Identity Property in Mathematics

An identity element is a special number in a set with respect to an operation, which leaves any element unchanged when the operation is applied.

Definition

For a set with operation $\ast$, the identity element $e$ satisfies:

This is read as: There exists an element e in the set of real numbers (ℝ) such that for all elements a in the same set, the operation satisfies: a ∗ e = e ∗ a = a.

In other words, e is a special element — called the identity element — that leaves any element a unchanged when used in the operation.

The key point here is that the identity element also has to be a part of the set. Remember that we define and describe identities with reference to two things, a set and an operation. Not all operations have the same identity and some sets do not contain the identity element for the set in question.

Common Identities

• Additive Identity: 0 (since $a + 0 = a$)
• Multiplicative Identity: 1 (since $a \times 1 = a$)
• Matrix Multiplicative Identity: Identity Matrix $I$ such that $AI = IA = A$

Key Point

Natural Numbers have a multiplicative identity but not an additive identity because 0 is not part of of the natural number set.

Build the Identity Equations