Inverse Property of Multiplication

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

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.

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