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| "Veritasium" (YT) - "The Big Misconception About Electricity" ? |
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| penfold:
--- Quote from: HuronKing on March 24, 2022, 08:33:59 pm ---[...] I stand by what I've said - does j have physical relevance? Yes. :) --- End quote --- Engineer to engineer, yes, I'll accept that, never really disputed the "relevance" and all though I described complex only as a mathematical nicety, it's a hella useful one! The rest is just a pedantic aside... I may have read too much into the word "relevance"... but hey... I've been strict enough with aetherist about definitions, measurability, etc, I'm just on high alert haha. But yeah, I think that because a VNA can work out the phase in terms of I and Q using only electronic interpretations of mathematical functions... I'd be happy (not that my sense of happiness is relevant... but a happy penfold doesn't argue pedantry as much) at a stretch to call it (i, j etc) a (shudder) measurable quantity... just not a pure one and not unique... but more than sufficient for 99% of professional engineering and 85% of research and a valuable link between "reality according to test equipment" and useful algebra. Dimensionality is a tricky one. In the sense that you could have y as a function of x in the field of real numbers and plot y vs x on a cartesian graph, it isn't strictly multi-dimensioned until you assign a metric and some "vector-y-ness" to x and y, it's the same even as far as the complex field until you find an alternative representation (say as a matrix) cue Hamilton. But... Hamilton showing that complexs can also be represented as row and column 'vectors' or matrices (see wiki page). It's then when sqrt(-1) starts to lose physical significance (to me at least) or to no more nor less significant than any other vector notation. Yeah, it's still the same number in disguise, but doesn't necesarily need to be sqrt(-1), or even an orthonormal basis, so long as it spans the space. |
| bsfeechannel:
--- Quote from: penfold on March 24, 2022, 03:45:44 am ---My verbal description remains purely verbal and not at all physical, it just describes a physical object. --- End quote --- Your DNA is a description of you. But it is a description that can replicate itself and even build an entire you. We can encode your DNA sequence using the letters ACGT. It'll describe you uniquely. It'll be purely verbal, but once decoded to assemble the actual nucleic acids it represents, it'll be an functional polymer. So, is math the encoding of the "DNA" of the universe? That's what David Hilbert and his program aimed to ascertain until Kurt Gödel screwed it all up. --- Quote ---Maths is a descriptive language in which the natural phenomena are described, from those descriptions we can hypothesize, test, and refine new theories... the phenomena, including the big bang, relativity, quantum, etc all existed before humans and maths... yet that curiously happened. The language in which these descriptions are encoded - since it can be communicated verbally... is not exclusively physical. --- End quote --- I would say math is perhaps language minus contradiction. Since it doesn't admit paradoxes, it is a convenient tool to describe things for which ambiguities would be inadmissible. I like Al-Khwarizmi's preface when he introduced algebra to the world in 850. The fondness for science [...] has encouraged me to compose a short work on Calculating by Completion and Reduction [a.k.a algebra], confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, law suits, or trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned. So that's what math is all about: making life easier and less ambiguous. |
| bsfeechannel:
--- Quote from: adx on March 24, 2022, 02:26:38 pm ---This is all a bit silly - it started with a gentile troll about i vs j, then we're now back to arguments over half-arsed engineering. --- End quote --- You reduced engineers to mere solder monkeys (no Cartesian coordinates, no vectors, no y, no functions). What do you expect? --- Quote ---Uh uh. We'll have the reality of the industrial engineer, --- End quote --- Your reality may vary, then. Because in my reality of an engineer in the industry, knowledge of math and physics count a lot. --- Quote --- what you're complaining about is not misconceptions, but work. --- End quote --- You don't get it. What you're advocating creates engineers who can't see beyond a limited set of "best practices" or rules of thumb. Heck, before I was an engineer, I was a technician. And even in our formal training in electronics during high school we were taught to apply the Cartesian coordinate system (that we had learned in middle school) to interpret the measurements of scopes, plotters, and whatnot and complex numbers to analyze AC circuits. Your point is inexcusable. --- Quote ---So run with the reality, and stop assuming students need to "study" Cartesian coordinate systems (why?!) and teach the concepts. --- End quote --- But, but, but, the Cartesian coordinate system is a concept. --- Quote ---All this mathematics and (dare I say it) physics, does no good. --- End quote --- I weep for the future. --- Quote ---I was going to let you have that one, --- End quote --- Of course you were. Look at the venerable Rigol DS1052E. See the X and Y markings near CH1 and CH2, respectively? And how about the MATH button? Adding, subtracting, multiplying channels and FFT-ing, as far as I know are math operations and functions. CRT scopes even had a Z axis for controlling the trace intensity. So, ¯\_(ツ)_/¯ --- Quote ---Once again, this is about sqrt(-1), not vectors. That expensive thing you showed is called a vector network analyser, not a really complex mathematical network analyser. --- End quote --- Complex numbers form a real vector space. That's how you visualize them. |
| SiliconWizard:
Complex numbers are just vectors in R², with the property: i² = -1. You can write i as the (0, 1) vector, and the multiplication as a generalization of the cross-product of two vectors. Actually, i² = -1 (or: (0, 1)x(0,1) = (-1, 0)) comes naturally from the generalized cross-product in R². |
| SandyCox:
--- Quote from: SiliconWizard on March 25, 2022, 03:40:28 am ---Complex numbers are just vectors in R², with the property: i² = -1. You can write i as the (0, 1) vector, and the multiplication as a generalization of the cross-product of two vectors. Actually, i² = -1 (or: (0, 1)x(0,1) = (-1, 0)) comes naturally from the generalized cross-product in R². --- End quote --- The concept of a cross product is only defined for three-dimensional vectors. The complex numbers form a commutative ring, more specifically a field and a complete metric space. So calling it a vector space is confusing. It still is a vector space, but with more properties. So let's call it a field. |
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