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 zero (multiplicatively) involves the familiar rule: you cannot divide by zero.

Back to '1.6 Inverse Property of Multiplication'