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kerja kursus additional mathematics 2011Before, shop-fitting primarily contains installing surfaces, shelves and additional essential accessories that have been needed to keep and show the retailer's products and was regarded significantly less significant to companies with
Whether you're new to algebra or just looking for a refresher, Algebra Success in 20 Minutes a Day offers a 20-step lesson plan that provides quick and thorough instruction in practical, critical skills. Stripped of unnecessary math jargon but bursting with algebra essentials, this extensive guide covers all vital algebra skills, including combining like terms, solving quadratic equations, polynomials, and beyond. This proven study aid is completely revised with updated lessons and exercises that give students and workers alike the algebra skills they need to succeed. Algebra Success in 20 Minutes a Day also includes: Hundreds of practice exercises, including word problems Application of algebra skills to real-world (and real-work) problems A diagnostic pretest to help pinpoint strengths and weaknesses Targeted lessons with crucial, step-by-step practice in solving algebra problems A helpful posttest to measure progress after the lessons Glossary, additional resources, and tips for preparing for important standardized or certification tests
Author Info Abstract This book is designed to fill the gap between introductory undergraduate texts and advanced texts for graduate students. Its comprehensive coverage ensures that readers understand both the 'how' and the 'why' of econometrics, as it explains not only the mathematical techniques for econometric problem-solving but also the mathematical foundations of the discipline. Developed with careful pedagogical methodology throughout, the text makes full use of empirical examples and includes appendices providing 'ready reference' and refresher courses on basic mathematics, as well as further material for the more advanced student11164495111644
9780812026214ithmetic the Easy Everybody uses arithmetic on virtually a daily basis, and this book serves as a handy brush-up for general readers while it also helps students master basic skills that they need before moving up to high-school-level math and beyond. It reviews addition, subtraction, multiplication, and division, then moves on to calculating with fractions, decimals, and percentages. A concluding chapter reviews units of measurement and word problems. Chapters are filled with short practice exercises, all of which are answered at the back of the book. The book features many tables, charts, and line illustrations. Barron's Easy Way books introduce a variety of academic and practical subjects to students and general readers in clear, understandable language. Ideal as self-teaching manuals for readers interested in learning a new career-related skill, these books have also found widespread classroom use as supplementary texts and brush-up test-preparation guides. Subject heads and key phrases that need to be learned are set in a second
Geometry for Dummies - 2nd edition Summary: Geometry proofs may trip up more students than any other single topic in all of high school math. Geometry For Dummies, 2nd Edition, tackles this problem head on, providing proven strategies for solving geometry proofs when students are stumped. Students need help with getting a handle on what seems to them to be a totally foreign and mysterious process. This book presents a dozen powerful strategies that make proofs much easier for the students who struggle with them. This book cont...show moreains dozens of examples of places in a proof where a student is likely to get stuck and then provide tips for how to get unstuck. Mark Ryan has a proven ability to explain concepts in a way that gives students the clearest, easiest, and best way of understanding a concept. For example, instead of routinely listing the properties of various quadrilaterals (four-sided figures) as most geometry books do, relying on rote memory for student learning, Geometry For Dummies 2nd Edition explains how these properties (and others) can be understood and learned other than by rote memory, in a way that fosters understanding. The second edition covers many of the same concepts, such as deductive and inductive reasoning, lines, angles, arcs, circles, multi-sided figures, substitution principles, pi, area and slope, inequalities, rates and rations, and more. The new edition also includes detailed explanations of how to work example problems, pinpointing areas that can trick students into misunderstanding the true nature of the problem530470089466 Your purchase benefits those with developmental disabilities to live a better quality of life. some wear on edges and corners minimal cover wear minimal stains on edges minimal wear on pa...show moreges
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From Newton's Law of Gravity to the Black-Scholes model used by bankers to predict the markets, equations, are everywhere - and they are fundamental to everyday life. In this book, the author sets out seventeen equations that have altered the course of human history. It is also an exploration - and explanation - of life on earth. In its updated second edition, this book explores computational methods for problems arising in the areas of classical analysis, approximation theory, and ordinary differential equations, among others. Includes new exercises, and a complete solutions manual. Helping Children Learn to Love Their Most Hated Subject--And Why It's Important for America by: Jo Boaler $26.86 $29.85 inc GST $24.42 $27.14 ex GST ISBN: 9780670019526 The U.S. is rapidly falling behind the rest of the developed world in terms of math education. In this straightforward and inspiring book, Boaler presents concrete solutions to help reverse this trend, including classroom approaches, essential strategies for students, and advice for parents. Making good decisions under conditions of uncertainty requires an appreciation of the way random chance works. In this Very Short Introduction, John Haigh provides a brief account of probability theory; explaining the philosophical approaches, discussing probability distributions, and looking its applications in science and economics. In this Very Short Introduction, Jacqueline Stedall explores the rich historical and cultural diversity of mathematical endeavour from the distant past to the present day, using illustrative case studies drawn from a range of times and places; including early imperial China, the medieval Islamic world, and nineteenth-century Britain. Leads us on a journey through five revolutions in geometry, from the Greek concept of parallel lines to the notions of hyperspace. This title reveals how simple questions anyone might ask about space in the living room or in some other galaxy have been the hidden engine of science's highest achievements. Provides plain-English explanations of the most challenging aspects of trig, plus numerous practice problems, and their easy-to-follow solutions. This helpful guide is the next best thing to a personal trigonometry tutor!--
the fundamentals of discrete mathematics with DISCRETE MATHEMATICS FOR COMPUTER SCIENCE with Student Solutions Manual CD-ROM! An increasing number of computer scientists from diverse areas are using discrete mathematical structures to explain concepts and problems and this mathematics text shows you how to express precise ideas in clear mathematical language. Through a wealth of exercises and examples, you will learn how mastering discrete mathematics will help you develop important reasoning skills that will continue to be useful throughout your career. Customer Reviews Most Helpful Customer Reviews on Amazon.com (beta) Amazon.com: 3 reviews 1 of 1 people found the following review helpful A text for the masochistic learnerJan. 14 2014 By Jwheezy - Published on Amazon.com Format: Hardcover This was a required textbook for a course at my university. My professor pulled all the homework from the ends of each chapter. This part of the book is one of my biggest gripes. The reading sections of this book pack a large amount of material in a brief page or two for each section followed by homework exercises. The exercise sections have are about as long as the actual information sections, meaning they are packed with questions. This would be a positive for this book except the questions aren't similar, so the included CD with the odd problems solved will often be of little help because question 3 will be a completely different sort of problem than question 4. Since each problem is so unique, you'll often be left dealing with problems that are considerably more complex than anything found in the reading sections of the text. If you are using the questions of this book for homework, be prepared to use google extensively. As an example, the book may explain how to perform an operation on 2 sets of numbers. Then in the homework, it will ask you to perform the same operation on 5 sets abstract sets without ever explaining how to go about doing that. I ended up receiving an A in the course, but that was after spending ~8 hours for each 10-14 question homework. Most of that time was spent on the internet trying to learn the material from whatever sites I could find. The reading sections of this text are an excersize in frustration. In one of the explanations for a concept in the book, the author literally uses the phrase "from [problem], it is obvious that the answer is [answer]." That was the entire explanation on the topic. A textbook should never say the phrase "from X, it is obvious that Y" if the whole section is supposed to be telling you how to find Y from X in the first place. This is an introductory text into formal logic, proofs, and set mathematics. Yet, you'll often find that the author skips steps in his solutions which may be obvious to someone familiar with the material but that is obviously not the target of this text. There is an occasional table for reference which doesn't explain what the relationship between anything on the table is (I'm looking at you, Table of Commonly Used Tautologies....). This book covers a great number of topics in a fairly small book, for a textbook that is. However, this book suffers from a lack of depth necessary to reach its potential. If you have a choice, skip this text. If, like me, you are required to use this text.... Google everything and god help you. Extremely poor organization.Jan. 13 2014 By Dan G. - Published on Amazon.com Format: Hardcover Verified Purchase This book has an extremely poor organization of information. It's like the authors just threw a bunch of information at the book without thinking about how a student has to go through learning the mathematical concepts. The only reason I have to use this book is because a professor from my university was one of the authors. Get another book on discrete mathematics if you want to really learn the material. 1 of 8 people found the following review helpful Great TextbookSept. 7 2011 By mfox - Published on Amazon.com Format: Hardcover Verified Purchase This textbook was the exact same one I needed for class and was MUCH cheaper than buying from the school store. It was even in better condition than what was advertised! I would definitely recommend this book.
Mathematical Equations Math Suga is a complete utility to calculate mathematical equations, lets you to calculate mathematical equations with various features. Math Suga is a complete utility to calculate mathematical equations, lets you to calculate mathematical equations with various features. A techincal documentation program with the following features 1. Writing Equations 2. Graphing Equations 3.... A puzzle where you delve into a mathematical maze for answers to simple arithmetic equations. A puzzle where you delve into a mathematical maze for answers to simple arithmetic equations. You must navigate the underground maze using numbers as clues while digging a tunnel for your math worm to get home. Mark the tunnel by completing... A&G Grapher III can draw any 2D or 3D mathematical equation. Equations can be of any complexity, and with a proper setting you donít even need to start an equation with y= or z= because this graphing software is programmed to handle complex equations such as y+xy = sinx. Along with this, a number of tools are provided to analyze or modify the plotted graphs. Some of the tools included in this graphing calculator are rotation, matrix... This is a small easy to use calculator for most windows operating systems. This is a small easy to use calculator for most windows operating systems. This calculator is simple, it is not a scientific one and can be used for basic mathematical equations. This calculator is made by and it...MathCast is an equation editor, an software that allows you to input mathematical equations. MathCast is an equation editor, an software that allows you to input mathematical equations. These equations can be used in written documents and webpages. The equations can be rendered graphically to the screen, to picture files, or to MathML -... A mathematical puzzle where equations are all mixed up. Find the correct equations to solve the puzzle. Equations run in various formats like boxes, crosses, diagonals, mazes and go in all directions. A fun way to reinforce math or just use your logic. MathToWeb is a command-line authoring tool that converts mathematical expressions written in AMS-Latex to presentation MathML. MathToWeb is a command-line authoring tool that converts mathematical expressions written in AMS-Latex to presentation MathML. It allows the author to write web pages in XHTML (just a strict form of HTML) containing equations in LaTeX rather than...GlassCalc is a simple calculator with extensive support for mathematical expressions. GlassCalc is a simple calculator with extensive support for mathematical expressions. Instead of using buttons, it has a text-only input and keeps a full history of expressions and results. It also has full support for user-defined functions and...
Discrete Mathematics 9780131593183 ISBN: 0131593188 Edition: 7 Pub Date: 2008 Publisher: Prentice Hall Summary: This textbook provides an accessible introduction to discrete mathematics, using an algorithmic approach that focuses on problem-solving techniques. Each chapter has a special section dedicated to showing students how to attack and solve problems. Johnsonbaugh, Richard is the author of Discrete Mathematics, published 2008 under ISBN 9780131593183 and 0131593188. Seven hundred twenty three Discrete Mathematic...s textbooks are available for sale on ValoreBooks.com, one hundred thirty five used from the cheapest price of $106.79, or buy new starting at $151
Product Description This supplemental DVD is designed to be used along with the corresponding ACE Math PACE 1103 for Grade 9 (sold-separately). Twelve lessons are included and cover intersection and union, linear inequalities, systems of linear inequalities, conjunctions and disjunctions, Venn diagrams, and more. Lessons range in time from approximately two minutes through twelve minutes in length
Find a Leicester, MA CalculusI have covered everything from the history of the Bible to each of the writers, also highlighting the themes such as sovereignty, mankind's integrity and fulfillment of prophecy. It addresses several types of equations such as first Order Differential Equations such as Linear Equations, Separabl
Arithmetic to Algebra through Pre-Calculus; Mathematics Texts for High School and College WELCOME TO ACTION MATHEMATICS PRODUCING TEXTBOOKS THAT FOSTER CRITICAL THINKING, UNDERSTANDING, AND LEARNING IN ORDER TO HELP STUDENTS TO BE SUCCESSFUL IN ARITHMETIC, GEOMETRY, ALGEBRA, PRE-CALCULUS, AND OTHER MATHEMATICS IS OUR TOP PRIORITY WE ALSO SPECIALIZED IN RELEARNING MATH LEARNING AND RELEARNING MATH IS VERY IMPORTANT IN THE NEW WORLD ECCONOMY THE MATH TEXTS IN THE CRITICAL THINKING APPROACH TO LEARNING MATH ARE RECOMMENDED BY SECONDARY SCHOOLS, HOMESCHOOLERS, AND SELF STUDY STUDENTS For all students wanting to learn mathematics; this site provides free help in arithmetic, free help in geometry, free help in algebra, and free help in pre-calculus topics in mathematics. We cannot learn math for you, but Dr. Del and Dr. Mel will try to make it easier for you to learn. See the Free Math Help Page in our Site Map CLASS STUDYING FROM M.T.E.'S PRE-CALCULUS HOME OF THE CRITICAL THINKING APPROACH TO LEARNING MATHEMATICS M.T.E. LTD publishes ARITHMETIC TO ALGEBRA THROUGH PRE-CALCULUS series of texts; these textbooks and materials are for mathematics courses normally taught in high school (7th to 12th grades), but are also taught in colleges and universities. The mathematics in this series is presented in an understandable way in order to maximize success in math courses. This series mentioned above is titled: "The Critical Thinking Approach to Mathematics." The materials and texts in this math series contain special sets of exercises designed to foster Critical Thinking and Problem Solving. These sets are called "Exercises for Understanding" and "Problems for Problem Solving". The Texts and Materials of M.T.E. Ltd publishers are designed to Foster Reading, Understanding, and Computation -- not memory. BECAUSE OF THE EMPHASIS ON UNDERSTANDING AND LEARNING, THIS SERIES OF TEXTS IS EXCELLENT FOR TEACHING MATHEMATICS The Company motto is "Striving for Student Understanding and Success in the learning of Arithmetic, Geometry, Algebra. or Pre-Calculus". Student Understanding is defined as a student being able to state, in their own words, the meaning of the math concept being taught. As a customer service MTE has collected information about the CLEP mathematics examinations and has organized this information for those interested in CLEP. Find out how to make some extra money for you or your math department. The Critical Thinking Approach to Mathematics Series of texts was specifically created to facilitate the understanding of mathematics. The series is for home school, high school, college, or university students; or any individual who needs to learn or relearn math. Few humans totally understand math; however, college capable students can and should understand more math than most do after graduating from high school. The Series Critical Thinking Approach to Learning Mathematics of M.T.E. includes the following: Competency Arithmetic Geometric & Measurement Topics Beginning Algebra (1st year algebra) Intermediate Algebra (2nd year algebra) Pre-Calculus (Advanced algebra and trigonometry) FIVE TEXTBOOKS IN M.T.E.'S CRITICAL THINKING APPROACH TO MATHEMATICS SERIES MANY OF OUR CUSTOMERS ARE CONCERNED WITH AND WANT MORE INFORMATION ON ONE OR MORE OF THE FOLLOWING TOPICS: THE BUTTONS WILL CONNECT YOU TO ADDITIONAL INFORMATION ABOUT THESE TOPICS. is an excellent Series of textbooks, which has been used for over 15 years with thousands of students to successfully improve math vocabulary and the understanding of mathematics. This series has been used in classrooms, home schooling, math labs, tutorial or self-study situations, or industrial employee improvement programs. Click on the appropriate text in the site map, located on the left, for more information about each text package. These texts are based on the results of nearly 20 years of research concerned with determining why students in high schools and college memorize math, but fail to understand the concepts; thus eliminating any chances of taking university level math courses needed for degrees that lead to higher paying positions. Obtaining a textbook to help a student learn math and get better grades. THE TEXTS IN THE M.T.E. SERIES: THE STAIRWAY TO CALCULUS. The emphasis used in the writing of the Critical Thinking Approach to Mathematics Series is that of understanding mathematics. These texts feature new and different types of exercises to foster better understanding and retention of the mathematics learned as well as the ability to read math and math notation. Increasingly, more and more parents are discovering that their student who used to enjoy math now hates math because of the excessive memory approach used in traditional texts. Consequently, students using the materials of M.T.E. get better math grades and do not need to retake high school math in the college or university remedial programs as do most students using the traditional approach. This saves both time and money for the student while attending college. Satisfaction is fully guaranteed on all M.T.E. products. For more about this guarantee click the "Information about the Critical Thinking Approach to Mathematics Page" in the site map located at the top left of this page. Helping students understand the importance of problem solving and critical thinking. Everyone in all occupations is concerned with problem solving; this is particularly true in mathematics. Problem Solving is Applied Critical Thinking. Business and industry seek people who are problem solvers; consequently, they most often get the better higher paying positions. Students taught critical thinking improve their ability to think for themselves and are better problem solvers. Critical Thinking is logical and productive thinking that is capable of extending ideas, creating new ideas, and synthesizing patterns of thought. Critical Thinking is a type of disciplined reasoning. Educational Critical Thinking must be self-directed, self-disciplined, self-monitored, and self-corrective. The definition used for critical thinking in Critical Thinking Approach to Mathematics texts was "The ability to ask questions and to seek answers toward the positive solution of a problem or situation." Understanding Algebra is the key to passing a university level mathematics course and this, in turn, is the key to obtaining a higher paying position in a student's future. Traditional texts contain more content than can reasonably be learned in a school term by a student; in the case of mathematics texts, "more is not better" it is simply more. In order to complete the text and pass required tests, students and teachers are forced to use memorization as the main mode of learning; memorized information is stored in short-term memory and is soon lost forever. Traditional texts contain so much information that there is not enough time to teach critical thinking and problem solving. More and more parents are stating that their student, who used to enjoy and like mathematics, now hates math because of the great amount of memory work involved in their traditional text. The Critical Thinking Approach to Learning Mathematics Series of M.T.E. contains all the necessary mathematics from arithmetic to algebra to pre-calculus, but eliminates the unnecessary and "frill" concepts in that course. It was determined that 30 Units (Chapters) would be required to adequately cover the mathematics from Arithmetic to Pre-Calculus to prepare a student sufficiently to enroll and pass a university level Calculus course. For more about this and other topics, click the "Information about the Critical Thinking Approach to Mathematics Page" in the site map located on the top left of this page. Helping students to think for themselves and understand math concepts instead of memorizing. In today's math education, students are taught to memorize math facts; because teachers and educators feel this is the best approach to have better results on standardized tests. In many cases, this is a necessary result for schools to obtain federal and state funds. Unfortunately, memorization of math facts not only does not guarantee the understanding of math concepts; it often has the opposite effect on learning these concepts. Memorization: Students are trying to memorize all necessary facts that are needed to pass the course. However, this is not possible for most students and such a course of action has no long-term benefits; hence causing the basic problem in mathematics today, lack of understanding. which results in poor performance or failure in succeeding math courses. Increasingly it is also causing students to hate math even though they use to enjoy math. This lack of preparation often causes the student to take the remedial math in college or university adding a year or two in college and additional tuition costs. "In many cases the successful completion of a university level math course is one of the requirements for obtaining degrees that can lead to higher paying positions after graduation. College capable students can and should understand more mathematics than most do after graduating from high school. The Critical Thinking Approach to Learning Mathematics Series of texts has been specifically created to facilitate the understanding of math. Make mathematics a life's partner; instead of a feared-required task. For more about this and other topics, click the "Information about the Critical Thinking Approach to Mathematics Page" in the site map located on the top left of this page. In many cases the successful completion of a university level math course is one of the requirements for obtaining degrees that can lead to higher paying positions after graduation. College capable students can and should understand more mathematics than most do after graduating from high school. The Critical Thinking Approach to Learning Mathematics Series of texts has been specifically created to facilitate the understanding of math. About the Company and the Authors. Materials and Technology for Education (M.T.E.) Ltd. was started in the fall of 1996. As an outgrowth of the research project, the authors were conducting trying to determine why high school graduates who had passed math courses with good grades could not pass the mathematics entrance exam for the university. Having gotten very successful results from the research and the materials written for the research project; the authors felt they needed to produce textbooks based on the materials used in the research; thus M.T.E. publishing company was formed. The main objective of the company was, and still is, to produce math textbooks that have a higher rate of student success than traditional texts. AUTHOR TEAM RESEARCH As mentioned above, the authors were asked by their university to research the problem of why did so many students having good to excellent grades in high school math courses fail the mathematics entrance exam for the university. After a couple of years of attempting various "Band-Aid" cures, it was decided that major overhaul was needed, particularly the curriculum, course materials, and teaching methods. The research project was called "Continuous Sequence in Basic Mathematics" (CSBM) and the results of CSBM are presented in the "Information About Critical Thinking Approach to Mathematics" page of this web site. After a few years there were as many as 3000 students a semester in CSBM. The materials used in CSBM were constantly revised to improve student success rate. Toward the end of CSBM, the materials and improved teaching allowed a rate of 72% of the students to make grades of 87.5%, aB+, or higher. Those graduates of CSBM who opted to take a university calculus course, over a 3-year period, made better grades (Ave. 3.04 to Ave. 2.43) than the students who were not part of the research. (The Mathematical Association of America. conducted this study). The research and materials for the "Critical Thinking Approach to Teaching Mathematics" have been cited positively by: MAA Visiting Committee, University Evaluation Committees, ERIC, The New York Times and This World Magazine. ABOUT THE AUTHORS. Melvin Poage, Ph. D. Mathematics - Michigan State University Carl Arendsen, Ph. D. Mathematics - Michigan State University Delano Wegener, Ph. D. Mathematics - Ohio University that some time overlap occurs since the authors occasionally held two occupations at the same time.) The Text Critical Thinking Approach to Competency Arithmetic is recommended by Colleges, Universities, Secondary Schools, Home Schoolers, and Self Study Students because it emphasizes understanding, learning, and relearning of mathematical concepts as opposed to memorization. This review of Arithmetic text fosters Student Success and enables the Teaching ofWe have always provided a free consulting service to answer questions for all students wanting to learn mathematics. We cannot learn math for you, but we will try to make it easier for you to learn. In this context we are providing free help in arithmetic, free help in geometry, free help in algebra, and free help in pre-calculus topics in mathematics. ABOUT THE AUTHORS Delano Wegener, Ph. D. Mathematics - Ohio University Melvin Poage, Ph. D. Mathematics - Michigan State University The authors were asked by their university to research the problem of why did so many students having good to excellent grades in high school math courses fail the mathematics entrance exam for the university. The research project was called "Continuous Sequence in Basic Mathematics" (CSBM) and the results of CSBM lead to the development of the Critical Thinking Approach to Mathematics series of texts and materials There were as many as 3000 students a semester in CSBM. The materials used in CSBM were constantly revised to improve student success rate. Toward the end of CSBM, the materials and improved teaching allowed a rate of 72% of the students to make grades of 87.5%, aB+, or higher. Those graduates of CSBM who opted to take a university calculus course, over a 3-year period, made better grades (Ave. 3.04 to Ave. 2.43) than the students who were not part of the research. (The Mathematical Association of America. conducted this study). Dr. Delano Wegener now lives and works in St. Louis Missouri area and Dr, Melvin Poage now lives and works in Denver Colorado. They frequently talk or E-mail one another to discuss topics related to mathematics or the teaching of mathematics. It is interesting to note that the first name of the wifes of both professors is Esther. that some time overlap occurs since the authors occasionally held two occupations at the same time.) . HOW TO USE There are two types of help available for you to use. #1 DR DEL DR DEL has available a reference web-site that contains explanations for almost all of the difficult math concepts in mathematics courses that are taught before Calculus. To access this excellent reference use the Button Below to Open Dr Del's Help Link. In this menu the topics on the left are general and the ones on the right are course specific to his university classes. I recommend the Special topics in mathematics. also has references available and a space for you to ask direct questions. To use this service, first check the reference buttons below to see if your question is one of these topics. If not, complete the form below the reference boxes to ask a direct question. Select your problem type below. The Text Critical Thinking Approach to Competency Arithmetic is recommended by Colleges, Universities, Secondary Schools, Home Schoolers, and Self Study Students because it emphasizes understanding,learning, and relearning of mathematical concepts as opposed to memorization. This review of Arithmetic text fosters Student Success and enables the Teaching ofIf you wish to see another page on this web-site and the M.T.E. Site Map does not appear at the left of your screen Open the Link Shown Below: Each Section of this text is followed by four sets of exercises: Exercises for Vocabulary (to aid in reading and vocabulary in math), Exercises for Computation (regular drill and practice), Exercises for Understanding (to aid in building critical thinking processes), and a Self-Test. Each Unit (Chapter) is followed by Problems for Problem Solving (a collection of problems from science and industry applicable to the topics in that Unit). Because of the emphasis on understanding, this is an outstanding text for teaching arithmetic concepts. Available with Critical Thinking Approach to Competency Arithmetic is an Instructor's Manual and a Test Kit. Answers for some of the odd numbered exercises appear at the end of the student text while a set of complete answers appear in the Instructor's Manual for the text. The Instructor's Manual contains teaching and testing hints, a possible schedule, and all answers including those to the Exercises for Understanding and Problems for Problem Solving. The Test Kit contains a Placement Test, four Criteria Referenced Unit Tests (one for each Chapter), and a Final. All tests have been developed using the suggestions and guidance of the University of Chicago Testing Department. These multiple-choice tests are designed to test mastery by students of the concepts. ; STUDYING FROM COMPETENCY ARITHMETIC TESTIMONOMIALS and CUSTOMER COMMENTS "This is the way I wish I had originally learned arithmetic; I hate to admit that I never knew a definition for real numbers that students at this level could understand". "A colleague visited my class and decided that she wanted to try a pilot in her school using this text." "Our school has enjoyed great student success using your Critical Thinking Approach to Competency Arithmetic. I am pleased to inform you that we will continue this adoption." "We are excited to have been able to produce data about student success in this course. Student attitudes improved greatly which was the best result of using this text." The Text Critical Thinking Approach to Geometry & Measurement Topics is recommended by Colleges, Universities, Secondary Schools, Home Schoolers, and Self Study Students because it emphasizes understanding,learning, and relearning of mathematical concepts as opposed to memorization. This practical course in Geometry fosters Student Success and enables the Teaching of math concepts because of the extra amount of time allowed to master these needed concepts. Critical Thinking Approach to Geometry & Measurement Topics contains the curriculum for a year Geometry math course normally taught in 10th grade or in a review sequence in college. Many students struggle with geometry through no fault of their own. Over stress on proving theorems in geometry courses has caused many students to have a frustrating experienced that can result in a loss of self-confidence. Critical Thinking Approach to Geometry & Measurement Topics features a comprehensive, step-by-step approach that simplifies complex concepts by breaking them up into bite- size pieces, using high quality illustrations, and providing real-life examples. Geometric facts are presented first leaving the proofs until after the concepts are understood. The mathematics is presented in an understandable way in order to maximize success in this course. The Geometry & Measurement Topics Text in the Arithmetic to Algebra through Pre-Calculus Series contains all necessary topics needed to understand Beginning Algebra. The complete Table of Contents for this text is presented at the bottom of this web-page. If you wish to see another page on this web-site and the M.T.E. Site Map does not appear at the left of your screen Open the Link Shown Below: Each Section of Geometric and Measurement Topics is followed by 4 sets of exercises: Exercises for Vocabulary (for reading and vocabulary development), Exercises for Computation (regular drill and practice exercises), Exercises for Understanding (to develop critical thinking skills), and a Self-Test. The Instructor's Manual contains Problems for Problem Solving (a collection of problems from science and industry applicable to the topics in that Unit. Available with Critical Thinking Approach to Geometric and Measurement Topics Is an Instructor's Manual and a Test Kit. Answers for some of the odd numbered exercises appear at the end of the student text while the complete answers appear in the Instructor's Manual. Because of the emphasis on understanding, this is an outstanding text for teaching geometric concepts. The Instructor's Manual contains teaching and testing hints, possible schedule, the student copies for Problems for Problem Solving, and all answers to the Critical Thinking Exercises (Exercises for Understanding and Problems for Problem Solving). The Test Kit contains a Placement Test, 4 Criteria Referenced Unit Tests (one for each Unit), and a Final. All tests have been developed using the suggestions and guidance of the University of Chicago Testing Department. These multiple-choice tests are designed to test mastery of the content by students. DOING HOMEWORK FROM GEOMETRIC AND MEASUREMENT TOPICS TESTIMONIALS and CUSTOMER COMMENTS "I was very pleased to find a Unit on logic in this text; this very important topic is not included in traditional texts. Introducing mathematical logic (the same as that used in electronics to program computers) at this level not only gives students an understanding and use of a powerful tool in science, but gives them advanced preparation for life in the computer age." "I was not aware that College Entrance Exams and Calculus did not require a full year of geometry in order to be able to complete these successfully." "Teaching geometry from a practical viewpoint first makes a large difference in a student's ability to understand this subject. I was glad to learn that there is NO X-step proof; that proofs should only be as long as needed to convince the audience. The number of steps doesn't matter." "Our school has enjoyed great student success using Critical Thinking Approach to Geometric and Measurement Topics. The attitudes of our students improved which we had not expected." The Text Critical Thinking Approach to Beginning Algebra (First Year Algebra) is recommended by Colleges, Universities, Secondary Schools, Home Schoolers, and Self Study Students because it emphasizes understanding, learning, and relearning of mathematical concepts as opposed to memorization. This Algebra 1 textbook fosters Student Success and enables the Teaching of math concepts because of the extra amount of time allowed to master these needed concepts. Critical Thinking Approach to Beginning Algebra has been recommended as the text for Beginning Algebra by schools. homeschoolers, and for self study when learning and understanding Algebra is the main objective. Recommended for students want to understand and learn beginning algebra concepts as opposed to memorizing them. Available soon will be a new additional Chapter on "Introduction to Linear Algebra" on a separate PDF File. This topic is becoming critical for students needing to know about our changing world because of Quantum Mechanics. Critical Thinking Approach to Beginning Algebra contains the curriculum for a first year Algebra math course normally taught in 9th grade or in a review sequence in college. Many students struggle with algebra through no fault of their own. Algebra has its own unique language and set of rules, and the frustration experienced by algebra students can result in a loss of self-confidence. Critical Thinking Approach to Beginning Algebra features a comprehensive, step-by-step approach that simplifies complex concepts by breaking them up into bite-size pieces, using high quality illustrations, and providing real-life examples. The mathematics is presented in an understandable way in order to maximize success in this course. The Beginning Algebra Text in the Arithmetic to Algebra through Pre-Calculus Series contains all necessary topics needed to understand Probability and Statistics or Intermediate Algebra. The complete Table of Contents for this text is presented at the bottom of this web-page. If you wish to see another page on this web-site and the M.T.E. Site Map does not appear at the left of your screen Open the Link Shown Below: Each Section of Critical Thinking Approach to Beginning Algebra is followed by 4 sets of exercises: Exercises for Vocabulary (for reading and vocabulary development), Exercises for Computation (regular drill and practice exercises), Exercises for Understanding (to develop critical thinking skills), and a Self-Test. The Instructor's Manual contains Problems for Problem Solving (a collection of problems from science and industry applicable to the topics in that Unit). Because of the emphasis on understanding, this is an outstanding text for teaching algebra or algebraic concepts. Available with Critical Thinking Approach to Beginning Algebra is an Instructor's Manual and a Test Kit. Answers for some of the odd numbered exercises appear at the end of the text while the complete answers appear in the Instructor's Manual for the text. The Instructor's Manual contains teaching and testing hints, a possible schedule, the student copies for Problems for Problem Solving, and all answers including those to the Exercises for Understanding and Problems for Problem Solving. The Test Kit contains a Placement Test, 7 Criteria Referenced Unit Tests (one for each Unit), and a Final. All tests have been developed using the suggestions and guidance of the University of Chicago Testing Department. These multiple-choice tests are designed to test mastery by students of the content in the text. HOME SCHOOLING WITH BEGINNING ALGEBRA TESTIMONIALS and CUSTOMER COMMENTS "Some of the features of this text that we found particularly helpful to our students were the raised negative sign in polynomial subtraction - (polynomial) changed to -1(polynomial)." "Stress on equivalent equations made students more aware of multiple forms of the same equation, and far fewer homework questions about specific algorithms, instead, questions were asked about concepts involved." "I was very pleased that a "live person" was available to answer my questions at this web site." "The A, B, or C grades of our students improved to 62% from 43% and the dropout or did not attend class went down to 12% from 28%. Our students were also better prepared to work with literal equations." Text Critical Thinking Approach to Intermediate Algebra (Second Year Algebra) is recommended by Colleges, Universities, Secondary Schools, Home Schoolers, and Self Study Students because it emphasizes understanding, learning, and relearning of mathematical concepts as opposed to memorization. This Algebra 2 textbook fosters Student Success and enables the Teaching of math concepts because of the extra amount of time allowed to master these needed concepts. Critical Thinking Approach to Intermediate Algebra contains the curriculum for the second year algebra math course normally taught in 11th grade or in a review sequence in college. Many students struggle with algebra through no fault of their own. Second Year Algebra has its own unique language and set of rules, and the frustration experienced by algebra students can result in a loss of self-confidence. Critical Thinking Approach to Intermediate Algebra features a comprehensive, step-by-step approach that simplifies complex concepts by breaking them up into bite-size pieces, using high quality illustrations, and providing real-life examples. The Critical Thinking Approach to Intermediate Algebra has a unique approach to factoring quadratic equations that removes much of the guessing found in the traditional approach. The mathematics is presented in an understandable way in order to maximize success in this course. The Intermediate Algebra Text in the Arithmetic to Algebra through Pre-Calculus Series contains all necessary topics needed to understand Pre-Calculus or Advanced Algebra. The complete Table of Contents for this text is presented at the bottom of this web-page. If you wish to see another page on this web-site and the M.T.E. Site Map does not appear at the left of your screen Open the Link Shown Below: Each Unit in Critical Thinking Approach to Intermediate Algebra contains 4 sets of exercises: Exercises for Vocabulary (for reading and vocabulary development), Exercises for Computation (regular drill and practice exercises), Exercises for Understanding (to develop critical thinking skills), and a Self-Test. The Instructor's Manual contains Problems for Problem Solving (a collection of problems from science and industry applicable to the topics in that Unit). Because of the emphasis on understanding, this is an outstanding text for teaching algebra and algebraic concepts. Available with Critical Thinking Approach to Intermediate Algebra is an Instructor's Manual and a Test Kit. Answers for some of the odd numbered exercises appear at the end of the text while the complete answers appear in the Instructor's Manual for the text. The Instructor's Manual contains teaching and testing hints, possible schedule, the student copies for Problems for Problem Solving, and the answers including those to the Exercises for Understanding and Problems for Problem Solving. The Test Kit contains a Placement test, 7 Criteria Referenced Unit Tests (one for each Unit), and a Final. All tests have been developed using the suggestion and guidance of the University of Chicago Testing Department. These multiple-choice tests are designed to test mastery of the content by students. HOME SCHOOLING USING INTERMEDIATE ALGEBRA TESTIMONIALS and CUSTOMER COMMENTS "Students enjoyed the method of completing the square presented and were better prepared to understand and use the quadratic formula. Students had few problems with factoring and the technique for factoring trinomials with coefficients greater than one eliminated the usual problems encountered by students." "Students had an unusually good command of math vocabulary because of the Exercises for Vocabulary: students understood the content and were able to give examples when asking questions. The Mean MAA Test score rose from .459 to .607." "The methods for factoring quadratic trinomials is the best presentation I have ever seen. Students really understood what they were doing, consequently enjoyed this topic." "This company honestly knows what "customer service" is and what it means to an individual customer." Text Critical Thinking Approach to Pre-Calculus is recommended by Colleges, Universities, Secondary Schools, Home Schoolers, and Self Study Students because it emphasizes understanding, learning, and relearning of mathematical concepts as opposed to memorization. This Advanced Algebra and Trigonometry textbook fosters Student Success and enables the Teaching of math concepts because of the extra amount of time allowed to master these needed concepts. The Pre-Calculus Text in the Arithmetic to Algebra through Pre-Calculus Series contains the Advanced or College Algebra and Trigonometry needed to understand calculus. The mathematics is presented in an understandable way in order to maximize success in this course. Critical Thinking Approach to Pre-Calculus contains the curriculum for Advanced or College Algebra and Trigonometry; a math course normally taught in 12th grade or in a review sequence in college. This text contains four Chapters on algebraic functions, two Chapters on Trigonometry, and two Chapters on Circular Functions. If you wish to see another page on this web-site and the M.T.E. Site Map does not appear at the left of your screen Open the Link Shown Below: Each Section in Pre-Calculus is followed by 4 sets of exercises: Exercises for Vocabulary, Exercises for Computation, Exercises for Understanding, and a Self-Test. The Instructor's Manual contains Problems for Problem Solving (a collection of problems from science and industry applicable to the topics in that Unit). Because of the emphasis on understanding, this is an outstanding text for teaching concepts associated with pre-calculus. Available with Critical Thinking Approach to Pre-Calculus is an Instructor's Manual and a Test Kit. Answers for some of the odd numbered exercises appear at the end of the text while the complete answers appear in the Instructor's Manual for the text. The Instructor's Manual contains teaching and testing hints, possible schedule, the student copies for Problems for Problem Solving, and all answers including those to the Exercises for Understanding and Problems for Problem Solving The Test Kit contains a Placement Test, 7 Criteria Referenced Unit Tests (one for each Chapter) and a Final. All tests have been developed using the suggestions and guidance of the University of Chicago Testing Department. These multiple-choice tests are designed to test mastery of the content by students. SELF-STUDY USING PRE-CALCULUS TESTIMONIALS and CUSTOMER COMMENTS "Our students do not have as good a mathematics background as most students when they start our Pre-Calculus course. I was concerned that they would not be able to understand these concepts and not be very successful. However, because of the good explanations and other types of exercises they not only understood the concepts but also enjoyed the course. "We had difficulty covering the text, but we are going to adopt the first two algebra texts in this series and that will make a big difference in the future because our students will be better prepared for pre-calculus." "The presentation of functions is the best I have ever seen. Students really understood what they were doing and could discuss these concepts with each other and other instructors." "The presentation of trigonometry, especially the identities, is outstanding" This Series (Arithmetic to Algebra through Pre-calculus) from MTE was designed to maximize student understanding and reduce the dependence on rote memory. The Series, which emphasizes, A Critical Thinking Approach to Learning Mathematics includes the following: THE TEXTS IN THE M.T.E. SERIES: THE STAIRWAY TO CALCULUS. The Critical Thinking Approach to Learning Mathematics Series of MTE are based on the results of nearly 20 years of research concerned with determining why students in high schools and college memorize math, but fail to understand most of the concepts. Unfortunately memorized concepts are stored in the student's short-term memory and are soon forgotten. This circumstance causes students to fail or get low grades in math on college or university entrance exams, eliminating any chance of taking a university level math course needed for a degree that leads to higher paying positions. Satisfaction is guaranteed on all M.T.E. products. If you are not satisfied with a M.T.E. product you have purchased, contact the publisher, in writing, within 6 weeks of the purchase date. M.T.E. Ltd. 3095 South Trenton Street Denver, Colorado 80231 - 4164 In your letter, state the full name of the product and the date of purchase. You will receive a permission to return this product as soon as the information is verified (usually by return mail). You must pay the return shipping cost to M.T.E. If the product is in excellent condition, you will receive credit on your credit card or a check for the full refund. If the product is not in excellent condition, it will be returned to the sender. THE SECRETS OF LEARNING MATH There are essentially two secrets to learning mathematics; one is concerned with the content and the other with teaching. There is far too much content at each grade level, and every few years the amount of content is increased. Under these conditions, students are urged to memorize concepts without understanding in order to maintain an acceptable grade level on tests. Research has determined that, for most math students, the memorization of concepts without understanding is stored in the short-term memory and soon forgotten. The first secret to student success in math is to reduce the amount of content at each level so that there is time to learn and understand the concepts. The second secret is one concerned with teaching. I have often heard that in the history of education, the "best" education took place in the one-room schoolhouse. If this is true, then it is because in a one-room schoolhouse education often came about as a result of older students teaching younger ones; both students gain from this experience. When a student is able to explain in his or her own words a math concept, that student understands the concept. The second secret to student success in learning math is to have students understand math concepts and demonstrate their understanding the concept by explaining it in their own words to someone else. Conclusions from the Research There seems to be two main factors that could be corrected in our teaching of math courses that would go a long way toward eliminating the current problem. These two factors are memorization and acceleration. Memorization: Students are trying to memorize all necessary facts that are needed to pass the course. Unfortunately, this is not possible for most students and such a course of action has no long-term benefits: hence causing the basic problem in mathematics. Acceleration: We are attempting to teach too much too fast in math courses. This, in turn, causes students to rely on memorization as their best hope to pass the course. We do not give them the proper amount of time to understand concepts, which reduces the need for memory. STUDYING FROM GEOMETRIC AND MEASUREMENT TOPICS RESEARCH INDICATES THREE CHANGES The research results indicated that certain changes were needed to correct and eliminate the problem of memorization in mathematics courses. Procedures for correcting these problems are presented in the Critical Thinking Approach to Learning Mathematics Series of M.T.E. 1. Need to restructure and reduce curriculum Over the years traditional texts have constantly added more and more content to each course causing instructors to "jam" too much content in too short a time. Traditional courses present an impossible number of concepts to be learned in one term, forcing students to memorize rather than understand concepts; this, in turn, forces much of the following term to be spent in review. The Critical Thinking Approach to Learning Mathematics Series of M.T.E. contains all the necessary mathematical topics for each course, but eliminates the unnecessary and "frill" concepts in that course. It was determined that 30 Units (Chapters) would be required to adequately cover the mathematics from Arithmetic to Pre-Calculus to prepare a student sufficiently to enroll and pass Calculus or another university level mathematics course. The 30 Units were developed from a survey of university faculty of the least understood critical topics which professors expected students to know before they attempted to enroll in calculus. Units were subdivided into Sections, and then Objectives for each Section were developed. These texts were written using the Objectives as a guide. 