The facts on file dictionary of mathematics

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 PKG 10 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 students of mathematics, but we hope that it will also be helpful to other science students and to anyone interested in science Facts On File also publishes dictionaries in a variety of disciplines, including biology, chemistry, forensic science, marine science, 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 headwords 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 formulae, 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 Appendixes I Symbols and Notation 245 II Symbols for Physical Quantities 248 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 measurement 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-dimensional 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 synonym 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 In geometry, the lower side of a triangle, or other plane figure, or the lower face of a pyramid or other solid In a number system, the number of different symbols used, including zero viii Cross-references These are references within an entry to other entries that may give additional useful information Cross-references are indicated in two ways When the word appears in the definition, it is printed in small capitals: Abelian group /ă-beel-ee-ăn / (commutative 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’, usually 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 circumference 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, enclosed 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 headwords that are two-word phrases not have pronunciations if the component words are pronounced elsewhere in the dictionary ix Appendix II Symbols for Physical Quantities acceleration angle a θ, φ, α , β, etc angular acceleration angular frequency, 2πf angular momentum angular velocity area breadth circular wavenumber density diameter distance energy force frequency height kinetic energy length mass moment of force α ω L ω A b k ρ d s, L W, E F f, ν h Ek, T l m M moment of inertia momentum period potential energy power pressure radius reduced mass relative density solid angle thickness time torque velocity viscosity volume wavelength wavenumber weight work 248 I p τ Ep, V P p r µ d Ω, ω d t T v η V λ σ W W, E Appendix III Areas and Volumes Plane figures Figure Dimensions Perimeter Area triangle sides a, b, and c, angle A a+b+c ½bc.sinA square side a 4a a2 rectangle sides a and b 2(a + b) a×b kite diagonals c and d parallelogram sides a and b distances c and d apart 2(a + b) a.c or b.d circle radius r 2πr πr2 ellipse axes a and b 2[(a2 + b2)/2] ab ẵc ì d Solid figures Figure Dimensions Area Volume cylinder radius r, height h 2πr(h + r) πr2h cone base radius r, slant height l, height h πrl πr2h/3 sphere radius r 4πr2 4πr3/3 249 Appendix IV Expansions sin x x/1! – x3/3! + x5/5! – x7/7! + x9/9! – cos x – x2/2! + x4/4! – x6/6! + x8/8! – ex + x/1! + x2/2! + x3/3! + x4/4! + x5/5! + sinh x x + x3/3! + x5/5! + x7/7! + x9/9! + cosh x + x2/2! + x4/4! + x6/6! + x8/8! + loge(1+x) x – x2/2 + x3/3 – x4/4 + x5/5 – loge(1–x) –x – x2/2 – x3/3 – x4/4 – x5/5 – (1 + x)n + nx + n(n – 1)x2/2! + n(n – 1)(n – 2)x3/3! + for |x| < f(a + x) f(a) + xf′(a) + (x2/2!)f′′(a) + (x3/3!)f′′′(a) + (x4/4!)f′′′′(a) + where f′(a) denotes the first derivative, f′′(a) the second derivative, etc f(x) f(0) + xf′(0) + (x2/2!)f′′(0) + (x3/3!)f′′′(0) + (x4/4!)f′′′′(0) + 250 Appendix V Derivatives x is a variable, u is a function of x, and a and n are constants Function f(x) Derivative df(x)/dx x ax a axn naxn–1 eax aeax loge x 1/x loga x (1/x)loge a cos x –sin x sin x cos x tan x sec2 x cot x –cosec2 x sec x tan x.sec x cosec x –cot x.cosec x cos u –sin u.