The Number Mysteries: A Mathematical Odyssey through Everyday Life (MacSci)
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THE NUMBER MYSTERIES
THE NUMBER MYSTERIES
A Mathematical Odyssey through Everyday Life
MARCUS DU SAUTOY
THE NUMBER MYSTERIES
Copyright © Marcus du Sautoy, 2011.
All rights reserved.
First published in 2010 by Fourth Estate, an imprint of HarperCollins UK First Published in the United States in 2011 by
PALGRAVE MACMILLAN®–a division of St. Martin’s Press LLC, 175
Fifth Avenue, New York, NY 10010.
Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world.
Palgrave® and Macmillan® are registered trademarks in the United States, the United Kingdom, Europe and other countries.
ISBN 978-0-230-11384-8
Library of Congress Cataloging-in-Publication Data
Du Sautoy, Marcus.
The number mysteries : a mathematical odyssey through everyday life / Marcus du Sautoy.
p. cm.
Includes index.
ISBN 978-0-230-11384-8 (pbk.)
1. Mathematics—Miscellanea. I. Title.
QA99.D8 2011
510—dc22
2011000535
A catalogue record of the book is available from the British Library.
Design by Letra Libre, Inc.
First Palgrave Macmillan edition: May 2011
10 9 8 7 6 5 4 3 2 1
Printed in the United States of America.
CONTENTS
A Note on Websites
Introduction
ONE
The Curious Incident of the Never-Ending Primes
TWO
The Story of the Elusive Shape
THREE
The Secret of the Winning Streak
FOUR
The Case of the Uncrackable Code
FIVE
The Quest to Predict the Future
Picture Credits
Index +
A NOTE ON WEBSITES
There are references throughout the book to external websites. All of these sites can be accessed in the usual way, by typing the address into a web browser. Alternatively, if you have a smartphone, you can use it to scan the QR codes printed by each website. QR codes are two-dimensional grids of black and white pixels and act like a bar code. On a QR code–compatible smartphone, you will need to download a QR reader first. To scan the code, launch the reader and hold the camera lens over the bar code in a sufficient amount of light. Most Android smartphones come with a built-in facility to scan QR codes. Those who own an iPhone can download a free QR reader from the App Store. Just enter the words QR reader into the search box. Blackberry phones running Blackberry Messenger 5.0 (or more recent versions) can also scan QR codes using the “Scan a Group Barcode” option in the Blackberry Messenger menu. A QR reader won’t just be useful for enhancing your reading of the book. Originating in Japan, QR codes are being used increasingly across the globe on posters, as boarding passes for flights, and even on T-shirts to facilitate arranging a date with the person wearing the T-shirt.
INTRODUCTION
Is climate change a reality? Will the solar system suddenly fly apart? Is it safe to send your credit card number over the Internet? How can I beat the casino?
Ever since we’ve been able to communicate, we’ve been asking questions, trying to make predictions about what the future holds, and negotiating the environment around us. The most powerful tool that humans have created to navigate the wild and complex world we live in is mathematics.
From predicting the trajectory of a football to charting the population of lemmings, from cracking codes to winning at Monopoly, mathematics has provided the secret language to unlock nature’s mysteries. But mathematicians don’t have all the answers. There are many deep and fundamental questions we are still struggling to answer.
In each chapter of The Number Mysteries, I will take you on a journey through the big themes of mathematics, and at the end of each chapter, I reveal a mathematical mystery that no one has yet been able to solve. These are some of the great unsolved problems of all time.
But solving one of these conundrums won’t just bring you mathematical fame—it will also bring you an astronomical fortune. An American businessman, Landon Clay, has offered a prize of a million dollars for the solution to each of these mathematical mysteries. You might think it strange that a businessman should want to hand out such big prizes for solving mathematical puzzles. But he knows that the whole of science, technology, the economy, and even the future of our planet relies on mathematics.
Each of the five chapters of this book introduces you to one of these million-dollar puzzles.
Chapter 1, “The Curious Incident of the Never-Ending Primes,” takes as its theme the most basic object of mathematics: numbers. I will introduce you to the primes—the most important numbers in mathematics but also the most enigmatic. A mathematical million awaits the person who can unravel their secrets.
In chapter 2, “The Story of the Elusive Shape,” we take a journey through the world’s weird and wonderful shapes: from dice to bubbles, from tea bags to snowflakes. Ultimately, we tackle the biggest challenge of them all—what shape is our universe?
Chapter 3, “The Secret of the Winning Streak,” will show you how the mathematics of logic and probability can give you the edge when it comes to playing games. Whether you like playing with Monopoly money or gambling with real cash, mathematics is often the secret to coming out on top. But there are some simple games that still fox even the greatest minds.
Cryptography is the subject of chapter 4, “The Case of the Uncrackable Code.” Mathematics has often been the key to unscrambling secret messages. But I will reveal how you can use clever mathematics to create new codes that let you communicate securely across the Internet, send messages through space, and even read your friend’s mind.
