Mathematical foundations Books

Book SynopsisOriginally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics in intuitionism such as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work.Table of ContentsFrontispiece L. E. J. Brouwer; Editorial preface; 1. Historical introduction and fundamental notions; 2. General properties of species, spread directions, spreads and spaces; 3. Order; 4. Precision analysis of the continuum; 5. The bunch theorem; Appendix; Notes; References; Index.

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Book SynopsisOver the last 45 years, Boolean theorem has been generalized and extended in several different directions and its applications have reached into almost every area of modern mathematics; but since it lies on the frontiers of algebra, geometry, general topology and functional analysis, the corpus of mathematics which has arisen in this way is seldom seen as a whole. In order to give a unified treatment of this rather diverse body of material, Dr Johnstone begins by developing the theory of locales (a lattice-theoretic approach to 'general topology without points' which has achieved some notable results in the past ten years but which has not previously been treated in book form). This development culminates in the proof of Stone's Representation Theorem.Table of ContentsPreface; Advice to the reader; Introduction; 1. Preliminaries; 2. Introduction to locales; 3. Compact Hausdorff spaces; 4. Continuous real-valued functions; 5. Representations of rings; 6. Profiniteness and duality; 7. Continuous lattices; Bibliography; Index of categories; Index of other symbols; Index of definitions.

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Book SynopsisA Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence, with the first being essentially self-contained, and are accessible to graduate students with a good background in mathematics. Volume 1, which is devoted to general concepts, can be used for advanced undergraduate courses on category theory. After introducing the terminology and proving the fundamental results concerning limits, adjoint functors and Kan extensions, the categories of fractions are studied in detail; special consideration is paid to the case of localizations. The remainder of the first volume studies various 'refinements' of the fundamental concepts of category and functor.Trade Review"...Not only is this the most comprehensive book ever written on category theory, it is by far the best written...What the author has understood is that one cannot understand this subject without lots of examples..." Gian-Carlo Rota, The Bulletin of Mathematics Books"...these volumes will be of enormous value to graduate students in pure or applied category theory." Martin Hyland, Mathematical ReviewsTable of ContentsIntroduction; 1. The language of categories; 2. Limits; 3. Adjoint functors; 4. Generators and projectives; 5. Categories of fractions; 6. Flat functors and Cauchy completeness; 7. Bicategories and distributors; 8. Internal category theory; Bibliography; Index.

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Book SynopsisHigher-dimensional category theory draws its inspiration from areas as diverse as topology, quantum algebra, mathematical physics, logic, and theoretical computer science. This is the first book on the subject and lays its foundations, appealing to graduate students and researchers who wish to become acquainted with this modern branch of mathematics.Table of ContentsPart I. Background: 1. Classical categorical structures; 2. Classical operads and multicategories; 3. Notions of monoidal category; Part II. Operads. 4. Generalized operads and multicategories: basics; 5. Example: fc-multicategories; 6. Generalized operads and multicategories: further theory; 7. Opetopes; Part III. n-categories: 8. Globular operads; 9. A definition of weak n-category; 10. Other definitions of weak n-category; Appendices: A. Symmetric structures; B. Coherence for monoidal categories; C. Special Cartesian monads; D. Free multicategories; E. Definitions of trees; F. Free strict n-categories; G. Initial operad-with-contraction.

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Book SynopsisAn integrated approach to the theory of nonnegative matrices, emphasising connections with game theory, optimisation, mathematical programming, mathematical economics and statistics. The minimal prerequisites make this accessible to new graduate students.Trade Review"It is a great work; great by its dimensions, written with extreme love and care, concentrating the knowledge of a generation which was supreme in the history of matrix theory. It is a very illuminating and highly readable exposition of interesting topics which are of great relevance both to theory and applications." Mathematical Reviews Clippings 98hTable of ContentsPreface; 1. Perron-Frobenius theory and matrix games; 2. Doubly stochastic matrices; 3. Inequalities; 4. Conditionally positive definite matrices; 5. Topics in combinatorial theory; 6. Scaling problems and their applications; 7. Special matrices in economic models; References; Index; Author index.

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Book SynopsisAssuming only a basic knowledge of manifolds theory, algebra, and measure theory, this 1998 book is a systematic and largely self-contained introduction to Margulis-Zimmer theory. It should appeal to anyone interested in Lie theory, differential geometry and dynamical systems.Trade ReviewReview of the hardback: 'This text might well become a standard one …'. G. Pilz, Internationale Mathematische NachrichtenTable of ContentsPreface; 1. Topological dynamics; 2. Ergodic theory - part I; 3. Smooth actions and Lie theory; 4. Algebraic actions; 5. The classical groups; 6. Geometric structures; 7. Semisimple Lie groups; 8. Ergodic theory - part II; 9. Oseledec's theorem; 10. Rigidity theorems; Appendix: Lattices in SL(n, R); References; Index.

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Book SynopsisThe purpose of this book is to introduce the basic ideas of mathematical proof to students embarking on university mathematics. The emphasis is on helping the reader in understanding and constructing proofs and writing clear mathematics. This is achieved by exploring set theory, combinatorics and number theory, topics which include many fundamental ideas which are part of the tool kit of any mathematician. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. Over 250 problems include questions to interest and challenge the most able student as well as plenty of routine exercises to help familiarize the reader with the basic ideas.Trade Review'The book is written with understanding of the needs of students …' European Mathematical SocietyTable of ContentsPart I. Mathematical Statements and Proofs: 1. The language of mathematics; 2. Implications; 3. Proofs; 4. Proof by contradiction; 5. The induction principle; Part II. Sets and Functions: 6. The language of set theory; 7. Quantifiers; 8. Functions; 9. Injections, surjections and bijections; Part III. Numbers and Counting: 10. Counting; 11. Properties of finite sets; 12. Counting functions and subsets; 13. Number systems; 14. Counting infinite sets; Part IV. Arithmetic: 15. The division theorem; 16. The Euclidean algorithm; 17. Consequences of the Euclidean algorithm; 18. Linear diophantine equations; Part V. Modular Arithmetic: 19. Congruences of integers; 20. Linear congruences; 21. Congruence classes and the arithmetic of remainders; 22. Partitions and equivalence relations; Part VI. Prime Numbers: 23. The sequence of prime numbers; 24. Congruence modulo a prime; Solutions to exercises.

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Book SynopsisThis tract presents an exposition of methods for testing sets of special functions for completeness and basis properties, mostly in L2 and L2 spaces. The emphasis on methods of testing and their applications will also interest scientists and engineers engaged in fields such as the sampling theory of signals in electrical engineering and boundary value problems in mathematical physics.Table of ContentsPreface; 1. Foundations; 2. Orthogonal sequences; 3. Non-orthogonal sequences; 4. Differential and integral operators; Appendix; Bibliography; Subject index.

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Book SynopsisPractical Foundations collects the methods of construction of the objects of twentieth-century mathematics. Although it is mainly concerned with a framework essentially equivalent to intuitionistic Zermelo-Fraenkel logic, the book looks forward to more subtle bases in categorical type theory and the machine representation of mathematics. Each idea is illustrated by wide-ranging examples, and followed critically along its natural path, transcending disciplinary boundaries between universal algebra, type theory, category theory, set theory, sheaf theory, topology and programming. Students and teachers of computing, mathematics and philosophy will find this book both readable and of lasting value as a reference work.Trade ReviewReview of the hardback: 'This is a fascinating and rewarding book … each chapter has several pages of subtle, provocative and imaginative exercises. In summary, it is a magnificent compilation of ideas and techniques: it is a mine of (well-organised) information suitable for the graduate student and experienced researcher alike.' Roy Dyckhoff, Bulletin of the London Mathematical SocietyTable of Contents1. First order reasoning; 2. Types and induction; 3. Posets and lattices; 4. Cartesian closed categories; 5. Limits and colimits; 6. Structural recursion; 7. Adjunctions; 8. Algebra with dependent types; 9. The quantifiers.

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Book SynopsisThis 2006 book was the first of its kind to specifically address the comprehensive introduction to the mathematical principles needed by modern social scientists. The material introduces basic mathematical principles necessary to do analytical work in the social sciences. The core purpose is to present fundamental notions in standard notation and standard language.Trade Review"Ultimately, math is a language, and Jeff Gill's new book is a comprehensive guide to its grammar, syntax, and vocabulary. The book covers essential topics for advanced social science research and will be an excellent teaching, reference, and self-study tool." --Scott Desposato, University of California, San Diego"The body of topics that Gill covers is both appropriate and desirable for any primer in mathematics. The use of substantive examples is really outstanding, and makes this a uniquely attractive commodity for political science faculty who would be teaching a mathematical research/methods class over the course of a semester." --Alan Wiseman, Ohio State UniversityTable of Contents1. The basics; 2. Analytic geometry; 3. Linear algebra: vectors, matrices and operations; 4. Linear algebra continued: matrix structure; 5. Elementary scalar calculus; 6. Additional topics in scalar and vector calculus; 7. Probability theory; 8. Random variables; 9. Markov chains.

