The Mathematics Lover's Companion: Masterpieces for Everyone

by Edward R. Scheinerman

Overview

Twenty-three mathematical masterpieces for exploration and enlightenment

“A mind-broadening experience.”—Paul J. Campbell, Mathematics Magazine

How can a shape have more than one dimension but fewer than two? What is the best way to elect public officials when more than two candidates are vying for the office? Is it possible for a highly accurate medical test to give mostly incorrect results? Can you tile your floor with regular pentagons? How can you use only the first digit of sales numbers to determine if your accountant is lying? Can mathematics give insights into free will?
 
Edward Scheinerman, an accomplished mathematician and enthusiastic educator, answers all these questions and more in this book, a collection of mathematical masterworks. In bite-sized chapters that require only high school algebra, he invites readers to try their hands at solving mathematical puzzles and provides an engaging and friendly tour of numbers, shapes, and uncertainty. The result is an unforgettable introduction to the fundamentals and pleasures of thinking mathematically.

Product Details

ISBN-13:	9780300255393
Publisher:	Yale University Press
Publication date:	01/19/2021
Pages:	296
Sales rank:	677,908
Product dimensions:	6.40(w) x 8.90(h) x 0.90(d)

About the Author

Edward Scheinerman is professor of applied mathematics and vice dean for engineering education at Johns Hopkins University. He has twice won the Mathematical Association of America’s Ford Award for excellent mathematical writing.

