| They're the same thing in the sense they have the same roots. The most confusing thing about complex numbers is the language. First you're told negative numbers can't have roots, then you're told they so can too, but you have to call the roots "complex" or "imaginary." This sets up cognitive dissonance which can be harder to deal with than the math. (What even is an "imaginary number"? What are those words supposed to mean?) In reality complex numbers are a way of moving from the number line to a number circle. (Which eventually generalises to a 3-sphere when you get to quaternions.) That's all they are. Instead of linear arithmetic - which is about combining magnitudes in one dimension - you can now do arithmetic that combines magnitudes with rotations. The extra dimension makes it possible to solve equations with solutions that don't exist on the basic number line. It also makes it easier to do calculations that combine magnitude with phase - which includes pretty much anything that rotates or processes linear combinations of sine waves, and which a straight vector tuple can't handle. If someone had told me this when I was learning complex numbers the cognitive dissonance wouldn't have hurt quite as much. |
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]