(du/dx) sin u cos u.(du/dx) tan u sec2 u.(du/dx) loge u (1/u)(du/dx) sin–1 (x/a) 1/√(a2 – x2) cos–1 (x/a) –1/√(a2 – x2) tan–1 (x/a) a/(a2 + x2) 251 Appendix VI Integrals x is a variable and a and n are constants Note that a constant of integration C should be added to each integral Function f(x) Integral Ύf(x)dx a ax x x2/2 xn xn+1/(n + 1) 1/x loge x eax eax/a loge ax xloge ax – x cos x sin x sin x –cos x tan x loge (cos x) cot x loge (sin x) sec x loge (sec x + tan x) cosec x loge (cosec x – cot x) 1/√(a2 – x2) sin–1 (x/a) –1/√(a2 – x2) cos–1 (x/a) a/(a2 + x2) tan–1 (x/a) 252 Appendix VII Trigonometric Formulae Addition formulae: sin(x + y) = sinx cosy + cosx siny sin(x – y) = sinx cosy – cosx siny cos(x + y) = cosx cosy – sinx siny cos(x – y) = cosx cosy + sinx siny tan(x + y) = (tanx + tany)/(1 – tanx tany) tan(x – y) = (tanx – tany)/(1 + tanx tany) Double-angle formulae: sin(2x) = sinx cosx cos(2x) = cos2x – sin2x tan(2x) = 2tanx/(1 – tan2x) Half-angle formulae: sin(x/2) = ±√[(1 – cosx)/2] cos(x/2) = ±√[(1 + cosx)/2] tan(x/2) = sinx/(1 + cosx) = (1 – cosx)/sinx Poduct formulae: sinx cosy = ½[sin(x + y) + sin(x – y)] cosx siny = ½[sin(x + y) – sin(x – y)] cosx cosy = ½[cos(x + y) + cos(x – y)] sinx siny = ½[cos(x – y) – cos(x + y)] 253 Appendix VIII Conversion Factors Length To convert into multiply by inches feet yards miles nautical miles nautical miles kilometers kilometers meters meters meters meters meters meters kilometers kilometers miles miles nautical miles inches feet yards 0.0254 0.3048 0.9144 1.60934 1.85200 1.15078 0.621371 0.539957 39.3701 3.28084 1.09361 Area To convert into multiply by square inches square inches square feet square yards square miles square miles acres acres square centimeters square meters square meters square meters square meters square kilometers square centimeters square meters square meters square meters square kilometers acres square meters square miles square inches square feet square yards acres square miles square miles 6.4516 6.4516 × 10–4 9.2903 × 10–2 0.836127 2.58999 640 4046.86 1.5625 × 10–3 0.155 10.7639 1.19599 2.47105 × 10–4 3.86019 × 10–7 0.386019 Volume To convert into multiply by cubic inches cubic inches cubic feet cubic feet cubic yard gallon (US) gallon (US) gallon (US) liters cubic meters liters cubic meters cubic meters liters cubic meters gallon (UK) 1.63871 × 10–2 1.63871 × 10–5 28.3168 0.0283168 0.764555 3.785438 3.785438 × 10–3 0.83268 254 Appendix VIII Conversion Factors Mass To convert into multiply by pounds pounds hundredweight (short) hundredweight (short) tons (short) tons (short) kilograms kilograms kilograms tonnes tonnes tonnes kilograms tonnes kilograms tonnes kilograms tonnes pounds hundredweights (short) tons (short) pounds hundredweights (short) tons (short) 0.453592 4.53592 × 10–4 45.3592 0.0453592 907.18 0.90718 2.204623 0.022046 1.1023 × 10–3 2204.623 22.0462 0.90718 The short ton is used in the USA and is equal to 2000 pounds The short hundredweight (also known as the cental) is 100 pounds The long ton, which is used in the UK, is equal to 2240 pounds (1016.047 kg) The long hundredweight is 112 pounds (50.802 kg) long ton equals 20 long hundredweights Force To convert into multiply by pounds force pounds force pounds force pounds force poundals poundals