Multiplicative Inverse
2026-02-28 01:26 Diff
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Last updated on December 10, 2025

The multiplicative inverse of a number is another number that, when multiplied with the original number, always results in 1. In this article, we will be discussing multiplicative inverse and its applications.

What is Multiplicative Inverse?

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A number’s reciprocal is its multiplicative inverse. The multiplicative inverse of a number 'n' is written as \(\frac{1}{n} \). Here, 1 becomes the numerator, and the number becomes the denominator.

When a number is multiplied by its reciprocal, the result will always be 1. 

Multiplying the number ‘n’ with its reciprocal: \(n \times \frac{1}{n} = 1 \)
For example, let’s take the number 4

According to the multiplicative inverse property:

n =  4

\(\frac{1}{n} = \frac{1}{4} \)

Therefore, \(n \times \frac{1}{n} = 1 \;\Rightarrow\; 4 \times \frac{1}{4} = 1 \) 

Hence, multiplying 4 by its reciprocal gives 1 as the final result.

Multiplicative Inverse Property

The multiplicative inverse property says that when you multiply a number by its reciprocal, the result will always be 1. In the image, you can see that 1/n is the reciprocal of n, and their product gives 1.

For example, imagine you have five apples, and you want to split them into five groups of 1 apple each. To do that, you divide the apples by 5. Dividing a number by itself is the same as multiplying it by its reciprocal. So, 5 ÷ 5 = 5 × ⅕ = 1.

Here, ⅕ is the multiplicative inverse of 5.

How to Find Multiplicative Inverse ?

The multiplicative inverse is found by dividing 1 by the given number. Now let’s learn how to find the multiplicative inverse. Follow the steps given below:

Step 1: Write the given number as a fraction. For example, if the number is 7, write it as \(\frac{7}{1} \).

Step 2: Now switch the numerator and denominator. So 71 will become 17. Now the numerator is 1 and the denominator is 7

Step 3: Now multiply both the fractions obtained in Step 1 and Step 2 to make sure that the product is 1

\(\frac{7}{1} \times \frac{1}{7} = \frac{7}{7} = 1 \)

Hence, the multiplicative inverse of 7 is \(\frac {1}{7}\) and their product is always 1.

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How to find Multiplicative Inverse of Integers ?

An integer is a number that can be positive or negative. It can never be a decimal or fraction.

For positive integers, the product of the number and its reciprocal is always 1. Similarly, the product of a negative integer with its reciprocal is also 1. 

Let the negative integer be -n.

The multiplicative inverse for -n will be \(\frac{1}{-n} \)


Multiplying \((-n) \times \frac{1}{-n} \) gives the product as 1

For example, take -9. The multiplicative inverse for \(-9 \text{ is } \frac{1}{-9} \)

Multiplying \(-9 \text{ is } \frac{1}{-9} \) gives the product of 1. 

How to Find Multiplicative Inverse of a Fraction ?

Let \(\frac{m}{n}\) be the fraction. The multiplicative inverse of m/n will be \(\frac{n}{m}\). 

Here, both m and n cannot be zero.

Multiplying \(\frac{m}{n} \times \frac{n}{m} \) will give the product as 1.

For example, take the fraction \(\frac{5}{10} \). The multiplicative inverse will be \(\frac{10}{5} \)

Multiplying \(\frac{5}{10} \times \frac{10}{5} \) gives \(\frac{50}{50} = 1 \). Both numerator and denominator simplify to 1.

How to Find Multiplicative Inverse of a Mixed Fraction ?

To find the multiplicative inverse of a mixed fraction, first convert the given mixed fraction into an improper fraction. After converting, change the position of the fraction upside down to get the multiplicative inverse.

Let’s take 4\(\frac{1}{2} \) as the mixed fraction.

Converting 4 \(\frac{1}{2} \) into an improper fraction results in \(\frac{9}{2} \)

Since the improper fraction is \(\frac{9}{2} \), its multiplicative inverse is \(\frac{2}{9} \)

Multiplying the fractions \(\frac{9}{2} \) and \(\frac{2}{9} \) will give 1 as the product:

\(\frac{9}{2} \times \frac{2}{9} = 1 \)

How to find Multiplicative Inverse of 0 ?

