[mokus]

I found so much of math I had learned previously in the “it’s weird but this works if you take it on faith” sense was suddenly blatantly obvious after learning some abstract algebra. I wish I had learned that stuff way earlier.

In the case of complex numbers, I find the “paradox” disappears when you think of it in terms of fields abstractly. To put it maybe a bit overly simply - instead of focusing on the idea of “square roots of negative numbers”, instead step back and consider that number-like operations make sense for things that aren’t numbers at all in the traditional sense. One particularly useful example is 2d vectors, which you can add in the usual sense and “multiply” in polar form by multiplying “r” and adding “theta”. It turns out that these vectors with these operations act a LOT like numbers, and it also turns out that that weird multiply operation is actually super useful. One easy interpretation is combined scale and rotation transforms, with “multiplication” implementing composition.

Once you do that, it also turns out that solving equations like “what transformation composed with itself equals a 2x scaling with 180° rotation?” also make sense (i.e. “solve x^2 = -2”), and when you solve polynomials in this new system you get more solutions than you did for regular numbers. And that the thing you just invented IS the field of “complex numbers”.

[Sorry for the verbosity and probably poor organization, I’m in a bit of a hurry IRL and didn’t have time to edit it down. I did edit a bit for clarity and to fix typos, etc., though]