Inverse Property of Multiplication

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

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

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

INVERSE PROPERTIES

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

$a+(−a)=0$

$-a \text { is the additive inverse of a.}$

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

$a \cdot \frac {1}{a} = 1$

$\frac {1}{a} \text { is the multiplicative inverse of a.}$

Source: Rice University, https://openstax.org/books/prealgebra/pages/7-4-properties-of-identity-inverses-and-zero
This work is licensed under a Creative Commons Attribution 4.0 License.

EXAMPLE 7.35

Find the multiplicative inverse:

(a) $9$

(b) $-\frac {1}{9}$

(c) $0.9$

TRY IT 7.69

(a) $5$

(b) $-\frac{1}{7}$

(c) $0.3$

TRY IT 7.70

(a) $18$

(b) $-\frac{4}{5}$

(c) $0.6$

Exercise 7.35

To find the multiplicative inverse, we find the reciprocal.

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

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

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

TRY IT 7.69

(a) $\frac{1}{5}$

(b) $-7$

(c) $\frac{10}{3}$

TRY IT 7.70

(a) $\frac{1}{18}$

(b) $-\frac{5}{4}$

(c) $\frac{5}{3}$