Multiplicative Inverse
In this article, we will introduce the concept of the multiplicative inverse. Multiplicative inverses are used to simplify mathematical expressions. The word 'inverse' implies something opposite/contrary in effect, order, position, or direction. A number that nullifies the impact of a number to identity 1 is called a multiplicative inverse. Let us get into detail for more clarity of the concept.
What is Multiplicative Inverse?
The multiplicative inverse of a number is defined as a number which when multiplied by the original number gives the product an identity 1. The multiplicative inverse of 'a' is denoted by a^{1} or 1/a. In other words, when the product of two numbers is 1, they are said to be multiplicative inverses of each other. The multiplicative inverse of a number is defined as the division of 1 by that number. It is also called the reciprocal. Multiplicative Inverse Property: The product of a number and its multiplicative inverse is 1.
For example, let's consider 5 apples. Now, divide the apples into five groups of 1 each. To make them into groups of 1 each, we need to divide them by 5. Dividing by the number itself is equivalent to multiplying by the multiplicative inverse of the number. Hence, 5÷5=5×1/5=1. Here, 1/5 is the multiplicative inverse of 5.
Multiplicative Inverse of a Natural Number
Natural numbers are counting numbers starting from 1. The multiplicative inverse of a natural number a is 1/a.
Examples
 3 is a natural number. If we multiply 3 by 1/3, the product is 1. Therefore, the multiplicative inverse of 3 is 1/3.
 Similarly, the multiplicative inverse of 110 is 1/110.
Multiplicative Inverse of a Negative Number
Just as for any positive number, the product of a negative number and its reciprocal must be equal to 1. Thus, the multiplicative inverse of any negative number is it's reciprocal. For example, (6)×(1/6)=1, therefore, the multiplicative inverse of 6 is 1/6.
Let us consider a few more examples for a better understanding.
Multiplicative Inverse of a Unit Fraction
A unit fraction is a fraction with the numerator 1 If we multiply a unit fraction 1/x by x, the product is 1. The multiplicative inverse of a unit fraction 1/x is x.
Examples
 The multiplicative inverse of the unit fraction 1/7 is 7. If we multiply 1/7 by 7, the product is 1. 1/7×7=1.
 The multiplicative inverse of the unit fraction 1/50 is 50. If we multiply 1/50 by 50, the product is 1. 1/50×50=1.
Multiplicative Inverse of a Fraction
The multiplicative inverse of a fraction x/y, (x,y ≠ 0) is y/x. x/y × y/x=1
Examples
 The multiplicative inverse of 2/7 is 7/2. If we multiply 2/7 by 7/2, the product is 1. 2/7×7/2=1.
 The multiplicative inverse of 76/43 is 43/76. If we multiply 76/43 by 43/76, the product is 1. 76/43×43/76=1.
Multiplicative Inverse of a Mixed Fraction
To find the multiplicative inverse of a mixed fraction, converting the mixed fraction into an improper fraction, then determine its reciprocal. For example, the multiplicative inverse of \(3\dfrac{1}{2}\)
 Step 1: Convert \(3\dfrac{1}{2}\) to an improper fraction, that is 7/2.
 Step 2: Find the reciprocal of 7/2, that is 2/7. Thus, the multiplicative inverse of \(3\dfrac{1}{2}\) is 2/7.
Multiplicative Inverse of Complex Numbers
To find the multiplicative inverse of complex numbers and real numbers is quite difficult as you are dealing with rational expressions, with a radical (or) square root in the denominator part of the expression, which makes the fraction a bit complex.
Now, the multiplicative inverse of a complex number of the form a + \(i\)b, such as 3+\(i\)√2, where the 3 is the real number and \(i\)√2 is the imaginary number. In order to find the reciprocal of this complex number, multiply and divide it by 3\(i\)√2, such that: (3+\(i\)√2)(3\(i\)√2/3\(i\)√2) ^{ }= 9 + \(i\)^{2}2/3\(i\)√2 = 9 + (1)2/3\(i\)√2 = 92/3\(i\)√2 = 7/3\(i\)√2. Therefore, 7/3\(i\)√2 is the multiplicative inverse of 3+\(i\)√2
Also, the multiplicative inverse of 3/(√21) will be (√21)/3. While finding the multiplicative inverse of any expression, if there is a radical present in the denominator, the fraction can be rationalized, as shown for a fraction 3/(√21) below,
 Step 1: Remove the radical in the denominator. Multiply the entire fraction by the conjugate. \(\frac{3}{\sqrt{2}1} \times \frac{\sqrt{2}1}{\sqrt{2}1}\)
 Step 2: Solve. \(\frac{3 \sqrt{2}+1}{2 \cdot 1}\)
 Step 3: Simplify to the lowest form. 3(√2+1)
Modular Multiplicative Inverse
The modular multiplicative inverse of an integer p is another integer x such that the product px is congruent to 1 with respect to the modulus m. It can be represented as: px \(\equiv \) 1 (mod m). In other words, m divides px  1 completely. Also, the modular multiplicative inverse of an integer p can exist with respect to the modulus m only if gcd(p, m) = 1
In nutshell, the multiplicative inverses are as follows:
Type  Multiplicative Inverse  Example 

