Mathematics for the Liberal Arts

DONALD BINDNER, PhD, is Assistant Professor in the Department of Mathematics and Computer Science at Truman State University.

MARTIN J. ERICKSON, PhD, is Professor of Mathematics in the Department of Computer Science at Truman State University and the author or coauthor of numerous books on mathematical topics.

JOE HEMMETER, PhD, is a freelance software developer and a past professor of mathematics at both the University of Delaware and Truman State University.

Preface v

I Mathematics in History

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

2 The Growth of Mathematics to 1600

2.1 China

2.2 India

2.3 Islam

2.4 European Mathematics Awakens

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

II Two Pillars of Mathematics

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 The Fundamental Theorem of Calculus

4.14 The Riemann Integral

4.15 Signed Areas and Other Integrals

4.16 Application: Rocket Science

4.17 Infinite Sums

4.18 Exponential Growth and Doubling Times

4.19 Beyond Calculus

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

A Answers to Selected Exercises

B Suggested Reading

Index

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