= Identity
{disambiguate=mathematics}
{wiki}
= Mathematical identity
{synonym}
A <mathematical identity> is an <equation> that holds for every assignment in a specified <domain of a function>. For example, the <distributive property> gives $(x+1)^2=x^2+2x+1$ for every <real number> $x$. An identity can replace one <mathematical expression> by another without imposing a new restriction on that domain.
The domain remains essential. The equality $(x^2-1)/(x-1)=x+1$ holds for every real $x\ne1$, but its left side is undefined at $x=1$. Cancelling a factor does not retrospectively define the original <mathematical expression> at the excluded input. An identity is different from an <identity element> of an <algebraic operation> or an <identity function>, which are separate concepts despite sharing a name.
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