Definition of Multiplicative Inverse
In math the multiplicative inverse of a number is another number which when multiplied by the original number gives \( 1 \) as the product. If ‘\( ~N ~\)’ is a natural number, the multiplicative inverse of ‘\( ~N~ \)’ will be \( \frac{1}{N} \) or \( N^{-1} \).
So the multiplicative inverse of a number is the reciprocal of that number.
For example, for \( N = 5 \) the reciprocal will be \( \frac{1}{5} \).
Now,
\( 5 \times \frac{1}{5} = 1 \)
So, \( \frac{1}{5} \) is the multiplicative inverse of \( 5 \).
Let us see how to find the multiplicative inverse of different types of numbers.
Multiplicative Inverse of Fractions
Any given fraction can be converted into its reciprocal by simply swapping the values of the numerator and denominator. For instance, the reciprocal of fraction \( \frac{5}{7} \) is \( \frac{7}{5} \).
\( \frac{5}{7} \times \frac{7}{5} = 1 \)
So \( \frac{7}{5} \) is the multiplicative inverse of \( \frac{5}{7} \).
Multiplicative Inverse of Mixed Numbers
In order to find the multiplicative inverse of a mixed number, the mixed number should first be converted to a proper fraction and then follow the same process that we use for fractions. Here’s an example. Find the multiplicative inverse of \( 5\frac{1}{6} \).
\( 5\frac{1}{6} \)
\( =\frac{31}{6} \) [Convert to an improper fraction]
\( \frac{6}{31} \) [Reciprocal]
\( \frac{31}{6} \times \frac{6}{31} = 1 \)
So , \( \frac{6}{31} \) is the multiplicative inverse of \( 5\frac{1}{6} \).
Multiplicative Inverse of Decimal Numbers
The multiplicative inverse of a decimal number is the reciprocal of that number.
Example:
For \( N = 0.5 \)
Reciprocal of \( N = \frac{1}{0.5} \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac{1}{\frac{50}{100}} \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac{100}{50} \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~= 2 \)
So, the multiplicative inverse of \( 0.5 \) is \( 2 \).
Rapid Recall
Solved Multiplicative Inverse Examples
Example 1: Find the multiplicative inverse of the fraction \( \frac{13}{19} \).
Solution:
The given number is \( \frac{13}{19} \) that is a fraction.
So its multiplicative inverse can be obtained by swapping its numerator and denominator, that is: \( \frac{19}{13} \)
We can also find the multiplicative inverse of \( \frac{13}{19} \) by using \( \frac{1}{N} \).
Here \( N = \frac{13}{19} \).
Multiplicative inverse of \( N = \frac{1}{N} \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac{1}{\frac{13}{19}} \) [Substitute \( N = \frac{13}{19} \)]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac{19}{13} \)
Therefore, the multiplicative inverse of the fraction \( \frac{13}{19} \) is \( \frac{19}{13} \).
Example 2: What will be the multiplicative inverse of the following numbers?
a) \( 5 \)
b) \( \frac{3}{7} \)
c) \( 2\frac{1}{5} \)
Solution:
a) The multiplicative inverse of \( 5 \) is \( \frac{1}{5} \).
b) The multiplicative inverse of \( \frac{3}{7} \) is \( \frac{7}{3} \).
c) The number \( 2\frac{1}{5} \) is a mixed fraction. Convert it into an improper fraction which is \( \frac{11}{5} \).
Now, the multiplicative inverse of \( \frac{11}{5} \) is \( \frac{5}{11} \).
Example 3: Sara’s home is \( \frac{3}{5} \) km away from her school. She can cover a distance of \( \frac{1}{3} \) km in a minute. How many minutes will Sara take to go back home from school?
Solution:
Given: The total distance from Sara’s home to school \( = \frac{3}{5} \) km
The distance Sara can cover in a minute \( = \frac{1}{3} \) km
The time taken by Sara to reach her home from school can be determined by dividing total distance by distance covered in one minute, that is:
\( = \frac{\text{Total distance}}{\text{Distance covered in a minute}} \)
\( = \frac{\frac{3}{5}}{\frac{1}{3}} \) [Substitute values]
\( = \frac{3}{5} \times 3 \) [Multiplicative inverse of \( \frac{1}{3} \) is \( 3 \)]
\( = \frac{9}{5} \) [Multiply]
\( = 1.8 \) [Divide]
Therefore Sara takes \( 1.8 \) minutes to get back home from school.
The multiplicative inverse of 0 is undefined as dividing any number by zero is not defined.
The multiplicative inverse property states that the product of any number and its multiplicative inverse will always be 1.
The multiplicative inverse of 1 is 1 itself.