1.6 Introduction and Key Insights

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.