## What is a Reciprocal?

A reciprocal is a fraction that has been flipped upside-down. The word comes from the Latin root word “reciprocus” which means “alternating” or “going backward and forward”. Ancient mathematicians chose this word because you have to switch or “alternate” the numerator and denominator to find the reciprocal of a fraction.

## Examples

The reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\).

The reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\) which can be simplified to \(2\).

The reciprocal of \(6\) is \(\frac{1}{6}\).

## Multiplicative Inverse Property

Reciprocals are also known as multiplicative inverses and every number has one. The Multiplicative Inverse Property says that if you multiply any number by its multiplicative inverse, the product is 1.

For example, the multiplicative inverse of \(\frac{2}{3}\) is \(\frac{3}{2}\). And when you multiply the original fraction by its multiplicative inverse, the product is 1.

\[\frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1\]

## Reciprocals of Whole Numbers

If you are ever asked to find the reciprocal of a whole number, just remember that whole numbers can be written as fractions. For example, the whole number \(8\) can be written as \(\frac{8}{1}\). After the whole number is written as a fraction, you can reverse the numerator and denominator to find the reciprocal \(\frac{1}{8}\).

## When Will I Use Reciprocals?

When you divide fractions, you will multiply the first fraction by the reciprocal of the second fraction.

When you solve equations, you can multiply both sides of the equation by the multiplicative inverse of a coefficient to solve for the variable.

When you divide rational expressions (fractions with polynomials), you will multiply the first rational expression by the reciprocal of the second.

When you are asked to find the slope of perpendicular lines, the slopes will be opposite reciprocals of each other.