Identity

There are at least two possible meanings in mathematics for the word identity.

Equations

An identity is "a general relationship which always holds, usually over some choice of variables." For example, $(x+1)^2=x^2+2x+1$ is an identity, since it holds regardless of choice of variable. We therefore sometimes write $(x+1)^2\equiv x^2+2x+1$ .

Abstract Algebra

Given a binary operation G on a set S, $G: S \times S \to S$ , an identity for G is an element $e\in S$ such that for all $a \in S$ , $G(e, a) = G(a, e) = a$ . For example, in the real numbers, if we take G to be the operation of multiplication $G(a, b) = a\cdot b$ , the number 1 will be the identity for G. If instead we took G to be addition ( $G(a, b) = a + b$ ), 0 would be the identity.

Identities in this sense are unique. Imagine we had two identities, $e$ and $e'$ , for some operation $G$ . Then $e = G(e, e') = e'$ , so $e = e'$ , and so e and e' are in fact equal.