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555718294920923 PETER J. BENTLEY THE BOOK OF NUMBERS THE SECRET OF NUMBERS AND HOW THEY CHANGED THE WORLD FIREFLY BOOKS 8 A FIREFLY BOOK Published by Firefly Books Ltd. 2008 Copyright © 2008 Peter J. Bentley All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the Publisher. Third printing, 2009 Publisher Cataloging-in-Publication Data (U.S.) Bentley, Peter J. The book of numbers / Peter J. Bentley. [272) p. : col. photos.; cm. Includes index. Summary: Explores the fascination with numbers and mathematics in every aspect of life from The DaVinci Code to the Enigma Code, as well as in the digital world. ISBN-13978-1-55407-361-0 ISBN-10 1-55407-361-8 1. Mathematics - Popular works. 2. Number theory - Popular works. I. Title. 510 dc22 OA93.B468 2008 Library and Archives Canada Cataloguing in Publication Bentley, Peter J., 1972- The book of numbers / Peter Bentley. Includes index. ISBN-13 978-1-55407-361-0 ISBN-10: 1-55407-361-8 1. Mathematics - Popular works. 2. Number theory Popular works. I. Title. OA93.B442008 510 C2007-904540-5 Published in the United States by Firefly Books (U.S.) Inc. PO. Box 1338, Ellicott Station Buffalo, New York 14205 Published in Canada by Firefly Books Ltd. 66 Leek Crescent Richmond Hill, Ontario L4B 1H 1 For Cassell Illustrated: Publishing Director: lain MacGregor Commissioning Editor: Laura Price Editor: Jenny Doubt Creative Director: Geoff Fennell Layout: Keith Williams Production: Caroline Alberti Cover photography: Corbis/Matthias Kulka/zefa back top left; /Joseph Sohm, Visions of America back bottom left; /Gustavo Tomsich back centre. Getty/Jeff Foott back top right. Science Photo Library back bottom left. Printed in China BEFORE THE BEGINNING 12 MUCH ADO ABOUT NOTHING 28 SMALL IS BEAUTIFUL 36 52 MURDERING IRRATIONALS 72 GOLDEN PHI 86 GOOD AND EVEN 112 THE GREATEST INVENTION 126 THE ETERNAL TRIANGLE 140 A SLICE OF PI 166 DECIMALIZATION 184 TRISKAIDEKAPHOBIA 194 AS FAST AS YOU CAN GO 216 THE NEVERENDING STORY I before the beginning I before the begonning We live in numbers, talk in numbers and watch numbers for entertainment. Numbers rule our lives, they wake us up, tell us where to go, how to get there and when to leave. Numbers are judges of all, they assess and compare with complete authority and impartiality. But numbers also lie; they may mean anything except the truth. Numbers can save our lives, and a love of the wrong kind of numbers can ruin us. Numbers can be our friends, our lifelines and our lucky charms. Numbers can also kill us. You are made of numbers. So am I. example the speed of light - seemed to be at Thousands of years ago, when there was the heart of the construction of our universe. no difference between science and religion, In those days it was widely believed that such numbers seemed to hold the key to under numbers pointed to the mysterious underlying standing the universe. They might not have design of God. An understanding of those trickled down in front of our eyes like a scene numbers would be like reading divine messages from The Matrix, but lurking inside many forms written into the fabric of existence. Those were important numbers that seemed too pioneers and adventurers who dared to explore common to be coincidental. The same ratios the uncharted territories of numbers were kept reappearing in nature, perhaps between the exploring the very substance of their world. diameter of a circle and its circumference, or in They were unpicking the details of life, the the curvature of seashells. The same geometric universe and everything. Their solutions were shapes and the numbers embedded within them not a single number, but a whole collection of kept being discovered in unlikely places, such as important numbers, as well as tools to the spacing of the planets of our solar system. manipulate those numbers. Even something as unlikely as a speed - for Today, science has taken over from religion. We still believe that our universe has hugely important numbers associated with it. We now BEGINNING know that they are the patterns visible in the woven tapestry from which everything is made. Some patterns in the tapestry are made from such thick threads that they catch the eye immediately: numbers such as 1t, e and 8. Some form the bulk of the material: numbers like 0, 1, 2, 3 and --./2. Some stand out like accidental spills on the fabric, such as 10 and 13. Other numbers and concepts, like c and 00, point to the size and shape of the tapestry. Some, such as i, are only visible as tenuous ripples of complexity that flow through the cloth. I before the beginning build complicated machines. We discovered the numbers of interacting waves, allowing us to understand music, the swing of pendulums and the bizarre properties of light. We learned how numbers describe position, speed and accelerations, enabling us to understand the motion of the planets and comprehend the planet we live on. We learned how numbers define time, space and the different sizes of infinity, enabling us to understand how the flow Those who explore the fundamental truths of time changes and how our universe began. of nature are now known by names such as Today we continue to learn the numbers that mathematicians, astronomers and physicists. affect subatomic particles, and those behind But however we label them, these people were complex systems such as economies, societies (and still are) explorers. They did not weave the and consciousness. These remarkable tapestry they examined. They did not invent their achievements have created our modern world numbers or mathematical ideas as a novelist of telephones, cars, music, computers and might invent a story. They searched for the truth, airplanes. They have enabled almost every and tried to explain it, inventing new languages modern device you use, the food you eat and of numbers simply to be able to write down their the job you do. Your entire lifestyle is shaped by discoveries. Some pursued this goal for science, our understanding of numbers. some for religion and some for fame. This book is about the explorers of numbers The explorers we will follow in this book and the inventors of mathematics. The were very clever and many have been called motivations and beliefs of these eccentric geniuses - but they were people too. They had characters are often surprising. However, more complicated lives, had arguments, failings and surprising than their discoverers, are the numbers successes. Galileo was a medical school drop themselves. out, Newton threatened to burn down his Albert Einstein once said, "There are two parents' house, Bernoulli stole his son's work, ways to live your life. One is as though nothing is Pascal was a bully and Einstein had a child out a miracle. The other is as though everything is a of wedlock. Some were murdered because of miracle." Numbers don't remove your ability to numbers. Others lost their sanity. Put them all in marvel at the world, they increase it. a room together and you'd probably be deafened Numbers are miraculous, as you're about to by the shouting. But they all had an appreciation find out. of numbers that made them exceptional. They came from allover the world, yet their language was universal. As their explanations improved, so did the language of numbers. Through our quirky pioneers we learned how numbers make shapes, angles and connections, enabling us to measure our land, to design and