2.Need to have more emphasis in reading math Students who have difficulty in math are often those who are not able to read their texts with understanding. These students cannot reconstruct math concepts after reading a math text and are handicapped by the inability to ask meaningful questions in class or be able to successfully review for upcoming tests. Exercises for Vocabulary were designed and inserted in the Critical Thinking Approach to Mathematics materials at the end of each Section to help students learn terminology, to read mathematics, to be able to use reference material, and to be able to reconstruct concepts as needed. The Texts in this series are written at an appropriate level with understandable explanations. 3. Need more emphasis on critical thinking and problem solving skills Students who have difficulty in math are often those who cannot use critical thinking to improve their reasoning. They usually need help to improve their problem solving skills. The ability to solve problems and think critically is a valuable asset in seeking employment. Exercises for Understanding containing discussion, explanation, and discovery exercises were designed and inserted in the Critical Thinking Approach to Mathematics materials at the end of each Section to improve critical thinking skills. If one is able to think critically, then problem solving is not a mystique. In addition, a set of Problems for Problem Solving is included in the Instructor's Manual. This collection of problems, gathered from science and industry, is designed to give students practice in working with "real life" problems and provide an opportunity to use newly learned math concepts in a problem solving situation. GETTING HELP FROM A TUTOR ON A LESSON FROM CRITICAL THINKING APPROACH TO PRE-CALCULUS 4. Need to continue emphasis on drill and practice Traditional texts have always contained an ample number of drill and practice exercises; this necessary aspect of learning math is continued in the Critical Thinking Approach to Mathematics materials by inserting Exercises for Computation and a Self Test at the end of each Section. When considering drill and practice exercises, the difference between the texts of this series and traditional texts are the use of an algorithm in alternate ways instead of just more and more exercises. (Homework should not be a form of punishment, but should be a learning experience.) WHAT IS CRITICAL THINKING? Everyone thinks at some level, it is our nature to do so; however, undirected thinking is often biased, distorted, nonproductive, illogical or partially uninformed. Because students have been using undirected thinking, they often feel they know all there is to know about thinking. Our challenge is to demonstrate to our students that they can extend their thinking abilities to include "educational critical thinking". They must do enough thinking to question, analyze, and extend data and concepts that affect their lives. Some of the CEO's in industry estimate that 90% of their future employees will be needed as thinkers, expediters, and communicators; to be able do this successfully students will have to read, write, and think. This ability will need to include math. MAKING DECISIONS IN BUSINESS ABOUT EMPLOYEE MATH EDUCATION< "Educational Critical Thinking" is logical and productive thinking that is capable of extending ideas, creating new ideas, and synthesizing patterns of thought. Educational Critical Thinking is a type of disciplined reasoning. Educational Critical Thinking must be self-directed, self-disciplined, self-monitored, and self-corrective. The definition used for critical thinking in the M.T.E.'s series of texts for the Critical Thinking Approach to Mathematics was: "The ability to ask questions and to seek answers toward the positive solution of a problem or situation." COMMITTEE ADOPTING CRITICAL THINKING APPROACH TO MATHEMATICS TEXTS ABOUT THE INSTRUCTOR'S MANUAL This Manual is designed to help Instructors, Teachers, and Tutors with the information they need to conduct a successful course. The Manual contains an Introduction with helpful teaching hints. A Unit by Unit commentary which includes: the purpose of each section, the Objectives of each Subsection, new vocabulary, answers to the Exercises for Understanding, the Solutions for the Problems for Problem Solving, a Complete set of answers to all exercises, and Student Copies for the Problems for Problem Solving. ABOUT THE TEST KIT University of Chicago Section of the Keller Plan played a major role in the initial development of the MTE test package. Since there were more than 1000 students enrolled each semester in the math program using the MTE Texts and because they all took a CRT Unit Test on the same day at the end of each Unit; it was necessary to be able to give and evaluate these tests over a weekend. In order to do this in such a short time period; it was necessary to use the multiple-choice format. On the other hand, we did not want the test to be "multiple-guess." It was in this connection that the University of Chicago staff were helpful enabling the development of valid, objective, efficient, and reliable tests. Each Test Kit contains a Placement Test , Criteria Reference Tests for each Unit, and a Final Test. The Placement Test can be used to determine if the student is properly placed in the series. ( The Critical Thinking Approach for Competency Arithmetic does not have a placement Test.) The Unit Criteria Referenced Tests will determine if the student has mastered the necessary content. The Final Test should be used as one part of a more comprehensive test with the other parts prepared by the instructor. The Test Kits have general instructions, a student copy for each test, and an answer sheet with other data. The tests for the MTE Texts were developed and improved over a seven-year period. During this period over 11,000 students took various forms of the tests and a complete computer analyses of all student test results were made to increase their validity and reliability. From the Publisher This team of authors has taken the time and spent a great amount of effort to develop a series of texts with successful student learning as the main goal. Using an organizational scheme based on the objectives students need to learn, the authors have incorporated understandable explanations; illustrations that clearly present concepts, and have included a variety of exercises that not only provide the necessary drill, but challenge students to think about math concepts. This series of texts is setting a new direction, marching counter to that of traditional texts which contain too much content; forcing students to memorize (not understand) as the only way to pass. This new direction in education taken by this team of authors is summed-up in their logo below. THE M.T.E. LOGO CLEP COLLEGE LEVEL ENTRANCE EXAMINATION PROGRAM ABOUT CLEP COLLEGE-LEVEL EXAMININATION PROGRAM(S) The College-Level Examination Program(s) or CLEP provides students of any age with the opportunity to demonstrate college-level achievement through a program of exams in undergraduate college courses. There are more than 2,900 accredited colleges and universities that grant credit and/or advanced standing for CLEP exams. To find the names and locations of these institutions granting credit for CLEP; search the CLEP College database . This data bank is located on the CLEP web-site at - Nearly half of these institutions also administer the exams at their own CLEP test centers. CLEP is one homeschooler-friendly testing option that provides a way for a student to earn the same amount of college credit that you would get if you took - and did well in - a semester- or year-long college course covering the same material. You can shorten your path to a college degree - with CLEP! Using the CLEP examinations it is possible for a student to "test-out" of a significant portion of an Associates Degree program before attending college. Many smaller universities accept passing CLEP test results toward their four-year degree programs. The CLEP program administered by the College Entrance Examination Board, offers standardized examinations in many college level areas. In Mathematics, examinations are offered in College Math, College Algebra, Trigonometry, College Algebra and Trigonometry, and Calculus. CLEP exams provide a possibility for students to achieve double credit for a course both in high school and college. This is particularly true as it is becoming quite apparent that the material being taught in the first two years of college mathematics at most universities and colleges is a "review of high school math". It seems practical to teach it right the first time and get college credit as well as save time and money. THE FOLLOWING BUTTONS WILL CONNECT YOU TO SOME IMPORTANT FACTS ABOUT CLEP. The CLEP College Mathematics exam was developed to cover material generally taught in a college course for non-mathematics majors and majors in other fields not requiring a knowledge of advanced mathematics. Nearly half of the exam requires the candidate to solve routine straightforward problems; the remainder involves solving non routine problems in which candidates must demonstrate their understanding of concepts. The exam includes questions on logic and sets, the real number system, functions and their graphs, probability and statistics, and topics from algebra. It is assumed that candidates are familiar with currently taught mathematics vocabulary, symbols, and notation. The exam places little emphasis on arithmetic calculations, and it does not contain any questions that require the use of a calculator. However, an online scientific calculator (non-graphing) is available during the exam as part of the testing software. The exam contains approximately 60 questions to be answered in 90 minutes. * The College Algebra examination covers material usually taught in a one-semester college course in algebra. About half the exam is made up of routine problems requiring basic algebraic skills; the remainder involves solving non routine problems in which candidates must demonstrate their understanding of concepts. The exam includes questions on basic algebraic operations; linear and quadratic equations, inequalities, and graphs; algebraic, exponential, and logarithmic functions; and miscellaneous other topics. It is assumed that the candidate is familiar with currently taught algebraic vocabulary, symbols, and notation. The exam places little emphasis on arithmetic calculations, and it does not contain any questions that require the use of a calculator. However, an online scientific calculator (non graphing) will be available during the exam. The Trigonometry examination covers material usually taught in a one-semester college course in trigonometry with primary emphasis on analytical trigonometry. More than half the exam is made up of routine problems requiring basic trigonometric skills; the remainder involves solving non routine problems in which candidates must demonstrate their understanding of concepts. The exam includes questions on trigonometric functions and their relationships; evaluation of trigonometric functions of positive and negative angles; trigonometric equations and inequalities; graphs of trigonometric functions; trigonometry of the triangle; and miscellaneous other topics. It is assumed that the candidate is familiar with currently taught trigonometric vocabulary and notation and with both radian and degree measure. The exam places little emphasis on arithmetic calculations. A calculator is not permitted on the first part of the exam, but an online scientific (non graphing) calculator will be available during the second part of the test. Some questions in the second part do require the use of a calculator. The examination contains 65 questions to be answered in 90 minutes. Part 1 contains 25 questions in 30 minutes and does not allow the use of a calculator. Part 2 contains 40 questions in 60 minutes and requires the use of an online scientific calculator. 15% Evaluation of trigonometry functions of angles with terminal sides in various quadrantants or on an axis, including positive and negative angles in both degrees and radians, also angles greater than 360&deg The College Algebra-Trigonometry examination covers material usually taught in a one-semester course that includes both algebra and trigonometry. Such a course is usually taken by students who have studied algebra and geometry in high school, but who need additional study of pre-calculus mathematics before enrolling in calculus and other advanced courses at the college level. Approximately half the test is made up of routine problems requiring basic algebraic and trigonometric skills; the remainder involves solving non routine problems in which candidates must demonstrate their understanding of concepts. The algebra part of the test includes questions on basic algebraic operations; linear, and quadratic equations; inequalities and graphs; algebraic, exponential, and logarithmic functions; and miscellaneous other topics. The trigonometry part of the test includes questions on trigonometric functions and their relationships, evaluations of trigonometric functions of positive and negative angles, trigonometric equations and inequalities, graphs of trigonometric functions, trigonometry of the triangle, and miscellaneous other topics. It is assumed that the candidate is familiar with currently taught algebraic and trigonometric vocabulary and notation with both radian and degree measure. The College Algebra-Trigonometry examination requires all of the knowledge and skills required by the separate examinations in College Algebra and Trigonometry. (See tables above for more detailed information.) The combined examination contains 63 questions to be answered in 90 minutes. There are three separately timed sections. Part 1 consists of 30 algebra questions in 45 minutes; an online scientific calculator is provided although there are no questions that require a calculator. Part 2 contains 13 questions to be answered in 15 minutes and does not allow the use of a calculator. Part 3 contains 20 questions to be answered in 30 minutes and requires the use of an online scientific calculator. Students ready for CLEP MATH EXAM CLEP EXAMINATION CREDIT Most colleges and universities grant credit for CLEP exams, but not all. There are 2,900 institutions that grant credit for CLEP and each of them sets its own CLEP policy; in other words, each institution determines for which exams credit is awarded, the scores required and how much credit will be granted. Therefore, before you take a CLEP exam, check directly with the college or university you plan to attend to make sure that grants credit for CLEP and review the specifics of its policy. Typically, a college lists all its academic policies, including CLEP policies, in its general catalog. You'll probably find the CLEP policy statement under a heading such as Credit-by-Examination, Advanced Standing, Advanced Placement, or External Degree Program. If you can't find this information, ask the admission or registrar's office for a copy of the college's credit-by-examination policy. Not all colleges award the same amount of CLEP credit for individual tests. Furthermore, some colleges place a limit on the total amount of credit you can earn through CLEP or other exams. Other colleges may grant you exemption but no credit toward your degree. Knowing several colleges' policies concerning these issues may help you decide which college to attend. If you think you can pass a number of CLEP exams, you may want to attend a college that will allow you to earn credit for all or most of them. CLEP web-site contains a data base of colleges of more than 2900 accredited colleges and universities that award credit for satisfactory scores on CLEP exams It is possible to do a Search in this web-site for both colleges and universities that accept CLEP credit and the test sites where an CLEP exam may be taken. MAIL CLEP P.O. Box 6600 Princeton, New Jersey 08541 - 6600 PHONE (800) 257 9558 CLEP Office Hours are 8 am to 6pm Monday - Friday (If ordering a CLEP Exam over the phone you will need to be prepared to pay by credit card. American Express, Master Card, Visa only) Most colleges publish the required scores for earning CLEP credit in their general catalog or in a brochure. The required score for earning CLEP credit may vary from exam to exam, so find out the minimum qualifying score for each exam you're considering. Getting credit for general requirements At some colleges, you may be able to apply your CLEP credit to the college's core curriculum requirements. For example, all students may be required to take at least six hours of humanities, six hours of English, three hours of mathematics, six hours of natural science, and six hours of social science, with no particular courses in these disciplines specified. In these instances, CLEP credit may be given as "6 hrs. English Credit" or "3 hrs. Math Credit" without specifying for which English or mathematics courses credit has been awarded. Find out before you take a CLEP exam what type of credit you can receive or whether you will be exempted from a required course but receive no credit. Some colleges won't grant credit for a CLEP exam if you've already attempted a college-level course closely aligned with that exam. For example, if you successfully completed English 101 or a comparable course on another campus, you'll probably not be permitted to receive CLEP credit in that subject also. Some colleges won't permit you to earn CLEP credit for a course that you failed. Colleges usually award CLEP credit only to their enrolled students. There are other stipulations, however, that vary from college to college. Here are some additional questions to keep in mind and ask your college: Do you need to formally apply for CLEP credit by completing and signing a form? Do you have to "validate" your CLEP score by successfully completing a more advanced course in the subject? Does the college require the optional free-response (essay) section for the examinations in Composition and Literature as well as the multiple-choice portion of the CLEP exam you're considering? Will you be required to pass a departmental test such as an essay, laboratory, or oral exam in addition to the CLEP multiple-choice exam? Knowing the answers to these questions ahead of time will permit you to schedule the optional free-response or departmental exam when you register to take your CLEP exam. At the time you take the exam, you can indicate in test software the college, employer, or certifying agency that you want to receive your CLEP test scores. There is no additional cost for this service ‹ your exam fee covers it. If you haven't decided by the time you take the test which institution you want to receive your scores, leave that item blank. OBTAINING CLEP TRANSCRIPTS If you did not indicate a score recipient institution at the time of your exam and you want to request your CLEP scores, you can do so by ordering a CLEP Transcript. This Transcript is a cumulative score report of all the CLEP exams you have taken and the scores you earned in the last 20 years. To obtain a CLEP Transcript: Download the Transcript Request Form (.pdf/35k). Requires Adobe Acrobat Reader (latest version recommended). Mail the completed form with your payment to CLEP to the address on the form. Or call (800) 257-9558 if ordering with a credit card. Please provide your name (at testing time), date of birth, social security number, exam title(s), test date, testing location/center, and where you want your scores to be sent.
ABA Math is a program to help autistic kids learn rote arithmetic facts. It is modeled after the Applied Behavioral Analysis methods pioneered by Ivor Lovaas at UCLA. It is intended for use under the direct supervision of a teacher or therapist Funiter (FUNction ITERation) is developed for educational purposes, generating graphs of several types for iteration of real and complex functions with comfortable switching between related types of graphs
're a Mathcad novice or veteran user, this first-of-its-kind book can help you quickly tap the awesome power of Mathcad, the world's most popular computer software for doing mathematics. Students and other new users are introduced to essential underlying concepts and key features of Mathcad in a user-friendly way--while the book's abundant sample problems from multiple disciplines, invaluable how-to tips, and accompanying CD-ROM exercises help experienced users discover and apply the power of previously unused Mathcad features to their daily work.
Descriptions and Ratings (1) Date Contributor Description Rating 24 Nov 2009 MathWorks Classroom Resources Team Northeastern University's computer-based discovery lab, which teaches computer programming and engineering concepts to freshmen by providing hands-on experience with test and measurement instrumentation, uses MATLAB as its programming environment. Using MATLAB as the programming language allows professors to introduce programming and engineering concepts in simple, observable, and sequential steps. MATLAB and various MathWorks toolboxes allow students to use a single, efficient software tool to control their instruments and analyze their data.
Maths To support students enrolled in maths and statistics courses, The Learning Centre provides a range of short videos online associated with mathematics and learning skills. Success in Maths for Statistics (SIMS) Topics include formulas, arithmetic, calculator, basic statistics, graphing. Complete the online readiness testing (UConnect username and password required) or complete the first CMA on your STA2300 course webpage, to self assess your knowledge and determine whether you need to attend this workshop. Completing the first CMA is part of your first assignment in STA2300.
Elementary Linear Algebra - 4th edition Summary: The text starts off using vectors and the geometric approach, plus, it features a computational emphasis. The combination helps students grasp the concepts. At the same time, it provides a challenge for mathematics majors5.61 +$3.99 s/h Good TGChavez La Mesa, CA Good VERY little pencil notes thru-out-still VERY USABLE-clean cover-We ship out FAST w/FREE tracking on this item-(Gotta have it fast? ) Expedited shipping AVAILABLE-(Personalized Service~Safe Pack...show moreaging~Expedited moves you to front of the line
Calculus of Variations with Applications Applications-oriented introduction to variational theory develops insight and promotes understanding of specialized books and research papers. ...Show synopsisApplications-oriented introduction to variational theory develops insight and promotes understanding of specialized books and research papers. Suitable for advanced undergraduate and graduate students as a primary or supplementary text. 1969 edition
11.5 Probability with the Fundamental Counting principle, Permutations, and Combinations 11.6 Events Involving Not and Or; Odds 11.7 Events Involving And; Conditional Probability 11.8 Expected Value 12. Statistics 12.1 Sampling, Frequency Distributions, and Graphs 12.2 Measures of Central Tendency 12.3 Measures of Dispersion 12.4 The Normal Distribution 12.5 Problem Solving with the Normal Distribution 12.6 Scatter Plots, Correlation, and Regression Lines 13. Mathematical Systems 13.1 Mathematical Systems 13.2 Rotational Symmetry, Groups, and Clock Arithmetic 14. Voting and Apportionment 14.1 Voting Methods 14.2 Flaws of Voting Methods 14.3 Apportionment Methods 14.4 Flaws of Apportionment Methods 15. Graph Theory 15.5 Graphs, Paths, and Circuits 15.2 Euler Paths and Euler Circuits 15.3 Hamilton Paths and Hamilton Circuits 15.4 Trees45855
Rates of Change and Limits Calculating Limits Using the Limit Laws Precise Definition of a Limit One-Sided Limits and Limits at Infinity Infinite Limits and Vertical Asymptotes Continuity Tangents and Derivatives 3. Differentiation The Derivative as a Function Differentiation Rules The Derivative as a Rate of Change Derivatives of Trigonometric Functions The Chain Rule and Parametric Equations Implicit Differentiation Related Rates Linearization and Differentials Estimating with Finite Sums Sigma Notation and Limits of Finite Sums The Definite Integral The Fundamental Theorem of Calculus Indefinite Integrals and the Substitution Rule Substitution and Area Between Curves 6. Applications of Definite Integrals Volumes by Slicing and Rotation About an Axis Volumes by Cylindrical Shells Lengths of Plane Curves Moments and Centers of Mass Areas of Surfaces of Revolution and The Theorems of Pappus Work Fluid Pressures and Forces Sequences Infinite Series The Integral Test Comparison Tests The Ratio and Root Tests Alternating Series, Absolute and Conditional Convergence Power Series Taylor and Maclaurin Series Convergence of Taylor Series; Error Estimates Applications of Power Series Fourier Series Double Integrals Areas, Moments and Centers of Mass Double Integrals in Polar Form Triple Integrals in Rectangular Coordinates Masses and Moments in Three Dimensions Triple Integrals in Cylindrical and Spherical Coordinates Substitutions in Multiple Integrals 16. Integration in Vector Fields Line Integrals Vector Fields, Work, Circulation, and Flux Path Independence, Potential Functions, and Conservative Fields Green's Theorem in the Plane Surface Area and Surface Integrals Parametrized Surfaces Stokes' Theorem The Divergence Theorem and a Unified Theory Appendices Mathematical Induction Proofs of Limit Theorems Commonly Occurring Limits Theory of the Real Numbers Complex Numbers The Distributive Law for Vector Cross Products Determinants and Cramer's Rule The Mixed Derivative Theorem and the Increment Theorem The Area of a Parallelogram's Projection on a Plane00
College Algebra: Enhanced with Graphing Utilities Michael Sullivan??? s time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing ...Show synopsisMichael Sullivan??? 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. The Enhanced with Graphing Utilities Series has evolved to meet today??? s course needs by integrating the usage of graphing calculator, active-learning, and technology in new ways to help students be successful in their course, as well as in their futureThe book is not that bad but the way it is organized leaves a lillte to be desired. I have found that the index is not correct when looking for particular items as they are off by a few pages and this has happened several times. Still, this is an OK book for my college
First Course in Calculus (Undergraduate Texts in Mathematics) for an Amazon Gift Card of up to £3.02, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more Book Description edition is similar in content to his book "Short Calculus", only about 3 times as big. Like "Short Calculus", it is intended as an introductory course - first year at University and possibly for good A' level students. The material is presented in a very clear and easy to understand. It starts right at the begining. The first two chapters contains preliminary material essential to understand Calculus. The first 4 parts of the book has the same chapters as "Short Calculus", but containing extra material. If you have done both A'level and Further A' level mathematics, then you will have seen most of the topics in these first 4 parts of the book. The 5th part concerns functions of several variables. You will typically only see this at University. Serge Lang was one of the main contributors to Nicholas Boubaki, and is both an eminent mathematician and teacher. So there is no surprise that the material contains rigour, even though the concepts are expressed so clearly and simply. Besides the clear explanations, there are some excellent proofs that are so much simpler than those I have seen in A'level texts. A good example is for addition formulae - cos(A+B)that only requires pythagoras theorem and a single identity that can itself be derived from pythagoras theorem for the proof. Each chapter contains numerous exercises. These start of very easy and gradually get more difficult. In the appendix at the end of the book appears the answers to many of the questions in these exercises. It is the perfect book for self-study. This has become one of my favourate introductory calculus texts. I highly recommend this book to all those readers interested in mathematics. Serge Lang's text does an effective job of teaching you the skills you need to solve challenging calculus problems, while teaching you to think mathematically. The text is principally concerned with how to solve calculus problems. Key concepts are explained clearly. Methods of solution are effectively demonstrated through examples. The challenging exercises reinforce the concepts, while enabling you to develop the skills required to solve hard problems. Answers to the majority of exercises (not just the odd-numbered ones) are provided in a hundred page appendix, making this text suitable for self-study. In some sections, such as related rates and max-min problems, Lang provides many fully worked out solutions. As effectively as Lang conveys the key concepts and teaches you how to solve problems, he does not neglect the subject's logical development. Topics are introduced only after their logical foundations have been laid. Results are derived. Theorems are proved when Lang feels that they will add to the reader's understanding. Through his exposition and his grouping of logically related exercises, Lang teaches the reader how a mathematician thinks about the subject. The book is divided into five sections: review of basic material, differentiation and elementary functions, integration, Taylor's formula and series, and functions of several variables. The heart of the course is the middle three sections. Most of the topics covered in the review of basic material should be familiar to most readers. However, it is still worth reading since there are challenging problems, properties of the absolute value function are derived from defining the absolute of a number as the square root of the square of the number, conic sections and dilations may be unfamiliar to some readers, and Lang views the material through the prism of a mathematician who knows what concepts are important for understanding higher mathematics. Lang introduces the derivative as the slope of a curve in order to motivate the introduction of the idea of a limit. Next, Lang teaches you techniques of differentiation and shows you how to use them solve applications such as related rate problems. After a detailed discussion of the sine and cosine functions, Lang introduces the Mean Value Theorem and illustrates how it can be used for curve sketching and solving for maxima or minima. Lang covers properties of inverse functions before concluding the section by defining the natural logarithm of x as the area under the curve y = 1/x between 1 and x and defining the exponential function f(x) = e^x as its inverse. The integral is introduced as the area under a curve, with the natural logarithm taken as the motivating example. Lang explains the relationship between integration and differentiation before introducing techniques of integration and their applications. Integration with respect to polar and parametric coordinates is introduced to expand the range of applications. The exercises introduce additional tricks that enable you to solve integrals that do not succumb to the basic techniques. A table of integrals is included on the inside of the book's front and back covers. Lang's demonstrates the power of differential and integral calculus through his discussion of approximation of functions through their Taylor polynomials. This chapter should also give you an idea of how your calculator calculates square roots and the values of trigonometric, exponential, and logarithmic functions. The behavior of series, including convergence and divergence tests, concludes the material on single variable calculus. The material on functions of several variables in the final section of the book is covered in somewhat greater detail in Lang's Calculus of Several Variables (Undergraduate Texts in Mathematics). Since the corresponding chapters in that text include additional sections on the cross product, repeated partial derivatives, and further techniques in partial differentiation and an expanded section on functions depending only on their distance from the origin, I chose to read these chapters in Lang's multi-variable calculus text. The material that is included here, on vectors, differentiation of vectors, and partial differentiation, should give the reader a solid foundation for a course in multi-variable calculus. I have some caveats. There are numerous errors, including some in the answer key. Some terminology is nonstandard, notably the use of bending up (down) for concave up (down), or missing, limiting the text's usefulness as a reference. In the chapter on Taylor polynomials, when Lang requests an answer accurate to n decimal places, what he really means is that the error in the answer should be less than 1/10^n, which is not the same thing. At one point, Lang claims that the Extreme Value Theorem, which he leaves unnamed, is obvious. I turned to the more rigorous texts Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (Second Edition) by Tom M. Apostol and Calculus by Michael Spivak, where I discovered proofs covering one and half pages of text of the Extreme Value Theorem and a preliminary result on which it depends that Lang does not state until an appendix much later in the book. Perhaps Lang meant the Extreme Value Theorem is intuitive. While I found much of the text to be clear, I sometimes found myself turning to Apostol's text for clarification when I read Lang's proofs. Despite my reservations, I think this text is well worth reading. Reading the text and working through the exercises gives you a good understanding of the key concepts and techniques in calculus, enables you to develop strong problem solving skills, prepares you well for more advanced mathematics courses, and gives you a sense of how mathematicians think about the subject. 13 of 13 people found the following review helpful 5.0 out of 5 starsshines without all the bling and flash24 Jun 2007 By tech book guy - Published on Amazon.com Format:Hardcover This book by the late Prof. Lang covers calculus in a clear and concise manner. I own more than a few calculus books and this book is one of my favorites. The book looks like a math book in that it is not a 1200 page glossy coloring book with multi-colored inserts on every page. I think that the style of this book is a hugh improvement over most of the books on the market. I think a student who buys this book along with a good calculus study guide would be very well set. 27 of 31 people found the following review helpful 4.0 out of 5 starsCalculus for beginning college students28 Aug 2002 By A Customer - Published on Amazon.com Format:Hardcover\|Verified Purchase I needed to bring my high school calculus up to speed for first year physics studies and found this to be the only book which covered the necessary ground. The material is presented in a thorough manner with the great majority of topics shown with proofs. The book is very well organized and there are abundant worked examples. Some problems are offered which deal with matters not covered in the text, but usually there is a worked example given among the answers. Lang deals with the material in a clear fashion so that the subject matter is usually not difficult to follow.On the negative side I can say that there is no human touch between the covers. His sole attempt at humor is an item following a list of problems in which he notes "relax". In the foreword he exhibits his firm belief that many freshmen arrive unprepared for college calculus, which may be true. But nowhere in the book is there a note of encouragement, so it cannot be described as reader friendly. Finally the index is pathetic--just three pages for a book of 624 pages, so that finding things can be frustrating. 12 of 12 people found the following review helpful 5.0 out of 5 starsPromotes real understanding of calculus26 Mar 2008 By Coleman Nee - Published on Amazon.com Format:Hardcover I had to take a refresher calculus course as a prerequisite to get into graduate school, but the assigned text (Edwards and Penney) was horrible. Like every other mass market calculus, it was filled with colorful diagrams and digressions on how to use calculators, but little in the way of explanation. Fortunately I found Lang's calculus in the university book store and it cured all of my problems. Unlike the bloated E&P, Lang's book is clear and concise. E&P covers more material to be sure, but for the essentials nothing beats Lang. After reading this book calculus became easy for me again. Which is as it should be, since calculus is a surprisingly simple subject if expalined well. 5 of 5 people found the following review helpful 3.0 out of 5 starsGood book, not great14 Feb 2010 By W. Ghost - Published on Amazon.com Format:Hardcover The book is OK, but I wouldn't say it's great. There are lots of exercises that ask you to do simple symbolic manipulation so you'll remember rules -- but there are too few exercises that require the reader to actually think harder and be creative. The explanations are often shallow and not as stimulating as they could be, in my opinion. Some examples of sections that I think are not well written are the one about implicit differentiation (the discussion is too short and not clear, and there are less exercises in this section than in others); the one about rate of change (some examples are boring, like "find the rate of change of the area of a circle given the rate of change of its diameter"; he does not make it clear that he's always derives with relation to time and that, for example, the radius and height of a cylinder should be understood as functions of time, so there's a feeling of sloppiness). It's a good book,anyway. Now, it becomes a really great book when compared to the colorful, flashy books available today.
Tagged Questions If you were going to teach you kids programming and asked me what book to use as a guide, I would recommend you either Java programming for kids or Python for kids. But what if I want to teach kidsI can't seem to wrap my head around why we should use $cm^2$ for area. According to my textbook we use it for converting units of area but I don't understand how $1cm$ is any different from $1cm^2$. ...This is going to be an annoying question, but I have to ask it as it is annoying me. I once read a book on infinity that was written by an American female maths writer. She was very easy to read and aI am wondering if there is a name for these kind of matrices and if they are interesting or not? Do they even exist? Let $A$ be a $n\times n$ matrix with elements $a_{ij}$. $A= (a_{ij})_{i,j\in\{1,When I first learn about the logarithm function $\log$ or $\ln$. My professor said that $\log x$ is a function that when we derive we get the inverse function $1/x$. This $\log$ becomes very popular ... I'm a high school student who is considering doing an undergraduate in engineering. However my the long term plan is to pursue math at a higher level. I want to do engineering at undergrad because I ... I was told that If I reject the null hypothesis than my CI will not include the population mean. However the problem that I just did has a population mean of 2.72 and a lower CI of 2.5 and an upper CI ... The problem sets that you usually get in a university course is a small fraction of the exercises in your textbook. Which raises a question: do you need to solve all the exercises from your textbook? ... I have a normal Hill function of: $y = \dfrac{x^\lambda}{h^\lambda + x^\lambda}$; where $\lambda$ is Hill coefficient, and $h$ indicates the infection point. I am concerning if we could add another ... I'm in year 10 and have a B grade - yet still manage to mess it up when I face a question regarding decimals. For example: Round 8.647 to one decimal place. Would this be 8.7 or 8.6 as I have never ... I'm having some difficulty understanding 'Linear Homogeneous Recurrence Relations' and 'Inhomogeneous Recurrence Relations', the notes that we've been given in our discrete mathematics class seem to ... Someone else got a 68% on their first account and an 83% on their second account. At a minimum, what value do they need on the third account so that their overall bank dividenda (mean) is at least an ... I don't know if its appropriate question here...but I anyway want to try... I have an algorithm in which I have a finite data in which each element is assumed as an element of a metric space with a ... Ironically, I'm on a mathematics-based forum using the terms "variables" and "differentiate" in entirely non-mathematical ways, but what I mean by this question is as follows: I'm self-studying and ... I have one question please...I solved it in this way..so I am not sure..that is it right or not? and If I solved it in the wrong way..so would like to know about the correct way and method to solve ...
eIMACS, the distance learning division of the Institute for Mathematics & Computer Science (IMACS), is dedicated to delivering the highest quality interactive math and computer science classes for gifted and talented secondary school students. University-level, online courses include Advanced Mathematical Logic, University Computer Science, AP Computer Science, and Test Prep for the AP Computer Science and AP Calculus exams. eIMACS is based in Plantation, Florida, and serves students from across the US and around the world, including homeschoolers, students without access to AP Computer Science, and students who want to study advanced math and computer science at their own pace. For high school level courses for talented middle school-aged children, consider Elements of Mathematics: Foundations (EMF). 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The Cartoon Guide to Calculus About the Book "In Gonick's work, clever design and illustration make complicated ideas or insights strikingly clear." —New York Times Book Review Larry Gonick, master cartoonist, former Harvard instructor, and creator of the New York Times bestselling, Harvey Award-winning Cartoon Guide series now does for calculus what he previously did for science and history: making a complex subject comprehensible, fascinating, and fun through witty text and light-hearted graphics. Gonick's The Cartoon Guide to Calculus is a refreshingly humorous, remarkably thorough guide to general calculus that, like his earlier Cartoon Guide to Physics and Cartoon History of the Modern World, will prove a boon to students, educators, and eager learners everywhere. Book Description A complete—and completely enjoyable—new illustrated guide to calculus Master Educator and Librarian Resources Critical Praise "How do you humanize calculus and bring its equations and concepts to life? Larry Gonick's clever and delightful answer is to have characters talking, commenting, and joking-all while rigorously teaching equations and concepts and indicating calculus's utility. It's a remarkable accomplishment-and a lot of fun." —Lisa Randall, Professor of Physics, Harvard University, and author of Knocking on Heaven's Door Gonick is to graphical expositions of advanced materials as Newton or Leibniz is to calculus. The difference is that Gonick has no rival. —Xiao-Li Meng, Whipple V. N. Jones Professor of Statistics and Department Chair, Harvard University Larry Gonick's sparkling and inventive drawings make a vivid picture out of every one of the hundreds of formulas that underlie Calculus. Even the jokers in the back row will ace the course with this book. —David Mumford, Professor emeritus of Applied Mathematics at Brown University and recipient of the National Medal of Science I always thought that there are no magic tricks that use calculus. Larry Gonick proves me wrong. His book is correct, clear and interesting. It is filled with magical insights into this most beautiful subject. —Persi Diaconis, Professor of Mathematics, Stanford It has no mean derivative results about the only derivatives that matter…. A spunky tool-toting heroine called Delta Wye seems the perfect role model for our next generation. —Susan Holmes, Professor of Statistics, Stanford A creative take on an old, and for many, tough subject…Gonick's cartoons and intelligent humor make it a fun read. —Amy Langville, Recipient of the Distinguished Researcher Award at College of Charleston and South Carolina Faculty of the Year
Loci Browse Articles This interactive Geogebra applet allows exploration of a linear transformation in terms of images of a set of points. The Geogebra interface allows dragging of points and vectors to make for versatile explorations of basic linear algebra ideas. Suggested activities are included. This interactive, browser-based game provides students with a great activity for the "reform calculus" problem of finding "function-derivative pairs" based only on graphs. Several variations are available. The dihedral group D8 (sometimes called D4), the group of symmetries of the square, is one of the simplest finite groups. In these three mathlets, we will explore different aspects of this symmetry group to get a better understanding of its structure.
