# Identity (mathematics)

For other senses of this word, see identity.

In mathematics, the term identity has several important uses:

• An identity is an equality that remains true even if you change all the variables that are used in that equality.[1][2]

An equality in mathematical sense is only true under more particular conditions. For this, the symbol ≡ is sometimes used (note, however, that the same symbol can also be used for a congruence relation as well.)

• In algebra, an identity or identity element of a set S with an operation is an element which, when combined with any element s of S, produces s itself. In a group (an algebraic structure), this is often denoted by the symbol ${\displaystyle e}$.[3]
• The identity function (or identity map) from a set S to itself, often denoted ${\displaystyle \mathrm {id} }$ or ${\displaystyle \mathrm {id} _{S}}$, such that ${\displaystyle \mathrm {id} (x)=x}$ for all x in S.[4]
• In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. It is often denoted by the symbol ${\displaystyle I}$.[3]

## Examples

### Identity relation

A common example of the first meaning is the trigonometric identity

${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1\,}$

which is true for all real values of ${\displaystyle \theta }$ (since the real numbers ${\displaystyle {\mathbb {R}}}$ are the domain of both sine and cosine), as opposed to

${\displaystyle \cos \theta =1,\,}$

which is only true for certain values of ${\displaystyle \theta }$ in a subset of the domain.

### Identity element

The concepts of "additive identity" and "multiplicative identity" are central to the Peano axioms. The number 0 is the "additive identity" for integers, real numbers, and complex numbers. For the real numbers, for all ${\displaystyle a\in {\mathbb {R}},}$

${\displaystyle 0+a=a,\,}$
${\displaystyle a+0=a,\,}$ and
${\displaystyle 0+0=0.\,}$

Similarly, The number 1 is the "multiplicative identity" for integers, real numbers, and complex numbers. For the real numbers, for all ${\displaystyle a\in {\mathbb {R}},}$

${\displaystyle 1\times a=a,\,}$
${\displaystyle a\times 1=a,\,}$ and
${\displaystyle 1\times 1=1.\,}$

### Identity function

A common example of an identity function is the identity permutation, which sends each element of the set ${\displaystyle \{1,2,\ldots ,n\}}$ to itself.

## Comparison

These meanings are not mutually exclusive; for instance, the identity permutation is the identity element in the set of permutations of ${\displaystyle \{1,2,\ldots ,n\}}$ under composition.

## References

1. "The Definitive Glossary of Higher Mathematical Jargon". Math Vault. 2019-08-01. Retrieved 2020-08-13.
2. "Identity - Math Open Reference". www.mathopenref.com. Retrieved 2020-08-13.
3. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-13.
4. Weisstein, Eric W. "Identity Map". mathworld.wolfram.com. Retrieved 2020-08-13.