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.



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  \(x^{-1}\)


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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