Hans Gilde’s weblog

Why a negative times a negative is a positive

Posted in mathematics by Hans on September 30, 2009

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 $-3$ really mean?

The answer: A negative value, let’s say $-x$, is exactly a value where $x+(-x)=0$. That’s formally an additive inverse.

Now let’s look at $(-x)(-y)$. If we can believe that $(-x)(-y)=-(-(xy))$ then we’re in good shape.

What does $-(-(xy))$ mean? Per above, it means the number that, when added to $-(xy)$, equals zero. And that is obviously $xy$.

13 Responses

1. Brent said, on September 30, 2009 at 2:27 pm

Ah, nice! And the reason that $(-x)y = -(xy)$ is because of the distributive property: $(-x)y + xy = (-x + x)y = 0y = 0$.

• Hans said, on October 2, 2009 at 8:41 am

Yes, although I would explain it by rearranging multiplication, because many people are comfortable seeing that done:

$(-x)y=(-1x)y=-1xy=-(xy)$

• Brent said, on October 2, 2009 at 9:08 am

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.

• Hans said, on October 2, 2009 at 9:51 am

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.

2. Julian Hyde said, on October 1, 2009 at 2:09 pm

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

• Hans said, on October 2, 2009 at 9:45 am

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.

• Hans said, on October 2, 2009 at 11:35 am

By the way, I’d be interested to hear a price-of-apples explanation for negative-negative multiplication.

3. [...] 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 [...]

4. Richard Veryard said, on October 20, 2009 at 6:14 pm

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

• Hans said, on October 20, 2009 at 7:50 pm

Heh heh. To quote that blog post:

“I thought that was an interesting way of looking at it.”

5. robert white said, on March 17, 2011 at 7:21 am

I JUST WANT2 KNO WHO MADE UP NEGATIVE + NEGATIVE = POSITIVE AND UNTIL I GET MII ANSER IM GETTIN BAK ON DIS WEBSIOTE EVR AGAIN

6. robert white said, on March 17, 2011 at 7:23 am

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

7. addi said, on May 20, 2011 at 9:05 am

this is to complicating!