About Reciprocal in Math
Defining Reciprocal
The reciprocal of a number is 1 upon the number, that is, the reciprocal of “\(a\) is \(\frac{1}{a}\)”. It is also called the multiplicative inverse. Reciprocal is also written as the number raised to the power -1, that is, the reciprocal of “\(x\) is \(x^{-1}\) or \(\frac{1}{x}\)”.
Reciprocal of a fraction is found by interchanging the numerator and denominator of the fraction.
For example, the reciprocal of \(\frac{5}{4}\) is \(\frac{4}{5}\).
Reciprocal of a mixed number is obtained by first converting the mixed number into an improper fraction and then interchanging the numerator and denominator.
For example, to find the reciprocal of \(3\frac{2}{5}.\)
\(3\frac{2}{5}\) is first written as \(\frac{17}{5}\). Then the reciprocal of \(\frac{17}{5}\) is \(\frac{5}{17}\).
The reciprocal of a negative number will be a negative number, that is, the reciprocal of -5 is -\(\frac{1}{5}\).
Now, the reciprocal of a decimal is obtained by taking inverse of the decimal, that is, the reciprocal of 1.25 is \(\frac{1}{1.25}\).
Or, we can convert the decimal into a fraction, and then find its reciprocal.
So,
Therefore, the reciprocal of 1.25 is \(\frac{1}{1.25}\) or \(\frac{4}{5}\) or 0.8.
Solved Reciprocal Examples
Example 1: Find the reciprocal of 5 and 8
Solution:
Given numbers are 5 and 8.
We know that the reciprocal of a number is 1 upon the number itself.
Reciprocal of 5 is \(\frac{1}{5}\)
Reciprocal of 8 is \(\frac{1}{8}\)
Example 2: Find the reciprocal of \(5\frac{1}{3}\)
Solution:
\(5\frac{1}{3}\) is a mixed number.
Firstly convert the mixed number into an improper fraction and then find its reciprocal.
\(5\frac{1}{3}=\frac{\left(5\ \times\ 3\right)\ +\ 1}{3}\)
= \(\frac{16}{3}\)
Reciprocal of \(\frac{16}{3}\) is \(\frac{3}{16}\).
Therefore the reciprocal of \(5\frac{1}{3}\) is \(\frac{3}{16}\).
Example 3: Find the value of \(\frac{5}{2}+\frac{7}{4}-\frac{1}{3}\) and then find its reciprocal.
Solution:
\(\frac{5}{2}+\frac{7}{4}-\frac{1}{3}=\frac{5\times6}{2\times6}+\frac{7\times3}{4\times3}-\frac{1\times4}{3\times4}\)
= \(\frac{30}{12}+\frac{21}{12}-\frac{4}{12}\)
= \(\frac{30\ +\ 21\ -\ 4}{12}\)
= \(\frac{47}{12}\)
The reciprocal of \(\frac{47}{12}\) is \(\frac{12}{47}\).
Therefore, the reciprocal of \(\frac{5}{2}\ +\frac{7}{4}\ -\frac{1}{3}\) is \(\frac{12}{47}\) .
Division by zero is not defined. Hence, reciprocal of 0 = \(\frac{1}{0}\) which cannot be defined. Hence the reciprocal of zero is undefined.
Real numbers include positive numbers, zero and negative numbers.
We can have the reciprocal of positive numbers and negative numbers, but the reciprocal for zero is not defined.
When a number is multiplied by its reciprocal, we get the value as 1.
Example: \(9\ \times\ \frac{1}{9}\ =\ 1\)