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"Veritasium" (YT) - "The Big Misconception About Electricity" ?
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TimFox:
I posted this before, in another thread.  Prof. Fano had a subtle sense of humor, but this anecdote is a good parable about the application of "i" and "j".

The late Professor Ugo Fano at the University of Chicago was giving a lecture on how to compute macroscopic quantities with quantum mechanics.
His example was electrical polarization in a dielectric as a function of frequency.
He set up the equations for a "perturbation" calculation, which involved the Hamiltonian (energy) of the E-field interacting with bound electrons.
He then expressed the external E-field as a Fourier expansion, an integral over frequency w of terms  E(w) exp(iwt) .
A theoretically-minded student in the front row objected, "Dr Fano, that Hamiltonian is not Hermitian!", by which he meant that energy is real-valued, but the individual terms in the integral were complex.
Dr Fano replied by erasing "i" and replacing it with "j", proclaiming that now it was Hermitian.
SandyCox:

--- Quote from: TimFox on March 25, 2022, 02:39:09 pm ---
--- Quote from: SandyCox on March 25, 2022, 11:26:15 am ---
--- Quote from: SiliconWizard on March 25, 2022, 03:40:28 am ---Complex numbers are just vectors in R², with the property: i² = -1. You can write i as the (0, 1) vector, and the multiplication as a generalization of the cross-product of two vectors. Actually, i² = -1 (or: (0, 1)x(0,1) = (-1, 0)) comes naturally from the generalized cross-product in R².

--- End quote ---

The concept of a cross product is only defined for three-dimensional vectors.

The complex numbers form a commutative ring, more specifically a field and a complete metric space. So calling it a vector space is confusing. It still is a vector space, but with more properties. So let's call it a field.

--- End quote ---

A cross-product of two vectors gives another vector as the product.
The scalar product (or inner product or dot product) of two vectors gives a scalar as the product.

--- End quote ---
Yes. But the cross product is only defined for three-dimensional vectors and the Complex numbers are two dimensional. Furthermore, the cross product of a vector with itself is zero. This is not what we want for the complex numbers.

The product of Complex numbers, as ordered pairs, is defined as
(a, b)(c, d)  = (ac-bd, ad+bc)

In general vectors can only be multiplied by scalars. If the vector space has an inner product, it is referred to as an inner product space.
TimFox:
Yes
HuronKing:
Fascinating discussion so far.  ;D


--- Quote from: adx on March 25, 2022, 04:04:02 pm ---First, my concern over sqrt(-1) in electrical engineering, penfold has it right: "and the j is an operator rather than a quantity ... it is stretching it a bit far to say that it is a physical quantity".
--- End quote ---

Did I say it was a physical quantity? Please show me (I've tried to find where I might've implied that but I don't see it). j is not an Ohm. But it is a representation of phase-shift in Ohms and a damn good one. Is that not physically relevant?


--- Quote ---I don't think it is any sort of tautology to say mathematical concepts are not real, if one then goes on and asserts that some part has physical relevance. Not all engineers are naturals at maths and can easily identify where that link appears (ie goes from nothing to something without explanation). Some people here seem to be struggling with it too - perhaps from over-familiarity.
--- End quote ---

You might as well be arguing that multiplication has no 'physical relevance' to engineering because you could just add the numbers up... like, yes? What is your point? Should we count on our fingers and toes because applying math makes us feel dumb?  ;)

I've said, many times, that engineers can and do get confused by this. And there are some engineers better at it than others. None of that is an excuse. There are way more problems I can solve quickly and efficiently with multiplication than I can with addition - even though multiplication is just an extension of addition.


--- Quote ---Does the 'value' sqrt(-1) have innate physical relevance for anything like phasors (or even quantum mechanical wavefunctions)? In other words, would these engineering uses suffer some fatal breakdown if they were replaced by two 'ordinary' numbers without some extra special property added? I genuinely didn't know as a student, although I slowly learned they are simply 'hack vectors' and more akin to polar to Cartesian conversion than some mysterious fact of mathematics. (But whether mathematics has more of a reality of its own is a different and much more interesting question.)
--- End quote ---

Complex numbers ARE ordinary numbers. In point of fact, what the heck IS an 'ordinary' number? That's not a formal definition. What is that?


