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Now in its fifth edition, A Mathematics Sampler presents mathematics as both science and art, focusing on the historical role of mathematics in our culture. It uses selected topics from modern mathematics—including computers, perfect numbers, and four-dimensional geometry—to exemplify the distinctive features of mathematics as an intellectual endeavor, a problem-solving tool, and a way of thinking about the rapidly changing world in which we live. A Mathematics Sampler also includes unique LINK sections throughout the book, each of which connects mathematical concepts with areas of interest throughout the humanities. The original course on which this text is based was cited as an innovative approach to liberal arts mathematics in Lynne Cheney's report, "50 HOURS: A Core Curriculum for College Students", published by the National Endowment for the Humanities.
William P. Berlinghoff is visiting professor of mathematics at Colby College. Kerry E. Grant is professor of mathematics at Southern Connecticut State University. Dale Skrien is professor of computer science at Colby College
Chapter 1 Preface Chapter 2 To the Student Part 3 Chapter 1: Problems and Solutions Chapter 4 What Is Mathematics? Chapter 5 Problem Solving Chapter 6 It All Adds Up Chapter 7 The Mathematical Way of Thinking Chapter 8 Topics for Papers Chapter 9 For Further Reading Part 10 Chapter 2: Mathematics of Patterns: Number Theory Chapter 11 What Is Number Theory? Chapter 12 Divisibility Chapter 13 Counting Divisors Chapter 14 Summing Divisors Chapter 15 Proper Divisors Chapter 16 Even Perfect Numbers Chapter 17 Mersenne Primes Chapter 18 LINK: Number Theory and Cryptography Chapter 19 Topics for Papers Chapter 20 For Further Reading Part 21 Chapter 3: Mathematics of Axiom Systems: Geometries Chapter 22 What is Geometry? Chapter 23 Euclidean Geometry Chapter 24 Euclid and Parallel Lines Chapter 25 Axiom Systems and Models Chapter 26 Consistency and Independence Chapter 27 Non-Euclidean Geometries Chapter 28 Axiomatic Geometry and the Real World Chapter 29 LINK: Axiom Systems and Society Chapter 30 Topics for Papers Chapter 31 For Further Reading Part 32 Chapter 4: Mathematics of Chance: Probability and Statistics Chapter 33 The Gamblers Chapter 34 The Language of Sets Chapter 35 What Is Probability? Chapter 36 Counting Processes Chapter 37 LINK: Counting and the Genetic Code Chapter 38 Some Basic Rules of Probability Chapter 39 Conditional Probability Chapter 40 LINK: Probability and Marketing Chapter 41 What Is Statistics? Chapter 42 Central Tendency and Spread Chapter 43 Distributions Chapter 44 Generalization and Prediction Chapter 45 LINK: Statistics in the Psychology of Learning Chapter 46 Topics for Papers Chapter 47 For Further Reading Part 48 Chapter 5: Mathematics of Infinity: Cantor's Theory of Sets Chapter 49 What Is Set Theory? Chapter 50 Infinite Sets Chapter 51 The SIze of N Chapter 52 Rational and Irrational Numbers Chapter 53 A Different Size Chapter 54 Cardinal Numbers Chapter 55 Cantor's Theorem Chapter 56 The Continuum Hypothesis Chapter 57 The Foundations of Mathematics Chapter 58 LINK: Set Theory and Metaphysics Chapter 59 Topics for Papers Chapter 60 For Further Reading Part 61 Chapter 6: Mathematics of Symmetry: Finite Groups Chapter 62 What Is Group Theory? Chapter 63 Operations Chapter 64 Some Properties of Operations Chapter 65 The Definition of a Group Chapter 66 Some Basic Properties of Groups Chapter 67 Subgroups Chapter 68 Lagrange's Theorem Chapter 69 Lagrange's Theorem Proved [Optional] Chapter 70 Groups of Symmetries Chapter 71 LINK: Groups in Music and in Chemistry Chapter 72 Topics for Papers Chapter 73 For Further Reading Part 74 Chapter 7: Mathematics of Space and Time: Four-Dimensional Geometry Chapter 75 What Is Four-Dimensional Geometry? Chapter 76 One-Dimensional Space Chapter 77 Two-Dimensional Space Chapter 78 Three-Dimensional Space Chapter 79 Four-Dimensional Space Chapter 80 Cross Sections Chapter 81 Cylinders and Cones [Optional] Chapter 82 LINK: 4-Space in Fiction and in Art Chapter 83 Topics for Papers Chapter 84 For Further Reading Part 85 Chapter 8: Mathematics of Connection : Graph Theory Chapter 86 What Is Graph Theory? Chapter 87 Some Basic Terms Chapter 88 Edge Paths Chapter 89 Vertex Paths Chapter 90 Crossing Curves Chapter 91 Euler's Formula Chapter 92 Looking Back Chapter 93 LINK: Diagraphs and Project Management Chapter 94 Topics for Papers Chapter 95 For Further Reading Part 96 Chapter 9: Mathematics of Machines: Computer Algorithms Chapter 97 What Is a Computer? Chapter 98 The Traveling Salesman Problem Chapter 99 The Speed of a Computer Chapter 100 Algorithms and Sorting Chapter 101 Comparing Algorithms Chapter 102 Complexity Analysis Chapter 103 NP-Completeness Chapter 104 Implications of NP-Completeness Chapter 105 LINK: Algorithms, Abstraction, and Strategic Planning Chapter 106 Topics for Papers Chapter 107 For Further Reading Part 108 APPENDICES Part 109 Appendix A: Basic Logic Chapter 110 Statements and Their Negations Chapter 111 Conjunctions and Disjunctions Chapter 112 Conditionals and Deduction Chapter 113 Topics for Papers Chapter 114 For Further Reading Part 115 Appendix B: A Brief History of Mathematics Chapter 116 Preliminary Thoughts Chapter 117 From the Beginning to 600 B.C. Chapter 118 600 B.C. to A.D. 400 Chapter 119 400 to 1400 Chapter 123 The Fifteenth and Sixteenth Centuries Chapter 124 The Seventeenth Century Chapter 125 The Eighteenth Century Chapter 126 The Nineteenth Century Chapter 127 The Twentieth Century Chapter 128 Topics for Papers Chapter 129 For Further Reading Part 130 Appendix C: Literacy in the Language of Mathematics