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Last updated on 17 September 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.
A number’s reciprocal is its multiplicative inverse. The multiplicative inverse of a number 'n' is written as 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 × 1/n = 1
For example, let’s take the number 4
According to the multiplicative inverse property:
n = 4
1/n = 1/4
Therefore, n × 1/n = 1 → 4 × 1/4 = 1
Hence, multiplying 4 by its reciprocal gives 1 as the final result.
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 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
7/1 × 1/7 = 7/7 = 1
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 should be 1. Similarly, the product of a negative integer with its reciprocal should also be 1.
Let the negative integer be -n. The multiplicative inverse for -n will be 1/-n
Multiplying -n × 1/(-n) gives the product as 1
For example, take -9. The multiplicative inverse for -9 is 1/-9
Multiplying -9 × 1/-9 gives the product as 1
Let m/n be the fraction. The multiplicative inverse of m/n will be n/m.
Here, both m and n cannot be zero.
Multiplying m/n × n/m will give the product as 1.
For example, take the fraction 5/10. The multiplicative inverse will be 10/5
Multiplying 5/10 × 10/5 gives 50/50 = 1. Both numerator and denominator simplify to 1.
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 1/2 as the mixed fraction.
Converting 4 1/2 into an improper fraction results in 9/2
Since the improper fraction is 9/2, its multiplicative inverse is 2/9
Multiplying the fractions 9/2 and 2/9 will give 1 as the product:
9/2 × 2/9 = 1
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 1/0, but it is not defined.
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 1/Z, which is 1/(a + ib)
For example, take 3 + i√2
In 3 + i√2:
3 is the real part, and i√2 is the imaginary part.
The multiplicative inverse of 3 + i√2 is 1/ 3 + i√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 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)
→ 1/(a + ib) × (a - ib)/(a - ib) = (a - ib)/(a² + b²)
Using the identity (a + ib)(a − ib) = a² − (ib)² and i² = −1, we solve the denominator as
a2 + b2
Step 3: Simplify to the simplest form
The modular multiplicative inverse of the number q is another number p, such that ‘q × p’ will always be 1 (remainder) when divided by ‘x’
It is represented as q × p ≡ 1 (mod x)
This means that ‘x’ can completely divide q × p - 1
For example, let’s find the modular inverse of 3 mod 7
We write the expression as q × p ≡ 1 (mod x)
Here, q is 3 and x is 7. We have to find p.
Therefore, the expression is written as 3 × p ≡ 1 (mod 7)
This means that the result should be when 3 is multiplied by some number ‘p’ when divided by 7.
Now we need to check for values of p. Start with number 1.
Here, p is 5 because 3 × 5 = 15. When 15 is divided by 7, the remainder is 1.
Therefore, the modular multiplicative inverse of 3 mod 7 is 5
To master finding the multiplicative inverse, follow the given tips and tricks.
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:
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.
Find the multiplicative inverse of -25
1/-25
For a negative number, its multiplicative inverse will always be in the form 1/-n, where -n is the negative number. Therefore, the multiplicative inverse of -25 is 1/-25
What is the multiplicative inverse of 1 2/3 ?
3/5
First, convert the mixed fraction 1 2/3 into an improper fraction. Convert 1 2/3 to improper fraction: (1 × 3 + 2)/3 = 5/3. Then reciprocal: 3/5.
What is the modular multiplicative inverse of 3 mod 11?
4 is the modular multiplicative inverse of 3 mod 11
The expression that satisfies the modular inverse is:
a × b ≡ 1 (mod x).
Here, we need to find ‘b’. Applying the values of ‘a’ and ‘x’ in the expression, we get
a × b ≡ 1 (mod x) as 3 × b = 1 (mod 11)
Here, the value of ‘b’ is 4.
3×4=12≡1 (mod 11)
Therefore, the modular multiplicative inverse of 3 mod 11 is 4
What is the multiplicative inverse of 25%?
The multiplicative inverse of 25% is 4
First, convert 25% to an improper fraction.
25% = 25100 = 14
Since the improper fraction is 14, the multiplicative inverse is 4
Find the multiplicative inverse of 500 and convert it into a decimal
Multiplicative inverse is 1/500. The decimal form of 1/500 = 0.002.
Multiplicative inverse is the reciprocal of the given number. Therefore, the multiplicative inverse of 500 is 1/500 and its decimal form is 0.0002
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.
: She loves to read number jokes and games.