1Cademy - Finding the Multiplicative Inverse of Integers, Fractions, and Decimals

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Finding the Multiplicative Inverse of Integers, Fractions, and Decimals

To find the multiplicative inverse (reciprocal) of a number, determine the value that, when multiplied by the original number, yields $1$ . The method depends on the form of the number:

Whole number: The multiplicative inverse of $9$ is $\frac{1}{9}$ , because $9 \cdot \frac{1}{9} = 1$ . In general, the reciprocal of any nonzero integer $n$ is $\frac{1}{n}$ .
Negative fraction: The multiplicative inverse of $-\frac{1}{9}$ is $-9$ . Flipping the numerator and denominator of $-\frac{1}{9}$ gives $-\frac{9}{1} = -9$ , and indeed $\left(-\frac{1}{9}\right)(-9) = 1$ . Notice that the sign of the reciprocal matches the sign of the original number.
Decimal: To find the multiplicative inverse of $0.9$ , first convert the decimal to a fraction: $0.9 = \frac{9}{10}$ . Then flip the fraction to obtain the reciprocal: $\frac{10}{9}$ . Therefore the multiplicative inverse of $0.9$ is $\frac{10}{9}$ .

In every case, the strategy is the same: express the number as a fraction $\frac{a}{b}$ and then swap the numerator and denominator to get $\frac{b}{a}$ . The product of a number and its multiplicative inverse is always $1$ .

Updated 2026-04-21

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OpenStax Elementary Algebra

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