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Identity element
 
 
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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....
, an identity element (or neutral element) is a special type of element of a set with respect to 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....
 on that set. It leaves other elements unchanged when combined with them. This is used for 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....
s and related concepts
Magma (algebra)

In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a Set M equipped with a single binary operation M × M ? M....
.

The term identity element is often shortened to identity when there is no possibility of confusion; we do so in this article.

Let (S,*) be a set S with a binary operation * on it (known as a magma
Magma (algebra)

In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a Set M equipped with a single binary operation M × M ? M....
).






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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....
, an identity element (or neutral element) is a special type of element of a set with respect to 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....
 on that set. It leaves other elements unchanged when combined with them. This is used for 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....
s and related concepts
Magma (algebra)

In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a Set M equipped with a single binary operation M × M ? M....
.

The term identity element is often shortened to identity when there is no possibility of confusion; we do so in this article.

Let (S,*) be a set S with a binary operation * on it (known as a magma
Magma (algebra)

In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a Set M equipped with a single binary operation M × M ? M....
). Then an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.

An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). The distinction is used most often for sets that support both binary operations, such as ring
Ring (mathematics)

In mathematics, a ring is a type of algebraic structure. There is some variation among mathematicians as to exactly what properties a ring is required to have, as described in detail below....
s. The multiplicative identity is often called the unit in the latter context, where, unfortunately, a unit
Unit (ring theory)

In mathematics, a unit in a ring R is an invertible element of R, i.e. an element u such that there is a v in R withThat is, u is an invertible element of the multiplicative monoid of R....
 is also sometimes used to mean an element with a multiplicative inverse.

Examples

setoperationidentity
real number
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...
s
 (addition)
0
0 (number)

0 is both a number and the numerical digit used to represent that number in numeral system. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures....
real number
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...
s		· (multiplication)1
1 (number)

1 is a number, number names, and the name of the glyph representing that number.It represents a single entity, the unit of counting or measurement....
real number
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...
s		ab (exponentiation)1 (right identity only)
m-by-n matrices
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
 + (addition)matrix of all zeroes
Zero matrix

In mathematics, particularly linear algebra, a zero matrix is a matrix with all its entries being 0 . Some examples of zero matrices areThe set of m×n matrices with entries in a ring K forms a ring ....
n-by-n square matrices
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
 · (multiplication)matrix with 1 on diagonal
and 0 elsewhere
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....
all functions
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 from a set M to itself		° (function composition)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....
all functions
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 from a set M to itself		* (convolution
Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operator on two function s f and g, producing a third function that is typically viewed as a modified version of one of the original functions....
)		d (Dirac delta)
character strings, lists concatenation empty string
Empty string

In computer science and formal language theory, the empty string is the unique string of String #Formal_theory zero. It is denoted with "?" or sometimes ?....
, empty list
extended real numbers minimum/infimum +8
extended real numbers maximum/supremum -8
subsets of a set M n (intersection) M
sets ? (union) (empty set
Empty set

In mathematics, and more specifically set theory, the empty set is the unique Set having no members. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced....
)
boolean logic
Boolean logic

Boolean algebra is a logical calculus of logical values, developed by George Boole in the late 1830s. It resembles the algebra of real numbers as taught in high school, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of conjun...
 		 ? (logical and) ? (truth)
boolean logic
Boolean logic

Boolean algebra is a logical calculus of logical values, developed by George Boole in the late 1830s. It resembles the algebra of real numbers as taught in high school, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of conjun...
 		 ? (logical or) ? (falsity)
compact surfaces # (connected sum) Sē
only two elements * defined by
e * e = f * e = e and
f * f = e * f = f		both e and f are left identities,
but there is no right identity
and no two-sided identity


Properties

As the last example shows, it is possible for (S,*) to have several left identities. In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity, then they are equal and there is just a single two-sided identity. To see this, note that if l is a left identity and r is a right identity then l = l * r = r. In particular, there can never be more than one two-sided identity. If there were two, e and f, then e * f would have to be equal to both e and f.

It is also quite possible for an algebra to have no identity element. The most common examples of this are the dot product
Dot product

In mathematics, the dot product, also known as the scalar product, is an operation which takes two vector over the real numbers R and returns a real-valued scalar quantity....
 and cross product
Cross product

In mathematics, the cross product is a binary operation on two vector s in a three-dimensional Euclidean space that results in another vector which is orthogonal to the plane containing the two input vectors....
 of vector
Vector

Vector may refer to:...
s. In the former case the lack of an identity element is related to the fact that the elements multiplied are vectors but the product is a scalar
Scalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
. With cross products the absence of an identity element is related to the fact that the direction of any nonzero cross product is always orthogonal to any element multiplied - so that it is not possible to obtain a vector in the same direction as the original.

See also

  • Absorbing element
    Absorbing element

    In mathematics, an absorbing element is a special type of element of a Set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself....
  • Inverse element
    Inverse element

    In mathematics, the idea of inverse element generalises the concepts of additive inverse, in relation to addition, and Multiplicative inverse, in relation to multiplication....
  • Additive inverse
    Additive inverse

    In mathematics, the additive inverse, or opposite, of a number n is the number that, when addition to n, yields 0 .The additive inverse of F is denoted −F....
  • Monoid
    Monoid

    In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element....
  • Unital
    Unital

    In mathematics, an Algebra over a field is unital if it contains a multiplicative identity element , i.e. an element 1 with the property 1x = x1 = x for all elements x of the algebra....
  • Quasigroup
    Quasigroup

    In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division " is always possible....