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Book SynopsisThe is an introduction to simple type theory, exploring the relationship between proof and calculation. Each of its 52 sections ends with a set of exercises, some 200 in total. These are designed to help the reader get to grips with the subject. An appendix contains complete solutions to them.Trade Review'A well-written introduction to proof theory and its connections with computability.' Leon Harkleroad, Zentralblatt für Mathematik'… recommended for the student or researcher who's been exposed to bits and pieces of the Curry-Howard correspondence, but wants a sharper idea of the big picture and is willing to work through the exercises to see how the details fit together. Simmons has succeeded in pulling together the main fruits of the correspondence for simple types in a single text. … It can't be emphasized enough that the great thing about this book is its many well-chosen completely solved exercises. This alone makes it a valuable text, especially for self-study.' ACM SIGACT NewsTable of ContentsIntroduction; Preview; Part I. Development and Exercises: 1. Derivation systems; 2. Computation mechanisms; 3. The typed combinator calculus; 4. The typed l-calculus; 5. Substitution algorithms; 6. Applied l-calculi; 7. Multi-recursive arithmetic; 8. Ordinals and ordinal notation; 9. Higher order recursion; Part II. Solutions: A. Derivation systems; B. Computation mechanisms; C. The typed combinator calculus; D. The typed l-calculus; E. Substitution algorithms; F. Applied l-calculi; G. Multi-recursive arithmetic; H. Ordinals and ordinal notation; I. Higher order recursion; Postview; Bibliography; Commonly used symbols; Index.

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Book SynopsisIn this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra. Starting with intuitive descriptions of mathematically and physically common phenomena, it leads up to a precise specification of the Category of Sets. Suitable for advanced undergraduates and beginning graduate students.Trade Review"...the categorical approach to mathematics has never been presented with greater conviction than it has in this book. The authors show that the use of categories in analyzing the set concept is not only natural, but inevitable." Mathematical Reviews"To learn set theory this way means not having to relearn it later.... Recommended." ChoiceTable of ContentsForeword; 1. Abstract sets and mappings; 2. Sums, monomorphisms and parts; 3. Finite inverse limits; 4. Colimits, epimorphisms and the axiom of choice; 5. Mapping sets and exponentials; 6. Summary of the axioms and an example of variable sets; 7. Consequences and uses of exponentials; 8. More on power sets; 9. Introduction to variable sets; 10. Models of additional variation; Appendices; Bibliography.

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Book SynopsisOne of the most remarkable recent occurrences in mathematics is the refounding, on a rigorous basis, of the idea of infinitesimal quantity, a notion which played an important role in the early development of the calculus and mathematical analysis. In this new edition basic calculus, together with some of its applications to simple physical problems, are presented through the use of a straightforward, rigorous, axiomatically formulated concept of 'zero-square', or 'nilpotent' infinitesimal - that is, a quantity so small that its square and all higher powers can be set, literally, to zero. The systematic employment of these infinitesimals reduces the differential calculus to simple algebra and, at the same time, restores to use the âœinfinitesimalâ methods figuring in traditional applications of the calculus to physical problems - a number of which are discussed in this book. This edition also contains an expanded historical and philosophical introduction.Trade Review'This might turn out to be a boring, shallow book review: I merely LOVED the book...the explanations are so clear, so considerate; the author must have taught the subject many times, since he anticipates virtually every potential question, concern, and misconception in a student's or reader's mind.' MAA Reviews'John Bell has done a first rate job in presenting an elementary introduction to this fascinating subject ... I recommend it highly.' J. P. Mayberry, British Journal for the Philosophy of ScienceTable of ContentsIntroduction; 1. Basic features of smooth worlds; 2. Basic differential calculus; 3. First applications of the differential calculus; 4. Applications to physics; 5. Multivariable calculus and applications; 6. The definite integral: Higher order infinitesimals; 7. Synthetic geometry; 8. Smooth infinitesimal analysis as an axiomatic system; Appendix; Models for smooth infinitesimal analysis.

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Book SynopsisExplores fundamental concepts in arithmetic. This book begins with the definitions and properties of algebraic fields. It then discusses the theory of divisibility from an axiomatic viewpoint, rather than by the use of ideals. It also gives an introduction to p-adic numbers and their uses, which are important in modern number theory.Table of ContentsCh. I Algebraic Fields 1 Ch. II Theory of Divisibility (Kronecker, Dedekind) 33 Ch. III Local Primadic Analysis (Kummer, Hensel) 71 Ch. IV Algebraic Number Fields 141 Amendments 223

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Book SynopsisJohn Napier (1550-1617) is celebrated today as the man who invented logarithms--an enormous intellectual achievement that would soon lead to the development of their mechanical equivalent in the slide rule: the two would serve humanity as the principal means of calculation until the mid-1970s. Yet, despite Napier's pioneering efforts, his life andTrade Review"John Napier fills a gap concerning an important, and often ignored, chapter of mathematical history."--George Szpiro, Nature "In this engaging book, we learn more about Napier the mathematician, the religious zealot, the person."--Devorah Bennu, The Guardian, Grrl Scientist "Edinburgh born John Napier, the inventor of logarithms, is in danger of fading into the shadows of the scientific landscape. In the new book John Napier: Life, Logarithms, and Legacy, Julian Havil does a marvelous job of bringing Napier back into the spotlight."--Stephanie Blanda, American Mathematical Society blog "I'm sure after reading this entertaining and enjoyable book, Napier will climb some rungs on your ladder of famous mathematicians."--A. Bultheel, European Mathematical Society "Havil ... gives a rich history of Napier's involvement in the Protestant reformation, his introduction of logarithms, and his legacy."--Choice "With this book, the author continues his impressive series of illuminating, accessible monographs on the history of mathematics."--Bart J. I. Van Kerkhove, Mathematical Review "This book fills a clear gap in published work on Napier and is likely to be the standard point of departure for those interested in his life and work for some years to come."--Mark McCartney, London Mathematical Society Newsletter "It is clearly a very interesting book."--Ernesto Nungesser, Irish Math Society Bulletin "Havil's attention to detail is without equal in the opinion of this reviewer."--John A. Adam, ScotiaTable of ContentsAcknowledgments xv Introduction 1 Chapter One Life and Lineage 8 Chapter Two Revelation and Recognition 35 Chapter Three A New Tool for Calculation 62 Chapter Four Constructing the Canon 96 Chapter Five Analogue and Digital Computers 131 Chapter Six Logistics: The Art of Computing Well 155 Chapter Seven Legacy 179 Epilogue 207 Appendix A Napier's Works 209 Appendix B The Scottish Science Hall of Fame 210 Appendix C Scotland and Conflict 211 Appendix D Scotland and Reformation 216 Appendix E A Stroll Down Memory Lane 220 Appendix F Methods of Multiplying 229 Appendix G Amending Napier's Kinematic Model 232 Appendix H Napier's Inequalities 233 Appendix I Hos Ego Versiculos Feci 236 Appendix J The Rule of Three 238 Appendix K Mercator's Map 250 Appendix L The Swiss Claimant 264 References 270 Index 275

