# Starting with transformations for complex numbers

I’m searching for a way to teach complex numbers that starts with transformations and then brings in Algebra from there. What I’m envisioning is that we’ll become fluent in the arithmetic of transformations with rotations, translations and dilations and that the kicker will be that arithmetic with i gives a shockingly succinct way to say all that.
In other words, in most approaches that I’ve seen, the kicker with complex numbers is that we learn to be more imaginative and open to weirdness. I like that, but I think I would like it more if the focus was on the shocking efficacy of algebra for representing rotations and dilations.
I think that this is going to be a mess with the Regents class that I teach, simply because I’m not investing in all the tools that I would need to actually pull this off. Today I had everything going great. Kids were developing the regular 1, i, -1, -i cycle with 90 degree rotations represented by (0, 1)^n. It was all great, and right then I felt this pressure to introduce i, and then I did, and in a lot of ways it undercut what I had spent the last half hour building.
What I’d really like to do is to murk around in right triangle trigonometry for a while before introducing i. I’d like to define a class of transformations T_(a,b) that uses dilations and rotations to map the point (1, 0) to (a, b). I’d like to develop a general rule for the composition of these transformations, and then, only then, I’d like to say: “Wait. What if we made the following formal moves: i will represent (0, 1), (1, 0) will just be 1, and, in general, (a, b) will be a + bi?”
Then, I think, we could have a revalatory moment for these kids, where everything just falls together in an absurdly clean way…as long as you’re willing to swallow that i^2 = -1.
At that point, I don’t think that I’d have to ask kids to swallow the idea that i^2 = -1. I think that they’d demand it.
Anyway, that’s what I’d like to try to do. I’m not sure that I’ll be able to pull it off, but I’d like to give it a shot.