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"Veritasium" (YT) - "The Big Misconception About Electricity" ?
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adx:
First clean up some unreplieds...


--- Quote from: HuronKing on March 25, 2022, 07:20:57 pm ---
--- 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?

--- End quote ---

Real numbers from my context. Avoiding formal definitions was part of my point. I don't care if mathematicians (or you) say complex numbers are a single number or not, I can define any number as an algebraic construction, but going on to say the box set of Star Wars is "an ordinary number" would be silly.


--- Quote from: HuronKing on March 25, 2022, 07:20:57 pm ---
--- 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.'

--- End quote ---

You lost me there, despite knowing what you mean. I would say sines are more physical as a feature of 2D geometry and oscillations. Multiplication is more of an abstract tool relating to quantities - but could be physical. Lateral is geometry. My question is about sqrt(-1), not the complex plane.


--- Quote from: HuronKing on March 25, 2022, 07:20:57 pm ---
--- 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.

--- End quote ---

That was in response to bsfeechannel being particularly appeal to authority adjacent. Your story about Charles Steinmetz and Edith Clarke is nice, but them being good at what they did has nothing to do with my question as far as I can see.
adx:

--- Quote from: HuronKing on March 31, 2022, 05:15:22 pm ---
--- Quote from: adx on March 31, 2022, 03:16:16 pm ---But "what's to say" isn't a proof. And we are clear in our claim that 90° = sqrt(-1), or rotation is "connected directly to solutions of x^2+1 = 0" - it's an extraordinary claim, unscientific in its boldness coming from historical ideas of something no one ever really worked out (to my knowledge). (In this sense perhaps mathematics is to engineering what the pre-science medicine is to modern medicine - full of ideas (many good) but isn't science?)
--- End quote ---

Gauss and others worked it all out for us. In fact, some of the most brilliant minds in human history turned their attention towards this. It's the basis of the Fundamental Theorem of Algebra. It's not so mysterious, really.

--- End quote ---

Well I guess I just don't believe. If someone like me insists on being an ignoramus who won't or can't understand (I can't be expected to tell the difference), and you are limited to 'appeal to authority adjacent' claims because there is no trivial proof, then in the absence of launching into full time study I can just remain skeptical. It's not a carload of students trying to get to the top of a hill (then not drive off it). I can still use I and Q, and I can pretend j doesn't mean anything beyond how it gets used.


--- Quote from: HuronKing on March 31, 2022, 05:15:22 pm ---
--- Quote ---It could be fundamental, or it could be we set ourselves up for a trick and believe this illusion means more than it does.
--- End quote ---

I'm content that it's not an illusion since the mathematics has tremendous predictive power in physics and engineering.

--- End quote ---

A working illusion will also have "tremendous predictive power", so I'm not content.

I don't know what that predictive power is anyway. If you mean frequency domain analysis, a 'frequency' must describe amplitude and phase, it’s direct and obvious. It doesn't need some [inflationary language trigger warning] ridiculous number system to describe it, just 2 reals, or even an unsigned magnitude and direction. That direction's 0 has the reference direction you needed.

I (sort of) regret leaving out ", or both" from my claim above (the false dichotomy sounded more dramatic).

I had some ideas, but I don't think it will help.

It just seems awfully convenient that when multiplying by -1 gives an infinite frequency oscillator ( (-1)^n creates problems ), that it is possible to define an in-between situation (literally i*i=-1) of pathologically orthogonal numbers to create a quadrature oscillation (circularly polarised) to do the job and represent any phase. It's like we half made it up for the purpose that any sane person would call a vector. The other half seems fundamental. Of course complex numbers can be visualised on a plane, because we designed them that way (Cartesian coordinates). The question is whether imaginary numbers deserve to be "an axis", or just happen to work that way because we think they should. Imaginary numbers are dreamed up from fanciful mathematical impossibilities (x^2 is non-physical for -ve x: we can't have negative length). It seems to have more in common with rotation in 3D than any 2D geometrical construction. Sus picious. Might be best to make a tinfoil hat, or if I already have one, make a roll of foil from it so I can make another one later on (metal fatigue and infinite patience permitting).


--- Quote from: HuronKing on March 31, 2022, 05:15:22 pm ---
--- Quote ---And that's possibly all I need to say on it without knowing more. I have learned why complex numbers have fundamental physical relevance, but also why they might not.
--- End quote ---

This is progress.  :D

--- End quote ---

Except that leaves me in a clearer version of where I started out - knowing how it's used and vaguely why, but not accepting it.