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Book SynopsisThe mathematics of ancient Egypt was fundamentally different from our math today. Contrary to what people might think, it wasn't a primitive forerunner of modern mathematics. This title provides an introduction to the intuitive and often-surprising art of ancient Egyptian math.Trade Review"Count Like an Egyptian would make an excellent addition to math classrooms at many different levels. Reimer includes problems in the text and solutions in the back of the book, so the reader can practice techniques and get a feel for exactly how the system works as they go through the book. The mathematics is basic enough to be helpful for children learning fractions or multiplication for the first time, but it's also different enough from the methods most of us know that adults will get a lot out of it as well."--Evelyn Lamb, Scientific American "History lovers will gain much more than just insight into the Egyptian mind-set. The author interleaves mathematical exposition with short essays on Egyptian history, culture, geography, mythology--all, like the rest of the book, beautifully illustrated... For a lively and inquiring mind the book has a good deal to offer. It is well written, lavishly illustrated, and just awfully interesting. The book is a pleasure to hold, to browse, and to read."--Alexander Bogomolny, Cut the Knot "You get the feeling that David Reimer must be a pretty entertaining teacher. An associate professor of mathematics at the College of New Jersey, he has taken on the task of explaining ancient math systems by having you use them. And though it's not easy, he manages to lead you, step by step, through a hieroglyphic based calculation of how many 10-pesu loaves of bread you can make from seven hekat of grain."--Nancy Szokan, Washington Post "An interesting combination of history, ancient literature and mythology, arithmetic puzzles and mathematics, and lavishly illustrated with numerous colour diagrams, this engaging book is unusual, thought-provoking and just plain fun to read."--Devorah Bennu, GrrlScientist, The Guardian "Count Like an Egyptian is a beautifully illustrated and well-written book... Reimer's overriding goal is to demonstrate that Egyptian fraction arithmetic is fascinating, versatile, and well suited for whatever calls fractions into existence... By working through the material Reimer patiently and gently presents, the reader will have a more thorough understanding and appreciation of how Egyptian scribes made the calculations needed to administer an empire bent on building pyramids and granaries, surveying flooded riverside property, digging irrigation basins, and rationing or exchanging bread and beer supplies amongst its gangs of workers... This book should find a home in libraries used by middle school and high school mathematics teachers. It also provides a good resource for mathematics education professors and their students on the college level as they explore historical beginnings of mathematical ideas, make cultural comparisons, and develop interdisciplinary connections."--Calvin Jongsma, MAA Reviews "An interesting combination of history, ancient literature and mythology, arithmetic puzzles and mathematics, and lavishly illustrated with numerous colour diagrams, this engaging book is unusual, thought-provoking and just plain fun to read."--GrrrlScientist "This amusing popular introduction to an uncommon subject is a mental adventure that sheds new light on the thought processes of a lost civilization and will appeal both to those who enjoy mathematical puzzles and to Egyptophiles."--Edward K. Werner, Library Journal "In general I really like this book and believe it is, if not necessarily a must for all Egyptophiles, then definitely one to put on the wish list as an interesting addition to your bookshelf... It is fun way of working through complicated and yet practical mathematics which makes the Rhind Papyrus come alive and gives an insight into the logical brain of ancient Egyptian scribes."--Charlotte Booth, charlottesegypt.com "Reimer succeeds very well in transferring his enthusiasm tor the Egyptian system to the reader. The reactions from his students who were used tor a try-out are claimed to be positive. But even if you do not want to graduate as an Egyptian scribe, you may be charmed by the witty Egyptian system and you will be delighted by the colourful illustrations and Reimer's entertaining account of it all."--A. Bultheel, European Mathematical Society "Count Like an Egyptian takes the reader step-by-step through the ancient Egyptian methods, which are surprisingly different from our own, and yet, in the capable hands of author David Reimer, surprisingly understandable. This lovely book has fun illustrations to demonstrate the various operations, basic geometry, and other tasks faced by the scribes... This book is a pleasure to read and makes Egyptian math a pleasure to learn."--Gretchen Wagner, San Francisco Book Review "The book is intended to be used as a teaching tool and includes practice examples for the student. It would be difficult to imagine a work that more effectively covers this aspect of the ancient civilization."--JPP, Ancient Egypt "David Reimer succeeds in keeping the mathematics in Count Like an Egyptian clever and light, raising this book into a rare category: a coffee table book that is serious and fun."--Robert Schaefer, New York Journal of Books "This volume is ideal for anyone, and I truly mean anyone, young or old, mathematician, student or teacher, who wants to learn how the ancient Egyptians did mathematics... This book has all the Egyptian mathematics a general mathematician, teacher or student could ever want to learn. In particular it would be a perfect resource for a schoolteacher, elementary through lower division college. The material is presented in a direct and accessible manner."--Amy Shell-Gellasch, CSHPM Bulletin "Overall this is a didactic and well written book, with many important illustrations, with some incursions in the mathematics of other ancient cultures."--European Mathematical Society "With Reimer's guidance, motivating stories, and lighthearted remarks, readers can become facile with Egyptian algorithms and the insights they reveal... Valuable for all readers looking for a guided of an alternative to traditional school arithmetic and the torpor that algorithmic training causes."--Choice "[T]his book is a worthwhile read for anyone interested in seeing exactly how ancient Egyptians dealt with mathematics. It will help put our present algorithms into perspective as simply one of many possible algorithms one could use to perform arithmetic operations."--Victor J. Katz, Mathematical Reviews Clippings "[Reimer] ... set himself to understand and explain the ancient methods, and the result is an approachable, thorough and lavishly-produced book."--Owen Toller, Mathematical Gazette "Count like an Egyptian is a beautifully glossy and colourful book; the presentation of hieroglyphs is particularly well done, and fully interated into the surrounding text... This book has given me a new perspective on day-to-day arithmetic."--Christopher Hollings, Mathematics Today "This is a wonderful book, very well written, filled with illustrations on every page, witty, addressing anyone interested in grade school arithmetic."--Victor V. Pambuccian, Zentralblatt MATH "Count Like an Egyptian is important for anyone interested in alternative algorithms... If you want to roll up your sleeves and learn some new mathematics, this is the book for you."--Michael Manganello, Mathematics Teacher "An engaging and beautifully illustrated book that deals with the basics of ancient Egyptian mathematics, set in the wider context of other ancient mathematical systems."--Corinna Rossi, Aestimatio "A great approach and a dedicated effort. One hopes the book will reflect that persistence and it does... This is a book that comes recommended, for anyone who wants to know where our current basis of mathematics comes from through to those with an interest in maths and history."--Gordon Clarke, Gazette of the Australian Mathematical SocietyTable of ContentsPreface vii Introduction ix Computation Tables xi 1 Numbers 1 2 Fractions 13 3 Operations 22 4 Simplification 55 5 Techniques and Strategies 80 6 Miscellany 121 7 Base-Based Mathematics 144 8 Judgment Day 182 Practice Solutions 209 Index 235

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Book SynopsisTrade Review"Offers a clear and beautiful progression from addition to modern number theory."--Math-Blog "The authors did a remarkable job in making some aspects of modern number theory very accessible to readers with only a minimal knowledge of mathematics, say a student who had a first calculus course. However, also mathematicians who do not have number theory as their main focus will enjoy this book."--Adhemar Bultheel, European Mathematical Society "Ash and Gross do a masterful job of leading students from finite sums to modular forms and to the forefront of modern number theory... This is an excellent piece of mathematical writing."--Choice "[A]n accessible and fun introduction to modular forms... [Summing It Up] is engaging and conversational, without losing accuracy or essential rigor."--Dominic Lanphier, American Mathematical MonthlyTable of Contents*Frontmatter, pg. i*CONTENTS, pg. vii*PREFACE, pg. xi*ACKNOWLEDGMENTS, pg. xv*INTRODUCTION: WHAT THIS BOOK IS ABOUT, pg. 1*CHAPTER 1. PROEM, pg. 11*CHAPTER 2. SUMS OF TWO SQUARES, pg. 22*CHAPTER 3. SUMS OF THREE AND FOUR SQUARES, pg. 32*CHAPTER 4. SUMS OF HIGHER POWERS: WARING'S PROBLEM, pg. 37*CHAPTER 5. SIMPLE SUMS, pg. 42*CHAPTER 6. SUMS OF POWERS, USING LOTS OF ALGEBRA, pg. 50*CHAPTER 7. INFINITE SERIES, pg. 73*CHAPTER 8. CAST OF CHARACTERS, pg. 96*CHAPTER 9. ZETA AND BERNOULLI, pg. 103*CHAPTER 10. COUNT THE WAYS, pg. 110*CHAPTER 11. THE UPPER HALF-PLANE, pg. 127*CHAPTER 12. MODULAR FORMS, pg. 147*CHAPTER 13. HOW MANY MODULAR FORMS ARE THERE?, pg. 160*CHAPTER 14. CONGRUENCE GROUPS, pg. 179*CHAPTER 15. PARTITIONS AND SUMS OF SQUARES REVISITED, pg. 186*CHAPTER 16. MORE THEORY OF MODULAR FORMS, pg. 201*CHAPTER 17. MORE THINGS TO DO WITH MODULAR FORMS: APPLICATIONS, pg. 213*BIBLIOGRAPHY, pg. 225*INDEX, pg. 227

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Book SynopsisTrade Review"Having a healthy skepticism toward numbers and giving readers the tools to think about math more logically is the purpose of this easily read, slight book. Brian W. Kernighan adroitly distills complex issues. His tone is more that of a mellow friend breaking down a concept that flummoxes you rather than an Ivy League professor expounding on the elegance of numbers."---Jacqueline Cutler, NJ.com"Numbers, graphs and statistics can often be misleading and misrepresented. In Millions, Billions, Zillions: Defending Yourself in a World of Too Many Numbers, Kernighan provides the reader with an entertaining and useful guide to avoid becoming a victim of number abuse."---Ben Rothke, RSA Conference"I can wholeheartedly recommend reading this book, because of the infectious way the author describes his interaction with numbers."---J. Herret, International Mathematical News"This is a must-read for anyone looking to cure their “number numbness”"---Tibi Puiu, ZME Science

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Book SynopsisTrade Review"Nahin has written a beautifully clear, fascinating book on a topic which is truly vital to so many areas of science and I would recommend anyone who enjoys puzzle solving and having new tools to tackle old (or new) problems should read it."---Jonathan Shock, Mathemafrica

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Book SynopsisContains translations of papers on propositional satisfiability and related logical problems which appeared in ""Problemy Sokrashcheniya Perebora"", published in Russian in 1987.Table of ContentsAlgorithmics of $NP$-hard problems by R. I. Freidzon Algorithmics of propositional satisfiability problems by E. Ya. Dantsin Semantics of S. Yu. Maslov's iterative method by V. Ya. Kreinovich Ergodic properties of Maslov's iterative method by M. I. Zakharevich Anomalous properties of Maslov's iterative method by I. V. Melichev Possible nontraditional methods of establishing unsatisfiability of propositional formulas by Yu. V. Matiyasevich Dual algorithms in discrete optimization by G. V. Davydov and I. M. Davydova Models, methods, and modes for the synthesis of program schemes by A. Ya. Dikovsky Effective calculi as a technique for search reduction by M. I. Kanovich Lower bounds of combinatorial complexity for exponential search reduction by N. K. Kossovskii On a class of polynomial systems of equations following from the formula for total probability and possibilities for eliminating search in solving them by An. A. Muchnik S. Maslov's iterative method: 15 years later (freedom of choice, neural networks, numerical optimization, uncertainty reasoning, and chemical computing) by V. Kreinovich.

