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This is veering really close to the question of "is mathematics physical?" and that's a big question!
Hang on, that's not a very big question, and the answer is relatively simple. Maths itself is not physical, or it is only as physical as any language in which you can express logic, it's conceptual. The links between that language and quantities defined within is also defined and there is an observable consistency between the results of additive processes in 'nature' and in the mathematical system etc... hence why one should always include units against any number with physical significance because that defines the process by which one takes the number on paper and stacks calibrated metre-rules end-on-end to reach a distance. It's all defined, we're safe.
Okay, okay, now you hang on!
The question I've been answering, from adx, is this one,
Is there any place in engineering, anywhere, where sqrt(-1) has any physical relevance at all? The only place I've ever seen it doing something useful (beyond being an arcane convenience for mathematicians) is in a Feynman lecture where it quasi-continuously described a wave function inside and out of an energy well or something (I can't find it now).
If you're suggesting that sqrt(-1) has no physical relevance because MATHEMATICS has no physical relevance... then yea... okay let's go with that, sqrt(-1) has no physical relevance because it's part of mathematics which inherently has no physical relevance.... it's kind of a tautology and one I don't find that terribly helpful for 1) engineering students or 2) actual engineers trying to devise logical frameworks to relate phenomena to a method of describing and predicting them.
I read adx's question as, if we use sqrt(-1) in our engineering calculations, what does it mean? Does it have a physical meaning? Or is it just something used to torture students with 'claptrap' and is useless for all us manly-men practically practical-minded engineers? That's a valid question but totally independent of the philosophical question 'is mathematics physical' which I don't really care about (I mean I do, but not here).
Now in answering adx's question, in engineering, does sqrt(-1) have physical meaning? The answer is a resounding YES!!!
Just because people got confused by bad pedagogy in school (I'm included in that) or there are specific engineering lines of work or problem solving techniques that don't use sqrt(-1) is TOTALLY independent of my answers to his question.
Does sqrt(-1) have physical meaning in engineering? Yes. Steinmetz proved it (and I hope references to 'waffley' texts aren't in reference to Steinmetz's treatise). And Edith Clarke literally wrote the book on AC Power Analysis. She was hired as the first woman electrical engineer in the USA in an age of extreme sexism by General Electric to solve power problems stumping their engineers - some of these were problems no one else could figure out:
https://www.google.com/books/edition/Circuit_Analysis_of_A_C_Power_Systems/JB4hAAAAMAAJ?hl=en&gbpv=1&printsec=frontcoverI mean it... she literally solved problems no one else could figure out by using hyperbolic functions and complex impedances. She is a big reason our long-distance energy grid can even exist:
Steady-state stability in transmission systems calculation by means of equivalent circuits or circle diagramshttps://ieeexplore.ieee.org/document/6534694You can read the paper here:
https://speakingwhilefemale.co/wp-content/uploads/2020/09/Clarke_Transmission.pdfAnd I've already shared MY experience in RF engineering and antenna design that sqrt(-1) has tremendous physical meaning and application. If complex phasors and impedances and sqrt(-1) is all worthless claptrap for engineers - then don't use AnSys HFSS simulation software and stay away from RF, I guess?
If the response to all this is 'nuh uh, I've never needed it..." well then... fine. Good for you. But don't be deluded into thinking that other engineers aren't using it and ascribing physical meaning to it all the time and changing the world. BTW I used complex impedances last night in my class explaining the origins of harmonics in motors and how to interpret a 3-phase phasor diagram like you'd see on a Keysight Power Analyzer:
I really don't think I need to provide more examples of the 'physical relevance' of sqrt(-1). Take it or leave it. The power engineers and RF engineers are quite happy with it.
One last thing about sqrt (-1), I REFUSE to let ourselves be biased against the attribution of physical meaning to sqrt(-1) because freaking Rene Descartes decided to be a smartass and call them 'imaginary numbers' as if they were 'less real' than other numbers in mathematics (which has been argued that NONE of the other numbers in math are real/physical either so the distinction is irrelevant).
So adx, I submit to you that your issue with the 'physicality' of sqrt(-1) is because of an idiotic naming convention.
This is a mess and as a teacher/working engineer/former student who also got confused, I hate it. Screw you Descartes and screw all the math teachers in the intervening centuries who perpetuated this tragedy of a ridiculous name. That bastard Descartes didn't even know how to take a derivative or do a surface integral (he did help us get there though). He shouldn't be allowed to confuse students for centuries because of his antiquated philosophical biases.
At least when the Big Bang got a derogatory name it was kinda cool sounding... but it also has confused people about what cosmologists actually think about it.