multiplicative inverse

If the additive inverse undoes a number by getting back to 0, the multiplicative inverse undoes it by getting back to 1. The multiplicative inverse of a number is its reciprocal — flip the fraction upside down — and multiplying the two together gives exactly 1.

Formally, the multiplicative inverse of a nonzero number a is 1/a, because a × (1/a) = 1. For a fraction you simply swap numerator and denominator: the reciprocal of 3/5 is 5/3, and the reciprocal of 4 (which is 4/1) is 1/4. The number 1 is its own reciprocal, and so is −1.

There is one number with no multiplicative inverse: zero. No number times 0 can ever equal 1, which is another face of the rule that you cannot divide by zero. The existence of reciprocals is what makes division possible — dividing by a number is the same as multiplying by its reciprocal, so x ÷ a = x × (1/a).