Identity Element

In mathematics, an identity element is any mathematical object that, when applied by an operation such as addition or multiplication, to another mathematical object such as a number leaves the other object unchanged. The two most familiar examples are 0, which when added to a number gives the number; and 1, which is an identity element for multiplication. The identity element is sometimes also called a neutral element; and sometimes shortened to simply the term identity.

More formally, an identity element is defined with respect to a given operation and a given set of elements. For example, 0 is the identity element for addition of integers; 1 is the identity element for multiplication of real numbers. From these examples, it is clear that the operation must involve two elements, as addition does; and not a single element, as such operations as taking a power.

In equation format, let (T,*) be a set T (all natural numbers: {0, 1, 2, 3, .. .}) associated with a binary operation * (such as addition or multiplication, where * represents + or x, respectively). Suppose, f and a are elements of T. Then, f is an identity element, if f * a = a and a * f = a, for all a in T. As an example, if a = 2, f = 0, and * = +, then 0 + 2 = 2 and 2 + 0 = 2. Therefore, 0 is an identity element.

Sometimes a set does not have an identity element for some operation. For example, the set of even numbers has no identity element for multiplication, although there is an identity element for addition. Most mathematical systems require an identity element. For example, a group of transformations could not exist without an identity element; that is, the transformation that leaves an element of the group unchanged.

See also Identity property; Set theory.