What is Multiplicative Inverse?

The word meaning of ‘inverse’ is something opposite in effect. So, the multiplicative inverse of a number is a number that nullifies the impact of the number to identity 1. Thus the multiplicative inverse of a number is a number by which the multiplication results in 1.

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That is, number b is the multiplicative inverse of the number a, if a × b = 1.

For example: Here is a group of 7 dimes.

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To make them into groups of 1 each, we need to divide it by 7. The division is the reverse process of multiplication. Dividing by a number is equivalent to multiplying by the reciprocal of the number.

Thus, 7 ÷7=7 × 1⁄7=1.

Here,1⁄7is called the multiplicative inverse of 7. Similarly, the multiplicative inverse of 13 is1⁄13.

Another word for multiplicative inverse is ‘reciprocal’. It comes from the Latin word ‘reciprocus’ which means returning.

Multiplicative inverse of a natural number

In the given image to make the unit groups of 8 stars, we need to divide it by 8.

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8 ÷8=8 × 18=1

Thus, the multiplicative inverse of 8 is1⁄8.

In general, if a is a natural number, the multiplicative inverse or reciprocal of a is1⁄a.

Multiplicative inverse of a unit fraction

To make a unit fraction, say1⁄4to 1, we need to add it 4 times. Or in other words, multiply1⁄4by 4. Thus, the multiplicative inverse of1⁄4is 4.

In general, the multiplicative inverse or reciprocal of unit fraction1⁄xis x.

Multiplicative inverse of a fraction

By what number should we multiply the fraction3⁄4to get 1?

3⁄4× ?=1

By the properties of equality, if we multiply or divide both sides of an equation by the same number the equation remains true. So, by multiplying the equation by 4 and dividing it by 3 on both sides gives us

3⁄4× ? × 4 ÷3 =1×4 ÷3

3⁄4× ? ×4⁄3=4⁄3

Canceling the common terms:

1× ? =4⁄3

Thus, the multiplicative inverse of3⁄4is4⁄3.

The multiplicative inverse or reciprocal of a fractiona⁄bisb⁄a.

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Fun facts:

The multiplicative inverse of zero is not defined.The multiplicative inverse of 1 is 1 itself.