PROFESSIONAL GUIDES Dictionary of Mathematics Terms Third Edition • More than 800 terms related to algebra, geometry, analytic geometry, trigonometry, probability, statistics, logic, and calculus • An ideal reference for math students, teachers, engineers, and statisticians • Filled with illustrative diagrams and a quick-reference formula summary Douglas Downing, Ph.D. Dictionary of Mathematics Terms Third Edition Dictionary of Mathematics Terms Third Edition Douglas Downing, Ph.D. School of Business and Economics Seattle Pacific University Dedication This book is for Lori. Acknowledgments Deepest thanks to Michael Covington, Jeffrey Clark, and Robert Downing for their special help. © Copyright 2009 by Barron’s Educational Series, Inc. Prior editions © copyright 1995, 1987. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means without the written permission of the copyright owner. All inquiries should be addressed to: Barron’s Educational Series, Inc. 250 Wireless Boulevard Hauppauge, New York 11788 www.barrronseduc.com ISBN-13: 978-0-7641-4139-3 ISBN-10: 0-7641-4139-2 Library of Congress Control Number: 2008931689 PRINTED IN CHINA 987654321 CONTENTS Preface vi List of Symbols ix Mathematics Terms 1 Appendix 381 Algebra Summary 381 Geometry Summary 382 Trigonometry Summary 384 Brief Table of Integrals 388 PREFACE Mathematics consists of rigorous abstract reasoning. At first, it can be intimidating; but learning about math can help you appreciate its great practical usefulness and even its beauty—both for the visual appeal of geometric forms and the concise elegance of symbolic formulas expressing complicated ideas. Imagine that you are to build a bridge, or a radio, or a bookcase. In each case you should plan first, before begin- ning to build. In the process of planning you will develop an abstract model of the finished object—and when you do that, you are doing mathematics. The purpose of this book is to collect in one place ref- erence information that is valuable for students of math- ematics and for persons with careers that use math. The book covers mathematics that is studied in high school and the early years of college. These are some of the gen- eral subjects that are included (along with a list of a few entries containing information that could help you get started on that subject): Arithmetic: the properties of numbers and the four basic operations: addition, subtraction, multiplication, division. (See also number, exponent, and logarithm.) Algebra: the first step to abstract symbolic reasoning. In algebra we study operations on symbols (usually let- ters) that stand for numbers. This makes it possible to develop many general results. It also saves work because it is possible to derive symbolic formulas that will work for whatever numbers you put in; this saves you from hav- ing to derive the solution again each time you change the numbers. (See also equation, binomial theorem, qua- dratic equation, polynomial, and complex number.) Geometry: the study of shapes. Geometry has great visual appeal, and it is also important because it is an vi vii example of a rigorous logical system where theorems are proved on the basis of postulates and previously proved theorems. (See also pi, triangle, polygon, and polyhedron.) Analytic Geometry: where algebra and geometry come together as algebraic formulas are used to describe geometric shapes. (See also conic sections.) Trigonometry: the study of triangles, but also much more. Trigonometry focuses on six functions defined in terms of the sides of right angles (sine, cosine, tangent, secant, cosecant, cotangent) but then it takes many sur- prising turns. For example, oscillating phenomena such as pendulums, springs, water waves, light waves, sound waves, and electronic circuits can all be described in terms of trigonometric functions. If you program a com- puter to picture an object on the screen, and you wish to rotate it to view it from a different angle, you will use trigonometry to calculate the rotated position. (See also angle, rotation, and spherical trigonometry.) Calculus: the study of rates of change, and much more. Begin by asking these questions: how much does one value change when another value changes? How fast does an object move? How steep is a slope? These prob- lems can be solved by calculating the derivative, which also allows you to answer the question: what is the high- est or lowest value? Reverse this process to calculate an integral, and something amazing happens: integrals can also be used to calculate areas, volumes, arc lengths, and other quantities. A first course in calculus typically cov- ers the calculus of one variable; this book also includes some topics in multi-variable