Multiplicative Inverse
The multiplicative inverse is defined as the reciprocal of a given number. It is used to simplify mathematical expressions. The word 'inverse' implies something opposite/contrary in effect, order, position, or direction. A number when multiplied to its multiplicative inverse results in 1.
What is Multiplicative Inverse?
The multiplicative inverse of a number is defined as a number that when multiplied by the original number gives the product as 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 of the number. The multiplicative inverse formula says that the product of a number and its reciprocal is 1.
There are different types of numbers like natural numbers, fractions, unit fractions, negative numbers, etc. Let us understand the multiplicative inverse formula for each type of number.
Natural numbers are counting numbers starting from 1. The multiplicative inverse of a natural number a is 1/a. For example, 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 Integers
Finding the multiplicative inverse of positive integers is the same as natural numbers (explained above). Just like positive integers, the product of a negative number and its reciprocal must be equal to 1. Thus, the multiplicative inverse of any negative number is its reciprocal. For example, (6) × (1/6) = 1, therefore, the multiplicative inverse of 6 is 1/6. Note that the multiplicative inverse of a negative number is always negative. And, in the multiplicative inverse of a negative integer, the negative sign will attach to the numerator, and not with the denominator.
Multiplicative Inverse of a Fraction
The multiplicative inverse of a fraction a/b is b/a because a/b × b/a = 1 when (a,b ≠ 0). For example, 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).
A unit fraction is a fraction with the numerator 1. If we multiply a unit fraction 1/x by x, the product is 1. Thus, 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 Mixed Fraction
To find the multiplicative inverse of a mixed fraction, convert the mixed fraction into an improper fraction, then determine its reciprocal. For example, let us find 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.
It is interesting to note that the multiplicative inverse of a mixed number is always a proper fraction whose value is less than 1.
Multiplicative Inverse of 0
As per the definition of multiplicative inverse, it is the number that when multiplied to the original number results in 1 as the product. But with 0, we know that the product of 0 with any number is always 0. So, the multiplicative inverse of 0 does not exist.
We can also understand this using the properties of division which states that the division of any number by 0 is not defined. The multiplicative inverse of 0 can be written as 1/0, but its value is not defined. So, it does not exist.
Multiplicative Inverse Property
The multiplicative inverse property states that the product of a number with its reciprocal is always equal to 1. Look at the image given below where 1/n is the multiplicative inverse of the number n and 1 is the product.
For example, let us 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 a number by itself is equivalent to multiplying it by its multiplicative inverse. Hence, 5 ÷ 5 = 5 × 1/5 = 1. Here, 1/5 is the multiplicative inverse of 5.
How to Find Multiplicative Inverse?
The multiplicative inverse of a number is also known as its reciprocal. It is very easy to find the multiplicative inverse of a number using the following steps:
 Step 1: Divide the given number by 1.
 Step 2: Write it in the form of a fraction. Say, the reciprocal of a is 1/a.
 Step 3: Simplify and get the answer.
Let us find the multiplicative inverse of 2/3. The first step is to divide it by 1, which will result in 1/(2/3) = 3/2. Therefore, the reciprocal of 2/3 is 3/2.
Multiplicative Inverse of Complex Numbers
Complex numbers of the form Z = a + ib, such as Z = 3+i√2, where 3 is the real number and i√2 is the imaginary number. The multiplicative inverse of a complex number Z is 1/Z. The reciprocal of this complex number is 1/3+i√2. It can be simplified by multiplying and dividing it by 3i√2, such that: (3i√2)/(3+i√2)(3i√2) ^{ }= (3i√2)/(9i^{2}2) = (3i√2)/(9+2) = (3i√2)/11. Therefore, 3/11  i√2/11 is the multiplicative inverse of 3+i√2.
Follow the steps given below to find the multiplicative inverse of a complex number a + ib:
 Step 1: Write the reciprocal in the form of 1/(a+ib).
 Step 2: Multiply and divide this number by the conjugate of (a+ib).
 Step 3: Apply the following formulae: (a + b)(a  b) = a^{2}  b^{2}, and i^{2} = 1.
 Step 4: Simplify to the lowest form.
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 a 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 and Tricks:
 The multiplicative inverse of a fraction can be obtained by flipping 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 x^{1}.
 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
 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.
☛ Related Topics:
Multiplicative Inverse Examples

Example 1: A pizza is sliced into 8 pieces. Tom keeps 3 slices of the pizza at the counter and leaves the rest on the table for his 3 friends to share. What is the portion that each of his friends gets? Do we apply multiplicative inverse here?
Solution:
Since Tom ate 3 slices out of 8, it implies he ate 3/8^{th} part of the pizza.
The pizza left out = 1  3/8 = 5/8
5/8 to be shared among 3 friends ⇒ 5/8 ÷ 3.
We take the multiplicative inverse of the divisor to simplify the division.
5/8 ÷ 3/1
= 5/8 × 1/3
= 5/24
Answer: Each of Tom's friends will be getting a 5/24 portion of the leftover pizza.

Example 2: The total distance from Mark's home to school is 3/4 of a kilometer. He can ride his cycle 1/3 kilometer in a minute. In how many minutes will he reach his school from home?
Solution:
Total distance from home to school = 3/4 km
Distance covered in a minute = 1/3 km
The time taken to cover the total distance = total distance/distance covered in a minute
= 3/4 ÷ 1/3
The multiplicative inverse of 1/3 is 3.
3/4 × 3 = 9/4 = 2.25 minutes
Answer: Time taken to cover the total distance by Mark is 2.25 minutes.

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 reciprocal and check if the product is 1.
(9/10) × (10/9) = 1.
Answer: The multiplicative inverse of 9/10 is 10/9.
FAQs on Multiplicative Inverse
What is the Meaning of Multiplicative Inverse?
The multiplicative inverse of any number is another number that when multiplied by the original number gives the product as 1. For example, the multiplicative inverse of 2 is 1/2. It is also known as the reciprocal of a number.
What is the Difference between Reciprocal and Multiplicative Inverse?
Reciprocal and multiplicative inverse mean the same in mathematics. When the product of two numbers is 1, then the numbers are said to be reciprocals or multiplicative inverses of each other.
How to Calculate Multiplicative Inverse?
To find the multiplicative inverse of a number, we divide it by 1. So, the multiplicative inverse of x is 1/x.
What is the Multiplicative Inverse of 9?
If we multiply 9 by 1/9, the product is 1. Therefore, the multiplicative inverse of 9 is 1/9.
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 itself.
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 Rational Number?
The multiplicative inverse of a rational number is its reciprocal. The multiplicative inverse of any rational number, x/y, where x,y ≠ 0 is y/x. For example, the multiplicative inverse of 2/3 is 3/2. We just flip the numerator and denominator to find the multiplicative inverse.
What is the 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.
Why do we Use Multiplicative Inverse?
In math, the multiplicative inverse is used to simplify expressions. One major application of multiplicative inverse is while solving division problems. While dividing two numbers, we multiply the reciprocal of the divisor to the dividend. For example, 2 ÷ 4 = 2 × 1/4 = 1/2.
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 Multiplicative Inverse Modulo?
The modular multiplicative inverse of an integer a is another integer x such that the product ax is congruent to 1 with respect to the modulus m. It can be represented as: ax \(\equiv \) 1 (mod m). The multiplicative inverse of “a modulo m” exists if and only if a and m are relatively prime, i.e. gcd(a, m) = 1.
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