"Veritasium" (YT) - "The Big Misconception About Electricity" ?

[TimFox]

The German mathematician Kronecker famously said "Natural numbers were created by God, everything else is the work of men."
In that context, "everything else" includes zero, negative integers, rational fractions, irrational numbers, etc., since "natural numbers" in mathematics means the set of positive (non-zero) integers.
https://www.cantorsparadise.com/kronecker-god-and-the-integers-28269735a638

[HuronKing]

Some people are of the opinion that only the natural numbers (excluding 0) have "physical meaning". (whatever "physical meaning" might be?)
Does 0 have physical meaning?
Do the negative numbers have physical meaning?
What about sqrt(2) which is an irrational number?

[penfold]

[...]
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.
[...]

That's not at all what I'm saying. I'm saying that only real numbers directly relate to the physical world because they are so defined. The imaginary unit we attach to reactance is an artifact from the mathematical analysis that is used to describe and represent it in terms of sine waves. I don't for a second dispute that from the real values of measured quantities a result in terms of an imaginary unit can be arrived at, be presented, and is useful (immensely so in linear circuits)... but it isn't a physical quantity, in that case, it is an interpretation of real physical measurements represented in such a way that is closer to the maths and the j is an operator rather than a quantity. I'm still not disputing your statement as far as the 'relevance' or usefulness of imaginary quantities... but it is stretching it a bit far to say that it is a physical quantity... a point you may have been missing from adx's side of the argument.

So as far as undergraduate teaching goes, it's a perfectly fair approach to present reactance as an imaginary quantity with physical relevance because spice and a VNA will tell you it is. But, just, it's not the end of the story, Fourier and Laplace aren't the only transforms, and the simplified view of complex reactance falls over in non-linear systems.

[penfold]

Some people are of the opinion that only the natural numbers (excluding 0) have "physical meaning". (whatever "physical meaning" might be?)
Does 0 have physical meaning?
Do the negative numbers have physical meaning?
What about sqrt(2) which is an irrational number?

Extending the real numbers to an algebraic structure in which the square root of minus 1 exists is brilliant. Furthermore, all polynomials can be factored into monomials. How great is that?

The field of Complex numbers is not the same as the vector space of two-dimensional vectors over the real numbers. The multiplication is different. j isn't a unit since its square isn't equal to j. 1 is the unit of the Complex numbers.

In my opinion, trying to assign "physical meaning" to mathematical concepts only works for very simple problems. Just trust the Mathematics and look at what the theory tells you. Sometimes our intuition fails horribly. Trust the math.

The formalization of maths, was a surprisingly recent occurrence, at least with sets, groups, and categories defining algebras and arithmetics from a truly axiomatic basis. You've got to bear in mind that there are several things which the figure '1' represents: being the multiplicative identity, the successor function, and the start of the number line, all having different philosophical interpretations (historically leading to disputes over their significance) but were defined as equal/equivalent by Russel and Whitehead (1920s? I forget the date). With the modern definitions, those disputes are moot and/or based on outdated origins.

"Trust the math" is a very valuable phrase that I'm glad to see. Because of the physical significance of numbers and quantities and difficulties in finding relationships between them is largely what pushed Grassman, Hamilton, and Clifford vector and geometric algebras back in favor of the wishy-washy i,j,k vector calculus operators. It took Clifford algebra many years to resurface as a better mathematical representation for EM in relativistic and quantum theories. In geometric algebras, not only do i,j, and k become kinda bendy changeable vectors, but you also have to contemplate the ideas of planes and cubes as vectors and their physical representation must be viewed through a metric, potentially on a topological manifold... the maths is beautiful (I have low standards) but it stops making sense at the moment you attempt to visualize it.

[TimFox]

Although we can probably agree that the voltage variable V(t) is real-valued in electrical engineering, in Quantum Mechanics the physical wave function must be complex (in the mathematical sense of real and imaginary components). 
This was impressed upon me in college Quantum Mechanics classes, since a non-complex wave function would not have enough degrees of freedom, and the time-dependent Schrödinger equation explicitly starts with i.
https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation
(I tried without success to cut and paste the equation from that Wikipedia article:  see the section "Preliminaries")

[HuronKing]

[...]
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.
[...]

That's not at all what I'm saying. I'm saying that only real numbers directly relate to the physical world because they are so defined. The imaginary unit we attach to reactance is an artifact from the mathematical analysis that is used to describe and represent it in terms of sine waves. I don't for a second dispute that from the real values of measured quantities a result in terms of an imaginary unit can be arrived at, be presented, and is useful (immensely so in linear circuits)... but it isn't a physical quantity, in that case, it is an interpretation of real physical measurements represented in such a way that is closer to the maths and the j is an operator rather than a quantity. I'm still not disputing your statement as far as the 'relevance' or usefulness of imaginary quantities... but it is stretching it a bit far to say that it is a physical quantity... a point you may have been missing from adx's side of the argument.