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Book SynopsisContains the proceedings of the first DIMACS workshop. This work covers topics including multicommodity flows, graph matchings and colorings, the traveling salesman problem, integer programming and complexity theory. It is suitable for researchers in combinatorics and combinatorial optimization.Table of ContentsMatrix cones, projection representations, and stable set polyhedra by L. Lovasz and A. Schrijver A generalization of Lovasz's $\theta$ function by G. Narasimhan and R. Manber On cutting planes and matrices by A. M. H. Gerards Random volumes in the $n$-cube by M. E. Dyer, A. Furedi, and C. McDiarmid Test sets for integer programs, $\forall \exists$ sentences by R. Kannan Solvable classes of generalized traveling salesman problems by S. N. Kabadi and R. Chandrasekaran Handles and teeth in the symmetric traveling salesman polytope by D. Naddef On the complexity of branch and cut methods for the traveling salesman problem by W. Cook and M. Hartmann Existentially polytime theorems by K. Cameron and J. Edmonds The width-length inequality and degenerate projective planes by A. Lehman On Lehman's width-length characterization by P. D. Seymour Applications of polyhedral combinatorics to multicommodity flows and compact surfaces by A. Schrijver Vertex-disjoint simple paths of given homotopy in a planar graph by A. Frank and A. Schrijver On disjoint homotopic paths in the plane by A. Frank On the complexity of the disjoint paths problem (extended abstract) by M. Middendorf and F. Pfeiffer The paths-selection problem by M. Middendorf and F. Pfeiffer Planar multicommodity flows, max cut, and the Chinese postman problem by F. Barahona The cographic multiflow problem: An epilogue by A. Sebo Exact edge-colorings of graphs without prescribed minors by O. Marcotte On the chromatic index of multigraphs and a conjecture of Seymour, (II) by O. Marcotte Spanning trees of different weights by A. Schrijver and P. D. Seymour.

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Book SynopsisWant to sharpen your mathematical wits? If so, then Mathematical Conundrums is for you. Daily Telegraph enigmatologist, Barry R. Clarke, presents over 120 fiendish problems that will test both your ingenuity and persistence. Between these covers are puzzles in geometry, arithmetic, and algebra (there is even a section for computer programmers). And, for the smartest readers who wish to stretch their mind to its limits, a selection of engaging logic and visual lateral puzzles is included. Although no puzzle requires a greater knowledge of mathematics than the high school curriculum, this collection will take you to the edge. But are you equal to the challenge? Features High-school level of mathematics is the only pre-requisite Variety of algebraic, route-drawing, and geometrical conundrums Hints section for the lateral puzzles Warm-up excercises to sharpen the wits Full solutions to every problem Barry R. Clarke has published over 1,500 puzzles in The Daily Telegraph and has contributed enigmas to New Scientist, The Sunday Times, Readerâs Digest, The Sunday Telegraph, and Prospect magazine. His book Challenging Logic Puzzles Mensa has sold over 100,000 copies. As well as a PhD in Shakespeare Studies, Barry has a masterâs degree and academic publications in quantum physics. He is now working on a revised theory of the hydrogen atom. Other skills include mathematics tutor, filmmaker, comedy-sketch writer, cartoonist, computer programmer, and blues guitarist! For more information please visit http://barryispuzzled.com.Table of Contents1. Introduction. 2. Mind sharpeners. 3. Geometry. 4. Arithmetic. 5. Algebra. 6. Programmable Puzzles. 7. Logic. 8. Visual-lateral

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Book SynopsisExpanding Mathematical Toolbox: Interweaving Topics, Problems, and Solutions offers several topics from different mathematical disciplines and shows how closely they are related. The purpose of this book is to direct the attention of readers who have an interest in and talent for mathematics to engaging and thought-provoking problems that should help them change their ways of thinking, entice further exploration and possibly lead to independent research and projects in mathematics. In spite of the many challenging problems, most solutions require no more than a basic knowledge covered in a high-school math curriculum.To shed new light on a deeper appreciation for mathematical relationships, the problems are selected to demonstrate techniques involving a variety of mathematical ideas. Included are some interesting applications of trigonometry, vector algebra and Cartesian coordinate system techniques, and geometrical constructions and inversion in solving mechanical engineering problems and in studying models explaining non-Euclidean geometries.This book is primarily directed at secondary school teachers and college professors. It will be useful in teaching mathematical reasoning because it emphasizes how to teach students to think creatively and strategically and how to make connections between math disciplines. The text also can be used as a resource for preparing for mathematics Olympiads. In addition, it is aimed at all readers who want to study mathematics, gain deeper understanding and enhance their problem-solving abilities. Readers will find fresh ideas and topics offering unexpected insights, new skills to expand their horizons and an appreciation for the beauty of mathematics.Trade ReviewThis is a collection of mathematical theory and exercises organized in chapters, each one devoted to a different topic (geometrical constructions, Euclidean vectors, inequalities, trigonometry, etc.). The format of each chapter is a general introduction about the topic including some theory followed by a, usually short, problem formulation, and an extensive discussion of the solution which may include more theory and proofs, triggering new exercises etc. Emphasis is on the interaction of topics that are usually treated separately in classical text books. The assumed mathematical knowledge of the reader is at the level of secondary schools or beginning university.The problems can look astonishingly difficult at first sight, baffling the reader. Applying straightforward methods, is usually not the way to solve them. The proposed solution may illustrate some clever trick that makes the problem easy to solve. So the problem formulation is supposed to trigger curiosity and stimulate to look for the appropriate key to crack it. The diversity of the topics and the kind of problems, gives it the allure of a puzzle book. Thinking outside the box and recognizing patterns is often more important than the mathematical prerequisites. The latter are provided anyway, including proofs, either in the solution or in the appendix. There are also excursions beyond the exercises, like for example a short introduction to non-Euclidean geometry.So this is an unusual mixture of theory and exercises. The latter are certainly not of the drilling type, some even reach the level of mathematical Olympiads. Most problems have some geometrical aspect, even the algebraic ones, because these problems are easier to understand by a general public. This implies that calculus and analysis are more in the background. It is an excellent book to prepare for mathematics Olympiads, and teachers may find here inspiration for their lessons.Reviewer: Adhemar Bultheel (Leuven)MSC:00A09Popularization of mathematics51-01Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometryKeywords:geometry; algebraic geometry; trigonometry; Euclidean vector space; puzzles; math Olympiad; inequalitiesThis is a collection of mathematical theory and exercises organized in chapters, each one devoted to a different topic (geometrical constructions, Euclidean vectors, inequalities, trigonometry, etc.). The format of each chapter is a general introduction about the topic including some theory followed by a, usually short, problem formulation, and an extensive discussion of the solution which may include more theory and proofs, triggering new exercises etc. Emphasis is on the interaction of topics that are usually treated separately in classical text books. The assumed mathematical knowledge of the reader is at the level of secondary schools or beginning university.The problems can look astonishingly difficult at first sight, baffling the reader. Applying straightforward methods, is usually not the way to solve them. The proposed solution may illustrate some clever trick that makes the problem easy to solve. So the problem formulation is supposed to trigger curiosity and stimulate to look for the appropriate key to crack it. The diversity of the topics and the kind of problems, gives it the allure of a puzzle book. Thinking outside the box and recognizing patterns is often more important than the mathematical prerequisites. The latter are provided anyway, including proofs, either in the solution or in the appendix. There are also excursions beyond the exercises, like for example a short introduction to non-Euclidean geometry.So this is an unusual mixture of theory and exercises. The latter are certainly not of the drilling type, some even reach the level of mathematical Olympiads. Most problems have some geometrical aspect, even the algebraic ones, because these problems are easier to understand by a general public. This implies that calculus and analysis are more in the background. It is an excellent book to prepare for mathematics Olympiads, and teachers may find here inspiration for their lessons.Reviewer: Adhemar Bultheel (Leuven)MSC:00A09Popularization of mathematics51-01Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometryKeywords:geometry; algebraic geometry; trigonometry; Euclidean vector space; puzzles; math Olympiad; inequalitiesTable of Contents1. Beauty in Mathematics. 2. Euclidean Constructions. 3. Inversion and its properties. 4. Using Geometry for Algebra. Classic Mean Averages’ geometrical interpretations. 5. Using Algebra for Geometry. 6. Trigonometrical explorations. 7. Euclidean Vectors. 8. Cartesian coordinates in problem solving. 9. Inequalities Wonderland. 10. Guess and Check game.

