Identity (mathematics)

equation that is satisfied for all values of the variables
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 .[3]
  • The identity function (or identity map) from a set S to itself, often denoted or , such that 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 .[3]

Examples change

Identity relation change

A common example of the first meaning is the trigonometric identity


which is true for all real values of   (since the real numbers   are the domain of both sine and cosine), as opposed to


which is only true for certain values of   in a subset of the domain.

Identity element change

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  


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


Identity function change

A common example of an identity function is the identity permutation, which sends each element of the set   to itself.

Comparison change

These meanings are not mutually exclusive; for instance, the identity permutation is the identity element in the set of permutations of   under composition.

Related pages change

References change

  1. "The Definitive Glossary of Higher Mathematical Jargon". Math Vault. 2019-08-01. Retrieved 2020-08-13.
  2. "Identity - Math Open Reference". Retrieved 2020-08-13.
  3. 3.0 3.1 "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-13.
  4. Weisstein, Eric W. "Identity Map". Retrieved 2020-08-13.

Other websites change