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I

identity An equation that states that two mathemati-

cal expressions are equal for all possible values of any

variables that occur in them is called an identity. For

example, the

DIFFERENCE OF TWO SQUARES

formula

x2– y2= (x– y)(x+ y) and the trigonometric equation

sin2θ+ cos2θ= 1 are identities. Sometimes the symbol ≡

is used instead of = to indicate that the statement is an

identity. It is read as “identically equal to.”

A numerical statement, one involving no variables

at all, might still be called an identity if it illuminates

something interesting about the numbers involved. For

example, the following relationship could be called an

identity because of the unexpected appearance of the

number π:

See also

EQUATION

;

IDENTITY ELEMENT

.

identity element (identity, neutral element) An ele-

ment of a set that, combined with any other element of

the set, leaves that element unchanged is called an iden-

tity element. More precisely, given a

BINARY OPERATION

“*” on a set S, we say that ein Sis an identity element if:

a*e = a

e*a = a

for all elements aof the set.

For example, the number zero is an identity ele-

ment for the set of real numbers under the operation of

addition: we have that a+ 0 = a= 0 + afor all numbers

a. With respect to multiplication, 1 is an identity ele-

ment, since a×1 = a= 1 ×afor all numbers a. (The

number zero is sometimes called an “additive identity”

and the number 1 a “multiplicative identity.”)

In matrix theory the

IDENTITY MATRIX

is an identity

element under multiplication for the set of all square

matrices of a particular size. In

SET THEORY

, the

EMPTY

SET

is an identity under the operation of taking union.

Any

GROUP

is required to have an identity element.

It is impossible for a mathematical system to pos-

sess two different identity elements with respect to the

same binary operation. If, for instance, eand fare both

identity elements for a set S, then e*f equals e(since fis

an identity element), and it also equals f(since eis an

identity element). Thus eand fare the same.

In some mathematical theories it is desirable to dis-

tinguish between a “left identity,” that is an element eL

such that eL*a = afor all elements a, and a “right iden-

tity” eRthat satisfies a*eR= afor all a. Of course, if the

theory satisfies the

COMMUTATIVE PROPERTY

, then eLand

eRare equal and represent an identity for the system.

See also

INVERSE ELEMENT

.

identity matrix (unit matrix) A square

MATRIX

with

each main diagonal element equal to 1 and all other

entries equal to zero is called an identity matrix. Such a

matrix is usually denoted Ior Id, or even Idnif the