Monday, October 5, 2009

beautiful negatives

teacher (emphatically): A double negative makes a positive, but a double positive can never make a negative!
student (lazily, from the back of class): yeah, yeah...

I am not sure where this topic started, but The Number Warrior has collected up pointers to most of the "minus times minus is plus" posts that have been cropping up here.

So far I haven't read about what I thought was the usual visualization (but it has probably been mentioned, burried in some comments): multiplying by -1 is a counter-clockwise rotation by 180 around zero.

The problem with this image is that it takes us off the number-line and has us floating in space for a moment. But this is exactly right - the space we are floating in is actually the field of complex numbers, and seeing "multiplying by -1" as "counter clockwise rotation by 180" is the visualization that corresponds to "the most beautiful equation in the world" $e^{i\pi} = -1$. In the same way, a rotation by 90 degrees corresponds to a multiplication by $i$, and multiplying by $i$ twice gets us to the same place as multiplying by -1, which makes sense, since $i^2=-1$. It has to be counter-clockwise in order to give the imaginary axis its usual direction.

Saying that multiplying by -1 corresponds to a 180 degree rotation does not offer an explanation for why a negative times a negative is positive, but it provides a way of seeing it that is consistent with how we visualize other operations (complex multiplication). If you actually said "this is why a negative times a negative is a positive" someone could easily ask, "why does complex multiplication involve rotation at all?" If anything, this visualization is a way of providing a comfortable introduction to the idea that complex multiplication involves a rotation.

Although "negative times negative is positive" seems obvious once you've gotten used to it, putting double negation in a more general context is interesting. If we always expect not not a to equal a, we are relying on the law of the excluded middle, which not everyone accepts all the time.