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ENCYCLOPEDIA OF Mathematics James Tanton, P .D. h Encyclopedia of Mathematics Copyright © 2005 by James Tanton, Ph.D. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval systems, without permission in writing from the publisher. For information contact: Facts On File, Inc. 132 West 31st Street New York NY 10001 Library of Congress Cataloging-in-Publication Data Tanton, James Stuart, 1966– Encyclopedia of mathematics/James Tanton. p. cm. Includes bibliographical references and index. ISBN 0-8160-5124-0 1. Mathematics—Encyclopedia. I. Title. QA5.T34 2005 510′.3—dc22 2004016785 Facts On File books are available at special discounts when purchased in bulk quantities for businesses, associations, institutions, or sales promotions. Please call our Special Sales Department in New York at (212) 967-8800 or (800) 322-8755. You can find Facts On File on the World Wide Web at http://www.factsonfile.com Text design by Joan M. Toro Cover design by Cathy Rincon Illustrations by Richard Garratt Printed in the United States of America VB Hermitage 10 9 8 7 6 5 4 3 2 1 This book is printed on acid-free paper. C ONTENTS Acknowledgments v Introduction vi A to Z Entries 1 Feature Essays: “History of Equations and Algebra” 9 “History of Calculus” 57 “History of Functions” 208 “History of Geometry” 226 “History of Probability and Statistics” 414 “History of Trigonometry” 510 Appendixes: Appendix I Chronology 539 Appendix II Bibliography and Web Resources 546 Appendix III Associations 551 Index 552 A CKNOWLEDGMENTS My thanks to James Elkins for reading a substantial portion of the manuscript, for his invaluable comments, and for shaping my ideas in writing a number of specific entries. Thanks also go to Frank K. Darm- stadt, executive editor at Facts On File, for his patience and encourage- ment, and to Jodie Rhodes, literary agent, for encouraging me to pursue this project. I also wish to thank Tucker McElroy and John Tabak for tak- ing the time to offer advice on finding archives of historical photographs, and the staff of The Image Works for their work in finding photographs and granting permission to use them. But most of all, thanks to Lindy and Turner for their love and support, always. v I NTRODUCTION Mathematics is often presented as a large collection of disparate facts to be absorbed (memorized!) and used only with very specific applications in mind. Yet the development of mathematics has been a journey that has engaged the human mind and spirit for thousands of years, offering joy, play, and creative invention. The Pythagorean theorem, for instance, although likely first developed for practical needs, provided great intellec- tual interest to Babylonian scholars of 2000 B.C.E., who hunted for extraordinarily large multidigit numbers satisfying the famous relation a2+ b2 = c2. Ancient Chinese scholars took joy in arranging numbers in square grids to create the first “magic squares,” and Renaissance scholars in Europe sought to find a formula for the prime numbers, even though no practical application was in mind. Each of these ideas spurred further questions and further developments in mathematics—the general study of Diophantine equations, semi-magic squares and Latin squares, and public- key cryptography, for instance—again, both with and without practical application in mind. Most every concept presented to students today has a historical place and conceptual context that is rich and meaningful. The aim of Facts On File’s Encyclopedia of Mathematics is to unite disparate ideas and provide a sense of meaning and context. Thanks to the encyclopedic format, all readers can quickly find straightforward answers to questions that seem to trouble students and teachers alike: • Why is the product of two negative numbers positive? • What is π, and why is the value of this number the same for all circles? • What is the value of πfor a shape different than a circle? • Is every number a fraction? • Why does the long-division algorithm work? • Why is dividing by a fraction the same as multiplying by its reciprocal? • What is the value of ii? • What is the fourth dimension? vi Introduction vii This text also goes further and presents proofs for many of the results discussed. For instance, the reader can find, under the relevant entries, a proof to the fundamental theorem of algebra, a proof of Descartes’s law of signs, a proof that every number has a unique prime factorization, a proof of Bretschneider’s formula (generalizing Brahmagupta’s famous formula), and a derivation of Heron’s formula. Such material is rarely presented in standard mathematical textbooks. In those instances where the method of proof is beyond the scope of the text, a discussion as to the methods behind the proof is at least offered. (For instance, an argument is presented to show how a formula similar to Stirling’s formula can be obtained, and the discussion of the Cayley-Hamilton theorem shows that every matrix satisfies at least some polynomial equation.) This encyclopedia aims to be satisfying to those at all levels of interest. Each entry contains cross-references to other items, providing the opportunity to explore further context and related ideas. The reader is encouraged to browse. As a researcher, author, and educator in mathematics, I have always striven to share with my students the sense of joy and enthusiasm I expe- rience in thinking about and doing mathematics. Collating, organizing, and describing the concepts a high-school student or beginning college- level student is likely to encounter in the typical mathematics curriculum, although a daunting pursuit, has proved to be immensely satisfying. I have enjoyed the opportunity to convey