In this chapter we will introduce the concept of reciprocal of rational number with examples.
We will also tell you the method to find the reciprocal of any given rational number.
Multiplicative inverse of rational number
The reciprocal of any rational number is also called multiplicative inverse.
It is produced by switching the position of numerator and denominator of given rational number.
Let a/b is the given rational number.
To get the reciprocal, change the numerator and denominator position.
Hence, the reciprocal is b/a.
Features of reciprocal number
The reciprocal when multiplied with the original rational number produce number 1.
For example;
Let 11/3 be the given rational number.
It’s reciprocal is given as 3/11.
Multiply both the numbers we get;
\mathtt{\Longrightarrow \ \frac{11}{3} \times \frac{3}{11}}\\\ \\ \mathtt{\Longrightarrow 1}
Conclusion
Multiplication of rational number with its reciprocal always results in number 1.
Multiplicative inverse of Negative Rational numbers
The reciprocal of negative rational number is also a negative rational number.
Note that while taking the reciprocal only the position of numerator and denominator change, the sign of the number remains the same.
For example;
Find the reciprocal of -6/11.
Interchanging the numerator and denominator numbers, we get -11/6.
Hence, \mathtt{\frac{-6}{11}} reciprocal is \mathtt{\frac{-11}{6}} .
Does every rational number has multiplicative inverse ?
NO !!
Rational number 0 does not have any multiplicative inverse.
Let me help you understand this.
Number 0 can be written as \mathtt{\frac{0}{1}} .
Now if you interchange the position of numerator & denominator, you will get \mathtt{\frac{1}{0}} .
The any number with 0 in denominator is “not defined” .
\mathtt{\frac{1}{0} \Longrightarrow not\ defined}
Hence, rational number 0 has no reciprocal.
Inverse of Reciprocal
When we take the inverse of reciprocal, we will return back to the original number.
For example;
Let a/b be the given rational number.
The reciprocal of a/b is b/a.
Now if you take reciprocal of b/a, you will get a/b, which is an original number.
Rational number that are equal to their reciprocal
There are only two rational numbers 1 & -1 whose value is equal to its reciprocal.
For example;
Consider rational number 1.
Number 1 can be written as \mathtt{\frac{1}{1}}
For reciprocal, let us interchange the position of numerator and denominator.
We get;
Hence, reciprocal of 1 is the same number 1.
Similar is the case with -1.
Conclusion
The number 1 & -1 have value equal to its reciprocal.
I hope you understood the above concepts. Let us solve some problems for better clarity.
Reciprocal of Rational number – Solved problems
(01) Find the reciprocal of below rational numbers.
(i) 5/3
(ii) -2/7
(iii) -7
(iv) 0
(v) 6/-13
(vi) 100/21
(vii) -11
Solution
We know that reciprocal can be found by switching the position of numerators and denominators.
(i) 5/3
Reciprocal is 3/5
(ii) -2/7
Reciprocal is -7/2
(iii) -7
The given number can be written as -7/1.
So the reciprocal is -1/7.
(iv) 0
Reciprocal of number 0 is not defined.
(v) 6/-13
Reciprocal is -13/6
(vi) 100/21
Reciprocal is 21/100
(vii) -11
The number can be written as -11/1.
Reciprocal is -1/11.
(02) Find the reciprocal of given rational numbers.
(i) \mathtt{\frac{2}{3} \times \frac{7}{9}}
(ii) \mathtt{\frac{-1}{5} \times \frac{-6}{13}}
(iii) \mathtt{\frac{-11}{3} \times \frac{1}{4}}