So as far as undergraduate teaching goes, it's a perfectly fair approach to present reactance as an imaginary quantity with physical relevance because spice and a VNA will tell you it is. But, just, it's not the end of the story, Fourier and Laplace aren't the only transforms, and the simplified view of complex reactance falls over in non-linear systems.

It is only because Descartes defined it that way... 

'Real' vs 'imaginary' are completely made up terms from a 17th century mathematician that have nothing to do with whether something, say, 'exists.'
I ascribe NO importance to those terms, at all, because those terms have no relevance for us other than history. Descartes couldn't understand how sqrt(-1) would have physical relevance or meaning (just like mathematicians before him couldn't understand how 0 or negatives had physical relevance)... but that doesn't mean we can't. Remember, he relied on geometric proofs for everything. He DID NOT KNOW calculus. 

His 'real' has nothing to do with what you or I can consider to be 'real.' I can't stress this enough.
https://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf
https://plato.stanford.edu/entries/descartes-mathematics/

And some more on this:
https://www.nature.com/articles/s41567-019-0581-x.pdf?origin=ppub

The statement of Euler's Formula is much more compelling theorem about the deep relationship between physical quantities and the 'imaginary' number. Descartes didn't understand complex numbers. Euler got closer to understanding them, and Gauss slam dunked it with Hamilton.

And the part underlined in your quote is exactly what Steinmetz, heh, 'real'ized. Oscillating voltages are sinusoids with some magnitude over time. Sinusoids can be represented by a complex number. Thus the oscillating voltages and the response of components can be represented by the complex numbers.

And here is where my brain is melting down:
If these sinusoids are physical quantities, how is the j description of them not a physical quantity?

Or, in more specific terms, how is a sinusoidal voltage with time-shift physical but not that same voltage written in terms of j? 

I'm going to be a little silly here, but if mathematics is just a language for describing physical things, then this is like saying words for 'rock' in English are 'real' words but words for 'rock' in French are 'imaginary' because I can't conceive of anyone who would might find it easier to speak French. Take that Descartes! 

The pedagogy of teaching complex numbers needs to change. Stuff like this is a good start:
https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

Numbers are 2-dimensional. Yes, it’s mind bending, just like decimals or long division would be mind-bending to an ancient Roman. (What do you mean there’s a number between 1 and 2?). It’s a strange, new way to think about math.

We asked “How do we turn 1 into -1 in two steps?” and found an answer: rotate it 90 degrees. It’s a strange, new way to think about math. But it’s useful. (By the way, this geometric interpretation of complex numbers didn’t arrive until decades after i was discovered).

[HuronKing]

This video is interesting and has a quote from Gauss on the subject I've never seen before. Apparently he shares my contempt for the idiotic 'imaginary' naming convention (I swear I'd never seen it before):

That this subject has been hitherto surrounded by mysterious obscurity is to be attributed largely to an ill-adapted notation. If for example +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question.

Carl Friedrich Gauss
 Gauß, Werke, Bd. 2, S. 178.

Free your minds! Gauss besieges you! 

Lateral units is also a brilliant alternative name. I'm going to start finding ways to use it.

[TimFox]

Your opinion that Descartes gave an unfortunate name to i is reasonable, and the connotations of the word "imaginary" certainly have led some to think that i is icky.
However, this reminds me of the endless discussions, especially from people new to the field, that the historical assignment of + and - to charges and voltages is backwards, since electrons are -, and that we should all change to the poster's preferred way and re-name all of our equations and equipment.  While we are at it, we should also reverse red and black terminals on our voltmeters to agree with the new normal.

[penfold]

[...]
And here is where my brain is melting down:
If these sinusoids are physical quantities, how is the j description of them not a physical quantity?

Or, in more specific terms, how is a sinusoidal voltage with time-shift physical but not that same voltage written in terms of j? 
[...]

Okay, I can see how that might confuse you, think about how you might measure phase on an oscilloscope. One option is to measure the time difference between common events in the wave and relate that as a fraction to the wave's period. On the other hand, a VNA works by performing a basis transformation of the wave to a new orthonormal pair of signals bases, i.e. from time-voltage to voltage-voltage components at a defined frequency, it then shoves a j in front of one component and hey presto. You can call them I and Q, e1 and e2, i and j, or real and imaginary. It is just a mathematical nicety that complex numbers neatly represent 2d vectors, its not fundamental or especially general. Euler's formula extended beyond complex numbers to be the exponent of matrices in general really further highlights that the complex "scalar plus imaginary" isn't a unique form and that, say, time and voltage could be described by any pair of orthogonal basis vectors in an arbitrarily dimensioned system.

