Cover image for Strange curves, counting rabbits, and other mathematical explorations
Title:
Strange curves, counting rabbits, and other mathematical explorations
Author:
Ball, Keith M., 1960-
Personal Author:
Publication Information:
Princeton, N.J. : Princeton University Press, [2003]

©2003
Physical Description:
xiii, 251 pages : illustrations ; 24 cm
General Note:
Includes index.
Language:
English
ISBN:
9780691113210
Format :
Book

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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

Prefacep. xi
Acknowledgementsp. xiii
Chapter 1 Shannon's Free Lunchp. 1
1.1 The ISBN Codep. 1
1.2 Binary Channelsp. 5
1.3 The Hunt for Good Codesp. 7
1.4 Parity-Check Constructionp. 11
1.5 Decoding a Hamming Codep. 13
1.6 The Free Lunch Made Precisep. 19
1.7 Further Readingp. 21
1.8 Solutionsp. 22
Chapter 2 Counting Dotsp. 25
2.1 Introductionp. 25
2.2 Why Is Pick's Theorem True?p. 27
2.3 An Interpretationp. 31
2.4 Pick's Theorem and Arithmeticp. 32
2.5 Further Readingp. 34
2.6 Solutionsp. 35
Chapter 3 Fermat's Little Theorem and Infinite Decimalsp. 41
3.1 Introductionp. 41
3.2 The Prime Numbersp. 43
3.3 Decimal Expansions of Reciprocals of Primesp. 46
3.4 An Algebraic Description of the Periodp. 48
3.5 The Period Is a Factor of p - 1p. 50
3.6 Fermat's Little Theoremp. 55
3.7 Further Readingp. 56
3.8 Solutionsp. 58
Chapter 4 Strange Curvesp. 63
4.1 Introductionp. 63
4.2 A Curve Constructed Using Tilesp. 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 Originalp. 73
4.6 A Computer Programp. 75
4.7 A Gothic Friezep. 76
4.8 Further Readingp. 79
4.9 Solutionsp. 80
Chapter 5 Shared Birthdays, Normal Bellsp. 83
5.1 Introductionp. 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 Curvep. 93
5.6 The Area under a Normal Curvep. 100
5.7 Further Readingp. 105
5.8 Solutionsp. 106
Chapter 6 Stirling Worksp. 109
6.1 Introductionp. 109
6.2 A First Estimate for n!p. 110
6.3 A Second Estimate for n!p. 114
6.4 A Limiting Ratiop. 117
6.5 Stirling's Formulap. 122
6.6 Further Readingp. 124
6.7 Solutionsp. 125
Chapter 7 Spare Change, Pools of Bloodp. 127
7.1 Introductionp. 127
7.2 The Coin-Weighing Problemp. 128
7.3 Back to Bloodp. 131
7.4 The Binary Protocol for a Rare Abnormalityp. 134
7.5 A Refined Binary Protocolp. 139
7.6 An Efficiency Estimate Using Telephonesp. 141
7.7 An Efficiency Estimate for Blood Poolingp. 144
7.8 A Precise Formula for the Binary Protocolp. 147
7.9 Further Readingp. 149
7.10 Solutionsp. 151
Chapter 8 Fibonacci's Rabbits Revisitedp. 153
8.1 Introductionp. 153
8.2 Fibonacci and the Golden Ratiop. 154
8.3 The Continued Fraction for the Golden Ratiop. 158
8.4 Best Approximations and the Fibonacci Hyperbolap. 161
8.5 Continued Fractions and Matricesp. 165
8.6 Skipping down the Fibonacci Numbersp. 169
8.7 The Prime Lucas Numbersp. 174
8.8 The Trace Problemp. 178
8.9 Further Readingp. 181
8.10 Solutionsp. 182
Chapter 9 Chasing the Curvep. 189
9.1 Introductionp. 189
9.2 Approximation by Rational Functionsp. 193
9.3 The Tangentp. 202
9.4 An Integral Formulap. 207
9.5 The Exponentialp. 210
9.6 The Inverse Tangentp. 213
9.7 Further Readingp. 214
9.8 Solutionsp. 215
Chapter 10 Rational and Irrationalp. 219
10.1 Introductionp. 219
10.2 Fibonacci Revisitedp. 220
10.3 The Square Root of dp. 223
10.4 The Box Principlep. 225
10.5 The Numbers e and [pi]p. 230
10.6 The Irrationality of ep. 233
10.7 Euler's Argumentp. 236
10.8 The Irrationality of [pi]p. 238
10.9 Further Readingp. 242
10.10 Solutionsp. 243
Indexp. 247