Mathematics » Properties of Real Numbers » Properties of Identity, Inverses, and Zero

# Using the Inverse Properties of Addition and Multiplication

## Using the Inverse Properties of Addition and Multiplication

 What number added to 5 gives the additive identity, 0? $$5+\_\_\_\_\_=0$$ What number added to −6 gives the additive identity, 0? $$-6+\_\_\_\_\_=0$$

Notice that in each case, the missing number was the opposite of the number.

We call $$-a$$ the additive inverse of $$a.$$ The opposite of a number is its additive inverse. A number and its opposite add to $$0,$$ which is the additive identity.

What number multiplied by $$\frac{2}{3}$$ gives the multiplicative identity, $$1?$$ In other words, two-thirds times what results in $$1?$$

 $$\frac{2}{3}·\_\_\_=1$$

What number multiplied by $$2$$ gives the multiplicative identity, $$1?$$ In other words two times what results in $$1?$$

 $$2·\_\_\_=1$$

Notice that in each case, the missing number was the reciprocal of the number.

We call $$\frac{1}{a}$$ the multiplicative inverse of $$a\left(a\ne 0\right)\text{.}$$ The reciprocal of a number is its multiplicative inverse. A number and its reciprocal multiply to $$1,$$ which is the multiplicative identity.

We’ll formally state the Inverse Properties here:

### Definition: Inverse Properties

Inverse Property of Addition for any real number $$a,$$

$$\begin{array}{}\hfill & a+(−a)=0 & \\ \hfill -a\text{ is the} & \mathbf{\text{additive inverse}} & \text{of }a.\hfill \end{array}$$

Inverse Property of Multiplication for any real number $$a\ne 0,$$

$$\begin{array}{}\hfill & a·\frac{1}{a}=1 & \\ \hfill \frac{1}{a}\text{ is the} & \mathbf{\text{multiplicative inverse}} & \text{of }a.\hfill \end{array}$$

## Example

Find the additive inverse of each expression:

$$13$$

$$-\frac{5}{8}$$

$$\phantom{\rule{0.2em}{0ex}}0.6$$.

### Solution

To find the additive inverse, we find the opposite.

The additive inverse of $$13$$ is its opposite, $$-13.$$

The additive inverse of $$-\frac{5}{8}$$ is its opposite, $$\frac{5}{8}.$$

The additive inverse of $$0.6$$ is its opposite, $$-0.6.$$

## Example

Find the multiplicative inverse:

$$9\phantom{\rule{0.2em}{0ex}}$$

$$-\frac{1}{9}\phantom{\rule{0.2em}{0ex}}$$

$$0.9$$.

### Solution

To find the multiplicative inverse, we find the reciprocal.

The multiplicative inverse of $$9$$ is its reciprocal, $$\frac{1}{9}.$$

The multiplicative inverse of $$-\frac{1}{9}$$ is its reciprocal, $$-9.$$

To find the multiplicative inverse of $$0.9,$$ we first convert $$0.9$$ to a fraction, $$\frac{9}{10}.$$ Then we find the reciprocal, $$\frac{10}{9}.$$