I'm going to be a little silly here, but if mathematics is just a language for describing physical things, then this is like saying words for 'rock' in English are 'real' words but words for 'rock' in French are 'imaginary' because I can't conceive of anyone who would might find it easier to speak French. Take that Descartes! 

There's a very fine line between silliness and ignorance... I do hope you were on the right side, it just doesn't read like you were.

[HuronKing]

I think we're getting closer so I'm going to hone in on this remark.

It is just a mathematical nicety that complex numbers neatly represent 2d vectors, its not fundamental or especially general.

Because our algebraic number system is at most 2-dimensional (as far as I know - I'm not a mathematician so you may know more than me). 'Imaginary' numbers are not any less 'real' than the 'real' numbers... if the real numbers are ascribed to have any meaning themselves that is.

That's why to me asking if j has any physical relevance (remember, this is the question that started all this) is like asking if -1 has any physical relevance... or the sine function, or the exponential e. Like, yes? Obviously? But maybe not so obviously because I was confused by it as a student, my students get confused by it, and even working engineers get confused by it. Hehe. 

And the power in j is in representing phase shifts very conveniently. Are there other ways to do it? Yea, of course, but that was not what was asked.

I tried to make the analogy with a question about power supplied by a voltage source being a negative quantity when the source is actually ABSORBING power is like a time-shifted voltage source being that same voltage source multiplied by an 'imaginary/lateral' quantity equivalent to the time-shift. My experience has been that people get so used to seeing power always expressed as a positive quantity that they need to be reminded power can be negative depending on the perspective. I'm sorry that the analogy was not well-received - I think it is a useful analogy that has helped my students. 

There's a very fine line between silliness and ignorance... I do hope you were on the right side, it just doesn't read like you were.

Of course I just wanted to get another dig at Descartes. 

But in going back to reread what I wrote some pages ago, I am still trying to find wherever I may have erred.

I stand by what I've said - does j have physical relevance? Yes. 

[penfold]

[...]
I stand by what I've said - does j have physical relevance? Yes. 

Engineer to engineer, yes, I'll accept that, never really disputed the "relevance" and all though I described complex only as a mathematical nicety, it's a hella useful one! The rest is just a pedantic aside... I may have read too much into the word "relevance"... but hey... I've been strict enough with aetherist about definitions, measurability, etc, I'm just on high alert haha. But yeah, I think that because a VNA can work out the phase in terms of I and Q using only electronic interpretations of mathematical functions... I'd be happy (not that my sense of happiness is relevant... but a happy penfold doesn't argue pedantry as much) at a stretch to call it (i, j etc) a (shudder) measurable quantity... just not a pure one and not unique... but more than sufficient for 99% of professional engineering and 85% of research and a valuable link between "reality according to test equipment" and useful algebra.

Dimensionality is a tricky one. In the sense that you could have y as a function of x in the field of real numbers and plot y vs x on a cartesian graph, it isn't strictly multi-dimensioned until you assign a metric and some "vector-y-ness" to x and y, it's the same even as far as the complex field until you find an alternative representation (say as a matrix) cue Hamilton.

But... Hamilton showing that complexs can also be represented as row and column 'vectors' or matrices (see wiki page). It's then when sqrt(-1) starts to lose physical significance (to me at least) or to no more nor less significant than any other vector notation. Yeah, it's still the same number in disguise, but doesn't necesarily need to be sqrt(-1), or even an orthonormal basis, so long as it spans the space.

[bsfeechannel]

My verbal description remains purely verbal and not at all physical, it just describes a physical object.

Your DNA is a description of you. But it is a description that can replicate itself and even build an entire you. We can encode your DNA sequence using the letters ACGT. It'll describe you uniquely. It'll be purely verbal, but once decoded to assemble the actual nucleic acids it represents, it'll be an functional polymer.

So, is math the encoding of the "DNA" of the universe? That's what David Hilbert and his program aimed to ascertain until Kurt Gödel screwed it all up.

Maths is a descriptive language in which the natural phenomena are described, from those descriptions we can hypothesize, test, and refine new theories... the phenomena, including the big bang, relativity, quantum, etc all existed before humans and maths... yet that curiously happened. The language in which these descriptions are encoded - since it can be communicated verbally... is not exclusively physical.

I would say math is perhaps language minus contradiction. Since it doesn't admit paradoxes, it is a convenient tool to describe things for which ambiguities would be inadmissible.

I like Al-Khwarizmi's preface  when he introduced algebra to the world in 850.

The fondness for science [...] has encouraged me to compose a short work on Calculating by Completion and Reduction [a.k.a algebra], confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, law suits, or trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned.

So that's what math is all about: making life easier and less ambiguous.

[bsfeechannel]

This is all a bit silly - it started with a gentile troll about i vs j, then we're now back to arguments over half-arsed engineering.

You reduced engineers to mere solder monkeys (no Cartesian coordinates, no vectors, no y, no functions).



