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"Veritasium" (YT) - "The Big Misconception About Electricity" ?
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hamster_nz:
I'm really surprised that nobody has mentioned changes of basis vectors, changes of coordinate systems, translations, transforms, and of ways of changing your 'perspective' and 'view' on the same underlying system. Lagrangian mechanics vs Hamiltonian mechanics vs Newton's laws of motion, and so on.

In some cases sqrt(-1) has a physical meaning - for example, in a physical system it could be energy 'transformed' into a value that you can't measure in the units you are working with - you might be measuring displacement/distance and imaginary quantity might be energy stored in a spring, or in system's momentum. In electrical system the 'imaginary' unit might be current, if you are working with voltages.

And then quite often, the simplest coordinate system to analyze a system isn't the most obvious one, or maybe the coordinate system doesn't have any physical interpretation at all.

Why are people devolving to discussions of "negative apples"? next it will be "I've got zero Ferraris in the garage! How is that possible? I've never had a Ferrari in the garage".

And yes you can measure a negative length, you just need to be careful about defining your basis vectors.

Or what I am left wondering is this: Why are people acting like 11 year old nerds? Has the world gone crazy?
adx:

--- Quote from: SandyCox on April 05, 2022, 07:04:59 am ---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?

--- End quote ---

Yes, I agree with all of that. I'm hoping it helps illustrate my point - if they are the same, then what purpose is served by drawing on a 'contested' (at least students have a lot of problems with it) concept, being sqrt(-1)? Or why require a definition j*j = -1 to generate a dimension that was right there in front of us all along? Is that dimension innately and clearly defined by j*j = -1, or have people extended it by axiom, at least to some extent? If so, is that mapping (phasor to complex number) merely (or partly) synthetic, and if so, is it right to say that this mapping (in itself) is physical?

It is clear to me that people interpret this in different ways, so it is potentially impossible to convey.
HuronKing:

--- Quote from: penfold on April 05, 2022, 08:23:43 am ---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.

--- End quote ---

I'm frustrated that after pages and pages of conversation on this point we return RIGHT BACK TO SQUARE ONE. This assignment of impossibility to a definition without even considering all the assumptions that are made about the consideration of the impossibility. Really, that objection is the same thing pre-Medieval mathematicians did say about negative numbers.

Whereas the complex numbers (what Gauss called lateral numbers) are just another type of representation of numbers we deal with all the time.

How can I have negative current? Negative power consumption? Negative dollars in my bank account? Heck, I can even have negative areas in solutions to integrals.

These all assume implicitly that there is a DIRECTION to the quantity. As you say, a negative length implies a direction to the quantity. I didn't say anything about owning negative apples... but someone is owed some apples - that we both see.  ;)

So, if I can assign a forwards and backwards direction to a quantity (positive and negative)... why is it 'OMGZ IMPOSSIBLE MAN!' to assign... rotational direction to the quantity? Rotation isn't just forwards and backwards, but all the places in between. That's all sqrt(-1) means. I know you know that - but after many replies and seeing adx's latest comment (where he asks yet again what the point is of sqrt[-1]) I just shrug now.  :-//
SandyCox:
This discussion is really more about mathematical philosophy than engineering. For engineers, the subject of mathematical philosophy doesn't put bread on the table.

Here's another interesting one:

Lets say that we have a rectangle with width w and height h. Does its area remain the same if we turn the rectangle through 90 degrees? If it does, then we have "proven" commutativity for the real numbers, i.e. w*h = h*w.

I tend not to worry too much about these type of questions and rather focus on making my designs work. I need to be able to factor polynomials over the complex numbers to design filters and control loops. I need the poles and zeroes of transfer functions. I need the Fourier transform for the frequency domain perspective. I need theorems from Complex analysis. So I tend not to worry about the philosophical meaning of j. It works and that's good enough for me.
HuronKing:
Early on in this discussion the relevance of sqrt(-1) in a quantum wave function got a passing mention.

What's fun is that Paul Dirac was initially puzzled by the appearance of apparently 'negative energy states' in his relativistic solutions involving the Klein-Gordon equation. He ploughed ahead anyway and ended up mathematically discovering positrons (later experimentally verified):
https://quantummechanics.ucsd.edu/ph130a/130_notes/node478.html
https://quantummechanics.ucsd.edu/ph130a/130_notes/node490.html
https://quantummechanics.ucsd.edu/ph130a/130_notes/node504.html

Thanks @TimFox for sharing this website in another thread. There's good stuff here.  :D
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