this post was submitted on 16 Sep 2024
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[–] [email protected] 18 points 3 months ago (2 children)

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.

[–] [email protected] 12 points 3 months ago (1 children)

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.

[–] [email protected] 1 points 3 months ago (1 children)

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.

[–] [email protected] 4 points 3 months ago (1 children)

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.

[–] [email protected] 0 points 3 months ago

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'.

[–] [email protected] 7 points 3 months ago (1 children)

Ever since I went down a particularly nasty rabbit hole and came out with a tenuous grasp on quaternions, imaginary numbers started feeling very simple, familiar and logical.

[–] [email protected] 2 points 3 months ago (1 children)

Yeah. The thing that made me "get" quaternions was thinking about clocks. The hands move around in a 2d plane. You can represent the tips position with just x,y. However the axis that they rotate around is the z axis.

To do a n dimensional rotation you need a n+1 dimensional axis. So to do a 3D rotation you need a 4D axis. This is bassicly a quat.

You can use trig to get there in parts but it requires you to be careful to keep your planes distinct. If your planes get parallel you get gimbal lock. This never happens when working with quats.

[–] [email protected] 2 points 3 months ago

I still maintain that quats are the closest you can get to an actual lovecraftian horror in real life. I mean, they were carved into a stone bridge by a crazy mathematician in a fit of madness. How more lovecraftian can you get?