So, I’m a PhD student in CS, and I was feeling a bit embarrassed about being rusty on my calculus and various aspects of high-school algebra and analysis-adjacent stuff. I haven’t needed to think about complex numbers for almost fifteen years now. I definitely had a decent grasp of them at the time, but a few things like the interpretation of multiplication and the meaning of the complex conjugate always felt more like “dumb accidents” shaking out of the algebra of the multiplication, at least the way I was taught in high school.

I’ve also spent some time working through an abstract algebra textbook on my own, when I’m not super busy with grad school. I like group theory a whole lot, but working with fairly straightforward groups when I make my own examples, I never really truly grasped why group actions and representation theory are a big deal. The correspondences I could work out were fairly obvious ones.

Anyway, I was laying awake in bed a few nights ago, and was trying to remember the geometric interpretation of the multiplication of complex numbers. The next day I sat down in a coffee shop with a pencil and a notebook and started playing around. Once I noted the similarity between multiplying by i and a specific linear transform, I realized I could construct an isomorphism between C under multiplication and a specific subgroup of GL(2,R), and was delighted to discover I’d worked out the matrix-based definition of complex numbers on my own (I had never seen or heard of it!).

Thinking about multiplying complex numbers in terms of composing linear transforms (in particular the Abelian subgroup of GL(2,R) consisting of rotation and scaling) immediately cleared up a whole bunch of things for me. The intuition of the relationship between the algebraic and geometric view of the complex conjugate shakes out immediately, which made me happy, and made the definition of the multiplicative inverse seem much less arbitrary than 19-year-old me felt it was. Also, I was like “ohhh that’s what representation theory is about.”

At another level, I learned not to be embarrassed about reviewing seemingly elementary stuff, because revisiting them from a more sophisticated perspective is surprisingly fruitful!