alternating series 11

In modern times the subject of algebra has been

widened to include

ABSTRACT ALGEBRA

,

GROUP THE

-

ORY

, and the study of alternative number systems

such as

MODULAR ARITHMETIC

.

BOOLEAN ALGEBRA

looks at the algebra of logical inferences, matrix alge-

bra the arithmetic of

MATRIX

operations, and vector

algebra the mechanics of

VECTOR

operations and

VECTOR SPACE

s.

An algebraic structure is any set equipped with one

or more operations (usually

BINARY OPERATION

s) satis-

fying a list of specified rules. For example, any group,

RING

,

FIELD

, or vector space is an algebraic structure. In

advanced mathematics, a vector space that is also a

field is called an “algebra.”

See also

BRACKETS

;

COMMUTATIVE PROPERTY

;

DIS

-

TRIBUTIVE PROPERTY

;

EXPANDING BRACKETS

;

FUNDA

-

MENTAL THEOREM OF ALGEBRA

;

HISTORY OF EQUATIONS

AND ALGEBRA

(essay);

ORDER OF OPERATION

.

algebraic number A number is called algebraic if it

is the root of a

POLYNOMIAL

with integer coefficients.

For example, (1/2) (5 + √

—

13) is algebraic since it is a

solution to the equation x2– 5x+ 3 = 0. All

RATIONAL

NUMBERS

are algebraic (since a fraction a/b is the solu-

tion to the equation bx – a= 0), and all square, cube,

and higher roots of integers are algebraic (since n

√

—

ais a

solution to xn– a= 0).

At first thought it seems that all numbers are alge-

braic, but this is not the case. In 1844 French mathe-

matician J

OSEPH

L

IOUVILLE

made the surprising

discovery that the following number, today called

“Liouville’s constant,” cannot be a solution to any inte-

ger polynomial equation:

Numbers that are not algebraic are called “transcen-

dental.”

In 1873 French mathematician Charles Hermite

(1822–1901) proved that the number eis transcenden-

tal, and, nine years later in 1882 German mathematician

C

ARL

L

OUIS

F

ERDINAND VON

L

INDEMANN

established

that πis transcendental. In 1935 Russian mathematician

Aleksandr Gelfond (1906–68) proved that any number

of the form abis transcendental if aand bare both alge-

braic, with adifferent from 0 or 1, and birrational.

(Thus, for example, 2√

–

3is transcendental.)

The German mathematician G

EORG

C

ANTOR

(1845–1918) showed that the set of algebraic numbers

is

COUNTABLE

. As the set of real numbers is uncount-

able, this means that most numbers are transcendental.

The probability that a real number chosen at random is

algebraic is zero. Although it was proven in 1929 that

eπis transcendental, no one to this day knows whether

or not ππis algebraic.

In analogy with algebraic numbers, a

FUNCTION

y= f(x) is called “algebraic” if it can be defined by a

relation of the form

pn(x)yn+ pn– 1(x)yn– 1+…+p1(x)y+ p0(x) = 0

where the functions pi(x) are polynomials in x. For

example, the function y= √

—

xis an algebraic function,

since it is defined by the equation y2– x= 0. A tran-

scendental function is a function that is not algebraic.

Mathematicians have shown that trigonometric, loga-

rithmic, and exponential functions are transcendental.

See also

CARDINALITY

.

algorithm An algorithm is a specific set of instruc-

tions for carrying out a procedure or solving a mathe-

matical problem. Synonyms include “method,”

“procedure,” and “technique.” One example of an

algorithm is the common method of

LONG DIVISION

.

Another is the E

UCLIDEAN ALGORITHM

for finding the

GREATEST COMMON DIVISOR

of two positive integers.

The word algorithm is a distortion of “al-Khw–

arizm

–

ı,”

the name of a Persian mathematician (ca. 820) who

wrote an influential text on algebraic methods.

See also

BASE OF A NUMBER SYSTEM

; M

UHAMMAD

IBN

M–

US

–

AAL

-K

HW

–

ARIZM

–

ı.

alternating series A

SERIES

whose terms are alternately

positive and negative is called an alternating series. For

example, the G

REGORY SERIES

is

an alternating series, as is the (divergent) series: 1 – 1 + 1

– 1 + 1 – 1 +… Alternating series have the form

, with each aipositive

number.