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Complex numbers are about rotation. That’s the only odd insight in Euler’s equation. Once you get that, it all falls into place (and in fact starts to explain a
by garmaine 7y ago
Complex numbers are about rotation. That’s the only odd insight in Euler’s equation. Once you get that, it all falls into place (and in fact starts to explain all the use of complex numbers in engineering).
- xaedes 7y agoHm does that mean you could say the complex quantum probability is just a probability over rotations?
- tim333 7y agoWell you add up all the rotation like stuff and square the result to get the probability. In the book QED Feynman describes it as adding little arrows to explain it to a lay audience. In physical terms it's probably most like adding waves and then the square of the amplitude or wave energy is the probability but not quite - no one really knows what the actual reality is.
- baq 7y agoKinda sorta, imagine that probability has a direction. Interference is then about alignment of these directions.
- posterboy 7y agowhat does electric engineering have to do with rotation? Generators perhaps run in circles, an inductor is a coil often enough, but a capacitor?
- mikorym 7y agoElectromagnetism concerns movement of what appear to be moving circles. If you roll a circle with a pen marker it draws a sine or cosine curve. If you try to draw the spiral of a light wave you get perpendicular sine and cosine curves. Light is electromagnetic radiation and in turn both electricity and magnetism are fundamentally connected forces.
- posterboy 7y agothere is no "spiral of a light wave" lol. If anything, I'd think of transversal movement. Ironically I cannot say that nobody has ever seen a photon. But nobody has ever seen an electron. There could be a model in which electrons move transversal (at mm/s), mediated by photons, but I have not looked that closely into it. It's not part of the usual theories. if you attach a pen to a hoop and move the hoop along a wall, you do not get a wave, you get (edit: what looks like but isn't) sharp discontinuity where the pen touches the ground (https://google.com/search?q=cos%28x-sin%28x%29%29 https://google.com/search?q=cos%28x-sin%28x%29%29)
- garmaine 7y agoCircular polarized light is a spiral, at least in terms of how we normally draw the vector annotation.
- rimliu 7y agoYou may say that AC has a lot to do with rotation.
- garmaine 7y agoEM is about rotating fields. Phase is very fundamentally connected with rotation. These are where complex numbers show up.
- creato 7y ago> Complex numbers are about rotation. Why? (I'm not denying that it is, just pointing out that like the person I first replied to, you are jumping over the interesting part here and stating interesting conclusions like they are definitions.)
- garmaine 7y agoAs I said, "that's the only odd insight." I'm not claiming it's obvious, it's not. but once explained it makes sense and the rest falls in place. Here's a better resource than I can type up: https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ https://betterexplained.com/articles/a-visual-intuitive-guid...
- jacobolus 7y agoA complex number “is”† a quotient of two Euclidean vectors in a plane. That is, Z = u\v is the object which when you multiply it by one vector u, scales and rotates that into another vector v. We can write uZ = uu⁻¹v = v. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf https://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf https://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf † More precisely, the algebra of complex numbers is structurally the same as the algebra of quotients of planar vectors.