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Book SynopsisCan you really keep your eye on the ball? How is massive data collection changing sports?Sports science courses are growing in popularity. The author's course at Roanoke College is a mix of physics, physiology, mathematics, and statistics. Many students of both genders find it exciting to think about sports. Sports problems are easy to create and state, even for students who do not live sports 24/7. Sports are part of their culture and knowledge base, and the opportunity to be an expert on some area of sports is invigorating. This should be the primary reason for the growth of mathematics of sports courses: the topic provides intrinsic motivation for students to do their best work.From the Author:The topics covered in Sports Science and Sports Analytics courses vary widely. To use a golfing analogy, writing a book like this is like hitting a drive at a driving range; there are many Trade ReviewThe book is written at a level that is accessible to a large audience. It contains a small number of applications that make use of calculus; otherwise, only a high school level mathematics background is required. Furthermore, one can easily skip over those sections that require calculus and still have plenty of accessible material to read.Sports Math is well written and easy to read. The book should appeal to anyone interested in the quantitative aspects of athletics. Each chapter of the books ends with a fairly large number of exercises and also pointers to further reading. Thus, the book could be used not only as a textbook for a course but also as a nice resource for student projects.~Mathematical Reviews, 2017Minton presents a textbook based on the current status of a sport science course that has evolved since he began teaching it in 1988. He offers a sample of topics that he knows something about and finds interesting, and hopes that instructors and students will find the book useful. His topics are projectile motion, rotational motion, sports illusions, collisions, ratings systems, voting systems, saber- and other metrics, randomness in sports, sports strategies, and big data and beyond.~ProtoView, 2017 This work discusses how mathematics is used to analyze popular American sports like football, baseball, and basketball. Minton (mathematics, Roanoke College) has based this book on several of his undergraduate courses. The book covers two major aspects: the physics involved in sports (e.g., the motion of a ball) and the statistics used to make probabilistic ratings of performance and success. The beginning chapters consider topics from mechanics, such as “Projectile Motion,” “Rotational Motion,” and “Collisions.” The rest of the text is devoted to statistics used in sports ratings and analysis, with many examples from specific games played in the big leagues or by major colleges. The material covered is selective and quirky; the level of analytical mathematics and statistics ranges from simple to advanced, including calculus, matrixes, and game theory. Each chapter has solved examples and end-of-chapter questions, problems, and suggestions for projects. There are pictures and graphs interspersed throughout the text. The book is not suitable as a standard text in any conventional course—it will best serve as a supplement.--N. Sadanand, Central Connecticut State University 2018The book is written at a level that is accessible to a large audience. It contains a small number of applications that make use of calculus; otherwise, only a high school level mathematics background is required. Furthermore, one can easily skip over those sections that require calculus and still have plenty of accessible material to read.Sports Math is well written and easy to read. The book should appeal to anyone interested in the quantitative aspects of athletics. Each chapter of the books ends with a fairly large number of exercises and also pointers to further reading. Thus, the book could be used not only as a textbook for a course but also as a nice resource for student projects.~Mathematical Reviews, 2017Minton presents a textbook based on the current status of a sport science course that has evolved since he began teaching it in 1988. He offers a sample of topics that he knows something about and finds interesting, and hopes that instructors and students will find the book useful. His topics are projectile motion, rotational motion, sports illusions, collisions, ratings systems, voting systems, saber- and other metrics, randomness in sports, sports strategies, and big data and beyond.~ProtoView, 2017This work discusses how mathematics is used to analyze popular American sports like football, baseball, and basketball. Minton (mathematics, Roanoke College) has based this book on several of his undergraduate courses. The book covers two major aspects: the physics involved in sports (e.g., the motion of a ball) and the statistics used to make probabilistic ratings of performance and success. The beginning chapters consider topics from mechanics, such as “Projectile Motion,” “Rotational Motion,” and “Collisions.” The rest of the text is devoted to statistics used in sports ratings and analysis, with many examples from specific games played in the big leagues or by major colleges. The material covered is selective and quirky; the level of analytical mathematics and statistics ranges from simple to advanced, including calculus, matrixes, and game theory. Each chapter has solved examples and end-of-chapter questions, problems, and suggestions for projects. There are pictures and graphs interspersed throughout the text. The book is not suitable as a standard text in any conventional course—it will best serve as a supplement.--N. Sadanand, Central Connecticut State University 2018Table of ContentsProjectile Motion. Rotational Motion. Sports Illusions. Collisions. Rating Systems. Voting Systems. Saber- and Other Metrics. Randomness in Sports. In-Game Strategies. Predictive Analytics.

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Book SynopsisA Stitch in Line: Mathematics and One-Stitch Sashiko provides readers with instructions for creating hitomezashi items with minimum outlay. The reader is guided through the practical steps involved in creating each design, and then the mathematics which underpins it is explained in a friendly, accessible way. This is a fantastic book for anyone who is interested in recreational mathematics and/or fibre arts and can be a useful resource for teaching and learning mathematical concepts in a fun and engaging format. Features Numerous full-colour photographs of hitomezashi stitch patterns which have been mathematically designed. Suitable for readers of all mathematical levels and backgrounds no prior knowledge is automatically assumed. A compressed encoding for recording and designing hitomezashi patterns to be stitched or drawn. Accessible explanations and explorations of mathematical concepts inherent in, or illustr

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Book SynopsisTeaching and Learning Mathematics Online, Second Edition continues to present meaningful and practical solutions for teaching mathematics and statistics online. It focuses on the problems observed by mathematics instructors currently working in the field who strive to hone their craft and share best practices with the community. The book provides a set of standard practices, improving the quality of online teaching and the learning of mathematics. Instructors will benefit from learning new techniques and approaches to delivering content. New to the Second Edition Nine brand new chapters Reflections on the lessons of COVID-19 Explorations of new technological opportunities.

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Book SynopsisString art is a well-known and popular activity that uses string, a board, and nails to produce artistic images (although there are variations that use different modalities). This activity is beloved because simple counting rules are used to create beautiful images that can both adorn walls and excite young minds. The downside of this highly tactile activity is that it is quite time-consuming and rigid. By contrast, electronic string art offers much more flexibility to set up or change nail locations and counting rules, and the images created from those changes change instantaneously.Electronic String Art: Rhythmic Mathematics invites readers to use the author's digital resources available on the ESA website to play with the parameters inherent in string art models while offering concise, accessible explanations of the underlying mathematical principles regarding how the images were created and how they change. Readers will have the opportunity to creaTable of ContentsPart I. Preliminary Issues. 1. Introduction and Overview. 2. How Polygons are Drawn. 3. String Art Basics. 4. Issues involving Commonality. 5. Cycles. 6. Alternative ways to Obtain an Image. 7. Levels of Subdivision Points. 8. Shape-Shifting Polygons. 9. An Overarching Question. 10. Functionally Modified String Art files. 11. A sampling of Image Archetypes. 12. n = P images. 13. 60-Second Images. 14. Challenge Questions for Part II. 15. Centered-Point Flowers. 16. Double Jump Models. 17. Four Color Clock Arithmetic. 18. Larger Jump Set Models. 19. Busting out of our Polygonal Constraint. 20. Challenge Questions for Part III. 21. Basic Properties of Numbers. 22. Angles in Polygons and Stars. 23. Modular Arithmetic. 24. Modular Multiplicative Inverses, MMI. 25. A Guide to the Web Model. 26. Suggestions for Mathematics Teachers.

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Book SynopsisThis book was developed to address a need. Quantitative Literacy courses have been established in the mathematics curriculum for decades now. The students in these courses typically dislike and fear mathematics, and the result is often a class populated by many students who are unmotivated and uninterested in the material. This book is a text for such a course; however, it is focused on a single idea that most students seem to already have some intrinsic interest in and is written at an accessible level. It covers the basic ideas of discrete probability and shows how these ideas can be applied to familiar games (roulette, poker, blackjack, etc.). The gambling material is interweaved through the book and introduced as soon as the necessary mathematics has been developed. Throughout, mathematical formalism and symbolism have been avoided, and numerous examples are provided. The book starts with a simple definition of probability, goes through some basic

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Book SynopsisNumber Systems: A Path into Rigorous Mathematics aims to introduce number systems to an undergraduate audience in a way that emphasises the importance of rigour, and with a focus on providing detailed but accessible explanations of theorems and their proofs.The book continually seeks to build upon students' intuitive ideas of how numbers and arithmetic work, and to guide them towards the means to embed this natural understanding into a more structured framework of understanding. The authorâs motivation for writing this book is that most previous texts, which have complete coverage of the subject, have not provided the level of explanation needed for first-year students. On the other hand, those that do give good explanations tend to focus broadly on Foundations or Analysis and provide incomplete coverage of Number Systems.Features Approachable for first year undergraduates, but still of interest to more advanced students and postgraduates Does not merely present definitions, theorems and proofs, but also motivates them in terms of intuitive knowledge and discusses methods of proof Draws attention to connections with other areas of mathematics Plenty of exercises for students, both straightforward problems and more in-depth investigations Introduces many concepts that are required in more advanced topics in mathematicsNew to the second edition Complete solutions to all exercises, and hints for the in-depth investigations Extensive changes to chapters 4 and 5, including defining integral domains as distinct from commutative rings, a more complete discussion of irreducibles, primes and unique factorisation, and more topics in elementary number theory A completely revised chapter 8, giving a more coherent account of quadratic rings and their unique (or non-unique) factorisation properties A thorough correction of typos and errors across all chapters Updates to the bibliography

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Book SynopsisThis text is the fifth and final in the series of educational books written by Israel Gelfand with his colleagues for high school students. These books cover the basics of mathematics in a clear and simple format - the style Gelfand was known for internationally. Gelfand prepared these materials so as to be suitable for independent studies, thus allowing students to learn and practice the material at their own pace without a class. Geometry takes a different approach to presenting basic geometry for high-school students and others new to the subject.  Rather than following the traditional axiomatic method that emphasizes formulae and logical deduction, it focuses on geometric constructions. Illustrations and problems are abundant throughout, and readers are encouraged to draw figures and move them in the plane, allowing them to develop and enhance their geometrical vision, imagination, and creativity. Chapters are structured so that only certaiTrade Review“This book is intended to engage the reader visually, tactilely, and kinesthetically. … It has a good set of material to enliven more traditional geometry instruction. … There are problems and exercises throughout. The exercises are accompanied by solutions.” (MAA Reviews, October 10, 2020) Table of ContentsPoints and Lines: A Look at Projective Geometry.- Parallel Lines: A Look at Affine Geometry.- Area: A Look at Symplectic Geometry.- Circles: A Look at Euclidean Geometry.