Table of Contents

Preface xv

Joy xv

Overview xvii

How to read a mathematics book xviii

About the cover xviii

Acknowledgments xix

Prelude: Theorem and Proof 1

The last words 4

Part I Number 5

1 Prime Numbers 7

The integers 7

Factoring 9

How many? 10

A constructive approach 13

A different proof 14

Two infamous problems 16

Application to cryptography 17

2 Binary 19

When in Rome 19

Unary 20

A middle ground 21

Calculation 22

Extensions 25

3 0.999999999999 … 27

The meaning of decimals 27

Decimal numbers with infinitely many digits 28

Let's be reckless 30

4 <$$$> 33

Rational numbers 33

The diagonal of a square 34

Beyond the rational 35

Constructible numbers 38

Playing in tune 40

5 i 44

Another square-root conundrum 44

Imaginary numbers 46

Complex numbers 47

The Fundamental Theorem of Algebra 50

6 π 52

What is π? 52

Transcendence* 55

Relatively prime 56

7 e 60

Leonhard Eider 60

An "interesting" number 61

The deranged hat check clerk 64

The average gap between primes 66

A miraculous equation 69

8 ∞ 71

Sets 71

Infinite sets of unequal size 75

Transfinite numbers 80

Weirdness in setland 81

9 Fibonacci Numbers 86

Squares and dominoes 86

The Fibonacci numbers 88

Sums of Fibonacci numbers 89

Proof by induction 91

Combinatorial proof 93

Ratios of Fibonacci numbers and the golden mean 97

10 Factorial! 102

Books on the shelf 102

Is there a formula? 104

A puzzle 107

What is 0!? 107

11 Benford's Law 109

Wild measurements 110

Multiplication tables 112

Catching crooks cooking books 115

Refining the problem with scientific notation 115

Yards or feet? 117

What's logs got to do with it? 120

Collecting the loose ends 123

12 Algorithm 124

Sorting 125

Greatest common divisor 131

Least common multiple 136

Part II Shape 141

13 Triangles 143

It all adds up to 180 143

Area 145

Centers 148

Lurking equilateral triangles 151

14 Pythagoras and Fermat 155

The Pythagorean Theorem 155

Absolute value of complex numbers 158

Pythagorean triples 159

Fermat's Last Theorem 162

15 Circles 165

A precise definition 165

Art equation 166

Triangles right inside 167

Ptolemy's Theorem 167

Packing 168

Kissing circles 170

Pascal's Hexagon Theorem 173

16 The Platonic Solids 175

Polyhedra 175

Euler's polyhedral formula 178

Is that all there is? 183

Archimedian solids 187

17 Fractals 190

Sierpinski's triangle 190

Between dimensions 192

Box counting 194

The dimension of Sierpinski's triangle 198

Pascal and Sierpinski 201

The Koch snowflake 202

18 Hyperbolic Geometry 204

Euclid's Postulates 204

What is a line? 207

An entire plane inside a disk 210

Implications 211

Part III Uncertainty 215

19 Nontransitive Dice 217

A game of two dice 218

A challenger 219

Triumph of the underdog 220

Further examples 220

20 Medical Probability 223

Conditional probability* 226

21 Chaos 230

Functions 231

Iterating the logistic map 233

From order to chaos 237

The Collatz 3x + 1 problem 242

22 Social Choice and Arrow's Theorem 245

Two-party elections 245

Elections with more than two candidates 249

The independence of irrelevant alternatives 255

23 Newcomb's Paradox 259

Newcomb's game 259

Don't money on the table! 262

Greed doesn't pay 263

Conflict and resolution 265

Computer as the Chooser 267

Further Reading 269

Index 271

Editorial Reviews

"Twenty-three masterworks whose analysis Scheinerman presents with rigor and accessibility."—James Ryerson, New York Times Book Review

“A mind-broadening experience.”—Paul J. Campbell, Mathematics Magazine

"This delightful book is designed to allow non-mathematicians to share in the joy of mathematical proofs. . . . While every mathematician will be familiar with most of the proofs, the clear exposition can be enjoyed for its own sake."—Frederick Norwood, Mathematical Reviews Clippings

“Scheinerman’s writing style and wit have led to a book that is both inviting and engaging, with a selection of topics that should appeal to a wide audience.”—American Mathematical Monthly

“[The Mathematics Lover’s Companion] is a mind-broadening experience, of the sort that could be a basis for a liberal arts course in mathematics, and one that would also be particularly beneficial for potential elementary teachers.”—Paul J. Campbell, Mathematics Magazine

"An accessible presentation of some of the most compelling and beautiful ideas of mathematics. Scheinerman demonstrates the diversity and liveliness of the subject in a friendly and inviting way."—Susan Jane Colley, Oberlin College and Editor, The American Mathematical Monthly

"A great read for anyone who wants to learn what math really is, no prerequisites required. And those of us in the field are reminded of what first drew us to it."—Maria Chudnovsky, Princeton University and 2012 MacArthur Fellow

"In this well-written book, the reader is taken on a fascinating journey across a broad landscape of beautiful mathematics."—Ron Graham, author of Magical Mathematics

"A first-rate mathematician and expositor, Scheinerman takes us on a guided tour of great mathematical ideas. Written with clarity and humor, The Mathematics Lover's Companion will appeal to a wide audience."—Arthur Benjamin, author of The Magic of Math: Solving for x and Figuring Out Why

"A clear and lively exposition that will entice non-mathematicians into the world of mathematics."—Anna Lachowska, coauthor of Beautiful, Simple, Exact, Crazy: Mathematics in the Real World

From the Publisher

…The Mathematics Lover's Companion…includes the Fibonacci sequence as one of 23 masterworks whose analysis Scheinerman presents with rigor and accessibility (assuming you have some taste for formal systems) for nonmathematicians.

The New York Times Book Review - James Ryerson

“Twenty-three masterworks whose analysis Scheinerman presents with rigor and accessibility.”—James Ryerson, New York Times Book Review

New York Times Book Review - James Ryerson

“A clear and lively exposition that will entice non-mathematicians into the world of mathematics.”—Anna Lachowska, coauthor of Beautiful, Simple, Exact, Crazy: Mathematics in the Real World

Ann Lachowska

“An elegant sampler of many beautiful and interesting mathematical topics. This could become one of the best books available for a popular audience interested in what mathematics really is.”—Jayadev Athreya, University of Washington

Jayadev Athreya

“A first-rate mathematician and expositor, Scheinerman takes us on a guided tour of great mathematical ideas. Written with clarity and humor, The Mathematics Lover's Companion will appeal to a wide audience.”—Arthur Benjamin, The Magic of Math: Solving for x and Figuring Out Why

Arthur Benjamin

“In this well-written book, the reader is taken on a fascinating journey across a broad landscape of beautiful mathematics.”—Ron Graham, author of Magical Mathematics

Ron Graham

“An accessible presentation of some of the most compelling and beautiful ideas of mathematics. Scheinerman demonstrates the diversity and liveliness of the subject in a friendly and inviting way.”—Susan Jane Colley, Oberlin College and Editor, The American Mathematical Monthly

Susan Colley

“A great read for anyone who wants to learn what math really is, no prerequisites required. And those of us in the field are reminded of what first drew us to it.”—Maria Chudnovsky, Princeton University and 2012 MacArthur Fellow

Maria Chudnovsky