# Identity *Identity* is a property of binary operations that is satisfied if there exists an element $a_0$ such that for every element in $a$ the following holds: $ a_0 \circ a = a \circ a_0 = a $ (this is an oversimplification that is however true in most cases, see [[#Right and Left Identities]]) Those operations are said to have an *Identity Element*. For multiplication the identity element is 1, for addition it is 0. Division and subtraction do not have identity elements. ## Right and Left Identities The identity property can be split into two distinct properties, depending on whether the identity element is applied on the left or the right side of the operator. If defined this way, we can say that subtraction has right identity, 0: $x - 0 = x$. Similarly, division's right identity is 1. Even if a binary operation has different right and left identity elements, it is still said to satisfy the identity property.