Question

All the integers are rational numbers. Justify the statement?

Answer 1

Hint: In order to make a relation between rational numbers and integers, we should first know what is rational number and what are integers. Integers are all whole numbers that can be marked on the number line, negative and positive numbers. Whereas, rational numbers are numbers which can be expressed in the form of $\dfrac{p}{q}$ where $q \ne 0$.

Complete step-by-step answer:
To verify the statement that “All the integers are rational numbers”, let’s take an example of any two integers, suppose $ - 3$ and $8$.
For the integer $ - 3$, we can write it as $\dfrac{{ - 3}}{1}$ which is exactly equal to $ - 3$, because we know that any number divided by $1$, returns the same value. And, $\dfrac{{ - 3}}{1}$ is in the form of $\dfrac{p}{q}$ where $1 \ne 0$. That means the integer $ - 3$ is a rational number.
Similarly, for the other number $8$, it can also be written as $\dfrac{8}{1}$, which will give $8$ because of the above reason and we gain a rational number.
And, this shows that every integer can be expressed as a rational number.

Therefore, the statement “All the integers are rational numbers” is absolutely true.

Note: But the vice versa of the statement is not true that is “All rational numbers are integers” is not true. Let’s take an example by taking a rational number $\dfrac{3}{2}$ where $2 \ne 0$. But when we solve/divide it further, we get $\dfrac{3}{2} = 1.5$ that is a decimal not an integer.
Though the reverse of the statement is not true.
Hence, “All the integers are rational numbers” but “All rational numbers are not integers”.