When a set possesses an identity element for a given operation, the mathematical system of the set and operation is said to possess the identity property for that operation. An identity element is defined as 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.
For addition, the identity property is called the additive identity. For multiplication, it is called the multiplicative identity.
Generally, the set of all functions of a variable over the real numbers has the identity element, or identity function, I(x) = x. In other words, if f (x) is any function over the real numbers, then f (I(x)) = I(f (x)) = f (x).
For the case of addition, as a simple example, 0 + a = a, where a is any number; so 0 is the identity number because any value for a remains unchanged in the operation for addition. In a simple example for multiplication, 1 x b = b, where b is any number; so 1 is the identity number because any value for b remains unchanged in the operation for multiplication.
See also Function; Identity element.