Multiplicative Inverse

Multiplicative inverse of a number is another number that, when multiplied by the original number, results in the identity element for multiplication, which is 1. In other words, for a non-zero number a, its multiplicative inverse is denoted as a⁻¹, and it satisfies the equation: a⋅a^-1= 1. We can also define multiplicative inverse as the reciprocal of a number. A number when multiplied with its own multiplicative inverse(reciprocal), then we get 1. In this article, we will learn about multiplicative inverse their definition, multiplicative inverse of natural numbers, fraction, unit fraction, mixed fraction, and complex numbers.

Table of Content

What is Multiplicative Inverse?
Multiplicative Inverse of Natural Number
Multiplicative Inverse of Fraction
Multiplicative inverse of Mixed Fraction
Multiplicative Inverse of Complex Numbers

What is Multiplicative Inverse?

Multiplicative inverse meaning is a reciprocal of a number which is denoted as a^-1. Let there is a number X its multiplicative inverse is represented as 1/X or X^-1. The English meaning of “inverse” is “opposite”. It’s important to note that the multiplicative inverse is not defined for zero because there is no number you can multiply by 0 to get 1. Let us understand the examples of multiplicative inverse.

Examples for multiplicative inverse:

2^-1 or 1/2 is a multiplicative inverse because 2 × 1/2 = 1.
5^-1 or 1/5 is a multiplicative inverse because 5 × 1/5 = 1.
4^-1 or 1/4 is a multiplicative inverse because 4 × 1/4 = 1.

Multiplicative Inverse Definition

Multiplicative inverse is defined as the reciprocal of a number and is represented as 1/n or n^-1 where n is a number.

Multiplicative Inverse Property

The identity of multiplicative inverse states that every nonzero number a has a unique multiplicative inverse, denoted as a^-1, such that the product of a and its multiplicative inverse is equal to the multiplicative identity that is 1.

Mathematically it is written as:

a . a^-1= 1

Multiplicative Inverse of Natural Number

The multiplicative inverse of a natural number N i.e. {1,2, 3, …, n} will be 1/N. Example: The multiplicative inverse of 1 is 1. One important thing that we have to remember is that we cannot find multiplicative inverse of zero (0).

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Multiplicative Inverse of Fraction

We can also find the multiplicative inverse of a fraction. We have to find the reciprocal of a fraction in order to get its multiplicative inverse. Let there is a fraction i.e. X/Y then its multiplicative inverse of X/Y will be Y/X because when Y/X is multiplies by X/Y then the result will be 1.

Examples of Multiplicative Inverse of Fraction are:

The multiplicative inverse of 2/3 is 3/2 because when we multiply 2/3 with 3/2 we get 1.
The multiplicative inverse of 5/6 is 6/5 because when we multiply 5/6 with 6/5 we get 1.
The multiplicative inverse of 7/6 is 6/7 because when we multiply 7/6 with 6/7 we get 1.

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Multiplication Inverse of Unit Fraction

The multiplicative inverse of unit fraction can also be find out by reciprocal. Unit fraction includes numbers like 1/2, 1/3, 1/5, etc. The examples of multiplicative inverse of a unit fraction are:

The multiplicative inverse of 1/5 will be 5.
The multiplicative inverse of 1/3 will be 3.
The multiplicative inverse of 1/7 will be 7.

Multiplicative inverse of Mixed Fraction

In order to find multiplicative inverse of mixed fraction we have to solve the mixed fraction into simple fraction and then we reciprocal the simple fraction.

Examples of multiplicative inverse of Mixed Fraction are as follows:

The simple fraction of mixed fraction 4(3/2) will be 11/2 and its multiplicative inverse will be 2/11.
The simple fraction of mixed fraction 2(1/2) will be 5/2 and its multiplicative inverse will be 2/5.
The simple fraction of mixed fraction 4(3/2) will be 11/2 and its multiplicative inverse will be 2/11.

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Multiplicative Inverse of Complex Numbers

The multiplicative inverse of a complex number a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1), is given by (a + bi)^-1 or 1/ (a + bi).

To find the multiplicative inverse, you often multiply the numerator and denominator by the conjugate of the complex number. The conjugate of a + bi is a − bi. So, the multiplicative inverse becomes 1/ (a + bi) × (a-bi)/ (a + bi) = (a-bi)/ (a² + b²).

Following are the examples of Multiplicative Inverse of Complex Numbers:

Multiplicative inverse of 2 + 3i will be 1/ (2 + 3i) × (2 – 3i) / (2 – 3i) = 2-3i/13.
Multiplicative inverse of -1 + 2i will be 1/ (-1 + 2i) × (-1-2i)/(-1-2i) = -1-2i/5.

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Multiplicative Inverse – Solved Examples

Example 1: What is the Multiplicative Inverse of 7?

Solution:

The multiplicative Inverse of 7 is 1/7

Example 2: What is the Multiplicative Inverse of 2-3i?

Solution:

Multiplicative inverse of 2-3i is 1/(2-3i)

z^-1 = 1/(2-3i) × (2 + 3i)/(2 + 3i)

Solving Above we get

z^-1 = (2 + 3i)/13

Example 3: Find the Multiplicative Inverse of -2/7.

Solution:

The Multiplicative Inverse of -2/7 is -7/2

Multiplicative Inverse – Practice Questions

1. Find the multiplicative inverse of the following:

(i) Multiplicative Inverse of 2

(ii) Multiplicative Inverse of 5

(iii) Multiplicative Inverse of 2 + 3i

2. Find the multiplicative inverse of 3/7.

3. Find the multiplicative inverse of 4+3i, where i is the imaginary unit.

4. Calculate the multiplicative inverse of -8.

5. Find the multiplicative inverse of 1/17.

Multiplicative Inverse – FAQs

1. What is Multiplicative Inverse?

Multiplicative inverse is nothing but a reciprocal of a number which is denoted as a^-1. Let there is a number X its multiplicative inverse is represented as 1/X or X^-1.