this post was submitted on 12 Aug 2023
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Up to the introduction of quantum mechanics imaginary numbers where only ever a theoretical tool and any calculation in electromagnetism, mechanics or even relativity can be done without them.
Also, any measurement you can make will always result in real numbers because there is no logical interpretation for imaginary measurements (a speed of 2+i m/s doesnt really make sense)
Bro, are you not aware of the Fourier transform?!? Electrical impedance? Wtf???
I said that any calculation in electrodynamics CAN be done without imaginary numbers, I never said that it would be the most common or convenient way of doing things.
If you use a different form of solution to maxwells equations, electrical impedance can totally be expressed as just another real property. Fourier transform also is not necessary to solve maxwells equations or any other physical systems. It just might make it significantly easier and more convenient.
Obviously imaginary numbers existed and where used way before quantum mechanics was a thing but they werent technically necessary in physics because they never appeared in the equations of fundamental theories (Maxwells equations, general relativity, newtonian mechanics)
Yes, and one CAN integrate by taking paper cuttings and dispense entirely with the idea of infinity.
I was just trying to make an argument that imaginary numbers were technically not necessary and thus it makes historical sense that they werent seen as something 'real'. Im not trying to get people to stop using them ;)
Eh, this is not worth your time or mine to argue about. Let’s move on. Also, I take your point.
Imaginary numbers are indeed poorly named. They are not much more imaginary than members of ℝ.
It's all fine... except for the part where reality has a √-1 component.
Well, in AC circuits, having √3̅+√-̅1̅ A of current makes as much sense as having 2 amps with a 30° phase shift. It's just easier notation for calculations - Cartesian coordinates for what would otherwise be polar.
That's BS notation. If you want Cartesian, just use 3i+1j, no need for some impossible √-1 that you then redefine some operations for, just so it becomes orthogonal to R.
You might want to look up geometric algebra for a better geometric interpretation of complex numbers than the complex plane with a "real" and "imaginary" axis
The nice thing about 𝑖 = √-̅1̅ is that you don’t need to redefine any operations for it, ℐ𝓂 is “automatically” orthogonal to ℛℯ.
Yeah but quantum mechanics is just magic.