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.

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