--- Quote from: HuronKing on March 31, 2022, 05:15:22 pm ---
--- Quote ---Bah, my answer to this part was to be a quote (I thought from Gauss) saying the true nature of the imaginary numbers remains elusive. Can't find it anywhere.
--- End quote ---

You're probably referring to the 'shadow of shadows' quote which should be weighted in its context. Gauss was tackling Euler's Identity in his doctoral dissertation to prove the Fundamental Theorem of Algebra.

--- End quote ---

Maybe, but I thought it had "elusive" in it. The shadows of shadows thing reminds me of my "banana numbers" which I had planned to invoke in roo-tons esque style to explain the cancellation of sqrt(-1) in that epic math duel video which was a better explanation better than Wikipedia. Best left undescribed even if only to avoid long sentences.


--- Quote from: HuronKing on March 31, 2022, 05:15:22 pm ---Me: "Okay class, now that we've learned about real numbers, let's now learn about imaginary numbers."
Student: "Wait, why are we learning fake math?"
Me: "No, it's real math."
Student: "But you said it's imaginary."
Me: "Not really, the better name is complex numbers."
Student: "Oh God no! Why do we need to learn complicated math?"
Me: "It's not complicated. It's complex."
Student: "Yea! That's what I said. Math is stupid. You're making me learn complex imaginary math that I'll never use. Blegh."

--- End quote ---

How did you get that recording of me?! The chances are remote - the one and only maths class I ever bothered turning up to, only to be hurfed out after 5 mins from suffering a hypnic jerk and throwing my bic fluoro pink pen clear across the lecture theatre, delivering that unmistakable clatter as it hit the wood of the sound diffusor panel wall. Not my finest moment*, still, where's that recycled roll of foil.

(* Was actually some boring as all hell circuit analyis class of utter theory (S domain stuff?), and I didn't get hurfed out, just awoke to a couple of hundred incredulous eyes each beaming forth an accusing stare, once I had worked out what was happening from my complex plane induced (therefore involuntary) nap, I had to suffer through the indignity of trying to find my pink pen in that same silence only punctured with things like "do you have a spare pe... oh, ok, where did it go?" shuffle klonk klonk "found it" clatter "oops" clatter CLATTER "sorry dropped it again" muffled scream "sorry I thought that was the foot of the chair" shuffle shuffle shuffle "sorry" and so on until the lecturer asked me "are you quite finished?", to which I could only answer "I think so". Glad all that embarrassment is behind me. Maths wasn't in the big lecture theatres, so the pen couldn't have gone very far.)

Must admit, if the complex plane hadn't been invented, I would be going "ooh ooh you can plot it like this". I just can't accept something that doesn't make sense to me and never seemed to have any practical relevance as that dsprelated article bemoans (quote below).


--- Quote from: HuronKing on March 31, 2022, 05:15:22 pm ---Lulz - the suggested additional reading is Charles Steinmetz' Theory and Calculation of Electrical Apparatus where that icky j appears on page 2:
https://www.google.com/books/edition/Theory_and_Calculations_of_Electrical_Ap/UjEKAAAAIAAJ?hl=en&gbpv=0

--- End quote ---

Hardly dents my argument (or really evidence) that some references and areas of engineering use phasors without sqrt(-1), or even j. I'm not saying j isn't widely used.

Whereas "Unfortunately DSP textbooks often define the symbol j and then, with justified haste, swiftly carry on with all the ways that the j operator can be used to analyze sinusoidal signals. Readers soon forget about the question: What does j = √-1 actually mean?" is my point (even to some extent the "unfortunately")...


--- Quote from: HuronKing on March 31, 2022, 05:15:22 pm ---Maybe take your investigations beyond Wikipedia?  ::)
https://www.dsprelated.com/showarticle/192.php

I'm amused by this site also taking the great pains to explain how the j is unfortunately named and glossed over too quickly when it is taught. In any case, You can thank Euler for making sines and cosines equivalent to j rotations.

--- End quote ---

I saw that article when looking for guidance on imaginary numbers. Quite neat and a good explanation, but on sqrt(-1) is again 'appeal to authority adjacent' (especially with all that stuff about Herr Euler, Gauss' brilliant introduction of the complex plane and comparison to Einstein - welcome analogies and hyperbole in this context, but not proof (also incorrect)). It describes how it behaves, light on what it actually means. e^(j(pi)/2) = jsin(pi/2) = j doesn't show j = pi/2 as an argument of sin obviously, but a means to scale the 'fake' vector j. It could be because the derivative of eix is ieix and that's how oscillators (and circles) are made (thereby defining the behaviour of i - fakely, or it could be the 'proof of concept' that I sought). As an engineer and mathematics weakling I am more interested in how things work rather than bathing in the glory and beauty of unquestioningly received wisdoms (pretty much the entire theme of your point) so I can't fall for the Klaptrapp conspiracy and it's time to despool that roll of foil once again: Never trust the math.