--- Quote ---I too read the bit about Gauss suggesting "lateral" and thought that might have helped set the pedagogical direction for engineering uses, but I have no problem with the word "imaginary" or the reason it was originally used, especially if this lateralness is not truly innate (ie, an illusion).
--- End quote ---

Lateral is an expression of the rotation of the quantity. It is as 'physical' as multiplication is 'physical' as the sine function is 'physical.'


--- Quote ---"Waffley texts" I meant anything that is used as or perhaps is an "argument from authority" fallacy (per Wikipedia), eg Steinmetz says so so it must be true. Steinmetz says it is a handy trick, so if I read that right, it is an answer to my question that sqrt(-1) has no direct / special / innate physical relevance (because it is a handy trick).
--- End quote ---

To hell with that. I never appeal to authority. The only reason I or anyone else gives a damn about Charles Steinmetz and Edith Clarke is that they taught engineers all over the world how to use complex numbers to solve problems that stumped EVERYONE ELSE in the engineering industry until they came along. The proof is in their work and the results their analysis produced - nothing else. I've linked their works and plenty of other things to learn about it. The rest is up to you.


--- Quote ---My issue with the 'physicality' of sqrt(-1) is that it is so meaningless in engineering and unrelated to its original reason for being, that it allows what is really two numbers to be called one, and that is all it is used for (and to conjure up sine waves). I don't have an issue with negative numbers because they are not two numbers masquerading as one; the sign bit (unitary minus operator) has a genuine reason for being. I don't have an issue with vectors because they don't masquerade as one quantity. As penfold illuminated for me, a phasor is effectively a de-glorified scope screenshot or v vs t plot, for repetitive sinewaves - the entire signal. Complexians would call that "a number".
--- End quote ---

This is nonsense.   :D
The j has every reason to exist the same way negative sign operators do. Gauss demonstrated that. That your brain refuses to accept it (as evidenced by words like 'masquerade' 'conjure up sine waves' etc) is something else. You're saying you still think 'imaginary' number means it's not 'existing' or that it's a fake, a fiction of some kind. That is NOT TRUE. No more a fiction than negative numbers or sine functions, which you're apparently fine with, so whatever.  :-X


--- Quote ---I've already posted what I think about zero etc, but I have no problem ascribing some potential physicality to all real numbers, because they embody the principles of proportionality (linearity), repeatability, measurement, divisibility etc - even noise. I have never seen the "beauty" in mathematics (I can't even begin to understand what that means), but I think A/D converters are wonderful things.
--- End quote ---

See, you're still restricted by Descartes' idiotic naming convention. I can assign ALL of those same properties to the complex j numbers. In fact, I do, all the time. I can measure the impedance of a capacitor. Don't tell me it isn't physical... I can see it and its effects on my circuits! I can literally define the power consumption of a circuit as S = VI* = P + jQ volts-amps. Why is this so impossible or non-physical?

I'm not citing waffle-y texts at you. I'm citing actual engineering practices. You can take them or leave them.
https://www.electronics-tutorials.ws/accircuits/power-triangle.html

Go take issue with Keysight. Surely they have no idea about the lack of physicality of the j in their impedance analyzers  >:D
https://www.keysight.com/us/en/assets/7018-06840/application-notes/5950-3000.pdf

Keysight Impedance Measurement Handbook:
https://assets.testequity.com/te1/Documents/pdf/keysight/impedance-measurement-handbook.pdf


--- Quote ---I think i is icky, because +-sqrt(-1) is wholly less useful than +-sqrt(+1), yet multiplying by either has the same type of effect (an arbitrary phase shift, eg 90 or 180 deg). Let us not forget that i is composed of the multiplicative identity and unitary minus. Hmm, it's getting late, better head off before I say something I'll agree with.

--- End quote ---

Then you're not an AC power engineer. There is nothing shameful about not being an AC power engineer. But you're not - so don't proclaim sqrt(-1) has no meaningful/practical usefulness in engineering. That's just plain wrong. And I can't believe I'm on an engineering forum trying to convince other engineers about how useful j is (well maybe I should believe it - I did say it can be confusing).  :(
TimFox:
In 12th grade, we defined "hairy numbers" as those that did not "come out even".
"Ordinary" is in the eye of the beholder, but mathematics for many years now has defined "natural", "real", "complex", "rational", "transcendental", etc. numbers rigorously.
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