A mathematical identity is an equation which is true regardless of the input of the variable. For example, the equation x/2 = x times 0.5 is an identity. Suppose we input the number 1. We get 1/2 = 1 x 0.5. This is true. Suppose we enter the number 6. We get 6/2 = 6 x 0.5. This is also true. Suppose we enter 102,084/2 = 102,084 x 0.5. This is also true! In these interesting equations, regardless of the input of the variable, the equation is true.
This is quite different from (though also somewhat similar to) our previously discussed notion of a conditional equation. In the case of a conditional equation, the equation is true only on the condition of the input of one specific variable. In the case of an identity, however, no matter what number you enter into the variable, the equation is true. It is also a concept related to our previously discussed concept of an inconsistent equation. In some sense, it is the complete opposite of an inconsistent equation, since in the case of an inconsistent equation, the equation is absolutely never true regardless of the value entered into the variable.