Here below is an exchange between me, the perplexed, and Dr Math. Read it, and please if you can show me why -a*-b MUST NOT equal ONLY either +ab or -ab. What else can -a*-b be equal to? >[Question] >Dear Dr Math I long ago wrote you asking how to prove the convention, >- * - = + and you replied with the webpage which illustrates the point >and ends with these remarks. > >" For example, if we adopted the convention that (-1)(-1) = -1, the >distributive property of multiplication wouldn't work for negative >numbers: > > (-1)(1 + -1) = (-1)(1) + (-1)(-1) > > (-1)(0) = -1 + -1 > > 0 = -2 > >As Sherlock Holmes observed, "When you have excluded the impossible, >whatever remains, however improbable, must be the truth." > >Since everything except +1 can be excluded as impossible, it follows >that, however improbable it seems, (-1)(-1) = +1. > >[Difficulty] >Well that just confuses me more, and I cannot get it straight in my >mind, besides there has to be another more cogent way which is not >hard to understand. > >[Thoughts] >I suppose that - a * - b = - ab. > >But - a * - b can only = either + ab or -ab. > >And self evidently + a * -b = -ab > >From the supposition above and the last proposition >-a * -b = + a * -b > >dividing across by -b > >then + a = -a ; this however is false, > >Therefore -a * -b MUST BE = +ab. > >This proof works for me, I hope it works for other people too. > >Thank You This is enough for your own purposes, which is to see why -a * -b is NOT -ab. I wouldn't call it a proof, because it is not really clear that -a * -b has to be only ab or -ab; also, the method of contradiction seems like overkill here. But it's fine if your intent is not to really prove it, but to convince yourself that it makes sense.