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### Summary

### 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.

### Author Notes

Keith Ball is Professor of Mathematics at University College London

### Reviews 1

### Choice Review

Ball (mathematics, University College London) has prepared a recreational math book with enough heft to give its intended audience a series of mental workouts, ranging from the rough equivalent of a stroll to the corner mailbox to a hard mile run. The writing style is open and engaging, and though the author does not go into full details, enough are provided to give a good flavor to each line of problems, and even to provoke further questions. An adequate preparation for reading (and, hopefully, working through) the book would be a good background in high school mathematics; some knowledge of calculus is helpful for perhaps at most 25 percent of the material. Most useful would be curiosity and the willingness to sit down and do the work. Chapter threads include coding and information (a la Shannon); counting (Pick's theorem); Fermat's Little Theorem and the periods of the decimal representations of 1/p for p prime; Fibonacci numbers; and Stirling's formula. Each chapter has a collection of problems, their solutions, and suggestions for further reading. ^BSumming Up: Highly recommended. General readers; lower-division undergraduates. D. Robbins Trinity College (CT)

### Table of Contents

Preface | p. xi |

Acknowledgements | p. xiii |

Chapter 1 Shannon's Free Lunch | p. 1 |

1.1 The ISBN Code | p. 1 |

1.2 Binary Channels | p. 5 |

1.3 The Hunt for Good Codes | p. 7 |

1.4 Parity-Check Construction | p. 11 |

1.5 Decoding a Hamming Code | p. 13 |

1.6 The Free Lunch Made Precise | p. 19 |

1.7 Further Reading | p. 21 |

1.8 Solutions | p. 22 |

Chapter 2 Counting Dots | p. 25 |

2.1 Introduction | p. 25 |

2.2 Why Is Pick's Theorem True? | p. 27 |

2.3 An Interpretation | p. 31 |

2.4 Pick's Theorem and Arithmetic | p. 32 |

2.5 Further Reading | p. 34 |

2.6 Solutions | p. 35 |

Chapter 3 Fermat's Little Theorem and Infinite Decimals | p. 41 |

3.1 Introduction | p. 41 |

3.2 The Prime Numbers | p. 43 |

3.3 Decimal Expansions of Reciprocals of Primes | p. 46 |

3.4 An Algebraic Description of the Period | p. 48 |

3.5 The Period Is a Factor of p - 1 | p. 50 |

3.6 Fermat's Little Theorem | p. 55 |

3.7 Further Reading | p. 56 |

3.8 Solutions | p. 58 |

Chapter 4 Strange Curves | p. 63 |

4.1 Introduction | p. 63 |

4.2 A Curve Constructed Using Tiles | p. 65 |

4.3 Is the Curve Continuous? | p. 70 |

4.4 Does the Curve Cover the Square? | p. 71 |

4.5 Hilbert's Construction and Peano's Original | p. 73 |

4.6 A Computer Program | p. 75 |

4.7 A Gothic Frieze | p. 76 |

4.8 Further Reading | p. 79 |

4.9 Solutions | p. 80 |

Chapter 5 Shared Birthdays, Normal Bells | p. 83 |

5.1 Introduction | p. 83 |

5.2 What Chance of a Match? | p. 84 |

5.3 How Many Matches? | p. 89 |

5.4 How Many People Share? | p. 91 |

5.5 The Bell-Shaped Curve | p. 93 |

5.6 The Area under a Normal Curve | p. 100 |

5.7 Further Reading | p. 105 |

5.8 Solutions | p. 106 |

Chapter 6 Stirling Works | p. 109 |

6.1 Introduction | p. 109 |

6.2 A First Estimate for n! | p. 110 |

6.3 A Second Estimate for n! | p. 114 |

6.4 A Limiting Ratio | p. 117 |

6.5 Stirling's Formula | p. 122 |

6.6 Further Reading | p. 124 |

6.7 Solutions | p. 125 |

Chapter 7 Spare Change, Pools of Blood | p. 127 |

7.1 Introduction | p. 127 |

7.2 The Coin-Weighing Problem | p. 128 |

7.3 Back to Blood | p. 131 |

7.4 The Binary Protocol for a Rare Abnormality | p. 134 |

7.5 A Refined Binary Protocol | p. 139 |

7.6 An Efficiency Estimate Using Telephones | p. 141 |

7.7 An Efficiency Estimate for Blood Pooling | p. 144 |

7.8 A Precise Formula for the Binary Protocol | p. 147 |

7.9 Further Reading | p. 149 |

7.10 Solutions | p. 151 |

Chapter 8 Fibonacci's Rabbits Revisited | p. 153 |

8.1 Introduction | p. 153 |

8.2 Fibonacci and the Golden Ratio | p. 154 |

8.3 The Continued Fraction for the Golden Ratio | p. 158 |

8.4 Best Approximations and the Fibonacci Hyperbola | p. 161 |

8.5 Continued Fractions and Matrices | p. 165 |

8.6 Skipping down the Fibonacci Numbers | p. 169 |

8.7 The Prime Lucas Numbers | p. 174 |

8.8 The Trace Problem | p. 178 |

8.9 Further Reading | p. 181 |

8.10 Solutions | p. 182 |

Chapter 9 Chasing the Curve | p. 189 |

9.1 Introduction | p. 189 |

9.2 Approximation by Rational Functions | p. 193 |

9.3 The Tangent | p. 202 |

9.4 An Integral Formula | p. 207 |

9.5 The Exponential | p. 210 |

9.6 The Inverse Tangent | p. 213 |

9.7 Further Reading | p. 214 |

9.8 Solutions | p. 215 |

Chapter 10 Rational and Irrational | p. 219 |

10.1 Introduction | p. 219 |

10.2 Fibonacci Revisited | p. 220 |

10.3 The Square Root of d | p. 223 |

10.4 The Box Principle | p. 225 |

10.5 The Numbers e and [pi] | p. 230 |

10.6 The Irrationality of e | p. 233 |

10.7 Euler's Argument | p. 236 |

10.8 The Irrationality of [pi] | p. 238 |

10.9 Further Reading | p. 242 |

10.10 Solutions | p. 243 |

Index | p. 247 |