Multiplicative Inverse

The multiplicative inverse of a number is a number which when multiplied with the original number equals one. Here, the original number must never be equal to 0. The multiplicative inverse of a number X is represented as X⁻¹ or 1/X. The multiplicative inverse of a number is also referred to as its reciprocal.

In this article, we shall be learning in detail about the multiplicative inverse. The topic has been simplified by the experts at Vedantu to make your learning experience engaging and fun. So, let’s get started.

Multiplicative Inverse Example

• Multiplicative Inverse of One:

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

• Multiplicative Inverse of Zero:

The multiplicative inverse of zero does not exist. This is because 0xN=0 and 1/0 is undefined.

• Multiplicative Inverse of a Natural Number:

The multiplicative inverse of a natural number X is X⁻¹ or 1/X. For example, the multiplicative inverse of 256 is 1/256 because 256x1/256=1.

• Multiplicative Inverse of a Negative Number:

The multiplicative inverse of a natural number -Y is -Y⁻¹ or 1/-X. For example, the multiplicative inverse of -8 is 1/-8 because -8x1/-8=1.

• Multiplicative Inverse of A Fraction:

The multiplicative inverse of a fraction x/y is y/x. In case the fraction is a unit fraction, then its multiplicative inverse will be the value present in the denominator. For example, the multiplicative inverse of 5/6 is 6/5 and the multiplicative inverse of 1/9 is 9.

\[\frac{4}{7}\times \frac{7}{4}\] = 1

\[\frac{2}{3}\times \frac{3}{2}\] = 1

Multiplicative Inverse Property 

The multiplicative inverse property states that a number P, when multiplied with its multiplicative inverse, gives the result as one.

Px1/P=1

Examples:

2x1/2=1

3/4x 4/3=1

How to Find a Multiplicative Inverse?

The easiest trick to finding the multiplicative of any rational number (except zero) is just flipping the numerator and denominator.

We can also find the multiplicative inverse by using a linear equation as follows. In the below equation y is the unknown multiplicative inverse.

8/9 * y = 1

y= 1/ (8/9)

y=1*(9/8)

y=9/8

Multiplicative Inverse Of A Complex Number 

The multiplicative inverse of any complex number x+yi is 1/(x+yi). In this multiplicative inverse, x and y are rational numbers and i is radical.

In this case, we must always remember to rationalise the multiplicative inverse. Our final answer should not contain any radicals in the denominator.

Rationalisation:

• To rationalise, multiply the numerator and denominator of 1/(x+yi) with (x-yi). This will give you (x-yi)/(x²-(yi)²).

• When we perform this operation using numbers instead of variables, we will get a constant whole number in the denominator and radicals in the numerator. At this step, our multiplicative inverse is rationalised. 

Problems: Find the Multiplicative Inverse

Problem 1: What is the reciprocal of 105/7.

Solution: The reciprocal of 105/7 is 7/105.

If we further simplify. We get,

7/105 = 1/15

So, the reciprocal of 15 is 1/15, because, 15 × 1/15 = 1.

Hence it satisfies the reciprocal property.

Problem 2: Find the reciprocal of Y²

Solution: The reciprocal of y² is 1/y² or y⁻²

Verification:  y² × y⁻² = 1

1 = 1

Did you know?

If X⁻¹ or 1/X is the multiplicative inverse of X, then X is the multiplicative inverse of X⁻¹ or 1/X. This is due to the commutative property of multiplication, which states that the result does not change if the order of numbers is changed.

One is called the multiplicative identity because when multiplied by itself, it gives itself as the result. In other words, 1 is the reciprocal of itself. This can be written as 1x1=1.

Conclusion

Hence multiplicative inverse of a number is a number which when multiplied with the original number equals one. Here, the original number must never be equal to 0. The multiplicative inverse of a number X is represented as X-1 or 1/X. The article discusses all the important and relevant information that will build strong concepts in students.