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.
In the arithmetic of integers, a negative number represents a debt, while a positive number represents a fortune. When we multiply two negative numbers, we are essentially looking at the removal of a debt.
For example, if someone takes away four of your debts, where each debt is worth ₹3 (represented as -3), the multiplication is represented as (-3) × (-4). Taking away a debt makes you richer because you no longer owe that amount.
Since four debts of ₹3 total ₹12, removing them effectively increases your wealth by ₹12. Thus, the product of two debts becomes a fortune, proving that a negative times a negative equals a positive: (-3) × (-4) = +12.
Explanation
The textbook uses Brahmagupta's concept of fortunes (positive numbers) and debts (negative numbers) to explain integer multiplication. A negative times a negative is a positive because multiplying by a negative represents the removal or taking away of a debt. When a debt is removed, one's financial standing improves, resulting in a positive fortune. The hint provided in the text illustrates this perfectly: taking away 4 debts of ₹3 each (-3 × -4) leaves you ₹12 richer (+12).