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.
Author(s): Keith M. Ball
Publisher: Princeton University Press
Year: 2003
Language: English
Pages: 268
Tags: Математика;Популярная математика;
Cover......Page 1
Title......Page 4
Copyrights......Page 5
Contents......Page 8
Preface......Page 12
Acknowledgements......Page 14
1.1 The ISBN Code......Page 18
1.2 Binary Channels......Page 22
1.3 The Hunt for Good Codes......Page 24
1.4 Parity-Check Construction......Page 28
1.5 Decoding a Hamming Code......Page 30
1.6 The Free Lunch Made Precise......Page 36
1.7 Further Reading......Page 38
1.8 Solutions......Page 39
2.1 Introduction......Page 42
2.2 Why Is Pick’s Theorem True?......Page 44
2.3 An Interpretation......Page 48
2.4 Pick’s Theorem and Arithmetic......Page 49
2.5 Further Reading......Page 51
2.6 Solutions......Page 52
3.1 Introduction......Page 58
3.2 The Prime Numbers......Page 60
3.3 Decimal Expansions of Reciprocals of Primes......Page 63
3.4 An Algebraic Description of the Period......Page 65
3.5 The Period Is a Factor of p −1......Page 67
3.6 Fermat’s Little Theorem......Page 72
3.7 Further Reading......Page 73
3.8 Solutions......Page 75
4.1 Introduction......Page 80
4.2 A Curve Constructed Using Tiles......Page 82
4.3 Is the Curve Continuous?......Page 87
4.4 Does the Curve Cover the Square?......Page 88
4.5 Hilbert’s Construction and Peano’s Original......Page 90
4.6 A Computer Program......Page 92
4.7 A Gothic Frieze......Page 93
4.8 Further Reading......Page 96
4.9 Solutions......Page 97
5.1 Introduction......Page 100
5.2 What Chance of a Match?......Page 101
5.3 How Many Matches?......Page 106
5.4 How Many People Share?......Page 108
5.5 The Bell-Shaped Curve......Page 110
5.6 The Area under a Normal Curve......Page 117
5.7 Further Reading......Page 122
5.8 Solutions......Page 123
6.1 Introduction......Page 126
6.2 A First Estimate for n!......Page 127
6.3 A Second Estimate for n!......Page 131
6.4 A Limiting Ratio......Page 134
6.5 Stirling’s Formula......Page 139
6.6 Further Reading......Page 141
6.7 Solutions......Page 142
7.1 Introduction......Page 144
7.2 The Coin-Weighing Problem......Page 145
7.3 Back to Blood......Page 148
7.4 The Binary Protocol for a Rare Abnormality......Page 151
7.5 A Refined Binary Protocol......Page 156
7.6 An Efficiency Estimate Using Telephones......Page 158
7.7 An Efficiency Estimate for Blood Pooling......Page 161
7.8 A Precise Formula for the Binary Protocol......Page 164
7.9 Further Reading......Page 166
7.10 Solutions......Page 168
8.1 Introduction......Page 170
8.2 Fibonacci and the Golden Ratio......Page 171
8.3 The Continued Fraction for the Golden Ratio......Page 175
8.4 Best Approximations and the Fibonacci Hyperbola......Page 178
8.5 Continued Fractions and Matrices......Page 182
8.6 Skipping down the Fibonacci Numbers......Page 186
8.7 The Prime Lucas Numbers......Page 191
8.8 The Trace Problem......Page 195
8.9 Further Reading......Page 198
8.10 Solutions......Page 199
9.1 Introduction......Page 206
9.2 Approximation by Rational Functions......Page 210
9.3 The Tangent......Page 219
9.4 An Integral Formula......Page 224
9.5 The Exponential......Page 227
9.6 The Inverse Tangent......Page 230
9.7 Further Reading......Page 231
9.8 Solutions......Page 232
10.1 Introduction......Page 236
10.2 Fibonacci Revisited......Page 237
10.3 The Square Root of d......Page 240
10.4 The Box Principle......Page 242
10.5 The Numbers e and π......Page 247
10.6 The Irrationality of e......Page 250
10.7 Euler’s Argument......Page 253
10.8 The Irrationality of π......Page 255
10.9 Further Reading......Page 259
10.10 Solutions......Page 260
Index......Page 11
C......Page 264
G......Page 265
N......Page 266
S......Page 267
W......Page 268
