Encyclopedia Of Mathematics – James Tanton

Place the tourists from the first bus into the rooms given by the powers of 3: 3, 9, 27,… Place the tourists from the second bus into the rooms given by the powers of 5: 5, 25, 125,… Next use all rooms numbered the powers of 7, of 11, of 13, and so on, down along the list of all the PRIME numbers. As, according to there are infinitely many primes, and no two powers of different primes are the same, this allocation scheme does the trick.

See also CARDINALITY; FUNDAMENTAL THEOREM OF Hindu-Arabic numerals The numeral system we use today is called the Hindu-Arabic numeral system. Using a base-10 PLACE-VALUE SYSTEM, numbers are expressed via combinations of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, organized so as to represent groupings of powers of 10. For instance, the number 574 repre- sents the five groups of 100, seven groups of 10, and four single units.

This numerical system originated from India around 600 C.E., almost in the exact same form as we use it today. The system was transmitted to the Arabs two centuries later as they worked to translate the San- skrit works on astronomy into Arabic. The Arab math- ematician MUHAMMAD AL-KHW–ARIZM–ı (ca. 800) wrote an influential treatise describing the Hindu numeral system, and used it in his famous book Hisab al-jabr w’al-muq¯abala (Calculation by restora- tion and reduction), from whose title the modern word algebra is derived. As Western scholars began translat- ing the Arabic texts into Latin, word of the efficient numeration system spread across Western Europe.

Ital- ian scholar FIBONACCI (ca. 1170–1250) avidly pro- moted their use. By the end of the 17th century, the Hindu-Arabic numeral system completely replaced the cumbersome system of ROMAN NUMERALS that were the standard in Europe for over 1,500 years. See also BASE OF A NUMBER SYSTEM; DECIMAL REP- Hippasus of Metapontum See PYTHAGORAS. Hippocrates of Chios (ca.

470–410 B.C.E.) Greek Geometry Born in Khíos (Chios), Greece, Hippo- crates is remembered as the first mathematician to have found the area of a curved figure, namely, that of a LUNE. He also made first steps toward properly analyz- ing the problem of SQUARING THE CIRCLE. Very little is known of Hippocrates’ life.

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. CONTENTS 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 v ACKNOWLEDGMENTS 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.

vi 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.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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