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, \(\frac{10}{9}.\)
This lesson is part of:
Properties of Real Numbers