Multiplicative Inverse: Definition, Properties & Examples

The multiplicative inverse is defined as the number that, when multiplied with another number, will equal the number one.​ It is also known as the reciprocal of the given value.

• The multiplicative inverse is always the identity of the value used for the computation of numbers. 
• The word reciprocal is derived from the latin word reciprocus.
• It is used to simplify all the mathematical representations. 
• When you multiply an original number with a reciprocal of a number, the original number must never be equal to 0. 
• For instance, the multiplicative inverse of a number X is represented as X^-1 or 1/X. 
• It can be used for various mathematical domains and numbers.
• To calculate the multiplicative inverse of a real number then, divide the required number by 1.

Read More: Probability

Key Terms: Multiplicative inverse, Complex Numbers, Natural Numbers, Reciprocal, Modulo of multiplicative inverse, Domains, Real Numbers

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What is Multiplicative Inverse?

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A multiplicative inverse of a number is another number, which, when multiplied by 1, gives the original number itself. In simple words, the multiplicative inverse is 1/n for any number (n).

• A multiplicative inverse and the reciprocal of a number both are the same.
• It is a technique in mathematics that helps simplify complex mathematical problems.
• It is a pair of numbers that is when multiplied with another number is similar to 1.

It can be more explained with the following example:

• If a number 8 is given so the multiplicative inverse of the number 8 is 1/8.
• It is because when you multiply 8 & 1/8 with each other the answer would be 1.
• So the meaning of a multiplicative inverse is a number that contradicts the consequence of a number for identifying 1.

The video below explains this:

Quadratic Equations Detailed Video Explanation:

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Multiplicative Inverse Examples 

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Various examples of multiplicative inverse are as follows:

Multiplicative Inverse of One

The multiplicative inverse of one is one. This is only because 1x1=1.

Multiplicative Inverse of Zero

The multiplicative inverse of zero does not exist because 0 x N=0 and 1/0 are undefined.

Multiplicative Inverse of a Natural Number

The multiplicative inverse of a natural number X is X^-1 or 1/X. For instance, the multiplicative inverse of 237 is 1/237 because 237 x 1/237 = 1.

Multiplicative Inverse of a Negative Number

The multiplicative inverse of a negative number -Y is -Y^-1 or 1/-X. For instance, the multiplicative inverse of -7 is 1/-7 because -7 x 1/-7 = 1.

Multiplicative Inverse of a Fraction

Here, if the fraction is a unit fraction, then its multiplicative inverse will be the value that is present in the denominator. For instance, the multiplicative inverse of 5/7 is 7/5, and the multiplicative inverse of 1/8 is 8.

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Properties of Multiplicative Inverse

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The property of multiplicative inverse clarifies that if you multiply a number with its reciprocal or the multiplicative inverse, you will always get 1 as an answer.

• Consider a number p then P × 1 / P = 1.

For Examples:

• 4 × 1 / 4 =1

• 4 / 5 × 5 / 4 =1

Read More: Baye’s Theorem

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How to Find the Multiplicative Inverse?

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To calculate the multiplicative inverse of a number, try to solve some examples.

Unit fraction

If a fraction has a numerator 1, it will be called a unit fraction. It can be calculated as follows:

• Consider multiplication of  1/9 by 9, the product is 1.
• 1 / 9 × 9=1.

Mixed fraction

To calculate the multiplicative inverse of a mixed fraction, first, you need to convert the mixed one into a proper fraction and determine its reciprocal.

• For Examples: calculate the mixed fraction of 4 ¹/₃
• First, we need to convert it into a proper fraction
• 4 × 3 + 1=13 / 3
• Now we need to find there reciprocal of 13 / 3
• So it will be 3 / 13.

and, multiplicative inverse of 4 ¹/₃ is 3 ¹/₃

Read More: Maxima and Minima

Natural numbers

The natural numbers are all the counting numbers that start from the number 1. The multiplicative inverse of the natural numbers is 1 / a. 

• It can be found out as 1 / a × a=1
• For Examples: if we need to find out the multiplicative inverse of 7,9
• The multiplicative inverse of 1 / 7 × 7 = 1
• The multiplicative inverse of 1 / 9 × 9 = 1.

Thus, it is clear that the reciprocal of all the natural numbers is 1.

Complex numbers

The sum of an original & an imaginary saber is called a Complex number. It is expressed in a different form that is: a+ci. Where ‘a is an original number, on the other side, ‘ci’ is a complex number.

• So it is clear that a complex number is a combination of two numbers is a complex number.
• But one number should be original & the other one should be an imaginary one.
• Examples: z = 3 – 2i
• z=3+2i
• z² =32+-2i²
• 9 + 4 =13

Read More: Trigonometry Values

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Modulo of Multiplicative Inverse

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A modular is a part of arithmetic; the modular of the multiplicative inverse of numbers is a number x such that the product bx is congruent to 1 concerning the modular m. This can be written as s: bx =1 mod m.

• It is the standard way of writing it, which shows that m evenly divides the quantity box-1. 
• The modular multiplicative inverse is different from the other multiplicative inverses.

It can be understood by two different methods, which are as follows:

First method

• Consider two integers numbers' m'. Now, find out the modular multiplicative inverse of 'b' under the modulo 'm'.
• The Modular of an integer' x' such that
• ax=1mod m

Second method

• If b & m are coprime, then you can also find multiplicative inverse modulo with the help of The Extended Euclidean Algorithm.
• By the Extended Euclidean Algorithm, it takes numbers to say 'b' & 'a' find their gcd & also find 'x' & 'y'.
• Now, to find the Reciprocal of 'b' under 'm', we need to substitute b=m in the above formula. As we know, m & b are relatively prime, so the value of gcd would be taken as 1.
• ax + my =1
• And if we take modulo on both sides, All we get is
• ax + my = 1 mod m

Read More: Perimeter and Area of Circle

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Solved Examples of Multiplicative Inverse

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Example 1: What is the multiplicative inverse of -5?

Solution: As we know, the reciprocal of -5 is -1 / 5

• Number x Multiplicative inverse = 1
• (-5) x (-1/5) = 1
• 1 = 1

Therefore, the multiplicative inverse of -5 is -1 / 5.

Example 2: What is the reciprocal of 7 / 53?

Solution: Multiplicative inverse of 7/53 = (1 / 7) / (1 / 53)

• Required inverse is : 53 / 7
• Number x Multiplicative inverse = 1
• (7/53) x (53/7) = 1

Therefore, the solution is 53/7.

Example 3: Find the reciprocal of x². 

Solution: The reciprocal of x² is 1 / x² or x^-2

• x² × x^-2 = 1
• 1 = 1

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Things to Remember

• The multiplicative inverse for number 1 is one, but for number 0, it can not be defined.
• Multiplicative inverse is also known as reciprocal inverse.
• To calculate the multiplicative inverse for a mixed fraction, you need to convert it into a proper fraction.
• Always remember when a number is multiplied by its multiplicative inverse, the result is always 1.
• The multiplicative inverse is also called a reciprocal number.

Read More: Differential Equations