poundals poundals dynes dynes dynes dynes kilograms force kilograms force kilograms force kilograms force newtons newtons newtons newtons newtons kilograms force dynes poundals newtons kilograms force dynes pounds force newtons kilograms force pounds force poundals newtons dynes pounds force poundals kilograms dynes pounds force poundals 4.44822 0.453592 444822 32.174 0.138255 0.031081 13825.5 0.031081 10–5 1.01972 × 10–6 2.24809 × 10–6 7.2330 × 10–5 9.80665 980665 2.20462 70.9316 0.101972 100000 0.224809 7.2330 255 Appendix VIII Conversion Factors Work and energy To convert into multiply by British Thermal Units British Thermal Units British Thermal Units kilowatt-hours kilowatt-hours kilowatt-hours calories calories calories joules joules joules joules joules electronvolts ergs joules calories kilowatt-hours joules calories British Thermal Units joules kilowatt-hours British Thermal Units calories kilowatt hours British Thermal Units electron volts ergs joules joules 1055.06 251.997 2.93071 × 10–4 3600000 859845 3412.14 4.1868 1.16300 × 10–6 3.96831 × 10–3 0.238846 2.7777 × 10–7 9.47813 × 10–4 6.2418 × 1018 107 1.6021 × 10–19 10–7 Pressure To convert into multiply by atmospheres bars pounds per square inch pounds per square inch pounds per square inch kilograms per square meter kilograms per square meter kilograms per square meter pascals pascals* pascals pascals 101325 100000 68894.76 kilograms per square meter atmospheres 703.068 pascals 9.80661 pounds per square inch atmospheres 1.42234 × 10–3 kilograms per square meter pounds per square inch atmospheres 0.101972 pascals pascals *1 pascal = newton per square meter 256 0.068046 9.67841 × 10–5 1.45038 × 10–4 9.86923 × 10–6 Appendix IX Powers and Roots n n2 n3 √n 16 25 27 64 125 1.000 1.414 1.732 2.000 2.236 1.000 1.260 1.442 1.587 1.710 10 36 49 64 81 100 216 343 512 729 000 2.449 2.646 2.828 3.000 3.162 1.817 1.913 2.000 2.080 2.154 11 12 13 14 15 121 144 169 196 225 33` 728 197 744 375 3.317 3.464 3.606 3.742 3.873 2.224 2.289 2.351 2.410 2.466 16 17 18 19 20 256 289 324 361 400 096 913 832 859 000 4.000 4.123 4.243 4.359 4.472 2.520 2.571 2.621 2.668 2.714 21 22 23 24 25 441 484 529 576 625 261 10 648 12 167 13 824 15 625 4.583 4.690 4.796 4.899 5.000 2.759 2.802 2.844 2.844 2.924 26 27 28 29 30 676 729 784 841 900 17 576 19 683 21 952 24 389 27 000 5.099 5.196 5.292 5.385 5.477 2.962 3.000 3.037 3.072 3.107 31 32 961 024 29 791 32 768 5.568 5.657 3.141 3.175 257 √n Appendix IX Powers and Roots n n2 n3 √n 33 34 35 089 156 225 35 937 39 304 42 875 5.745 5.831 5.916 3.208 3.240 3.271 36 37 38 39 40 296 369 444 521 600 46 656 50 653 54 872 59 319 64 000 6.000 6.083 6.164 6.245 6.325 3.302 3.332 3.362 3.391 3.420 41 42 43 44 45 681 764 849 936 025 68 921 74 088 79 507 85 184 91 125 6.403 6.481 6.557 6.633 6.708 3.448 3.476 3.503 3.530 3.557 46 47 48 49 50 116 209 304 401 500 97 336 103 823 110 592 117 649 125 000 6.782 6.856 6.928 7.000 7.071 3.583 3.609 3.634 3.659 3.684 51 52 53 54 55 601 704 809 916 025 132 651 140 608 148 877 157 464 166 375 7.141 7.211 7.280 7.348 7.416 3.708 3.733 3.756 3.780 3.803 56 57 58 59 60 136 249 364 481 600 175 616 185 193 195 112 205 379 216 000 7.483 7.550 7.616 7.681 7.746 3.826 3.849 3.871 3.893 3.915 61 62 63 64 65 721 844 969 096 225 226 981 238 328 250 047 262 144 274 625 7.810 7.874 7.937 8.000 8.062 3.936 3.958 3.979 4.000 4.021 258 √n Appendix IX Powers and Roots n n2 n3 √n 66 67 68 69 70 356 489 624 761 900 287 496 300 763 314 432 328 509 343 000 8.124 8.185 8.246 8.307 8.367 4.041 