I have a confession to make. Even though I graduated from MIT a long time ago, I’ve never really “gotten” imaginary numbers — till now. I understood the arithmetic of them, but I could never get a real sense of why. This gave me a lot of trouble in my electrical engineering courses. Today I was reading Edgar Rice Burroughs’ space opera Pirates of Venus, which has this amusing line: “I saw that argument was useless and said no more; there is no use arguing with a man who can multiply anything by the square root of minus one.” That got me thinking about my old frustration.
I’d gotten as far as thinking that imaginary numbers would be less confusing if we called them “orthogonal numbers” instead; they’re no more imaginary than real numbers, but it’s useful to conceive of them as a number line that’s orthogonal to the primary number line. A complex number is a vector in the two-dimensional plane, and addition and subtraction are no problem. But why should the square root of -1 be i? The standard answer is so that you can solve equations that otherwise would have no solution. But why do they need solutions? There’s no solution to dividing a number by zero, and we don’t invent a new class of numbers just so we can do that. (Infinities aren’t numbers.)
Doing some web searches, I came across a page which provided the key: Multiplication by i really means rotation by 90°! Look at the results of repeated multiplication by i on the complex plane and it works. Multiply 1 by i and you get i (1 on the vertical axis). Multiply again by i and you get -1 on the horizontal axis. Keep going and you get –i (-1 on the vertical axis) and then you come a full 360° back to 1.
But wait! If you want to rotate just 45°, shouldn’t that be multiplication by i/2? That gives you a result at 90°, not 45°. But then I realized that multiplying by i/2 doesn’t mean half as much rotation; it means half as much of the rotated result. To rotate 45°, you must need to multiply by the square root of i. What is this square root? Using ordinary algebra, it comes out as (i + 1) / sqrt(2); and using ordinary trigonometry, this is the 45° position on the circle.
It’s annoying, though, that this scheme requires an exponential progression of complex numbers to describe a linear progression of rotation. So the log of a complex number ought to be useful, since it describes a quantity which is proportional to the amount of rotation. What base log would work most cleanly?
Well, we can relate a complex number on the unit circle to an angle θ by the formula cos(θ) + i sin(θ). That’s basic trigonometry; read 1 as x and i as y if it helps. Then θ is the log that we’re looking for. Radians are easier to work with than degrees, so the log of a full circle’s rotation should come to 2π, or any integer multiple of it. It seems that logs of this kind won’t be unique. So we want a base, let’s call it x, such that xθi = cos(θ) + i sin(θ). Go through some messy calculations, and x comes out to be about 2.71828. Mathematicians like to call it e. So it all comes together, and the strange-looking equation “e2πi = 1″ makes physical sense as the description of a full circle of rotation.
It’s really not that (pardon the pun) complex an idea, and I’m sure some of you are flabbergasted that I never got it, but somehow it had never been explained to me properly. (And some of you were lost by the second paragraph. That’s OK; it balances out.) I suspect a lot of people would get it more easily if imaginary numbers were called “orthogonal numbers” or almost anything else, and if multiplication by an imaginary number were explained as rotation rather than as satisfying a mathematician’s need for completeness. Then you’ve got a tool for dealing with real physical systems like the interaction of wave phases, rather than a toy.