Question

 Is zero a rational number? Can you write it in the form of ${\displaystyle \frac{p}{q}}$, where p and q are integers and $q\ne 0$?

Answer 1

Hint: Try writing various fractions such that 0 is in the numerator and it is divided by some integer q such that $q\ne 0$. If all such numbers exist then zero is a rational number.

Complete answer:

As we know a rational number is a number which is represented in the form of ${\displaystyle \frac{a}{b}}$ , where $b\ne 0$ and a and b have not any common factors except 1.

Then it can be represented as a fraction of two integers.

For example ${\displaystyle \frac{10}{15}}$ 

As we see this a fraction but not written in lowest form of fraction so first convert this fraction into lowest form of fraction.

$\Rightarrow {\displaystyle \frac{10}{15}}={\displaystyle \frac{2\times 5}{3\times 5}}$

Now cancel out the common terms we have,

$\Rightarrow {\displaystyle \frac{10}{15}}={\displaystyle \frac{2}{3}}$

So this fraction converts into a rational number where ($3\ne 0$) and has no common factors except 1.

Now consider the given number zero (0).

As we know 0 is an integer.

So when we divide 0 by any integer except itself the value is 0.

So 0 is written as ${\displaystyle \frac{0}{q}}={\displaystyle \frac{p}{q}}$ where p and q both are integers and (P = 0, $q\ne 0$) and it is written in lowest form of fraction (i.e. it has no common factors except 1).

Therefore 0 is a rational number.

Note: In the definition of rational numbers such that ${\displaystyle \frac{p}{q}},q\ne 0$. It is defined that q should not be equal to zero because if it is not so, we can have a fraction of the form finite divided by 0, which will be nothing but not-denied. Hence this condition is imposed to take into consideration only defined fractions.