Q1.
Why does a negative times a negative equal a positive? Think of it in terms of action and debt. If a negative number represents a debt, then multiplying by a negative number represents the removal of that debt. (Hint: If someone takes away (–) four of your debts that are each worth ₹3 (that is, –3), you are effectively ₹12 richer! Therefore, (–3) × (– 4) = +12.)
Answer
Because removing a debt makes you richer — and “removing” is what the second negative sign does.
one debt of ₹3 → −3
taking away 4 such debts → (−3) × (−4)
your position improves by ₹12 → +12
taking away 4 such debts → (−3) × (−4)
your position improves by ₹12 → +12
The book's picture is convincing, but the rule is also forced by the arithmetic we already accept. Watch what the distributive law demands:
(−3) × [4 + (−4)] = (−3) × 0 = 0
so (−3) × 4 + (−3) × (−4) = 0
we already know (−3) × 4 = −12
therefore −12 + (−3) × (−4) = 0
hence (−3) × (−4) = +12
so (−3) × 4 + (−3) × (−4) = 0
we already know (−3) × 4 = −12
therefore −12 + (−3) × (−4) = 0
hence (−3) × (−4) = +12
Why it happens: once we insist that a × 0 = 0 and that p(q + r) = pq + pr must keep holding for negative numbers, the value of (−3) × (−4) is no longer a matter of choice. Any other answer would break the distributive law. So “minus times minus is plus” is not a convention invented to be neat — it is the only value that keeps the whole system consistent.
Check it yourself: run the same argument on (−1) × (−1). From (−1)(1 + (−1)) = 0 you get −1 + (−1)(−1) = 0, so (−1)(−1) = 1.