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

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TimFox:

--- Quote from: SandyCox on April 05, 2022, 01:58:43 pm ---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.

--- End quote ---

At a more elementary level of usefulness, I have mentioned using a two-phase lock-in amplifier to measure "I" and "Q" components of a signal coherent to the reference signal.  From these two voltages (bipolar real numbers), I can treat the values as either (1.  Real and imaginary components) or (2.  Computed magnitude and phase angle).  Elementary complex algebra gives me the relationship between these two representations of the two values.  When adding values, it is simpler to use the real and imaginary components.  When multiplying values, it is simpler to use magnitude and phase.  Doing the computation in either way should give the same result, for example multiplying the complex current times the complex impedance to get the complex voltage.

adx:
Not implying anything by not having time to reply to everything or one right now, but I'll try these 2:


--- Quote from: HuronKing on April 05, 2022, 01:43:20 pm ---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.

--- End quote ---

It has effects far beyond what you might assume - I have only thought that through properly (or with any real interest) recently, and in part because of one of penfold's earlier posts (about the chickens) and your earlier power supply example. You might not have negative power consumption if it defined as consumption. Money is a fiction so you don't have anything beyond some person or corporate's idea. You can have a virtual negative area. Many concepts break, change or become otherwise ill-defined. It's not just "direction".


--- Quote from: HuronKing on April 05, 2022, 01:43:20 pm ---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.  :-//

--- End quote ---

Rotation isn't just a direction, but a quantity on its own. By your argument negating a natural number (say) is still a quantity - you can have more or less of it. Quantities can be offset. Many quantities are relative measurements, where zero has no special meaning. Angle is another quantity. When you multiply by sqrt(-1) to claim it is rotated (or kind of 'subnegated'), you start breaking more features like with some of the negatives, and you imply a whole new degree of freedom. That's why you need 2 real numbers in a complex number to promote to 2D. You seem to think sqrt(-1) is a fundamental property of all numbers so it is always there. To me no, you need to explicitly build a vector. If you go on arbitrarily adding parameters and degrees of freedom, you end up with the box set of Star Wars, and calling that a quantity.

HuronKing:

--- Quote from: adx on April 05, 2022, 04:24:43 pm ---You seem to think sqrt(-1) is a fundamental property of all numbers so it is always there. To me no, you need to explicitly build a vector. If you go on arbitrarily adding parameters and degrees of freedom, you end up with the box set of Star Wars, and calling that a quantity.

--- End quote ---

BECAUSE IT IS ALWAYS THERE. IT IS A FUNDAMENTAL PROPERTY OF ALL NUMBERS.
https://en.wikipedia.org/wiki/Complex_number


--- Quote ---Complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales."
--- End quote ---
— R. Penrose

Get educated, man - complex numbers make up ALL THE NUMBERS:
https://www.mathsisfun.com/sets/number-types.html

adx:
Wo, no. I see what's going on here. I'm going to go off and not think about it.

TimFox:
18th and 19th century mathematics is something that happened to other people.

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