The multiplicative inverse of a number is its reciprocal, which when multiplied by the number gives 1.

The multiplicative inverse of 0 is not possible because multiplying any number by 0 will always be 0. The multiplicative inverse of 0 is written as \(\frac{1}{0} \), but it is not defined.

How to Find Multiplicative Inverse of Complex Numbers ?

Complex numbers are made of two parts, a real part (any number) and an imaginary part (i). Let Z be a complex number, where \(Z = a + ib \).

Here, ‘a’ is the real part, and ‘ib’ is the imaginary part. The multiplicative inverse of Z is \(\frac{1}{Z} \), which is 

For example, take 3 + i√2

In \(3 + i\sqrt{2} \):

3 is the real part, and \(i\sqrt{2} \) is the imaginary part.

The multiplicative inverse of \(3 + i\sqrt{2} \) is \(\frac{1}{3 + i\sqrt{2}} \)

Now, to find the multiplicative inverse of complex numbers, follow the steps given below:

Step 1:  Let the complex number \(Z = a + ib \). The reciprocal form of the given complex will be \(\frac{1}{a + ib} \)

Step 2: Take the conjugate of \((a+ib)\), which is \((a-ib)\). We take the conjugate to remove the imaginary part by multiplying and dividing the inverse with \((a-ib) \)

→ \(\frac{1}{a + ib} \times \frac{a - ib}{a - ib} = \frac{a - ib}{a^2 + b^2} \)

Using the identity \((a + ib)(a - ib) = a^2 - (ib)^2 \quad \text{and} \quad i^2 = -1 \), we solve the denominator as
\(a^2 + b^2\)

Step 3: Simplify to the simplest form

\(\frac {1}{z} = \frac {a}{a^2 + b^2} - \frac {b}{a^2 + b^2}\)

What is Modular Multiplicative Inverse ?

The modular multiplicative inverse of a number p is another number x such that their product px leaves a remainder of 1 when dividend by m. This is written as,

\(px ≡ 1 (mod m)\)

This means that m perfectly divides px - 1.

A modular multiplicative inverse of p exists only when p and m are coprime, that is, gcd (p, m) = 1.
 

Type Multiplicative Inverse Example Natural number For a natural number x, its inverse is 1/x The multiplicative inverse of 4 is 1/4 Integer For any non-zero integer x, the inverse is 1/x The multiplicative inverse of -4 is -1/4 Fraction For a fraction x/y where, x and y are non-zero, the inverse is y/x The multiplicative inverse of 2/7 is 7/2 Unit Fraction For a unit fraction 1/x, the inverse is x The multiplicative inverse of 1/20 is 20

Important Notes

  • A number’s multiplicative inverse is also known as its reciprocal.
     
  • When you multiply a number by its reciprocal, the result is always 1.
     
  • Only non-zero numbers have a multiplicative inverse.
     
  • The multiplicative inverse of a fraction is found by swapping its numerator and denominator.
     
  • For a unit fraction like 1/x, the multiplicative inverse is simply x.

Tips and Tricks for Multiplicative Inverse

To master finding the multiplicative inverse, follow the given tips and tricks.
 

  • To find the inverse, switch the numerator and denominator. For the number 3, the multiplicative inverse will be 1/3.
     
  • If you are asked to find the inverse of 0, remember that it can never be written in its reciprocal form.
     
  • Always convert the mixed fraction into an improper fraction before finding the multiplicative inverse.
     
  • Check your final answer by multiplication: After finding the multiplicative inverse, multiply it by the original number. The result should always be 1.
     
  • Use in algebraic expressions carefully: When finding the inverse of variables or expressions, treat them the same way as numbers, flip the numerator and denominator.
     
  • Parents can use everyday examples, such as dividing snacks, to help children understand reciprocals more easily.
     
  • Teachers can use visual objects and classroom activities to help students practice finding multiplicative inverses.
     
  • Children should think of the multiplicative inverse as the partner that, when multiplied, makes 1.

Common Mistakes and How to Avoid Them in Multiplicative Inverse

Children might find it confusing while solving problems using the multiplicative inverse, leading to incorrect results. We will now discuss some mistakes a child can make, also the solutions to overcome them.

