The Square Root of 2 A Dialogue Concerning a Number and a Sequence
The square root of 2 is a fascinating number – if a little less famous than such mathematical stars as pi, the number e, the golden ratio, or the square root of –1. (Each of these has been honored by at least one recent book.) Here, in an imaginary dialog
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The Square Root of 2 A Dialogue Concerning a Number and a Sequence
David Flannery
COPERNICUS BOOKS An Imprint of Springer Science+Business Media
in Association with Praxis Publishing, Ltd.
© 2006 Praxis Publishing Ltd. 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. Published in the United States by Copernicus Books, an imprint of Springer Science+Business Media. Copernicus Books Springer Science+Business Media 233 Spring Street New York, NY 10013 www.springeronline.com Library of Congress Control Number: 2005923268
Manufactured in the United States of America. Printed on acid-free paper.
9 8 7 6 5 4 3 2 1 ISBN-10: 0-387-20220-X ISBN-13: 978-0387-20220-4
Why, sir, if you are to have but one book with you on a journey, let it be a book of science. When you have read through a book of entertainment, you know it, and it can do no more for you; but a book of science is inexhaustible. . . . —James Boswell Journal of a Tour to the Hebrides with Samuel Johnson
Contents
Prologue
ix
Chapter 1
Asking the Right Questions
1
Chapter 2
Irrationality and Its Consequences
37
Chapter 3
The Power of a Little Algebra
75
Chapter 4
Witchcraft
121
Chapter 5
Odds and Ends
191
Epilogue
249
Chapter Notes
251
Acknowledgments
255
vii
Prologue
You may think of the dialogue you are about to read, as I often did while writing it, as being between a “master” and a “pupil”—the master in his middle years, well-versed in mathematics and as devoted and passionate about his craft as any artist is about his art; the pupil on the threshold of adulthood, articulate in speech, adventuresome of mind, and enthusiastically receptive to any knowledge the more learned teacher may care to impart. Their conversation—the exact circumstances of which are never described—is initiated by the master, one of whose tasks is to persuade his disciple that the concept of number is more subtle than might first be imagined. Their mathematical journey starts with the teacher guiding the student, by way of questions and answers, through a beautifully simple geometrical demonstration (believed to have originated in ancient India), which establishes the existence of a certain number, the understanding of whose nature is destined to form a major part of the subsequent discussion between the enquiring duo. Strong as the master’s motivation is to have the younger person glimpse a little of the wonder of mathematics, stronger still is his desire to see that his protégé gradually becomes more and more adept at mathematical reasoning so that he may experience the pure pleasure to be had from simply “finding things out” for himself. This joy of discovery is soon felt by the young learner, who having embarked upon an exploration, is richly rewarded when, after some effort, he chances upon a sequence of numbers that he surmises is inextrica
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