What is a reciprocal? For example, why is the reciprocal of 8 equal to 0.125?

In order to take a reciprocal of a number all you need to do is divide the number into 1. That is:

Reciprocal = 1 ÷ your number

Reciprocals can be useful in calculus and working out the gradient of a normal line.

Example 1

Take the reciprocal of 8.

Reciprocal = 1 ÷ your number

= 1 ÷ 8

= 1/8 or 0.125

Example 2

Take the reciprocal of 3/5.

Reciprocal = 1 ÷ your number

= 1 ÷ 3/5

= 5/3 or 1.7 to 1 decimal place.

Notice in this example, if you take the reciprocal of the fraction all you need to do is flip the fraction over.

Example 3

Take the reciprocal of 1/9

Reciprocal = 1 ÷ your number

= 1 ÷ 1/9

= 9

Example 4

Take the reciprocal of -4

Reciprocal = 1 ÷ your number

= 1 ÷ -4

= -1/4 or -0.25

Notice that the answer is still negative if you take the reciprocal of a negative number.

Example 5

Take the reciprocal of 0.16

Reciprocal = 1 ÷ your number

= 1 ÷0.16

=0.0625

Example 6

Take the reciprocal of √3

Reciprocal = 1 ÷ your number

= 1 ÷ √3

= 0.577 rounded to 3 significant figures.

Example 7

Take the reciprocal of 3/10. Also take the reciprocal of your answer. What do you notice?

Reciprocal = 1 ÷ your number

= 1 ÷ 3/10

= 10/3 or 3.3 recurring.

All you need to do now is take the reciprocal of your answer.

Reciprocal = 1 ÷ your number

1 ÷ 10/3

= 3/10 or 0.3

So if you take the reciprocal again you end up with your original number.

Summary of taking reciprocals

In order to take a reciprocal just divide the number into 1. If you are taking the reciprocal of a fraction then just turn the fraction upside down so that the numerator and denominator swap over.