Real-Life Applications of Multiplicative Inverse

Multiplicative inverse is not just used in daily life but also in professional fields. Given below are some real-life applications of the multiplicative inverse:

  • We use multiplicative inverse in cryptography to enable secure encryption and decryption.
     
  • In finance, multiplicative inverse helps calculate the reciprocal exchange rates for currency conversions.
     
  • It is also used in scaling to adjust the given quantities proportionally.
     
  • Measurements for the conversion of units are done with the help of multiplicative inverse.
     
  • In computer graphics and physics simulations to calculate inverse transformations, such as undoing rotations, scaling, or other linear operations we apply multiplicative inverse.

Download Worksheets

Problem 1

Find the multiplicative inverse of -25?

Okay, lets begin

\(\frac{1}{-25} \)

Explanation

For a negative number, its multiplicative inverse will always be in the form \(\frac{1}{-n} \), where -n is the negative number. Therefore, the multiplicative inverse of -25 is \(\frac{1}{-25} \).

Well explained 👍

Problem 2

What is the multiplicative inverse of 1 2/3 ?

Okay, lets begin

\(\frac{3}{5} \)

Explanation

First, convert the mixed fraction 1 \(\frac{2}{3} \) into an improper fraction. Convert 1 2/3 to improper fraction: \(\frac{1 \times 3 + 2}{3} = \frac{5}{3} \). Then reciprocal: \(\frac{3}{5} \).

Well explained 👍

Problem 3

What is the modular multiplicative inverse of 3 mod 11?

Okay, lets begin

4 is the modular multiplicative inverse of 3 mod 11.

Explanation

The expression that satisfies the modular inverse is: 

\(a \times b \equiv 1 \pmod{x} \). 

Here, we need to find ‘b’. Applying the values of ‘a’ and ‘x’ in the expression, we get

\(a \times b \equiv 1 \pmod{x} \quad \text{as} \quad 3 \times b \equiv 1 \pmod{11} \)

Here, the value of ‘b’ is 4.

\(3 \times 4 = 12 \equiv 1 \pmod{11} \)

Therefore, the modular multiplicative inverse of 3 mod 11 is 4.

Well explained 👍

Problem 4

What is the multiplicative inverse of 25%?

Okay, lets begin

The multiplicative inverse of 25% is 4.

Explanation

First, convert 25% to an improper fraction.

25% = 25100 = 14

Since the improper fraction is 14, the multiplicative inverse is 4.

Well explained 👍

Problem 5

Find the multiplicative inverse of 500 and convert it into a decimal.

Okay, lets begin

Multiplicative inverse is \(\frac{1}{500} \). The decimal form of \(\frac{1}{500} = 0.002 \).

Explanation

Multiplicative inverse is the reciprocal of the given number. Therefore, the multiplicative inverse of 500 is \(\frac{1}{500} \) and its decimal form is 0.0002.

Well explained 👍

FAQs on Multiplicative Inverse

1.What is the multiplicative inverse of 1?

The multiplicative inverse of 1 will always be 1

2.What is the inverse of a decimal number?

First, convert the given decimal into an improper fraction and then take its reciprocal to get the multiplicative inverse.

3.How can we verify the multiplicative inverse?

If the given number and its reciprocal give the product as 1, we can verify that the inverse is correct.

4.Write the multiplicative inverse of the unit fraction.

Unit fractions are those with 1 as the numerator. Therefore, the inverse of 8 is \(\frac{1}{8} \)

5.Why does zero not have a multiplicative inverse?

It is because no number multiplied by 0 gives 1 as the product.

6.How can I help my child understand the concept of multiplicative inverse at home?

Start with simple numbers, show that the multiplicative inverse of a number x is \(\frac {1}{x}\) because multiplying them gives 1. For example, \(2 \times \frac{1}{2} = 1 \).
Use visual aids, like number lines or coins, and play interactive games using flash cards and quizzes.

7.How does understanding multiplicative inverse improve my child’s overall math skills?

Understanding multiplicative inverse strengthens a child’s skills in fractions, decimals, and division, making calculations easier. It also builds a foundation for algebra and problem-solving in real-life math situations.

Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.