Can you explain why we need $q \neq 0$ in the definition of a rational number $\frac{p}{q}$? [Page No. 47]

Because $\frac{p}{q}$ means $p$ divided by $q$ , and division by zero has no meaning.

Suppose $q = 0$ and we wanted $\frac{p}{0}$ to equal some number $n$ . By the meaning of division that would require $p = n \times 0$ . But $n \times 0 = 0$ for every $n$ . So, if $p$ is not 0, no value of $n$ works (no answer at all); and if $p$ is 0, then every number works (no single answer). Either way there is no sensible value, so we must forbid $q = 0$ .