UNIV 1330 - Beginning Algebra Undergraduate Bulletin Course Description This course introduces students to fundamental algebraic concepts in preparation for Intermediate Algebra. Concepts covered include word problems, fractions, graphing, linear equations and inequalities, factoring, operations with real numbers, and polynomials. Lecture and activity based instruction. Entering students with an ACT score below 19 in Math are required to register for this course or UNIV 1340 Intermediate Algebra during their first enrollment period. Once enrolled, students must enroll in UNIV 1330 during each subsequent enrollment period until they earn a course grade of C or higher, after which they must enroll in UNIV 1340 the following semester. Students enrolling in this course may have only three total attempts between this course and UNIV 1340 to complete their remediation requirements in Math. The grade in this course will not be used to compute semester and cumulative grade point averages. The course does not count toward any degree. Fall, Spring, Summer. Students are required to pass this course with a grade of "C" or higher. To learn more about this or any University College course, please contact:
6-6-13Developmental Mathematics by Trigsted, Bodden, and Gallaher is the first online, completely "clickable" combined Prealgebra, Beginning Algebra, and Intermediate Algebra text to take full advantage of MyMathLab's features and benefits. Kirk Trigsted saw marked improvements in student learning when he started teaching with MyMathLab, but he noticed that most students started their assignments by going directly to the MyMathLab homework exercises without consulting their textbook. This inspired Kirk to write a true eText, built within MyMathLab, to create a dynamic, seamless learning experience that would better meet the needs and expectations of his students. Completely clickable and fully integrated–the Trigsted eText is designed for today's learners. Developmental Mathematics is also available to be packaged with two printed resources to provide additional support for you: The eText Reference is a spiral-bound, printed version of the eText that provides a place for you to do practice work and summarize key concepts from the online videos and animations. In addition to the benefits it provides you, the eText Reference is also a nice resource for those instructors that prefer a printed text for class preparation. The Guided Notebook is an interactive workbook that guides you through the course by asking you to write down key definitions and work through important examples for each section of the eText. This resource is available in a three-hole-punched, unbound format to provide the foundation for a personalized course notebook. You can integrate your class notes and homework notes within the appropriate section of the Guided Notebook. Instructors can customize the Guided Notebook files found within MyMathLab. This is the MyMathLab Student Access Kit only, and does not include the supplementary materials listed above. Product Details ISBN-13: 9780321880161 Publisher: Pearson Publication date: 6/20/2013 Edition number: 1 Product dimensions: 6.00 (w) x 8.80 (h) x 0.20 (d) Meet the Author Kirk Trigsted teaches mathematics at the University of Idaho and has been director of the Polya Mathematics Center since its inception in 2001. Kirk has taught with MyMathLab for many years, and has contributed to the videos for several Pearson books. Kirk is also actively involved with the National Center for Academic Transformation (NCAT). Kevin Bodden is a professor of mathematics at Lewis & Clark Community College where he has taught since 1999. He holds a master's degree in mathematics from Southern Illinois University at Edwardsville and a master's degree in engineering from Purdue University. He has authored or co-authored ancillary material for numerous textbooks ranging from basic college math to calculus and statistics. He has contributed videos for several of these textbooks and has authored math content on grant projects for the Illinois Community College Board. Kevin is married with three children and is actively involved in their school and extracurricular activities. In his spare time, he enjoys soccer, camping, and geocaching. Randy Gallaher is a professor of mathematics at Lewis & Clark Community College, where he has taught since 1997. Prior to this position, Randy taught high school and middle school mathematics for five years in Missouri. He holds a master's degree in mathematics from Southeast Missouri State University and has completed additional graduate coursework at both Missouri State University and the University of Illinois at Urbana-Champaign. He has coauthored ancillary materials for numerous math and statistics textbooks and has worked as a math author on several grant projects for the Illinois Community College Board. Randy is married with three children and spends most evenings actively involved in their activities. In his limited free time, he loves to fish the small rivers and streams of southern
In the first page they find a determinate and use it to see what values make a matrix invertable, then they use determinate properties to answer a series of questions. Then they work a vector equation. I can definately recognize linear algebra since I took it twice.
Pre-Algebra Demystified [NOOK Book] ... More About This Book anxious. With Pre-Algebra Demystified, you master the subject one simple step at a time -- at your own speed. This unique self-teaching guide offers problems at the end of each chapter, quizzes to pinpoint weaknesses, and a 100-question final exam to reinforce the entire book. This is a fast and entertaining self-teaching course that's specially designed to reduce anxiety. Get ready to: * Build a working knowledge of whole numbers, fractions, decimals, ratios, and percents * Grasp word problems and underlying math concepts * Learn basic equations and operations with integers * Construct a solid foundation for algebra, geometry, statistics, and business math Related Subjects Meet the Author Allan G. Bluman taught mathematics and statistics in high school, college, and graduate school for 39 years. He received his Ed.D. from the University of Pittsburgh and has written three mathematics textbooks published by McGraw-Hill. Mr. Bluman is the recipient of "An Apple for the Teacher Award" for bringing excellence to the learning environment and the "Most Successful Revision of a Textbook" award from McGraw-Hill. His biographical also record appears in Who's Who in American Education, Fifth 28, 2010 Pre Algebra The book was not for myself. I was told by the user that the book was of a high degree of assistance. As a matter of fact it did increase test scores. 2 out of 2 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted June 20, 2008 Great Review Deciding to change careers and go back to school for an engineering degree. Its been 20 years since I last thought about this stuff. I thought it best to go back as far as I could on the mathematics. This book was an excellent starting point. I agree that there isn't a lot of explanation, but simple facts and examples are all that is needed. Like I said, it has been over 20 years for me, but I no problem following along. I hope the rest of the series is this good. 2 out of 2 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted September 16, 2005 great book never have taken algerbra (adult student) this book was great for teaching yourself could be have a little more detailed explanation but all in all pretty good learning book 2 out of 2 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Welcome to the Department of Mathematics and Statistics. We offer a wide variety of undergraduate and graduate degree programs designed for students with diverse career or higher educational goals. Our faculty members maintain active research programs in the fields of combinatorics, algebra, analysis, applied mathematics and applied statistics. In a nod toward the unity of mathematics, we offer the following question—whose answer requires several of the above fields, as well as geometry: A collection of small waves are travelling through shallow water and happen to collide. What happens next? The first half of the above sentence is governed by the famous KdV equations. (Jerry Bona, UIC, spoke at our colloquium about these waves not long ago.) The second half of the above sentence is governed by cells in the totally positive part of the Grassmannian and plabic graphs. (Dr. Lauve can tell you more about this aspect of the theory of totally positive matrices.) See our Faculty Research page for a list of local people to ask for more details, or consult the original sources:
This is a free online textbook designed for the Advanced Algebra instructor. According to the author, he "developed a set of... see more This is a free online textbook designed for the Advanced Algebra instructor. According to the author, he "developed a set of in-class assignments, homework and lesson plans, that work for me and for other people who have tried them. The complete set comprises three separate books that work together:•The Homework and Activities Book contains in-class and homework assignments that are given to the students day-by-day." "•The" target=״_blank״ Concepts Book provides conceptual explanations, and is intended as a reference or review guide for students; it is not used when teaching the class." •The" target=״_blank״ Teacher's Guide provides detailed lesson plans; it is your guide to how the author "envisioned these materials being used when I created them (and how I use them myself) " target=״_blank״ Instructors should note that this book probably contains more information than you will be able to cover in a single school year." This is the Conceptual Explanations part of Kenny Felder's course in Advanced Algebra II. It is intended for students to... see more This is the Conceptual Explanations part of Kenny Felder's course in Advanced Algebra II. It is intended for students to read on their own to refresh or clarify what they learned in class. This text is designed for use with the "Advanced Algebra II: Homework and Activities" ( and the "Advanced Algebra II: Teacher's Guide" ( collections to make up the entire course. Algebra textbook is a college-level, introductory textbook that covers the important subject of Algebra -- one of the basic building blocks of studies in higher mathematics. Boundless works with subject matter experts to select the best open educational resources available on the web, review the content for quality, and create introductory, college-level textbooks designed to meet the study needs of university students.This textbook covers:The 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 SolvingGraphs, Functions, and Models -- Graphing, Functions: An Introduction, Modeling Equations of Lines, Functions Revisited, Algebra of Functions, TransformationsFunctions, Equations, and Inequalities -- Linear Equations and Functions, Complex Numbers, Quadratic Equations, Functions, and Applications, Graphs of Quadratic Functions, Further Equation Solving, Working with Linear InequalitiesPolynomial and Rational Functions -- Polynomial Functions and Models, Graphing Polynomial Functions, Polynomial Division; The Remainder and Factor Theorems, Zeroes of Polynomial Functions and Their Theorems, Rational Functions, Inequalities, Variation and Problem SolvingExponents and Logarithms -- Inverse Functions, Graphing Exponential Functions, Graphing Logarithmic Functions, Properties of Logarithmic Functions, Growth and Decay; Compound InterestSystems 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 FractionsConic Sections -- The Parabola, The Circle and the Ellipse, The Hyperbola, Nonlinear Systems of Equations and InequalitiesSequences, Series and Combinatorics -- Sequences and Series, Arithmetic Sequences and Series, Geometric Sequences and Series, Mathematical Inductions, Combinatorics, The Binomial Theorem, Probability' " Algebra for College Students is designed to be used as an intermediate level text for students who have had some prior... see more " Algebra for College Students is designed to be used as an intermediate level text for students who have had some prior exposure to beginning algebra in either high school or college. This text explains the why's of algebra, rather than simply expecting students to imitate examples
ALS Math Courses Math Courses Math 010A - Whole Number, Fractions, Decimals -- Math 010A students will review whole number skills and learn to compute with fractions and decimals. Students will learn effective math study strategies and demonstrate increased confidence in their ability to study and use mathematics. MTH 010A is intended for students who need to strengthen their basic math skills before moving on to the faster pace of MTH 020. See the term schedule of classes for a list of available courses or the online catalog for full course descriptions. Math 010T - Whole Numbers, Fractions, Decimals: FOR WOMEN-- This class has the same course objectives as Math 010A, but is focused on needs and experiences of women learning math basic skills. Designed originally to provide support for Women in Transition students, ALS has opened up seats for women who are not in Women in Transition. Many Math10T students enjoy the women only learning environment and go on to register for a section of Math20 just for women. More Information, Opportunities and Frequently Asked Questions I used to know this! But I can't remember now. What resources could I use to review Math 10 skills? 1) Review the videos and try the practice exercises BEFORE you take your math placement test and see if you can test at a higher level -- or-- 2) If you have already taken your math placement test and wonder if you could place higher with some review, spend some time reviewing and re-take the placement test. Where can I find more options for review? You may also review by completing practice problems in the Math Placement Prep online review site. Once there, login, or click on the "Login as a guest" button. There you will also find resources to help you prepare for the Math tests. If you have questions about the Math Placement Prep online review site, please contact the Math Division.
Discrete Transition to Advanced Mathematics - 04 edition Summary: As the title indicates, this text is intended for courses aimed at bridging the gap between lower level mathematics and advanced mathematics. The transition to advanced mathematics presented is discrete since continuous functions are not studied. The text provides a careful introduction to techniques for writing proofs and a logical development of topics based on intuitive understanding of concepts. The authors utilize a clear writing style and a wealth of examples t...show moreo develop an understanding of discrete mathematics and critical thinking skills. Including more topics than can be covered in one semester, the text offers innovative material throughout, particularly in the last three chapters (e.g. Fibonacci Numbers and Pascal's Triangle). This allows flexibility for the instructor and the ability to teach a deeper, richer course3440518515.19
Modern Geometries - 5th edition Summary: This comprehensive, best-selling text focuses on the study of many different geometries -- rather than a single geometry -- and is thoroughly modern in its approach. Each chapter is essentially a short course on one aspect of modern geometry, including finite geometries, the geometry of transformations, convexity, advanced Euclidian geometry, inversion, projective geometry, geometric aspects of topology, and non-Euclidean geometries. The Fifth Edition reflects the re...show morecommendations of the COMAP proceedings on "Geometry's Future," the NCTM standards, and the Professional Standards for Teaching Mathematics. ...show less Introduction to Geometry. Development of Modern Geometries. Introduction to Finite Geometries. Four-Line and Four-Point Geometries. Finite Geometries of Fano and Young. Finite Geometries of Pappus and Desargues. Other Finite Geometries. 2. GEOMETRIC TRANSFORMATIONS. Introduction to Transformations. Groups of Transformations. Euclidean Motions of the Plane. Sets of Equations for Motions of the Plane. Applications of Transformations in Computer Graphics. Properties of the Group of Euclidean Motions. Motions and Graphics of Three-Space. Similarity Transformations. Introduction to the Geometry of Fractals and Fractal Dimension. Examples and Applications of Fractals. The Philosophy of Constructions. Constructible Numbers. Constructions in Advanced Euclidean Geometry. Constructions and Impossibility Proofs. Constructions by Paper Folding and by Use of Computer Software. Constructions with Only One Instrument. 6. THE TRANSFORMATION OF INVERSION. Basic Concepts. Additional Properties and Invariants under Inversion. The Analytic Geometry of Inversion. Some Applications of Inversion. Foundations of Euclidean and Non-Euclidean Geometries. Introduction to Hyperbolic Geometry. Ideal Points and Omega Triangles. Quadrilaterals and Triangles. Pairs of Lines and Area of Triangular Regions. Curves. Elliptic Geometry. Consistency; Other Modern Geometries. 1997 Hardcover69.2177.66 +$3.99 s/h Good TextbookBarn Woodland Hills, CA 05343518
Day 54: Tuesday 4/29: Handed out last Graded Work, and gave students time in lab to work on that, or to continue working with practice exercises from Chapter 12. Day 53: Monday, 4/28: Went through the concept of a line integral, and showed students how to develop the integral first as the concept, and then to work with this concept to get to a point where we have an integral that we can actually work with. Worked through the example from the Mathematica material by hand so that students could get a sense of the process of creating a line integral. Day 52: Thursday 4/24: Had students work on Mathematica to create Vector Fields and curves in vector fields after briefly discussing the material. Talked about ideas as students created the material on Mathematica. We will talk more fully about line integrals on Monday. Day 51: Wednesday, 4/23: Worked more on triple integrals. Did more work with examples from section 12.8. Day 50: Tuesday, 4/22: Worked on problem 24 from section 12.7 with triple integrals in rectangular coordinates. Began problem 12 from section 12.8 that we first set up in cylindrical coordinates. Asked students to come in tomorrow with the integral set up in rectangular coordinates. Day 48: Thursday, 4/17: Developed the conversion formulas and the dV component for both cylindrical and spherical coordinates. We will work on as many problems as possible next Monday and Tuesday. Day 47: Wednesday, 4/16: Started discussing triple integrals in rectangular coordinates. Emphasized that we enter the solid region of integration parallel to an axis, and then project the solid region into the remaining plane and revert to techniques of double integrals. Worked through exercise 16 from 12.7. This problem required remembering some information from earlier in the course in terms of finding the equation of a plane. Day 45: Monday, 4/14: Had students do Evaluations at the beginning of the class. spent 2nd half of class answering questions for the test tomorrow. Day 44: Thursday, 4/10: Spent time working on some integration, as well as reviewing some key ideas about regions of integration, and how we determine the limits of integration. Day 43: Wednesday 4/9: Had students work on numbers 8 , 12 and 28 from section 12.4. Also asked them to work on problem 40 and 52 from page 825 Day 42: Tuesday, 4/8: Finished problem 14, and did two more problems with students. Told students to complete the integration. I will bring the answers to class tomorrow. Day 41: Monday, 4/7: Began section 12.4 - Polar coordinates and double integrals with polar coordinates. Developed the concept that dA is r dr dtheta. We would use polar coordinates either because of the boundary region, or because of the integrand. Started exercise 14, but didn't get to finish it. Day 40: Thursday, 4/3: Finished section 12.3 - Worked on another example. Day 39: Wednesday, 4/2: Began section 12.3 - setting up definite integrals over non rectangular regions in the plane. Discussed that depending on the region or depending on the integrand, we might find it better to work in dx dy order, or dy dx order. Worked through an example. Day 38: Tuesday, 4/1: Students continued work on GW 8 while I worked with groups as they had questions. Extended due date to Monday, 4/7. Day 37: Monday, 3/31:Began Chapter 12. Reviewed how we began considering definite integrals in Calculus 1. Extended ideas to definite integrals for functions of two variables. Completed an example of a double integral. Day 36: Thursday, 3/27: Worked with groups as students had questions on their two assigned problems. Day 35: Wednesday 3/26: Worked through a boundary problem, helping students see that this was just an extension of the extreme value theorem from Calculus 1. Showed that we had to consider 3 different scenarios... general critical points, critical points on the boundaries, and endpoints. Assigned students two problems that they need to work on for tomorrow. Tomorrow students will meet in pairs to work through more ideas. Graded Work 8 lays out the process. Began section 11.7 - optimization. Discussed why we would want to consider the second order Taylor Polynomial in terms of determining the behavior we would have for a function at a critical point. Mapped out the three options.... paraboloid, saddle, or a cylinder. Considered a basic quadratic in two variables, and showed how we would compute the discriminant... the value that determines which of the three scenarios we have. Day 32: Thursday 3/20: Students continued their work on GW 7. Again, I answered questions for students as they encountered them. Day 32: Wednesday 3/19: Presented the Chain Rule (Section 11.5) and completed an example showing both how to generate the formula, and then how to apply it to get the final answer. Day 31: Tuesday 3/18: Students worked in the lab on GW 7. I worked with individuals as questions arose. Day 30: Monday, 3/17: Handed back tests with answer key. Talked briefly about the tests. Results were not great. Students MUST commit to doing the practice exercises if they want to do well in the course. Collected GW 6. Continued work with the gradient vector. Reviewed the key concepts of the gradient vector, using a sheet with a contour diagram on it to help reinforce that the gradientt vector lives in the domain plane. Found the gradient vector for 4 points. Drew the points and the vectors on the contour diagram. Discussed major concepts. Looked at the gradient vector field, and discussed what it told us about the surface. At the end of class, handed out Mathematica Instructions for working with gradient vectors, as well as GW 7. Students will have the opportunity to work on this material in the computer lab tomorrow. Strongly suggested that students at least read through the instructions tonight so they have a sense of what they will be workig on. Day 29: Thursday 3/6Test on Unit 3. Happy Spring Break!! Day 28: Wednesday 3/5 Hand back GW 5, and spent some time discussing the assignment. Students need to concentrate on doing their best work for the graded work. Extended the due date for GW 6 until after the break so that students would make sure it really was their best work! Answered review questions for test tomorrow. Day 27: Tuesday 3/4Teacher out sick. Day 26: Monday 3/3Worked on considering the gradient vector,and why it was important. Showed that the maximum rate of increase is in the direction of the gradient vector, and showed that the gradient vector is perpendicular to the level curve through the point (a, b). Emphasized that the gradient vector is in the domain plane. Did an example, and discussed how we would interpret the answer. We always interpret derivatives as change in output per 1 unit change in input - or in this case in distance traveled in a particular direction. Day 25: Thursday, 2/27: Began section 11.6 - Directional Derivatives. Showed how we could replace the difference in output values in our limit definition of derivative with the differential to see that the directional derivative was essentially the dot product of a vector which we called the gradient, and a unit vector in the direction we wanted to move. Worked on two examples from the text book to compute directional derivatives. While we worked on the first example, established that this was essentially a three step process: 1) Make sure you have a unit vector in direction you want to move. 2) Find the gradient vector at a particular point, 3) Form the dot product of the two vectors. Test on Unit 2 will be on Thursday, 3/6 Day 24: Wednesday, 2/26:Picked up where we left off. Discussed that the increment was the actual change in function values, and the difference in output values along the tangent plane was called the differential. By local linearity, the differential is a good approximation for the increment as long as our point is close to the point of tangency. Completed an example that asked for finding the increment and differential for a particular function. Day 23: Tuesday, 2/25: Collected HW 5. Discussed what GW 6 entailed. Worked on section 11.4 - tangent planes and linear approximations. Had students recall information about the tangent line and local linearity from calc 1, and Euler's method from Calc 2. Used that information to help create the equation of the tangent plane for functions of two variables. Talked about using this equation to create a form that showed the difference in output values along the tangent plane. Will continue with these ideas tomorrow. Day 22: Monday, 2/24: Teacher had to be at a meeting during class time. Sent students an email asking them to work through the tutorial instructions on Partial Derivatives. When they have completed this work, it should be submitted in the Angel Drop box. Here is the assignment for Graded Work 6. I will collect GW 5 on Tuesday, and we will continue with our work in the chapter. Day 21: Thursday, 2/20: Asked if there were any questions on GW 5. No one had questions. Worked on several examples of finding the first order partial derivatives after reviewing what we had discussed on Wednesday. Extended ideas into the second order partials, and showed that there were 4 second order partial derivatives. Through one of the examples we worked on in class, showed that the mixed order partials were actually equal. This will always be the case as long as we have the same number of each type of variable, though we cannot currently prove that. Day 20: Wednesday, 2/19: Handed back GW 5 with answers. Talked briefly about the homework. The explanations are absolutely essential...and take time and energy to make sure that the "whys" are answered. Had students individually recapture what they remembered about derivatives of functions of one variable. Had students share ideas with each other. With student input, reviewed key concepts of derivative of f(x). Used this information to springboard into the concept of partial derivatives with respect to x and y. Developed the geometric and algebraic concepts for the partial derivative of f with respect to x. Worked through an example of finding the partial derivatives. Day 19: Thursday, 2/13: Snow Day Students should work on the instructions for graphing in 3 space (see link on day 18). These need to be submitted in Angel, Learning Modules, Graded Work, GW 5 - Preliminary Tutorial, and will count as 5 points towards GW 5. Day 18: Wednesday, 2/12: Discussed another function completely using a Mathematica generated Handout. Considered domain and range from an analytical point of view. Considered the cross sections. Considered the contour diagram. Used this information to fully discuss the function behavior. Students should complete the work from these instructions as 5 points towards their next homework grade. Day 17: Tuesday, 2/11: Continued discussion of the function we worked with yesterday. Talked about and computed the rates of change, and discussed what that told us about the function. Created the contour curves (level curves) for z = 0, 2, 4, 6. Graphed these, and observed the behavior of the contour diagram. Day 16: Monday, 2/10: Handed back tests and answers. Emphasized how important it is to make sure that students read through the notes, and see how the notes pretain to the practice exercises. Encouraged students to do a Test Analysis for Success Strategy Points. Started Section 9. 6. Emphasized the domain of a function of two variables is a set of ordered pairs in the plane. Did an example of a finding a domain. Began looking at cross sections of a function to help create a "wire framework" for the surface. We will continue to work tomorrow. Day 12: Monday, 2/3: Handed back quizzes with pink answer key. Reminded students that they could do a quiz analysis for Success Strategy Points. Check on my web site for further details for this option. It is NOT just redoing the questions (since you now have the answers!), but rather analyzing what went right or wrong. Handed out answers to GW 2 (yellow), and encouraged students to look at the answers carefully in terms of understanding what I am looking for in terms of written explanations. Discussed a strategy for working on exercise 19 in section 9.4 on page 661. Reviewed quickly the equation of a line in space, and showed how we could move from the parametric form to the symmetric form. Showed how to derive the equation of a plane given a point in the plane and a vector perpendicular to the plane. Did a quick example of writing the equation of a plane given a point and a perpendicular vector. Worked on exercise 24 on page 671. Students should finish up this question for tomorrow, and need to work on some of the practice exercises for this section. Reminder: Test on Unit 1 on Thursday, 2/6. Day 11: Thursday 1/30: Discussed how to create an equation for a line. Tried to have students see that all the key information was in the beginning of section 9.5, and used the picture from the book, and read through the discussion, actively showing students how to go back and forth from text to picture to help understand what is written. Ended with an example of finding the parametric equations for a line given two specific points. Students should work on exercises 1 - 19 odd. I will cover the rest of the material in the section early next week. Test on Unit 1 will be on Thursday, 2/6. Last half of class, students took a quiz on the basics from sections 1 to 4. Day 10: Wednesday 1/29: Discussed again the importance of having the beginning points of the vectors coinciding when you are determining the angle between the two vectors. Reviewed how you could be getting the angle which is the exterior angle of a triangle rather than the angle you are intending to get. Showed how to use the geometric definition of dot product to find the angle between the two vectors. This is the formula that Mathematica is actually employing when you use VectorAngle command. Reviewed how to determine if two vectors were parallel. There are 3 different approaches that you can use: 1) scalar multiples, 2) using ideas from dot product, 3) Using ideas from cross products. Showed that the magnitude of the cross product was the area of the parallelogram formed by the two vectors as sides. Emphasized how important it is for students to be reading the examples in the text book, and making sure they are completing the practice exercises and asking essential questions. Day 9: Tuesday, 1/28: Handed back GW 2, and asked students to make sure they shared the results with the group. Checked that students had complete the work from the sheet of instructions for Mathematica with vectors. this will add 5 points to the GW 3 assignment, so that the assignment will truly be out of 40 points, rather than 35. Students worked on GW 3, and I offered help as questions arose. Reminder: Quiz is on Thursday! Day 8: Monday, 1/27: Answered a question from the practice exercises (# 23 from section 9.2) and extended the results. Handed out GW 3, and emphasized that I needed students to have completed the work on the yellow sheet, and show me that they had completed that before they could begin the work on GW 3 in lab tomorrow. Continued discussion about Dot Products, and discussed how to use dot products to find projections of one vector on another. Emphasized that we had a scalar component, and that we then multiplied this scalar component by a unit vector in the same direction as a. Day 7: Thursday 1/23: Answered a question from the practice exercises. Presented the geometric definitions for Dot product and cross product. (Section 9.3 and 9.4) Presented algebraic approaches to create both of these products. Emphasized that the dot product is a scalar, while the cross product is a vector. Worked with students on a specific example for each. Emphasized that if two vectors are perpendicular (orthogonal, or normal), then the dot product of the two vectors should be 0. Students should work on the beginning level practice exercises for these sections. I will present more material from each of these sections on Monday. Handed out instructions for using Mathematica for vectors (yellow sheet ) extra copies in the folder outside my door. Students need to complete this work by Tuesday. The next turn in homework will involve using the patterns laid out in this material. Day 6: Wednesday 1/22: Collected GW 2. Presented material from section 9-2: Vectors. Reiterated that students should be reading the textbook after I presented material in class, and working with the practice exercises for that section. Remember: Practice Exercises are listed by section on the purple Practice Exercise sheet. Day 5: Tuesday 1/21: Hand out new Success Strategy Grade sheet (pink) with holes punched correctly! Extras are in the bin outside my office. Handed out How to Save to the Angel Toolbox instructions (gold). Hand out Issue with Automatic Positioning (gold) for Mathematica. Hand out my updated schedule (see my webpage). I had to move one of my office hours. Discussed information from section 9.1 in the text book. Asked students to read the text book, and to work on practice exercises. I will hand out Practice Exercises for Unit 1 tomorrow. I could not get to the copy machine this morning! Day 4: Thursday 1/16: Work in lab on GW 2. Showed students how to create a table of values. Reinforced that students need to make their presentation clear, with a minimum of command lines. Function definition and solve commands should be included in the body of the report, but the definitions to create graphs and tables should be moved to a "Background Page". Homework: Continue to work together on project. No class on Monday. GW 2 is due at the beginning of class on Wednesday. Day 3: Wednesday 1/15: Discussed textbook, and encouraged students to consider options. We ARE using the 2010 edition of the textbook, as listed on the syllabus. Had students check and make sure they have appropriate material in their blue books. Gave them a few minutes to meet a few more people. Gave about 15 minutes to meet with their group for GW 2, and make a plan for getting the most accomplished tomorrow. Tomorrow, students will be given the time to work in the lab on the assignment. Started discussing some of the key ideas in Chapter 9 section 1. Discussed inputs, outputs, points on graph, axes, and Right hand rule for 3 space. We will continue this discussion next week. Homework: Students need to continue to reclaim their Mathematica skills ( or gain them if they have not used Mathematica before). Students need to focus on how they will explain the material for GW 2 so that they can blend their voices with the time in the computer lab tomorrow. Day 2: Tuesday 1/14: Put students in groups of 3 at the computer based on whether they had said they had No, Low, Medium or High Mathematica skills. Hand out GW 2 (green), and have students begin working on it - with 2 using the computers, and one "supervising". The end goals are to have all students become familiar with how to use Mathematica for the skills listed on the assignment, and to work together to blend their "voices" into one coherent assignment. Each group will be submitting one assignment. Had students fill out Course Information cards. Had students meet some other students in class, and begin to discuss Remembering Some Calculus. (yellow) Students should have notes in the blue books provided for each of these questions. Homework: Students should complete working on the questions on the Remembering Calculus sheet in their blue book, and bring that material with them tomorrow. Students need to also review the core Mathematica material that they should know listed at the top of the yellow sheet.
More About This Book Overview STUDENT TESTED AND APPROVED! If you suffer from math anxiety, then sign up for private tutoring with Bob Miller! Do mathematics and algebraic formulas leave your head spinning? If so, you are like hundreds of thousands of other students who face math-especially, algebra-with fear. Luckily, there is a cure: Bob Miller's Clueless series! Like the teacher you always wished you had (but never thought existed), Bob Miller brings knowledge, empathy, and fun to math and pre-algebra. He breaks down the learning process in an easy, non-technical way and builds it up again using his own unique methods. Meant to bridge the gulf between the student, the textbook, and the teacher, Basic Math and Pre-Algebra for the Clueless is packed with all the latest information you need to conquer basic math and pre-algebra, including: "I am always delighted when a student tells me that he or she hated math … but taking a class with me has made math understandable … even enjoyable." Now it's your turn. Sharpen your #2 pencils, and let Bob Miller show you how to never be clueless again! Bob Miller was a lecturer in mathematics at City College of New York for more than 30 years. He has also taught at Westfield State College and Rutgers. His principal goal is to make the study of mathematics both easier and more enjoyable
DVD Features: Rated: G Run Time: 20 minutes Released: June 9, 2009 Originally Released: 2009 Label: am productions, llc Encoding: Region 1 (USA & Canada) Audio: Dolby Digital 2.0 Stereo - English Product Description: An engaging teaching aid for algebra teachers, this program explores the practical application of variables and equations with the use of a graphic calculator, leading viewers through a series of real-world examples where the concepts can be used, like the biology of honey bee colonies, and the forging of rivers through geological landscapes. The program leads viewers through the keystrokes involved in each example, and uses animations to illustrate ideas
Find a Doral, FL Science ...I have successfully trained young athletes who have developed in collegiate and professional athletes. Finite math is very similar to Math modelling which is the act of creating functions or equations that describe a given application or situation. Finite Math also has Matrix Algebra, Probability, Statistics, and logic.
Infographics For Dummies is a comprehensive guide to creating data visualization with viral appeal. Written by the founder of Infographic World, a New York City based infographic agency, and his top designers, the book focuses on the how-to of data, design, and distribution to create stunning, shareable infographics. Step-by-step instruction allows you to handle data like a pro, while creating eye-catching graphics with programs like Adobe Illustrator and Photoshop. The book walks you through the different types of infographics, explaining why they're so effective, and when they're appropriate. Ninety percent of the information transmitted to your brain is visual, so it's important to tickle the optic nerves to get people excited about your data. Infographics do just that. Much more exciting than a spreadsheet, infographics can add humor, interest, and flash while imparting real information. Putting your... 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Fundamentals of Precalculus - 2nd edition Summary: ''Fundamentals of Precalculus'' is designed to review the fundamental topics that are necessary for success in calculus. Containing only five chapters, this text contains the rigor essential for building a strong foundation of mathematical skills and concepts, and at the same time supports students' mathematical needs with a number of tools newly developed for this revision. A student who is well acquainted with the material in this text will have the necessary skills, understanding,...show more and insights required to succeed in calculus. ...show less Chapter 3 Trigonometric Functions 3.1 Angles and Their Measurements 3.2 The Sine and Cosine Functions 3.3 The Graphs of the Sine and Cosine Functions 3.4 The Other Trigonometric Functions and Their Graphs 3.5 The Inverse Trigonometric Functions 3.6Right Triangle Trigonometry 3.7Identities 3.8Conditional Trigonometric Identities 3.9The Law of Sines and the Law of Cosines 03215069793116 +$3.99 s/h Good Campus_Bookstore Fayetteville, AR Used - Good Hardcover. TEXTBOOK ONLY41.98 +$3.99 s/h 45.4746.90 +$3.99 s/h Good Penntext Downingtown, PA Tear in spine. May have minimal notes/highlighting, minimal wear/tear. Please contact us if you have any Questions. $110.063215069
Any student of linear algebra will welcome this textbook, which provides a thorough treatment of this key topic. Blending practice and theory, the book enables the reader to learn and comprehend the standard methods, with an emphasis on understanding how they actually work. At every stage, the authors are careful to ensure that the discussion is no more complicated or abstract than it needs to be, and focuses on the fundamental topics. The book is ideal as a course text or for self-study. Instructors can draw on the many examples and exercises to supplement their own assignments. End-of-chapter sections summarize the material to help students consolidate their learning as they progress through the book. As in previous editions, the focus in INTRODUCTORY ALGEBRA remains on the Aufmann Interactive Method (AIM). Students problems, helps them build their confidence and eventually master the concepts. To this point, simplicity plays a key factor in the organization of this edition, as in all other editions. All lessons, exercise sets, tests, and supplements are organized around a carefully-constructed hierarchy of objectives. This "objective-based" approach not only serves the needs of students, in terms of helping them to clearly organize their thoughts around the content, but instructors as well, as they work to design syllabi, lesson plans, and other administrative documents. The Eighth Edition features a new design, enhancing the Aufmann Interactive Method and the organization of the text around objectives, making the pages easier for both students and instructors to follow. Dr. Carleen Eaton guides you through Algebra 1 with captivating lessons honed from teaching math and science for over 10 years. This course meets or exceeds all state standards and is essential to those having trouble with Algebra in high school or college. Carleen's upbeat teaching style and real world examples will keep you engaged while learning. She covers everything in Algebra 1 from Linear Expressions to Systems of Equations and Rational Expressions. Along the way she has received multiple "Teacher of the Year" awards and rankings as one of the top instructors in California. Dr. Eaton received her M.D. from the UCLA School of Medicine. Algebra I is one of the most critical courses that students take in high school. Not only does it introduce them to a powerful reasoning tool with applications in many different careers, but algebra is the gateway to higher education. Students who do well in algebra are better prepared for college entrance exams and for college in general, since algebra teaches them how to solve problems and think abstractly—skills that pay off no matter what major they pursue.
Numerical Mathematics and Computing 9780495114758 ISBN: 0495114758 Pub Date: 2007 Publisher: Thomson Learning Summary: Authors Ward Cheney and David Kincaid show students of science and engineering the potential computers have for solving numerical problems and give them ample opportunities to hone their skills in programming and problem solving. The text also helps students learn about errors that inevitably accompany scientific computations and arms them with methods for detecting, predicting, and controlling these errors. A more t...heoretical text with a different menu of topics is the authors' highly regarded NUMERICAL ANALYSIS: MATHEMATICS OF SCIENTIFIC COMPUTING, THIRD EDITION. Cheney, Ward is the author of Numerical Mathematics and Computing, published 2007 under ISBN 9780495114758 and 0495114758. Three hundred seventeen Numerical Mathematics and Computing textbooks are available for sale on ValoreBooks.com, one hundred twelve used from the cheapest price of $11.85, or buy new starting at $116
Hi, I would really appreciate it someone who has the ib math higher data booklet and could send it to me. I have been searching everywhere and this seems to be the only place where I can get it. Thank you very much ! Hey, Can somebody please send this to me too.. I need it desperately right now! For HL maths please. Thanks. Also, If anyone has any relavent notes or anything for any of the following IB subjects, could u please help and send them over as well. I am just starting with IB, and would love as much help and advice as u would provide me with. Thanks again. The subjects I've chosen are as follows: HL: Economics, Maths and English SL: French B, Psychology, Biology, Physics and Extended Essay, TOK, and CAS.
About: Advanced Algebra II: Teacher's Guide Metadata Name: Advanced Algebra II: Teacher's Guide ID: col10687 Language: English (en) Summary: This is the Teacher's Guide for Kenny Felder's course in Advanced Algebra II. This guide is not an answer key for the homework problems: rather, it is a day-by-day guide to help the teacher understand how the author envisions the materials being used. This text is designed for use with the "Advanced Algebra II: Conceptual Explanations" ( and the "Advanced Algebra II: Homework and Activities" ( collections to make up the entire course.
Introductory Algebra-Text - 8th edition Summary: Lial/Hornsby/McGinnis's Introductory Algebra, 8e, gives students the necessary tools to succeed in developmental math courses and prepares them for future math courses and the rest of their lives. The Lial developmental team creates a pattern for success by emphasizing problem-solving skills, vocabulary comprehension, real-world applications, and strong exercise sets. In keeping with its proven track record, this revision includesan effective new design, many new exe...show morercises and applications, and increased Summary Exercises to enhance comprehension and challenge students' knowledge of the subject matter0321279212 Item in good condition and ready to ship2.29 +$3.99 s/h Good TXTBookSales1 Evansville, IN58 +$3.99 s/h Good BookCellar-NH Nashua, NH 0321279212 Has heavy shelf wear, but still a good reading copy. Name written on inside of cover A portion of your purchase of this book will be donated to non-profit organizations. We are a tested ...show moreand
12,019including precalculus precalculus
More About This Textbook Overview This book offers multiple interconnected perspectives on the largely untapped potential of elementary number theory for mathematics education: its formal and cognitive nature, its relation to arithmetic and algebra, its accessibility, its utility and intrinsic merits, to name just a few. Its purpose is to promote explication and critical dialogue about these issues within the international mathematics education community. The studies comprise a variety of pedagogical and research orientations by an international group that, collectively, make a compelling case for the relevance and importance of number theory in mathematics education in both pre-K-16 settings and mathematics teacher education. Topics variously engaged include: understanding particular concepts related to numerical structure and number theory; elaborating on the historical and psychological relevance of number theory in concept development; attaining a smooth transition and extension from pattern recognition to formative principles; appreciating the aesthetics of number structure; exploring its suitability in terms of making connections leading to aha! insights and reaching toward the learner's affective domain; reexamining previously constructed knowledge from a novel angle; investigating connections between technique and theory; utilizing computers and calculators as pedagogical tools; and generally illuminating the role number-theory concepts could play in developing mathematical knowledge and reasoning in students and teachers. Overall, the chapters in this book highlight number theory-related topics as a stepping stone from arithmetic toward generalization and algebraic formalism, and as a means forproviding intuitively grounded meanings of numbers, variables, functions, and proofs. Number Theory in Mathematics Education: Perspectives and Prospects is of interest to researchers, teacher educators, and students in the field of mathematics education, and is well suited as a text for upperlevel mathematics education
effectively learn the most important factors to be considered when answering questions in a maths exam. You will be aware of the most common mistakes in maths exams. You will gain a good knowledge of first and second order differential equations and second derivatives. You will learn about kinematics including acceleration, distance and velocity. You will understand Newton's laws of motion, Newton's laws of cooling, and Euler's method for solution of differential equations. This course will teach you the resolution of forces and how to calculate vectors. Modules in Advanced Mathematics 2 This free online course covers how to answer questions in examinations basic exam techniques such dividing by fractions and using negative signs differential equations such as first order and second order differential equations and rates of change . kinematics such as velocity and acceleration motion such as motion under a constant force and motion under a variable force chemical concentrations and chemical reactions resolution of forces such as static friction, and angle of friction vectors such as parametric equations differentiation and antidifferentiation of
This book grew out of a public lecture series, Alternative forms of knowledge construction in mathematics, conceived and organized by the first editor, and held annually at Portland State University from 2006. Starting from the position that mathematics is a human construction, implying that it cannot be separated from its historical, cultural, social,... more... Are you having trouble in finding Tier II intervention materials for elementary students who are struggling in math? Are you hungry for effective instructional strategies that will address students' conceptual gap in additive and multiplicative math problem solving? Are you searching for a powerful and generalizable problem solving approach that... more... This book provides fundamental knowledge in the fields of attosecond science and free electron lasers, based on the insight that the further development of both disciplines can greatly benefit from mutual exposure and interaction between the two communities. With respect to the interaction of high intensity lasers with matter, it covers ultrafast... more... How do you make mathematics relevant and exciting to young children? How can mathematics and literacy be combined in a meaningful way? How can stories inspire the teaching and learning of mathematics? This book explores the exciting ways in which story can be used as a flexible resource to facilitate children?s mathematical thinking. It looks... more... Applied mathematics connects the mathematical theory to the reality by solving real world problems and shows the power of the science of mathematics, greatly improving our lives. Therefore it plays a very active and central role in the scientific world.This volume contains 14 high quality survey articles — incorporating original results and... more... A complete training package lets you learn Adobe Illustrator CC at your own speed Adobe Illustrator is the leading drawing and illustration software used to create artwork for a variety of media. This book-and-DVD package provides 13 self-paced lessons that get you up to speed on the latest version of Illustrator (Creative Cloud). Step-by-step... more... This new volume introduces readers to the current topics of industrial and applied mathematics in China, with applications to material science, information science, mathematical finance and engineering. The authors utilize mathematics for the solution of problems. The purposes of the volume are to promote research in applied mathematics and computational... more... ICM 2010 proceedings comprises a four-volume set containing articles based on plenary lectures and invited section lectures, the Abel and Noether lectures, as well as contributions based on lectures delivered by the recipients of the Fields Medal, the Nevanlinna, and Chern Prizes. The first volume will also contain the speeches at the opening and closing... more...
