Multiplicative Inverse - Definition, Formula, Examples and Solved Questions

The multiplicative inverse is a key idea in mathematics that helps us “undo” multiplication. For any non-zero number, its multiplicative inverse is the number which, when multiplied with it, gives the result 1. In other words, every non-zero number has a “reciprocal” that cancels it out during multiplication. This concept also known as the reciprocal and is widely used in fractions, algebraic expressions, and solving equations. In this guide, you will learn the meaning of the multiplicative inverse, how to find it, and how to apply it step by step to simplify fractions and solve mathematical problems easily and correctly.

Table of Contents

• What is Multiplicative Inverse?
• Multiplicative Inverse of Natural Numbers
• Multiplicative Inverse of Fractions
• Multiplicative Inverse of Negative Numbers
• Multiplicative Inverse of Complex numbers
• Difference Between Multiplicative Inverse and Multiplicative Identity
• Solved Examples on Multiplicative Inverse

 

What is Multiplicative Inverse?

The multiplicative inverse of a non-zero number a is the number which, when multiplied by a, gives the product 1. It is written as 1/a or  a−1 . It is also called the reciprocal of a and 1 is the multiplicative identity.
So the multiplicative inverse of 5 is 1/5, because 5 × 1/5 = 1. The multiplicative inverse of 2/3 is 3/2, because 2/3 × 3/2 = 1.
Zero does not have a multiplicative inverse. There is no number you can multiply 0 by to get 1, since 0 × any number = 0.
If 'a' is any non-zero number:
The multiplicative inverse of a = 1/a =  a−1 and a × (1/a) = 1

 

Multiplicative Inverse of Natural Numbers

For any natural number N ( 1, 2, 3, 4, 5 … and so on), the multiplicative inverse is simply 1/N.
Let us look at some examples of multiplicative inverse of natural numbers

 

Multiplicative Inverse of Fractions

To find the multiplicative inverse of a fraction, flip it, i.e, swap the numerator and the denominator. This flipped fraction is the reciprocal.

Multiplicative inverse of p/q = q/p
Verification: (p/q) × (q/p) = pq/qp = 1

• Unit fraction: Unit fractions are fractions with numerator 1, like 1/5. Their inverse is the denominator itself. Multiplicative inverse of 1/n is n.

• Mixed Fractions: To find miltiplicative inverse of mixed fractions convert to an improper fraction first, then flip numerator and denominator.

Let us look at some examples of multiplicative inverse of fractions

 

Multiplicative Inverse of Negative Numbers

Negative numbers follow the same basic rule, but the negative sign stays in the inverse. The reciprocal of a negative number is also negative.

Multiplicative inverse of (−a) = −(1/a) = −1/a.

Let us look at some examples of multiplicative inverses of negative numbers.

 

 

Multiplicative Inverse of Complex numbers

Complex numbers have the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit where i^{2}= −1. To find the multiplicative inverse of an imaginary number we nedd to find its conjugate.

Multiplicative Inverse of (a + ib) = (a − ib) / ( a2+b2) 
In general: 1/(a+bi) × (a−bi)/(a−bi) = (a−bi)/( a2+b2)

For example: Find the multiplicative inverse of (3 + i4) 

Multiply the numerator and denominator by the conjugate (3 − 4i):

1/(3+i4) × (3−i4)/(3−i4) = (3−i4) / (3² + 4²)

= (3−i4) / (9+16) = (3−i4) / 25

Multiplicative Inverse = (3−i4)/25 = 3/25 − i4/25

 

Difference Between Multiplicative Inverse and Multiplicative Identity

 

Solved Examples on Multiplicative Inverse 

Example 1: Are 4/7 and 7/4 multiplicative inverses of each other?
Solution: Multiply: 4/7 × 7/4 = 28/28 = 1
Therefore, 4/7 and 7/4 multiplicative inverses of each other

Example 2: Find the multiplicative inverse of 4\frac{3}{5}.
Solution: Convert: 435 = (4×5 + 3)/5 = 23/5
The multiplicative inverse of 23/5 is 5/23.
Therefore, the multiplicative inverse of  435 is 5/23

Example 3: Find the multiplicative inverse of x².
Solution: Multiplicative inverse of  x2=1/x2=x−2
Verify: x2×x−2=x(2−2)=x0=1

Example 4: Find the multiplicative inverse of (1 + 2i).
Solution: Multiply by the conjugate (1 − 2i) in the numerator and denominator.

= (1 + 2i) ×   (1 − 2i) / (1 − 2i) = (1 − 2i) / (1² + 2²) = (1 − 2i) / 5

The multiplicative inverse of (1 + 2i) is  (1 − 2i) / 5