• 1.6: Inverse Property of Multiplication

    The inverse property of multiplication tells us that almost every real number has a multiplicative inverse. Since we treat the multiplicative identity, 1, as a neutral element, we can cancel numbers (multiplicatively). For example, the multiplicative inverse of 2 is the number \frac{1}{2}; this follows since 2\times \frac{1}{2}=1.

    Note that we would not be able to access multiplicative inverses like \frac{1}{2} if we only use integers (fractions like \frac{1}{2} are not whole numbers)! We need fractions or rational numbers to be able to cancel numbers multiplicatively.

    To cancel a number a multiplicatively, we always multiply by 1/a. While it is correct to call \frac{1}{a} the multiplicative inverse of a, we also call it the reciprocal of a (just like how we call the additive inverse of a number its negative). Unlike additive inverses, not every real number has a multiplicative inverse: zero is the one special number we cannot cancel (multiplicatively). The reason we cannot invert 0 (multiplicatively) involves the familiar rule you cannot divide by 0 (see section 1.8 below).