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Reciprocals

The reciprocal or multiplicative inverse of a number x is the number which, when multiplied by x, gives 1. So, the product of a number and its reciprocal is always 1. This is sometimes called the property of reciprocals. Note that the number 0 has no reciprocal.

Defining reciprocals

In mathematics, the reciprocal of any number is one divided by that number. For any number n, the reciprocal will be $\frac{1}{n}$ . If the given number is multiplied by its reciprocal, we get the value 1.

Example 1

The reciprocal of 7 is $\frac{1}{7}$ , which we can show by multiplying 7 and $\frac{1}{7}$

$7×\frac{1}{7}=1$

Other definitions of reciprocals:

• They are also called the multiplicative inverse.
• You can think of it as "turning the number upside down".
• They are also found by interchanging the numerator and denominator.
• All real numbers have a reciprocal except 0.
• The product of a number and its reciprocal is always equal to 1.
• Generally, a reciprocal is written as $\frac{1}{x}$ or ${x}^{\mathrm{-1}}$ .
• Reciprocals are also expressed by the number raised to the power of negative 1 and can be found for fractions and decimals as well.
• If you take the reciprocal of a number twice, the reciprocal is its own inverse operation.

Reciprocal of a number

The reciprocal of a number is defined as one over that number.

Example 2

The reciprocal of 5 is $\frac{1}{5}$ because $5×\frac{1}{5}=1$ .

Reciprocal of negative numbers

For any negative number $-x$ , the reciprocal can be found by writing the inverse of the given number with a minus sign along with that.

Example 3

The reciprocal of $-17$ is $\frac{-1}{17}$ .

Reciprocal of a fraction

The reciprocal of a fraction can be found by interchanging the values of the numerator and denominator.

Example 4

The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ .

Reciprocal of a mixed fraction

In order to find a reciprocal for a mixed fraction, first convert it into an improper fraction and perform the operation.

Example 5

Find the reciprocal of the mixed fraction $4\frac{1}{2}$ .

The first step is to convert the mixed fraction into an improper fraction.

$4\frac{1}{2}=\frac{9}{2}$

Now you perform the same operation to find the reciprocal by flipping the numerator and denominator.

The reciprocal for $\frac{9}{2}$ is $\frac{2}{9}$ .

So the reciprocal for $4\frac{1}{2}$ is $\frac{2}{9}$ .

Reciprocal of a decimal

The reciprocal of a decimal is the same as it is for the reciprocal of a number defined by one over the number.

Example 6

The reciprocal of a number $\frac{1}{x}$ .

So, the reciprocal of the decimal $.75=\frac{1}{.75}$ .

$\frac{1}{\left(\frac{75}{100}\right)}=\frac{1}{\left(\frac{3}{4}\right)}=\frac{4}{3}=1.3333\text{...}$

Application of reciprocal numbers

One of the main applications of reciprocal numbers is in the division of fractions. To divide the first fraction by the second fraction, the result can be found by multiplying the first fraction with the reciprocal of the second fraction.

Example 7

$\frac{2}{5}$ divided by $\frac{7}{5}$

The reciprocal of the second fraction is $\frac{7}{5}$ .

So $\frac{2}{5}$ divided by $\frac{7}{5}=\frac{2}{5}×\frac{5}{7}=\frac{2}{7}$

The answer to $\frac{2}{5}$ divided by $\frac{7}{5}$ is $\frac{2}{7}$ .

Two rules for reciprocals

The two most important rules for reciprocals are:

• For a number $x$ , the reciprocal will be $\frac{1}{x}$ , which can also be written as ${x}^{-1}$ . For example, if $9$ is the number, ${9}^{-1}$ will be the reciprocal.
• For a fraction $\frac{x}{y}$ , the reciprocal will be $\frac{y}{x}$ . For example, if $\frac{3}{7}$ is the given fraction, then its reciprocal will be $\frac{7}{3}$ .