Two numbers are reciprocals, or multiplicative inverses, of each other when their product is the multiplicative identity. For example, the reciprocal of $9$ is $\frac{1}{9}$ because their product is $1.$ The reciprocal of a number $a$ can be found by dividing $1$ by $a.$

Shortcuts exist to find the reciprocals of specific types of numbers such as natural numbers, integer numbers, fractions, and decimals.

Type	Reciprocal	Example
Natural number $a$	$\frac{1}{a}$	The reciprocal of $2$ is $\frac{1}{2}.$
Integer numbers $a,$ $a\ne 0$	$\frac{1}{a}$	The reciprocal of $\text{-}6$ is $\text{-}\frac{1}{6}.$
Fraction $\frac{a}{b},$ $b\ne 0$	$\frac{b}{a}$	The reciprocal of $\frac{3}{2}$ is $\frac{2}{3}.$
Decimal $a$	$\frac{1}{a}$	The reciprocal of $0.2$ is $\frac{1}{0.2}.$

Finding the reciprocal of a mixed number is like finding the reciprocal of a fraction. However, the mixed number must first be written as an improper fraction.