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.
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.
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}⁄_{7} is called the multiplicative inverse of 7. Similarly, the multiplicative inverse of 13 is ^{1}⁄_{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.
8 ÷8=8 × 18=1
Thus, the multiplicative inverse of 8 is ^{1}⁄_{8}.
In general, if a is a natural number, the multiplicative inverse or reciprocal of a is ^{1}⁄_{a}.
Multiplicative inverse of a unit fraction
To make a unit fraction, say ^{1}⁄4 to 1, we need to add it 4 times. Or in other words, multiply ^{1}⁄_{4} by 4. Thus, the multiplicative inverse of ^{1}⁄_{4} is 4.
In general, the multiplicative inverse or reciprocal of unit fraction ^{1}⁄_{x} is x.
Multiplicative inverse of a fraction
By what number should we multiply the fraction ^{3}⁄_{4} to 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 of ^{3}⁄_{4} is ^{4}⁄_{3}.
The multiplicative inverse or reciprocal of a fraction ^{a}⁄_{b} is ^{b}⁄_{a}.
Fun facts:
