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~Umbers the BOO K O F Umbers TH E BOOK OF ~umbers THE BOO K o F umbers John H. Conway • Richard K. Guy c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a re­ trieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Published in the United States by Copernicus, an imprint of Springer-Verlag New York, Inc. Copernicus Springer-Verlag New York, Inc. 175 Fifth Avenue New York, NY lOOlO Library of Congress Cataloging in Publication Data Conway, John Horton. The book of numbers / John Horton Conway, Richard K. Guy. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8488-8 e-ISBN-13: 978-1-4612-4072-3 DOl: 10.l007/978-1-4612-4072-3 1. Number theory-Popular works. I. Guy, Richard K. II. Title. QA241.C6897 1995 512'.7-dc20 95-32588 Manufactured in the United States of America. Printed on acid-free paper. 9 8 765 4 Preface he Book ofNumbers seems an obvious choice for our title, since T its undoubted success can be followed by Deuteronomy,Joshua, and so on; indeed the only risk is that there may be a demand for the earlier books in the series. More seriously, our aim is to bring to the inquisitive reader without particular mathematical background an ex­ planation of the multitudinous ways in which the word "number" is used. We have done this in a way which is free from the formality of textbooks and syllabuses, so that the professional mathematician can also glean important information in areas outside her own speciality, and correspondingly enrich her teaching. The uses of the word number are diverse, but we can identify at least three separate strands. The development of number (usually written in the singular), continually adapting and generalizing itself to meet the needs of both mathematics and its increasing variety of applications: the counting numbers, zero, fractions, negative num­ bers, quadratic surds, algebraic numbers, transcendental numbers, infinitesimal and transfinite numbers, surreal numbers, complex numbers, quatemions, octonions. Then there is the special study of the integers, Gauss's higher arithmetic, or the theory of numbers (usually written in the plural), which overlaps the more recent area of enumerative combinatorics. Special sets or sequences of numbers: the prime numbers, Mersenne v ~ THE BOOK OF NUMBERS and Fermat numbers, perfect numbers, Fibonacci and Catalan num­ bers, Euler and Eulerian numbers, Bernoulli numbers. Finally there is a host of special numbers: Ludolph's 'IT, Napier's e, Euler's 'Y, Feigenbaum's constant, algebraic numbers which arise in specific contexts ranging from the diagonal of a square or of a regular pentagon to the example of degree 71 which arises from the apparently simple "look and say" sequences. Although mathematics is traditionally arranged in logical se­ quences, that is not the way the human brain seems to work. While it is often useful to know parts of earlier chapters in the book when reading some of the later ones, the reader can browse at will, picking up nuggets of information, with no obligation to read steadily from cover to cover. Chapter 1 describes number words and symbols and Chapter 2 shows how many elementary but important facts can be discovered "without using any mathematics." Chapters 3 and 4 exhibit several sets of whole numbers which can manifest themselves in quite dif­ ferent contexts, while Chapter 5 is devoted to the "multiplicative building blocks," the prime numbers. Fractions, or "algebraic num­ bers of degree one," are dealt with in Chapter 6 and those of higher degree in Chapter 7. So-called "complex" and "transcendental" num­ bers are explained in Chapters 8 and 9. The final chapter is devoted to the infinite and the infinitesimal, to surreal numbers which con­ stitute an extremely large, yet infinitely small, subclass of the most recent development of number, the values of combinatorial games. The color graphics in Chapter 2 are the work of Kenny and Andy Guy, using Brian Wyvill's Graphicsland software at the Graphics Lab­ oratory in the Computer Science Department at The University of Calgary. Three of the figures in Chapter 4 are reproduced from Prze­ myslaw Prusinkiewicz and Aristid Lindenmayer, The Algorithmic Beauty of Plants, Springer-Verlag, New York 1990, by kind permis­ sion of the authors and publishers. The first figure is due to D.R. Fowler, the second to D.R.F. and P. Prusinkiewicz and the third to D.R.F. and A. Snider. We thank Andrew Odlyzko and Peter Renz for reading a much earlier draft of the book, and for making many useful suggestions. We PRE F ACE vii are also grateful for the polite and