The World of Numbers
Fortunes and debts
Use Brahmagupta's idea of fortunes and debts to work with negative numbers.
Brahmagupta asked: if 5 – 5 = 0, what is 3 – 5? He answered from trade and daily life.
He called positive numbers fortunes (dhana): money you have. He called negative numbers debts (ṛiṇa): money you owe. Moving to the left of zero on the number line gives the negative numbers.
The positive whole numbers, their negatives and zero together make the integers, written ℤ, from the German word Zahlen, 'numbers'.
His rules, in money language:
- fortune + fortune = fortune: 5 + 4 = 9;
- debt + debt = debt: (–5) + (–4) = –9 (owe ₹5, borrow ₹4 more, now owe ₹9);
- a fortune or a debt minus zero is unchanged;
- debt × fortune = debt: (–3) × 4 = –12;
- debt × debt = fortune: (–3) × (–4) = 12.
"Why is a debt times a debt a fortune?" Multiplying by a negative means taking away. If someone takes away four of your debts of ₹3 each, you are ₹12 richer. So (–3) × (–4) = +12.
Subtracting a negative works the same way: 10 – (–5) = 15. Cancelling a ₹5 debt leaves you ₹5 better off.
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