## Using the Inverse Properties of Addition and Multiplication

Contents

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, \(\frac{10}{9}.\)