patient persistence of Jerry Lyons, liesl Gibson and Henry Krell, and for the general competence of the Springer staff. We are proud to be among the early contributors to what is yet another successful Springer series. Princeton and Calgary John H. Conway and Richard K. Guy Contents PREFACE v 1 THE ROMANCE OF NUMBERS 1 2 FIGURES FROM FIGURES: DOING ARITHMETIC AND ALGEBRA BY GEOMETRY 27 3 WHAT COMES NEXT? 63 4 FAMOUS FAMILIES OF NUMBERS 91 5 THE PRIMACY OF PRIMES 127 6 FURTHER FRUITFULNESS OF FRACTIONS 151 7 GEOMETRIC PROBLEMS AND ALGEBRAIC NUMBERS 181 8 IMAGINING IMAGINARY NUMBERS 211 9 SOME TRANSCENDENTAL NUMBERS 237 10 INFINITE AND INFINITESIMAL NUMBERS 265 INDEX 301 ix v.J \ ...: The Romance of Numbers NUMBER WORDS Throughout history, number and numbers have had a tremendous influence on our culture and on our language. Thousands of words are obviously associated with numbers, for example, a monologue is a speech by 1 person; a bicycle has 2 wheels; a tripod is a stool with 3 legs; a quadruped is an animal with 4 legs; a pentathlon consists of 5 athletic events; a sextet is a piece for 6 musicians; a heptagon is a 7-comered figure; an octopus has 8 "feet"; a nonagenarian has lived for 9 decades; and decimal means counting in lOs; but in many other cases the connections, once just as vivid, have been obscured by the passage of time and changes in meaning. Have you ever realized just how many words are associated with "number" itself? This comes from an Indo-European root meaning "share" or "portion" and seems to have been originally associated with the division of land. "Nimble" refers to one who is quick to take his share; your "nemesis" was originally your portion of Fate; and 1 2 THE BOOK OF NUMBERS "numb" means "seized" or "taken." A "nomad" is one who wanders about in search of pasture land. There are many technical "nom" words: "binomial" (two-numbered), "astronomy" (numbering or al­ lotting the stars), "economy," "autonomy," and so on. In German we still have the word nebmen, to take, with imper­ ative "nimm." English once had much the same verb, but by Shake­ speare's day the word "nim" usually referred to illegal taking or pil­ fering. In Henry V, Corporal Nym is a doubtful character that one might well suspect of nimming. In our century "nim" has become the name of a take-away game. Latin and Greek nomisma was a coin, and this has given us "nu­ mismatics," the love of coins, and "nummulite," a coin-shaped fossil. "Supernumerary" now often means "redundant," but originally su­ pernumerary officials were those in addition to the number laid down by the regulations. HICKORY, DICKORY, DOCK­ EENY, MEENY, MINY, Mo There are several systems of words for numbers that survived for special purposes in out-of-the-way places. The particular sequence of words varies both with the part of the country and the purpose for which they were used. One such system is wan, twan, tetbera, metbera, pimp, setbera, letbera, bovera, dovera, dick, wanadick, twanadick, tetberadick, metberadick, pimpdick, setberadick, letberadick, boveradick, doveradick, bumjit, wanabumjit, .... Such rustic sequences appear in many countries. They are usually highly corrupted versions of the standard number systems of ancient languages. Hickory, dickory, dock. The mouse ran up tbe clock. The clock struck one; The mouse ran down. Hickory, dickory, dock. THE ROMANCE OF NUMBERS 3 Probably "hickory," "dickory," and "dock" are the words for "eight," "nine," and "ten" in one of these systems (compare "hov­ era, dovera, dick"), while "eeny, meeny, miny, mo" mean "one, two, three, four" in another. Most languages have interesting features about their number words, which repay study. We content ourselves with exhibiting those in Welsh. The alternative forms are for masculine and feminine, while the parenthetical forms are elisions in front of certain letters. There are separate ways of saying 50, 70, and 90. 1 un 16 un ar bymtheg 2 dau, dwy 17 dau, dwy ar bymtheg 3 toi, tair 18 deunaw 4 pedwar, pedair 19 pedwar, pedair ar bymtheg 5 pump (pum) 20 ugain 6 chwech (chwe) 30 deg ar hugain 7 saith 40 dugain 8 wyth 50 deg ar deugain; hanner cant 9 naw 60 trigain 10 deg 70 trigain a deg; deg ar thrigain 11 un ar ddeg 80 pedwar ugain 12 deuddeg 90 pedwar ugain a deg; deg a 13 toi, tair ar ddeg phedwar ugain 14 pedwar, pedair ar ddeg 100 cant 15 pymtheg Note that it is less duodecimal than many languages, which have sep­ arate words for eleven and twelve. It thus lends itself easily to modem decimalization. There are signs of bases 5 and 20, as well as 10.
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