8.1.B use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution; 8.1.C select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems; 8.3.B compare and contrast the attributes of a shape and its dilation(s) on a coordinate plane; and 8.3.C use an algebraic representation to explain the effect of a given positive rational scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of dilation. 8.4.A use similar right triangles to develop an understanding that slope, m, given as the rate comparing the change in y-values to the change in x-values, (y2 - y1)/ (x2 - x1), is the same for any two points (x1, y1) and (x2, y2) on the same line; 8.7.B use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders; 8.8.D use informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. 8.9 The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations. The student is expected to identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations. 8.10.B differentiate between transformations that preserve congruence and those that do not; 8.10.C explain the effect of translations, reflections over the x- or y-axis, and rotations limited to 90°, 180°, 270°, and 360° as applied to two-dimensional shapes on a coordinate plane using an algebraic representation; and 8.11.B determine the mean absolute deviation and use this quantity as a measure of the average distance data are from the mean using a data set of no more than 10 data points; and 8.11.C simulate generating random samples of the same size from a population with known characteristics to develop the notion of a random sample being representative of the population from which it was selected. 8.12 The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. 8.12.C explain how small amounts of money invested regularly, including money saved for college and retirement, grow over time; 8.12.E identify and explain the advantages and disadvantages of different payment methods; 8.12.F analyze situations to determine if they represent financially responsible decisions and identify the benefits of financial responsibility and the costs of financial irresponsibility; and 8.12.G estimate the cost of a two-year and four-year college education, including family contribution, and devise a periodic savings plan for accumulating the money needed to contribute to the total cost of attendance for at least the first year of college.
Hello all! Creating and submitting a perfect solution is something that takes some practice. I am going to make this a little easier over the next few weeks by supplying examples of worked out solutions that are constructed very well! We are going to start with math and do a new subject each week. I have displayed two high quality solutions below and have highlighted the key components. Although some of these highlighted parts are specific to math, most are general rules for authoring. If you ever need a reminder or a little help setting up your solution, reference this! First Example This first picture details how to correctly insert a title, the authors ID, and the date of creation. Writing the steps in complete sentences is very important and including a delimiter between each step is required. There should always be a blank space before and after the delimiter. Equations should always be aligned to the left after inserting one tab. In addition, each final answer in a multi-part questions must be boxed. Times New Roman (size 12) should be used throughout the entire solution. Again, each final answer should be boxed. Finally, there should be no delimiter at the end of the solution and your final answers should be boxed. Second Example This first picture shows how to correctly insert a title, authors ID and date created. There should always be a delimiter after this information and a vertical bar between the authors ID and date created. If important information is given in the problem, it must be clearly restated in the worked out solution. Again, delimiters should be used to separate steps of the solutions. You should always clearly outline what will be included in the rest of the solution. Tables should be created on your computer and centered in the middle of the page. Formulas should be defined explicitly. Times New Roman (size 12) should be used throughout, the final answer should be boxed, and there should be no delimiter at the end of the solution. Students also read…
Agile Mind's Precalculus course is focused on improving educational opportunities for students in mathematics. This course provides the tools and support that teachers need to ensure high achievement among their students. The course includes student practice, review, and test preparation activities and embedded formative assessments.
Complex Analysis(Google eBook) This unusually lively textbook on complex variables introduces the theory of analytic functions, explores its diverse applications and shows the reader how to harness its powerful techniques. Complex Analysis offers new and interesting motivations for classical results and introduces related topics that do not appear in this form in other texts. Stressing motivation and technique, and containing a large number of problems and their solutions, this volume may be used as a text both in classrooms and for self-study. Topics covered include: The complex numbers; functions of a complex variable; analytic functions; line integrals and entire functions; properties of entire functions and of analytic functions; simply connected domains; isolated singularities; the residue theorem and applications; contour integral techniques; conformal mapping and the riemann mapping theorem; maximum-modulus theorems for unbounded domains; harmonic functions; forms of analytic functions; analytic continuation; the gamma and zeta functions; application to other areas of mathematics. For this second edition, the authors have revised some of the existing material and have provided new exercises and solution Popular passages Page 1 - In the geometric representation, he wrote, one finds the "intuitive meaning of complex numbers completely established and more is not needed to admit these quantities into the domain of arithmetic.‎
״MATH GRAPHING XL is a powerful & unique 1D graphing calculator to graph mathematical expressions of arbitrary... see more ״MATH GRAPHING XL is a powerful & unique 1D graphing calculator to graph mathematical expressions of arbitrary complexity. It can help students improve their math skills by developing some visual intuition of mathematical expressions or advanced users who need some scientific capabilities only available with expensive desktop software.MATH GRAPHING XL provides the following functionalities:- Multiple expressions with quasi-unlimited number of variables can be combined to produce simple or complex formulas,- Interactive sliders can be created to visually investigate the role of important parameters on the graphical representation of the formulas,- Trace mode to display coordinates and derivative of marker on selected curve,- Solver tool to solve y = f(x) where x or y are unknown, or find local/global minima/maxima/extrema (NEW!),- Formula graphs can be saved to the device's Photos Album, or the formulas be exported through email with embedded graphs,- A list of favorite formulas can be created for editing or archival purposes,- Several formulas can be plotted simultaneously in different colors and styles,- Customizable graph and formula appearance: axes labels, title, curve color and style, ticks number, grid, wallpaper, font, font size, etc.- Single-precision calculator supporting variables, multiple expressions, and on-the-fly evaluation (NEW!).Example of mathematical expression:# Gabor functionx = 0:1:100sigma = 1:50; freq = 0:0.1; phase = 0:180u = cos(2pi(x-50)freq+phasepi/180)v = e^(-((x-50)^2)/(2sigma^2))y = uvplot(u,'lr'); plot(v,'lg'); plot(y,'lw')xlabel('time'); ylabel('amplitude')title('Gabor')״Cost of this app is $9.99 ״Designed for people of all ages, Math Flash Cards Subtraction is an app that allows the user to practice simple basic... see more ״Designed for people of all ages, Math Flash Cards Subtraction is an app that allows the user to practice simple basic subtraction facts or extend the users ability to work out complex subtraction subtraction skills, this just might be the app for you! * Multiplication is an app that allows the user to practice simple basic... see more ״Designed for people of all ages, Math Flash Cards Multiplication is an app that allows the user to practice simple basic multiplication facts or extend the users ability to work out complex multiplication multiplication skills, this just might be the app for you!Features** - Over 1,000,000 math problems to solve!-Math !!! by Math Pentagon has one of the largest collection of Math worksheets on iPad. It is a structured math learning... see more ״Math !!! by Math Pentagon has one of the largest collection of Math worksheets on iPad. It is a structured math learning program that engages students to practice math worksheets, exercises & teacher assignments. Teachers can capture results in real-time, review and understand student progress, thereby saving time, paper and enhancing productivity.The app covers several Math topics for Grade 4, Grade 5, Grade 6, Grade 7 and Grade 8. The course content includes Algebra, Geometry, Statistics, Decimals, Fractions, Percent, Ratios, Trigonometry, Arithmetic and more!!!With Math Pentagon,* Students can practice and solve all Math problems on the iPad* Teachers can assign worksheets to their student circle.* Teachers can review completed assignments and give immediate feedback.* Teachers/Students can track progress, worksheet history and scores.* Student-specific reports help Teachers understand progress, and prepare for further classroom teaching.* Students can learn LIVE from their teachers through the built-in LIVE Learning Center.* Students can score points and collect rewards.״First 25 questions you solve are free, and then buy a subscription starting from $5.99.
Precalculus prepares students for a first course in Calculus, as well as introducing topics that will be needed in other Mathematics courses. In preparation for a first course in Calculus, polynomial and rational functions are graphed, conic sections are analyzed, and limits are illustrated wit
First Course in Mathematical Modeling 9780495011590 ISBN: 0495011592 Edition: 4 Pub Date: 2008 Publisher: Cengage Learning Summary: Offering a solid introduction to the entire modeling process, A FIRST COURSE IN MATHEMATICAL MODELING, 4th Edition delivers an excellent balance of theory and practice, and gives you relevant, hands-on experience developing and sharpening your modeling skills. Throughout, the book emphasizes key facets of modeling, including creative and empirical model construction, model analysis, and model research, and provides m...yriad opportunities for practice. The authors apply a proven six-step problem-solving process to enhance your problem-solving capabilities -- whatever your level. In addition, rather than simply emphasizing the calculation step, the authors first help you learn how to identify problems, construct or select models, and figure out what data needs to be collected. By involving you in the mathematical process as early as possible -- beginning with short projects -- this text facilitates your progressive development and confidence in mathematics and modeling. Giordano is the author of First Course in Mathematical Modeling, published 2008 under ISBN 9780495011590 and 0495011592. Three hundred ninety five First Course in Mathematical Modeling textbooks are available for sale on ValoreBooks.com, one hundred twenty one used from the cheapest price of $113.48, or buy new starting at $266 business day. Expedited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions.[less]
Buy Used Textbook eTextbook We're Sorry Not Available New Textbook We're Sorry Sold Out More New and Used from Private Sellers Starting at $54 Elementary Statistics: Picturing the World, Fifth Edition, offers our most accessible approach to statistics-with more than 750 graphical displays that illustrate data, students are able to visualize key statistical concepts immediately. Adhering to the philosophy that students learn best by doing, this book relies heavily on examples#x13;25% of the examples and exercises are new for this edition. Larson and Farber continue to demonstrate that statistics is all around us and that it#x19;s easy to understand. #xA0; MyMathLabprovides a wide range of homework, tutorial, and assessment tools that make it easy to manage your course online. #xA0;
How to Ace the Rest of Calculus : The Streetwise Guide : Including Multi-Variable Calculus - 01 edition Summary: Do you remember being hopelessly confused in calculus class? Afterwards, you asked your brainy friend over a cup of coffee, "What was going on in that class?" Your friend then explained it all to you in five minutes flat, making it crystal clear. "Oh," you said, "is that all there is to it?" Later, you wished that friend was around to explain all the lectures to you. The original How to Ace Calculus played the role of that fri...show moreend for a first-semester calculus class. Now meet your new buddy, How to Ace the Rest of Calculus: The Streetwise Guide. Written by three gifted teachers, it provides humorous and highly readable explanations of the key topics of second and third semester calculus--such as sequences and series, polar coordinates and multivariable calculus--without the technical details and fine print that would be found in a formal text. Funny, irreverent, and flexible, How to Ace the Rest of Calculus shows why learning calculus can be not only a mind-expanding experience but also fantastic fun. ...show less Our feedback rating says it all: Five star service and fast delivery! We have shipped four million items to happy customers, and have one MILLION unique items ready to ship today! $4.866.7825 +$3.99 s/h New kaybook1 Curtisville, PA 0716741741 New publisher overstock $8.50 +$3.99 s/h VeryGood Already Read Used Books Alexandria, VA Mild bumping, rubbing, and wear from handling to corners and spine; Mild tanning to page edges; Spine straight and tight; Free USPS track and confirm on US orders $11.03 +$3.99 s/h New PaperbackshopUS Secaucus, NJ New Book. Shipped from US within 4 to 14 business days. Established seller since 2000 $11.66
More About This Textbook Overview This book covers mathematics of finance, linear algebra, linear programming, probability, and descriptive statistics, with an emphasis on cross-discipline principles and practices. Designed to be reader-friendly and accessible, it develops a thorough, functional understanding of mathematical concepts in preparation for their application in other areas. Each chapter concentrates on developing concepts and ideas followed immediately by developing computational skills and problem solving. Two-part coverage presents a library of elementary functions and finite mathematics. For individuals looking for a view of mathematical ideas and processes, and an illustration of the relevance of mathematics to the real world. Illustrates relevance of mathematics to the real world. Editorial Reviews From The Critics This college textbook is for a one-term course for students who have had one-and-a-half years of high school algebra or the equivalent. the book is designed to give students substantial experience in modeling and solving real-world problems, to increase their understanding of the applicability of mathematics to everyday life. the text contains over 260 completely worked problems, each followed by a similar matched problem for the student to work. The exercise sets have a range of degree of difficulty, to ensure all levels of students are challenged but also able to experience success. Activities are included to encourage verbalization of mathematical concepts, results and processes. Specific changes made to the ninth edition are not stated. Annotation c. Book News, Inc., Portland, OR (booknews.com) Booknews Written for students with a background in high school algebra, this text explains finite mathematics with an emphasis on applications to the business and finance worlds. Topics covered include linear inequalities, probability, data description, and Markov chains. Annotation c. by Book News, Inc., Portland, Or. Systems of Linear Inequalities in Two Variables. Linear Programming in Two Dimensions—A Geometric Approach. A Geometric Introduction to the Simplex Method. The Simplex Method: Maximization with Problem Constraints of the Form . The Dual; Minimizationwith Problem Constraints of the Form . Maximization and Minimization with Mixed Problem Constraints. Chapter 5 Review. Chapter 5 Group Activities. Preface any calculator system to be substantially error-free. For any errors remaining, the authors would be grateful if they were sent to: Karl E. Byleen, 9322 W Garden Court, Hales Corners,Introduction any calculator system tm be substantially error-free. For any errors remaining, the authors would be grateful if they were sent to: Karl E. Byleen, 9322 W Garden Court, Hales Corners,, 2014 received this book in such bad condition I called and complained received this book in such bad condition I called and complained. Then retuned the book for a credit this was back on 1/13 that I called and returned it shortly there after. Today I made my 3rd call to customer service because they keep tiring to charge my credit card for the full amount of the book. The first manager I got to day said pay the bill or we will turn you in to collections. can I give a -5 stars. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Advanced Algebra (Mathematical Modeling) In this course, the primary focus is the creation of linear and exponential models to represent rates of growth involving a variety of situations, particularly human population growth. The course addresses the following focus questions: How do rates of change relate to the steepness of a graph? How do we use graphs to interpret data? What is the definition of slope and how does it relate to real-life situations? How do we represent slope graphically and algebraically? What is the relationship between slope and a derivative? What real-life situations illustrate exponential growth? How do we represent exponential functions in terms of different bases? How do we convert an exponential function to a base of e? How can we adjust an exponential function to fit it to a set of data? In addition, the following standards are addressed and assessed throughout the unit: Evaluating average rates of change Understanding the relationship between the rate of change of a function and the appearance of its graph Using exponential functions to model real-life situations Developing an algebraic definition of slope Understanding the significance of a negative slope Seeing average speed as the slope of a secant line Developing the concept of the derivative of a function at a point Seeing that the derivative of a function at a point is the slope of the tangent line at that point Finding numerical estimates for the derivatives of functions at specific points Observing that the rate of change in population is proportional to the population Discovering any exponential function can be expressed another number as the base Learning that the value e is the same number as the special base for exponential functions Strengthening an understanding of logarithms Using an exponential function to fit a curve to numerical data Summarizing ideas about linear and exponential growth Readings: "Small World, Isn't It?" - Interactive Mathematics Program "Population: 7 Billion" - National Geographic Magazine "Fatima's Story" - ZPG Population Education Program Research: Students research population factors that affect the population growth in their selected countries from various sources considering both quantitative and qualitative data. Media Used: "World in the Balance" - NOVA DVD "Aftermath: Population Zero" - National Geographic Interim Assessments: Students explore mathematical patterns in world population data and make forecasts to extend the patterns. Students organize their thinking in writing to present to the instructor and to their peers. Students analyze graphical representations of linear and exponential sequences and make inferences based on the behavior of the graph. Additionally, students interpret statements about the graphs of different sequences, justifying their thinking in writing. Students construct an addition (linear) and a multiplicative (exponential) model representing new computer sales. Students graph the data, calculate the discrepancies between their mathematical models and the actual data, and then interpret which model is better for forecasting future sales. Students then forecast sales into the future and explain some reasons why forecasting future sales would be important to the company. Significant Assignments: Students learn and analyze how scaling can affect the visual representation of graphs through the creation of several graphs utilizing the same data set. Students find "addition" and "multiplication" growth numbers to increase a given number to a specified target in a specific number of steps. Utilizing a mathematical model, students utilize growth numbers to calculate intermediate values and then extend the model into the future or conversely into the past. Students represent sequences algebraically working with the concepts of inverse operations, reciprocals, multiples, and powers. Students graph addition (linear) and multiplicative (exponential) sequences and make comparisons including how the value of the growth number affects the shape of the graph. Students construct secant lines and tangent lines on the graph of an exponential sequence and differentiate between them. Significant Activities or Projects: Students research demographic data about their native countries and complete a population pyramid representing the age groups. Students interpret their population pyramids by making inferences about factors (social, religious, political, cultural) that could impact the shape of their pyramid. Students construct and analyze growth spirals representative of addition (linear) and multiplicative (exponential) functions. Students display and present their work on chart paper, indicating visual characteristics of each type of spiral. Students calculate the discrepancy between mathematical models representing the population growth of the United States and interpret which model fits actual data more closely, and then make resulting forecasts. Sample PBATs: Native Country Population Project: Students tabulate the actual population data of their native country using the time period from 1950-2000. Students calculate the rate of growth under both a linear and exponential model and compare the resulting forecasts to assess which is closer to the actual population data both numerically and graphically. Students construct the secant lines and tangent line on their graphs and predict the rate of change at a specific year. Students then use the graphing calculator to find an exponential regression model to compare to their prediction. Students also use an exponential base of e, to forecast future population data and to predict when their country if continuing at its current rate of growth would be completely filled with people. Students also reflect on the social, political, and cultural factors that may affect their mathematical forecasts.
... More About This Book computational applications. A number of results such as optimized version of the Buchberger algorithm are presented in textbook format for the first time. This book requires no prerequisites other than the mathematical maturity of an advanced undergraduate and is therefore well suited for use as a textbook. At the same time, the comprehensive treatment makes it a valuable source of reference on Gr bner bases theory for mathematicians, computer scientists, and others. Placing a strong emphasis on algorithms and their verification, while making no sacrifices in mathematical rigor, the book spans a bridge between mathematics and computer science. Editorial Reviews Booknews The centerpiece of Grobner basis theory is the Buchberger algorithm, the importance of which is explained, as it spans mathematical theory and computational applications. This comprehensive treatment is useful as a text and as a reference for mathematicians and computer scientists and requires no prerequisites other than the mathematical maturity of an advanced undergraduate. Strong emphasis is placed on algorithms and their verification, with no sacrifices in mathematical rig
MATH 151 - Calculus I Slope tells something about the rate of change in a line. This is an extremely useful concept, but has the shortcoming of only being applicable to lines. In Calculus I this problem is overcome by the derivative, essentially a concept of slope that can be applied to functions other than lines. Armed with the derivative, we can answer questions about the rate of change of many functions, allowing us to find maxima or minima of functions, study velocity and acceleration of physical bodies, chemical reactions and population growth. We can graph complex curves and describe the relative efficiency of rival computer algorithms. Indeed, the calculus provides a universal language to precisely describe and compute rates of growth and corresponding changes in amount. MATH 152 - Calculus II Nearly everyone knows that the area of a circle is πr², and so on. But few think about where these formulas come from. In Calculus II we use the concept of the integral to study the area under curves. This naturally generalizes to the study of volumes of solids in space. But this same concept, combined with the derivative (from Calculus I) can be used in many unexpected and powerful ways. Quantities as diverse as the GNP (gross national product) and total run time of a computer program can be described as an area under a curve on a graph. Calculus II provides tools to compute these quantities and relate them to the functions that describe their rates of change. It is possible for infinitely many numbers to sum to a finite value. For example, it can be shown that 1+½+¼+...=2. The integral and derivative are used as tools to help us understand such infinite series. In turn, these series help us to understand several functions better. For example, and can be written as infinitely long polynomials and can be approximated reasonably well by, say, polynomials of degree four or five. Math 203 - History of Mathematics Often, when we learn mathematics, we learn it without the story of who developed it, and when and why. In the History of Mathematics, we look at the stories behind the mathematics. These stories take us to many places on the earth and through a long period of time. We begin about 4000 years ago with the ancient civilizations of Egypt and Mesopotamia, where there was already a good deal of mathematics known, particularly algebra and the art of computation. We also explore the early mathematical discoveries of China and India. Next we go to the amazing flowering of mathematics that occurred in ancient Greece: geometry, astronomy, trigonometry and much more. We see some more development of algebra with the Arabic mathematicians of Medieval times; in fact our word, algebra, is from the Arabic. We next move to Europe to see algebra in Italy, analytic geometry in France, logarithms in Scotland and the beginnings of calculus almost everywhere. We follow the development of calculus and see how it changed from around 1600 to around 1800. We then look at the surprising story of non-Euclidean geometry in the 1800's. We can only survey more recent discoveries briefly because they are more difficult and there are so many of them. We study the biographies of a number of mathematicians, and look at the special problems encountered by women mathematicians. Many of the students in this class intend to become mathematics teachers so we examine the histories of specific areas of mathematics taught in the schools, such as number systems, algebra, geometry and trigonometry. MATH/STAT 242 - Introduction to Mathematical Statistics As the title suggests, we will apply mathematical techniques to develop some of the fundamental ideas of statistics. So just what is statistics? Statistics is the art of extracting patterns from data. This might consist of summarizing complicated data, whether numerically, graphically or by constructing a simple mathematical model that connects pieces of data to one another. Whereas mathematics uses a language of certainty, theorems and proofs, statistics has developed precisely to deal with uncertainty, estimates, bounds and probabilities. In this course we will examine answers to several important questions in statistics. How do you describe a data set so as to capture its 'center' and its variation? This will lead to topics such as the mean and the variance of a sample. What is probability and how do we model it mathematically? This will lead to the classical distributions: binomial, Poisson, exponential and normal. How do you decide whether your preconceptions about a large population are in agreement with the data obtained from a sample? This will lead us to confidence intervals and hypothesis testing. Throughout the course, we will see that statistics is much more than just the application of mathematical techniques. We will see that, before we can apply the mathematics, we must have good data and reasonable models. After we have done our mathematical analysis, we must still decide whether we have enough certainty to make conclusions. In short, we will be the lawyers, judge and jury in the court of data analysis. We will apply the techniques of algebra and calculus to investigate probability, to develop models and to explore their properties and understand why some estimation techniques have better properties than others do. We will apply Minitab statistical software to real world data sets and to simulated data sets. Successful completion of MATH 151 is a prerequisite for this course. This course is cross-listed under both mathematics and statistics. Students can take this course for the mathematics major and minor, the statistics minor and the actuarial science minor. MATH 245 - Discrete Structures The possession of logical reasoning skills is essential for anyone interested in computer science. In this class, students enhance these skills by studying a variety of mathematical topics related to the study of computer science, which may include propositional logic, set theory, relations, functions, combinatorics, graph theory, and applications of these topics. Students also learn proof-techniques such as induction (a "domino" technique that allows one to prove that a statement relating to a variable n is true for all positive integers n) and proof by contradiction (in which one proves a desired result by showing that if it isn't true, nonsensical things happen), thereby increasing their mathematical maturity and their ability to make reasoned arguments, prerequisites for programming. Topics vary from term to term, and may depend on student interest. Here is a sample of things students may learn in this class: (1) The logical difference between the statements, Not all people have red hair and All people do not have red hair; (2) How to show that 1+2+3+...+n= n(n+1)/2, for any positive integer n; (3) How to show that the set of integers and the set of rational numbers have the same "size", but the set of real numbers is "bigger"; (4) How to compute the probability of getting a royal flush in poker. The course is intended primarily for computer science majors and math majors. Note: though this class has Math 152 as a prerequisite, to ensure the mathematical preparedness of its students, its material is not directly related to that learned in the calculus sequence. MATH 253 - Calculus III Most things are related to more than just one factor. For example, your minimum monthly credit card payment depends on the total you owe and your interest rate. The amount you actually pay depends on the minimum payment due and the amount you have available to pay. The growth rate of a deer population depends on the size of the population, its age distribution, the food supply and predation. The pressure exerted by gas in a cylinder depends on the amount of gas, its temperature and the volume of the cylinder. Other functions may only depend on one variable, but give an output that is more than just one number. For example, a person traveling around the world has, at any given time, a latitude and a longitude (and perhaps an altitude too if s/he is in an airplane). Thus position can be considered a function of time but it cannot be represented by a single value; it must be given as a doublet (or triplet) of numbers representing latitude and longitude (and altitude). Such a doublet or triplet can be represented as a vector. Calculus III extends the ideas of Calculus I and II by considering derivatives and integrals of functions with more than one variable, or of vector-valued functions. Along the way, other possible coordinate systems (such as polar coordinates) are discussed. MATH 317 - Introduction to Proof in Mathematics In mathematics we accept a statement as true only if we have a proof that it is true. Since the method of proof is so basic to mathematics, anyone who seriously wants to learn mathematics beyond a fairly elementary level must be able to understand proofs and be reasonably proficient at constructing them. The purpose of this class is to teach you how to understand proofs and to develop your skills at constructing proofs. Skill at proving develops over a long period of time; this class is only a beginning. The best way to learn to do proofs is to do them, so you will be given plenty of opportunity to practice proving things. We will begin with an introduction to logic. Logic is a tool that we will use to analyze proofs to see if they are correct and to help us to construct proofs. We will practice writing proofs in a number of areas of mathematics: set theory, including infinite sets, inequalities and functions. We will study the whole numbers using mathematical induction. In addition to the usual lecture format, a good deal of class time will be spent with students presenting their proofs to the class or constructing proofs together. MATH 321 - Geometry The geometry most of us learned in high school is based on Euclid's famous 5 Postulates and works well for describing things in or on a flat surface. However, the surface of our world is not flat and any pilot or ship's navigator must understand the rules of spherical geometry. The discovery of two-dimensional non-Euclidean geometries early in the nineteenth century by Gauss, Bolyai and Lobachevski allowed us to ask for the first time, "Could the geometry of the three-dimensional universe in which we live also be non-Euclidean?" The work of Riemann and, later, Minkowski provide a geometric structure for Einstein's theory of relativity and modern theories of cosmology where the ultimate collapse or expansion of the universe is related to the curvature of space itself. The discovery of two-dimensional non-Euclidean geometries also initiated a momentous shift in our view of the entire mathematical enterprise. The question of axiomatic foundations raised by the non-Euclidean geometries now pervades all branches of the subject and forms the acid test of mathematical validity. This class examines the foundations of geometry that lead to Euclid's geometry in the plane and to other possible geometries, most notably spherical and hyperbolic, and concentrates on exploring the rules of geometric logic that are universal. MATH 331 - Linear Algebra Why algebra? Algebra was invented because of the limitations of our geometric intuition. In applications ranging from business to engineering to the social sciences, it is often useful to work with data that naturally correspond to points in the plane, or in three-dimensional space, or even in fifty-dimensional space. Certainly we could draw pictures or build models to avoid algebra for points in the plane or in three-dimensional space, but what pictures or models could help us to "see" in fifty dimensions? This obstacle motivates the development of vectors and the development of algebraic rules and techniques for manipulating them. In this course we pursue two intimately related subjects: matrix theory and linear algebra. Matrix theory is concerned with vectors and matrices. Vectors are the n-dimensional generalizations of the ordered pairs representing points in the plane. We will investigate how our geometric concepts naturally imbed in algebraic concepts. We will learn how the geometry of lines and planes, lengths and angles is replaced by systems of equations and operations on vectors. Further, we will see how systems of equations can be analyzed in terms of the properties of a single algebraic object: the matrix. Linear algebra is the study of sets of vectors and how operations on individual vectors can be applied to entire sets. Linear algebra is the abstraction of the fundamental properties displayed by vectors and matrices. This abstraction allows us to use the knowledge and skills developed working with vectors and matrices to answer questions about the behavior of wave functions in Fourier analysis or about the nature of solutions to important families of differential equations. This course is very different from calculus. In calculus there are relatively fewer theoretical ideas, and most of the course is devoted to applying those ideas and the associated techniques to specific computations. In MATH 331 students learn a large variety of new ideas and, while calculations are important, they are primarily tools for understanding the examples that motivate the theory. Consequently much of the work in this course is focused on explaining why certain relationships between ideas are true or why certain sets have specified properties rather than on simply producing a slope or an integral or a number. Calculus is a prerequisite for this course primarily because students rarely have adequate facility with mathematical thinking, working with equations, working extensively with symbols, thinking about exceptions or using technical language-prior to completing the calculus sequence. MATH/STAT 342 - Probability and Statistical Theory This is a continuation of MATH/STAT 242 (Introduction to Mathematical Statistics, previously MATH/STAT 341). In this class, students will expand their basic knowledge from MATH/STAT 242 into broader and deeper probability and statistics theory. For instance, students will learn about conditional distributions of multiple random variables, limiting distributions, moment generating functions and higher moments than mean and variance. Students will learn more methods for testing statistical hypotheses, such as the two-sample T test, the F-test and non-parametric methods. There will also be an introduction to analysis of variance (ANOVA). To insure that students learn more than just theoretical ideas a term project applying class knowledge to solving real world problems is usually assigned. Minitab will be used for the data analysis. Students are required to complete MATH/STAT 242 prior to enrolling in this class. MATH/STAT 342 is cross listed under mathematics and statistics. Students can take it for the mathematics major (or minor), the statistics minor and the actuarial science minor. MATH/STAT 348 - Applied Regression Analysis and ANOVA Regression analysis of data is a powerful statistical tool that is widely used in biology, psychology, management, engineering, medical research, government and many other fields. It provides a technique for building a reasonable mathematical model that relates the mean value of a response (e.g., profit) to various independent variables or predictors (e.g., advertising budgets, size of inventory, etc.). Any prediction or estimation based on a random sample of data will contain a certain unknown error. In this course, students will learn various methods to build a best regression model for a given set of data under certain constraints so that the error is minimized. When the relation between the dependent and independent variables is linear, we call it linear regression. Students will also learn about nonlinear regression, where there can be a nonlinear relationship (such as quadratic or exponential). Real world problem solving skills are emphasized. Minitab is used extensively for the data exploration and data analysis. A term project (with open topics) is normally assigned for students to explore knowledge beyond the classroom. Students are required to complete MATH/STAT 341 prior to enrolling in this class. MATH/STAT 348 is cross-listed under mathematics and statistics. Students can take it for the mathematics major (or minor), the statistics minor and the actuarial science minor. MATH 351 - Differential Equations Differential equations are a powerful tool in constructing mathematical models for the physical world. Their use in industry and engineering is so widespread and they perform so well that they are among the most successful of modeling tools. For example, a cup of hot coffee is initially at and is left in a room with an ambient temperature of . Suppose that initially it is cooling at a rate of per minute. Then the model for the cup's temperature is . This is an example of a differential equation. We are interested in predicting the temperature, T, of the coffee at any time t. We can also ask, "How long does it take the coffee to cool to a temperature of, say, MATH 356 - Numerical Analysis When one pushes the square root button on a calculator to compute the square root of 2, one should ask, "How does the calculator do it?" Numerical analysis deals with implementing numerical methods to answer questions like this one. While numerical methods have always been useful, since the invention of computers, the role of numerical methods in scientific research has become essential. No modern applied mathematician, physical scientist or engineer can be properly trained without some understanding of numerical methods. There is more involved here than just knowing how to use the methods. One needs to know how to analyze their accuracy and efficiency. Numerical analysis is a broad and challenging mathematical activity, whose central theme is the effective constructability of various kinds of approximations. MATH 411 - The Mathematics of Risk In this course, which is central to the financial mathematics major, we examine how the formulae that populate finance books are developed. We investigate the relationship between income and expense streams through time, and the present value of an investment with those cash flows. We will investigate what calculus can tell us about the sensitivity of that valuation to changes in market interest rates, and how securities can be designed to make the securities insensitive to small changes. We will develop the basic theory of geometric brownian motion for the pricing of securities such as stocks, and we will develop the binomial tree model for pricing derivative securities such as call options. We will learn about Lagrange multipliers, and use them to understand Markowitz optimal portfolio theory. While it would be very helpful to have a basic understanding of stocks and bonds before starting this course; it is much more important to have a solid command of the big ideas of calculus - rates of change, accumulation of changes, optimization, partial derivatives - and to have a solid command of basic probability and expected values. MATH 433 - Abstract Algebra If you can tell time, you already know some abstract algebra: you just don't know you know it! Suppose you have lunch every day at 1:00pm. Then you'll have lunch at 1:00pm today and at 1:00pm tomorrow. We just called both of those times '1:00pm', but they're not really the same moment in time, since they're occurring on different days! It turns out they both can be thought of as representatives of a coset of in ; this coset, in turn, is an element of the factor group . Huh, you ask? What's a coset? What's ? What's ? What's a factor group?! Take this class and find out! Abstract algebra is the study of algebraic structures such as groups, rings and fields. (You don't know what these objects are yet, but if you take this class you will!) You encounter such objects everywhere in math: the coordinate plane is an example of a group; the set of all matrices over the real numbers is an example of a ring; the set of all real numbers is a field. By studying these structures abstractly, we can give one proof for many results that hold for wildly different objects, instead of proving each result for each object separately. Abstract algebra is a beautiful and powerful area of mathematics and it is an essential part of any mathematics curriculum. It has applications in many sciences, from physics to chemistry, in addition to having extremely important uses in areas such as cryptography. While the concepts in this class require minimal prerequisite knowledge of topics such as calculus, this class is heavily proof-based and requires a large amount of mathematical maturity. The ability to write grammatically and make logical arguments is extremely important, while the ability to differentiate will be of little, if any, use. Conceptual understanding, not a calculator, is at the heart of this course! MATH/EDUC 446 - Mathematics in Secondary Education This course has been designed for prospective teachers of middle school and high school mathematics and reflects the recommendations of the National Council of Teachers of Mathematics (NCTM). The following excerpt is from the NCTM Principles and Standards book: "The Teaching Principle" Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. Teachers need to know and use 'mathematics for teaching' that combines mathematical knowledge and pedagogical knowledge. They must be information providers, planners, consultants and explorers of uncharted mathematical territory. They must adjust their practices and extend their knowledge to reflect changing curricula and technologies and to incorporate new knowledge about how students learn mathematics. They also must be able to describe and explain why they are aiming for particular goals." The course takes the art of teaching through a series of motivational ideas suitable for many grade levels and abilities and includes a discussion of activities, materials and manipulatives suitable for classroom use. Problem solving and heuristics is a major theme in the course. Other topics covered include cooperative learning, questioning techniques, technology, lesson planning, homework options, mini-discovery lessons and technology lessons. MATH 455 - Mathematical Analysis Why does calculus work? In this course we examine the foundations of calculus. What properties of the real numbers distinguish them from the rational numbers? What role do these differences play in the development of such fundamental concepts as limits and convergence? What does continuity really mean, and why do we need it? Along the way, we will study sequences, series and limits, first of numbers, and then of functions. One consequence of our study will be a better appreciation of the central role of power series in many of the results of calculus. This course is strongly recommended for anyone considering a graduate degree in pure or applied mathematics, statistics, theoretical physics or operations research. Surprisingly, a deep understanding of the theoretical underpinnings of calculus is necessary to make progress in such applied areas as optimization, numerical analysis, financial modeling, probability and differential equations. This course is almost entirely focused on formal definitions and rigorous proofs. Students are encouraged to have as much exposure to proofs as possible prior to enrolling in this course.
