I was taught this as a rule to memorise and it has bothered me since. Every explanation I find is either "two wrongs make a right", which is not mathematics, or a debt analogy that stops working the moment I push on it.
Is there a reason it has to be positive, or was it simply chosen because it is convenient?
Accepted Answer
It was chosen, but not freely. It is forced if you want the distributive law to keep working, and that law is worth more than the sign rule.
Take a(b + c) = ab + ac, and set a = −1, b = 1, c = −1:
−1 × (1 + (−1)) = −1 × 1 + (−1) × (−1)
The left side is −1 × 0, which is 0. The right side is −1 plus whatever (−1)(−1) is.
So 0 = −1 + (−1)(−1), which means (−1)(−1) must be 1. Anything else and the distributive law breaks.
That is the real answer. You could define it the other way, but then arithmetic stops being consistent and almost every formula you know fails.
The debt analogy is where most people come unstuck, so it is worth saying why it fails rather than repeating it.
"Taking away a debt makes you richer" explains subtracting a negative. It does not explain multiplying two negatives, because there is no second debt being taken away, and people quietly slide from one to the other.
Analogies are good for remembering a rule you already trust. They are not good for justifying it, and this is the clearest case of that I know.
Pattern version, if you want to see it rather than prove it. Look at −3 times each of 3, 2, 1, 0: you get −9, −6, −3, 0, going up by 3 each time. Carry on to −1 and −2 and the pattern gives 3 and 6.
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