Strange curves, counting rabbits, and other mathematical explorations

268 Pages·2003·5.37 MB·English

by  Keith M. Ball

Strange Curves, Counting Rabbits,
and Other Mathematical Explorations
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Strange Curves,
Counting Rabbits, and
Other Mathematical Explorations
Keith Ball
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright © 2003byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,
41WilliamStreet,Princeton,NewJersey08540
IntheUnitedKingdom:PrincetonUniversityPress,
3MarketPlace,Woodstock,OxfordshireOX201SY
Allrightsreserved
LibraryofCongressCataloguing-in-PublicationData
Ball,KeithM.,1960–
Strangecurves,countingrabbits,andothermathematical
explorations/KeithBall.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN0-691-11321-1(acid-freepaper)
1.Mathematics—Popularworks.I.Title.
QA93.B342003
510—dc21 2003047183
BritishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
ThisbookhasbeencomposedinLucida
TypesetbyT&TProductionsLtd,London
Printedonacid-freepaper(cid:1)∞
www.pupress.princeton.edu
PrintedintheUnitedStatesofAmerica
10987654321
Formyparents
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Contents
Preface xi
Acknowledgements xiii
Chapter One
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
1.7 Further Reading 21
1.8 Solutions 22
Chapter Two
Counting Dots 25
2.1 Introduction 25
2.2 Why Is Pick’s Theorem True? 27
2.3 An Interpretation 31
2.4 Pick’s Theorem and Arithmetic 32
2.5 Further Reading 34
2.6 Solutions 35
Chapter Three
Fermat’s Little Theorem and Infinite Decimals 41
3.1 Introduction 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
vii
CONTENTS
3.5 The Period Is a Factor of p−1 50
3.6 Fermat’s Little Theorem 55
3.7 Further Reading 56
3.8 Solutions 58
Chapter Four
Strange Curves 63
4.1 Introduction 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
4.8 Further Reading 79
4.9 Solutions 80
Chapter Five
Shared Birthdays, Normal Bells 83
5.1 Introduction 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
5.7 Further Reading 105
5.8 Solutions 106
Chapter Six
Stirling Works 109
6.1 Introduction 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
6.6 Further Reading 124
6.7 Solutions 125
viii
CONTENTS
Chapter Seven
Spare Change, Pools of Blood 127
7.1 Introduction 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
7.9 Further Reading 149
7.10 Solutions 151
Chapter Eight
Fibonacci’s Rabbits Revisited 153
8.1 Introduction 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
8.9 Further Reading 181
8.10 Solutions 182
Chapter Nine
Chasing the Curve 189
9.1 Introduction 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
9.7 Further Reading 214
9.8 Solutions 215
ix