Inverse Property of Multiplication

Use the Inverse Properties of Addition and Multiplication

What number multiplied by

$\frac {2}{3}$ gives multiplicative identity,

$1$ ? In other words, two-thirds times what results in

$1$ ?

$\frac {2}{3} \cdot \text{____} =1$

We know

$\frac{2}{3} \cdot \frac{3}{2}=1$

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

$2 \cdot \text{____} =1$

We know

$2 \cdot \frac{1}{2}=1$

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

We call $\frac {1}{a}$ the multiplicative inverse of $a(a≠0)$ . 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
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