The Facts On File DICTIONARY of MATHEMATICS The Facts On File DICTIONARY of MATHEMATICS Fourth Edition Edited by John Daintith Richard Rennie The Facts On File Dictionary of Mathematics Fourth Edition Copyright © 2005, 1999 by Market House Books Ltd 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 For Library of Congress Cataloging-in-Publication Data, please contact Facts On File, Inc. ISBN 0-8160-5651-X 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 Compiled and typeset by Market House Books Ltd, Aylesbury, UK Printed in the United States of America MP PKG10987654321 This book is printed on acid-free paper. PREFACE This dictionary is one of a series designed for use in schools. It is intended for stu- dents of mathematics, but we hope that it will also be helpful to other science stu- dents and to anyone interested in science. Facts On File also publishes dictionaries in a variety of disciplines, including biology, chemistry, forensic science, marine sci- ence, physics, space and astronomy, and weather and climate. The Facts On File Dictionary of Mathematics was first published in 1980 and the third edition was published in 1999. This fourth edition of the dictionary has been extensively revised and extended. The dictionary now contains over 2,000 head- words covering the terminology of modern mathematics. A totally new feature of this edition is the inclusion of over 800 pronunciations for terms that are not in everyday use. A number of appendixes have been included at the end of the book containing useful information, including symbols and notation, symbols for physical quantities, areas and volumes, expansions, derivatives, integrals, trigonometric for- mulae, a table of powers and roots, and a Greek alphabet. There is also a list of Web sites and a bibliography. A guide to using the dictionary has also been added to this latest version of the book. We would like to thank all the people who have cooperated in producing this book. A list of contributors is given on the acknowledgments page. We are also grateful to the many people who have given additional help and advice. v ACKNOWLEDGMENTS Contributors Norman Cunliffe B.Sc. Eric Deeson M.Sc., F.C.P., F.R.A.S. Claire Farmer B.Sc. Jane Farrill Southern B.Sc., M.Sc. Carol Gibson B.Sc. Valerie Illingworth B.Sc., M.Phil. Alan Isaacs B.Sc., Ph.D. Sarah Mitchell B.A. Roger Adams B.Sc. Roger Picken B.Sc. Janet Triggs B.Sc. Pronunciations William Gould B.A. vi CONTENTS Preface v Acknowledgments vi Guide to Using the Dictionary viii Pronunciation Key x Entries A to Z 1 Appendixes I. Symbols and Notation 245 II. Symbols for Physical 248 Quantities III. Areas and Volumes 249 IV. Expansions 250 V. Derivatives 251 VI. Integrals 252 VII. Trigonometric Formulae 253 VIII. Conversion Factors 254 IX. Powers and Roots 257 X. The Greek Alphabet 260 XI. Web Sites 261 Bibliography 262 vii GUIDE TO USING THE DICTIONARY The main features of dictionary entries are as follows. Headwords The main term being defined is in bold type: absolute Denoting a number or measure- ment that does not depend on a standard reference value. Plurals Irregular plurals are given in brackets after the headword. abscissa (pl. abscissas or abscissae) The horizontal or x-coordinate in a two-dimen- sional rectangular Cartesian coordinate system. Variants Sometimes a word has a synonym or alternative spelling. This is placed in brackets after the headword, and is also in bold type: angular frequency (pulsatance) Symbol: ω The number of complete rotations per unit time. Here, ‘pulsatance’ is another word for angular frequency. Generally, the entry for the syn- onym consists of a simple cross-reference: pulsatance See angular frequency. Abbreviations Abbreviations for terms are treated in the same way as variants: cosecant (cosec; csc) A trigonometric function of an angle equal to the reciprocal of its sine .... The entry for the synonym consists of a simple cross-reference: cosec See cosecant. Multiple definitions Some terms have two or more distinct senses. These are numbered in bold type base 1. In geometry, the lower side of a triangle, or other plane figure, or the lower face of a pyramid or other solid. 