Chapter 5 is about what we would all love to be able to do: “The Quest to Predict the Future.” I will explain how the equations of mathematics are the best fortune-tellers. They predict eclipses, explain why boomerangs come back, and ultimately tell us what the future holds for our planet. But some of these equations we still can’t solve. The chapter ends with the problem of turbulence, which affects everything from David Beckham’s free kicks to the flight of an airplane, yet is still one of mathematics’ greatest mysteries.
The mathematics I present ranges from the easy to the difficult. The unsolved problems that conclude each chapter are so difficult that no one knows how to solve them. But I am a great believer in exposing people to the big ideas of mathematics. We get excited about literature when we encounter Shakespeare or Steinbeck. Music comes alive the first time we hear Mozart or Miles Davis. Playing Mozart yourself is tough, and Shakespeare can often be challenging, even for the experienced reader. But that doesn’t mean that we should reserve the work of these great thinkers for the cognoscenti. Mathematics is just the same. So if some of the mathematics feels tough, enjoy what you can and remember the feeling of reading Shakespeare for the first time.
At school, we are taught that mathematics is fundamental to everything we do. In these five chapters, I want to bring mathematics to life to show you some of the great mathematics we have discovered to date. But I also want to give you the chance to test yourself against the biggest brains in history, as we look at some of the problems that remain unsolved. By the end, I hope you will understand that mathematics really is at the heart of all that we see and do.
THE CURIOUS INCIDENT OF THE NEVER-ENDING PRIMES
1, 2, 3, 4, 5 . . . it seems so simple: add 1, and you get the next number. Yet without numbers, we’d be lost. Arsenal vs. Manchester United—who won? We don’t know. Each team scored lots of goals. Want to look something up in the index of this book? Well, the bit about winning the National Lottery is somewhere in the middle of the book. And the lottery itself? Hopeless without numbers. It’s quite extraordinary how fundamental the language of numbers is to negotiating the world.
Even in the animal kingdom, numbers are fundamental. Packs of animals will base their decision to fight or flee on whether their group is outnumbered by a rival pack. Their survival instinct depends in part on a mathematical ability, yet behind the apparent simplicity of the list of numbers lies one of the biggest mysteries of mathematics.
2, 3, 5, 7, 11, 13 . . . These are the primes, the indivisible numbers that are the building blocks of all other numbers—the hydrogen and oxygen of the world of mathematics. These protagonists at the heart of the story of numbers are like jewels studded through the infinite expanse of numbers.
Yet despite their importance, prime numbers represent one of the most tantalizing puzzles we have come across in our pursuit of knowledge. Knowing how to find the primes is a total mystery because there seems to be no magic formula that gets you from one to the next. They are like buried treasure—and no one has the treasure map.
In this chapter, we will explore what we do understand about these special numbers. In the course of this journey, we will discover how different cultures have tried to record and survey primes and how musicians have exploited their syncopated rhythm. We will find out why the primes have been used to try to communicate with extraterrestrials and how they have helped to keep things secret on the Internet. At the end of the chapter, I unveil a mathematical enigma about prime numbers that will earn you a million dollars if you can crack it. But before we tackle one of the biggest conundrums of mathematics, let us begin with one of the great numerical mysteries of our time.
WHY DID BECKHAM CHOOSE THE NUMBER 23 SHIRT?
When David Beckham joined the Real Madrid soccer team in 2003, there was a lot of speculation about why he’d chosen to play in the number 23 shirt. It was a strange choice, many thought, since he’d been playing in the number 7 shirt for England and Manchester United. The trouble was that on the Real Madrid team, the number 7 shirt was already being worn by Raúl, and the Spaniard wasn’t about to move over for this glamour-boy from England.
Many different theories were put forth to account for Beckham’s choice, and the most popular was the Michael Jordan theory. Real Madrid wanted to break into the American market and sell lots of replica shirts to the huge US population. But soccer is not a popular game in the States. Americans like baseball and basketball—games that end with scores like 100 to 98 and in which there’s invariably a winner. What’s the point of a game that goes on for 90 minutes and can end with no side scoring or winning?
With this theory in mind, Real Madrid did its research and found that the most popular basketball player in the world was definitely Michael Jordan, the Chicago Bulls’ most prolific scorer. Jordan sported the number 23 shirt for his entire career. All Real Madrid had to do was put 23 on the back of a soccer shirt, cross their fingers, and hope that the Jordan connection would work its magic and that they would break into the American market.
Others suggested a more sinister theory. Julius Caesar was assassinated by being stabbed in the back 23 times. Was Beckham’s choice for his back a bad omen? Still others thought that maybe the choice was connected with Beckham’s love of Star Wars (Princess Leia was imprisoned in Detention Block AA23 in the first Star Wars movie). Or was Beckham a secret member of the Discordianists, a modern cult that reveres chaos and has a cabalistic obsession with the number 23?
But as soon as I saw Beckham’s number, a more mathematical solution immediately came to mind. 23 is a prime number. A prime number is a number that is divisible only by itself and 1. 17 and 23 are prime because they can’t be written as two smaller numbers multiplied together, whereas 15 isn’t prime because 15 = 3 × 5. Prime numbers are the most important numbers in mathematics because all other whole numbers are built by multiplying primes together.