Introductory Combinatorics Focusing on the core material of value to students in a wide variety of fields, this book presents a broad comprehensive survey of modern ...Show synopsisFocusing on the core material of value to students in a wide variety of fields, this book presents a broad comprehensive survey of modern combinatorics at an introductory level. The author begins with an introduction of concepts fundamental to all branches of combinatorics in the context of combinatorial enumeration. Chapter 2 is devoted to enumeration problems that involve counting the number of equivalence classes of an equivalence relation. Chapter 3 discusses somewhat less direct methods of enumeration, the principle of inclusion and exclusion and generating functions. The remainder of the book is devoted to a study of combinatorial structures.Hide synopsis Description:Good. Hardcover. May include moderately worn cover, writing,...Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780121108304-4-0-3 Orders ship the same or next business day. Expedited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions. ISBN: 9780121108304
Real Analysis 9781852333140 ISBN: 1852333146 Publisher: Springer Verlag Summary: Understanding the concepts and methods of real analysis is an essential skill for every undergraduate mathematics student. Written in an easy-to-read style, Real Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. Combining an informal style with precision mathematics, Real Analysis covers all the key topics with f...ully worked examples and exercises with solutions. Featuring: * Sequences and series - considering the central notion of a limit * Continuous functions * Differentiation * Integration * Logarithmic and exponential functions * Uniform convergence * Circular functions All these concepts and techniques are deployed in examples in the final chapter to provide the student with a thorough understanding of this challenging subject. Howie, John M. is the author of Real Analysis, published under ISBN 9781852333140 and 1852333146. Five hundred twenty five Real Analysis textbooks are available for sale on ValoreBooks.com, one hundred five used from the cheapest price of $29.25, or buy new starting at $34
Overview Designed to supplement an ordinary differential equations textbook, this book focuses on the features of Mathematica that are useful for analyzing differential equations to deepen the reader's understanding. More About This Book Overview Designed to supplement an ordinary differential equations textbook, this book focuses on the features of Mathematica that are useful for analyzing differential equations to deepen the reader's understanding. Editorial Reviews Booknews Uses the mathematical software system to introduce numerical methods, geometric interpretation, symbolic computation, and qualitative analysis. Assumes no prior experience with Mathematica and includes instructions for using it on Macintosh, Windows, NeXT, and the X Window System. Contains a glossary and sample notebook solutions
...Some topics may require a review of related basic skills. I've taught high school and college algebra for over 10 years. Algebra 2 was one of the first high school math classes I taught and I continue to teach the same concepts in collegeQuantitatively and qualitatively, the student will describe the process of solutions and characteristics of solutions. Thermodynamic relationships will be investigated. Students will explore the factors that affect the rates of a reaction and apply them to the theory of dynamic equilibrium
Course Syllabus Course Description Elementary draw conclusions from data. The course introduces the student to applications in engineering, business, economics, medicine, education, the sciences, and other related fields. The use of technology (computers or graphing calculators) will be required in certain applications. Texts, Materials, and Plug-ins Texts These course materials are designed for use with Collaborative Statistics by Barbara Illowsky and Susan Dean. This text may be used online or can be downloaded in PDF format at no cost through the Connexions website, or you may choose to purchase a low-cost printed copy using the "Order Printed Copy" link provided on the collection home page linked above. Instructors wishing to customize this textbook can do so by creating a Connexions account. Connexions accounts are free and allow users to rip, mix, and burn content by updating modules and creating custom collections of educational content. Please see the Connexions website to learn more about Connexions and how you can use it to customize your students' learning experience at absolutely no cost. Materials Required Calculator: The TI-83 calculator is required. There are many examples that use the TI-83 calculator and contain the calculator instructions. YOU WILL BE TAUGHT HOW TO USE THE CALCULATOR IN THE COURSE LESSONS. Throughout the course, you will be given instructions for the TI-83 Calculator. Labs and projects make use of the TI graphing calculator and may be done individually or in groups of up to four. Please download the TI-83 calculator guidebook from this TI-83 Site. Follow the links for "TI-83 Plus Silver Edition" OR "TI-83 Plus" and use the "Guidebooks" link. Homework and Suggested Grading The purpose of homework is to help you learn the material in the course. You learn the most and do your best if you do the homework problems. You are expected to do the chapter PRACTICE in the workbook before attempting the homework. The answers to the Practice are in the back of the workbook. Then do the assigned odd numbered homework problems in the text and check those answers in the back of the text. Table 1 Homework Total Points Lowest Points (out of 700) for: Percentage Exams (3 @ 100 points each) 300 A: 630 90-100% Quizzes (12 @ 10 points each, 3 lowest dropped) 90 B: 546 78-89% Labs (2 @ 30 points each) 60 C: 462 66-77% Projects (2 @ 75 points each) 150 D: 385 55-65% Final Exam 100 F: Below 385 0-54%
Basic Technical Mathematics - 2nd edition Summary: * A non-rigourous, accurate presentation of precalc topics as applied to the technologies.* Examples are worked in great detail; second color is used to explain concepts, and margin annotations explain the steps in the examples; cumulative review exercises
Geometry Guide illustrates every geometric principle, formula, and problem type tested on the GMAT to help you understand and master the intricacies of shapes, planes, lines, angles, and objects. 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 83 "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 Geometry question bank.
978-81-219-2398-9Hindi1. ... This book is a part of a series of three books written to provide complete coverage of the NCERT science syllabus forclass9 prescribed by Central Board of Secondary Education. ... CHAPTER1: THE SENTENCE CHAPTER 2: THE NOUN CHAPTER 3: ... ... Chapter9 Diary entry on a memorable day in Nominalisation III ... Prove that a linear equation in two variables has infinitely many solutions and justify their being written as ordered pairs of ... Mathematics -8 NCERT Mathematics -IX NCERTCLASS- NURSERY Mathematics ( R.S. Agrawal ... ... and the final chapter demonstrates that while many questions still remain unanswered, ... of the NCERT and is based on the Continuous and Comprehensive Evaluation (CCE) ... Mridul Hindi Pathmala 0 Mridul Hindi Pathmala 1 Mridul Hindi Pathmala 2 Mridul Hindi Pathmala 3
Featuring humor, easy-to-understand explanations, and silly illustrations, Life of Fred is guaranteed to make your math studies come alive! Each text is written as a novel, including a hilarious story line based on the life of Fred Gauss. As Fred encounters the need for math during his daily exploits, he learns the methods necessary to solve his predicaments – plus loads of other interesting facts! Filled with plenty of solved examples, each book is self-teaching and reusable – perfect for families full of learners. Introduce your students to Fred today and see how his fun, lighthearted approach to learning is revolutionizing mathematics! Life of Fred: Beginning Algebra Expanded Edition covers the following concepts: Finite/Infinite Numbers Natural numbers Whole numbers Integers Adding signed numbers Ratios Multiplying signed numbers Proportions Inequalities in the integers Continued Ratios Adding like terms Rectangles Trapezoids Sectors Symmetric Law of Equality Order of operations Solving Equations Rational numbers Set builder notation Distance-rate-time problems Distributive law Reflexive Law of Equality Proof of the distributive law Coin problems Age problems Transposing Solving systems of equations by elimination Work problems in two unknowns Graphs Plotting points Averages Graphing linear equations Graphing any equation Solving systems of equations by graphing Solving systems of equations by substitution Inconsistent and Dependent systems of equations Factorial function Areas and volumes Commutative laws Negative exponents Multiplying polynomials Solving quadratic equations by factoring Common factors factoring Easy trinomial factoring Difference of squares factoring Grouping factoring Harder trinomial factoring Solving fractional equations Simplifying rational expressions Adding and subtracting rational expressions Multiplying and dividing rational expressions Solving pure quadratic equations Square roots Pythagorean theorem Real numbers Irrational numbers Fractional exponents Solving radical equations Rationalizing the denominator Quadratic equations in everyday life Solving quadratic equations by completing the square Quadratic formula Long division of polynomials Functions Slope Finding slope of a line from its equation Slope-intercept form of a line Range of a function Fast way to graph y = mx + b Fahrenheit-Celsius conversions Graphing inequalities Why you can't divide by zero Absolute value Solving inequalities in one unknown. Product: Life Of Fred: Beginning Algebra Expanded Edition - Grades 8-10 Vendor: Z Twist Books Minimum Grade: 8th Grade Maximum Grade: 10th Grade Weight: 2.73 pounds Length: 10.25 inches Width: 7 inches Height: 1.5 inches Subject: Math Curriculum Name: Life of Fred Learning Style: Auditory, Visual Teaching Method: Charlotte Mason, Unschooling There are currently no reviews for Life Of Fred: Beginning Algebra Expanded Edition - Grades 8-10.
Elementary Algebra: Concepts and Applications The goal of Elementary Algebra: Concepts and Applications, 7e is to help today's students learn and retain mathematical concepts by preparing them ...Show synopsisThe goal of Elementary Algebra: Concepts and Applications, 7e is to help today's students learn and retain mathematical concepts by preparing them for the transition from skills-oriented elementary
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, This article originally appeared in the 04
Use Wolfram\|Alpha to Solve Calculus Problems and…... Use Wolfram\|Alpha to Solve Calculus Problems and… Everything Else. Wolfram\|Alpha is like Google on crack. However, it is not technically a search engine; it is a "computational knowledge" engine. They use a huge collection of trustworthy, built-in data to get the user the information or knowledge they are looking for. When you search for an item, Wolfram\|Alpha gives you all of the relevant knowledge they have on that specific search query. For example, here is the results for the search "when did the Beatles break up?" Not only do you get the date the Beatles broke up, you also get how long away that date is from today and other noteworthy events that occurred on the same day. Here is another example, for the search "carbon footprint driving 536 miles at 32mpg" that tells you the amount of fuel consumed and the amount of c02 and carbon emitted. Because Wolfram\|Alpha is just retrieving answers from its huge database of information and formulas, you have to be specific and ask non-opinionated questions. For example, the website does not know which Lil Wayne song is the best. However, it does know things that are not opinions, like the nutritional facts of 10,000 big macs and how many planes are currently flying directly over you. I find Wolfram\|Alpha to be better than Google when I am quickly looking for specific answers. I just typed in "Countries that border France" on both Wolfram\|Alpha and Google. Wolfram\|Alpha quickly showed me a list of the 8 countries and a map with of France with its bordering countries highlighted. Google on the other hand sent me over to Yahoo Answers… Other than a fun search engine, Wolfram\|Alpha can also be used as a highly effective tool for college. Like the title mentions, the knowledge engine can in fact solve any calculus problem. It can easily solve any math problem thrown its way, from a basic algebra problem to whatever this is. Wolfram\|Alpha can also be used for many other college courses such as biology, astronomy, history, etc. As Wolfram\|Alpha can be kind of confusing and hard to get the hang of at first, I suggest going through this short tour and looking at some examples to help give you a better sense of how to use it. Even if you find it a little bit confusing at first, keep trying because Wolfram\|Alpha really is a great way to "hack college."
Differential Equations and Integrals How Much Water Is In Crater Lake?part of Pedagogy in Action:Partners:Spreadsheets Across the Curriculum:Geology of National Parks:Examples Heather Lehto, Department of Geology, University of South Florida Spreadsheets Across the Curriculum module/Geology of National Parks course. Students calculate an answer from a bathymetric map by summing volumes of vertical prisms. Archimedes and Pipart of Spreadsheets Across the Curriculum:General Collection:Examples Christina Stringer Spreadsheets across the Curriculum Activity. Student build spreadsheets that allow them to estimate pi using the same iterative process as Archimedes. Radioactive Decay and Popping Popcorn -- Understanding the Rate Lawpart of Spreadsheets Across the Curriculum:General Collection:Examples Christina StringerUniversity of South Florida, Tampa FL 33620 Spreadsheets Across the Curriculum module. Students build spreadsheets to forward model an example of exponential decay and interpret the meaning of the decay constant. Understanding Mortgage Paymentspart of Spreadsheets Across the Curriculum:General Collection:Examples Jody Murphy Spreadsheets across the Curriculum module. Students build Excel spreadsheets to calculate monthly mortgage payments and evaluate how much of their payment is applied to the principle and interest. Maximize the Volume of a Box: Exploring Polynomial Functionspart of Spreadsheets Across the Curriculum:General Collection:Examples Nasser Dastrange Spreadsheets Across the Curriculum module. Students build spreadsheets to find the maximum volume of an open-top box by cut-and-folding a sheet of cardboard.
Mathematics for Engineers and Technologists 9780750655446 ISBN: 0750655445 Publisher: Elsevier Science & Technology Books Summary: Features real-world examples, case studies, assignments and knowledge-check questions throughout Introduces key mathematical methods in practical engineering contexts *Bridges the gap between theory and practice Mathematics for Engineers and Technologists provides all the essential mathematical information an engineering student needs in preparation for real-life engineering practice. The authors present the subjec...t from an engineering systems perspective - a uniquely student centred approach which introduces the need for the techniques under discussion, before introducing the techniques themselves. With easily accessible material introduced through case studies, assignments and knowledge-check questions, this book is designed to bridge the gap between academic study and vocational application. The interactive style of the book brings the subjects to life, and activities and case studies keep the mathematics firmly rooted in the context of real life engineering practice, rather than focussing on theory alone. Fox, Huw is the author of Mathematics for Engineers and Technologists, published under ISBN 9780750655446 and 0750655445. Four hundred eighty seven Mathematics for Engineers and Technologists textbooks are available for sale on ValoreBooks.com, ninety eight used from the cheapest price of $4.69, or buy new starting at $64.01 This book is carefully designed to be used on a wide range of introductory courses at first degree and HND level in the U.K., with content mat [more] This item is printed on demand. This book is carefully designed to be used on a wide range of introductory courses at first degree and HND level in the U.K., with content matched to a variety of first year degree modules from IEng and other BSc Engineeri.[less]
Written for students, engineers, and researchers, this book presents an introduction to mathematics by solving problems ranging from very simple to complex. Topics covered include numerical analysis as well as statistics. MATLAB is used throughout the book to solve numerous examples. In addition, a supplemental set of MATLAB M-files is available for download.
Algebra 2, Student Edition From the first day your students begin to learn the vocabulary of algebra until the day they take final exams and standardized tests, these programs ...Show synopsisFrom the first day your students begin to learn the vocabulary of algebra until the day they take final exams and standardized tests, these programs strengthen student understanding and provide the tools students need to succeed 2A 3E. Description:Fair. This book is in acceptable condition, Ex-library with some...Fair. This book is in acceptable condition, Ex-library with some writings and jacket is worn and taped but the book is still usable. Same day shipping
1 00:00:00,000 --> 00:00:02,560 PROFESSOR: Welcome to class again. 2 00:00:02,560 --> 00:00:05,560 This time it's not Professor Forney it's me, so my name is 3 00:00:05,560 --> 00:00:06,810 Ralf Koetter. 4 00:00:06,810 --> 00:00:11,650 5 00:00:11,650 --> 00:00:14,150 You guys have some substantial chalk here at MIT. 6 00:00:14,150 --> 00:00:20,130 7 00:00:20,130 --> 00:00:24,390 And I'm visiting here from the University of Illinois, so 8 00:00:24,390 --> 00:00:26,860 Professor Forney thought I could teach this class here. 9 00:00:26,860 --> 00:00:31,530 10 00:00:31,530 --> 00:00:31,722 All right, let's see. 11 00:00:31,722 --> 00:00:35,070 So I understand that last time, last Wednesday, you went 12 00:00:35,070 --> 00:00:40,060 through all the finite field stuff, meaning, so you know 13 00:00:40,060 --> 00:00:46,080 what that would mean, the finite field. 14 00:00:46,080 --> 00:00:48,620 There's p elements, p to the m elements. 15 00:00:48,620 --> 00:00:53,190 Whatever q you have here, is a power of a prime in order to 16 00:00:53,190 --> 00:00:54,440 be a field. 17 00:00:54,440 --> 00:00:56,710 18 00:00:56,710 --> 00:01:02,740 So this one, as a notation, is a ring of polynomials. 19 00:01:02,740 --> 00:01:06,010 You've seen that too. 20 00:01:06,010 --> 00:01:09,170 So I assume you know everything about finite fields 21 00:01:09,170 --> 00:01:12,660 that you will need to know here, at least, except for one 22 00:01:12,660 --> 00:01:14,460 more theorem which Professor Forney told 23 00:01:14,460 --> 00:01:17,870 me he did not cover. 24 00:01:17,870 --> 00:01:21,192 And this is the fundamental theorem of algebra. 25 00:01:21,192 --> 00:01:23,630 I have to write a little bit smaller with this thing here, 26 00:01:23,630 --> 00:01:26,626 otherwise I'll run out. 27 00:01:26,626 --> 00:01:27,876 AUDIENCE: [UNINTELLIGIBLE PHRASE] 28 00:01:27,876 --> 00:01:29,950 29 00:01:29,950 --> 00:01:33,440 PROFESSOR: Oh, I know, that's probably better. 30 00:01:33,440 --> 00:01:34,690 Better. 31 00:01:34,690 --> 00:01:40,540 32 00:01:40,540 --> 00:01:53,175 With the algebra, at least that's what it's often called, 33 00:01:53,175 --> 00:01:57,420 and really, about 60 percent of all the proofs in algebra 34 00:01:57,420 --> 00:01:59,520 eventually boil down to this here. 35 00:01:59,520 --> 00:02:30,510 And what it says is, polynomial of degree m, f beta 36 00:02:30,510 --> 00:02:46,490 equals zero, at most, m of beta. 37 00:02:46,490 --> 00:02:48,310 At least, that's one way to formulate it. 38 00:02:48,310 --> 00:02:49,560 Let me see. 39 00:02:49,560 --> 00:02:52,420 40 00:02:52,420 --> 00:02:54,400 So that's fine. 41 00:02:54,400 --> 00:02:57,490 So what it says is a polynomial of degree m has at 42 00:02:57,490 --> 00:03:00,560 most m roots. 43 00:03:00,560 --> 00:03:02,910 Once you all have seen that, probably one way or another, 44 00:03:02,910 --> 00:03:06,250 but because of its importance, I want to 45 00:03:06,250 --> 00:03:07,500 emphasize it once more. 46 00:03:07,500 --> 00:03:11,140 47 00:03:11,140 --> 00:03:14,670 Do we need a proof of this? 48 00:03:14,670 --> 00:03:17,940 In true MIT spirit we do. 49 00:03:17,940 --> 00:03:23,460 And the proof would go something like this. 50 00:03:23,460 --> 00:03:27,020 You look at problem number one in your homework assignment, 51 00:03:27,020 --> 00:03:30,630 and from problem number one, I could prove that here, too, 52 00:03:30,630 --> 00:03:32,280 but since it's in the homework, I won't. 53 00:03:32,280 --> 00:03:35,790 54 00:03:35,790 --> 00:03:38,170 You can write the following given any beta. 55 00:03:38,170 --> 00:03:42,810 56 00:03:42,810 --> 00:03:57,440 Write f of x as f of x is equal to plus alpha. 57 00:03:57,440 --> 00:04:01,064 58 00:04:01,064 --> 00:04:05,340 Alphas are the field so that's by some sort of long division 59 00:04:05,340 --> 00:04:07,754 you get to that. 60 00:04:07,754 --> 00:04:09,480 That's what I'm not going to prove. 61 00:04:09,480 --> 00:04:17,710 Then f of beta is equal zero is the same as saying that 62 00:04:17,710 --> 00:04:19,160 alpha is equal to zero. 63 00:04:19,160 --> 00:04:29,570 So if either is a root of the polynomial, zero, it follows 64 00:04:29,570 --> 00:04:39,464 that f of x has this thing here as a factor, this h of x, 65 00:04:39,464 --> 00:04:52,010 x minus beta, where because of the degree properties of 66 00:04:52,010 --> 00:05:01,740 polynomials, h of x is m minus 1. 67 00:05:01,740 --> 00:05:05,740 And so the rest follows by induction. 68 00:05:05,740 --> 00:05:08,580 So basically, then we can prove that this polynomial 69 00:05:08,580 --> 00:05:10,870 has, at most, m minus 1 roots, and so on. 70 00:05:10,870 --> 00:05:13,450 And you can descend this route, and so the rest follows 71 00:05:13,450 --> 00:05:14,700 by induction. 72 00:05:14,700 --> 00:05:18,480 73 00:05:18,480 --> 00:05:33,180 In particular we can say if f of x has m distinct roots beta 74 00:05:33,180 --> 00:05:51,040 one, beta m, then it factors completely into the linear 75 00:05:51,040 --> 00:05:53,310 factors like this. 76 00:05:53,310 --> 00:05:59,490 So I just wanted to quickly state the fundamental theorem 77 00:05:59,490 --> 00:06:03,380 of algebra, since we need it in a proof later on, and I 78 00:06:03,380 --> 00:06:04,630 think you didn't go through it. 79 00:06:04,630 --> 00:06:09,660 80 00:06:09,660 --> 00:06:09,760 OK. 81 00:06:09,760 --> 00:06:13,080 So last time, you learned everything about fields, 82 00:06:13,080 --> 00:06:17,010 finite fields, extension fields, so chapter eight is 83 00:06:17,010 --> 00:06:18,750 pretty much what we have to cover now. 84 00:06:18,750 --> 00:06:22,310 85 00:06:22,310 --> 00:06:25,120 What is the whole idea of chapter eight? 86 00:06:25,120 --> 00:06:41,190 It's linear codes, codes, MDS codes, and redundant codes. 87 00:06:41,190 --> 00:06:44,520 Oh, by the way, do you have any questions about this here? 88 00:06:44,520 --> 00:06:45,020 That in any way? 89 00:06:45,020 --> 00:06:46,270 It's pretty straight, right? 90 00:06:46,270 --> 00:06:48,350 91 00:06:48,350 --> 00:06:52,280 OK, so I understand in chapter six or so, you had already 92 00:06:52,280 --> 00:06:53,876 linear codes over the binary fields. 93 00:06:53,876 --> 00:06:56,540 94 00:06:56,540 --> 00:07:05,610 So let's just define codes over a larger field, formally, 95 00:07:05,610 --> 00:07:12,380 a linear code C of length n. 96 00:07:12,380 --> 00:07:31,180 97 00:07:31,180 --> 00:07:39,620 No subspace of Fn. 98 00:07:39,620 --> 00:07:41,050 So whatever the field is. 99 00:07:41,050 --> 00:07:43,890 So F could be any extension field, could be the binary 100 00:07:43,890 --> 00:07:49,080 field, so it really generalizes a definition of 101 00:07:49,080 --> 00:07:51,670 code, of what a linear code is. 102 00:07:51,670 --> 00:07:54,330 OK, so it's a subspace. 103 00:07:54,330 --> 00:07:57,690 What can be derived from that? 104 00:07:57,690 --> 00:08:00,580 Since it's a subspace, it's a group. 105 00:08:00,580 --> 00:08:05,440 And then we can derive minimum distance properties. 106 00:08:05,440 --> 00:08:09,620 So let's first define it again, since it's slightly 107 00:08:09,620 --> 00:08:11,865 different than the definition for binary codes. 108 00:08:11,865 --> 00:08:21,310 109 00:08:21,310 --> 00:08:32,480 Between Fn, say Fqn. 110 00:08:32,480 --> 00:08:35,530 So I denote the vectors with an underscore. 111 00:08:35,530 --> 00:08:38,840 I think in the notes, it's boldface notation, so 112 00:08:38,840 --> 00:08:41,835 translate that online as I go here. 113 00:08:41,835 --> 00:08:53,920 The distance between two words x and y, given as dx, the 114 00:08:53,920 --> 00:09:00,786 number of positions that x_i is unequal to y_i. 115 00:09:00,786 --> 00:09:02,540 AUDIENCE: What's the subscript? 116 00:09:02,540 --> 00:09:04,230 PROFESSOR: There, a q. 117 00:09:04,230 --> 00:09:06,650 Oh, this is another thing I should warn you about. 118 00:09:06,650 --> 00:09:10,890 My handwriting is bound to deteriorate during class. 119 00:09:10,890 --> 00:09:13,830 So I usually start out reasonably okay, towards the 120 00:09:13,830 --> 00:09:15,570 end of the class it's -- 121 00:09:15,570 --> 00:09:18,370 I tell my students to throw little pieces of chalk at me 122 00:09:18,370 --> 00:09:23,710 when it gets too bad and I'm not facing them, so please 123 00:09:23,710 --> 00:09:25,400 just say something if it gets too bad. 124 00:09:25,400 --> 00:09:28,720 125 00:09:28,720 --> 00:09:30,595 So distance is defined as that, quickly. 126 00:09:30,595 --> 00:09:33,770 127 00:09:33,770 --> 00:09:36,290 So it doesn't really matter what the values are here. 128 00:09:36,290 --> 00:09:39,700 The x_i and the y_i could assume different values. 129 00:09:39,700 --> 00:09:44,050 It's a somewhat coarse measure for the real, the difference 130 00:09:44,050 --> 00:09:49,520 between code words, or difference between words. 131 00:09:49,520 --> 00:09:52,560 Why do you think I say it's coarse? 132 00:09:52,560 --> 00:09:53,965 In digital communications in particular? 133 00:09:53,965 --> 00:09:58,760 134 00:09:58,760 --> 00:09:59,670 Good question, right? 135 00:09:59,670 --> 00:10:04,600 In the end, we want to map that into a modulation scheme. 136 00:10:04,600 --> 00:10:06,870 In the end, we want to map our codes that we are deriving 137 00:10:06,870 --> 00:10:08,210 here into modulation schemes. 138 00:10:08,210 --> 00:10:09,630 In the end, we want to embed them into 139 00:10:09,630 --> 00:10:11,610 some Euclidean space. 140 00:10:11,610 --> 00:10:15,060 Now, different elements of our alphabet we will map to 141 00:10:15,060 --> 00:10:17,900 different elements in Euclidean space. 142 00:10:17,900 --> 00:10:22,650 So basically, approximating their distance relation in 143 00:10:22,650 --> 00:10:25,410 Euclidean space, which we are really interested in with the 144 00:10:25,410 --> 00:10:30,630 Hamming distance here is pretty coarse, but we can do, 145 00:10:30,630 --> 00:10:31,290 so we do that. 146 00:10:31,290 --> 00:10:32,540 It's an approximation, at least. 147 00:10:32,540 --> 00:10:35,660 148 00:10:35,660 --> 00:10:36,145 That clear? 149 00:10:36,145 --> 00:10:37,770 All set? 150 00:10:37,770 --> 00:10:38,285 All right. 151 00:10:38,285 --> 00:10:38,855 AUDIENCE: [UNINTELLIGIBLE PHRASE] the 152 00:10:38,855 --> 00:10:39,926 Hamming distance [UNINTELLIGIBLE] the same as 153 00:10:39,926 --> 00:10:42,450 the Euclidean distance? 154 00:10:42,450 --> 00:10:44,035 PROFESSOR: Well, it depends on the modulation scheme. 155 00:10:44,035 --> 00:10:45,970 It very much depends on the modulation scheme. 156 00:10:45,970 --> 00:10:54,890 If you have a 8-PSK scheme, where you would label, put in 157 00:10:54,890 --> 00:10:58,730 the words, here, with three bit symbols, or with the 158 00:10:58,730 --> 00:11:05,900 symbol from F8, then it's definitely different. 159 00:11:05,900 --> 00:11:08,690 It's definitely different. 160 00:11:08,690 --> 00:11:10,460 So if you do anti-polar signaling, then 161 00:11:10,460 --> 00:11:13,970 it's directly reflected. 162 00:11:13,970 --> 00:11:17,510 OK, I'm starting to digress already. 163 00:11:17,510 --> 00:11:22,930 164 00:11:22,930 --> 00:11:32,440 So just for completeness, minimum distance, minimum 165 00:11:32,440 --> 00:12:01,540 Hamming, of a code subset Fqn is d as a minimum code of dxy, 166 00:12:01,540 --> 00:12:04,682 and they have to be different, it's the same as before. 167 00:12:04,682 --> 00:12:08,250 168 00:12:08,250 --> 00:12:10,340 So now if I claim that -- 169 00:12:10,340 --> 00:12:15,610 170 00:12:15,610 --> 00:12:21,400 so the minimum distance of a code is also given by the 171 00:12:21,400 --> 00:12:36,870 minimum between 0 and x in the code 0 and x, and this is 172 00:12:36,870 --> 00:12:46,260 minimum of the Hamming weight of x, and you could 173 00:12:46,260 --> 00:12:50,080 do 0 x in the code. 174 00:12:50,080 --> 00:12:51,440 So that's all old stuff. 175 00:12:51,440 --> 00:12:55,850 I just write it down so we get started here. 176 00:12:55,850 --> 00:13:00,050 Is that clear, from the group property, why 177 00:13:00,050 --> 00:13:01,680 this would be true? 178 00:13:01,680 --> 00:13:05,990 So if you just take this, we can add basically x to both x 179 00:13:05,990 --> 00:13:08,890 and y, just translating the whole relation 180 00:13:08,890 --> 00:13:10,380 to somewhere else. 181 00:13:10,380 --> 00:13:13,380 So in particular, we translate it here, once we have it here, 182 00:13:13,380 --> 00:13:16,890 than the distance between 0 and x is just the weight. 183 00:13:16,890 --> 00:13:18,660 OK. 184 00:13:18,660 --> 00:13:21,140 So far, so good. 185 00:13:21,140 --> 00:13:22,410 Now what is next? 186 00:13:22,410 --> 00:13:25,060 Generate a matrix. 187 00:13:25,060 --> 00:13:26,710 This is not really in the notes, but 188 00:13:26,710 --> 00:13:27,960 I think it's important. 189 00:13:27,960 --> 00:13:35,100 190 00:13:35,100 --> 00:13:39,160 So see, the code here is a subspace. 191 00:13:39,160 --> 00:13:42,190 It's a linear space, so it has a generator, it has 192 00:13:42,190 --> 00:13:44,210 generators, k generators. 193 00:13:44,210 --> 00:13:59,793 So let g1 be k, write this off the code. 194 00:13:59,793 --> 00:14:02,430 195 00:14:02,430 --> 00:14:07,080 So as a basis of the vector space, that this would be a 196 00:14:07,080 --> 00:14:09,610 basis of the vector space, any basis would be fine here. 197 00:14:09,610 --> 00:14:12,710 198 00:14:12,710 --> 00:14:27,370 Then C may be defined as all the x in Fqn such that x is 199 00:14:27,370 --> 00:14:29,260 sum over -- 200 00:14:29,260 --> 00:14:30,510 what do I call it -- 201 00:14:30,510 --> 00:14:32,440 202 00:14:32,440 --> 00:14:47,050 fi gi, where fi is in Fq. 203 00:14:47,050 --> 00:14:56,690 And the reason I introduce this, we can -- 204 00:14:56,690 --> 00:14:58,900 this is just the definition of a space, right? 205 00:14:58,900 --> 00:14:59,450 That's clear. 206 00:14:59,450 --> 00:15:08,620 So if you have these generators, you find a 207 00:15:08,620 --> 00:15:13,866 generator matrix, uh-oh, it already starts. 208 00:15:13,866 --> 00:15:15,116 Let me -- 209 00:15:15,116 --> 00:15:24,420 210 00:15:24,420 --> 00:15:42,940 matrix g which contains, as a m matrix 211 00:15:42,940 --> 00:15:48,680 containing the rows gi. 212 00:15:48,680 --> 00:15:52,320 So the i-th row in the generator matrix is just gi. 213 00:15:52,320 --> 00:16:12,540 Then you also can write as x is equal to f times g, f 214 00:16:12,540 --> 00:16:20,040 element Fqk, or just the same statement as this one, so 215 00:16:20,040 --> 00:16:21,290 nothing has happened. 216 00:16:21,290 --> 00:16:25,280 217 00:16:25,280 --> 00:16:28,260 So basically, the reason I did that, I wanted to introduce 218 00:16:28,260 --> 00:16:34,350 the term generator matrix, which is sort of important. 219 00:16:34,350 --> 00:16:52,220 And one more property of this orthogonal complement 220 00:16:52,220 --> 00:16:54,750 of C, of the code. 221 00:16:54,750 --> 00:16:57,270 So what does that mean? 222 00:16:57,270 --> 00:17:09,030 So the orthogonal complement of the code you could write as 223 00:17:09,030 --> 00:17:22,140 Fqn such that sum of x_i y_i is equal to 0. 224 00:17:22,140 --> 00:17:29,210 The sum is obviously over the field for all y in the code. 225 00:17:29,210 --> 00:17:32,560 226 00:17:32,560 --> 00:17:37,084 What's the dimension of this, of the orthogonal complement? 227 00:17:37,084 --> 00:17:38,240 AUDIENCE: n minus k. 228 00:17:38,240 --> 00:17:41,400 PROFESSOR: n minus k, clearly, because we have ambient space 229 00:17:41,400 --> 00:17:46,330 is n dimensional, we impose k linear constraints on this, by 230 00:17:46,330 --> 00:17:50,830 the k generators, so the k dimensions of, take note, by 231 00:17:50,830 --> 00:17:52,680 the generators drop out. 232 00:17:52,680 --> 00:17:58,310 So the dimension of the orthogonal 233 00:17:58,310 --> 00:18:00,910 complement is n minus k. 234 00:18:00,910 --> 00:18:06,100 235 00:18:06,100 --> 00:18:08,400 So what else do we need to say about this? 236 00:18:08,400 --> 00:18:13,390 237 00:18:13,390 --> 00:18:22,910 C is called the dual code for this reason. 238 00:18:22,910 --> 00:18:25,510 239 00:18:25,510 --> 00:18:26,470 C is called dual code. 240 00:18:26,470 --> 00:18:29,300 In particular, it's a code that's a linear space. 241 00:18:29,300 --> 00:18:32,160 It's a subspace of Fqn again, it's a code. 242 00:18:32,160 --> 00:18:36,130 So it's just as nice a code as C at this 243 00:18:36,130 --> 00:18:37,380 point in time at least. 244 00:18:37,380 --> 00:18:42,740 245 00:18:42,740 --> 00:18:45,110 So it's called a dual code. 246 00:18:45,110 --> 00:18:48,785 To C, if it is a code, it has a generator matrix. 247 00:18:48,785 --> 00:18:51,400 248 00:18:51,400 --> 00:19:04,630 Let h be a generator matrix for C dual. 249 00:19:04,630 --> 00:19:07,470 So in particular, we could define C dual now, for 250 00:19:07,470 --> 00:19:10,810 example, by the equivalent of this relation here. 251 00:19:10,810 --> 00:19:17,700 But because it's a dual code, we now also can define the 252 00:19:17,700 --> 00:19:34,420 original code in an equivalent way such that x times h 253 00:19:34,420 --> 00:19:37,990 transpose is 0. 254 00:19:37,990 --> 00:19:43,330 We could define our original code C either as the image of 255 00:19:43,330 --> 00:19:49,580 a matrix g, of a generator matrix g, or as a kernel of a 256 00:19:49,580 --> 00:19:52,470 parity-check matrix h. 257 00:19:52,470 --> 00:19:55,440 So h is a ...WRITING ON BOARD... 258 00:19:55,440 --> 00:20:03,790 259 00:20:03,790 --> 00:20:07,960 for C. So that's all pretty much straight linear algebra, 260 00:20:07,960 --> 00:20:12,340 and I'm sure you've seen that in many different places. 261 00:20:12,340 --> 00:20:13,660 Any questions about any of this? 262 00:20:13,660 --> 00:20:17,962 263 00:20:17,962 --> 00:20:19,396 AUDIENCE: So the addition of the dual 264 00:20:19,396 --> 00:20:20,840 [UNINTELLIGIBLE PHRASE] 265 00:20:20,840 --> 00:20:22,430 the summation [UNINTELLIGIBLE PHRASE] 266 00:20:22,430 --> 00:20:24,994 equals 0 for all [INAUDIBLE] other than x, right? 267 00:20:24,994 --> 00:20:26,260 [UNINTELLIGIBLE PHRASE] 268 00:20:26,260 --> 00:20:28,170 PROFESSOR: Oh no, no, no, it doesn't have to be 269 00:20:28,170 --> 00:20:29,740 different from x. 270 00:20:29,740 --> 00:20:35,460 If y is in the code, if y is in C, then x has to be 271 00:20:35,460 --> 00:20:37,020 orthogonal to it. 272 00:20:37,020 --> 00:20:38,850 They can be the same vector, in particular, if you have 273 00:20:38,850 --> 00:20:41,630 binary vectors, an even made binary vector is 274 00:20:41,630 --> 00:20:43,560 orthogonal to itself. 275 00:20:43,560 --> 00:20:45,820 It's a little bit odd, but that's the 276 00:20:45,820 --> 00:20:47,070 magic of finite fields. 277 00:20:47,070 --> 00:20:51,100 278 00:20:51,100 --> 00:20:51,275 OK. 279 00:20:51,275 --> 00:20:51,830 Good. 280 00:20:51,830 --> 00:20:53,430 So these are codes, now we could stop. 281 00:20:53,430 --> 00:20:57,100 We have defined the object, and obviously it exists, 282 00:20:57,100 --> 00:21:02,720 because we could just write something down and it exists. 283 00:21:02,720 --> 00:21:06,050 So once we have defined it, the next question is, what 284 00:21:06,050 --> 00:21:09,120 sort of codes do exist? 285 00:21:09,120 --> 00:21:13,140 So that's what we're going to do next. 286 00:21:13,140 --> 00:21:16,650 287 00:21:16,650 --> 00:21:17,900 First, question one. 288 00:21:17,900 --> 00:21:30,000 289 00:21:30,000 --> 00:21:36,880 Codes do, what type of codes do exist? 290 00:21:36,880 --> 00:21:40,130 So which codes do you know? 291 00:21:40,130 --> 00:21:42,920 AUDIENCE: [INAUDIBLE] 292 00:21:42,920 --> 00:21:45,800 PROFESSOR: You know Reed-Muller codes, you know 293 00:21:45,800 --> 00:21:49,330 probably sporadic binary codes that are out there. 294 00:21:49,330 --> 00:21:51,900 295 00:21:51,900 --> 00:21:53,430 These are all binary codes. 296 00:21:53,430 --> 00:21:58,220 So what type of codes exist over larger fields? 297 00:21:58,220 --> 00:22:01,600 298 00:22:01,600 --> 00:22:03,070 Many, many classes. 299 00:22:03,070 --> 00:22:07,400 There exists the equivalent of the Reed-Muller codes, there 300 00:22:07,400 --> 00:22:10,910 exist QRE Reed-Muller codes, and there exist generalized 301 00:22:10,910 --> 00:22:13,690 Reed-Muller codes, and, and, and, and, and. 302 00:22:13,690 --> 00:22:19,700 But we are interested in a very special class today, 303 00:22:19,700 --> 00:22:23,080 which is MDS codes. 304 00:22:23,080 --> 00:22:35,475 It stands for Maximum Distance Separable. 305 00:22:35,475 --> 00:22:38,520 306 00:22:38,520 --> 00:22:40,020 It's a strange name. 307 00:22:40,020 --> 00:22:42,350 There's no particular reason for MDS. 308 00:22:42,350 --> 00:22:51,440 309 00:22:51,440 --> 00:22:52,690 But, let's see what we can do with that. 310 00:22:52,690 --> 00:22:55,400 311 00:22:55,400 --> 00:22:58,420 What type of codes do exist? 312 00:22:58,420 --> 00:23:04,640 So we have parameters of codes -- oh, I think you write the 313 00:23:04,640 --> 00:23:06,740 curly bracket, right -- 314 00:23:06,740 --> 00:23:09,730 n, k and d. 315 00:23:09,730 --> 00:23:14,040 So that would mean a code of length n, dimension k, and 316 00:23:14,040 --> 00:23:16,140 distance d. 317 00:23:16,140 --> 00:23:20,450 And let me add something to it a q, if you want to emphasize 318 00:23:20,450 --> 00:23:21,925 that this is a query field. 319 00:23:21,925 --> 00:23:24,810 320 00:23:24,810 --> 00:23:29,140 So are all numbers possible here? 321 00:23:29,140 --> 00:23:36,345 What do we have, a 20, 19, 17 code over, I 322 00:23:36,345 --> 00:23:39,390 don't know, over F8. 323 00:23:39,390 --> 00:23:41,580 Is this possible? 324 00:23:41,580 --> 00:23:42,830 What would you think? 