Question

$\dfrac{p}{q}$ is a rational number when:
A. p = 0, q $\ne $ 0
B. p = 0, q = 0
C. p $\ne $ 0, q = 0
D. p = 1, q = 0

Answer 1

Hint: Try to recall the definition of rational and irrational numbers. Think over the fact that 0 is also an integer.

Complete step-by-step answer:
Before moving to the options, let us talk about the definitions of rational numbers followed by irrational numbers.
So, rational numbers are those real numbers that can be written in the form of $\dfrac{p}{q}$ such that both p and q are integers and q $\ne $ 0. In other words, we can say that the numbers which are either terminating or recurring when converted to decimal form are called rational numbers. All the integers fall under this category.
Now, moving to irrational numbers.
Those real numbers which are non-terminating and non-recurring are termed as irrational numbers.
The roots of the numbers which are not perfect squares fall under the category of irrational numbers. $\pi \text{ and }e$ are also the standard examples of irrational numbers.
Now let us move to the solution to the above question.
First consider option (a) p = 0, q $\ne $ 0, here p and q both are integers and q $\ne $ 0 so it is a rational number.
Therefore, option (a) is the correct answer.
However, let us discuss the other options as well.
As for being rational q must not be equal to 0.
Therefore, we can say that option (b), (c), and (d) do not represent a rational number.

Note: We can also think as 0 is an integer and thus a rational number as well. Further 0 can be written as $\dfrac{0}{1}$ as well, making option (a) the correct answer.