Mathematics for the Liberal Arts

Preface xi PART I MATHEMATICS IN HISTORY 1 The Ancient Roots of Mathematics 3 1.1 Introduction 3 1.2 Ancient Mesopotamia and Egypt 7 1.3 Early Greek Mathematics: The First Theorists 20 1.4 The Apex: Third Century Hellenistic Mathematics 42 1.5 The Slow Decline 58 2 The Growth of Mathematics to 1600 73 2.1 China 74 2.2 India 89 2.3 Islam 102 2.4 European Mathematics Awakens 120 3 Modern Mathematics 139 3.1 The 17th Century: Scientific Revolution 140 3.2 The 18th Century: Consolidation 156 3.3 The 19th Century: Expansion 170 3.4 The 20th and 21st Centuries: Explosion 193 3.5 The Future 217 II TWO PILLARS OF MATHEMATICS 4 Calculus 221 4.1 What Is Calculus? 221 4.2 Average and Instantaneous Velocity 222 4.3 Tangent Line to a Curve 226 4.4 The Derivative 232 4.5 Formulas for Derivatives 235 4.6 The Product Rule and Quotient Rule 241 4.7 The Chain Rule 248 4.8 Slopes and Optimization 253 4.9 Applying Optimization Methods 259 4.10 Differential Notation and Estimates 266 4.11 Marginal Revenue, Cost, and Profit 270 4.12 Exponential Growth 276 4.13 Periodic Functions of Trigonometry 287 4.14 The Fundamental Theorem of Calculus 293 4.15 The Riemann Integral 297 4.16 Signed Areas and Other Integrals 301 4.17 Application: Rocket Science 306 4.18 Infinite Sums 311 4.19 Exponential Growth and Doubling Times 317 4.20 Beyond Calculus 321 5 Number Theory 323 5.1 What Is Number Theory? 323 5.2 Divisibility 324 5.3 Irrational Numbers 329 5.4 Greatest Common Divisors 331 5.5 Primes 336 5.6 Relatively Prime Integers 339 5.7 Mersenne and Fermat Primes 343 5.8 The Fundamental Theorem of Arithmetic 345 5.9 Diophantine Equations 350 5.10 Linear Diophantine Equations 354 5.11 Pythagorean Triples 358 5.12 An Introduction to Modular Arithmetic 361 5.13 Congruence 366 5.14 Arithmetic with Congruences 370 5.15 Division with Congruences; Finite Fields 374 5.16 Fermat's Last Theorem 381 5.17 Unfinished Business 383 A Answers to Selected Exercises 385 B Suggested Reading 401 Index 405