Reciprocal

We have studied what is a fraction earlier too. Let’s recall it again, if a whole thing is divided into equal parts then each part is said to be a fraction. A fraction is formed up of two parts - numerator and denominator.


Expressed as -  \[\frac{\text{numerator}}{\text{denominator}}\]. 


Now let us study reciprocal meaning. Reciprocal is also called as multiplicative inverse. Reciprocals are used to make the equations easier to solve.


The reciprocal of a fraction is obtained by inverting the fraction i.e interchanging the numerator and the denominator. In simple words reciprocal means flipping upside down.


In this article, we will study reciprocal definition, how to find the reciprocal of numbers, fractions, mixed fractions, decimal numbers, and reciprocal examples.


(image will be uploaded soon)


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.


(image will be uploaded soon)


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.


For example: 

 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: \[\frac{6}{7}\]= \[\frac{7}{6}\]

  1. -7

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

  1.  \[4\frac{9}{2}\]

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


=  \[\frac{17}{2}\]


=  \[\frac{2}{17}\]

  1. 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. \[6\frac{8}{2}\]

FAQ (Frequently Asked Questions)

1. What is the Difference Between a Reciprocal of a Number and the Opposite of the Number?

Answer: Reciprocal of a number x is 1/x.


And the opposite of the number x is -x.


Hence by the reciprocal rule when you multiply a reciprocal with its original number we get the result as 1.


While when you add the opposite to its original number we get the result is 0.


Example: 

Reciprocal of 7 is 1/7


7 × 1/7 = 1


Opposite of 7 is -7


7 + (-7) = 0

2. What are Fractions?

Answer: If a whole thing is divided into equal parts then each part is said to be a fraction. In other words, a fraction is a part of the whole. A fraction is formed up of two parts - numerator and denominator.


Expressed as -  numerator/denominator.


The numerator tells us how many of the total parts are taken and the denominator tells us the total number of equal parts.


For example, in ¼ the top number ‘1’ is said to be the numerator and ‘4’ is said to be the denominator. Here we can say that 1 is taken from 4 equal whole parts. Fractions are very important to understand as it is used in our daily life.

3. Why are Reciprocals Useful when Dividing Rational Expressions?

Answers: Dividing rational numbers is a difficult task. So we use reciprocals to divide the rational numbers. While dividing the rational number we take the reciprocal of the second number and then multiply it with the first number.


Example: Divide 6/27 ÷ 3/18


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


6/27 ÷ 3/18


Taking reciprocal of second number 3/18 as 18/3


= 6/27 × 3/18


Simplify the fractions into a simpler form


= 4/3