Inverse Property of Multiplication — Definition & Examples

A (nonzero) number multiplied by its reciprocal equals $1$

If you write a number as a fraction, that number's reciprocal is a fraction with the numerator and the denominator switched around. For example, the reciprocal of $\frac{4}{5}$ is $\frac{5}{4}$. Some general rules to keep in mind:

-If the fraction has a $0$ in the numerator, the reciprocal is undefined.

-If the fraction has a $0$ in the denominator, the fraction itself is undefined. Thus it can't have a reciprocal.

-Any other fraction has exactly one reciprocal.

Finally, the reciprocal of a reciprocal is the original fraction. This means if you flip a fraction's numerator and denominator, and then flip them back, you get the number you started with.

$$\frac{4}{5} \rightarrow \frac{5}{4} \rightarrow \frac{4}{5}$$

$\frac{4}{5} \times \frac{5}{4} = \frac{4 \times 5}{5 \times 4} = \frac{20}{20} = 1$

This will be the case any time you multiply reciprocals by each other. You're essentially doing the same calculation for the numerators and the denominators. Hence the numerator and the denominator of the result will always be the same. This means that the result will always be $1$.

In the case of the problem above, we're using $4 \times 5$ and $5 \times 4$. By the commutative property, these are the same. In fact, they both equal $20$.

Whole Numbers

Whole numbers have reciprocals too! Remember that you can always turn a whole number into an improper fraction by choosing a denominator and then making the numerator your whole number times the new denominator. The simplest way to do that is to just make your denominator $1$. For example, we can write $5$ as a fraction by just deciding that the denominator is $1$ and putting $5$ on top: $$5 = \frac{5}{1}$$

This is valid because dividing any number by $1$ produces the original number. We've written a whole number as a fraction with the same value.

The reciprocal of $\frac{5}{1}$ is $\frac{1}{5}$. All you have to do is switch the denominator (in this case, making it $1$ instead of $5$) and the numerator (changing $5$ to $1$).

$$\frac{5}{1} \times \frac{1}{5} = \frac{5\times1}{1\times5} = \frac{5}{5} = 1$$

The answer is $1$, so $\frac{5}{1}$ and $\frac{1}{5}$ are reciprocals of each other!

A number times its reciprocal equals $1$

A number's reciprocal is just the number, but upside-down. For example, the reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$ and the reciprocal of $\frac{1}{5}$ is $\frac{5}{1}$ (or just $5$). This process can be reversed. That is, the reciprocal of a reciprocal is the original number. For instance,

$$\frac{3}{4} \rightarrow \frac{4}{3} \rightarrow \frac{3}{4}$$

When you multiply fractions together, you multiply the numerators and the denominators separately. Recall a number's reciprocal's denominator is the original number's numerator. As a result, you're basically doing the same multiplication problem twice, and you'll get the same number each time: $$\frac{3}{4} \times \frac{4}{3} =\frac{3\times4}{4\times3}= \frac{12}{12}$$

When fractions have the same number in their numerator and denominator, they're always equal to one: $$\frac{12}{12} = 1$$

The one exception is $\frac{0}{0}$. We're not allowed to divide by zero.

(Video) Inverse Property of Addition and Multiplication

by William Chermak

Here's another video by William Chermak. This one covers the inverse property of multiplication. It also covers the inverse property of addition. However, in this explanation we'll focus on the multiplication part.

$\large{\text{Summary}}$

The inverse property of multiplication states that any number multiplied by its multiplicative inverse will equal $1$. $$\boxed{\frac ab\times\frac ba=1}$$

In other words, we are canceling fractions out. We start with one fraction and multiply it by its reciprocal. This is one of many times you might see multiplication of two fractions.

Just be careful. We can't do this if $a$ or $b$ equals $0$, because that means the above involves dividing by zero.

This is great! We have a way to turn fractions into a $1$. And all we have to do is find the original number's reciprocal.

Examples

Example 1: If we take $5$ and multiply it by its inverse, $\frac15$, we get: $$5\times\frac15=\frac55=\boxed{1}$$

Example 2: Perhaps we want to solve the equation $x \times \frac{2}{5} = 2$. We want to isolate the $x$, so we divide both sides by the fraction next to the $x$. This is the same as multiplying both sides by the reciprocal of $\frac{2}{5}$:

$$\begin{align} x \times \frac{2}{5} &= 2 \\ x \times \frac{2}{5} \times \frac{5}{2} &= 2 \times \frac{5}{2} \\ x \times \frac{\cancel{2}}{\cancel{5}} \times \frac{\cancel{5}}{\cancel{2}} &= \cancel{2} \times \frac{5}{\cancel{2}} \\ x &= \boxed{5} \end{align}$$

As an aside, if our equation contains a mixed number, we can still solve it this way. We just have to turn it into an improper fraction first. Also, in Example 2 we can check our answer by plugging $x$ back into the equation. It looks like this:

$$x \times \frac{2}{5} = 5 \times \frac{2}{5} = \cancel{5} \times \frac{2}{\cancel{5}} = 2$$