I'm being called back in:
I should have left it "no proof trivial enough to satisfy me", it seems when justifiably hindered by that situation you will resort to an appeal to authority.
Look man, math is hard. I'm not writing a math textbook in a forum post to 'satisfy you'. I've linked you to videos and articles (including super-intro level videos) for you to get educated on the proofs and the history behind those proofs. It's not appeal to authority. I am not saying 'so and so said so so therefore it's right.' I'm saying there are copious resources at many levels of competency to build up your knowledge. Here is yet another one:
If you're too lazy to try or you have a cognitive/religious resistance to it (see my remarks below) - that's on you. I'm only going to spoonfeed you so much.
Steinmetz does use them to solve a non-phasor (transient) equation which has complex roots (damped oscillation), but he is then at great pains to say they should produce a real result - closer, but an artefact of using algebra to do calculations. Clarke is likewise careful to ensure there is no overly strong buy-in to a mathematical concept of complex numbers, and she treats vectors and complex quantities (being a complex of two quantities) almost as equivalent. Both authors use it as a tool at a time when analytical solutions were an enormous optimisation.
Translation:
"Steinmetz and Clarke use them to solve physical problems but I'm not convinced those numbers have physical meaning."
Whatever dude.
That is the look I am going for.
Wait, are you saying you want to raise pre-medieval mathematical concepts that were summarily rejected in the last 200 years?
Gawd, 18th and 19th century mathematics really is something that happened to other people.
Not now, then. Twice I provided (well you did for one) examples of the concept not being used or necessary, both times you were up on my case trying to argue around the facts. The piece I left off that comment (again to try to make it shorter probably errantly) was "You don't need to try to prove it is." - you're grasping at straws, I would have thought needlessly.
Because whatever you're trying to argue on this point is stupid whataboutism. Arguing about places where complex numbers are not necessary to solve the problem is utterly immaterial to complex numbers themselves or what they are used for. It's a ridiculous tangent.
That's because there seems to be no solution but you believe there is.
So now I know that you do think sqrt(-1) is a fundamental property of all numbers, I can understand it (and your subsequent reply) in context. Negation is no more than a 180 degree rotation, half the complex nature that all numbers possess, with the other half ready should it be needed. That's a matter of belief.
And with one fell swoop you have reduced mathematics in engineering and physics to a matter of religion.
I guess Paul Dirac spoke a magical spell and willed positrons into existence with complex numbers.
One I don't seem to share, due to insufficient evidence. I don't think it's impossible or implausible or even something I should believe against. It's just that when faced with fanciful notions like "how many sheep do you have" -> "oh, about 34 + j0", I am entitled to remain skeptical. Perfectly entitled.
And there we have it - the extent of your mathematical desire is counting sheep on your fingers and toes. Are you a shepherd or an electrical engineer?
I've said over and over and OVER again that not everyone needs the complex numbers to solve their equations nor does every equation necessitate writing the complex form (even if it is always there, hiding, lurking, waiting to pounce on you!
).
However, voltages, currents, and power are not sheep... and your conflation of voltage, current and power with sheep in an electrical engineering forum is deeply disturbing.
The definition of power is
S =
VI* = P + jQ VA. That's a complex number. Whine about it all you want - that's how it is. And that's the simplest form of it - it gets worse with unbalanced systems or non-linear systems.
https://en.wikipedia.org/wiki/AC_powerWhat you are NOT entitled to is to tell other engineers they don't require complex numbers as you do here:
I'm starting to wonder if any modern engineering requires that sort of mathematics. Engineers don't tend to use it directly that often, some not at all. Basic arithmetic on a computer seems to be enough for the 21st century. So "other people" might be 20th century engineers.
I literally told you, as one example, that you can't even use HFSS Antenna E&M Simulation Software without inputting variables in terms of complex numbers. Did you forget? It was a bunch of pages ago, here it is (p.12 for example):
http://www.ece.uprm.edu/~rafaelr/inel6068/HFSS/HFSS_Antenna_v2015_v1/workshop_instructions_trainee/ANSYS_HFSS_Antenna_W03_1_Post_Processing.pdfThis is an entire field of engineering whose automated software cannot even be used without a deep understanding of complex numbers to even input the values for calculation. Have you ever even heard of an S-parameter?
https://en.wikipedia.org/wiki/Scattering_parametersI've never used this software but sure seems like a good understanding of complex numbers is would be handy here, especially for Optimal Power Flow Analysis (given that you need to know what the kVAR and VA units all mean in the various fields):
https://etap.com/docs/default-source/qa-documentation/etap-getting-started.pdfIt's funny to me how hand-wavy you are about "oh computers will do it all" when the computer simulation is only as smart as the engineer who uses it. Garbage in, garbage out, as they say.
PS
I want to reiterate, as I said pages ago, I'm not terribly interested in metaphysical questions like "does mathematics physically exist." That's a question for philosophers.
What I am interested in, from the question posed pages and pages ago, is can these numbers be assigned physical meaning for solving actual problems. I've never seen the number "3" so arguing about whether it exists or not is pointless here - let alone arguing whether sqrt(-1) exists...
But I have seen 3 apples and I have seen complex power. And if our descriptions of those phenomena be but an illusion, they are a damned good illusion, because somehow the illusion has predictive power for revealing phenomena we've never seen before. That makes it fantastically relevant and physical.