this post was submitted on 16 Sep 2024
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Imaginary numbers are the proof that even in mathematics you can discover stuff even though you don't understand what you have found. Complex numbers encode rotation.
Yup. When you have a circuit that is not purely resistive the inductive or capacitive load causes the voltage and current to not be in phase. It looks like ohms law is being violated. However the missing part of the energy is in the imaginary component to be returned latter.
But that is hardly a 'natural occurence' of complex numbers - it just turned out that they were useful to represent the special case of harmonic solutions because of their relationship with trig functions.
No. It's more what the previous poster said about encoding rotation. It's just not a xyz axes. It's current, charge, flux as axes. The trig is how you collapse the 3d system into a 2d or 1d projection. You lose some information but it's more useful from a spefic reference.
Without complex numbers you can't properly represent the information.
The natural representation would be the transient solution u(t) or i(t). Harmonic solutions are merely a special case, for which it turned out complex numbers were useful (because of the way they can represent rotation). They certainly serve a purpose there, but imo this is not an instance of 'complex numbers appearing in nature'.