Question

What is the rational number that does not have a reciprocal?
(a) 0
(b) 1
(c) 4
(d) –4

Answer 1

Hint: We start solving the problem by recalling the definition of a rational number as the number that is in the form of $ \dfrac{p}{q} $, where p and q are integers and $ q\ne 0 $. We then write the given integers in options to rational form by taking the denominator as 1. We then take the reciprocal of the obtained rational numbers and check whether those reciprocals satisfy the property of rational numbers to get the required answer.

Complete step by step answer:
According to the problem, we need to find the rational number that doesn’t have a reciprocal.
Let us first recall the definition of a rational number.
We know that the rational number is defined as a number that is in the form of $ \dfrac{p}{q} $ , where p and q are integers and $ q\ne 0 $.
We know that an integer ‘I’ can be written as $ \dfrac{I}{1} $, which is the rational form of an Integer.
Let us write the integers given in each option in the rational form. We then take the reciprocal of them to check whether that reciprocal satisfies the property of rational numbers.
Now, let us check option (a).
We have given integer 0, which can be written as $ \dfrac{0}{1} $ in rational form.
We know that reciprocal of a rational number $ \dfrac{a}{b} $ is defined as $ \dfrac{b}{a} $ .
So, the reciprocal of $ \dfrac{0}{1} $ is $ \dfrac{1}{0} $ , we can see that the denominator is zero which contradicts the property of rational numbers.
 $ \therefore $ Option (a) does not have a reciprocal.
Now, let us check option (b).
We have given integer 1, which can be written as $ \dfrac{1}{1} $ in rational form.
Now, the reciprocal of $ \dfrac{1}{1} $ is $ \dfrac{1}{1} $ , we can see that the obtained reciprocal satisfies the property of rational numbers.
 $ \therefore $ Option (b) does have a reciprocal.
Now, let us check option (c).
We have given integer 4, which can be written as $ \dfrac{4}{1} $ in rational form.
Now, the reciprocal of $ \dfrac{4}{1} $ is $ \dfrac{1}{4} $ , we can see that the obtained reciprocal satisfies the property of rational numbers.
 $ \therefore $ Option (c) does have a reciprocal.
Now, let us check option (d).
We have given integer –4, which can be written as $ \dfrac{-4}{1} $ in rational form.
Now, the reciprocal of $ \dfrac{-4}{1} $ is $ \dfrac{1}{-4} $ , we can see that the obtained reciprocal satisfies the property of rational numbers.
 $ \therefore $ Option (d) does have a reciprocal.
 $ \therefore $ The correct option for the given problem is an option (a).

Note:
 We should know that every integer is part of the system of rational numbers. Whenever we get this type of problem, we first try to recall the definition of the number system given in the problem. We should know a number of the form $ \dfrac{a}{0} $ is not a rational number and is undefined. Similarly, we can expect problems to find the rational number that is equal to its reciprocal.