4.062 4.082 4.102 4.121 71 72 73 74 75 041 184 329 476 5.625 357 911 373 248 389 017 405 224 421 875 8.426 8.485 8.544 8.602 8.660 4.141 4.160 4.179 4.198 4.217 76 77 78 79 80 776 929 084 241 400 438 976 456 533 474 552 493 039 512 000 8.718 8.775 8.832 8.888 8.944 4.236 4.254 4.273 4.291 4.309 81 82 83 84 85 561 724 889 056 225 531 441 551 368 571 787 592 704 614 125 9.000 9.055 9.110 9.165 9.220 4.327 4.344 4.362 4.380 4.397 86 87 88 89 90 396 569 744 921 100 636 056 658 503 681 472 704 969 729 000 9.274 9.327 9.381 9.434 9.487 4.414 4.431 4.448 4.465 4.481 91 92 93 94 95 281 464 649 836 025 753 571 778 688 804 357 830 584 857 375 9.539 9.592 644 9.695 9.747 4.498 4.514 4.531 4.547 4.563 96 97 98 99 100 216 409 604 801 10 000 884 736 912 673 941 192 970 299 000 000 9.798 9.849 9.899 9.950 10.000 4.579 4.595 4.610 4.626 4.642 259 √n Appendix X The Greek Alphabet A B Γ ∆ E Z H Θ I K Λ M α β γ δ ε ζ η θ ι κ λ µ alpha beta gamma delta epsilon zeta eta theta iota kappa lambda mu N Ξ O Π P Σ T Υ Φ X Ψ Ω 260 ν ξ ο π ρ σ τ υ φ χ ψ ω nu xi omikron pi rho sigma tau upsilon phi chi psi omega Appendix XI Web Sites Organizations: American Mathematical Society www.ams.org Mathematical Association of America www.maa.org International Mathematical Union www.mathunion.org London Mathematical Society www.lms.ac.uk Mathematics Foundation of America www.mfoa.org General resources: Mathematics on the Web www.ams.org/mathweb Mathematics WWW Virtual Library euclid.math.fsu.edu/Science/math.html MathGate Homepage www.mathgate.ac.uk Mathematical Resources www.ama.caltech.edu/resources.html Math-Net Links www.math-net.de/links/ show?collection=math Mathematics Resources on the WWW mthwww.uwc.edu/wwwmahes/ homepage.htm History and biography: History of Mathematics Archive www-history.mcs.st-and.ac.uk/history 261 Bibliography Bittinger, Marvin L Basic Mathematics 9th ed New York: Addison Wesley, 2002 Boyer, Carl B and Isaac Asimov A History of Mathematics 2nd ed New York: Wiley, 1991 Courant, Richard, Herbert Robbins, and Ian Stewart What Is Mathematics?: An Elementary Approach to Ideas and Methods 2nd ed Oxford, U.K.: Oxford University Press, 1996 Devlin, Keith The Language of Mathematics : Making the Invisible Visible New York: Owl Books, 2000 Dunham, William Journey Through Genius: The Great Theorems of Mathematics London: Penguin Books, 1991 Hall, James W and Brian A Mercer Beginning & Intermediate Algebra New York: McGraw-Hill, 2002 Johnson, David B and Thomas A Mowry Mathematics : A Practical Odyssey 5th ed Pacific Grove, CA: Brooks Cole, 2003 Jurgensen, Ray and Richard G Brown Geometry New York: Houghton Mifflin, 2000 Martin-Gay, K Elayn Prealgebra 4th ed New York: Prentice Hall, 2003 Martin-Gay, K Elayn Intermediate Algebra 4th ed New York: Prentice, 2004 Penrose, Sir Roger The Road to Reality: The Mathematics and Physics of the Universe London: Vintage, 2002 Stewart, Ian Concepts of Modern Mathematics New York: Dover Publications, 1995 Stewart, Ian Does God Play Dice? The New Mathematics of Chaos 2nd ed London: Penguin, 1997 Stewart, James Calculus 5th ed Pacific Grove, CA: Brooks Cole, 2002 Tannenbaum, Peter Excursions in Modern Mathematics 5th ed New York: Prentice Hall, 2003 262 .. .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. .. indicate the results of one problem from the known results of the other analysis The branch of mathematics concerned with the limit process and the concept of convergence It includes the theory of. .. 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