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Book SynopsisMath really is for everyoneso let's prove it.You've heard it from kids, from friends, and from celebrities: I'm bad at math. It's a line that society tends to accept without examinationafter all, some people just aren't math people, right?Wrong. As we do with other essential skills, we need to expose the stereotypes, challenge the negative mindsets, and finally confront the systemic opportunity gaps in math education, and replace them with a new vision for what math is, who it's for, and who can excel at it. In this book you'll find Research on teacher and student mindsets and their effect on student achievement Audience-specific and differentiated tools, reflection questions, and suggested actions for educators at all levels of the system Examples from popular media, as well as personal stories and anecdotes Quotes, data-driven figures, and suggestions for deeper learning on all aspects of a Trade ReviewOne of the saddest comments we often hear is "I was never any good at math." People blame themselves or the math. Rarely do they blame the mismatch between their cognitive and emotional needs and how they were being taught. In this engaging book, Lidia Gonzalez shines a light on the cultural, curricular, and classroom realities that are the real culprits. -- Steve LeinwandInsight into why we need to change the narrative, "I’m bad at math!" So many moments of "Yes, you hit the nail on the head!" Authentic stories and compelling evidence reveal how our society continues to perpetuate this harmful myth. There are abundant resources to help stakeholders dismantle systemic barriers that persist in math and math education and reflection questions for education professionals. Awesome work! -- Shelly M. JonesBad at Math? creates the space to unpack people’s dispositions about mathematics. Many people dislike how mathematics is used to position them as either competent or incompetent. This book provides the content and context for people to unpack mathematics as the tool that helps us critique and understand the world. -- Robert Q. Berry IIIThis book was a pleasure to read and reread! Though the main discussion is mathematics, it should be a must-read for all preschool through higher education professionals. It’s well written, and deeply rooted research tells the story. The long overdue, honest discussion is chock full of inclusive history and timely strategies positioning us to move forward and do better! -- Michele R. DeanWhat a powerful and thought-provoking book! Gonzalez does a masterful job of addressing what is wrong with mathematics education currently and what can be done to make mathematics more accessible for more students, particularly those who are marginalized. Through these changes, we can help make it less socially acceptable for people to say they are bad at math. -- Kevin J. DykemaThis book truly breaks down cultural norms to build up a powerful vision of mathematics for everyone. The engaging and thought-provoking discussions are paired with rich examples and resources that collectively create a powerful message to help us change the way math is perceived and achieved in schools. An important book for all education stakeholders! -- Jennifer Bay-Williams"Lidia Gonzalez does a masterful job sharing stories of student experiences in math class with a vision for creating access to all opportunities for all students. The material is written in a way that is very accessible for different education stakeholders, or for anyone who has experienced schooling. Bad at Math?: Dismantling Harmful Beliefs that Hinder Equitable Mathematics Education brings great clarity to how students and entire communities have been marginalized and continue to face oppression in mathematics spaces, along with actionable steps educators can take to disrupt negative beliefs about who belongs in mathematics spaces. Guided by the research and work of many brilliant mathematics educators, such as Sunil Singh, Dr. Peter Liljedahl, Dr. Rochelle Gutiérrez and Hema Khodai, just to name a few, this book shares many truths that need to acknowledged and addressed, which leads to ongoing opportunities for critical self-reflection throughout the book. Much gratitude for an excellent book." -- Jordan RappaportTable of ContentsIntroduction Chapter 1: What Does it Mean to be Good at Math? Chapter 2: Beyond Numbers and Equations: What is mathematics Chapter 3: Mathematicians and Mathematicians in Training Chapter 4: We are All Math People Chapter 5: Identity in Mathematics Education Chapter 6: School Mathematics Chapter 7: Mathematics as Gatekeeper Chapter 8: Achievement Gaps or Opportunity Gaps? Chapter 9: Is the School System Broken? Chapter 10: Teaching Mathematics as a Political Act Chapter 11: Where do we go from Here?

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Book SynopsisFourier analysis aims to decompose functions into a superposition of simple trigonometric functions, whose special features can be exploited to isolate specific components into manageable clusters before reassembling the pieces. This two-volume text presents a largely self-contained treatment, comprising not just the major theoretical aspects (Part I) but also exploring links to other areas of mathematics and applications to science and technology (Part II). Following the historical and conceptual genesis, this book (Part I) provides overviews of basic measure theory and functional analysis, with added insight into complex analysis and the theory of distributions. The material is intended for both beginning and advanced graduate students with a thorough knowledge of advanced calculus and linear algebra. Historical notes are provided and topics are illustrated at every stage by examples and exercises, with separate hints and solutions, thus making the exposition useful both as a course Trade Review'[Fourier Analysis: Volume l - Theory is] fabulous … Constantin structures his exercise sets beautifully, I think: they are abundant and long, covering a spectrum of levels of difficulty; each set is followed immediately by a section of hints (in one-one correspondence); finally the hints sections are followed by very detailed and well-written solutions (also bijectively). Can there be any clearer homage to the maxim that to learn mathematics one has to get one's hands really dirty? To boot, attention to detail is ubiquitous: it's everywhere in Constantin's presentation of proofs and arguments, as well as examples, all throughout the narrative itself. The entire presentation is very much to the point and the student who works through this book will come out knowing some real mathematics very well.' Michael Berg, MAA ReviewsTable of Contents1. Introduction; 2. The Lebesgue measure and integral; 3. Elements of functional analysis; 4. Convergence results for Fourier series; 5. Fourier transforms; 6. Multi-dimensional Fourier analysis; 7. A glance at some advanced topics; Appendix: historical notes; References; Index.

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Book SynopsisThis volume commemorates the life, work and foundational views of Kurt Gödel by exploring the impact of his work on current research and its future implications not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science.Trade Review'This is a very useful volume that brings together aspects of Gödel's work that relates to logic and mathematics …' The Mathematical IntelligencerTable of ContentsPart I. Historical Context - Gödel's Contributions and Accomplishments: 1. The impact of Gödel's incompleteness theorems on mathematics Angus Macintyre; 2. Logical hygiene, foundations, and abstractions: diversity among aspects and options Georg Kreisel; 3. The reception of Gödel's 1931 incompletability theorems by mathematicians, and some logicians, to the early 1960s Ivor Grattan-Guinness; 4. 'Dozent Gödel will not lecture' Karl Sigmund; 5. Gödel's thesis: an appreciation Juliette C. Kennedy; 6. Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity, and Hilbert's program Solomon Feferman; 7. Computation and intractability: echoes of Kurt Gödel Christos H. Papadimitriou; 8. From the entscheidungsproblem to the personal computer - and beyond B. Jack Copeland; 9. Gödel, Einstein, Mach, Gamow, and Lanczos: Gödel's remarkable excursion into cosmology Wolfgang Rindler; 10. Physical unknowables Karl Svozil; Part II. A Wider Vision - the Interdisciplinary, Philosophical, and Theological Implications of Gödel's Work: 11. Gödel and physics John D. Barrow; 12. Gödel, Thomas Aquinas, and the unknowability of God Denys A. Turner; 13. Gödel's mathematics of philosophy Piergiorgio Odifreddi; 14. Gödel's ontological proof and its variants Petr Hájek; 15. The Gödel theorem and human nature Hilary Putnam; 16. Gödel, the mind, and the laws of physics Roger Penrose; Part III. New Frontiers - Beyond Gödel's Work in Mathematics and Symbolic Logic: 17. Gödel's functional interpretation and its use in current mathematics Ulrich Kohlenbach; 18. My forty years on his shoulders Harvey M. Friedman; 19. My interaction with Kurt Gödel: the man and his work Paul J. Cohen; 20. The transfinite universe W. Hugh Woodin; 21. The Gödel phenomena in mathematics: a modern view Avi Wigderson.

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Book SynopsisFirst published in 1931, this book is the second edition of a 1917 original. The text provides an account of the method of least squares, aiming to obtain the best interpretation of the results of experiment without consideration of the way in which these results are obtained.Table of Contents1. Errors of observation; 2. The law of error; 3. The case of one unknown; 4. Observations of a different weight; 5. The general problem of the adjustment of indirect observations involving several unknown quantities; 6. Evaluation of the most probable values of the unknowns, their weights and probable errors; 7. The adjustment of conditioned observations; 8. The rejection of observations; 9. Alternatives to the normal law of errors; 10. Correlation; 11. Harmonic analysis; 12. The periodogram; Appendices; Index.