Well that's very wearying so I thought a nice conciliatory response would wind down arguments - pity how some things just don't work out :).

You could well be right, I just don't know. Like you say, I need to go off and sort that out myself.

Better I go off and wash the tortuous mess that is my hair.
HuronKing:
I'm going to keep my replies short.


--- Quote from: adx on April 05, 2022, 04:23:31 am ---Well I guess I just don't believe. If someone like me insists on being an ignoramus who won't or can't understand (I can't be expected to tell the difference), and you are limited to 'appeal to authority adjacent' claims because there is no trivial proof, then in the absence of launching into full time study I can just remain skeptical. It's not a carload of students trying to get to the top of a hill (then not drive off it). I can still use I and Q, and I can pretend j doesn't mean anything beyond how it gets used.
--- End quote ---

Again, I never appeal to authority. I've provided ample resources to read from Steinmetz and Clarke down to YouTube level basic introductions. You've got the full gambit of resources at all levels of rigor available. Something something horse to water.


--- Quote ---The question is whether imaginary numbers deserve to be "an axis", or just happen to work that way because we think they should. Imaginary numbers are dreamed up from fanciful mathematical impossibilities (x^2 is non-physical for -ve x: we can't have negative length).
--- End quote ---

You have 3 apples, and want to take away 5 apples. So you have negative 2 apples.

NEGATIVE 2 APPLES? WHAT IS THIS SORCERY? This is just mathematical claptrap invented to compensate for made up problems and invent solutions.

How can you have negative apples? IMPOSSIBLE!!!!  >:D

These excuses make you sound like a pre-medieval mathematician.


--- Quote ---Hardly dents my argument (or really evidence) that some references and areas of engineering use phasors without sqrt(-1), or even j. I'm not saying j isn't widely used.
--- End quote ---

How is that even an 'argument' to have? Like, yes? What is even the point of what you're trying to say now?


--- Quote ---I saw that article when looking for guidance on imaginary numbers. Quite neat and a good explanation, but on sqrt(-1) is again 'appeal to authority adjacent' (especially with all that stuff about Herr Euler, Gauss' brilliant introduction of the complex plane and comparison to Einstein - welcome analogies and hyperbole in this context, but not proof (also incorrect)).
--- End quote ---

The article isn't trying to rigorously prove it - just explain what it is, what it means, and get on with it to do some engineering work. I even showed you Steinmetz's books that introduced all this! If you're saying it's too hard to read the proofs and the cursory introductions are just appealing to authority... That's on you, man and I've done all I can.
 
SandyCox:
Take the set F of all 50 Hz sinusoidal waveforms. Each element f of F is of the form:
f= A cos(100*pi*t + phi),

The phasor transform maps f onto the complex number A angle(phi).

In fact, the phasor transform is an isomorphism between the field F and the field of Complex number. This means the only difference between the two is a change of notation. So the two are exactly the same.

ADX must agree that F has meaning physical?
penfold:

--- Quote from: SandyCox on April 05, 2022, 07:04:59 am ---[...]
In fact, the phasor transform is an isomorphism between the field F and the field of Complex number. This means the only difference between the two is a change of notation. So the two are exactly the same.
[...]

--- End quote ---

I like that wording... I think additionally because that transform can be performed in the same "domain" as the measured value, i.e. using analog circuits responding only to voltages or currents that produce the real and imaginary components that it can therefore be of a similar physical significance as using a compass and straight-edge to measure something geometric.

In a general sense what it is that irks me a little about complex numbers and physical significance is that there aren't many measurements I can think of that are genuinely unequivocally negative in their measurement (without considering a direction or rate of change relative to something else) of which one must take a square root of.


--- Quote from: HuronKing on April 05, 2022, 04:52:43 am ---[...]
You have 3 apples, and want to take away 5 apples. So you have negative 2 apples.

NEGATIVE 2 APPLES? WHAT IS THIS SORCERY? This is just mathematical claptrap invented to compensate for made up problems and invent solutions.

How can you have negative apples? IMPOSSIBLE!!!!  >:D

These excuses make you sound like a pre-medieval mathematician.
[...]

--- End quote ---

I'm not sure that kind of response really helps. I mean, literally, how can I have negative apples?! Is that the number of apples that I must possess before I own zero? Where will these apples come from and to where will they go? In the sense of lengths, the negative implies a direction whereby we would still be counting a positive number of lengths in the backward direction... but I cannot own negative apples, I could owe a positive number of apples to a specific person perhaps. But the negative sign contains very little of the necessary information... hence negative numbers don't appear that often in accountancy.
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