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| adx:
--- Quote from: SiliconWizard on March 26, 2022, 06:45:05 pm ---I don't get what the problem is. Complex numbers are just a useful tool. Like most other tools we use. Their usefulness comes from the fact we get a lot more out of them than what defines them in the first place. If you think even the most mundane tool or model of reality we use is in fact more "real" than this, you're pretty deluded. If you have a problem with complex numbers, you should have a look at quaternions. You can also look at epsilon numbers. I find it interesting that some people would have no problem discussing hairy physics and convoluted quantum mechanics, yet find complex numbers "odd". :popcorn: Seriously, how abstract is math compared to modern theoretical physics? The latter is actually pure maths for the most part. And as bsfeechannel noted, there is no engineering without math anyway. --- End quote --- That (and the subsequent stuff about abstraction) is so tangential to what I'm saying (or asking). I know complex numbers are used as a tool. I was (partly) joking when I said "Engineers don't use j (or i)". My question about the "physical relevance" of sqrt(-1) (as a mathematical being) appears to have been too easy to take a different way, but there are only so many ways I can try to explain it without digging an impossibly deep hole (interesting to see how deep it goes!). I thought I was asking about the fundamental mathematical meaning of sqrt(-1)'s physical relevance to phasors or anything. Perhaps that was how it was taken? My formula x^2+1=0 above has no real solution. It states an impossibility. Imaginary numbers are an algebraic 'what if' to get around that, in the same way a negative number is a what if (what if I remove x units?). Neither imply possibility to a measurable value of x. No problem with either. Removing x units has clear (but not universal) physical relevance (perhaps I should have said mathematical relevance) to engineering measurements. Yes it's a tool as defined, but easy to explain why. Why does the algebraic 'what if' solution to x^2+1=0 have direct mathematical relevance to phasors? |
| bsfeechannel:
--- Quote from: adx on March 27, 2022, 01:06:24 am ---Removing x units has clear (but not universal) physical relevance (perhaps I should have said mathematical relevance) to engineering measurements. Yes it's a tool as defined, but easy to explain why. Why does the algebraic 'what if' solution to x^2+1=0 have direct mathematical relevance to phasors? --- End quote --- x² + 1 = 0 gives you a hint that what you're looking for is some x that is the side of a negative area. x can be neither a positive nor a negative number, because such numbers give you a positive area. So it is clear that x is in another dimension. Two dimensions form a plane. If instead of being added or removed from a single dimension, your x units are rotating in a plane--for instance, inside a generator--complex numbers seem adequate to describe your measurement. |
| adx:
Ok I'll buy it, despite the hint that the number breaks squaring (the negation operator is functional only once in the square) despite the algebra of the square being the source of the number in the first place. The question then remains whether this fundamental nature of the number (and complex plane) has direct relevance to phase of sine waves, or whether phasor analysis merely purloins the property of the complex plane as a "handy trick"? Negation breaks the summing property of addition, but my point is that has real relevance to positive as well as negative numbers. Claiming the same reality and physicality exists for imaginary vs real numbers isn't beyond imagination, but to me, claiming it is no less tenuous than negation in real numbers, seems like it might be false. We are taught (and people believe) it is as fundamental a law to engineering as electrons repelling. That has to be absolutely beyond question to be right. |
| penfold:
--- Quote from: adx on March 27, 2022, 01:06:24 am ---[...] Why does the algebraic 'what if' solution to x^2+1=0 have direct mathematical relevance to phasors? --- End quote --- I had half-baked a response to that earlier actually (hopefully that doesn't get taken as evidence of non-causality), I was pondering my own initial reaction to complex numbers from high-school maths. I think that 'what if?' solution is typical of most people's first exposure to complex numbers, demonstrating that there are still "some roots" to an apparently 1d problem. I initially just accepted that it was 'nice' and plodded along. By about first or second year EEE maths, when functions of a complex variable were introduced formally with power series (of a complex variable), residues, etc, it shed a little more light on things, at least to demonstrate that the function of x, for which we'd only ever assumed to be a function of a real value (and yet had complex roots... go figure) could actually be a function of a complex 'z=x+jy' which more naturally has a complex root, where f(z)=z2+1 is now a surface plot with height defined for values of x and y... only the height is complex but only goes completely to zero at +j and -j (i.e. y=+1,-1). If you were to draw cross-sections of the surface plot (as x2+1 is the cross-section at y=0) and the same function will look slightly different... you can even plot a cross-section at an angle where both x and y are varying... or any arbitrary function that links x and y in response to an arbitrary parameter (I'm too tired to wonder if that was relevant... could be Euler's formula with 'phase' as a parameter... really not sure where I'm heading with that). Looking at pole-zero responses of linear networks is where it began to make more sense to me. A circuit composed of a combination of real and imaginary impedances, in the Laplace domain will have moments (with respect to varying ω) with a tendency to head towards zero or infinity (as jω in the factors of the numerator or denominator cancels the imaginary part of the root)... except the real component of the root is not 'canceled' by jω and the root doesn't quite go to zero, nor does the transfer function get quite to zero or infinity. The resulting combination of real and imaginary values of the transfer function determines the phase shift of the output with respect to the input. |
| adx:
Only time for a partial reply for now: --- Quote from: HuronKing on March 25, 2022, 07:20:57 pm --- --- Quote from: adx on March 25, 2022, 04:04:02 pm ---First, my concern over sqrt(-1) in electrical engineering, penfold has it right: "and the j is an operator rather than a quantity ... it is stretching it a bit far to say that it is a physical quantity". --- End quote --- Did I say it was a physical quantity? Please show me (I've tried to find where I might've implied that but I don't see it). j is not an Ohm. But it is a representation of phase-shift in Ohms and a damn good one. Is that not physically relevant? --- End quote --- Tricky semantics. What I and I assume penfold were referring to was somewhere between a physical unit and representation as a tool. You said sqrt(-1) "has immense physical significance, just as 'zero' and 'negative' have immense physical significance" which I took to be that middle meaning. Saying j is physically relevant is different from saying sqrt(-1) is, to me. The latter being a very abstract mathematical concept, but j being defined as a practical tool by Steinmetz (yes, with overlap). sqrt(-1) is the first whole positive imaginary number (if there is such a thing) hence a quantity (of sorts), j is a rotation operator as defined by SandyCox in (a, b)(c, d) = (ac-bd, ad+bc) (with j as b or d). They happen to be algebraically identical. I've explained more since, but I hope that helps explain a bit better where I think I'm coming from. --- Quote from: HuronKing on March 25, 2022, 07:20:57 pm --- --- Quote ---I don't think it is any sort of tautology to say mathematical concepts are not real, if one then goes on and asserts that some part has physical relevance. Not all engineers are naturals at maths and can easily identify where that link appears (ie goes from nothing to something without explanation). Some people here seem to be struggling with it too - perhaps from over-familiarity. --- End quote --- You might as well be arguing that multiplication has no 'physical relevance' to engineering because you could just add the numbers up... like, yes? What is your point? Should we count on our fingers and toes because applying math makes us feel dumb? ;) --- End quote --- Yes, if it "adds" nothing practical or needs to be applied abstractly by some engineers who might then not know what they are doing as clearly. --- Quote ---I've said, many times, that engineers can and do get confused by this. And there are some engineers better at it than others. None of that is an excuse. There are way more problems I can solve quickly and efficiently with multiplication than I can with addition - even though multiplication is just an extension of addition. --- End quote --- And that's why. We don't want engineers getting confused on the job. I've 'moved the ruler along' n times to check a calculation, or tipped liquid into a measuring container to work out volume that could have been calculated. It's not what I meant anyway. My reply was referring to your suggestion that saying all numbers are imaginary (I'm paraphrasing) is a tautology because everyone knows that. For something like sqrt(-1), I don't know where it gets real. Time for one more before nie nies: --- Quote from: HuronKing on March 25, 2022, 07:20:57 pm ---I'm not citing waffle-y texts at you. I'm citing actual engineering practices. You can take them or leave them. https://www.electronics-tutorials.ws/accircuits/power-triangle.html --- End quote --- Although I've clarified more since, this is exactly what I don't have a problem with. j is defined only in the annotations on the diagrams as a 90 degree shift pictorially and as reactance. j doesn't appear in any of the body text or its formulae. The only hint as to what j might be (as a symbol) is mention of "which is the vector sum of the resistance and reactance". This is what I mean by things like "to the point they realise sqrt(-1) has no physical relevance, with j being the unit vector that I say it is". |
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