Natural Number x 
1/x  Multiplicative Inverse of 4 is 1/4 
x, x ≠ 0 
1/x  Multiplicative Inverse of 4 is 1/4 
Fraction x/y; x,y ≠ 0 
y/x  Multiplicative Inverse of 2/7 is 7/2 
Unit Fraction 1/x, x ≠ 0 
x  Multiplicative Inverse of 1/20 is 20 
Tips on Multiplicative Inverse
 The multiplicative inverse of a fraction can be obtained by interchanging the numerator and denominator.
 The multiplicative inverse of 1 is 1.
 The multiplicative inverse of 0 is not defined.
 The multiplicative inverse of a number x is written as 1/x or x1.
 The multiplicative inverse of a mixed fraction can be obtained by converting the mixed fraction into an improper fraction and determining its reciprocal.
Important Notes on Multiplicative Inverse
 The multiplicative inverse of a number is also called it's reciprocal.
 The product of a number and its multiplicative inverse is equal to 1.
 Multiplicative Inverse of a multiplicative inverse gives the original number. For example, multiplicative inverse of 1/5 is \(\dfrac{1}{\frac{1}{5}}\)=5
Related Topics
Solved Examples on Multiplicative Inverse

Example 1: A pizza is sliced into 8 pieces. Tom eats 3 slices of the pizza and leaves the rest. Determine the multiplicative inverse of the quantity of the pizza eaten by Tom.
Solution:
Since Tom ate 3 slices out of 8, it implies he ate 3/8 of the pizza. We need to determine the multiplicative inverse of 3/8. Interchange the numerator and the denominator to obtain the multiplicative inverse, which gives 8/3. Therefore, the multiplicative inverse of 3/8 is 8/3

Example 2: Daniel got 4 apples from the market. He ate \(1\tfrac{1}{2}\) out of them. Determine the multiplicative inverse of \(1\tfrac{1}{2}\)
Solution:
To determine the multiplicative inverse of \(1\tfrac{1}{2}\), convert the mixed fraction into an improper fraction. \(1\tfrac{1}{2}\) = 3/2. Now, find the multiplicative inverse of 3/2. The multiplicative inverse of 3/2 is 2/3. Therefore, the multiplicative inverse of \(1\tfrac{1}{2}\) is 2/3.

Example 3: Find the multiplicative inverse of 9/10. Also, verify your answer.
Solution:
The multiplicative inverse of 9/10 is 10/9. To verify the answer, we will multiply 9/10 with its multiplicative inverse and check if the product is 1. (9/10)×( 10/9)=1. Therefore, the multiplicative inverse of 9/10 is 10/9.
FAQs on Multiplicative Inverse
What is the Meaning of Multiplicative Inverse?
Multiplicative Inverse of any number is another number which when multiplied by the original number gives the product an identity.
What Is the Difference between Reciprocal and Multiplicative Inverse?
Reciprocal and multiplicative Inverse are the terms used in mathematics. When the product of two numbers is identity, then the numbers are said to be reciprocals or multiplicative inverses of each other. Hence, both mean the same thing. Just, 'multiplicative inverse' is a more general term.
What is the Multiplicative Inverse of 7?
If we multiply 7 by 1/7, the product is 1. Therefore, the multiplicative inverse of 7 is 1/7.
What is the Multiplicative Inverse of 1?
If we multiply 1 by 1, the product is 1. Therefore, the multiplicative inverse of 1 is 1.
What is the Multiplicative Inverse of 20?
If we multiply 20 by 1/20, the product is 1. Therefore, the multiplicative inverse of 20 is 1/20.
What is the Multiplicative Inverse of a Fraction?
The multiplicative inverse of a fraction is its reciprocal. The multiplicative inverse of any fraction, x/y, where x,y ≠ 0 is y/x.For example, the multiplicative inverse of 2/3 is 3/2.
What is Multiplicative Inverse Property?
The multiplicative inverse property states that the product of a number and its multiplicative inverse is always one. For example, 9×1/9=1.
What is the Multiplicative Inverse of 0?
The division by zero is not defined, therefore, the multiplicative inverse of 0 is undefined.
How to Find Modular Multiplicative Inverse?
The modular multiplicative inverse is an integer 'x' such that gcd(a, m) = 1. The multiplicative inverse of “a modulo m” exists if and only if a and m are relatively prime.