Free Articles: Identity (math)

2011-06-03

Identity (math)

Identity (mathematics)

For other uses, see Identity (disambiguation).

In mathematics, the term identity has several different important meanings:

Examples

Identity relation

A common example of the first meaning is the trigonometric identity

$\sin ^2 \theta + \cos ^2 \theta = 1\,$

which is true for all complex values of $θ$ (since the complex numbers $\Bbb{C}$ are the domain of sin and cos), as opposed to

$\cos \theta = 1,\,$

which is true only for some values of $θ$ , not all. For example, the latter equation is true when $\theta = 0,\,$ false when $\theta = 2\,$ .

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 $a\in\Bbb{R},$

$0 + a = a,\,$

$a + 0 = a,\,$ and

$0 + 0 = 0.\,$

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

$1 \times a = a,\,$

$a \times 1 = a,\,$ and

$1 \times 1 = 1.\,$

Identity function

A common example of an identity function is the identity permutation, which sends each element of the set $\{ 1, 2, \ldots, n \}$ to itself or $\{a_1,a_2, \ldots, a_n \}$ to itself in natural order.

Comparison

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

External links

A Collection of Algebraic Identities
EquationSolver – A webpage that can test a suggested identity and return a true/false "verdict".

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