# Reciprocal

## Reciprocal Definition

**Reciprocal** in Mathematics means any expression which when multiplied by another expression, gives unity (1) as a result. The reciprocal of any quantity is, one divided by that quantity. For a number a, it would be 1/a.

Example: Reciprocal of a number 7 is 1/7.

It has many other definitions too :

- It is also called the multiplicative inverse.
- It is similar to turning the number upside down.
- It is also found by interchanging the numerator and denominator.
- All the numbers have reciprocal except 0.
- The reciprocal of the product of a number is 1.

Generally, it is written as, **x=1/x or x = x ^{-1 }**for a number x.

- Examples: The reciprocals of 3 and 8 are 1/3 and 1 /8

Reciprocal of 3/4 is 4/3.

It is also expressed by the number raised to the power of negative one and can be found for fractions and decimal numbers too.

In math, when you take the reciprocal twice, you will get the same number what you started with.

- Example: T
**he reciprocal of 4 is 1/4.**When you repeat this step it becomes 4/1 or 4. Here you get the same number where you started with.

## Reciprocal of a Number

It is defined as one over that number.

Example: Find the reciprocal of 5

Solution: To find the solution, we will use x=1/x

Therefore, 5= 1/5

**The reciprocal of a function, f(x) = f(1/x)**

## Reciprocal of a Fraction

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

Example: Find the reciprocal of 2 / 3

Solution: To find the solution we will follow the following steps

The reciprocal of 2/3 is 3/2. (or) use the formula, x = 1/x, where 2/3=1/2/1/3

Therefore, the reciprocal of a fraction 2/3 is 3/2.

### Reciprocal of a Mixed Fraction

- In order to find the same for a mixed fraction, convert it into improper fractions and do the operation.

Consider a mixed fraction, 4(1/2).

First step is to convert a mixed fraction into an improper fraction.

4(1/2)= 1+1+1+1+1/2

= (2/2) + (2/2) + (2/2) + (2/2) + (1/2 )

= (2+2+2+2+1)/2

=9/2

4(1/2) = 9/2

Now, you get the fraction and do the same operation for finding the reciprocal by flipping the numerator and the denominator.

Therefore, the solution for 9/2 is 2/9.

## Reciprocal of a Decimal

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

Example: Find the reciprocal of a decimal 0.75

Solution: The reciprocal of a number, x=1/x

Therefore, 0.75= 1/0.75

An alternate method to find it is given below.

Consider the same example, 0.75.

First, you have to check whether the given decimal number is possible for converting into a fractional number. Here 0.75 is written as 3/4

Now, find the reciprocal of 3/4 which gives 4/3

When you verify both the solutions, it results in the same.

That is, 1/0.75 = 1.33 and

4/3 = 1.33

### Reciprocal Example

Numbers and their Reciprocals

2 | 1/2 |

9 | 1/9 |