In mathematics, the term identity has several important uses: For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...

• An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. For this, the symbol ≡ is sometimes used. (However, this can be ambiguous since the same symbol can also be used for a congruence relation.)
• In algebra, an identity or identity element of a set S with a binary operation is an element e which combined with any element s of S produces s.
• The identity function from a set S to itself, often denoted id or id_S, such that id(x) = x for all x in S.
• 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.

In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ... This article is about the branch of mathematics. ... For other uses, see identity (disambiguation). ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... 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. ... For the square matrix section, see square matrix. ...

Examples

Identity relation

A common example of the first meaning is the trigonometric identity In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

which is true for all real values of θ (since the real numbers are the domain of sin and cos), as opposed to In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

which is true only for some values of θ, not all. For example, the latter equation is true when , false when

See also list of mathematical identities. This page lists identities in the sense of mathematics, that is, identically true relations holding in algebra or between special functions. ...

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 In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of the natural numbers. ...

and

Similarly, The number 1 is the "multiplicative identity" for integers, real numbers, and complex numbers. For the real numbers, for all

and

Identity function

A common example of an identity function is the identity permutation, which sends each element of the set to itself. Permutation is the rearrangement of objects or symbols into distinguishable sequences. ...

Comparison

These meanings are not mutually exclusive; for instance, the identity permutation is the identity element in the set of permutations of under composition. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

External links

• EquationSolver - A webpage that can test a suggested identity and return a true/false "verdict".