# Why does -1 * -1 = +1 ?

I have been asked this by perplexed teachers, parents and kids:

Why did them silly mathematicians decide that the multiplication of two negative numbers is positive? What’s the logic?

The first thing is to acknowledge that this is a convention. We could define -1*-1 to be whatever we want. But there is good reason to decide it should be +1.

Let’s try to calculate:

-1 * ( 1 + -1)

Well, -1 is the negative of +1, which mean, by definition:

-1 + 1 = 0

So, -1 * (1+ -1) = -1 * 0

and multiplication by zero must be zero, so we have:

-1 * (1+ -1) = -1 * 0 = 0

But, if we open the parenthesis, we have:

-1 * (1 + -1) = -1*1 + -1*-1

now, multiplication by 1 is doing nothing, so -1*1 = -1, so we get:

-1 + -1*-1 = -1*1 + -1*-1 = -1*(1+ -1) = -1*0 = 0

hence, -1 is the negative of -1*-1, so the latter has to be +1.

What have we used? We used the definition of zero and 1, we used the definition of the negative of a number, and we used the law of distributivity (by which we can open the parenthesis).

We could decide, say, that -1*-1 = -1, but then we’d have to abandon one of the principles mentioned above, and that would make arithmetic rather unpleasant.

# Another proof that 0=2

Time for a math riddle. Haven’t done these in a while. Well, haven’t done any in this blog, when I come to think about it. OK. That was enough thinking. Let’s get down to biusiness.

Take a point on the complex plane. Take one which is on the unit circle:

$z=e^{i\theta}$

Now replace $\theta$ with $\phi = \frac{\theta}{2\pi}$. We get:

$z=e^{i\theta}=e^{2\pi\i\phi}$

Which by the simple laws of arithmetic gives us:

$z=e^{2\pi\i\phi}=\left(e^{2\pi\i}\right)^\phi=1^\phi=1$

So every point on the unit circle is 1!

As a simple consequence we get 1=-1. Add 1 on both sides and get 2=0.

QED.

Can you spot the error?