2. In a number system, the number of dif- ferent symbols used, including zero. viii Cross-references These are references within an entry to other entries that may give additional useful in- formation. Cross-references are indicated in two ways. When the word appears in the de- finition, it is printed in small capitals: Abelian group /ă-beel-ee-ăn / (commuta- tive group) A type of GROUP in which the elements can also be related to each other in pairs by a commutative operation. In this case the cross-reference is to the entry for ‘group’. Alternatively, a cross-reference may be indicated by ‘See’, ‘See also’, or ‘Compare’, usu- ally at the end of an entry: angle of depression The angle between the horizontal and a line from an observer to an object situated below the eye level of the observer. See also angle. Hidden entries Sometimes it is convenient to define one term within the entry for another term: arc A part of a continuous curve. If the cir- cumference of a circle is divided into two unequal parts, the smaller is known as the minor arc and… Here, ‘minor arc’ is a hidden entry under arc, and is indicated by italic type. The entry for ‘minor arc’ consists of a simple cross-reference: minor arc See arc. Pronunciations Where appropriate pronunciations are indicated immediately after the headword, en- closed in forward slashes: abacus /ab-ă-kŭs/ A calculating device consisting of rows of beads strung on wire and mounted in a frame. Note that simple words in everyday language are not given pronunciations. Also head- words that are two-word phrases do not have pronunciations if the component words are pronounced elsewhere in the dictionary. ix Pronunciation Key Bold type indicates a stressed syllable. In pronunciations, a consonant is sometimes dou- bled to prevent accidental mispronunciation of a syllable resembling a familiar word; for example, /ass-id/ (acid), rather than /as-id/ and /ul-tră- sonn-iks/ (ultrasonics), rather than /ul-tră-son-iks/. An apostrophe is used: (a) between two consonants forming a syllable, as in /den-t’l/ (dental), and (b) between two letters when the syllable might otherwise be mis- pronounced through resembling a familiar word, as in /th’e-ră-pee/ (therapy) and /tal’k/ (talc). The symbols used are: /a/ as in back /bak/, active /ak-tiv/ /ng/ as in sing /sing/ /ă/ as in abduct /ăb-dukt/, gamma /gam-ă/ /nk/ as in rank /rank/, bronchus /bronk-ŭs/ /ah/ as in palm /pahm/, father /fah-ther/, /o/ as in pot /pot/ /air/ as in care /kair/, aerospace /air-ŏ- /ô/ as in dog /dôg/ spays/ /ŏ/ as in buttock /but-ŏk/ /ar/ as in tar /tar/, starfish /star-fish/, heart /oh/ as in home /hohm/, post /pohst/ /hart/ /oi/ as in boil /boil/ /aw/ as in jaw /jaw/, gall /gawl/, taut /tawt/ /oo/ as in food /food/, croup /kroop/, fluke /ay/ as in mania /may-niă/ ,grey /gray/ /flook/ /b/ as in bed /bed/ /oor/ as in pruritus /proor-ÿ-tis/ /ch/ as in chin /chin/ /or/ as in organ /or-găn/, wart /wort/ /d/ as in day /day/ /ow/ as in powder /pow-der/, pouch /e/ as in red /red/ /powch/ /ĕ/ as in bowel /bow-ĕl/ /p/ as in pill /pil/ /ee/ as in see /see/, haem /heem/, caffeine /r/ as in rib /rib/ /kaf-een/, baby /bay-bee/ /s/ as in skin /skin/, cell /sel/ /eer/ as in fear /feer/, serum /seer-ŭm/ /sh/ as in shock /shok/, action /ak-shŏn/ /er/ as in dermal /der-măl/, labour /lay-ber/ /t/ as in tone /tohn/ /ew/ as in dew /dew/, nucleus /new-klee-ŭs/ /th/ as in thin /thin/, stealth /stelth/ /ewr/ as in epidural /ep-i-dewr-ăl/ /th/ as in then /then/, bathe /bayth/ /f/ as in fat /fat/, phobia /foh-biă/, rough /u/ as in pulp /pulp/, blood /blud/ /ruf/ /ŭ/ as in typhus /tÿ-fŭs/ /g/ as in gag /gag/ /û/ as in pull /pûl/, hook /hûk/ /h/ as in hip /hip/ /v/ as in vein /vayn/ /i/ as in fit /fit/, reduction /ri-duk-shăn/ /w/ as in wind /wind/ /j/ as in jaw /jaw/, gene /jeen/, ridge /rij/ /y/ as in yeast /yeest/ /k/ as in kidney /kid-nee/, chlorine /klor- /ÿ/ as in bite /bÿt/, high /hÿ/, hyperfine /hÿ- een/, crisis /krÿ-sis/ per-fÿn/ /ks/ as in toxic /toks-ik/ /yoo/ as in unit /yoo-nit/, formula /form- /kw/ as in quadrate /kwod-rayt/ yoo-lă/ /l/ as in liver /liv-er/, seal /seel/ /yoor/ as in pure /pyoor/, ureter /yoor-ee- /m/ as in milk /milk/ ter/ /n/ as in nit /nit/ /ÿr/ as in fire /fÿr/ x A 3 2 5 8 Abacus: the number 3258 is shown on the right-hand side. abacus /ab-ă-kŭs/ A calculating device abscissa /ab-siss-ă/ (pl.