Strange curves, counting rabbits, and other mathematical explorations / Keith Ball.

Format:

Book

Author/Creator:

Ball, Keith M., 1960-

Language:

English

Subjects (All):

Mathematics--Popular works.

Mathematics.

Genre:

Popular works.

Physical Description:

xiii, 251 pages : illustrations ; 24 cm

Place of Publication:

Princeton, N.J. : Princeton University Press, [2003]

Summary:

How does mathematics enable us to send pictures from space back to Earth? Where does the bell-shaped curve come from? Why do you need only 23 people in a room for a 50/50 chance of two of them sharing the same birthday? In Strange Curves, Counting Rabbits, and Other Mathematical Explorations, Keith Ball highlights how ideas, mostly from pure math, can answer these questions and many more. Drawing on areas of mathematics from probability theory, number theory, and geometry, he explores a wide range of concepts, some more light-hearted, others central to the development of the field and used daily by mathematicians, physicists, and engineers. Each of the book's ten chapters begins by outlining key concepts and goes on to discuss, with the minimum of technical detail, the principles that underlie them. Each includes puzzles and problems of varying difficulty. While the chapters are self-contained, they also reveal the links between seemingly unrelated topics. For example, the problem of how to design codes for satellite communication gives rise to the same idea of uncertainty as the problem of screening blood samples for disease. Accessible to anyone familiar with basic calculus, this book is a treasure trove of ideas that will entertain, amuse, and bemuse students, teachers, and math lovers of all ages.

Contents:

Chapter 1 Shannon's Free Lunch 1

1.1 The ISBN Code 1

1.2 Binary Channels 5

1.3 The Hunt for Good Codes 7

1.4 Parity-Check Construction 11

1.5 Decoding a Hamming Code 13

1.6 The Free Lunch Made Precise 19

Chapter 2 Counting Dots 25

2.2 Why Is Pick's Theorem True? 27

2.3 An Interpretation 31

2.4 Pick's Theorem and Arithmetic 32

Chapter 3 Fermat's Little Theorem and Infinite Decimals 41

3.2 The Prime Numbers 43

3.3 Decimal Expansions of Reciprocals of Primes 46

3.4 An Algebraic Description of the Period 48

3.5 The Period Is a Factor of p - 1 50

3.6 Fermat's Little Theorem 55

Chapter 4 Strange Curves 63

4.2 A Curve Constructed Using Tiles 65

4.3 Is the Curve Continuous? 70

4.4 Does the Curve Cover the Square? 71

4.5 Hilbert's Construction and Peano's Original 73

4.6 A Computer Program 75

4.7 A Gothic Frieze 76

Chapter 5 Shared Birthdays, Normal Bells 83

5.2 What Chance of a Match? 84

5.3 How Many Matches? 89

5.4 How Many People Share? 91

5.5 The Bell-Shaped Curve 93

5.6 The Area under a Normal Curve 100

Chapter 6 Stirling Works 109

6.2 A First Estimate for n! 110

6.3 A Second Estimate for n! 114

6.4 A Limiting Ratio 117

6.5 Stirling's Formula 122

Chapter 7 Spare Change, Pools of Blood 127

7.2 The Coin-Weighing Problem 128

7.3 Back to Blood 131

7.4 The Binary Protocol for a Rare Abnormality 134

7.5 A Refined Binary Protocol 139

7.6 An Efficiency Estimate Using Telephones 141

7.7 An Efficiency Estimate for Blood Pooling 144

7.8 A Precise Formula for the Binary Protocol 147

Chapter 8 Fibonacci's Rabbits Revisited 153

8.2 Fibonacci and the Golden Ratio 154

8.3 The Continued Fraction for the Golden Ratio 158

8.4 Best Approximations and the Fibonacci Hyperbola 161

8.5 Continued Fractions and Matrices 165

8.6 Skipping down the Fibonacci Numbers 169

8.7 The Prime Lucas Numbers 174

8.8 The Trace Problem 178

Chapter 9 Chasing the Curve 189

9.2 Approximation by Rational Functions 193

9.3 The Tangent 202

9.4 An Integral Formula 207

9.5 The Exponential 210

9.6 The Inverse Tangent 213

Chapter 10 Rational and Irrational 219

10.2 Fibonacci Revisited 220

10.3 The Square Root of d 223

10.4 The Box Principle 225

10.5 The Numbers e and [pi] 230

10.6 The Irrationality of e 233

10.7 Euler's Argument 236

10.8 The Irrationality of [pi] 238.

Notes:

Includes bibliographical references and index.

ISBN:

0691113211

OCLC:

52464848

Online:

Publisher description