calculus, such as partial derivatives and double integrals. (See also differential equation.) Probability and Statistics: the study of chance phe- nomena, and how that study can be applied to the analy- sis of data. (See also hypothesis testing and regression.) viii Logic: the study of reasoning. (See also Boolean algebra.) Matrices and vectors: See vector to learn about quan- tities that have both magnitude and direction; see matrix to learn how a table of numbers can be used to find the solution to an equation system with many variables. A few advanced topics are briefly mentioned because you might run into certain words and wonder what they mean, such as calculus of variations, tensor, and Maxwell’s equations. In addition, several mathematicians who have made major contributons throughout history are included. The Appendix includes some formulas from algebra, geometry, and trigonometry, as well as a table of integrals. Demonstrations of important theorems, such as the Pythagorean theorem and the quadratic formula, are included. Many entries contain cross references indicating where to find background information or further applica- tions of the topic. A list of symbols at the beginning of the book helps the reader identify unfamiliar symbols. Douglas Downing, Ph.D. Seattle, Washington 2009 LIST OF SYMBOLS Algebra ϭ equals is not equal Ϸ is approximately equal Ͼ is greater than Ն is greater than or equal to Ͻ is less than Յ is less than or equal to ϩ addition Ϫ subtraction # ϫ, multiplication Ϭ, / division 2 square root; radical symbol n 2 nth root ! factorial 1n 2 nCj, j number of combinations of n things taken j at a time; also the binomial theorem coefficient. nPj number of permutations of n things taken j at a time 0x 0 absolute value of x ∞ infinity ab 2 2 determinant of a matrix cd Greek Letters p pi (ϭ 3.14159...) ⌬ delta (upper case), represents change in d delta (lower case) ⌺ sigma (upper case), represents summation s sigma (lower case), represents standard deviation ix x u theta (used for angles) f phi (used for angles) m mu, represents mean e epsilon x chi r rho (correlation coefficient) l lambda Calculus ⌬x increment of x dy y¿, derivative of y with respect to x dx d2y y–, second derivative of y with respect to x dx2 0y partial derivative of y with respect to x 0x S approaches lim limit e base of natural logarithms; e = 2.71828. ∫ integral symbol Ύf1x2 dx indefinite integral b Ύ f1x2dx definite integral a Geometry ؠ degrees mٗ perpendicular Ќ perpendicular, as in ABЌDC l angle ᭝ triangle, as in ᭝ABC Х congruent xi ~ similar ʈ ʈ parallel, as៣ in AB CD ៣ arc, as in AB — line segment, as in 4 AB 4 line, as in AB! S ray, as in AB Vectors ʈ a ʈ length of vector a a # b dot product a ϫ b cross product ¥f gradient ¥ # f divergence ¥ ϫ f curl Set Notation { } braces (indicating membership in a set) ʝ intersection ʜ union л empty set Logic S implication, as in a S b (IF a THEN b) ~ p the negation of a proposition p ¿ conjunction (AND) (ٚ disjunction (OR IFF, 4 equivalence, (IF AND ONLY IF) ᭙x universal quantifier (means “For all x . .”) E existential quantifier (means “There exists an x . .”) 1 ABSOLUTE VALUE A ABELIAN GROUP See group. ABSCISSA Abscissa means x-coordinate. The abscissa of the point (a, b) in Cartesian coordinates is a. For con- trast, see ordinate. ABSOLUTE EXTREMUM An absolute maximum or an absolute minimum. ABSOLUTE MAXIMUM The absolute maximum point for a function y ϭ f (x) is the point where y has the largest value on an interval. If the function is differentiable, the absolute maximum will either be a point where there is a horizontal tangent (so the derivative is zero), or a point at one of the ends of the interval. If you consider all values of x (Ϫ∞ Յ x Յ ∞), the function might have a finite max- imum, or it might approach infinity as x goes to infinity, minus infinity, or both. For contrast, see local maximum. For diagram, see extremum. ABSOLUTE MINIMUM The absolute minimum point for a function y ϭ f (x) is the point where y has the smallest value on an interval. If the function is differentiable, then the absolute minimum will either be a point where there is a horizontal tangent (so the derivative is zero), or a point at one of the ends of the interval. If you consider all values of x (Ϫϱ Յ x Յϱ), the function might have a finite minimum, or it might approach minus infinity as x goes to infinity, minus infinity, or both. For contrast, see local minimum. For diagram, see extremum. ABSOLUTE VALUE The absolute value of a real number a, written as 0a 0 , is: 0a 0 ϭ a if a Ն 0 0a 0 ϭϪa if a Ͻ 0 Figure 1 illustrates the absolute value function. ACCELERATION 2 Figure 1 Absolute value function Absolute values are always positive or zero.