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

--- Quote from: HuronKing on March 24, 2022, 12:47:53 am ---[...]
This is veering really close to the question of "is mathematics physical?" and that's a big question!  :D

I'm saying that the terminology associated with 'imaginary' numbers is something we inherited from ancient mathematicians who didn't really know what they were dealing with - we got over it with negative numbers and zero, but sqrt(-1) is still something to be struggled with by students. I don't actually blame the ancient mathematicians - it's just unfortunate their prejudices about how to philosophically interpret these definitions have cursed students of today who hear something like 'imaginary numbers' versus 'real numbers' and assume these labels, by themselves, have something to do with physicality. They don't, at least in my opinion.  :)

In the case of electric circuits, we know the impedance of an inductor is Z = jwL and the impedance of a capacitor is Z = 1/jwC (thanks Steinmetz!)

Those impedances have physical effects and meaning on our circuits even though they have a weird looking j out in front. And while it is challenging to learn it's not so mysterious. As you said, it just means the incident current and incident voltage undergo a phase shift in time.

--- End quote ---

Hang on, that's not a very big question, and the answer is relatively simple. Maths itself is not physical, or it is only as physical as any language in which you can express logic, it's conceptual. The links between that language and quantities defined within is also defined and there is an observable consistency between the results of additive processes in 'nature' and in the mathematical system etc... hence why one should always include units against any number with physical significance because that defines the process by which one takes the number on paper and stacks calibrated metre-rules end-on-end to reach a distance. It's all defined, we're safe.

The middle bit was just a segue, I guess it comes across as prejudicial in how dismissive philosophers and mathematicians were, but it wasn't until mid-1800s to early 1900s when the definitions of maths were really put in place to settle the philosophical absurdity of a negative number of chickens (being difficult enough to convey the attribute of "owning no chickens" without also listing everything else you don't own, let alone being able to express such a severe lack of chickens to consider it negative). That is a hugely important split because the same principles of ratiocination that form(ed) mathematics also form the scientific method's basis. So without the formal definitions, the link between reality and squiggles on paper become incredibly tenuous and (logically) absurd.

The concept of reactance in terms of the imaginary unit is defined for sine-waves of fixed frequency, where the derivative preserves the wave shape (sin, cos, -sin, -cos etc) but is translated with respect to the parameter, the imaginary unit facilitates that translation by a simple multiplication... for sine-waves. The decomposition into sine-waves is nice like that for letting us do that, but it isn't unique; we could decompose a signal into squares, wavelet-type things or any other orthogonal basis and for each of those we would have a new complex-number-type-operator, it would just be a far bit more grotesque than an 'i'. In each of those cases, it would be representative of a transformation of the representation of voltage and not in itself a quantity.
adx:

--- Quote from: bsfeechannel on March 24, 2022, 01:29:46 am ---
--- Quote from: adx on March 23, 2022, 11:05:16 pm ---bsfeechannel: Seen your post come in. Yes j appears in a couple of places, as an annotation. I was thinking of cheap ass VNAs and Smith charts when I made my claim.

--- End quote ---

Let me get this straight. Because j doesn't appear explicitly on the VNA display, does it mean it is not there? Isn't the display representing a two-dimensional vector space? Aren't VNAs, VECTOR Network Analyzers?

Your point seems moot.

--- End quote ---

Yep. There's degrees, ratios, other numbers, the concept of a 2D vector space, but no j (a bit like expecting there to be a "y" on a plot of voltage vs current), and certainly no sqrt(-1).
bsfeechannel:

--- Quote from: adx on March 24, 2022, 01:43:15 am ---Yep. There's degrees, ratios, other numbers, the concept of a 2D vector space, but no j (a bit like expecting there to be a "y" on a plot of voltage vs current), and certainly no sqrt(-1).

--- End quote ---

And of course we should remove the study of Cartesian coordinate system from electronics engineering because y doesn't appear on the screen of any oscilloscope despite the fact that its display IS a Cartesian coordinate system and that any engineer who deserves to be called by that name must be capable of interpreting a Cartesian coordinate system. All of this because y doesn't appear on a plot of voltage vs current.

Gimme a break. Stop giving us that kind of crap only to justify your misconceptions and your precarious understanding of math and physics when applied to engineering.
adx:
That's not what I'm saying, and the fact you can't work out what I mean has been a surprising insight into the way the human mind works.

Actually no scratch that and I'll remain true to form; it is what I kind of mean (and of course that insight is no surprise to me). Yes, stop studying Cartesian coordinates and silly unit vector formulas. Forget about y. Ignore "functions". Teach the oscilloscope display for what it is.

Prove my misconception. Explain what is the direct relevance of sqrt(-1) to engineering without reference to waffley texts.
bsfeechannel:

--- Quote from: penfold on March 24, 2022, 01:42:31 am ---Hang on, that's not a very big question, and the answer is relatively simple. Maths itself is not physical, or it is only as physical as any language in which you can express logic, it's conceptual.
--- End quote ---

Well, if the universe appears to behave consistently logical in certain circumstances, math can provide a convenient description of what is going on and even help to predict future discoveries.

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