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

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TimFox:
The German mathematician Kronecker famously said "Natural numbers were created by God, everything else is the work of men."
In that context, "everything else" includes zero, negative integers, rational fractions, irrational numbers, etc., since "natural numbers" in mathematics means the set of positive (non-zero) integers.
https://www.cantorsparadise.com/kronecker-god-and-the-integers-28269735a638

HuronKing:

--- Quote from: SandyCox on March 24, 2022, 05:26:00 pm ---Some people are of the opinion that only the natural numbers (excluding 0) have "physical meaning". (whatever "physical meaning" might be?)
Does 0 have physical meaning?
Do the negative numbers have physical meaning?
What about sqrt(2) which is an irrational number?


--- End quote ---

penfold:

--- 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.

penfold:

--- Quote from: SandyCox on March 24, 2022, 05:26:00 pm ---Some people are of the opinion that only the natural numbers (excluding 0) have "physical meaning". (whatever "physical meaning" might be?)
Does 0 have physical meaning?
Do the negative numbers have physical meaning?
What about sqrt(2) which is an irrational number?

Extending the real numbers to an algebraic structure in which the square root of minus 1 exists is brilliant. Furthermore, all polynomials can be factored into monomials. How great is that?

The field of Complex numbers is not the same as the vector space of two-dimensional vectors over the real numbers. The multiplication is different. j isn't a unit since its square isn't equal to j. 1 is the unit of the Complex numbers.

In my opinion, trying to assign "physical meaning" to mathematical concepts only works for very simple problems. Just trust the Mathematics and look at what the theory tells you. Sometimes our intuition fails horribly. Trust the math.

--- End quote ---

The formalization of maths, was a surprisingly recent occurrence, at least with sets, groups, and categories defining algebras and arithmetics from a truly axiomatic basis. You've got to bear in mind that there are several things which the figure '1' represents: being the multiplicative identity, the successor function, and the start of the number line, all having different philosophical interpretations (historically leading to disputes over their significance) but were defined as equal/equivalent by Russel and Whitehead (1920s? I forget the date). With the modern definitions, those disputes are moot and/or based on outdated origins.

"Trust the math" is a very valuable phrase that I'm glad to see. Because of the physical significance of numbers and quantities and difficulties in finding relationships between them is largely what pushed Grassman, Hamilton, and Clifford vector and geometric algebras back in favor of the wishy-washy i,j,k vector calculus operators. It took Clifford algebra many years to resurface as a better mathematical representation for EM in relativistic and quantum theories. In geometric algebras, not only do i,j, and k become kinda bendy changeable vectors, but you also have to contemplate the ideas of planes and cubes as vectors and their physical representation must be viewed through a metric, potentially on a topological manifold... the maths is beautiful (I have low standards) but it stops making sense at the moment you attempt to visualize it.

TimFox:
Although we can probably agree that the voltage variable V(t) is real-valued in electrical engineering, in Quantum Mechanics the physical wave function must be complex (in the mathematical sense of real and imaginary components). 
This was impressed upon me in college Quantum Mechanics classes, since a non-complex wave function would not have enough degrees of freedom, and the time-dependent Schrödinger equation explicitly starts with i.
https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation
(I tried without success to cut and paste the equation from that Wikipedia article:  see the section "Preliminaries")

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