Take 105, for example. This number is clearly divisible by 5. So I can write 105 = 5 × 21. 5 is a prime number—an indivisible number—but 21 isn’t: I can write it as 3 × 7. So 105 can also be written as 3 × 5 × 7. But this is as far as I can go: I’m down to the primes, the indivisible numbers from which the number 105 is built. I can do this with any number since every number is either prime and indivisible or not prime and divisible by smaller numbers multiplied together.
Figure 1.1
The primes are the building blocks of all numbers. Just as molecules are built from atoms, such as hydrogen and oxygen or sodium and chlorine, numbers are built from primes. In the world of mathematics, the numbers 2, 3, and 5 are like hydrogen, helium, and lithium. That’s what makes them the most important numbers in mathematics. But they were clearly important to Real Madrid, too.
When I started looking a little closer at Real Madrid’s soccer team, I began to suspect that perhaps they had a mathematician on the bench. A little analysis revealed that at the time of Beckham’s move, all the Galácticos, the key players for Real Madrid, were playing in prime-number shirts: Carlos (the building block of the defense) wore number 3; Zidane (the heart of the midfield) was number 5; and Raúl and Ronaldo (the foundations of Real’s strikers) sported numbers 7 and 11, respectively. So perhaps it was inevitable that Beckham also got a prime number, one that he has become very attached to. When he joined LA Galaxy, he insisted on taking his prime number with him in his attempt to woo the American public with the beautiful game.
This may sound totally irrational coming from a mathematician, someone who is meant to be a logical analytical thinker. However, I also play in a prime-number shirt for my soccer team, Recreativo Hackney, so I felt some connection with the man in 23. My Sunday League team isn’t quite as big as Real Madrid and we didn’t have a 23 shirt, so I chose 17—a rather nice prime, as we’ll see later. But in our first season together, our team didn’t do particularly well. We play in the London Super Sunday League Division 2, and that season we finished rock bottom. Fortunately, this is the lowest division in London, so the only way to go was up.
But how were we to improve our league standing? Maybe Real Madrid was on to something—was there some psychological advantage to be had from playing in a prime-number shirt? Perhaps too many of us were in nonprimes, like 8, 10, or 15. The next season, I persuaded the team to change our gear, and we all played in prime numbers: 2, 3, 5, 7 . . . all the way up to 43. It transformed us. We got promoted to Division 1, where we quickly learned that primes last only for one season. We were relegated back down to Division 2, and we are now on the lookout for a new mathematical theory to boost our chances.
A Prime-Number Fantasy Football Game
Download the PDF file for this game from the Number Mysteries website (http://www.fifthestate.co.uk/numbermysteries/). Each player cuts out three Subbuteo-style players and chooses different prime numbers to write on their backs. Use one of the Euclid soccer balls from chapter 2 (page 64).
The ball starts with a player from team 1. The aim is to make it past the three players on the opponent’s team. The opponent chooses the first player to try to tackle team 1’s player. Roll the die. The die has six sides: white 3, white 5, and white 7, and black 3, black 5, and black 7. The die will tell you to divide your prime and the prime of your opponent’s player by 3, 5, or 7 and then take the remainder. If it is a white 3, 5, or 7, your remainder needs to match or beat the opposition. If it is black, you need to match or get less than your opponent.
To score, you must make it past all three players and then go up against a random choice of prime from the opposition. If at any point the opposition beats you, then possession switches
to the opposition. The person who has gained possession then uses the player who won to try to make it through the opposition’s three players. If team 1’s shot at the goal is missed, then team 2 takes the ball and gives it to one of its players.
The game can be played either against the clock or first to three goals.
SHOULD REAL MADRID’S GOALKEEPER WEAR THE NUMBER 1 SHIRT?
If the key players for Real Madrid wear primes, then what shirt should the goalkeeper wear? Or, put mathematically, is 1 a prime? Well, yes and no. (This is just the sort of math question everyone loves—both answers are right.) Two hundred years ago, tables of prime numbers included 1 as the first prime. After all, it isn’t divisible, since the only whole number that divides it is itself. But today, we say that 1 is not a prime because the most important thing about primes is that they are the building blocks of numbers. If I multiply a number by a prime, I get a new number. Although 1 is not divisible, if I multiply a number by 1, I get the number I started with, and on that basis, we exclude 1 from the list of primes and start at 2.
Clearly, Real Madrid wasn’t the first to discover the potency of the primes. But which culture got there first—the ancient Greeks? The Chinese? The Egyptians? It turns out that mathematicians were beaten to the discovery of the primes by a strange little insect.
WHY DOES AN AMERICAN SPECIES OF CICADA LIKE THE PRIME 17?
In the forests of North America, there is a species of cicada with a very strange life cycle. For 17 years, these cicadas hide underground doing very little except sucking on the roots of the trees. Then in May of the seventeenth year, they emerge at the surface en masse to invade the forest—up to a million of them per acre.
The cicadas sing away to one another, trying to attract mates. Together, they make so much noise that local residents often move out for the duration of this invasion every 17 years. Bob Dylan was inspired to write his song “Day of the Locusts” when he heard the cacophony of cicadas that emerged in the forests around Princeton when he was collecting an honorary degree from the university in 1970.