# 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 the choice of variable. Therefore, it is sometimes written $(x+1)^2\equiv x^2+2x+1$.

## 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 we instead 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.