Why Does Minus Times Minus Make Plus? // 37

An Age-Old Old-Age Problem The Emperor Scrumptius was born in 35 BC, and died on his birthday in AD 35. What was his age when he died?

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Answer on page 262

Why Does Minus Times Minus Make Plus? When we first meet negative numbers, we are told that multi- plying two negative numbers together makes a positive number, so that, for example, ðÀ2Þ6ðÀ3Þ ¼ þ6. This often seems rather puzzling.

The first point to appreciate is that starting from the usual conventions for arithmetic with positive numbers, we are free to define ðÀ2Þ6ðÀ3Þ to be anything we want. It could be À99, or 127p, if we wished. So the main question is not what is the true value, but what is the sensible value. Several different lines of thought all converge on the same result ­ namely, that ðÀ2Þ6ðÀ3Þ ¼ þ6. I include the þ sign for emphasis.

But why is this sensible? I rather like the interpretation of a negative number as a debt. If my bank account contains £­3, then I owe the bank £3. Suppose that my debt is multiplied by 2 (positive): then it surely becomes a debt of £6. So it makes sense to insist that ðþ2Þ6ðÀ3Þ ¼ À6, and most of us are happy with that. What, though, should ðÀ2Þ6ðÀ3Þ be? Well, if the bank kindly writes off (takes away) two debts of £3 each, I am £6 better off ­ my account has changed exactly as it would if I had deposited £þ6. So in banking terms, we want ðÀ2Þ6ðÀ3Þ to equal þ6.

The second argument is that we can't have both ðþ2Þ6ðÀ3Þ and ðÀ2Þ6ðÀ3Þ equal to þ6. If that were the case, then we could cancel the À3 and deduce that þ2 ¼ À2, which is silly.

The third argument begins by pointing out an unstated assumption in the second one: that the usual laws of arithmetic should remain valid for negative numbers. It proceeds by adding

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