 # Reciprocal (Multiplicative Inverse)

The reciprocal meaning is something that is the opposite of the given value. It is a natural fact that the product of the number and its reciprocal will always be identity value i.e. 1. Reciprocal in math is a common computation. In this article, students will learn this concept in an easy manner with suitable examples. Let us learn it!

## Reciprocal (Multiplicative Inverse)

In the number system theory of mathematics identity and inverse are the two essential computations. These two are related to each other. The multiplicative inverse of a given number n is the value represented as $$\frac{1}{n}$$. It is also termed as the reciprocal value of the number. Source: wikihow.com

### Reciprocal Definition:

The multiplicative inverse or reciprocal of a number x is simply denoted as $$x^{-1}$$ and computed as $$\frac{1}{x}$$. Thus:

Multiplicative inverse (x) = $$\frac{1}{x}$$ or  $$x^{-1}$$

Thus,

x × $$x^{-1}$$ = 1

It is also popular as the reciprocal of the number given. It must be noted that 1 is known as the multiplicative identity of any real number.

For example, the multiplicative inverse of 5 will be $$\frac{1}{5}$$.

Also, the reciprocal of zero is termed as $$\frac{1}{0}$$ i.e. $$\infty$$.

Thus the student can see that reciprocal mathematics is easy but important.

### Reciprocal Agreement Definition:

In this regard, one important term is the reciprocal agreement. Reciprocal agreement definition says that it is a reciprocal action or agreement which involves two terms who do the same thing to each other or agree to help each other in a similar way. It means if x is reciprocal of y then, y will also be the reciprocal of x. We can see that,

If x = $$\frac{1}{y}$$ then

y = $$\frac{1}{x}$$

Thus x and y will be reciprocal of each other.

### How to find reciprocal?

Reciprocal math is very easy for calculation. For many problems and applications, we can use this process. We can compute reciprocal as follows:

1. Reciprocal of a Number:

If x is an integer number such as 1,2,3,4,… , then the multiplicative inverse of x will be $$\frac{1}{x}$$.

For example, the multiplicative inverse of 9 is $$\frac{1}{9}$$.

1. Reciprocal of a Fraction:

If x is any rational number like $$\frac{p}{q}$$, then its reciprocal will be $$\frac{q}{p}$$.

For example, the multiplicative inverse of $$\frac{3}{4}$$ will be $$\frac{4}{3}$$.

Consider some more examples. The multiplicative inverse of 3 will be $$\frac{1}{3}$$, of $$\frac{-1}{3}$$ is -3, of 8 is $$\frac{1}{8}$$ and of 4/7 will be $$\frac{-7}{4}$$. But the multiplicative inverse of zero will be infinite, as $$\frac{1}{0}$$ is infinity. Also, whereas the multiplicative inverse of 1 will be 1 itself.

## Solved Examples for You

Q.1: Determine the reciprocals up to 30.

Solution : As we know that reciprocal of x = $$\frac{1}{x}$$. Thus we have the solution as in the given table.

 Value of x Reciprocal = $$\frac{1}{x}$$ 1 $$\frac{1}{1}$$ 2 $$\frac{1}{2}$$ 3 $$\frac{1}{3}$$ 4 $$\frac{1}{4}$$ 5 $$\frac{1}{5}$$ 6 $$\frac{1}{6}$$ 7 $$\frac{1}{7}$$ 8 $$\frac{1}{8}$$ 9 $$\frac{1}{9}$$ 10 $$\frac{1}{10}$$ 11 $$\frac{1}{11}$$ 12 $$\frac{1}{12}$$ 13 $$\frac{1}{13}$$ 14 $$\frac{1}{14}$$ 15 $$\frac{1}{15}$$ 16 $$\frac{1}{16}$$ 17 $$\frac{1}{17}$$ 18 $$\frac{1}{18}$$ 19 $$\frac{1}{19}$$ 20 $$\frac{1}{20}$$ 21 $$\frac{1}{21}$$ 22 $$\frac{1}{22}$$ 23 $$\frac{1}{23}$$ 24 $$\frac{1}{24}$$ 25 $$\frac{1}{25}$$ 26 $$\frac{1}{26}$$ 27 $$\frac{1}{27}$$ 28 $$\frac{1}{28}$$ 29 $$\frac{1}{29}$$ 30 $$\frac{1}{30}$$

Q.2 : Find the reciprocal of $$\frac{5}{8}$$.

Solution: let x = $$\frac{5}{8}$$.

Since, $$\frac{5}{8}$$ × $$\frac{8}{5}$$ = 1

Therefore, $$x^{-1}$$ = $$\frac{8}{5}$$.

Q.3: Find the reciprocal of $$a^{3}$$.

Solution: The reciprocal of $$a^{3}$$ will be:

$$\frac{1}{a^{3}}$$ , as $$\frac{1}{a^{3}}$$ × $$a^{3}$$ = 1.

Thus reciprocal of $$a^{3}$$ will be $$\frac{1}{a^{3}}$$ i.e. $$a^{-3}$$.

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