through the writing of this text, hopefully successfully, a continued sense of joy and delight in what mathematics can offer. Sadly, mathematics suffers from the ingrained perception that primary and secondary education of the subject should consist almost exclusively of an acquisition of a set of skills that will prove to be useful to students in their later careers. With the push for standardized testing in the public school system, this mind-set is only reinforced, and I personally fear that the joy of deep understanding of the subject and the sense of play with the ideas it contains is diminishing. For example, it may seem exciting that we can produce students who can compute 584 × 379 in a flash, but I am sad- dened with the idea that such a student is not encouraged to consider why we are sure that 379 × 584 will produce the same answer. For those students that may be naturally inclined to pause to consider this, I also worry about the response an educator would give upon receiving such a query. Is every teacher able to provide for a student an example of a system of arith- metic for which it is no longer possible to assume that a×b and b×a are always the same and lead a student through a path of creative discovery in the study of such a system? (As physicists and mathematicians have discov- ered, such systems do exist.) By exploring fundamental questions that chal- lenge basic assumptions, one discovers deeper understanding of concepts and finds a level of creative play that is far more satisfying than the perfor- mance of rote computation. Students encouraged to think this way have learned to be adaptable, not only to understand and apply the principles of a concept to the topic at hand, but also to apply those foundations and habits of mind to new situations that may arise. After all, with the current viii Introduction advances of technology in our society today, we cannot be sure that the rote skill-sets we deem of value today will be relevant to the situations and environments students will face in their future careers. We need to teach our students to be reflective, to be flexible, and to have the confidence to adapt to new contexts and new situations. I hope that this text, in some small ways, offers a sense of the creative aspect to mathematical thinking and does indeed gently encourage the reader to think deeply about concepts, even familiar ones. Encyclopedia of Mathematicscontains more than 800 entries arranged in alphabetical order. The aim of the historical notes, culture-specific arti- cles, and the biographical portraits included as entries, apart from providing historical context, is to bring a sense of the joy that mathematics has brought people in the past. The back matter of this text contains a timeline listing major accomplishments throughout the historical development of mathematics, a list of current mathematics organizations of interest to students and teachers, and a bibliography. A AAA/AAS/ASA/SAS/SSS Many arguments and proofs the AAS and ASA rules apply. This is sometimes called presented in the study of GEOMETRY rely on identify- the “HA congruence criterion” for right triangles.) ing similar triangles. The SECANT theorem, for c. The SAS rule:If two triangles have two sides of instance, illustrates this. Fortunately, there are a num- matching lengths with matching included angle, then ber of geometric tests useful for determining whether the two triangles are congruent. or not two different triangles are similar or congruent. The names for these rules are acronyms, with the let- The law of cosines ensures that the third side-lengths of ter A standing for the word angle, and the letter S for each triangle are the same, and that all remaining angles the word side. We list the rules here with an indica- in the triangles match. By the AAS and ASA rules, the tion of their proofs making use of the LAW OF SINES triangles are thus congruent. As an application of this and the LAWOFCOSINES: rule, we prove EUCLID’s isosceles triangle theorem: a. The AAA rule:If the three interior angles of one tri- The base angles of an isosceles triangle are angle match the three interior angles of a second tri- equal. angle, then the two triangles are similar. Suppose ABC is a triangle with sides AB and The law of sines ensures that pairs of corresponding AC equal in length. Think of this triangle as sides of the triangles have lengths in the same ratio. representing two triangles: one that reads BAC Also note, as the sum of the interior angles of any trian- and the other as CAB. These two triangles gle is 180°, one need only check that twocorresponding have two matching side-lengths with matching pairs of interior angles from the triangles match. included angles, and so, by the SAS rule, are congruent. In particular, all corresponding b. The AAS and ASA rules:If two interior angles and angles are equal. Thus the angle at vertex B of one side-length of one triangle match corresponding the first triangle has the same measure as the interior angles and side-length of a second triangle, corresponding angle of the second triangle, then the two triangles are congruent. namely, the angle at vertex C. By the AAA rule the two triangles are similar. Since a This result appears as Proposition 5 of Book I of pair of corresponding side-lengths match, the two trian- Euclid’s famous work THEELEMENTS. gles are similar with scale factor one, and are hence congruent. (Note that any two right triangles sharing a d. The SSS rule:If the three side-lengths of one triangle common hypotenuse and containing a common acute match the three side-lengths of a second triangle, angle are congruent: all three interior angles match, and then the two triangles are congruent. 1