325 00:23:42,830 --> 00:23:49,750 326 00:23:49,750 --> 00:23:51,482 No? 327 00:23:51,482 --> 00:23:52,430 AUDIENCE: [INAUDIBLE] 328 00:23:52,430 --> 00:23:54,100 PROFESSOR: It's not possible. 329 00:23:54,100 --> 00:23:56,160 It doesn't seem likely. 330 00:23:56,160 --> 00:24:01,430 What conflicts here, is the dimension and the distance. 331 00:24:01,430 --> 00:24:03,900 If you get a large dimension, in particular, if we would 332 00:24:03,900 --> 00:24:07,170 make this 20, what would that mean? 333 00:24:07,170 --> 00:24:09,840 It would mean we have to take the entire space. 334 00:24:09,840 --> 00:24:11,990 If you take the entire space, then the minimum 335 00:24:11,990 --> 00:24:13,780 weight word is 1. 336 00:24:13,780 --> 00:24:16,870 So this is possible. 337 00:24:16,870 --> 00:24:17,940 You know this is possible. 338 00:24:17,940 --> 00:24:22,240 If you drop this by 1, that seems very unlikely that we 339 00:24:22,240 --> 00:24:25,920 would get a 17 here. 340 00:24:25,920 --> 00:24:29,080 But what do we get here? 341 00:24:29,080 --> 00:24:30,330 2. 342 00:24:30,330 --> 00:24:31,920 343 00:24:31,920 --> 00:24:34,910 You get a 2 because that's what we can achieve with a 344 00:24:34,910 --> 00:24:37,290 single parity-check code. 345 00:24:37,290 --> 00:24:38,810 The parity-check code doesn't have to be 346 00:24:38,810 --> 00:24:41,820 restrained to binary. 347 00:24:41,820 --> 00:24:42,130 Why? 348 00:24:42,130 --> 00:24:43,520 Why would it be restrained to binary? 349 00:24:43,520 --> 00:24:46,850 350 00:24:46,850 --> 00:24:56,430 You could just, the set of all vectors let's define the 351 00:24:56,430 --> 00:25:03,900 single parity-check codes s p c, q, as the set of all 352 00:25:03,900 --> 00:25:11,165 vectors such that sum of the x_i is equal to 0. 353 00:25:11,165 --> 00:25:14,320 354 00:25:14,320 --> 00:25:18,380 So could we have a word of weight 1 in here? 355 00:25:18,380 --> 00:25:19,270 Obviously not, right? 356 00:25:19,270 --> 00:25:22,970 If it has a weight 1, how would it add up to 0? 357 00:25:22,970 --> 00:25:25,745 Because one position would never cancel 358 00:25:25,745 --> 00:25:27,070 with any other position. 359 00:25:27,070 --> 00:25:29,820 So the minimum weight is 2 here, and we get 360 00:25:29,820 --> 00:25:30,600 a distance of 2. 361 00:25:30,600 --> 00:25:32,940 So what's the next one? 362 00:25:32,940 --> 00:25:37,250 363 00:25:37,250 --> 00:25:41,740 It's tempting to say 3, right? 364 00:25:41,740 --> 00:25:47,190 3, but this is very much a question, now. 365 00:25:47,190 --> 00:25:50,710 Because this is not as easy to come by as a single 366 00:25:50,710 --> 00:25:53,060 parity-check. 367 00:25:53,060 --> 00:25:56,320 And that's what we're going to do next. 368 00:25:56,320 --> 00:26:04,010 We're going to define bounds on the maximum distance that a 369 00:26:04,010 --> 00:26:07,310 code can have altogether. 370 00:26:07,310 --> 00:26:12,210 OK, so let's do the following. 371 00:26:12,210 --> 00:26:14,750 372 00:26:14,750 --> 00:26:27,720 Which parameter is possible? 373 00:26:27,720 --> 00:26:28,970 OK. 374 00:26:28,970 --> 00:26:30,690 375 00:26:30,690 --> 00:26:34,270 So let's assume you have a code, an n,k,d code, and now 376 00:26:34,270 --> 00:26:38,720 we want to find a relation, a bound between n, k and d. 377 00:26:38,720 --> 00:26:39,970 How do we do this? 378 00:26:39,970 --> 00:26:43,220 379 00:26:43,220 --> 00:26:44,470 Any ideas? 380 00:26:44,470 --> 00:26:46,720 381 00:26:46,720 --> 00:26:48,740 Let a computer run for eternity and find 382 00:26:48,740 --> 00:26:49,790 all possible codes? 383 00:26:49,790 --> 00:26:51,190 No, no, no, no. 384 00:26:51,190 --> 00:26:53,256 We don't do this. 385 00:26:53,256 --> 00:26:56,770 We wouldn't get far. 386 00:26:56,770 --> 00:27:07,190 Let's assume we have an n,k,d code. 387 00:27:07,190 --> 00:27:10,910 388 00:27:10,910 --> 00:27:12,990 What does that mean? 389 00:27:12,990 --> 00:27:15,660 Well, let's write the code words all down in a huge 390 00:27:15,660 --> 00:27:26,790 matrix, so each row in this matrix 391 00:27:26,790 --> 00:27:28,240 corresponds to one code word. 392 00:27:28,240 --> 00:27:36,330 So this has a length n, this is q to the k, q is whatever 393 00:27:36,330 --> 00:27:41,200 the alphabet is of the code in question, and now we say it's 394 00:27:41,200 --> 00:27:43,640 an n,k,d code. 395 00:27:43,640 --> 00:27:49,660 What that means, it means, among other things is, say we 396 00:27:49,660 --> 00:27:57,850 delete, just punch out, d minus 1 positions. 397 00:27:57,850 --> 00:28:01,100 398 00:28:01,100 --> 00:28:04,450 We punch out d minus 1 positions of all code words 399 00:28:04,450 --> 00:28:07,020 and we look at the code that remains. 400 00:28:07,020 --> 00:28:08,350 You guys don't have colored chalk here, huh? 401 00:28:08,350 --> 00:28:12,020 402 00:28:12,020 --> 00:28:14,270 We look at the code that remains, it means we look at 403 00:28:14,270 --> 00:28:19,875 this part of the matrix. 404 00:28:19,875 --> 00:28:22,930 405 00:28:22,930 --> 00:28:24,180 Is that clear, what I'm doing here? 406 00:28:24,180 --> 00:28:26,440 407 00:28:26,440 --> 00:28:32,470 So if the code indeed had distance d, can there be any 408 00:28:32,470 --> 00:28:36,110 two rows equal in this part? 409 00:28:36,110 --> 00:28:38,150 Remember, we punch out all d minus 1. 410 00:28:38,150 --> 00:28:40,790 411 00:28:40,790 --> 00:28:45,160 Can there be any rows in this part that are equal? 412 00:28:45,160 --> 00:28:46,130 No, right? 413 00:28:46,130 --> 00:28:47,290 Couldn't be. 414 00:28:47,290 --> 00:28:50,360 They all have to be different. 415 00:28:50,360 --> 00:28:51,560 What does that mean? 416 00:28:51,560 --> 00:28:58,070 They all have to be different, but how many different tuples 417 00:28:58,070 --> 00:29:00,650 can we have in this part? 418 00:29:00,650 --> 00:29:10,200 Well, we have at most q to the n minus d minus 1. 419 00:29:10,200 --> 00:29:11,450 That's the length here. 420 00:29:11,450 --> 00:29:14,200 421 00:29:14,200 --> 00:29:17,650 This n minus d minus 1. 422 00:29:17,650 --> 00:29:18,900 Different tuples. 423 00:29:18,900 --> 00:29:32,350 424 00:29:32,350 --> 00:29:35,130 So how can we patch that together into a relation on 425 00:29:35,130 --> 00:29:36,380 the parameters? 426 00:29:36,380 --> 00:29:42,840 427 00:29:42,840 --> 00:29:46,820 It basically says, q, this is q to the k. 428 00:29:46,820 --> 00:29:51,460 q to the k is upper bounded by this. 429 00:29:51,460 --> 00:29:56,790 430 00:29:56,790 --> 00:29:58,040 It's upper bounded by this. 431 00:29:58,040 --> 00:30:00,740 432 00:30:00,740 --> 00:30:04,660 And let me take the logarithm on here, 433 00:30:04,660 --> 00:30:11,600 and we get this relation. 434 00:30:11,600 --> 00:30:14,265 That's a first incarnation of the tension that we get on 435 00:30:14,265 --> 00:30:16,950 code construction, on codes. 436 00:30:16,950 --> 00:30:18,270 And bound on this, at least. 437 00:30:18,270 --> 00:30:22,730 If you choose d large, the distance large, 438 00:30:22,730 --> 00:30:25,930 then k has to go. 439 00:30:25,930 --> 00:30:30,700 If you choose k large, the distance cannot be very large. 440 00:30:30,700 --> 00:30:34,620 So this is where we, for the first time, see this tension. 441 00:30:34,620 --> 00:30:38,210 And it's also important, I'm sorry that I run around like 442 00:30:38,210 --> 00:30:47,605 this here, n has to be at least k plus d, k 443 00:30:47,605 --> 00:30:48,960 plus d minus 1. 444 00:30:48,960 --> 00:30:51,590 445 00:30:51,590 --> 00:31:02,040 So here, you see this 28 in 3, it would just satisfy this. 446 00:31:02,040 --> 00:31:05,580 It would just satisfy this. 447 00:31:05,580 --> 00:31:06,830 So do we know it exists? 448 00:31:06,830 --> 00:31:09,180 449 00:31:09,180 --> 00:31:10,430 No. 450 00:31:10,430 --> 00:31:11,690 No, why would it? 451 00:31:11,690 --> 00:31:18,710 So far, we only have looked at this here, and so, if it would 452 00:31:18,710 --> 00:31:20,420 exist, it would have to satisfy that. 453 00:31:20,420 --> 00:31:23,110 But there's no reason to assume it exists. 454 00:31:23,110 --> 00:31:26,250 At the moment, at least. 455 00:31:26,250 --> 00:31:27,680 OK. 456 00:31:27,680 --> 00:31:29,180 This is called the Singleton bound. 457 00:31:29,180 --> 00:31:44,290 458 00:31:44,290 --> 00:32:02,570 Any code over any field, phi n, this relationship on the 459 00:32:02,570 --> 00:32:03,440 parameters. 460 00:32:03,440 --> 00:32:04,690 Good. 461 00:32:04,690 --> 00:32:11,040 462 00:32:11,040 --> 00:32:36,740 Any code satisfying and bound with equality is called MDS. 463 00:32:36,740 --> 00:32:39,750 So we have an MDS code if and only if it satisfies the 464 00:32:39,750 --> 00:32:42,070 Singleton bound with equality. 465 00:32:42,070 --> 00:32:44,670 That's the definition of MDS codes. 466 00:32:44,670 --> 00:32:48,280 And now it makes maybe a little bit more sense to talk 467 00:32:48,280 --> 00:32:51,530 about Maximum Distance Separable codes, well, in a 468 00:32:51,530 --> 00:32:55,020 sense, they have the maximum distance among all codes. 469 00:32:55,020 --> 00:32:58,300 You find all codes with the given n and k, if they're MDS, 470 00:32:58,300 --> 00:33:01,780 they have the maximum distance. 471 00:33:01,780 --> 00:33:05,720 OK, let's think about this a little here. 472 00:33:05,720 --> 00:33:09,080 473 00:33:09,080 --> 00:33:11,000 AUDIENCE: [INAUDIBLE] 474 00:33:11,000 --> 00:33:13,925 dependence on q [UNINTELLIGIBLE]? 475 00:33:13,925 --> 00:33:15,850 PROFESSOR: Yeah, there's a very strong dependence on q. 476 00:33:15,850 --> 00:33:17,800 The bound, not. 477 00:33:17,800 --> 00:33:20,610 The bound has no dependence on q. 478 00:33:20,610 --> 00:33:23,800 If the guys exist or not, that's very much 479 00:33:23,800 --> 00:33:24,680 dependent on q. 480 00:33:24,680 --> 00:33:26,185 We'll get to that. 481 00:33:26,185 --> 00:33:28,000 AUDIENCE: [INAUDIBLE] when the q is large, we have more 482 00:33:28,000 --> 00:33:30,090 options to [UNINTELLIGIBLE]? 483 00:33:30,090 --> 00:33:30,445 PROFESSOR: Absolutely. 484 00:33:30,445 --> 00:33:31,190 Absolutely. 485 00:33:31,190 --> 00:33:39,180 For binary, there's a very simple argument to show that 486 00:33:39,180 --> 00:33:43,200 there are no binary MDS codes except for the parity-check 487 00:33:43,200 --> 00:33:45,790 codes and the repetition codes and trivial code. 488 00:33:45,790 --> 00:33:53,290 489 00:33:53,290 --> 00:34:09,380 So say we have a binary code, a binary n,k,d code with a 490 00:34:09,380 --> 00:34:12,150 generator matrix. 491 00:34:12,150 --> 00:34:13,929 So what could the generator matrix be? 492 00:34:13,929 --> 00:34:22,159 There will be an identity part, and then there will be 493 00:34:22,159 --> 00:34:26,270 the rest of the generator matrix, and how could we 494 00:34:26,270 --> 00:34:30,489 possibly fill that in, in order to make it MDS? 495 00:34:30,489 --> 00:34:39,460 Because this is n, this is k, and we see in order to make it 496 00:34:39,460 --> 00:34:46,730 MDS, every single row has to have all entries equal to 1. 497 00:34:46,730 --> 00:34:50,250 Because if not all entries are equal to 1, here, then we 498 00:34:50,250 --> 00:34:53,780 immediately have exhibited a code word with a weight less 499 00:34:53,780 --> 00:34:57,560 than n minus k plus 1. 500 00:34:57,560 --> 00:35:01,420 So OK, we know the first row has to have all 1's. 501 00:35:01,420 --> 00:35:05,110 Because now, the weight of this row is exactly on the MDS 502 00:35:05,110 --> 00:35:05,950 [INAUDIBLE]. 503 00:35:05,950 --> 00:35:07,200 What about the next one? 504 00:35:07,200 --> 00:35:09,490 505 00:35:09,490 --> 00:35:11,080 The next one, same thing. 506 00:35:11,080 --> 00:35:13,880 507 00:35:13,880 --> 00:35:16,030 All entries have to be 1. 508 00:35:16,030 --> 00:35:17,500 But now we see the problem, right? 509 00:35:17,500 --> 00:35:20,050 Now we add those two guys, it should again be a code word, 510 00:35:20,050 --> 00:35:22,220 and we have a grade two code word. 511 00:35:22,220 --> 00:35:29,370 So this is, in a nutshell, to prove that there are no binary 512 00:35:29,370 --> 00:35:32,410 MDS codes except the trivial ones. 513 00:35:32,410 --> 00:35:43,390 So the trivial ones are n, n1, n, n minus 1, 2 and n1, n. 514 00:35:43,390 --> 00:35:45,920 These are the trivial ones. 515 00:35:45,920 --> 00:35:48,355 The space itself, so it's in a parity-check code, and the 516 00:35:48,355 --> 00:35:50,890 repetition code. 517 00:35:50,890 --> 00:35:53,510 These are the only binary MDS codes. 518 00:35:53,510 --> 00:35:57,270 And the argument is roughly there. 519 00:35:57,270 --> 00:35:58,520 OK, where was I? 520 00:35:58,520 --> 00:36:03,170 521 00:36:03,170 --> 00:36:05,000 Yeah, let's think about this a little bit more. 522 00:36:05,000 --> 00:36:07,370 And we are getting to exactly your question about the 523 00:36:07,370 --> 00:36:08,620 [UNINTELLIGIBLE]. 524 00:36:08,620 --> 00:36:11,380 525 00:36:11,380 --> 00:36:17,380 This here has to hold, this argument has to hold 526 00:36:17,380 --> 00:36:23,300 regardless of which d minus 1 positions we punch out. 527 00:36:23,300 --> 00:36:25,520 This argument has always to hold. 528 00:36:25,520 --> 00:36:29,100 Which means, think about it, it's an enormously strong 529 00:36:29,100 --> 00:36:32,030 combinatorial condition on the code. 530 00:36:32,030 --> 00:36:39,620 So you have a code, that means you have a code, you write it 531 00:36:39,620 --> 00:36:42,990 in a matrix like this, all the code words. 532 00:36:42,990 --> 00:36:49,000 You punch out an arbitrary collection of d minus 1one 533 00:36:49,000 --> 00:36:53,920 positions, and the rest, the remaining positions, have to 534 00:36:53,920 --> 00:36:58,100 make up the entire space here. 535 00:36:58,100 --> 00:37:02,910 The entire space Fqn minus to that right exponent. 536 00:37:02,910 --> 00:37:04,340 This is a very -- 537 00:37:04,340 --> 00:37:07,830 think about it, I mean, just writing down this is an 538 00:37:07,830 --> 00:37:11,290 enormously strong combinatorial condition. 539 00:37:11,290 --> 00:37:17,870 So that will actually lead to the codes existing only for a 540 00:37:17,870 --> 00:37:22,820 very, very special, for a subset of field sizes. 541 00:37:22,820 --> 00:37:26,360 In particular, like you said, we have to have enough freedom 542 00:37:26,360 --> 00:37:31,800 in the field size to fill up this matrix to satisfy this. 543 00:37:31,800 --> 00:37:34,360 544 00:37:34,360 --> 00:37:42,230 OK, before we get to that, before I say a word about the 545 00:37:42,230 --> 00:37:50,880 field size, let me formalize what I just said here, namely, 546 00:37:50,880 --> 00:37:54,850 that all of the other positions have to be exactly 547 00:37:54,850 --> 00:37:59,450 the q to the n minus d minus 1, different tuples. 548 00:37:59,450 --> 00:38:17,610 And the definition, let the code with q to the k, code 549 00:38:17,610 --> 00:38:30,190 words over alphabet Fq. 550 00:38:30,190 --> 00:38:35,410 551 00:38:35,410 --> 00:39:02,990 Let subset of the positions in C, i is called an information 552 00:39:02,990 --> 00:39:30,970 set if C constrained to i runs exactly through all the q to 553 00:39:30,970 --> 00:39:37,395 the k, runs through all the q to the k. 554 00:39:37,395 --> 00:39:46,060 555 00:39:46,060 --> 00:39:48,570 Fqk. 556 00:39:48,570 --> 00:39:53,300 So what it means is, you have a code, and you have a subset 557 00:39:53,300 --> 00:39:58,960 of positions, maybe this one, this one, this one, this one. 558 00:39:58,960 --> 00:40:03,090 This is a subset of positions if the code words. 559 00:40:03,090 --> 00:40:06,230 So if the matrix that remains after you take out the 560 00:40:06,230 --> 00:40:10,470 punctured columns, runs through all the q to the k 561 00:40:10,470 --> 00:40:14,298 elements of Fqk, then this is an information search. 562 00:40:14,298 --> 00:40:15,548 AUDIENCE: [INAUDIBLE] 563 00:40:15,548 --> 00:40:18,040 564 00:40:18,040 --> 00:40:21,870 PROFESSOR: Constrained to i, because i has size k. 565 00:40:21,870 --> 00:40:25,275 i is just -- its just about enough to describe every code 566 00:40:25,275 --> 00:40:31,830 word, if the restraint of C to the set would indeed be giving 567 00:40:31,830 --> 00:40:35,700 a unique vector for each code word. 568 00:40:35,700 --> 00:40:38,210 The reason to call it -- so this is the definition of 569 00:40:38,210 --> 00:40:40,220 information set. 570 00:40:40,220 --> 00:40:44,910 The reason to call it an information set, it's pretty 571 00:40:44,910 --> 00:40:45,620 straight, right? 572 00:40:45,620 --> 00:40:47,685 Why is it called an information set? 573 00:40:47,685 --> 00:40:53,740 574 00:40:53,740 --> 00:40:55,500 Because it's enough, right? 575 00:40:55,500 --> 00:40:56,250 Because it's enough. 576 00:40:56,250 --> 00:40:59,320 If you know exactly the value of a code word in these 577 00:40:59,320 --> 00:41:02,820 positions, then it is enough to recover 578 00:41:02,820 --> 00:41:04,706 the entire code word. 579 00:41:04,706 --> 00:41:07,210 When some genie tells you, gives you a code world which 580 00:41:07,210 --> 00:41:10,110 was corrupted by noise or something, but tells you, 581 00:41:10,110 --> 00:41:12,930 these k positions are OK. 582 00:41:12,930 --> 00:41:14,030 That's enough, that's all you need. 583 00:41:14,030 --> 00:41:15,050 That's an information set. 584 00:41:15,050 --> 00:41:17,420 You can recover the information from them. 585 00:41:17,420 --> 00:41:21,130 Actually, it is an application that pops up sometimes. 586 00:41:21,130 --> 00:41:23,870 That somehow, you get side information about some 587 00:41:23,870 --> 00:41:26,230 positions in the code word indeed being correct, and 588 00:41:26,230 --> 00:41:27,560 others not. 589 00:41:27,560 --> 00:41:30,340 And others you don't know about. 590 00:41:30,340 --> 00:41:35,755 So that's the information set, with respect to our MDS code. 591 00:41:35,755 --> 00:41:38,300 592 00:41:38,300 --> 00:41:45,180 So with respect to our MDS code, a corollary of the thing 593 00:41:45,180 --> 00:42:14,410 that involved any k positions in an MDS code, 594 00:42:14,410 --> 00:42:15,660 an information set. 595 00:42:15,660 --> 00:42:22,170 596 00:42:22,170 --> 00:42:25,380 So any k positions on information set. 597 00:42:25,380 --> 00:42:26,910 It's a really strong property. 598 00:42:26,910 --> 00:42:30,000 Really strong combinatorial property. 599 00:42:30,000 --> 00:42:33,420 OK, so far, so good. 600 00:42:33,420 --> 00:42:42,470 601 00:42:42,470 --> 00:42:46,970 This is so strong, this property, that we can say 602 00:42:46,970 --> 00:42:50,700 something about these codes even without even knowing if 603 00:42:50,700 --> 00:42:51,950 they exist. 604 00:42:51,950 --> 00:42:54,210 605 00:42:54,210 --> 00:42:57,320 So, so far, we have talked about these codes as if we 606 00:42:57,320 --> 00:42:59,080 knew they existed. 607 00:42:59,080 --> 00:43:01,890 Well, it's not entirely trivial, since we know those 608 00:43:01,890 --> 00:43:03,400 guys here exist. 609 00:43:03,400 --> 00:43:05,710 So it's not entirely empty, we're not out in 610 00:43:05,710 --> 00:43:08,250 cuckoo space, here. 611 00:43:08,250 --> 00:43:12,750 But do any other one exist, except for those? 612 00:43:12,750 --> 00:43:15,480 That's the question. 613 00:43:15,480 --> 00:43:16,470 We don't know that yet. 614 00:43:16,470 --> 00:43:19,920 We will show in a little while that they do, but we don't 615 00:43:19,920 --> 00:43:21,170 know that yet. 616 00:43:21,170 --> 00:43:23,960 617 00:43:23,960 --> 00:43:26,160 But the interesting part is that we can derive properties 618 00:43:26,160 --> 00:43:29,870 of those codes without even knowing they exist. 619 00:43:29,870 --> 00:43:31,470 And how do we do that? 620 00:43:31,470 --> 00:43:40,470 For example, we want to derive the following property, how 621 00:43:40,470 --> 00:44:00,620 many words of weight d exists in linear MDS code? 622 00:44:00,620 --> 00:44:01,960 One could ask that, right? 623 00:44:01,960 --> 00:44:05,280 If they exist, they're nice, and if they exist, we also 624 00:44:05,280 --> 00:44:09,820 want to know how many words do exist at minimum distance. 625 00:44:09,820 --> 00:44:12,460 Because that translates, again, directly into union 626 00:44:12,460 --> 00:44:17,060 bound arguments later on, and probability of error. 627 00:44:17,060 --> 00:44:18,210 So that's a good question. 628 00:44:18,210 --> 00:44:23,300 How many words of weight d exist in a MDS code? 629 00:44:23,300 --> 00:44:28,630 Let's call this n d, and we want to know how many. 630 00:44:28,630 --> 00:44:34,550 631 00:44:34,550 --> 00:44:37,000 So I'll let you think about this for a sec while I erase 632 00:44:37,000 --> 00:44:40,130 the board, and then somebody will tell me the answer. 633 00:44:40,130 --> 00:44:52,058 634 00:44:52,058 --> 00:44:53,984 So how can we think about this? 635 00:44:53,984 --> 00:45:00,290 636 00:45:00,290 --> 00:45:04,030 Let's try to do a similar argument as this one. 637 00:45:04,030 --> 00:45:18,780 Let's look at a single word, and let's assume that d 638 00:45:18,780 --> 00:45:23,600 positions, we ask the questions does there exist a 639 00:45:23,600 --> 00:45:27,730 code word within the first d positions? 640 00:45:27,730 --> 00:45:29,970 It is equivalent to the question, does there exist a 641 00:45:29,970 --> 00:45:35,410 code word that covers exactly all d positions? 642 00:45:35,410 --> 00:45:37,140 Any set of d positions. 643 00:45:37,140 --> 00:45:43,522 644 00:45:43,522 --> 00:45:44,480 AUDIENCE: 0 everywhere else? 645 00:45:44,480 --> 00:45:47,400 PROFESSOR: And 0 everywhere else. 646 00:45:47,400 --> 00:45:48,660 Why is that nice? 647 00:45:48,660 --> 00:45:51,880 If you could prove that, that there exists a word for all d 648 00:45:51,880 --> 00:45:57,800 positions, because then, we pretty much know what happens. 649 00:45:57,800 --> 00:46:02,837 Then we know that, well, if this is true, then there are n 650 00:46:02,837 --> 00:46:04,700 choose d ways to choose those d positions. 651 00:46:04,700 --> 00:46:08,210 652 00:46:08,210 --> 00:46:11,810 And then within those d positions, and since it's a 653 00:46:11,810 --> 00:46:16,690 linear code, we can multiply with the q minus 1 on the 654 00:46:16,690 --> 00:46:17,940 repeated element. 655 00:46:17,940 --> 00:46:20,200 656 00:46:20,200 --> 00:46:23,540 So if we can choose our d positions arbitrarily, then 657 00:46:23,540 --> 00:46:28,540 this is the number over words at distance d. 658 00:46:28,540 --> 00:46:31,540 So let's look at a word, and let's, without loss of 659 00:46:31,540 --> 00:46:33,515 generality, assume it's a first d positions. 660 00:46:33,515 --> 00:46:36,960 661 00:46:36,960 --> 00:46:39,770 So the first d positions. 662 00:46:39,770 --> 00:46:43,710 So in particular, these would be the first d minus 1 663 00:46:43,710 --> 00:46:58,200 positions, which would mean that this have length k. 664 00:46:58,200 --> 00:47:04,450 So if we have an MDS code, this is an information set. 665 00:47:04,450 --> 00:47:08,020 So if this is an information set, then we can fill up this 666 00:47:08,020 --> 00:47:11,880 thing with just about anything we want. 667 00:47:11,880 --> 00:47:18,330 So we choose this information set to be equal to 1. 668 00:47:18,330 --> 00:47:21,360 This is how we choose this information set, and by the 669 00:47:21,360 --> 00:47:24,840 property of MDS code, we are guaranteed that there exists a 670 00:47:24,840 --> 00:47:27,576 code word which is this part in the information set. 671 00:47:27,576 --> 00:47:30,590 672 00:47:30,590 --> 00:47:34,070 But we also are guaranteed it's a weight d word, right? 673 00:47:34,070 --> 00:47:37,240 The minimum distance is d, that means all of these m 674 00:47:37,240 --> 00:47:44,680 entries here, they must all be non-zero, in this part. 675 00:47:44,680 --> 00:47:47,370 Otherwise, it wouldn't have weight d. 676 00:47:47,370 --> 00:47:48,330 OK? 677 00:47:48,330 --> 00:47:49,435 And there we have it. 678 00:47:49,435 --> 00:47:51,180 That was all we needed to show. 679 00:47:51,180 --> 00:47:51,910 Right? 680 00:47:51,910 --> 00:47:56,160 Because now we have shown that there exists a word of weight 681 00:47:56,160 --> 00:47:59,840 d in the first d positions. 682 00:47:59,840 --> 00:48:01,090 Is that clear? 683 00:48:01,090 --> 00:48:14,650 684 00:48:14,650 --> 00:48:15,250 Let's try again. 685 00:48:15,250 --> 00:48:16,270 I will say the same words. 686 00:48:16,270 --> 00:48:17,575 Maybe it becomes clearer by that. 687 00:48:17,575 --> 00:48:22,710 688 00:48:22,710 --> 00:48:24,130 Let's look at a code word. 689 00:48:24,130 --> 00:48:27,790 This is a generic code word, at first, and we want to 690 00:48:27,790 --> 00:48:31,410 answer the question, does there exist a code word 691 00:48:31,410 --> 00:48:34,240 within, which has support only in the first d positions? 692 00:48:34,240 --> 00:48:36,890 693 00:48:36,890 --> 00:48:42,026 So does there exist a code word which is non-zero here, 694 00:48:42,026 --> 00:48:46,626 up to d, and which is zero everywhere else? 695 00:48:46,626 --> 00:48:49,225 That's the question we want to answer. 696 00:48:49,225 --> 00:48:55,575 697 00:48:55,575 --> 00:48:55,817 OK. 698 00:48:55,817 --> 00:48:57,930 Now here's what we do. 699 00:48:57,930 --> 00:48:59,700 We look at this road and say, you know what? 700 00:48:59,700 --> 00:49:04,890 Let's look at the last k positions, which have an 701 00:49:04,890 --> 00:49:08,380 overlap of 1 with this word here, 702 00:49:08,380 --> 00:49:10,130 because it's an MDS property. 703 00:49:10,130 --> 00:49:14,160 So we have this relation between n, k, and d. 704 00:49:14,160 --> 00:49:17,760 And since any k positions in the word are in information 705 00:49:17,760 --> 00:49:24,210 set, so we can choose whatever we want in this part, and this 706 00:49:24,210 --> 00:49:25,460 is what we choose. 707 00:49:25,460 --> 00:49:28,030 708 00:49:28,030 --> 00:49:33,050 By the property of MDS codes, this corollary, we are 709 00:49:33,050 --> 00:49:37,600 guaranteed there exists the code word which in the second 710 00:49:37,600 --> 00:49:41,180 half of the code word looks like this. 711 00:49:41,180 --> 00:49:44,570 And in the first half, it looks different. 712 00:49:44,570 --> 00:49:48,110 There's something else here, and I say, well it cannot have 713 00:49:48,110 --> 00:49:51,640 any 0 in here, because then it would have 714 00:49:51,640 --> 00:49:53,610 weighed less that d. 715 00:49:53,610 --> 00:49:56,330 So it has non-zeros here. 716 00:49:56,330 --> 00:50:00,780 So indeed, we have shown the existence of a code word which 717 00:50:00,780 --> 00:50:05,630 has non-zeroes in the first d positions. 718 00:50:05,630 --> 00:50:07,870 Very simple. 719 00:50:07,870 --> 00:50:10,320 And that was without loss of generality. 720 00:50:10,320 --> 00:50:14,310 You could make the same argument for any d positions. 721 00:50:14,310 --> 00:50:15,250 What have we shown? 722 00:50:15,250 --> 00:50:19,480 We have shown that indeed, we can choose any d positions in 723 00:50:19,480 --> 00:50:22,325 the code to support the minimum weight code 724 00:50:22,325 --> 00:50:25,540 word of weight d. 725 00:50:25,540 --> 00:50:29,440 This is how many ways we can choose this, then we have to 726 00:50:29,440 --> 00:50:32,840 multiply it with q minus 1. 727 00:50:32,840 --> 00:50:34,970 All non-zero field elements. 728 00:50:34,970 --> 00:50:37,550 The reason is, we might have chosen this, or we might have 729 00:50:37,550 --> 00:50:40,920 chosen omega or omega squared here, or just the multiples, 730 00:50:40,920 --> 00:50:42,812 the scalar multiples of it. 731 00:50:42,812 --> 00:50:47,320 732 00:50:47,320 --> 00:50:48,740 Interesting, right? 733 00:50:48,740 --> 00:50:54,230 This property, that any k positions on information set 734 00:50:54,230 --> 00:50:56,520 is really strong enough to prove the -- 735 00:50:56,520 --> 00:50:58,570 actually, it's strong enough to prove the entire weight 736 00:50:58,570 --> 00:51:00,980 distribution of an MDS code. 737 00:51:00,980 --> 00:51:03,175 AUDIENCE: [INAUDIBLE] 738 00:51:03,175 --> 00:51:04,425 [UNINTELLIGIBLE]? 739 00:51:04,425 --> 00:51:06,870 740 00:51:06,870 --> 00:51:10,030 PROFESSOR: No, no, no, why no, no, no, no. 741 00:51:10,030 --> 00:51:13,660 742 00:51:13,660 --> 00:51:16,190 So then you would get too much. 743 00:51:16,190 --> 00:51:21,000 If you write q minus 1 to the d, then you would want to 744 00:51:21,000 --> 00:51:22,920 multiply each position with a different value. 745 00:51:22,920 --> 00:51:26,590 746 00:51:26,590 --> 00:51:31,440 That would imply that there's more than one code word in the 747 00:51:31,440 --> 00:51:33,380 first d positions. 748 00:51:33,380 --> 00:51:36,790 More than one code word so that they are not scalar 749 00:51:36,790 --> 00:51:38,900 multiples of each other. 750 00:51:38,900 --> 00:51:42,060 If that would be true, then you could find a linear 751 00:51:42,060 --> 00:51:45,940 combination which is still 0 in this part, but has 752 00:51:45,940 --> 00:51:48,470 additional 0 here somewhere. 753 00:51:48,470 --> 00:51:50,470 But if that is true, then we don't have 754 00:51:50,470 --> 00:51:51,730 enough distance anymore. 755 00:51:51,730 --> 00:51:52,980 Then it's not an MDS code. 756 00:51:52,980 --> 00:51:56,630 757 00:51:56,630 --> 00:51:59,850 All right, so it's indeed q minus 1. 758 00:51:59,850 --> 00:52:02,870 Within each d positions, we have one dimensional space. 759 00:52:02,870 --> 00:52:06,184 It's just one dimension. 760 00:52:06,184 --> 00:52:07,434 AUDIENCE: [INAUDIBLE] 761 00:52:07,434 --> 00:52:09,940 762 00:52:09,940 --> 00:52:11,250 far off minimum weight code words? 763 00:52:11,250 --> 00:52:14,230 764 00:52:14,230 --> 00:52:15,690 PROFESSOR: Yeah, yeah, definitely. 765 00:52:15,690 --> 00:52:18,440 766 00:52:18,440 --> 00:52:20,665 Any other code must have less, so it would have less. 767 00:52:20,665 --> 00:52:23,950 768 00:52:23,950 --> 00:52:25,310 But every other code would have a 769 00:52:25,310 --> 00:52:26,560 smaller minimum distance. 770 00:52:26,560 --> 00:52:31,803 771 00:52:31,803 --> 00:52:34,629 AUDIENCE: [INAUDIBLE]. 772 00:52:34,629 --> 00:52:40,172 Suppose we let the last k minus 1 position zero, and the 773 00:52:40,172 --> 00:52:43,670 one before that, [UNINTELLIGIBLE PHRASE]. 774 00:52:43,670 --> 00:52:47,540 And you said that we can do it for any of [UNINTELLIGIBLE] 775 00:52:47,540 --> 00:52:48,490 total field? 776 00:52:48,490 --> 00:52:49,786 PROFESSOR: Sure. 777 00:52:49,786 --> 00:52:55,770 AUDIENCE: Since it's a linear code, some of those code words 778 00:52:55,770 --> 00:52:59,220 should be in the linear code, right? 779 00:52:59,220 --> 00:53:01,240 PROFESSOR: Sure. 780 00:53:01,240 --> 00:53:05,030 AUDIENCE: So because it's a field, also we are going to 781 00:53:05,030 --> 00:53:06,700 [INAUDIBLE] 782 00:53:06,700 --> 00:53:12,830 there exists an inverse [UNINTELLIGIBLE]? 783 00:53:12,830 --> 00:53:14,230 PROFESSOR: Absolutely. 784 00:53:14,230 --> 00:53:20,740 AUDIENCE: So if we add those two code words, we should have 785 00:53:20,740 --> 00:53:22,510 all zero, [UNINTELLIGIBLE] 786 00:53:22,510 --> 00:53:27,590 k minus 1, and have inverse at the one before. 787 00:53:27,590 --> 00:53:34,910 We get that code word which has a minimum weight, which is 788 00:53:34,910 --> 00:53:37,862 less that the one we have here? 789 00:53:37,862 --> 00:53:39,030 PROFESSOR: Good question. 790 00:53:39,030 --> 00:53:40,670 There is a trick out. 791 00:53:40,670 --> 00:53:43,640 There's a way out of this. 792 00:53:43,640 --> 00:53:45,610 Great argument. 793 00:53:45,610 --> 00:53:46,895 But there's a trick out. 794 00:53:46,895 --> 00:53:47,990 AUDIENCE: There's gotta be an upper bound 795 00:53:47,990 --> 00:53:49,390 PROFESSOR: No, no, there is a trick, there 796 00:53:49,390 --> 00:53:51,220 is a way out here. 797 00:53:51,220 --> 00:53:55,580 Namely, so let's put it like this. 798 00:53:55,580 --> 00:53:59,880 Right here we put in a 1, just for simplicity, let's assume 799 00:53:59,880 --> 00:54:01,550 all the other positions are also 1. 800 00:54:01,550 --> 00:54:04,160 801 00:54:04,160 --> 00:54:07,540 And then you say, this would be another code word, which 802 00:54:07,540 --> 00:54:11,300 has here an omega. 803 00:54:11,300 --> 00:54:12,300 I say, you know what? 804 00:54:12,300 --> 00:54:12,850 What's going to happen? 805 00:54:12,850 --> 00:54:15,330 All the other positions are going to be omega 2. 806 00:54:15,330 --> 00:54:18,110 807 00:54:18,110 --> 00:54:20,790 There's no way to combine these two guys to get an 808 00:54:20,790 --> 00:54:23,650 additional 0, unless you get all 0's. 809 00:54:23,650 --> 00:54:24,800 Unless you get to 0. 810 00:54:24,800 --> 00:54:26,250 That's what I said, it's a one-dimensional 811 00:54:26,250 --> 00:54:28,030 space in these positions. 812 00:54:28,030 --> 00:54:29,280 When it's a soft code . 813 00:54:29,280 --> 00:54:31,889 814 00:54:31,889 --> 00:54:33,139 AUDIENCE: [UNINTELLIGIBLE PHRASE] 815 00:54:33,139 --> 00:54:39,710 816 00:54:39,710 --> 00:54:43,870 PROFESSOR: It tells you that if you write down the minimum 817 00:54:43,870 --> 00:54:49,520 weight code words in the q minus 1 times d matrix, that 818 00:54:49,520 --> 00:54:53,620 is, you have a Latin square, basically. 819 00:54:53,620 --> 00:54:54,790 That's what it tells you. 820 00:54:54,790 --> 00:54:58,460 There's in no position, if you have anywhere in here an 821 00:54:58,460 --> 00:55:03,390 element alpha and element omega, the same omega pops up 822 00:55:03,390 --> 00:55:04,640 nowhere else. 823 00:55:04,640 --> 00:55:07,770 824 00:55:07,770 --> 00:55:12,410 There's ramifications of MDS codes in combinatorics left 825 00:55:12,410 --> 00:55:14,060 and right, so this would be a Latin square. 826 00:55:14,060 --> 00:55:18,390 827 00:55:18,390 --> 00:55:21,320 You know, you can learn a lot a lot about MDS codes if you 828 00:55:21,320 --> 00:55:23,750 think a little bit about that, and about combinatorics 829 00:55:23,750 --> 00:55:25,220 altogether. 830 00:55:25,220 --> 00:55:26,760 OK, where was I? 831 00:55:26,760 --> 00:55:29,608 So we know that's fun. 832 00:55:29,608 --> 00:55:38,250 And actually, in the homework, you going to do n d plus 1. 833 00:55:38,250 --> 00:55:40,900 834 00:55:40,900 --> 00:55:42,900 So the next one. 835 00:55:42,900 --> 00:55:49,100 But once you do n d plus 1, do all of them. 836 00:55:49,100 --> 00:55:52,370 In a sense it's just inclusion and exclusion from then on. 837 00:55:52,370 --> 00:55:54,430 The first one is sort of the toughest one, the rest is 838 00:55:54,430 --> 00:55:56,140 inclusion exclusion. 839 00:55:56,140 --> 00:56:00,610 And just for the heck of it, when you go home and do the 840 00:56:00,610 --> 00:56:02,950 homework, write them all out. 841 00:56:02,950 --> 00:56:06,910 It's a pretty looking formula, in the end. 842 00:56:06,910 --> 00:56:10,030 OK, so far, so good. 843 00:56:10,030 --> 00:56:12,955 So we have still talked about MDS codes without knowing if 844 00:56:12,955 --> 00:56:15,840 they exist. 845 00:56:15,840 --> 00:56:17,430 Except for the trivial ones here. 846 00:56:17,430 --> 00:56:21,250 847 00:56:21,250 --> 00:56:31,310 And the existence of MDS codes is actually not known for 848 00:56:31,310 --> 00:56:33,830 which parameters they exist. 849 00:56:33,830 --> 00:56:39,270 So I give you a research problem. 850 00:56:39,270 --> 00:56:56,580 The research problem is the main conjecture on MDS codes. 851 00:56:56,580 --> 00:57:00,320 852 00:57:00,320 --> 00:57:01,520 And it's always sort of tricky. 853 00:57:01,520 --> 00:57:04,330 When a research problem has a name, then 854 00:57:04,330 --> 00:57:07,590 that signifies danger. 855 00:57:07,590 --> 00:57:12,340 Then it means that it's not trivial. 856 00:57:12,340 --> 00:57:20,790 The question is, for which k d and q, for which sets of 857 00:57:20,790 --> 00:57:28,000 parameters n k d q, do MDS codes exist? 858 00:57:28,000 --> 00:57:44,250 And the conjecture this is that n k q, because in MDS 859 00:57:44,250 --> 00:57:46,930 code we can actually get rid of the d here. 860 00:57:46,930 --> 00:58:03,022 e, the longest length of an MDS code. 861 00:58:03,022 --> 00:58:17,150 The longest length of an MDS code, I mentioned k over an 862 00:58:17,150 --> 00:58:26,582 alphabet of size q. 863 00:58:26,582 --> 00:58:45,530 The conjecture is that n, k, d is less than q plus 1 for k at 864 00:58:45,530 --> 00:58:48,360 least 2, unless -- 865 00:58:48,360 --> 00:58:50,310 I always have to look that up -- 866 00:58:50,310 --> 00:58:53,394 867 00:58:53,394 --> 00:58:54,644 I think 2q. 868 00:58:54,644 --> 00:58:58,010 869 00:58:58,010 --> 00:59:07,480 And k plus 1 for k greater than q. 870 00:59:07,480 --> 00:59:11,350 871 00:59:11,350 --> 00:59:22,680 We talk about it in a second, except that n, there's 872 00:59:22,680 --> 00:59:29,380 a 3, 2 to the s. 873 00:59:29,380 --> 00:59:33,830 So if the alphabet is a power of 2, alphabet size an 874 00:59:33,830 --> 00:59:35,185 extension field of 2, basically. 875 00:59:35,185 --> 00:59:38,460 876 00:59:38,460 --> 00:59:49,640 q plus 2 and q minus 1 q to the s. 877 00:59:49,640 --> 00:59:52,740 q plus 2. 878 00:59:52,740 --> 00:59:55,680 OK, so this is the main conjecture on MDS codes. 879 00:59:55,680 --> 00:59:58,700 880 00:59:58,700 --> 01:00:03,970 Basically, it says that the length can essentially be as 881 01:00:03,970 --> 01:00:08,212 large as the alphabet size, but not larger. 882 01:00:08,212 --> 01:00:10,190 AUDIENCE: [INAUDIBLE] 883 01:00:10,190 --> 01:00:12,410 PROFESSOR: This q, yeah? 884 01:00:12,410 --> 01:00:13,490 Oh, yeah, n k q, sorry. 885 01:00:13,490 --> 01:00:15,650 It doesn't make sense otherwise. 886 01:00:15,650 --> 01:00:18,780 887 01:00:18,780 --> 01:00:22,210 So the lengths can be in the same order of magnitude as the 888 01:00:22,210 --> 01:00:23,140 alphabet size. 889 01:00:23,140 --> 01:00:27,700 That gives enough room, enough choices, to fill up this 890 01:00:27,700 --> 01:00:31,390 matrix with the information set, with the MDS property on 891 01:00:31,390 --> 01:00:34,240 the information sets. 892 01:00:34,240 --> 01:00:39,390 This is the parity-check code, this row is just taken out as 893 01:00:39,390 --> 01:00:40,640 a trivial code. 894 01:00:40,640 --> 01:00:43,970 895 01:00:43,970 --> 01:00:49,750 And then, the demon of mathematics conspired that 896 01:00:49,750 --> 01:00:51,940 this would also be true. 897 01:00:51,940 --> 01:00:55,810 So if you have an extension field of 2, and you want to 898 01:00:55,810 --> 01:01:00,720 give it a dimension three, MDS code, they exist for q plus 2. 899 01:01:00,720 --> 01:01:05,130 900 01:01:05,130 --> 01:01:05,241 Right. 901 01:01:05,241 --> 01:01:08,130 There are, of course, reasons for this, but they go pretty 902 01:01:08,130 --> 01:01:11,400 deep, why they exist for those parameters, and this is just 903 01:01:11,400 --> 01:01:14,150 mysterious. 904 01:01:14,150 --> 01:01:15,830 One can give reasons, so on another hand, 905 01:01:15,830 --> 01:01:17,080 it's just so, right. 906 01:01:17,080 --> 01:01:20,610 907 01:01:20,610 --> 01:01:22,870 There are exceptionally enough that they have names. 908 01:01:22,870 --> 01:01:29,370 The first one is the Hexacode, it's something with a 909 01:01:29,370 --> 01:01:42,230 generator matrix, and this goes over F4. 910 01:01:42,230 --> 01:01:49,110 So that's an MDS code of length six, so this is a n6, 911 01:01:49,110 --> 01:01:54,380 3, 4, MDS code over alphabet size 4. 912 01:01:54,380 --> 01:01:58,650 That's the first one, in that sequence here. 913 01:01:58,650 --> 01:02:00,790 Anyway. 