Dictionary of Mathematical Terms

411 Pages·2009·12.36 MB·English

Dictionary of
Mathematics
Terms

• 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.

PROFESSIONAL
GUIDES

Third Edition

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

9 8 7 6 5 4 3 2 1

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.)

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

viii

LIST OF SYMBOLS

Algebra

�
equals
�
is not equal
�
is approximately equal
�
is greater than
�
is greater than or equal to
�
is less than
�
is less than or equal to
�
addition
�
subtraction
�,
multiplication
�, /
division

square root; radical symbol

nth root
!
factorial
number of combinations of n things taken j at a
time; also the binomial theorem coefficient.

number of permutations of n things taken j at a
time

absolute value of x
∞
infinity

determinant of a matrix

Greek Letters
p
pi (� 3.14159...)
�
delta (upper case), represents change in
d
delta (lower case)
�
sigma (upper case), represents summation
s
sigma (lower case), represents standard
deviation

2 a
b
c
d 2

0x0

nPj

nCj,1n

j 2

2
n

2

#

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

derivative of y with respect to x

second derivative of y with respect to x

partial derivative of y with respect to x

S
approaches
lim
limit
e
base of natural logarithms; e = 2.71828.
∫
integral symbol

indefinite integral

definite integral

Geometry
�
degrees
m
�
perpendicular
�
perpendicular, as in

l
angle
�
triangle, as in �ABC
�
congruent

AB�DC

�

b

a
f1x2dx
�f1x2dx

0y
0x

y–, d2y

dx2

y¿, dy

dx

~
similar
parallel, as in
arc, as in AB
—
line segment, as in

4
line, as in AB
4

S
ray, as in

Vectors
length of vector a
dot product
cross product
¥f
gradient
¥ # f
divergence
¥ � f
curl

Set Notation
{ }
braces (indicating membership in a set)
�
intersection
�
union
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)
universal quantifier (means “For all x . . .”)
existential quantifier (means “There exists an
x . . .”)
E
�x

�

a � b
a # b
� a �

AB

!

AB

�
�
AB � CD
�

xi

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 � 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 (�∞ � x � ∞), 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 � 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 (�� � x � �), 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
, is:

Figure 1 illustrates the absolute value function.

0a0 � �a if a � 0

0a0 � a if a � 0

0a0

1
ABSOLUTE VALUE