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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.
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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 © 2003 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All rights reserved Library of Congress Cataloguing-in-Publication Data Ball, Keith M., 1960– Strange curves, counting rabbits, and other mathematical explorations/Keith Ball. p. cm. Includes bibliographical references and index. ISBN 0-691-11321-1 (acid-free paper) 1. Mathematics—Popular works. I. Title.
510—dc21
QA93.B34 2003 2003047183
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library This book has been composed in Lucida Typeset by T&T Productions Ltd, London ∞ Printed on acid-free paper www.pupress.princeton.edu
Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
For my parents
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Contents
Preface
xi
Acknowledgements Chapter One Shannon’s Free Lunch 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
The ISBN Code Binary Channels The Hunt for Good Codes Parity-Check Construction Decoding a Hamming Code The Free Lunch Made Precise Further Reading Solutions
Chapter Two Counting Dots 2.1 2.2 2.3 2.4 2.5 2.6
Introduction Why Is Pick’s Theorem True? An Interpretation Pick’s Theorem and Arithmetic Further Reading Solutions
Chapter Three Fermat’s Little Theorem and Infinite Decimals 3.1 3.2 3.3 3.4
Introduction The Prime Numbers Decimal Expansions of Reciprocals of Primes An Algebraic Description of the Period
xiii
1 1 5 7 11 13 19 21 22
25 25 27 31 32 34 35
41 41 43 46 48 vii
CONTENTS
3.5 3.6 3.7 3.8
The Period Is a Factor of p − 1 Fermat’s Little Theorem Further Reading Solutions
Chapter Four Strange Curves 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9
Introduction A Curve Constructed Using Tiles Is the Curve Continuous? Does the Curve Cover the Square? Hilbert’s Construction and Peano’s Original A Computer Program A Gothic Frieze Further Reading Solutions
Chapter Five Shared Birthdays, Normal Bells 5.1 5.2 5.3 5.4