So, what is all this stuff good for, anyway?

There was a special event going on today, and only five students were in Mr. Ambrose’s physics class. Instead of covering something that the majority would have to make up, he had a class activity with an air track – demonstrating elastic and inelastic collisions. The velocities of the cars on the track were measured with photogates, and we checked to see if momentum was conserved. We expected to see that a little bit of momentum would be lost, but that it would be close. Unfortunately, that wasn’t the case – the cars gained momentum. So, either we broke a law of physics, or else there was a large source of experimental error. We did find one big source – the air track wasn’t level. The cars were being accelerated by gravity, which messed up the results. Despite not being level, one set of results was within 25%, which isn’t bad for an ad-hoc setup.

In Ms. Colwell’s class, I gave a short presentation on the uses for imaginary numbers. They’re really not that hard to work with, given a little bit of practice, but conceptually they seem a bit remote from ordinary experience. Some of the students seem to be very “concrete” thinkers, and couldn’t quite grasp the meaning of imaginary numbers last week when they were learning about them. This is a common problem, I’d expect. I don’t remember what I thought when I first encountered them – I graduated from high school back in 1989, so it has been a little while – but several very prominent mathematicians had trouble with them. One girl’s comments last week echoed a quote from Leibniz - “I did not understand how… a quantity could be real, when imaginary or impossible numbers were used to express it.” She was having trouble with the idea that you could multiply a complex number by its complex conjugate and end up with something purely real. On the worksheet I gave them, I included a few quotes like this, and asked them for their views on complex numbers. There are no right or wrong answers – but I wanted to see them think about what the numbers mean.

In some contexts, they do have a real physical meaning, and that’s what the presentation was meant to show them. I chose two examples of places where imaginaries have real uses – electronic circuits, and control systems. In electrical engineering, use of imaginary numbers allows inductors and capacitors to be treated like resistors, and circuits to be described by algebraic equations rather than differential equations, which is a major simplification. In control systems, a particular system has a characteristic equation, and the real and imaginary parts of the roots tell different things about the system; the imaginary part tells how oscillatory it is – or as I described it to them, how “bouncy” it is. Obviously, in just a short time, I couldn’t go into any depth, but the goal was to let them know that these things do have uses and even physical meaning.

Next, they move on to powers. They’ve long been familiar with the very basics – squaring, cubing, etc. – but now they get to move on to negative and fractional powers. It’ll be interesting to see what kinds of conceptual difficulties this presents; again, it takes a simple concept that they can picture and extends it into a realm where it isn’t as easy to grasp.

I won’t be in on either Friday or next Monday, since they have a break – the students get a five-day weekend, and the teachers get four days, with a professional development day next Tuesday. So, my next post will most likely be a week from this Friday, when I’m back.