system
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human
How do imaginary numbers play into the real/physical world?

I was reading up on imaginary numbers and a source stated that they started out purely imaginary and for mathematical purposes and some stuff with Euler's equation (that I don't really understand except that it has to do with complex planes or something), but how does it govern physical things, like springs or circuits or other things with some sort of resonance. If it's just imaginary, how is it able to dictate/show non-imaginary things?
gpt
Real numbers are just as imaginary as Imaginary Numbers. It's all just made up stuff that can be used to talk about things very precisely. Someone just chose to call imaginary numbers "imaginary", it was a made up thing that just stuck that has absolutely nothing to do with the "realness" of these numbers.

Complex numbers (which have real and imaginary parts) describe 2D rotations. Just as regular numbers can describe length, scales, and things, complex numbers can describe rotations. The number i is a good way to talk about 90 degree rotations. The number sqrt(2)/2 + i\*sqrt(2)/2 is a good way to talk about 45 degree rotations. The number -i is a good way to talk about 270 degree rotations. In general, complex numbers can be used to talk about scaling things and rotating things simultaneously. Ie, the number 1+i is good at doing 45 degree rotations and scaling things up by a factor of sqrt(2).

This means that anything that rotates or oscillates can be described using complex (ie imaginary) numbers. So alternating currents are great at being described in this way, because they oscillate. Springs as well. Also things in Quantum Mechanics can be described in this way too. We often break up normal things into oscillatory components, and this is best done using complex numbers because of this. So if you want to find out the key tones that your voice uses, then it goes through math that uses complex numbers in order to pick this stuff out.

Much more generally, the complex numbers "complete" arithmetic, it has all the numbers we need. So in more abstract stuff, you absolutely cannot avoid it. You may be able to get around having to use complex numbers in the other stuff (it's more messy and more work, but you can do it), but you eventually can't avoid it. For instance, examining the behavior of systems that use feedback to adjust going forward, you basically have to use complex numbers because the way to analyze them forces you to need all types of numbers.

Don't be confused by the word "real" that we use for "Real numbers", they're just as made up, useful, and imaginary as "Imaginary numbers".