914 01:02:00,790 --> 01:02:02,180 Otherwise, we have this conjecture. 915 01:02:02,180 --> 01:02:05,850 If you solve this, you are going to be rich and famous, 916 01:02:05,850 --> 01:02:10,660 you're going to live in Hollywood, and 917 01:02:10,660 --> 01:02:11,760 maybe, maybe not. 918 01:02:11,760 --> 01:02:15,530 But you're going to be probably not rich, you're 919 01:02:15,530 --> 01:02:17,730 going to be famous about a couple of hundred people who 920 01:02:17,730 --> 01:02:23,160 know about this MDS conjecture, but very smart 921 01:02:23,160 --> 01:02:25,475 people have been looking for this for a long, long time. 922 01:02:25,475 --> 01:02:26,725 OK. 923 01:02:26,725 --> 01:02:29,356 924 01:02:29,356 --> 01:02:31,260 All right, 20 minutes left. 925 01:02:31,260 --> 01:02:34,840 So it's better we define, we make sure those codes exist. 926 01:02:34,840 --> 01:02:36,964 Do we have any question about this MDS conjecture? 927 01:02:36,964 --> 01:02:46,220 928 01:02:46,220 --> 01:02:50,070 OK, last 20 minutes, let's at least make sure 929 01:02:50,070 --> 01:02:51,410 those things exist. 930 01:02:51,410 --> 01:02:52,660 Reed-Solomon codes. 931 01:02:52,660 --> 01:03:00,550 932 01:03:00,550 --> 01:03:07,210 So Reed-Solomon codes cover this case. 933 01:03:07,210 --> 01:03:09,970 They are examples of codes which lie, which 934 01:03:09,970 --> 01:03:12,690 satisfy this equality. 935 01:03:12,690 --> 01:03:14,840 OK, so how do we define Reed-Solomon codes? 936 01:03:14,840 --> 01:03:20,780 937 01:03:20,780 --> 01:03:24,590 Now, just in a true mathematician spirit, write 938 01:03:24,590 --> 01:03:27,010 down consider the following. 939 01:03:27,010 --> 01:03:31,593 Consider the following code. 940 01:03:31,593 --> 01:03:37,510 941 01:03:37,510 --> 01:03:38,760 See? 942 01:03:38,760 --> 01:03:49,100 943 01:03:49,100 --> 01:04:01,135 Beta 0 beta q minus 1. 944 01:04:01,135 --> 01:04:09,808 945 01:04:09,808 --> 01:04:14,520 The beta i are the distinct field elements, the distinct 946 01:04:14,520 --> 01:04:17,310 elements in the finite field. 947 01:04:17,310 --> 01:04:21,075 f is a polynomial. 948 01:04:21,075 --> 01:04:24,590 949 01:04:24,590 --> 01:04:33,810 f is a polynomial, and the degree is less than k. 950 01:04:33,810 --> 01:04:34,950 OK, good. 951 01:04:34,950 --> 01:04:37,280 So we have defined a code. 952 01:04:37,280 --> 01:04:40,540 So what that means is we start from polynomials. 953 01:04:40,540 --> 01:04:46,496 The set of all polynomials of degree at most k. 954 01:04:46,496 --> 01:04:48,160 So what do we know about that set? 955 01:04:48,160 --> 01:04:50,520 It's a vector space, right? 956 01:04:50,520 --> 01:04:53,000 The set of all polynomials of degree at most k. 957 01:04:53,000 --> 01:04:55,640 We can add them to get a polynomial of degree at most 958 01:04:55,640 --> 01:04:58,320 k, we can multiply them with a scalar to get a polynomial 959 01:04:58,320 --> 01:05:00,390 with degree at most k. 960 01:05:00,390 --> 01:05:03,790 It's a vector space. 961 01:05:03,790 --> 01:05:10,820 So we take this vector space and evaluate for any element 962 01:05:10,820 --> 01:05:12,090 in that vector space. 963 01:05:12,090 --> 01:05:21,740 This element in all non-zero elements of the field and we 964 01:05:21,740 --> 01:05:22,270 get a code. 965 01:05:22,270 --> 01:05:31,160 We get a set of vectors, so we get a set of 966 01:05:31,160 --> 01:05:33,830 vectors, and that -- 967 01:05:33,830 --> 01:05:35,830 AUDIENCE: [INAUDIBLE] 968 01:05:35,830 --> 01:05:37,180 PROFESSOR: Yeah, I took all elements. 969 01:05:37,180 --> 01:05:38,310 Why not? 970 01:05:38,310 --> 01:05:40,260 Why not all elements? 971 01:05:40,260 --> 01:05:43,910 Strictly speaking, I should have taken one more in order 972 01:05:43,910 --> 01:05:46,070 to get the one here. 973 01:05:46,070 --> 01:05:47,800 We can talk about that in a sec. 974 01:05:47,800 --> 01:05:49,420 But this one more element would be -- 975 01:05:49,420 --> 01:05:53,980 976 01:05:53,980 --> 01:05:55,170 so it's a code. 977 01:05:55,170 --> 01:05:56,130 First of all, it's a code. 978 01:05:56,130 --> 01:05:56,370 Right? 979 01:05:56,370 --> 01:05:58,410 We all see it's a code. 980 01:05:58,410 --> 01:06:01,200 And once you see it's a code, we ask, what are the 981 01:06:01,200 --> 01:06:02,450 parameters? 982 01:06:02,450 --> 01:06:09,310 983 01:06:09,310 --> 01:06:10,560 The parameters. 984 01:06:10,560 --> 01:06:16,640 985 01:06:16,640 --> 01:06:22,440 So length, length is the easy one. 986 01:06:22,440 --> 01:06:25,450 Well, it's q. 987 01:06:25,450 --> 01:06:26,700 What is dimension? 988 01:06:26,700 --> 01:06:29,810 989 01:06:29,810 --> 01:06:33,660 Dimension of C. What's the dimension? 990 01:06:33,660 --> 01:06:38,810 991 01:06:38,810 --> 01:06:40,480 It's a little bit tricky, that question. 992 01:06:40,480 --> 01:06:45,020 993 01:06:45,020 --> 01:06:48,550 I actually, at Illinois, we have to 994 01:06:48,550 --> 01:06:51,560 take a class on teaching. 995 01:06:51,560 --> 01:06:53,840 How to become an effective teacher. 996 01:06:53,840 --> 01:06:58,070 And one of the things they told us is that if you ask a 997 01:06:58,070 --> 01:07:03,610 question, you have to wait for 12 seconds to get an answer. 998 01:07:03,610 --> 01:07:05,760 So what's the dimension? 999 01:07:05,760 --> 01:07:07,010 There you go. 1000 01:07:07,010 --> 01:07:12,560 1001 01:07:12,560 --> 01:07:16,700 This mapping, this mapping from a vector space to a 1002 01:07:16,700 --> 01:07:17,950 vector space. 1003 01:07:17,950 --> 01:07:20,170 1004 01:07:20,170 --> 01:07:22,070 This mapping, also called evaluation 1005 01:07:22,070 --> 01:07:25,300 map, is a linear map. 1006 01:07:25,300 --> 01:07:32,810 It's a linear map, meaning that, well, let's start 1007 01:07:32,810 --> 01:07:34,140 differently. 1008 01:07:34,140 --> 01:07:35,750 Let's start differently. 1009 01:07:35,750 --> 01:07:38,900 Do any two polynomials map to the same code word? 1010 01:07:38,900 --> 01:07:42,430 1011 01:07:42,430 --> 01:07:43,140 That you know. 1012 01:07:43,140 --> 01:07:46,760 That you cannot. 1013 01:07:46,760 --> 01:07:54,860 Are there any two codes, two polynomials, so are there f of 1014 01:07:54,860 --> 01:08:10,740 x, g of x, such that f of beta 0, so that they coincide in 1015 01:08:10,740 --> 01:08:13,520 all positions? 1016 01:08:13,520 --> 01:08:15,370 No, then they would be the same, right? 1017 01:08:15,370 --> 01:08:18,700 And the reason is because if there would be something like 1018 01:08:18,700 --> 01:08:25,880 that, then you could just look at h is f of x minus g of x, 1019 01:08:25,880 --> 01:08:29,130 which is just another polynomial of degree k. 1020 01:08:29,130 --> 01:08:33,100 And this would have to vanish in all positions. 1021 01:08:33,100 --> 01:08:40,950 If k is less than q, it could not possibly vanish in all 1022 01:08:40,950 --> 01:08:44,399 positions, because then the polynomial of degree k would 1023 01:08:44,399 --> 01:08:47,200 vanish in more than k positions. 1024 01:08:47,200 --> 01:08:48,720 Fundamental theorem of algebra. 1025 01:08:48,720 --> 01:08:50,430 The very beginning. 1026 01:08:50,430 --> 01:08:54,859 So the dimension of C is indeed k, the same as the 1027 01:08:54,859 --> 01:08:58,740 dimension of this vector space. 1028 01:08:58,740 --> 01:09:00,850 The dimension of the vector space of polynomials of the 1029 01:09:00,850 --> 01:09:03,000 degree k minus 1. 1030 01:09:03,000 --> 01:09:05,950 And the distance, if k is less than q, the 1031 01:09:05,950 --> 01:09:11,072 distance is equal to q. 1032 01:09:11,072 --> 01:09:18,920 The distance, what is it? 1033 01:09:18,920 --> 01:09:21,770 Same argument, roughly the same argument. 1034 01:09:21,770 --> 01:09:27,120 I think that's a linear code, so if it's a linear code, the 1035 01:09:27,120 --> 01:09:29,490 minimum distance of the code is the same as the minimum 1036 01:09:29,490 --> 01:09:32,580 weight of a non-zero word. 1037 01:09:32,580 --> 01:09:34,580 What's the minimum weight of a non-zero word? 1038 01:09:34,580 --> 01:09:37,200 1039 01:09:37,200 --> 01:09:41,319 These are polynomials of degree k minus 1. 1040 01:09:41,319 --> 01:09:43,920 What's the minimum weight of a non-zero word? 1041 01:09:43,920 --> 01:09:46,910 Well, we start out with the weight 1, and whenever the 1042 01:09:46,910 --> 01:09:52,300 polynomial evaluates to 0, one of the weights drops out. 1043 01:09:52,300 --> 01:09:55,470 So I claim the minimum distance as the minimum 1044 01:09:55,470 --> 01:10:12,100 weight, weight of the non-zero word, and this is n minus, 1045 01:10:12,100 --> 01:10:16,090 well, if any of these polynomials vanishes in all, 1046 01:10:16,090 --> 01:10:19,990 it vanishes in at most, k minus 1 positions. 1047 01:10:19,990 --> 01:10:25,210 At most, k minus 1 of these vectors here, of these 1048 01:10:25,210 --> 01:10:27,220 entries, is equal to 0. 1049 01:10:27,220 --> 01:10:31,970 So it drops by, at most, k minus 1. 1050 01:10:31,970 --> 01:10:35,260 Drops by at most, k minus 1. 1051 01:10:35,260 --> 01:10:36,510 And there we have it. 1052 01:10:36,510 --> 01:10:39,454 1053 01:10:39,454 --> 01:10:40,380 There we have it. 1054 01:10:40,380 --> 01:10:43,114 There we have, oh, this is q. 1055 01:10:43,114 --> 01:10:46,150 1056 01:10:46,150 --> 01:10:46,960 There we have it. 1057 01:10:46,960 --> 01:10:53,280 There we have that the minimum distance of the code satisfies 1058 01:10:53,280 --> 01:10:54,598 this equation. 1059 01:10:54,598 --> 01:10:56,470 AUDIENCE: [INAUDIBLE] 1060 01:10:56,470 --> 01:10:57,640 PROFESSOR: What? 1061 01:10:57,640 --> 01:10:58,230 AUDIENCE: The dimension? 1062 01:10:58,230 --> 01:10:58,760 PROFESSOR: The dimension. 1063 01:10:58,760 --> 01:11:01,900 So, it's the same argument, roughly. 1064 01:11:01,900 --> 01:11:07,410 So I say, the dimension, so let's just say the size of the 1065 01:11:07,410 --> 01:11:11,710 code is q to the k. 1066 01:11:11,710 --> 01:11:15,730 When is the size of the q to the k, if no two elements in 1067 01:11:15,730 --> 01:11:18,540 the space evaluate to the same code word? 1068 01:11:18,540 --> 01:11:21,432 But if two of them would evaluate to the same code 1069 01:11:21,432 --> 01:11:26,930 word, then we would less size than the vector space had. 1070 01:11:26,930 --> 01:11:30,320 But if two of them evaluate to the same code word, that means 1071 01:11:30,320 --> 01:11:37,070 this is true for all four positions. 1072 01:11:37,070 --> 01:11:41,160 Then we could define a polynomial h of the 1073 01:11:41,160 --> 01:11:42,990 degree k minus 1. 1074 01:11:42,990 --> 01:11:49,645 which disappears in more than k minus 1 positions. 1075 01:11:49,645 --> 01:11:52,510 I mean, all positions. 1076 01:11:52,510 --> 01:11:56,230 Cannot be, hence the size of the code is q to the k, so 1077 01:11:56,230 --> 01:11:58,370 this is a linear map, dimension is k. 1078 01:11:58,370 --> 01:12:01,660 1079 01:12:01,660 --> 01:12:03,120 OK. 1080 01:12:03,120 --> 01:12:03,660 So cool. 1081 01:12:03,660 --> 01:12:04,600 So we have it, right? 1082 01:12:04,600 --> 01:12:06,050 We have our MDS codes. 1083 01:12:06,050 --> 01:12:07,195 They exist. 1084 01:12:07,195 --> 01:12:08,080 Here they are. 1085 01:12:08,080 --> 01:12:10,610 They are Reed-Solomon codes. 1086 01:12:10,610 --> 01:12:15,840 Not all MDS codes are Reed-Solomon codes, but the 1087 01:12:15,840 --> 01:12:19,220 ones we are interested in, they are. 1088 01:12:19,220 --> 01:12:21,400 AUDIENCE: [INAUDIBLE] 1089 01:12:21,400 --> 01:12:25,000 PROFESSOR: Well, the distance is at least this, but the MDS 1090 01:12:25,000 --> 01:12:30,270 bounds is at most this, so it's equal to this. 1091 01:12:30,270 --> 01:12:43,720 But the MDS bounds, so the MDS bound has this is. 1092 01:12:43,720 --> 01:12:46,760 So with that. 1093 01:12:46,760 --> 01:12:52,140 So it's indeed, they lie exactly bang on to this. 1094 01:12:52,140 --> 01:12:54,840 There are MDS codes, Reed-Solomon codes. 1095 01:12:54,840 --> 01:12:55,910 So that is good. 1096 01:12:55,910 --> 01:12:57,612 So we know what they are. 1097 01:12:57,612 --> 01:13:03,280 So incidentally, where do you think this one more point is 1098 01:13:03,280 --> 01:13:04,980 that you would evaluate our polynomials in? 1099 01:13:04,980 --> 01:13:11,120 1100 01:13:11,120 --> 01:13:12,400 You've heard about projective geometries? 1101 01:13:12,400 --> 01:13:16,840 1102 01:13:16,840 --> 01:13:20,400 There's one more point, it's infinity. 1103 01:13:20,400 --> 01:13:24,940 You have, basically, if you look at the numbers, in order 1104 01:13:24,940 --> 01:13:27,410 to close it up, you want to add infinity to that, too. 1105 01:13:27,410 --> 01:13:32,020 1106 01:13:32,020 --> 01:13:37,690 In order to get this one more, this one addition in length, 1107 01:13:37,690 --> 01:13:40,450 you want to evaluate this also at infinity. 1108 01:13:40,450 --> 01:13:43,440 You will have opportunity to do that in the homework. 1109 01:13:43,440 --> 01:13:45,420 I looked at the homework and I was pleased to see this 1110 01:13:45,420 --> 01:13:47,220 problem there. 1111 01:13:47,220 --> 01:13:50,190 I hope you will be pleased, too. 1112 01:13:50,190 --> 01:13:53,780 OK, all right. 1113 01:13:53,780 --> 01:13:55,030 Any questions about this? 1114 01:13:55,030 --> 01:13:58,616 1115 01:13:58,616 --> 01:14:00,200 Let's see what else I wanted to say. 1116 01:14:00,200 --> 01:14:10,300 1117 01:14:10,300 --> 01:14:14,240 Because it just gives me a few minutes to talk about a few 1118 01:14:14,240 --> 01:14:28,570 properties of Reed-Solomon codes, a few properties of 1119 01:14:28,570 --> 01:14:29,820 Reed-Solomon codes. 1120 01:14:29,820 --> 01:14:31,890 1121 01:14:31,890 --> 01:14:34,035 And what did I want to say there? 1122 01:14:34,035 --> 01:14:46,430 1123 01:14:46,430 --> 01:14:59,620 On nested codes, so an RS code with parameters n k, maybe we 1124 01:14:59,620 --> 01:15:00,945 define them [UNINTELLIGIBLE] like this. 1125 01:15:00,945 --> 01:15:09,140 1126 01:15:09,140 --> 01:15:21,965 q is properly contained, k minus 1, minus 1. 1127 01:15:21,965 --> 01:15:26,730 1128 01:15:26,730 --> 01:15:31,840 This is pretty straight from the definition of RS codes. 1129 01:15:31,840 --> 01:15:34,850 1130 01:15:34,850 --> 01:15:39,480 The set of polynomials of degree at most k minus 1 1131 01:15:39,480 --> 01:15:41,990 contains the set of polynomials of degree at 1132 01:15:41,990 --> 01:15:44,170 most k minus 1. 1133 01:15:44,170 --> 01:15:51,270 So they are nested codes, property one. 1134 01:15:51,270 --> 01:15:54,030 1135 01:15:54,030 --> 01:15:55,960 You will see this is important, that they are 1136 01:15:55,960 --> 01:15:59,170 nested codes, for various constructions where 1137 01:15:59,170 --> 01:16:01,160 Reed-Solomon codes take part in later on. 1138 01:16:01,160 --> 01:16:04,000 1139 01:16:04,000 --> 01:16:15,990 A punctured RS code is again an MDS code. 1140 01:16:15,990 --> 01:16:20,050 1141 01:16:20,050 --> 01:16:22,460 Why is that so? 1142 01:16:22,460 --> 01:16:23,710 Why is that so? 1143 01:16:23,710 --> 01:16:27,610 1144 01:16:27,610 --> 01:16:30,510 Well, you see it? 1145 01:16:30,510 --> 01:16:33,670 1146 01:16:33,670 --> 01:16:36,490 Say if you puncture a Reed-Solomon code. 1147 01:16:36,490 --> 01:16:41,640 That means we just choose to not evaluate our code in this, 1148 01:16:41,640 --> 01:16:43,450 this position. 1149 01:16:43,450 --> 01:16:44,610 And this field element. 1150 01:16:44,610 --> 01:16:45,960 Well, we just drop that coordinate. 1151 01:16:45,960 --> 01:16:48,830 1152 01:16:48,830 --> 01:16:51,810 Does anything change in the arguments we have made? 1153 01:16:51,810 --> 01:16:58,450 Well, the length is now 1 less, the dimension, well, the 1154 01:16:58,450 --> 01:17:03,630 dimension is still the same, as long as k is not larger 1155 01:17:03,630 --> 01:17:05,680 than the length of the code. 1156 01:17:05,680 --> 01:17:10,650 The distance, still the same as the length, the distance is 1157 01:17:10,650 --> 01:17:16,240 at least the length minus the number of 0's. 1158 01:17:16,240 --> 01:17:18,630 So that equation still holds. 1159 01:17:18,630 --> 01:17:20,390 Well, but that's all we needed. 1160 01:17:20,390 --> 01:17:22,360 Still MDS code. 1161 01:17:22,360 --> 01:17:25,460 So there was really no -- it was not important. 1162 01:17:25,460 --> 01:17:28,640 It was not important if you took all field elements, or a 1163 01:17:28,640 --> 01:17:31,870 subset of the field elements with MDS property. 1164 01:17:31,870 --> 01:17:34,230 That has nothing to do with it. 1165 01:17:34,230 --> 01:17:36,220 In particular, we often in the end, we often 1166 01:17:36,220 --> 01:17:40,240 will drop the 0 element. 1167 01:17:40,240 --> 01:17:44,590 We often choose not to evaluate these polynomials in 1168 01:17:44,590 --> 01:17:48,270 the 0 of the field. 1169 01:17:48,270 --> 01:17:51,775 A punctured Reed-Solomon code is an MDS code. 1170 01:17:51,775 --> 01:17:56,030 1171 01:17:56,030 --> 01:17:58,350 So what else did I want to say about this? 1172 01:17:58,350 --> 01:18:01,801 1173 01:18:01,801 --> 01:18:04,270 What else did I want to say about this? 1174 01:18:04,270 --> 01:18:09,610 1175 01:18:09,610 --> 01:18:10,860 A generator matrix. 1176 01:18:10,860 --> 01:18:17,180 1177 01:18:17,180 --> 01:18:19,060 How would a generator matrix look like? 1178 01:18:19,060 --> 01:18:28,389 1179 01:18:28,389 --> 01:18:30,160 Yeah, how would it look like? 1180 01:18:30,160 --> 01:18:32,850 1181 01:18:32,850 --> 01:18:34,260 Basically, we can come from here, right? 1182 01:18:34,260 --> 01:18:37,085 We can take the generators of that space. 1183 01:18:37,085 --> 01:18:40,360 1184 01:18:40,360 --> 01:18:43,820 So basically, we say that one -- 1185 01:18:43,820 --> 01:18:52,070 1186 01:18:52,070 --> 01:19:02,780 generate the set of polynomials, that vector space 1187 01:19:02,780 --> 01:19:04,800 of polynomials with -- 1188 01:19:04,800 --> 01:19:09,910 1189 01:19:09,910 --> 01:19:12,060 so this is the basis of that vector space. 1190 01:19:12,060 --> 01:19:14,610 1191 01:19:14,610 --> 01:19:20,840 So if we map that basis, then we get a basis of the image of 1192 01:19:20,840 --> 01:19:22,090 the mapping. 1193 01:19:22,090 --> 01:19:24,200 1194 01:19:24,200 --> 01:19:27,610 And the mapping of that basis would give this. 1195 01:19:27,610 --> 01:19:32,080 So we evaluate the function 1 in all field elements -- 1196 01:19:32,080 --> 01:19:35,960 gives us 1. 1197 01:19:35,960 --> 01:19:39,410 We evaluate the function x in all field elements. 1198 01:19:39,410 --> 01:19:41,610 This gives us the next generator of the 1199 01:19:41,610 --> 01:19:43,220 Reed-Solomon code. 1200 01:19:43,220 --> 01:19:54,880 Well, 0 gives 0, 1 gives, oh, let's write like this. 1201 01:19:54,880 --> 01:19:57,160 We evaluate it in all field elements. 1202 01:19:57,160 --> 01:20:01,740 1203 01:20:01,740 --> 01:20:04,910 These are all the field elements. 1204 01:20:04,910 --> 01:20:23,103 The next one, and this goes up to beta -- 1205 01:20:23,103 --> 01:20:28,630 1206 01:20:28,630 --> 01:20:30,745 OK, so this would be a generator matrix. 1207 01:20:30,745 --> 01:20:34,150 1208 01:20:34,150 --> 01:20:35,320 That's fine. 1209 01:20:35,320 --> 01:20:45,030 So now, in order to make things a bit more interesting, 1210 01:20:45,030 --> 01:20:46,645 do you have to stop five minutes early? 1211 01:20:46,645 --> 01:20:47,895 We just started five minutes late? 1212 01:20:47,895 --> 01:20:50,980 1213 01:20:50,980 --> 01:20:54,435 OK then, I think that's over. 1214 01:20:54,435 --> 01:20:56,550 I think it's over. 1215 01:20:56,550 --> 01:21:02,150 One more thing for you guys to think about until you reach 1216 01:21:02,150 --> 01:21:06,250 home, then the rest you do next time. 1217 01:21:06,250 --> 01:21:18,820 So let beta 0 be equal to 0 beta 1, or beta i equal to 1218 01:21:18,820 --> 01:21:21,745 omega i minus 1 where omega is primitive in the field. 1219 01:21:21,745 --> 01:21:32,690 1220 01:21:32,690 --> 01:21:41,885 Then we can write the matrix v of omega. 1221 01:21:41,885 --> 01:22:03,310 1222 01:22:03,310 --> 01:22:07,410 I tend to see that the first k columns, the first k rows of 1223 01:22:07,410 --> 01:22:09,860 this matrix would be a generator matrix of a 1224 01:22:09,860 --> 01:22:10,830 Reed-Solomon code. 1225 01:22:10,830 --> 01:22:13,330 Of course it's the same as [UNINTELLIGIBLE]. 1226 01:22:13,330 --> 01:22:20,390 If we now delete the first position, we erase the first, 1227 01:22:20,390 --> 01:22:23,580 we puncture the first position all out, and we look at the 1228 01:22:23,580 --> 01:22:25,500 rest of the matrix. 1229 01:22:25,500 --> 01:22:26,370 This factor of the matrix. 1230 01:22:26,370 --> 01:22:30,180 Does this remind anybody of anything? 1231 01:22:30,180 --> 01:22:32,740 It's a DFT, it's a Fourier transform. 1232 01:22:32,740 --> 01:22:35,980 And that's what we start with next time. 1233 01:22:35,980 --> 01:22:41,780 So think about why this is a Fourier transform. 1234 01:22:41,780 --> 01:22:45,890 And maybe that's a nice analogy. 1235 01:22:45,890 --> 01:22:47,790 So we get the distance. 1236 01:22:47,790 --> 01:22:51,560 The distance is at least something, which means it's 1237 01:22:51,560 --> 01:22:52,360 not impulsive. 1238 01:22:52,360 --> 01:22:54,000 It's not a single 1 somewhere. 1239 01:22:54,000 --> 01:22:56,400 The vector that we get is not impulsive. 1240 01:22:56,400 --> 01:22:59,680 Maybe it has something to do with the bandwidth constraint 1241 01:22:59,680 --> 01:23:02,380 and the frequency domain. 1242 01:23:02,380 --> 01:23:03,990 That's what you have to think about on the way 1243 01:23:03,990 --> 01:23:05,690 home, and that's it. 1244 01:23:05,690 --> 01:23:06,940 Thanks so much. 1245 01:23:06,940 --> 01:23:15,114
Linear Algebra: A Modern Introduction David Poole's innovative book prepares students to make the transition from the computational aspects of the course to the theoretical by emphasizing vectors and geometric intuition from the start. Designed for a one- or two-semester introductory course and written in simple, "mathematical English" the book presents interesting examples before abstraction. This immediately follows up theoretical discussion with further examples and a variety of applications drawn from a number of disciplines, which reinforces the practical utility of the math, and helps students from a variety of backgrounds and learning styles stay connected to the concepts they are learning. Poole's approach helps students succeed in this course by learning vectors and vector geometry first in order to visualize and understand the meaning of the calculations that they will encounter and develop mathematical maturity for thinking abstractly
Buy Used $19.78 idea of a "category"--a sort of mathematical universe--has brought about a remarkable unification and simplification of mathematics. Written by two of the best-known names in categorical logic, Conceptual Mathematics is the first book to apply categories to the most elementary mathematics. It thus serves two purposes: first, to provide a key to mathematics for the general reader or beginning student; and second, to furnish an easy introduction to categories for computer scientists, logicians, physicists, and linguists who want to gain some familiarity with the categorical method without initially committing themselves to extended study. Editorial Reviews Review "Conceptual Mathematics provides an excellent introductory account to categories for those who are starting from scratch. It treats material which will appear simple and familiar to many philosophers, but in an unfamiliar way." Studies in History and Philosophy of Modern Physics Book Description The idea of a "category"--a sort of mathematical universe--has brought about a remarkable unification and simplification of mathematics. Written by two of the best known names in categorical logic, this is the first book to apply categories to the most elementary mathematics. Most Helpful Customer Reviews Lawvere and Schanuel have created a book at once accessible and stimulating at a great many levels. It discusses the concepts of Category Theory in a simulated "classroom" setting, addressing common questions of students at crucial points in the book. It also wanders in a care-free manner through an amazing number of topics. The book is interesting to non-mathematicians at a philosophical level, and to (beginning) mathematicians as an introduction to an exciting new area of mathematics. The authors have a great attitude, and offer great starting-points for investigation. I read it as a first year pure math undergraduate, and though it was at times at too low a level (the 'tests,' for instance, are very easy reviews of basic ideas), it never became boring. For me, it read 'like a novel' (and a page-turner, at that). My only gripe is the lack of an annotated "further reading" section, which would have rounded out the book. Many of the reviews evaluate the book from the perspective of graduate students in mathematics want to learn categories, and it's certainly the wrong choice for that purpose. If you think of this as a serious math textbook, then it fails in that goal: significant proofs are the exception rather than the rule; very few, and trivial, exercises; very lacking in depth. This is a great book because it provides a motivation for investigating categories. It helped me when I was in the position of hearing from a lot of places that subjects I was interested in often used category theory. I tried to read a few "real" books about category theory, and didn't get very far because they did not make the connections I was looking for. I accumulated three or four such books, all with bookmarks at about page 50 to 75. This book taught me relatively little about the theory of categories or the body of knowledge about them, but it provided a wealth of connections between categories and other topics, which made me better able to finish a couple of the real books and figure out what I needed to know there. My advice, if you're in anything like that situation, is to read this book. Just don't take it too seriously, and don't try to milk more out of it than is really there. Then go learn more about category theory from elsewhere. Highly intuitive introduction to this abstract, but highly practical area of mathematics with one glaring fault. First the good news. I have never seen a more carefully explained introduction into an area of mathematics. Many examples and explanations of the principles behind and applications of concept analysis. However, the glaring fault is organization. Details are given without adequate tie in to how they relate to others. The text bounces from one area to the next so it is easy to lose sight of the whole picture. On balance its strengths far outweigh its weaknesses so I recommend it without reservation. As a first introduction to Categories, this book is well written, clever, simple and very clear. However, I was disappointed with it. From the notoriety of the authors and the, yes, cool illustrations I assumed it would be a gem. However, it fell short. I've been toying with Category Theory for a few years, and every time I try to get into a book on Categories I get stumped at the notions of Functors and Natural Transformations. This book, however, dealt with neither at length, despite the fact that Category Theory originated around the notion of Natural Transformations in the first place. (As I understand it at least.) That said, there are many very cool passages in the book, including a functional analysis of a Chinese restaurant and an elegent exposition of Brouwer's Fixed Point Theorem. Still, for my purposes, I prefer Robert Goldblatt's "Topoi: The Categorical Analysis of Logig" and Michael Barr's "Category Theory for Computing Science". As both are intended for non Category Theorists, both build their presentations of Category Theory from sratch. Sadly, I think both are out of print. Not for the faint of heart, I'm told Saunders Mac Lane's "Categories for the Working Mathematician" is the classic. (It's on my wish list.) As a topic in itself, category theory should need not to wait until grad-level to be described just because that may be when category theory's power can really begin to be exploited, but unfortunately, most of the category theory books I have looked at presume that level of mathematics. Similar to what other reviewers noted, I would also say that this book demonstrates the potential of creating a good high-school/undergrad level intro to category theory. But unfortunately, that potential is not quite realized here. There are hokey intermittent "conversations with students", as a tool to describe ideas, that are more distraction than aid. Some of the examples given are rather condescending in their simplicity. Yet, at other times the authors seem to breeze through more difficult topics with little or no examples. And the organization seems erratic - there is no clear sense of a gameplan as to where they are leading the reader or how all the concepts fit together. Functors are surprisingly almost glossed over, as if they were relatively unimportant. There are exercises throughout the book, but with no answers provided, they are not really very helpful. Having said all that, with some focused effort on the reader's part, the ideas do come forth, and admittedly, the authors do cover a fairly broad spectrum of aspects of category theory. This is certainly a non-trivial topic to try and teach, and an introductory book cannot be faulted for not carrying every notion to the nth-degree of either breadth or depth. Category Theory is one of those topics that (to me) appears 'ho-hum' until you see it actually applied to various topics. The authors have necessarily had to perform a balancing act between describing concepts while not getting caught up in excessively complex examples.Read more ›
... More About This Book 618 fully solved problems to reinforce knowledge Concise explanations of all trigonometry concepts Updates that reflect the latest course scope and sequences, with coverage of periodic functions and curve graphing. Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time—and get your best test scores! Related Subjects Meet the Author Frank Ayres Jr., PhD, was formerly professor and head of the Department of Mathematics at Dickinson College, Carlisle, Pennsylvania. He is the author of eight Schaum's Outlines. Robert E. Moyer taught mathematics at Southwest Minnesota State University. He received his Doctor of Philosophy in Mathematics Education from the University of Illinois in 1974. From Southern Illinois University he received his Master of Science in 1967 and his Bachelor of Science in 1964, both in Mathematics
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Over 300 unusual problems, ranging from easy to difficult, involving equations and inequalities, Diophantine equations, number theory, quadratic equations, logarithms, more. Detailed solutions, as well as brief answers, for all problems are provided. This classic text and standard reference comprises all subjects of a first-year graduate-level course, including in-depth coverage of groups and polynomials and extensive use of categories and functors. 1989 edition. This groundbreaking monograph in advanced algebra addresses crossed products, which involve group and ring theory and the study of infinite group algebras, group-graded rings, and the Galois theory of noncommutative rings. 1989 edition. The problem-solving assumes theoretical and analytical skills, as well as algorithmic skills, coupled with a basic mathematical intuition. The concept of this problem book successfully supports the development of these skills of the solver and meanwhile What could be better than the bestselling Schaum's Outline series? For students looking for a quick nuts-and-bolts overview, it would have to be Schaum's Easy Outline series. Every ebook in this series is a pared-down, simplified, and tightly focused
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There are several threads here in which people have recommended sets of notes on relativity in which tensors are taught. In one of those sets of notes, that I particularly liked, the author Sean Carroll recommended the book by Frank Warner, on Differentiable manifolds and lie groups, as a "standard". I kind of like Michael Spivak's little book Calculus on manifolds, and his much longer series on Differential Geometry, say the first volume for starters. This is a mathematician talking, so I recommend getting some opinions from the physics experts too. Of course Carroll is presumably a physicist. Warnes book is nice because it also has an introduction to "Hodge theory" as I recall. Others here have recommended Tensor analysis on manifolds by Bishop and Goldberg, because it is not only a good classic text, but it is available in paper for a song. Look on the threads "What is a tensor", and Differential geometry lecture notes, and Math "Newb" Wants to know what a Tensor is, and others , for some free sites with downloadable material on tensors. I would warn you of one thing. I myself am primarily educated in the mathematics of manifolds and tensor bundles as in Spivak's calculus on manifolds. As you can see from numerous exchanges I have had with physicists on this forum I have great difficulty understanding what they are talking about. Thus i would suggest that it is not enough to understand only the mathematical concepts of manifolds and tensors, but one should go further and see these concepts in use either in differential geometry, or in physics. It is written by a high-school teacher, and oriented to physics application. However, it is quite rigourous enough (to me). It goes from the very beginning to advanced stuff in physics. Will_C #4 Sep8-04, 12:57 AM P: n/a Intro tensors book There are notes I highly recommended, because they are free on internet and starting at elementary level: 1) An Introduction To Tensors for Students of Physics and Engineering, by Joseph C. Kolecki 2) Quick Introduction to Tensor Analysis, by R.A. Sharipov 3) An Introduction to Tensor Analysis and Continuum Mechanics, by J.H. Heinbockel And I would like to thank the authors of these notes. Thank You! Will. Tide #5 Sep8-04, 01:57 AM Sci Advisor HW Helper P: 3,147 THE book on tensor calculus is by Synge and Schild (Tensor Calculus) but I don't know whether it's still in print. robphy #6 Sep8-04, 05:28 AM Sci Advisor HW Helper PF Gold P: 4,119 Quote by Tide THE book on tensor calculus is by Synge and Schild (Tensor Calculus) but I don't know whether it's still in print. Although Synge and Schild is great, I would have to say that Schouten's "Ricci Calculus" is THE book on tensor calculus... see also Schouten's "Tensor Analysis for Physicists" (Dover). rick1138 #7 Sep9-04, 12:18 AM P: 199 Schaum's Outline of Tensor Calculus is excellent, though it uses the classical, rather than modern notation. If you want to learn to calculate functions on curved spaces this is a good place to go. Another great resource is MTW's Gravitation, though a book on relativity, has some insightful sections on tensors.Of course, all the other books mentioned in this thread are good choices. mathwonk #8 Sep9-04, 02:57 PM Sci Advisor HW Helper P: 9,453 A modest proposal: I would suggest that one reason it is hard for physicists and mathematicians to communicate is that some physicists seem to continue to educate themselves via extremely old fashioned mathematics books, teaching a version of tensor calculus that is about 100 years old. I know Einstein used it, but the mathematics Einstein used is not sufficient to understand even today's physics. Modern physicists like (the real, departed) Feynman, and now Witten, have been not only creating new exciting physics, but also new mathematics, and are also inspiring mathematicians to try to catch up with the innovations they are bringing to both subjects. I have been a lecturer at the International Center for Theoretical Physics in Trieste (on Riemann surfaces, and theta functions on abelian varieties) and so I know a little about what at least some of today's physicists want to learn. In particular it is important to learn how geometry is linked to, and illuminated by, modern topology and analysis, such as deRham cohomology, and modern theory of partial differential operators. So I recommend, even if one learns the old versions of Ricci calculus, to also at least look at modern books like Spivak's differential geometry volumes 1 and 2. If that is too mathematical, I suggest trying Misner, Thorne and Wheeler, as recommended above. At some point one might want to look also at books such as the volume of proceedings "Lectures on Riemnann Surfaces" given at ICTP in 1987, pub. by World Scientific in 1989 (for background on string theory), and some works on algebraic geometry. Miles Reid's book is the easiest, and Shafarevich is very nice for a next course. To me many of the references recommended here, although excellent for what they are, nonetheless fall largely into the category: "primarily of historical interest". Feyman's work, in particular the theory of "Feynman integrals" has apparently led recently to the exciting mathematical topic of quantum cohomology, growing out of "quantum gravity" and other topics in which modern physicists lead the way. It seems to me at least, that these leaders are not using 100 year old mathematics, they are using mathematics that has not even been perfected yet. best regards, roy rick1138 #9 Sep10-04, 12:58 PM P: 199 Modern tensor notation is in my opinion superior, the problem is that many texts that teach it use abstraction as an excuse to avoid doing any actual calculations - in some cases modern methods step around tedious calculations in a profound and brilliant manner, but in the wrong hands modern tensor notation is obscure gibberish. A case in point is Darling's book - an excellent book overall, and one I recommend, but at times it is degenerates into storm of pretentious machinery. At one point he defines two objects, then proves that they are actually the same object to begin with. What is the point of that? At another point he gives that wrong formula for the wedge product of two forms, and error he would have found easily if he had actually tried to calculate anything with it. If there was a book using modern notation that was not divorced from the nuts and bolts of doing actual calculations I would recommend that, but unfortunately there are none. P.S. DeRham's book on cohomology is excellent - it is strangely obscure. Another excellent book on algebraic geometry is "An Invitation to Algebraic Geometry" by Karen Smith et. al. mathwonk #10 Sep11-04, 09:01 AM Sci Advisor HW Helper P: 9,453 You make an excellent point. It is certainly true that many authors of modern math books dwell excessively on the theory and omit useful calculations, in every subject. Now that you have raised this as a key virtue lacked by most modern treatments, maybe someone will recommend a book unknown to us which does have it. In the meantime perhaps your argument implies that physicists need to read both types of books, modern and classical. I have not read Darling, but can suggest a reason for giving two different definitions of the same object in some cases. Each definition may have different advantages, perhaps one is intuitively more appealing while the other admits easier calculation. Or one was used historically, while the other is of more modern acceptance. Just a guess. As I