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Book SynopsisWhy do we need the real numbers? How should we construct them? These questions arose in the nineteenth century, along with the ideas and techniques needed to address them. Nowadays it is commonplace for apprentice mathematicians to hear ''we shall assume the standard properties of the real numbers'' as part of their training. But exactly what are those properties? And why can we assume them? This book is clearly and entertainingly written for those students, with historical asides and exercises to foster understanding. Starting with the natural (counting) numbers and then looking at the rational numbers (fractions) and negative numbers, the author builds to a careful construction of the real numbers followed by the complex numbers, leaving the reader fully equipped with all the number systems required by modern mathematical analysis. Additional chapters on polynomials and quarternions provide further context for any reader wanting to delve deeper.Trade Review'Clearly this book is probing the fundamentals of mathematical analysis and will be useful as an extra reading for an introductory calculus course. It will certainly satisfy those readers who are looking for abstraction and who want to extract the maximal number of results from the minimal set of axioms. The historical elements on the side are entertaining … It is an excellent way to get in touch with the foundations of mathematics at a relatively elementary level.' Adhemar Bultheel, European Mathematical Society'Körner begins with historical anecdotes that illustrate how numbers have been used. He then proceeds to develop number systems axiomatically with careful definitions, theorems and proofs. His characteristic humor, digressions and historical insights are present throughout, but this is a rigorous treatment that seems - in some measure - to follow Dedekind's original approach.' Bill Satze, MAA Reviews'This is a comfortably paced introduction to the various notions of numbers … The relatively high level of abstraction of the second and third parts suggests that the ideal readership for this book is students who have taken at least some advanced undergraduate classes in mathematics. Classroom use would be possible in a philosophy of mathematics course …' M. Bona, Choice'The text is peppered with quotes from classical (ancient civilizations), modern (nineteenth and early twentieth centuries), and contemporary mathematical texts; classical, modern and current approaches to most of the subjects covered are offered; and practical applications of most of the materials introduced are carefully explained. As a consequence, there is a constant dialogue between past and present from both a theoretical and a practical point of view, which makes this book interesting from a conceptual point of view and useful as an introductory working tool.' Capi Corrales-Rodriganez, Mathematical Reviews'Fans of Körner's other books, like The Pleasures of Counting, will appreciate the conversational style of this book, which provides plenty of amusing footnotes, historical context, quotations and poems, while never relenting on the rigour of the mathematics.' Julia Collins, London Mathematical Society Newsletter'Where Do Numbers Come From? certainly adds to the pleasure of mathematics as well as narrating a journey that surely every mathematician should undertake at some stage. As such, I enthusiastically recommend it to all Gazette readers.' Nick Lord, The Mathematical GazetteTable of ContentsIntroduction; Part I. The Rationals: 1. Counting sheep; 2. The strictly positive rationals; 3. The rational numbers; Part II. The Natural Numbers: 4. The golden key; 5. Modular arithmetic; 6. Axioms for the natural numbers; Part III. The Real Numbers (and the Complex Numbers): 7. What is the problem?; 8. And what is its solution?; 9. The complex numbers; 10. A plethora of polynomials; 11. Can we go further?; Appendix A. Products of many elements; Appendix B. nth complex roots; Appendix C. How do quaternions represent rotations?; Appendix D. Why are the quaternions so special?; References; Index.

