Reciprocal: Definition, Rules, Solved Examples, Facts, FAQs

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What is a Reciprocal in Math?

The word “reciprocal’ comes from the Latin word “reciproc(us)”, meaning “returning” or “alternating.”

So, what is the meaning of reciprocal in math? How to find the reciprocal of a number? Simply divide 1 by the given number. In other words, we turn the number upside down, or interchange the numerator and denominator. 

Reciprocal

Definition of Reciprocal in Math

In math, the reciprocal of any quantity can be defined as 1 divided by that quantity.

How To Find Reciprocal

For any non-zero number “x,” the reciprocal will be 1x

If we multiply the given number by its reciprocal, we get the product 1. 

The reciprocal of x is also represented by x1.

Reciprocal of a number

Rules of Reciprocals

Let’s discuss two reciprocal rules.

  • For a number x, the reciprocal will be 1x and it can also can be written as x1

For example, if 10 is the number, then the reciprocal will be 110.

  • For a fraction xy, the reciprocal will be yx

For example, the reciprocal of 27 is 72.

What Is the Reciprocal of Zero?

We can find the reciprocal of any number, but we cannot apply the reciprocal condition on zero, since it will give an indefinite value. Division of any number by 0 is undefined. 

Thus, we can have a reciprocal for all real numbers except 0.

Reciprocal of a Natural Number

The reciprocal of a natural number is defined as the one divided by the given number.

Suppose we have to find the reciprocal of 5. 

We know that reciprocal of a number x is 1x

Therefore, reciprocal of a 5 is 15.

Reciprocal of Negative Numbers

For any negative number (x), we can find the reciprocal by writing the inverse of the given number with a minus sign along, 1x

Suppose we have to find the reciprocal of 8, we follow the steps given below: 

Step 1: We write the given number in the form of an improper fraction by writing the number 1 in the denominator. So, 8 can be written as 81.

Step 2: Interchange the numerator and denominator values. On interchanging in the example, we get 18. 

Step 3: Add a minus sign () to the resultant number. We get 18

Reciprocal of a Fraction

The reciprocal of a fraction is obtained by interchanging the numerator and the denominator.

Suppose we have to find the reciprocal of 45

Use the formula, reciprocal of x=1x,

Here, x=45 

Thus,  1x=145=54 

Therefore, the reciprocal of a fraction 45 is 54

Reciprocal of a Mixed Fraction

Suppose we have a mixed fraction 513

Step 1: Convert the mixed fraction into an improper fraction.

In this case, we get 163.

Step 2: Flip the numerator and denominator. 

In this case, we get 316

Thus, the reciprocal of 513 is 316

Reciprocal of a Decimal

We calculate the reciprocal of a decimal by one over the decimal number.

Method 1: 

Suppose we have to find a reciprocal of 0.25.

Step 1: Flip the numerator and the denominator.

In this case, we get 10.25.

Step 2: Change it to fractional form.

In this case, we get 10025=4

Step 3: If we get a whole number, we leave it till Step 2. If we get a fraction, we convert it into decimal by dividing the numerator by denominator.

Method 2:

Suppose we have to find the reciprocal of 0.52.

Step 1: Check whether the given decimal number is possible to convert into a fractional number. Here 0.52 is written as 52100=1325.

Step 2: Find the reciprocal of the fraction. Here, we get 2513.

Step 3: Convert the fraction into decimals. Here, we get 1.92.

Unity: Product of a Number and Its Reciprocal

If we multiply the number by the reciprocal of the number, we will get unity, that is, 1. 

Product of a number and its reciprocal

Some examples are given below: 

  • 5×15=1
  • 23×32=1
  • 50×150=1
  • 910×109=1

Applications of Reciprocal

The most important application of reciprocal is that it is used in division operation for fractions. If we want to divide the one fraction by the second fraction, we can find it by multiplying the first fraction with the reciprocal of the second fraction. 

Suppose we have to find 15÷75

Here, the first fraction is 15.

The second fraction is 75.

Thus, the reciprocal of the second fraction is 57.

Hence, 15÷75=15×57=17.

Facts about Reciprocal

  • Reciprocal of a number is also called its multiplicative inverse.
  • Reciprocal of a number is 1 divided by the number.

Solved Examples on Reciprocal

1. What is the reciprocal of 7?

Solution: The reciprocal of 7 is 17.

2. What is the reciprocal of 67? Verify your answer. 

Solution: The reciprocal of a fraction xy is yx. So, reciprocal of 67 is 76

Also, 67×76=1. Hence, verified.

3. What is the reciprocal of  5x3?

Solution:

Reciprocal of 5x3 is 35x.

4. A pizza is sliced into 8 pieces. Tom eats 3 slices of the pizza and leaves the rest. Determine the reciprocal of the quantity of the pizza left by Tom.

Solution: 

Total number of slices =8

Number of slices Tom ate =3

Number of slices left =83=5

Fraction of pizza left by Tom =58

Reciprocal =85

5. Stuart bought 4 apples from the market. He ate 212 apples from the 4 he bought. Determine the reciprocal of the fraction of the apples he ate.

Solution: Fraction of the apples he ate =212=52

Reciprocal =25

Practice Problems On Reciprocal

Reciprocal: Definition, Rules, Solved Examples, Facts, FAQs

Attend this quiz & Test your knowledge.

1

Which of the following statements is false?

Reciprocal of 5 is 15.
Reciprocal of 0 is not defined.
Reciprocal of 15 is 115.
Reciprocal of 19 is 9.
CorrectIncorrect
Correct answer is: Reciprocal of 15 is 115.
Reciprocal of 15 is 115.
2

The reciprocal of 0 is ______.

0
1
100
Not defined
CorrectIncorrect
Correct answer is: Not defined
The reciprocal of 0 is not defined.
3

Which of the following is the reciprocal of 21×5?

5×121
21×15
121×5
105
CorrectIncorrect
Correct answer is: 121×5
Reciprocal of 21 is 121 and reciprocal of 5 is 15. So, reciprocal of 21×5 is 121×5.
4

Identify the reciprocal of 0.075 from the following options.

13.3
340
10.025
13.3
CorrectIncorrect
Correct answer is: 13.3
0.075=751000=340 Reciprocal of 340 is 403=13.3
5

If the reciprocal of an algebraic expression is 2x2y, then the expression will be _______________.

12x2y
12x2y
12xy
2x2y
CorrectIncorrect
Correct answer is: 12x2y
Reciprocal of 2x2y is 12x2y.

Frequently Asked Questions On Reciprocal

The reciprocal of infinity is 0.

Reciprocal of 1 is 11=1. So, 1 has the same reciprocal as the number itself.

Inverse means opposite and reciprocal means equality. A reciprocal is a type of inverse but inverse is not necessarily a reciprocal.

The other name of reciprocal is multiplicative inverse.

The reciprocal of the reciprocal of a number is the number itself. For example: The reciprocal of 2 is 12 and the reciprocal of 12 is 112=2.