Reciprocal

Reciprocal Definition

In Mathematics, reciprocal meaning can be defined in the number of ways:

• The number \[\frac{1}{x}\], which multiplied by x gives product 1, known as a reciprocal of a number also called multiplicative inverse.

• The reciprocal of a number is one divided by that number example, reciprocal of x is \[\frac{1}{x}\].

• Reciprocal of a number is defined as turning the number upside down; an example reciprocal of \[\frac{x}{y}\] is \[\frac{y}{x}\].

• Reciprocal of a number can also be written as \[\frac{1}{n}\] or n-1.

Reciprocal of a Number

Reciprocal of a number is defined as the number divided by 1. Since a number can be written as a fraction with a denominator of one. For example

The number 3 can be written as \[\frac{3}{1}\]

and so reciprocal of \[\frac{3}{1} = \frac{1}{3}\]

Reciprocal Example: The reciprocal of 8.

From the rule of reciprocal x = \[\frac{1}{x}\]

We have, 8 = \[\frac{1}{8}\].

We get the same number if we again reciprocal the result.

Reciprocal of \[\frac{1}{8}\] is 8.

When we multiply the given number with its reciprocal we get the result as 1.

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Reciprocal of a Fraction

The reciprocal of a fraction is a fraction obtained by swapping or interchanging the values in the numerator and the denominator of the given fraction, just turn it upside down.

For Example 

The reciprocal of \[\frac{5}{8}\]

From the rule of reciprocal \[\frac{x}{y} = \frac{y}{x}\]

We have, \[\frac{5}{8} = \frac{8}{5}\]

Reciprocal of a Mixed Fraction

To find the reciprocal of the mixed fraction we have to first convert the mixed fraction into an improper fraction and then take its reciprocal means to turn it upside down.

 The reciprocal of \[2 \frac{1}{2}\]

First, convert the mixed fraction to improper fraction

\[2 \frac{1}{2}\] = \[\frac{(2\times 2 + 1)}{2}\]

= \[\frac{5}{2}\]

Now take reciprocal of \[\frac{5}{2}\] by flipping the numerator and denominator upside down i.e \[\frac{2}{5}\].

Reciprocal of a Decimal Number

The reciprocal of a decimal number is the same as in numbers defined by the number divided by 1.

For Example:

The reciprocal of a decimal number 0.50

From the reciprocal rule of a number, x = \[\frac{1}{x}\]

We have,  0.5= \[\frac{1}{0.5}\].

Application of Reciprocal

Reciprocal is used in division operation for fractions.

Example: Divide \[\frac{6}{27} \div \frac{18}{3}\].

Solution: For the division of two unlike fractions we have to take the reciprocal of the second number and then multiply them.

\[\frac{6}{27} \div \frac{18}{3}\]

Taking reciprocal of second number \[\frac{18}{3}\] as \[\frac{3}{18}\]

= \[\frac{6}{27} \times \frac{18}{3}\]

Simplify the fractions into a simpler form

= \[\frac{1}{27}\]

Rules for Reciprocal

1. For a number x, the reciprocal will be \[\frac{1}{x}\] or also can be written as x-1.

2. For a fraction \[\frac{x}{y}\] the reciprocal will be \[\frac{y}{x}\].

Facts:

• All the numbers except 0 have reciprocals. Because \[\frac{1}{0}\] is undefined.

• "Reciprocal" comes from the Latin word “reciprocus” meaning returning. Like going somewhere and then returning home again.

Solved Examples

Reciprocal examples

1. Find the reciprocal of the given numbers.

1. \[\frac{6}{7}\]

Solution: \[\frac{6}{7}\]= \[\frac{7}{6}\]

2. -7

Solution: - 7 = -\[\frac{1}{7}\]

3.  \[4\frac{9}{2}\]

Solution: \[4\frac{9}{2}\] = \[\frac{4\times 2 + 9}{2}\]

=  \[\frac{17}{2}\]

=  \[\frac{2}{17}\]

4. 6.7

Solution: 6.7 = \[\frac{1}{6.7}\]

Quiz Time

Write down the reciprocal of the following numbers,

1. 56 

2. -\[\frac{4}{7}\]

3. 0.005

4. \[6\frac{8}{2}\]