The Arithmetic of Integers

Transcript

The Arithmetic of Integers Brahmagupta’s rules for integers act exactly like money. Let’s look at them one by one Rule 1 - A fortune plus a fortune is a fortune Basically Positive + Positive = Positive A FORTUNE PLUS A FORTUNE IS A FORTUNE (Positive + Positive = Positive) Brahmagupta gave explicit rules for adding and multiplying these integers, which we wil still use exactly as he wrote them over 1,300 years ago: Two Money Bags: + Total: ₹900 (positive fortune) Two Gold Coins: Total: 9 Gold Coins (positive fortune)Rule 2 - A debt plus a debt is a debt Basically Negative + Negative = Negative Example: (–5) + (–4) = –9. If you owe ₹ 5 and borrow ₹ 4 more, you owe ₹ 9 A DEBT PLUS A DEBT IS A DEBT (Negative + Negative Negative) Two IOU Notes: Total: ₹90 (negative debt) Digging Pits: Total Depth: 9 feet (negative position)Rule 3 - A fortune minus zero is a fortune, a debt minus zero is a debt Basically Number – Zero = Number Doesn’t matter if Number is positive or negative ZERO'S EFFECT: NO CHANGE IN FORTUNE OR DEBT Fortune - Zero = Fortune (Positive - Zero Positive) You take away ₹0, so your fortune stays same! Debt Zero Debt (Negative - Zero Negative) You don't climb at all, so your debt depth is same!Rule 4 - The product of a debt and a fortune is a debt: Basically Negative × Positive = Negative Or Positive × Negative = Negative (order doesn’t matter) THE PRODUCT OF A DEBT AND A FORTUNE IS A DEBT (Negative * Positive = Negative) Repeat a Debt: Three debts of ₹10 total to ₹30 debt. Debt Over Time: Losing 5 coins every day for 4 days means 20 coins lost.Rule 5 - The product of two debts is a fortune Basically Negative × Negative = Positive Think of it in terms of action and debt. 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 THE PRODUCT OF TWO DEBTS IS A FORTUNE (Negative * Negative Positive) *The trickiest one. Canceling An IOU: Someone removes (negative 1 multiplier) your debt of 220. You have effectively gained a fortune of 220 ! Removing 4 Debts of ₹3 (original numbers): Imagine you are relieved of 4 sets of -₹3 debts. That's a net fortune of gained!