When a minus times a minus equals a plus

We all know from school that multiplying two negative numbers together gives a positive number, but can you think of a common-sense example of this rule in action that would convince someone who asked why?

I got asked why recently and I couldn’t.

Neither could anyone I asked. Plenty of examples involving mirrors and vectors etc. but nothing that didn’t sound rather like illusion and trickery. Nothing convincing.

So – after some thought, here’s an example that convinced the person who asked me. (Well, they say they’re convinced-ish, but I think that’s about as good as it’s going to get!)

This example is about getting two everyday, dependent variables that we can set a zero point on both and thus deal with the positive and negative values in both. Imagine I have a big bucket of sweets, and I have been giving you ten sweets a month for years.

How many more or less sweets do you have, six months from now?

Intuitively, you have 60 more sweets, and we can calculate that because you get +10 sweets/month and we want to know how many you have in 6 months;

10 x 6 = 60. (plus x plus = plus)

How about me? How many more or less sweets do I have, six months from now?

We know that I have 60 sweets less, because you have 60 sweets more. We can calculate this because I get -10 sweets/month;

-10 x 6 = -60. (minus x plus = minus)

That’s the easy ones done.

How many more or less sweets did you have, six months ago?

It should be easy to convince that if you have 60 sweets more in six months’ time, then you had 60 sweets less six months ago. We can calculate it using the same 10 sweets/month, but -6 months to go back in time.

10 x -6 = -60 (plus x minus = minus)

Finally, how many more or less sweets did I have, 6 months ago?

I still get -10 sweetsmonth. Just because we’re considering the past, you don’t start giving them to me or anything. In the case above, we used -6 to represent ‘six months ago’. So…

-10 x -6 = +60 (minus x minus = plus)

Which gives us the intuitively correct answer, that if I give you 10 sweets a month then I had 60 more sweets, six months ago.

It’s a tough one to argue with, because the answers are pretty obvious. Do you have a better way to explain why minus times minus equals a plus?