Regardless of the school you went to, if you did final year maths then you’ll relate to the following anecdote. Your stories might differ in specifics, but this is pretty much the archetype of what goes on in the classroom. (No doubt it also rings a bell if you’re a teacher.)
- In a maths class, I asserted that the intelligence quotient of the classmate I was talking to was zero.
- Said classmate replied that my IQ was zero POINT zero.
- Me: “Yours is zero point zero times ten to the minus INFINITY.”
- Him: “Yours is MINUS zero point zero times ten to the minus infinity.”
- Me: “Yours is minus zero point zero times ten to the minus infinity *i*.”
- Finally he concluded that my IQ was just i, a wholly imaginary quantity, and that at least his was real.
I tell this story as a gentle way to raise the topic of imaginary (and complex) numbers, which I’ve been musing on lately. I think my latest bout of musings were triggered by reading this, which I read via a chain of links that I can’t remember now, but would have included this.
The subject of my muse was why most attempts to explain the applications of complex numbers strike me as unilluminating.
Classroom banter doesn’t count as an application. Neither do pretty fractal patterns — they’re lovely, but like a game, the rules don’t have to mean anything. Neither is it remarkable that one abstraction can be used in the service of another abstraction, as in esoteric results from advanced mathematics. What counts are situations where it’s fruitful to take a quantity derived from real world measurements, and relate it to the square root of a negative number.
I never studied any applications of complex numbers myself, having not done any physics beyond first year university. But I’ve occasionally listened to, or read, people trying to explain how imaginary numbers are useful in their fields, and I almost invariably have no idea what they’re talking about. They might assert that imaginary numbers are useful for such-and-such, but in a way that leaves me no wiser about what said context has to do with squaring something and getting a negative result.
So it occurred to me that perhaps most explanations fail because they show the audience an imaginary-shaped peg without first cutting an imaginary-shaped hole. Let me illustrate the point with reference to negative numbers, and then relate it to imaginary/complex ones later on.
Once upon a time, negative numbers were widely regarded as an affront to common sense. Quantities less than nothing? Crazy. But applications soon became apparent, and those applications tended to fit very particular patterns. One can sensibly answer questions like: if you’re looking for a scenario that lends itself to being modelled with negative numbers, then what sort of characteristics should you look for?
In my metaphor, the answer to that question is the negative-shaped hole, and a particular scenario which has those characteristics is the negative-shaped peg. And here’s the answer: you look for a scenario where every quantity has a corresponding opposite. The opposite of having a particular speed is having the same speed in the opposite direction. The opposite of being so high off the ground is being so far under it (and, preferably, upside down). The opposite of having so much money is being so far in debt. And so on.
If you have a scenario in which various quantities have corresponding opposites, then you have a scenario that lends itself to being modelled with the aid of negative numbers. There’s your hole, and you’ll find any number of pegs to fit. Note also that negative numbers don’t measure anything mysterious; simply flip the coordinate system around and they become positive.
If we carry this pedagogical approach over to the imaginary case, the question becomes as follows. Suppose you’re looking for a scenario that lends itself to being modelled with imaginary numbers. What characteristics would you look for?
And for a pretty good answer, consider that in the world of imaginary numbers, there exists a function such that f(f(x)) = -x. In fact there are two such functions (multiplying by i and multiplying by -i), but the point is made by translating that formula into everyday language. It means there is something you can do to a quantity twice and always get back the opposite of what you started with.
That’s probably the simplest imaginary-shaped hole you can find. If you have a scenario in which there’s something you can do to a quantity twice and always get back the opposite of the quantity you started with, then you have a scenario that lends itself to being modelled with the aid of imaginary numbers. As for pegs, no problem. Try anything involving rotations and waves.
In the case of rotations, the thing you can do twice and end up with the opposite of what you started with is to rotate a quarter of the way around. In the case of waves, it’s to wait for a quarter of a wavelength to go by. Given such tools, if a physicist’s eyes roll at an elementary blunder in this blog post, you can calculate what that physicist will be looking at by the end of it. You can almost as simply calculate the same thing using old-fashioned real numbers, of course, but with a good i you can fit into a single formula what otherwise requires an algorithm. And you can show that the principle neatly carries over to finer intervals — less than a quarter of the whole.
I haven’t mentioned Euler’s equation — an oft-celebrated result of a very clever procedure that makes it meaningful to raise numbers to imaginary powers, and is typically taught in first year mathematics at university. The relevant consequence is that i can be re-written as e to the power of πi/2, and hence that i to the power of some number n is the same as e to the power of nπi/2 [note: π is pi — but my blog happens to use a font that doesn’t draw it very well]. Re-writing everything as exponents of e may make things more complicated for algebra, but simplifies things for calculus. But because I haven’t done any calculus since the 1990s (and wasn’t much good at it then), I won’t go there.
(Exponents of e are also neater if you measure angles in radians rather than quarter-revolutions, but I’m pretty sure the reason mathematicians measure angles in radians is because it leads to trigonometric functions having neat power series and hence Euler’s equation etc, so the causality is kind of inverted there.)
I’m definitely not going to mention quantum physics. If you’re interested, try a blog by someone who understands it. You didn’t come here for that.