General > General Technical Chat
"Veritasium" (YT) - "The Big Misconception About Electricity" ?
HuronKing:
--- Quote from: penfold on March 24, 2022, 05:44:51 pm ---
--- Quote from: HuronKing on March 24, 2022, 04:22:19 pm ---[...]
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.
[...]
--- End quote ---
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.
--- End quote ---
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! ;D
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/
--- Quote ---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).
--- End quote ---
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):
--- Quote ---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.
--- End quote ---
Carl Friedrich Gauss
Gauß, Werke, Bd. 2, S. 178.
Free your minds! Gauss besieges you! >:D
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:
--- Quote from: HuronKing on March 24, 2022, 07:03:29 pm ---[...]
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? ???
[...]
--- End quote ---
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.
--- Quote from: HuronKing on March 24, 2022, 07:03:29 pm ---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! ;D
--- End quote ---
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.
--- Quote from: penfold on March 24, 2022, 08:06:39 pm ---It is just a mathematical nicety that complex numbers neatly represent 2d vectors, its not fundamental or especially general.
--- End quote ---
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. :-[
--- Quote ---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.
--- End quote ---
Of course I just wanted to get another dig at Descartes. ;D
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. :)
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