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Book SynopsisTable of ContentsIntroduction 1 About This Book 1 Foolish Assumptions 2 Icons Used in This Book 2 Beyond the Book 3 Where to Go from Here 3 Unit 1: Getting Started with Basic Math & Pre-algebra 5 Chapter 1: Playing the Numbers Game 7 Inventing Numbers 8 Understanding Number Sequences 8 Evening the odds 8 Counting by threes, fours, fives, and so on 9 Getting square with square numbers 9 Composing yourself with composite numbers 10 Stepping out of the box with prime numbers 11 Multiplying quickly with exponents 12 Four Important Sets of Numbers 13 Counting on the counting numbers 13 Introducing integers 13 Staying rational 14 Getting real 14 Chapter 2: The Big Four Operations 15 The Big Four Operations 15 Adding things up: Addition 16 Take it away: Subtraction 16 A sign of the times: Multiplication 17 Doing math lickety-split: Division 18 Applying the Big Four Operations to Larger Numbers 18 Calculating stacked addition 18 Performing stacked subtraction 19 Calculating with stacked multiplication 21 Understanding long division 22 Unit 2: the Big Four Operations: Addition, Subtraction, Multiplication, and Division 25 Chapter 3: Counting on Success: Numbers and Digits 27 Knowing Your Place Value 28 Counting to ten and beyond 28 Telling placeholders from leading zeros 29 Reading long numbers 30 Close Enough for Rock ‘n’ Roll: Rounding and Estimating 30 Rounding numbers 30 Estimating value to make problems easier 32 Practice Questions Answers and Explanations 34 Whaddya Know? Chapter 3 Quiz 35 Answers to Chapter 3 Quiz 36 Chapter 4: Staying Positive with Negative Numbers 37 Understanding Where Negative Numbers Come From 38 Sign-Switching: Understanding Negation and Absolute Value 39 Addition and Subtraction with Negative Numbers 41 Starting with a negative number 41 Adding a negative number 41 Subtracting a negative number 42 Knowing Signs of the Times (and Division) for Negative Numbers 44 Practice Questions Answers and Explanations 47 Whaddya Know? Chapter 4 Quiz 51 Answers to Chapter 4 Quiz 52 Chapter 5: Putting the Big Four Operations to Work 55 Switching Things Up with Inverse Operations and the Commutative Property 56 Getting with the In-Group: Parentheses and the Associative Property 59 Distribution to lighten the load 61 Understanding Inequalities 63 Doesn’t equal (≠) 63 Less than (<) and greater than (>) 63 Less than or equal to (≤) and greater than or equal to (≥) 64 Approximately equals (≈) 64 Moving Beyond the Big Four: Exponents and Square Roots 65 Understanding exponents 66 Discovering your roots 67 Practice Questions Answers and Explanations 69 Whaddya Know? Chapter 5 Quiz 72 Answers to Chapter 5 Quiz 73 Unit 3: Getting a Handle on Whole Numbers 75 Chapter 6: Please Excuse My Dear Aunt Sally: Evaluating Arithmetic Expressions with PEMDAS 77 The Three E’s of Math: Equations, Expressions, and Evaluation 78 Seeking equality for all: Equations 78 Hey, it’s just an expression 78 Evaluating the situation 79 Putting the Three E’s together 79 Introducing Order of Operations (PEMDAS) 80 Expressions with only addition and subtraction 81 Expressions with only multiplication and division 81 Mixed-operator expressions 82 Handling Powers Responsibly 83 Prioritizing parentheses 84 Pulling apart parentheses and powers 85 Figuring out nested parentheses 86 Bringing It All Together: The Order of Operations 87 Practice Questions Answers and Explanations 89 Whaddya Know? Chapter 6 Quiz 98 Answers to Chapter 6 Quiz 99 Chapter 7: Turning Words into Numbers: Basic Math Word Problems 103 Dispelling Two Myths about Word Problems 104 Word problems aren’t always hard 104 Word problems are useful 104 Solving Basic Word Problems 105 Turning word problems into word equations 105 Plugging in numbers for words 109 Send in the clowns 109 Our house in the middle of our street 110 I hear the train a-comin’ 110 Solving More-Complex Word Problems 113 When numbers get serious 113 Too much information 115 Practice Questions Answers and Explanations 120 Whaddya Know? Chapter 7 Quiz 124 Answers to Chapter 7 Quiz 125 Chapter 8: Divisibility and Prime Numbers 127 Knowing the Divisibility Tricks 128 Counting everyone in: Numbers you can divide everything by 128 In the end: Looking at the final digits 128 Count it up: Checking divisibility by adding and subtracting digits 130 Less is more: Checking divisibility by subtracting 134 Cross-checking: Using multiple tests 135 Identifying Prime and Composite Numbers 136 Practice Questions Answers and Explanations 139 Whaddya Know? Chapter 8 Quiz 142 Answers to Chapter 8 Quiz 143 Chapter 9: Divided Attention: Factors and Multiples 145 Knowing Six Ways to Say the Same Thing 146 Understanding Factors and Multiples 146 Finding Fabulous Factors 148 Deciding when one number is a factor of another 148 Understanding factor pairs 148 Generating a Number’s Factors 149 Decomposing a Number into Its Prime Factors 150 Finding the Greatest Common Factor 151 Generating the Multiples of a Number 153 Finding the Least Common Multiple 153 Practice Questions Answers and Explanations 155 Whaddya Know? Chapter 9 Quiz 158 Answers to Chapter 9 Quiz 159 Unit 4: Fractions 161 Chapter 10: Understanding Fractions 163 Slicing a Cake into Fractions 164 Knowing the Fraction Facts of Life 165 Telling the numerator from the denominator 165 Flipping for reciprocals 166 Using ones and zeros 166 Mixing things up 167 Knowing proper from improper 167 Increasing and Reducing Terms of Fractions 169 Increasing the terms of fractions 170 Reducing fractions to lowest terms (simplifying fractions) 171 Converting between Improper Fractions and Mixed Numbers 174 Knowing the parts of a mixed number 174 Converting a mixed number to an improper fraction 175 Converting an improper fraction to a mixed number 176 Comparing Fractions with Cross-Multiplication 178 Working with Ratios and Proportions 180 Practice Questions Answers and Explanations 182 Whaddya Know? Chapter 10 Quiz 188 Answers to Chapter 10 Quiz 189 Chapter 11: Fractions and the Big Four Operations 191 Multiplying and Dividing Fractions 192 Multiplying numerators and denominators straight across 192 Doing a flip to divide fractions 194 Adding and Subtracting Fractions with the Same Denominator 196 Adding and Subtracting Fractions with Different Denominators 198 The easy case: Increasing the terms of one fraction 198 The difficult case: Increasing the terms of both fractions 200 Practice Questions Answers and Explanations 202 Whaddya Know? Chapter 11 Quiz 208 Answers to Chapter 11 Quiz 209 Chapter 12: Mixing Things Up with Mixed Numbers 213 Multiplying and Dividing Mixed Numbers 214 Adding Mixed Numbers 216 Adding mixed numbers that have the same denominator 216 Adding mixed numbers that have different denominators 217 Adding mixed numbers with carrying 217 Subtracting Mixed Numbers 220 Subtracting mixed numbers that have the same denominator 220 Subtracting mixed numbers that have different denominators 221 Subtracting mixed numbers with borrowing 222 Practice Questions Answers and Explanations 225 Whaddya Know? Chapter 12 Quiz 233 Answers to Chapter 12 Quiz 234 Unit 5: Decimals and Percents 241 Chapter 13: Getting to the Point with Decimals 243 Understanding Basic Decimal Stuff 244 Counting dollars and decimals 244 Identifying the place value of decimals 246 Knowing the decimal facts of life 247 Performing the Big Four Operations with Decimals 252 Adding decimals 253 Subtracting decimals 254 Multiplying decimals 256 Dividing decimals 257 Converting between Decimals and Fractions 262 Simple Decimal-Fraction Conversions 262 Changing decimals to fractions 264 Changing fractions to decimals 267 Practice Questions Answers and Explanations 271 Whaddya Know? Chapter 13 Quiz 279 Answers to Chapter 13 Quiz 280 Chapter 14: Playing the Percentages 285 Making Sense of Percentages 285 Dealing with Percentages Greater than 100% 286 Converting to and from Percentages, Decimals, and Fractions 287 Converting Percentages to Decimals 287 Changing Decimals to Percentages 288 Switching from Percentages to Fractions 288 Converting Fractions to Percentages 289 Solving Percentage Problems 290 Figuring out simple percent problems 291 Turning the problem around 292 Deciphering more-difficult percent problems 293 Putting All the Percent Problems Together 294 Identifying the three types of percent problems 294 Solving Percent Problems with Equations 295 Practice Questions Answers and Explanations 299 Whaddya Know? Chapter 14 Quiz 303 Answers to Chapter 14 Quiz 304 Chapter 15: Word Problems with Fractions, Decimals, and Percentages 307 Adding and Subtracting Parts of the Whole in Word Problems 308 Sharing a pizza: Fractions 308 Buying by the pound: Decimals 309 Splitting the vote: Percentages 309 Problems about Multiplying Fractions 310 Renegade grocery shopping: Buying less than they tell you to 310 Easy as pie: Working out what’s left on your plate 311 Multiplying Decimals and Percentages in Word Problems 313 To the end: Figuring out how much money is left 313 Finding out how much you started with 314 Handling Percent Increases and Decreases in Word Problems 316 Raking in the dough: Finding salary increases 316 Earning interest on top of interest 316 Getting a deal: Calculating discounts 317 Practice Questions Answers and Explanations 319 Whaddya Know? Chapter 15 Quiz 322 Answers to Chapter 15 Quiz 324 Unit 6: Reaching the Summit: Advanced Pre-algebra Topics 327 Chapter 16: Powers and Roots 329 Memorizing Powers and Roots 329 Remembering square numbers and square roots 330 Keeping track of cubic numbers and cube roots 330 Knowing a few powers of 2 and their related roots 331 Changing the Base 332 Negating a number raised to an exponent 332 Finding powers of negative numbers 332 Finding powers of fractions 333 Mixing negative numbers and fractions with exponents 333 Exponents of 0 and Negative Numbers 334 Exponents of 0 334 Negative exponents 335 Fractional Exponents 337 Exponents of 1 2 337 Exponents of 1 3 338 Exponents of 1 4 , 1 5 , 1 , and so forth 6 339 Other fractional exponents 339 Practice Questions Answers and Explanations 341 Whaddya Know? Chapter 16 Quiz 343 Answers to Chapter 16 Quiz 344 Chapter 17: A Perfect Ten: Condensing Numbers with Scientific Notation 347 First Things First: Using Powers of Ten as Exponents 348 Counting zeros and writing exponents 348 Exponential Arithmetic: Multiplying and Dividing Powers of Ten 350 Working with Scientific Notation 352 Writing in scientific notation 352 Understanding order of magnitude 354 Multiplying with scientific notation 355 Dividing with Scientific Notation 356 Practice Questions Answers and Explanations 357 Whaddya Know? Chapter 17 Quiz 360 Answers to Chapter 17 Quiz 361 Chapter 18: How Much Have You Got? Weights and Measures 363 Understanding Units 364 Adding and subtracting units 364 Multiplying and dividing units 364 Examining Differences between the English and Metric Systems 365 Looking at the English system 365 Looking at the metric system 369 Estimating and Converting between the English and Metric Systems 372 Estimating in the metric system 373 Converting units of measurement 375 Converting between English and Metric Units 377 Practice Questions Answers and Explanations 381 Whaddya Know? Chapter 18 Quiz 388 Answers to Chapter 18 Quiz 389 Chapter 19: Getting the Picture with Geometry 393 Getting on the Plane: Points, Lines, Angles, and Shapes 394 Making some points 394 Knowing your lines 394 Figuring the angles 395 Shaping things up 396 Getting in Shape: Polygon (and Non-Polygon) Basics 396 Closed Encounters: Shaping Up Your Understanding of 2-D Shapes 397 Polygons 397 Circles 399 Squaring Off with Quadrilaterals 400 Making a Triple Play with Triangles 403 Getting Around with Circle Measurements 405 Taking a Trip to Another Dimension: Solid Geometry 406 The many faces of polyhedrons 407 3-D shapes with curves 408 Building Solid Measurement Skills 409 Solving Geometry Word Problems 413 Working from words and images 413 Breaking out those sketching skills 415 Practice Questions Answers and Explanations 418 Whaddya Know? Chapter 19 Quiz 425 Answers to Chapter 19 Quiz 427 Chapter 20: Figuring Your Chances: Statistics and Probability 431 Gathering Data Mathematically: Basic Statistics 432 Understanding differences between qualitative and quantitative data 432 Working with qualitative data 433 Working with quantitative data 436 Looking at Likelihoods: Basic Probability 439 Figuring the probability 440 Oh, the possibilities! Counting outcomes with multiple coins 441 Practice Questions Answers and Explanations 444 Whaddya Know? Chapter 20 Quiz 447 Answers to Chapter 20 Quiz 449 Chapter 21: Setting Things Up with Basic Set Theory 451 Understanding Sets 451 Elementary, my dear: Considering what’s inside sets 452 Sets of numbers 454 Performing Operations on Sets 455 Union: Combined elements 455 Intersection: Elements in common 456 Relative complement: Subtraction (sorta) 457 Complement: Feeling left out 457 Practice Questions Answers and Explanations 459 Whaddya Know? Chapter 21 Quiz 461 Answers to Chapter 21 Quiz 462 Unit 7: the X-files: Introduction to Algebra 463 Chapter 22: Working with Algebraic Expressions 465 Seeing How X Marks the Spot 466 Expressing Yourself with Algebraic Expressions 466 Evaluating Algebraic Expressions 467 Knowing the Terms 470 Making the commute: Rearranging your terms 471 Identifying the coefficient and variable 472 Adding and Subtracting Like Terms 473 Identifying like terms 473 Adding and subtracting terms 474 Multiplying and Dividing Terms 475 Simplifying Expressions by Combining Like Terms 479 Removing Parentheses from an Algebraic Expression 481 Drop everything: Parentheses with a plus sign 481 Sign turnabout: Parentheses with a negative sign 481 Distribution: Parentheses with no sign 482 FOILing: Dealing with Two Sets of Parentheses 484 Practice Questions Answers and Explanations 487 Whaddya Know? Chapter 22 Quiz 495 Answers to Chapter 22 Quiz 496 Chapter 23: Solving Algebraic Equations 499 Understanding Algebraic Equations 500 Using X in Equations 500 Choosing among four ways to solve algebraic equations 501 The Balancing Act: Solving for x 503 Striking a balance 504 Using the Balance Scale to Isolate X 504 Rearranging Equations and Isolating x 506 Rearranging terms on one side of an equation 506 Moving terms to the other side of the equals sign 507 Removing parentheses from equations 509 Cross-multiplying 512 Practice Questions Answers and Explanations 515 Whaddya Know? Chapter 23 Quiz 525 Answers to Chapter 23 Quiz 526 Chapter 24: Tackling Algebra Word Problems 531 Solving Algebra Word Problems in Five Steps 531 Declaring a variable 532 Setting up the equation 533 Solving the equation 533 Answering the question 534 Checking your work 534 Choosing Your Variable Wisely 536 Solving More-Complex Algebraic Problems 539 Charting four people 539 Crossing the finish line with five people 540 Practice Questions Answers and Explanations 545 Whaddya Know? Chapter 24 Quiz 549 Answers to Chapter 24 Quiz 550 Chapter 25: Graphing Algebraic Equations 553 Graphing on the xy-Plane 554 Understanding the axes, the origin, and the quadrants 554 Plotting coordinates on the xy-plane 554 Graphing equations on the xy-plane 555 Understanding Linear Equations 559 Knowing the most basic linear equation 559 Changing the slope (m) 560 Changing the y-intercept (b) 561 Understanding slope-intercept form 562 Measuring the Slope of a Line 564 Estimating slope 564 Eyeballing slope on the xy-plane 566 Using the two-point slope formula 569 Graphing Linear Equations Using the Slope and y-intercept 572 Practice Questions Answers and Explanations 574 Whaddya Know? Chapter 25 Quiz 577 Answers to Chapter 25 Quiz 581 Index 583

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