What do you expect?

Uh uh. We'll have the reality of the industrial engineer,

Your reality may vary, then. Because in my reality of an engineer in the industry, knowledge of math and physics count a lot.

what you're complaining about is not misconceptions, but work.

You don't get it. What you're advocating creates engineers who can't see beyond a limited set of "best practices" or rules of thumb. Heck, before I was an engineer, I was a technician. And even in our formal training in electronics during high school we were taught to apply the Cartesian coordinate system (that we had learned in middle school) to interpret the measurements of scopes, plotters, and whatnot and complex numbers to analyze AC circuits.

Your point is inexcusable.

So run with the reality, and stop assuming students need to "study" Cartesian coordinate systems (why?!) and teach the concepts.

But, but, but, the Cartesian coordinate system is a concept.

All this mathematics and (dare I say it) physics, does no good.

I weep for the future.

I was going to let you have that one,

Of course you were. Look at the venerable Rigol DS1052E.



See the X and Y markings near CH1 and CH2, respectively? And how about the MATH button? Adding, subtracting, multiplying channels and FFT-ing, as far as I know are math operations and functions.

CRT scopes even had a Z axis for controlling the trace intensity.

So, ¯\_(ツ)_/¯

Once again, this is about sqrt(-1), not vectors. That expensive thing you showed is called a vector network analyser, not a really complex mathematical network analyser.

Complex numbers form a real vector space. That's how you visualize them.

[SiliconWizard]

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.

[SandyCox]

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.

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.

[SandyCox]

My verbal description remains purely verbal and not at all physical, it just describes a physical object.

Your DNA is a description of you. But it is a description that can replicate itself and even build an entire you. We can encode your DNA sequence using the letters ACGT. It'll describe you uniquely. It'll be purely verbal, but once decoded to assemble the actual nucleic acids it represents, it'll be an functional polymer.

So, is math the encoding of the "DNA" of the universe? That's what David Hilbert and his program aimed to ascertain until Kurt Gödel screwed it all up.

Maths is a descriptive language in which the natural phenomena are described, from those descriptions we can hypothesize, test, and refine new theories... the phenomena, including the big bang, relativity, quantum, etc all existed before humans and maths... yet that curiously happened. The language in which these descriptions are encoded - since it can be communicated verbally... is not exclusively physical.

I would say math is perhaps language minus contradiction. Since it doesn't admit paradoxes, it is a convenient tool to describe things for which ambiguities would be inadmissible.

I like Al-Khwarizmi's preface  when he introduced algebra to the world in 850.

The fondness for science [...] has encouraged me to compose a short work on Calculating by Completion and Reduction [a.k.a algebra], confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, law suits, or trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned.

So that's what math is all about: making life easier and less ambiguous.

The concept of dimensionality is clearly defined. It is the cardinality of a basis of the vector space. The dimensional of the real numbers is 1 and that of the complex numbers is 2.

Interestingly, there are exactly as many complex numbers as there are real numbers.

Complex analysis is much more powerful than real analysis.

[penfold]

My verbal description remains purely verbal and not at all physical, it just describes a physical object.

Your DNA is a description of you. But it is a description that can replicate itself and even build an entire you. We can encode your DNA sequence using the letters ACGT. It'll describe you uniquely. It'll be purely verbal, but once decoded to assemble the actual nucleic acids it represents, it'll be an functional polymer.

So, is math the encoding of the "DNA" of the universe? That's what David Hilbert and his program aimed to ascertain until Kurt Gödel screwed it all up.
[...]

That's an interesting example, it is also an awful example.

I would say math is perhaps language minus contradiction. Since it doesn't admit paradoxes, it is a convenient tool to describe things for which ambiguities would be inadmissible.

I like Al-Khwarizmi's preface  when he introduced algebra to the world in 850.

Yeah... you may want to update your reading material, maths has changed a fair amount since then.

[TimFox]

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.

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.

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.

[adx]

Oh, oh, where to start.

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".

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.

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.)

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).

"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).

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".

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.

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.

The rest can wait.

[bsfeechannel]

"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).

Steinmetz has no authority. His application of complex numbers to the analysis and design of AC circuits does. The authority comes from the logical soundness of its approach, the agreement with the facts and the solutions it brings, confirmed ad nauseam all over the world.

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).

So, because you see no meaning in i or j (probably because you're already refractory to math and physics) you say it should be forgotten altogether for the whole engineering, although an entire industry exists around the concept. But what do you suggest to replace it, to easily solve AC circuits? Are you some kind of new Steinmetz with an even better approach?

The rest can wait.

The rest is becoming impatient.

[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]

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.

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.

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.

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. 

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".

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?

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.

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.

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.)

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?

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).

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

"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).

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.

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".

This is nonsense.   
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. 

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.

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 
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

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.

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.