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, , as a neutral element, we can cancel numbers (multiplicatively). For example, the multiplicative inverse of is the number ; this follows since .
Note that we would not be able to access multiplicative inverses like if we only use integers (fractions like are not whole numbers)! We need fractions or rational numbers to be able to cancel numbers multiplicatively.
To cancel a number multiplicatively, we always multiply by . While it is correct to call the multiplicative inverse of , we also call it the reciprocal of (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 (multiplicatively) involves the familiar rule you cannot divide by (see section 1.8 below).
Read this section to see examples of how to apply the inverse property of addition. Focus on the examples in the boxes. Note that in the first multiplication example box, we can use the inverse property for fractions as well as whole numbers.
Watch this video for more examples of the inverse property of multiplication.