Mathematics for the liberal arts (2013)

PREFACE

Part I. MATHEMATICS IN HISTORY

Chapter 1. THE ANCIENT ROOTS OF MATHEMATICS

1.1 Introduction

1.2 Ancient Mesopotamia and Egypt

1.3 Early Greek Mathematics: The First Theorists

1.4 The Apex: Third Century Hellenistic Mathematics

1.5 The Slow Decline

Chapter 2. THE GROWTH OF MATHEMATICS TO 1600

2.1 China

2.2 India

2.3 Islam

2.4 European Mathematics Awakens

Chapter 3. MODERN MATHEMATICS

3.1 The 17th Century: Scientific Revolution

3.2 The 18th Century: Consolidation

3.3 The 19th Century: Expansion

3.4 The 20th and 21st Centuries: Explosion

3.5 The Future

Chapter 4. CALCULUS

4.1 What Is Calculus?

4.2 Average and Instantaneous Velocity

4.3 Tangent Line to a Curve

4.4 The Derivative

4.5 Formulas for Derivatives

4.6 The Product Rule and Quotient Rule

4.7 The Chain Rule

4.8 Slopes and Optimization

4.9 Applying Optimization Methods

4.10 Differential Notation and Estimates

4.11 Marginal Revenue, Cost, and Profit

4.12 Exponential Growth

4.13 Periodic Functions and Trigonometry

4.14 The Fundamental Theorem of Calculus

4.15 The Riemann Integral

4.16 Signed Areas and Other Integrals

4.17 Application: Rocket Science

4.18 Infinite Sums

4.19 Exponential Growth and Doubling Times

4.20 Beyond Calculus

Part II. TWO PILLARS OF MATHEMATICS

Chapter 5. NUMBER THEORY

5.1 What Is Number Theory?

5.2 Divisibility

5.3 Irrational Numbers

5.4 Greatest Common Divisors

5.5 Primes

5.6 Relatively Prime Integers

5.7 Mersenne and Fermat Primes

5.8 The Fundamental Theorem of Arithmetic

5.9 Diophantine Equations

5.10 Linear Diophantine Equations

5.11 Pythagorean Triples

5.12 An Introduction to Modular Arithmetic

5.13 Congruence

5.14 Arithmetic with Congruences

5.15 Division with Congruences; Finite Fields

5.16 Fermat’s Last Theorem

5.17 Unfinished Business

APPENDIX A. ANSWERS TO SELECTED EXERCISES

APPENDIX B. SUGGESTED READING