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’.
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’.