A Mathematics Sampler

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, "e;50 HOURS: A Core Curriculum for College Students"e;, published by the National Endowment for the Humanities.

Now in its fifth editionA 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.

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