recall, Spivak for example discusses the curvature tensor in progressive degrees of abstraction, starting from Riemann's original version, continuing up through modern incarnations. mathwonk #11 Sep12-04, 03:10 PM Sci Advisor HW Helper P: 9,453 What I think this comes down to is the dichotomy between calculating a quantity and understanding the meaning of that quantity. I claim that understanding allows calculation, but not vice versa. For instance, on page 14, of his nice notes on GR, Sean Carroll gives the transformation law, (1.51) in his numbering, for tensors and then says: "Indeed a number of books like to define tensors as collections of numbers transforming according to (1.51). While this is operationally useful, it tends to obscure the deeper meaning of tensors as geometric entities with a life independent of any chosen coordinate system." On page 15 he describes the scalar or dot product as a familiar example of a tensor of type (0,2). I am going to go out on a limb here and try to make a trivial calculation, beginning from a conceptual definition of a tensor of type (0,2) as a bilinear map from pairs of tangent vectors to numbers. I.e. I will try to derive the transformation law from the conceptual meaning. A simple example of such a tensor is a scalar product, i.e. a symmetric, bilinear mapping from pairs of tangent vectors to scalars. Such a thing is often denoted by brackets (or a dot) taking the pair of tangent vectors v,w to the number <v,w>. Now if f:M-->N is a differentiable mapping from one manifold M to another manifold N, such as a coordinate change, then one can pull back a scalar product from N to M using the derivative of f. I.e. if u,z are two tangent vectors at a point p of M, then applying the derivative of f to them takes them to 2 tangent vectors at the image point f(p) in N, where we can apply <,> to them. I.e. if <,> is the scalar product on N, then the pulled back scalar product f(<,>) acts on u,z by the obvious, only possible law: f(<u,z>) = <f'(u),f'(z)>, where f' is the derivative of f, given as a matrix of partials of f with respect to local coordinates in M and N. For example we could denote this matrix as f' = [dyi/dxj]. Now suppose we express the scalar product in N as a matrix, i.e. in local coordinates as A = [akl], sorry about the lack of subscripts. Imagine k and l are subscripts on a. Then if we want to express the pulled back scalar product as a matrix, we just see what it does to the vectors u,z as follows: f(<u,z>) = <f'(u),f'(z)> = [f'(u)] [A] [f'(z)] = [u]* [f']* [A] [f'] [z], where now everything is thought of as a matrix, and star means transpose of the matrix. Well since the matrix of partials f' is just [dyi/dxj], and A is [akl], we just multiply out the matrices to get the matrix of the pulled back scalar product as [f*(<,>)] Now this is exactly the transformation law Carroll calls (1.51) on page 14 of his notes and everyone else also calls the transformation law for a tensor of type or rank (0,2) in the various web sources given here and above. Notice too, if you can imagine my subscripts, that this satisfies the summation convention for subscripts. But I am not dependent on that because I know what it means, so i don't care whether I can see the subscripts or not, whereas someone dependent on seeing where the indices are may not be able to follow this. Anyone who knows conceptually what a tensor is would immediately realize that a homogeneous polynomial of degree d in the entries of a tangent vector, is a (symmetric) tensor of type (0,d), and that the components of the tensor are merely the coefficients of the polynomial (written as a non commutative polynomial, i.e. with a separate coefficient for xy and for yx). It follows of course that they transform via a d dimensional matrix of size n, where n is the dimension of the manifold, i.e. by a collection of n^d numbers. Subscript enthusiasts write this as a symbol like T, with d subscripts. That is an extremely cumbersome way to discuss tensors in my opinion, and leaves me at the mercy of the type setter, whereas knowing what they mean always bails me out eventually. I actually wrote a graduate algebra book, including linear and multilinear alkgebra once, and I discovered to my amusement that I could actually write down tensor products as matrices, and so on, just from the definitions, although I had never needed to do so before in my professional life. peace and love, roy mathwonk #12 Sep13-04, 02:14 PM Sci Advisor HW Helper P: 9,453 In 1996, at the end of a chapter on tensors in my graduate algebra notes, after writing out a calculation of the tensor product of two matrices, I wrote the following extremely naive remarks: "The complexity of this sort of calculation may be responsible for the fearsome reputation which "tensor analysis" once enjoyed. In ancient times, books on the topic were filled with lengthy formulas laden with indices. Learning the subject meant memorizing rules for manipulating those indices. Nowadays, confronted with the statement that such and such quantity is "a tensor", I hope we will understand this to mean simply the quantity has certain linearity properties with respect to each of its components. Of course skill in their use will still require an ability to calculate. In this regard, note that we are usually able to recover explicit calculations from our abstract approach, provided we always know exactly what the maps are that yield our isomorphisms. When we know the maps, a choice of bases gives us a calculation. Thus we must resist the tendency to remember only that certain modules are isomorphic, without knowing what the isomorphisms are. Fortunately the maps are virtually always the simplest ones we can think of." mathwonk #13 Sep13-04, 03:37 PM Sci Advisor HW Helper P: 9,453 I have just perused Spivak's volume 2, chapters 4,5, and 6, and want to recommend them extremely highly to everyone here who is interested in tensors and the relation between their classical and modern incarnations. I.e. this is not just a modern treatment, and not just a classical treatment, but he gives both treatments, with relations clearly drawn betwen the two. I.e. forget volume I, which may seem like a mathematician's indulgence, with its abstract definitions and modern treatments of manifolds. Go straight to the good stuff in volume II. The chapter headings are already enlightening: 1. is on curves, 2. is called "what they knew about surfaces before Gauss" and is only 8 pages long. 3. is on Gauss's theory of surfaces, and I mean Gauss's own version. Spivak presents Gauss's own work, Disquisitiones generales circa superficies curvas, and explains how to make sense of the original: "how to read Gauss", then explains how to state the results in modern terms. chapter 4. is a translation of Riemann's inaugural lecture on manifolds, including his generalization to n dimensions of Gauss' theory of curvature of surfaces. Basically, certain combinations of partial derivatives, now called Christoffel symbols, represent the coefficients of the "error term" needed to force the partial derivatives of a tensor of type (0,1), with respect to given local coordinates, to be themselves again a tensor. Then a certain combination of these Christoffel symbols defines the curvature tensor. In chapter 5 Spivak presents the classical Ricci calculus, subtitled "the debauch of indices", and proves computationally the "test case" that a manifold with zero curvature tensor is locally isometric to (flat) euclidean space. Then in chapter 6, Spivak presents the modern approach to a "connection", as simply a way of differentiating pairs of vector fields, linear over the functions in one of the variables, and obeying the Leibniz rule in the other variable. He then relates an abstract connection to a "classical connection", i.e. an expression analogous to the Christoffel symbols, but not necessarily arising from a metric, hence not necessarily symmetric. Nonetheless any connection leads again to a curvature tensor, which now is simply a certain commutator expression of derivatives. Spivak then reproves the basic test case, much more easily, using modern concepts. He continues to pursue the evolution of the concept of a connection through several modern versions, reproving the "test case" in all seven times, each time revealing more geometric content, as modern conceptual tools permit this. Thus Spivak presents a thoroughly classical treatment of connections and curvature, Ricci calculus, and Christoffel symbols, then shows how these concepts are viewed nowadays in simpler more conceptual terms. This seems to me the ideal place to learn to speak all these languages. In fact back in 1970, I once claimed that although I did not know what they meant, I did know that nowadays "Christoffel symbols" had become a triviality. Some people laughed at me for saying this, at which point I bet them I could prove my statement by learning what they meant in 5 minutes and then explain it to their satisfaction. I grabbed Spivak, opened volume 2, and persuaded them in a few minutes. This is truly a great book, which made a uniquely valuable contribution to understanding differential geometry. mathwonk #14 Sep14-04, 10:16 PM Sci Advisor HW Helper P: 9,453 To recommend Spivak again, I read this book (volume II) in one day almost 30 years ago, and have never consulted it again (except for the bet above) until yesterday for about 30 minutes. So it is not going to eat up a lot a lot of your time to give it the once over. It is so well written you can learn something from it very quickly. Although obviously in such a short time I did not come anywhere near mastering anything, still I feel I did learn something. Well after perusing the website of the publisher, I see the first and second editions are no longer available, and I am slightly disappointed to note that apparently the cover art has changed, and there are no longer strange animals on the front of voilume 5 waving flags and and marching in the name of "The generalized Gauss Bonnet theorem and what it means for mankind". You can never have too much nonsense in amthematics. The chapter on p.d.e. called "and now a word from our sponsor" remains however. rick1138 #15 Sep15-04, 11:40 PM P: 199 Which volume has the information on covariant derivatives? I am going to get my hands on a copy - the copy at the local tech school is the edition with the horrendeous typography, almost unreadable, which has kept me from a serious reading. mathwonk #16 Sep16-04, 12:29 AM Sci Advisor HW Helper P: 9,453 well I am going to guess it is volume 2. volume 1 is a treatise on general manifold theory, then vol 2 is the evolution of differentiual geometry since riemann and gauss, focussing on the curvature tensor. i do not know what a covariant derivative is, but the modern version of a connection is called there a koszul connection, and is simply a way of pairing two vector fields X,Y and getting another one called delta sub X (Y). it is linear in X over the ring of smooth functions, and obeys the leibniz rule in Y for multiplication by smooth functions. this is the modern version of christoffel symbols. I will just guess that this is a covariant derivative, but i can find out later. still i presume all the machinery of differential geometry is in that volume 2, because vol 3 is sort of classical examples like equations of hypersurfaces "codazzi equations etc" and I forget what vol 4 is, but 5 seems to be fancy stuff like gauss bonnet a la chern in n dimensions. we can probably see the contents of the chapters on the publish or perish website, i'll look and see. I also recall that volumes 1 and 2 were the actual content of the copurse mike taught, and 3,4,5 were added later, so surely he taught covariant derivatives in the course. but they might be already in vol 1 at the end. i only have vols 1 and 2 but they are at the office. ok i just looked on the publish or eprish website and here is the table of contents for volume 2 and you can see the words covariant derivatives in chapter 7, cartans theory of moving frames: a very nice clean discussion is in milnor's book, characteristic classes, joint with stasheff, in appendix C, p.289, "curvature, connections, and characteristic classes". he defines a "connection" on a bundle E over a manifold, as a C linear map from smooth sections of the given bundle E, to smooth sections of the tensor product of E with the cotangent bundle of the manifold. It is required that the map obey the leibniz rule. i.e. it takes the product of a function and a section to the product of the function times the image of the section, plus the tensor product of the section with the esterior derivative of the function. Then, he calls the image of the section under this mapping, the covariant derivative of the section. so covariant derivative is just another name for a connection. you get a wonderfully clear explanation in about 15 pages from milnor, of several basic points, and the link with classical connections. he then explains curvature and proves the gauss bonnet theorem connecting curvature with the euler characteristic. (Recall the polyhedral version of gauss bonnet: if we have a polyhedral surface and at each vertex define the curvature to be 2<pi> minus the sum of the angles of the polygons at that vertex, then the sum total over the curvatures over the polyhedron equals the euler characteristic, times 2<pi>.) gvk #18 Sep20-04, 02:32 PM P: 83 Being educated as a physicist, I understand many people who complain about "bourbaki" style of writing math textbooks, and I would not recommend to read the books by F. Warner and M. Spivak as a first introductory reading in modern geometry. (Spivak is only good to understand the historical line of development, but you have to have some background and being familiar with modern terminology for that.) In my opinion more or less suitable book, written by mathematicians for physicists and engineers, is Dubrovin, Novikov, Fomenko, Modern Geometry v. 1,2,3. This is three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics of 1st volume starts from curves and surfaces and include tensors and their differential calculus, vector fields, differential forms, the calculus of variations in one and several dimensions, and even the foundations of Lie algebra. So, the first volume would be enough for start. I looked in 2 and 3 v. and think its close to the front of modern geometry and definitly prepares for the reading more special books... The material of books is explained in simple and concrete language that is in terminology acceptable to physicists. There are some exercises, but should be more to get practical skills. If I will find the special problem book on modern geometry to accompanying this textbook, it would be excellent pair for any beginner.
Search Results MyWorkBook with Chapter Summaries for Introductory Algebra Through Applications About this title: MyWorkbook with Chapter Summaries provides extra practice exercises for every chapter of the text. This workbook can be packaged with the textbook or with the MyMathLab access kit and includes the following resources: Mathematically Speaking key vocabulary terms, and vocabulary practice problems Guided Examples with stepped-out solutions and similar Practice Exercises, keyed to the text by Learning Objective Additional exercises with ample space for students to show their work, keyed to the text by Learning Objective ... read more Note: This is a general synopsis. Each listing is described below. Your search:Home»Books»MyWorkBook with Chapter Summaries for Introductory Algebra Through Applications
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Mathcad for beginners Which system of computer mathematics is better or worse is the eternal question and requires hours of discussion. Therefore here we shall define the peculiarities of Mathcad, which can influence the user choice. Immense popularity and a huge number of users. Mathematical pack Mathcad is simple to use and easy to learn due to the adequacy of functions and operators in use to the traditional ways of writing mathematical expressions. A big number of printed materials and video courses on Mathcad. Mathcad is widely applied for solving problems in various fields of study. Textbooks on Mathcad in use for solving problems of higher mathematics, information science, theoretical mechanics, and strength of materials, etc. have been published. Mathcad can be successfully used both by school students for solving elementary mathematics problems and professional programmers, mathematicians, physicians in their scientific research. Possibility of export and import of data between Mathcad and other Windows-based applications. Continuous development and updating of Mathcad due to the efficient rise of new versions
Problem Solving in Mathematics Category: Mathematics Grade: 4 to 9 1. What is the purpose of Problem Solving needed, notes and suggestions, and in Mathematics? answers. The purpose of this approach is to emphasise  Laboratory activities: Investigations the use of problem solving skills in the involving mathematical applications. classroom.  Teaching mathematical concepts: In 2. With whom can it be used? addition to the sections in each book on The grade level books for grades 4-8 and 9 grade level content topics, there is an in- are suitable for use with students of all service guide self-servicing audiotapes for abilities. Additional problems appropriate for teachers. The principal can also be low achievers are contained in the Alternative provided with an audiotape to acquaint Problem Solving in Mathematics book. The him or her with the administrative activities in the Alternative Problem Solving responsibilities and support necessary for in Mathematics book differ from the regular implementing the program. books in the level of math computation and length of time needed for completion. They 4. What teaching procedures should be are generally appropriate for students in used with Problem Solving in grades 4-6. Mathematics? Specific procedures used vary according by 3. What is the format of Problem Solving activity. In general, the program emphasizes in Mathematics? the direct teaching and use of problem Each book is centered around grade- solving strategies. Examples of strategies appropriate topics, and includes the following used in the program are the sections: following:  Getting Started: Direct instruction of A. Problem Discovery, Formulation problem solving strategies.  State the problem in your own words  Clarify the problem through careful  Drill and Practice: Review activities for reading and by asking questions the beginning of the year. B. Seeking Information  Challenges: Problems of greater  Collect data needed to solve the problem difficulty appropriate for later in the year  Record solution possibilities or attempts after experience with problem solving. C. Analysing information  Teacher commentaries: Comments from  Eliminate extraneous information teachers on teaching objectives, problem  Make and/or use a systematic list or table solving skills pupils might use, materials D. Solve-Putting it together  Synthesis disabilities and students at risk for math  Make reasonable estimates failure.  Detect and correct errors References E. Looking back 1. Cawley, F. & Parmar, R.S. (1992).  consolidating Arithmetic programming for students with  Explain how you solved a problem disabilities: An alternative. Remedial and  Find another answer when more than one Special Education, 13, 6-18. is possible 2. Jitendra, A. & Xin, Y.P. (1997). Mathematical word-problem-solving F. Looking ahead-Formulating new instruction for students with mild problems disabilities and students at risk for math failure: A research synthesis. The Journal  Create new problems by varying a given of Special Education, 30, 412-438. one 3. Lane County Mathematics Project: Problem Solving in Mathematics. (1983). 5. In what types of settings is Problem Palo Alto, California: Dale Seymour Solving in Mathematics useful? Publications. This program is meant to be integrated into 4. Schaaf, O. (1984). Teaching problem- the regular curriculum. Lessons in the solving skills. Mathematics Teacher, 77, Problem Solving books can be used to replace 694-699. regular textbook pages. Also, regular 5. Schaaf, O. (1979). Introduction to the textbook pages might be used as practice for LCMP Mathematics Problem-Solving the problems presented in the Problem Programs. ERIC Document Reproduction Solving book. Service 6. To what extent has research shown Reviewed by: Naomi Slonim Problem Solving in Mathematics to be useful? Although no carefully controlled longitudinal studies have been conducted, evaluations by the authors have demonstrated positive effects of the program. These include gains on standardized achievement tests and special problem-solving skill tests. In addition, teachers have indicated that problem-solving strategies taught in the program generalize to other subjects such as Social Studies, Language Arts, and Science. The most significant gains occurred with exclusive use of the program and use as specified in teacher commentaries and in-service materials. In addition to program evaluation by the authors, research shows positive effects of mathematical word-problem solving/strategy instruction for students with
The bestselling author of Alex's Adventures in Numberland returns with a dazzling new book that turns even the most complex math into a brilliantly entertaining narrative. From triangles, rotations and power laws, to fractals, cones and curves, bestselling author Alex Bellos takes you on a journey of mathematical discovery with his signature... more... A simple, visual guide to helping children understand maths with Carol Vorderman Reduce the stress of studying maths and help your child with their homework, following Help Your Kids with Maths a unique visual guide which will demystify the subject for everyone. Updated to include the latest changes to the UK National Curriculum... more... This book contains the lectures presented at a conference held at Princeton University in May 1991 in honor of Elias M. Stein's sixtieth birthday. The lectures deal with Fourier analysis and its applications. The contributors to the volume are W. Beckner, A. Boggess, J. Bourgain, A. Carbery, M. Christ, R. R. Coifman, S. Dobyinsky, C. Fefferman, R.... more... Sun (food refrigeration and computerized food technology, University College-Dublin) provides information on the art and science of frozen foods for engineers and technologists working in research, development, and operations in the food industry. The handbook explains the basics of freezing, then focuses on freezing equipment and facilities and th more... You know mathematics. You know how to write mathematics. But do you know how to produce clean, clear, well-formatted manuscripts for publication? Do you speak the language of publishers, typesetters, graphics designers, and copy editors? Your page design-the style and format of theorems and equations, running heads and section headings, page breaks,... more... Includes 40 papers presented at the 4th ESERA conference held in The Netherlands, in August 2003. The papers presented at the conference deal with actual issues in the field, such as the learning of scientific concepts and skills, scientific literacy, informal science learning, science teacher education, modeling in science education. more...
The proper approach depends on your goals. If you are good and want to get better, that requires one technique. If you are intimidated by Math and you have always struggled, that requires a different method.
Product Description Get everything you need for a successful and pain-free year of learning math! This kit includes Saxon's Math 6/5 textbook, solutions manual, and tests/worksheets book, as well as the DIVE Math 6/5 CD-ROM. A balanced, integrated mathematics program that has proven itself a leader in the math teaching field, Math 6/5 covers advanced divisibility concepts; multiplication; integers; prime and composite numbers; powers; roots; probability; statistics; patterns and sequences; geometry and measurements; and ratios. The DIVE software teaches each Saxon lesson concept step-by-step on a digital whiteboard, averaging about 10-15 minutes in length; because each lesson is stored separately, you can easily move about from lesson-to-lesson as well as maneuver within the lesson you're watching. DIVE teaches the same concepts as Saxon, but does not use the problems given in the text; it cannot be used as a solutions manual. Product Reviews Be the first to write a review! Write Review Ask Christianbook \| Q: Which is better: the DIVE or the teaching tape tutorial? A: The item that would be better for you, depends upon your individual needs. In order to help you decide, summary descriptions for each of the products are listed below.DIVE provides a digital whiteboard presentation of the lesson and investigation concepts contained in the Saxon Math textbook without using the exact examples from the book. Problem set and test solutions are not included. All material is taught from a Christian worldview. These cannot be used without the Saxon textbooks. The Solutions Manual may be necessary for upper level math. Compatible with Windows or Mac.Teaching Tapes features live instruction by a state-certified teacher who explains and demonstrates each concept, example, practice problem and investigation. Problem set and test solutions are not included. Each lesson is approximately 10 to 15 minutes long including practice problem review. These DVDs cannot be used without the Saxon textbooks. The Solutions Manual may be necessary for upper level math. Can be played on any standard DVD player or in a DVD-Rom drive
9780495389613 ISBN: 0495389617 Edition: 4 Pub Date: 2008 Publisher: Cengage Learning Summary: Algebra can be like a foreign language. But one text delivers an interpretation you can fully understand. Building a conceptual foundation in the "language of algebra," iNTERMEDIATE ALGEBRA, 4e provides an integrated learning process that helps you expand your reasoning abilities as it teaches you how to read, write, and think mathematically. Packed with real-life applications of math, it blends instructional approac...hes that include vocabulary, practice, and well-defined pedagogy with an emphasis on reasoning, modeling, communication, and technology skills. The authors' five-step problem-solving approach makes learning easy. More student-friendly than ever, the text offers a rich collection of student learning tools, including Enhanced WebAssign online learning system. With ELEMENTARY AND INTERMEDIATE ALGEBRA, 4e, algebra makes sense! Tussy, Alan S. is the author of Elementary and Intermediate Algebra (with CengageNOW Printed Access Card), published 2008 under ISBN 9780495389613 and 0495389617. Five hundred thirty six Elementary and Intermediate Algebra (with CengageNOW Printed Access Card) textbooks are available for sale on ValoreBooks.com, two hundred eighty used from the cheapest price of $4.70, or buy new starting at $450495389617 Student Edition. No apparent missing pages. Heavy wear, wrinkling, creasing, Curling or tears on the cover and spine May be missing front or back cover. May have u [more] 0495389617 and binding my list this and this is the only difference from the standard version, the content is the same. Ships [more] ALTERNATE EDITION: Used - Hardcover. This is a TEACHERS EDITION copy. The cover and binding my list this and this is the only difference from the standard version, the content is the same. Ships in 24 hours or less![less] ISBN-13:9780495389613 ISBN:0495389617 Edition:4th Pub Date:2008 Publisher:Cengage Learning Valore Books is the top book store for cheap Elementary and Intermediate Algebra (with CengageNOW Printed Access Card) rentals, or used and new condition books available to purchase and have shipped quickly.
introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain). The presentation is almost entirely self-contained, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. No experience with analysis, geometry, number theory or topology is assumed. Within the context of linear algebra, K-theory organises and clarifies the relations among ideal class groups, group representations, quadratic forms, dimensions of a ring, determinants, quadratic reciprocity and Brauer groups of fields. By including introductions to standard algebra topics (tensor products, localisation, Jacobson radical, chain conditions, Dedekind domains, semi-simple rings, exterior algebras), the author makes algebraic K-theory accessible to first-year graduate students and other mathematically sophisticated readers. Even if your algebra is rusty, you can read this book; the necessary background is here, with proofs. More About the Author Inside This Book(Learn More) Browse and search another edition of this book. First Sentence Algebraic K-theory can be understood as a natural outgrowth of the attempt to generalize certain theorems in the linear algebra of vector spaces over a field to the wider context of modules over a ring.&nbspRead the first page Customer Reviews Most Helpful Customer Reviews on Amazon.com (beta) Amazon.com: 1 review 6 of 7 people found the following review helpful Didactic perfection9 Jun 2006 By Dr. Lee D. Carlson - Published on Amazon.com Format: Hardcover Verified Purchase Mathematics, particularly research mathematics, or mathematics that is close to the frontiers of research would be considerably easier to learn if mathematicians would both explain what they are going to do and explain what historical context motivates the problems or issues they are interested in. This would grant the needed insights and "intuitions" into the subject, which are absolutely necessary if one is to understand a particular mathematical topic in depth. In their papers, monographs, and textbooks, mathematicians could perhaps include at various places in the text some `fundamental insights' into the concepts that are being discussed. In addition, they could describe in detail what their goals are and what strategies they are going to use to solve the problems as they develop. Unfortunately not many works of mathematics are written in this manner, and so those who wish to learn a given topic must frequently undertake time-consuming literature searches and solve myriads of exercises and problem sets in order to gain the needed insight. This takes large blocks of time, and poses an even greater challenge for those, such as physicists, who not must not only master the physics but also the mathematical formalism used to frame physical theories. Physicists would welcome, and even in many cases grab at straws to learn mathematics in a way that they need to in order to push forward the frontiers of their own subject. This book though is very different, and is one of the best works of mathematics put in print in the last three decades. The author has given the reader a work that is not only mathematically rigorous but also fits the learning framework discussed above. There is no doubt that K-theory is a highly esoteric subject, but it can be learned much more easily by the study of this book. Within its covers there are myriads of fundamental insights that the author shares with the reader that make the learning of K-theory much more palatable and actually downright fun. It should not be thought however that the reader does not have to exercise a fair amount of cognition when wrestling with the intricacies of K-theory. This is true for K-theory as with other branches of mathematics, but those hungry for a true understanding of K-theory will deeply appreciate the author's efforts in this book. The goal of K-theory is to generalize linear algebra, the latter of which deals with linear transformations on vector spaces over fields. K-theory tries to find out, and make rigorous, what constructions in linear algebra carry over when the field is replaced by a ring R and the vector space is replaced by a module over this ring. The first issue that must be dealt with is that of the concept of dimension, which for the case of a vector space is well defined (it is the cardinality of its basis). But an R-module does not necessarily have a basis. So the strategy deployed by the author is find the R-modules that do have a dimension. To find out what it means for an R-module to have a basis, the author constructs R-modules that are generated by elements that are not "linearly related." These are called `free' R-modules and the generating set is called an `R-basis.' The author then characterizes free modules that have a finite basis. That this is a non-trivial exercise is proven by the fact that every nontrivial finite Abelian group is a Z-module, where Z is the ring of integers. In addition, every set is a basis of a free R-module so one must find `presentations' of R-modules. These allow the construction of R-modules satisfying certain needed properties. And then, as expected if one is to extend linear algebra, the author constructs `matrices' of R-linear maps between finitely generated free R-modules. However, a free R-module can have a basis with unequal cardinality, and therefore the author finds those rings R whose free modules have unique dimension and those over which every module is free. The needed property is called an `invariant basis number' and it turns out that most rings have an invariant basis number. But some finitely generated R-modules the author points out are too "small" to be free, and so he finds the appropriate generalization of free modules. These are the famous `projective modules' and are the objects on which algebraic K-theory is based. The designation of "projective" refers to the familiar notion of a projection in ordinary vector space theory, i.e. a linear, idempotent operator. The author describes projective modules as being the `direct summands' of free R-modules, and so to obtain the needed generalization of dimension he constructs Abelian monoids of R-modules under the direct sum operation. This involves finding a universal construction of an Abelian group from a semigroup and this leads to the famous Grothendieck group K0(R) of finitely generated projective R-modules. The finitely generated projective R-modules are `stably isomorphic' if they become equal in K0(R); they are `stably equivalent' if they become congruent in K0(R) modulo finitely generated free modules. The strategy then becomes that of adding R to two finitely generated R-modules to make them stably isomorphic (or stably equivalent). In addition, one must find out to what extent K0(R) determines whether a finitely generated projective R-module is free. This brings up the notion of an R-module being `stably free', and the author finds those stably free R-modules which are free. This involves the notion of the `matrix completion' of a ring and of "shortening" unimodular rows. The author also studies the connection of K0 with number theory, eventually showing that the projective class group is isomorphic to the ideal class group when R is a Dedekind domain. K0(R) is an element of a sequence of Abelian groups associated to each ring R. To find K1(R), the author finds an analog of the row operations in ordinary linear algebra. The elements of K1(R), the `Bass-Whitehead group' are row-equivalence classes of invertible matrices. A group homomorphism from K1(R) to a group G is the analog of the determinant in ordinary linear algebra, and is often called the `Whitehead-Bass' determinant. K1(R) can be thought of as the "abelianization" of the general linear group GL(R). The elements of K2(R) consist of the relations among the generators of the group of row operations on a matrix. The "standard" relations among these operations give the `Steinberg group', and K2(R) is the center of this group.
authors understand what it takes to be successful in mathematics, the skills that students bring to this course, and the way that technology can be used to enhance learning without sacrificing math skills. As a result, they have created a textbook with an overall learning system involving preparation, practice, and review to help students get the most out of the time they put into studying. In sum, Sullivan and Sullivan's Precalculus: Enhanced with Graphing Utilities gives students a model for success in mathematics. This is just the standalone book.
Navigating a Math Zone Class Making use of the Math Zone Students in the Math Zone are registered in the class MATH 1175, however they will be studying MATH 1103 Fundamental Math, MATH 1109 College Algebra, or MATH 1115 Pre-Calculus. Take the time to become familiar with your Math Zone class. Each week you are required to come in to the Math Zone to work through your individualized study plan. You'll read about how to solve problems using your e-text, and you'll be shown examples of how to work your way through math problems. If you get stuck, or lost, let the instructional staff know. They are there help. Take the time to resolve your problems. You are not penalized for asking for assistance, and your grade in the course is determined only by how well you do on your course work and exams. Students are provided lockers on entering the Math Zone where they must deposit cell phones and any other electronic devices, as well as any other items they prefer to leave in the locker. Cell phones and electronic devices are banned from being used in the Math Zone. Also, you are not allowed to drink or eat in the Math Zone. Please help us keep it clean for all users. Working on your study plan Each Math Zone class has its own plan of study. This describes all of the topics and sections that you have to cover in your e-text for that class. An outline of your study plan can be found by logging into Blackboard to access a copy of your class syllabus, or by logging into the Pearson MyLabsPlus website associated with your class. If you have any questions about what you need to do, please ask. Instead of attending lectures, each learning objective is segmented into short practice problems that provide an introduction to each topic. Some have accompanying videos. You have the opportunity to review worked examples and see sample solutions. You can then test yourself on whether you can work with those mathematical ideas covered in that lesson. If you are confident that you can go forward, you can take a short quiz, called a Quiz-Me. If you do well on these you earn Mastery Points, but most significantly, you make progress toward being able to take exams that chart your progress through the class. If you have difficulties with the material, you will be shown alternative approaches to explaining it; however if you continue to have difficulties, it is important that you ask for tutoring assistance. Indeed, at any point where you find yourself not understanding something, ask. It will save you time, and helps you to progress faster. Taking exams All exams must be taken in the Math Zone in a proctored setting. Before you take any exam, please let one of the Math Zone faculty know so that they can begin your testing session. You may use the desktop calculator to help you with your exam. Cell phones and other electronic devices are banned from the Math Zone at all times. Having them with your during an exam risks having the exam results being invalidated. Students who have 3 or more exam rule violations will be given an F grade for the course for the semester.

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