Q1.
Can you explain why we need q ≠ 0 in the definition of a rational number?
Answer
Because division by zero has no answer — and the symbol p/q means “the number which, multiplied by q, gives p”.
let p/q = x, so x × q = p
if q = 0 then x × 0 = p
but x × 0 = 0 for every x
if q = 0 then x × 0 = p
but x × 0 = 0 for every x
Now two separate disasters appear:
- If p ≠ 0, say 5/0, we would need x × 0 = 5. No number does that, so no value exists.
- If p = 0, that is 0/0, then x × 0 = 0 is true for every x. So every number would qualify, and the symbol names nothing in particular.
Why it happens: either there is no candidate or there are infinitely many, and in both cases p/q fails to name a single point on the number line. Allowing q = 0 would also wreck arithmetic: from 1 × 0 = 2 × 0 we could “cancel” the 0 and conclude 1 = 2.
Tip: the same restriction reappears in the division rule a/b ÷ c/d = ad/bc, which the chapter states only for b ≠ 0, d ≠ 0 and c ≠ 0 — the extra c ≠ 0 is there because c is about to become a denominator.