Identity (mathematics)

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Identity (mathematics)

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the term identity has several different important meanings:

mmon example of the first meaning is the trigonometric identity which is true for all real

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
values of (since the real numbers are the domain of sin and cos), as opposed to which is true only for some values of , not all. For example, the latter equation is true when , false when

See also list of mathematical identities

List of mathematical identities

This page lists identity in the sense of mathematics, that is, identically true relations holding in algebra or between special functions....
.

concepts of "additive identity" and "multiplicative identity" are central to the Peano axioms

Peano axioms

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Encyclopedia

In mathematics

Mathematics

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
Equality (mathematics)

Equality is the paradigmatic example of the more general concept of equivalence relations on a set: those binary relations which are reflexive relation, symmetric relation, and transitive relation....
which is true under more particular conditions. For this, the 'triple bar
Triple bar

The triple bar, =, is a symbol used in formal logic. It has the appearance of a "=" sign with a third line.Logically, it has a similar meaning to the if and only if coupler ?....
' symbol = is sometimes used. (However, this can be ambiguous since the same symbol can also be used with different meanings, for example for a congruence relation
Congruence relation

In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation....
.)
In algebra
Algebra

Algebra is a branch of mathematics concerning the study of structure , relation , and quantity. Together with geometry, mathematical analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics....
, an identity or identity element
Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
of a set S with a binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
· is an element e that, when combined with any element x of S, produces that same x. That is, for all x in S.

The identity function
Identity function

In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument....
from a set S to itself, often denoted or , s the function such that for all x in S. This function serves as the identity element in the set of all functions from S to itself with respect to function composition
Function composition

In mathematics, a composite function represents the application of one function to the results of another. For instance, the functions and can be composed by first computing a f and then applying a function g to the output of f....
.
In linear algebra
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
, the identity matrix
Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere....
of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. This matrix serves as the identity with respect to matrix multiplication
Matrix multiplication

In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. This article gives an overview of the various ways to perform matrix multiplication....
.

Examples

Identity relation

A common example of the first meaning is the trigonometric identity which is true for all real

Real number

List of mathematical identities

This page lists identity 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

Peano axioms

In mathematical logic, the Peano axioms, also known as the Dedekind?Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian people mathematician Giuseppe Peano....
. The number 0 is the "additive identity" for integers, real numbers, and complex numbers. For the real numbers, for all

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

Permutation

In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the element s of a set to other elements of the same set, i.e., exchanging elements of a set....
, which sends each element of the set to itself.

Comparison

These meanings are not mutually exclusive; for instance, the identity permutation is the identity element in the group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
of permutations of under composition

Function composition

In mathematics, a composite function represents the application of one function to the results of another. For instance, the functions and can be composed by first computing a f and then applying a function g to the output of f....
.

External links

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