Can you explain why we need ≠ in the definition of a rational number?

In the definition of a rational number $\dfrac{p}{q}$, we need q ≠ 0 because division by zero is undefined in mathematics.

If we allow q = 0, then $\dfrac{p}{0}$ would have no meaningful value, since there is no number that gives p when multiplied by 0 (any number multiplied by 0 gives 0).

For example :

⇒ $\dfrac{5}{0}$ would mean "what number multiplied by 0 gives 5?"

But, since any number × 0 = 0, no such number exists.

Also, $\dfrac{0}{0}$ is indeterminate as 0 × any number = 0.

Hence, the condition q ≠ 0 is necessary so that the rational number $\dfrac{p}{q}$ has a well-defined value.