Use the Inverse Properties of Addition and Multiplication

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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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