ones. Computing is quickly making much of geometry intriguing not only for philosophers and mathematicians, but also for scientists and engineers. What is the core set of topics that a practitioner needs to study before embarking on the design and implementation of a geometric system in a specialized discipline? This book attempts to find the answer. Every programmer tackling a geometric computing problem encounters design decisions that need to be solved. What may not be clear to individual programmers is that these design decisions have already been contemplated by others who have gone down some system design path only to discover (usually much later) that the design decisions that were made were lacking in some respect. This book reviews the geometric theory then applies it in an attempt to find that elusive "right" design.
Rich problems invite a variety of solution strategies. They provide an effective vehicle for accommodating a range of levels and learning styles. In this six-week online course participants will explore strategies for using problem solving to build skills and concepts for struggling students and to provide challenge for those who soar. After completing this course, you will be familiar with all the resources associated with your PoW membership access. You will be able to make an informed decision about how to start implementing PoWs with your students, and you'll have an idea of further steps to try when you are ready. NOTE: The next scheduled course is a "combination" version: one week summer intensive + three weeks in the fall. Assume that what is noted for Weeks 1-3 in the six-week version of the syllabus will be covered during Week 1 (summer) and what is noted for Weeks 4-6 will be covered in the fall. After you have become oriented to all of the possible ways to use the resources that come with a PoW membership and you have a foundation of mentoring skills, this course provides an opportunity to apply what you've learned with your own students and focus on the development of their mathematical thinking and problem solving, individually and collectively. In this course, we will collaboratively explore the fundamental ideas of calculus, including limits, derivatives, antiderivatives, and integrals, through the use of dynamic geometry software. While the course will cover a variety of calculus content, it is not a calculus course. This course is designed to enrich students' understanding of calculus ideas, to collaboratively explore these ideas with colleagues, and to engage in professional conversations about the implications of these developing experiences and technologies on the learning and teaching of calculus. In this course we will examine a continuum of student work from the Math Forum's Problems of the Week archive. Selected work will include a range of examples from "not knowing how to start" to "It's perfect. What could I possibly say to her?" We will consider specific ways to move students' thinking along this continuum. In this course you will have the opportunity to mentor student work submitted to the Math Forum from classrooms around the world. The goal is for you to become comfortable as well as successful in prompting students to develop sound mathematical practices through written feedback. After honing this skill with a diverse range of students, you might want to enroll in "Mentor Your Own: Supporting Strong Development of Mathematical Practices" to learn how to integrate written feedback into your own classroom practice. In this course, we'll explore ways in which the teaching and learning of math in grades 6 through calculus can be enhanced by appropriate use of dynamic mathematics software such as Sketchpad. We'll construct sketches from scratch and use existing activities and highly-authored sketches to cover topics in middle school, algebra, geometry, pre-calculus, and calculus (you don't need to know or remember calculus!). We will also learn about Sketchpad topics such as animation, action buttons, presentation sketches, transformations and iterations, and the construction and use of custom tools. Reg. price: $298 Sale price: $149 *You must successfully complete the course in order to receive the discount. No other offers apply; no refunds. Full memberships will expire 6.30.2014. Total cost savings equal $149. Offer expires 9.18.2013. Please email me with any questions. Suzanne Alejandre, Director of Professional Development
Book DescriptionEditorial Reviews From the Publisher The practical alternative to texts that are too long on theory and too short on techniques! In the first half of the text, the author presents a wealth of specific, step-by-step techniques and skills for teaching secondary mathematics. He includesFrom the Back Cover Key Benefit: The practical alternative to books that are too long on theory and too short on techniques! In the first half of the book, the author presents a wealth of specific, step-by-step techniques and skills for teaching secondary mathematics. Key Topics: Included areMore About the Author Alfred S. Posamentier (River Vale, NJ) is the dean of the School of Education and professor of mathematics education at Mercy College in Dobbs Ferry, NY. He was formerly the dean of the School of Education and professor of mathematics education at the City College of New York. He has published over forty-five books in the area of mathematics and mathematics education, including The Pythagorean Theorem: The Story of Its Power and Beauty; Math Charmers: Tantalizing Tidbits for the Mind; and (with Ingmar Lehmann) Mathematical Amazements and Surprises; The Fabulous Fibonacci Numbers; and Pi: A Biography of the World's Most Mysterious Number. Overall I found this book to be well written, informative, and practical. The main text covers the same information as general education courses but applies it to the mathematics classroom. My only complaint with the text is that it proposes solutions to problems or situations and then says that the teacher must decide what is best for the class. Yes, every class is different and the teacher knows the needs of the students better than an author, but while reading the book, I felt that the authors didn't want to committ to deciding what the best practice would be. The appendix of enrichment plans is the reason I am not reselling the book. It includes so many activities to try with all levels of math students that I feel the price of the book might well be justified. The book is a compilation of ideas and lists about effective teaching. I was unimpressed by some of the explanations by the author. He seems to be a little out of touch with what the curriculum and behavior management looks like in lower-income neighborhood schools. Easy reading because the book doesn't have much effective teaching inovation. I will not purchase another one of his books. This is an excelent text to use both in the college class room and everyday teaching. The text covers all developmental point of effective teaching. It puts to teacher in a positve position that makes you want to teach and students to learn. If you are looking to sharpen your skills in teaching Math, or any subject you need this text. If you follow it closely "IT WILL KEEP YOU OFF OF A PDP(Professional Development Plan)" For my math pedagogy classes we use this Posamentier's Teaching Secondary Math and Van de Walle's Elementary and Middle School Math. I like Posamentier's much more. I could imagine the ideas of this book applied to honors math classes and the math oriented high schools. Posamentier uses a lot of Geometry examples in this book and they seem very useful for teaching geometry. I think there are a lot of challenging problems that are good for high school students: Erastothene's measuring the circumference of the earth, Gauss' arithmetic series. exploring triangle medians with Geometer, Birthday Probability Problem. Posamentier also uses the Graphing Calculator a lot with this book- model train distance problem, medication, compound interest. And yes, solutions and how to set up the problem on the Graphing Calculator are given. I don't remember, but I think Posamentier does include a fair amount of problems on how to use Excel to math problems like simple interest and probability. Posamentier also provides math examples for motivating students: Tell a story, present a challenge, show the sequential knowledge, show a gap in student understanding. I feel that Posamentier knows his stuff, his tone in writing is concise and to the point. The book doesn't include it, but I would like to have seen how Posamentier would exactly teach a high school math lesson. Once again, there is a lot of geometry material presented in this book. I would have liked to have seen more trigonometry, there is a lack of examples on how to teach trig. I respect Posamentier, and I'm happy to have learned from his book. Great textbook. Many great examples on how to keep students engaged and provoke their own learning. Not being a super creative person, this math specific book allowed me to see examples of strategies that I didn't quite get how to transfer it to math when someone gave me an easy English example.
