Question

Is 0 a rational number? Can you write it in the form \[\dfrac{p}{q}\] where \[p\] and \[q\] are integers and \[q \ne 0\] ?

Answer 1

Hint: First, we will use the concept of a rational number to determine whether 0 is a rational number or not. If 0 is found out to be a rational number, then we will try to express it in \[\dfrac{p}{q}\] form.

Complete step-by-step answer:
First, let us have a look at the definition of a rational number.
A rational number is a number that can be expressed in the form of fraction \[\dfrac{p}{q}\] where \[p\] and \[q\] are integers and \[q \ne 0\]. Integer \[q\] can’t be equal to 0 because a fraction with denominator 0 is not defined. Some examples of rational numbers are \[\dfrac{1}{2}\], \[\dfrac{3}{2}\], \[\dfrac{{ - 5}}{6}\], \[\dfrac{{ - 4}}{3}\], \[\dfrac{{11}}{5}\], etc.
Now, let us verify if 0 satisfies the above definition :-
If we substitute 0 in place of \[p\] in the given definition, the result will be 0. \[q\] can be any other integer except 0.
0 can be represented in the form of a rational number in multiple ways. Some of the examples are \[\dfrac{0}{1}\],\[\dfrac{0}{2}\], \[\dfrac{0}{{ - 3}}\], \[\dfrac{0}{{ - 5}}\], \[\dfrac{0}{{11}}\], \[\dfrac{0}{{ - 16}}\], etc.
\[\therefore\] We can conclude that 0 is a rational number.

Note: Here, it is given that \[p\] and \[q\] are integers. Integers are the numbers which can be negative, positive or zero. But, as we have to write 0 in \[\dfrac{p}{q}\] form therefore \[q\] can be both a positive integer and a negative integer but \[q\] cannot be equal to 0 because \[\dfrac{0}{0}\] is not defined. We should also note that representation of 0 as a rational number is not unique as \[\dfrac{0}{1} = \dfrac{0}{2} = \dfrac{0}{{ - 3}} = \dfrac{0}{{ - 5}} = 0\] .