Why a negative times a negative is a positive
A few blogs have discussed an intuitive explanation of why multiplying two negatives produces a positive. That would be an explanation which makes sense to a non-mathematician.
The most intuitive argument from those blogs, to my mind, was from The Math Less Traveled and talks about a negative as a reflection about zero on the number line (image of this at The Number Warrior).
Here is how I’d explain it:
The first question to be asked is: What is a negative value? Like what does 
The answer: A negative value, let’s say 
Now let’s look at 
What does 
Ah, nice! And the reason that
is because of the distributive property: 
.
Yes, although I would explain it by rearranging multiplication, because many people are comfortable seeing that done:
Ah, good point. Although, of course, fundamentally the reason that (-x) = (-1)x is because of the distributive property! =) But I agree this is a nicer way to explain it.
Ha
My explanations are derived sort of like a stand up comic derives good jokes (which is by watching to see what makes people laugh). Having explained such things to many people, I observe which explanations produce a satisfied look, versus a confused or pretending-to-understand look.
Thus I don’t often explain things in the order that a good book would prove them.
It makes sense to me, but then I’m a mathematician. To convince a non-mathematician you have to get to the end of their arguments without tripping any of their switches. That means you need to make your argument without using any equations, axioms, or fancy words like ‘distributive’.
I think the best strategy is to appeal to people’s real world experience. People are capable of quite complex reasoning when are carrying out economic transactions, so my argument would invariably come down to something involving the price of apples.
I remember when I was about six I had serious doubts about whether negative numbers really existed. My father, who was an accountant, told me they didn’t really exist but they were a convenient way of dealing with things that were ‘borrowed’. I had the same doubts when, at fifteen, I first encountered imaginary numbers; then I remembered that negative numbers were just a convenience (and, for that matter, the rationals, reals, and even zero) and my fears went away.
Julian
I suppose that there are different audiences.
Some people may, as you were, be skeptical about whether certain numbers really exist. They may be looking for some philosophy or practical examples.
Others may wonder whether this property of multiplying negatives can really be seen from more basic properties of numbers, or if “that’s just how it works”.
My own ah-ha moment with numbers came when I heard that numbers, like words, don’t exist. They’re just tools we use to describe the world. So a negative or imaginary number doesn’t have to “exist” any more than “tension so think you can cut it with a knife” does.
By the way, I’d be interested to hear a price-of-apples explanation for negative-negative multiplication.
[...] a minus equal to a plus’? Math Less Traveled: Minus times minus is plus Hans Gilde’s weblog: Why a negative times a negative is a positive mathrecreation: beautiful negatives code monk: Minus times a minus is a [...]
The reason minus times minus makes a plus is because it’s a rotation. Dan Greene goes off into complex numbers, but also describes a student who finally got negative numbers.
http://exponentialcurve.blogspot.com/2006/12/next-lesson-intro-to-complex-numbers.html
Heh heh. To quote that blog post:
“I thought that was an interesting way of looking at it.”
I JUST WANT2 KNO WHO MADE UP NEGATIVE + NEGATIVE = POSITIVE AND UNTIL I GET MII ANSER IM GETTIN BAK ON DIS WEBSIOTE EVR AGAIN
I JUST WANT2 KNO WHO MADE UP NEGATIVE + NEGATIVE = POSITIVE AND UNTIL I GET MII ANSER IM NOT GETTIN BAK ON DIS WEBSITE EVR AGAIN
this is to complicating!