Rowman & Littlefield Publishers

Pages: 664
•

978-0-7425-0202-4 • Hardback • January 2001 • $98.00 • (£75.00)

978-1-4616-0876-9 • eBook • January 2001 • $93.00 • (£72.00)

**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

—**Accessible and Intriguing**: Provides liberal arts students with a broad appreciation of mathematics by exploring ways in which the artistic, aesthetic, intellectual, and humanistic aspects of mathematics are as important asits utility.

—**Consistent Pedagogy**: Each content chapter starts "from scratch" and proceeds to develop a significant mathematical idea, illustrating what mathematicians do in that area.

—**Flexible Organization**: the eight content chapters (chapters 2-9) may be covered in any combination and order. Some chapters have several natural stopping places to allow for introductory coverage as well as for in-depth exploration of the topic.

—**Connects the Disciplines**:*A Mathematics Sampler* provides brief, informal, and open-ended LINK sections at the end of each topic, connecting specific mathematical ideas with many other subject areas. The connections examined include art, fiction, music, philosophy, politics, psychology, social planning, and many more.

—**Unique Writing Exercises**: Spread throughout the book, these opportunities for writing reinforce mathematical content, and they provide suggestions for term papers.

—**An Instructor's Manual** written by William Berlinghoff is avail