Find a La Marque CalculusAll individuals have an innate understanding of mathematics. We constantly use math concepts in our everyday lives. The successful math student understands the relationship between our intuitive understanding of math and the concepts presented in a formal class
Algebra 1 - With CD (10 Edition) by Mary Jane Sterling Publisher Comments About the Contents: Pretest Helps you pinpoint where you need the most help and directs you to the corresponding sections of the book Topic Area Reviews Math Basics Numbers (Signed Numbers and Fractions) Linear Equations and Algebraic Fractions... (read more) Beyond Problem Solving & Comprehension by Arthur Whimbey Publisher Comments Building on the methods and techniques presented in Problem Solving and Comprehension, the authors show how their methods can be used to manage more difficult problems and situations. By providing step-by-step solutions for problems of figural reasoning,... (read more) Essentials of College Algebra, Alternate Edition by Lial Publisher Comments Essentials of College Algebra , Updated Edition, 1/e, has been specifically designed to provide a more compact and less expensive alternative to better meet the needs of colleges whose algebra courses do not include the more advanced topics. The authors... (read more) Technical Mathematics With Calculus 2ND Edition by John C Peterson Publisher Comments This version of Technical Mathematics with Calculus, 3E includes formal calculus concepts that are comprehensive in scope to help individuals prepare for technical, engineering technology, or scientific careers. Thorough coverage of precalculus topics... (read more) Mathematical Ideas by Charles Davi Miller Publisher Comments This manual provides solutions to odd-numbered exercises in the exercise sets, the Extensions, and the Appendix exercises, as well as solutions for all the Chapter Test exercises. Chapter summaries review key points in the text, providing extra examples,... (read more) Precalculus (with CD-ROM, Bca/Ilrn Tutorial, and Infotrac) by David Dwyer Publisher Comments Dwyer and Gruenwald's PRECALCULUS builds on the same features as their very successful COLLEGE ALGEBRA, Second Edition and COLLEGE ALGEBRA AND TRIGONOMETRY, Second Edition , while adding material to help students look ahead to calculus. The book is... (read more) Regular Algebra and Finite Machines (Dover Books on Mathematics) by John Horton Conway Publisher Comments World-famous mathematician John H. Conway based this classic text on a 1966 course he taught at Cambridge University. Geared toward graduate students of mathematics, it will also prove a valuable guide to researchers and professional mathematicians. His... (read more) Foundations of Geometry by C R Wylie Synopsis Geared toward students preparing to teach high school mathematics, this text explores the principles of Euclidean and non-Euclidean geometry and covers both generalities and specifics of the axiomatic method. 1964 edition.... (read more) Beginning Algebra by Julie Miller Publisher Comments Miller/O'Neill Beginning Algebra is an insightful text written by instructors who have first-hand experience with students of developmental mathematics. The authors have placed an emphasis on graphing, by including special sections called, "Connections... (read more) Trigonometry Refresher by A Albert Klaf Publisher Comments This classic text encompasses the most important aspects of plane and spherical trigonometry in a question-and-answer format. Its 913 specially selected questions appear with detailed answers that help readers refresh their trigonometry skills or clear... (read more) High School Algebra Tutor 2ND Edition by Rea Publisher Comments REAs High School Tutors is a series of useful and practical study guides. Each High School Tutor provides practice and understanding of various math, science, and history subjects, making them challenging and interesting. Hundreds of solved problems... (read more) Excursions in Modern Mathematics 5TH Edition by Peter Tannenbaum Publisher Comments This collection of "excursions" into modern mathematics is written in an informal, very readable style, with features that make the material interesting, clear, and easy-to-learn. It centers on an assortment of real-world examples and... (read more) Teach Yourself Algebra by P Abbott Publisher Comments "Teach Yourself Algebra "is a great introduction for learners having no prior experience with this ancient branch of mathematics. It acquaints readers with algebra and its basic components, such as equations, exponents, and indices. Then, using
Created by Eddie Fuller, Lang Moore and David Smith for the Connected Curriculum Project, the purpose of this module is to introduce the concept of vectors in space and explore their algebraic and geometric properties. ... This is a video about Niko Henderson, an engineer for Easton Sports. He uses science, mathematics, engineering and innovative testing to help produce some of the fastest bikes on the road. Treat your students to a rare... Created by David Smith for the Connected Curriculum Project, the purpose of this module is to apply linear algebra concepts to study the properties of sequences defined by difference equations. This is one within a... Created by Lawrence Moore and David Smith for the Connected Curriculum Project, the purpose of this module is to carry out an exploration of functions defined by data; to learn about data entry and plotting operations. ...

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