There are at least two possible meanings in mathematics for the word identity.
An identity is a general relationship which always holds, usually over some choice of variables. For example, is an identity, since it holds regardless of the choice of variable. Therefore, it is sometimes written .
Given a binary operation on a set , , an identity for is an element such that for all , . For example, in the real numbers, if we take to be the operation of multiplication (), the number will be the identity for . If we instead took to be addition (), would be the identity.
Identities in this sense are unique. Imagine we had two identities, and , for some operation . Then , so , and so and are in fact equal.