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:

- 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 \(\frac{1}{9}\).

- 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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