• @captainlezbian
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    32 hours ago

    It’s just dimensionally shifted. This is not only true, its truth is practical for electrical engineering purposes. Real and imaginary cartesians yay!

  • @[email protected]
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    3 hours ago

    Doesn’t this also imply that i == 1 because CB has zero length, forcing AC and AB to be coincident? That sounds like a disproving contradiction to me.

        • @captainlezbian
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          12 hours ago

          Yeah, 1 and i should be the same size. It’s 1 in the real dimension and 1 in the imaginary dimension creating a 0 but anywhere you see this outside pure math it’s probably a sinusoid

  • @someacnt_
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    13 hours ago

    Seems like one can maybe work with complex metric. Interesting idea

  • @[email protected]
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    279 hours ago

    This is why a length of a vector on a complex plane is |z|=√(z×z). z is a complex conjugate of z.

    • @[email protected]
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      44 hours ago

      I’ve noticed that, if an equation calls for a number squared, they usually really mean a number multiplied by its complex conjugate.

  • @[email protected]
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    15 hours ago

    Looks like a finite state machine or some other graph to me, which just happens to have no directed edges.

    • @Klear
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      116 hours ago

      After delving into quaternions, complex numbers feel simple and intuitive.

    • @affiliate
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      96 hours ago

      after you spend enough time with complex numbers, the real numbers start to feel wrong

      • @TeddE
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        22 hours ago

        Can we all at least agree that counting numbers are a joke? Sometimes they start at zero … sometimes they start at one …

    • enkers
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      2011 hours ago

      I never really appreciated them until watching a bunch of 3blue1brown videos. I really wish those had been available when I was still in HS.

      • @[email protected]
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        9 hours ago

        After watching a lot of Numberphile and 3B1B videos I said to myself, you know what, I’m going back to college to get a maths degree. I switched at last moment to actuarial sciences when applying, because it’s looked like a good professional move and was the best decision on my life.

  • @[email protected]
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    2611 hours ago

    Isn’t the squaring actually multiplication by the complex conjugate when working in the complex plane? i.e., √((1 - 0 i) (1 + 0 i) + (0 - i) (0 + i)) = √(1 + - i2) = √(1 + 1) = √2. I could be totally off base here and could be confusing with something else…

      • @candybrie
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        33 hours ago

        Considering we’re trying to find lengths, shouldn’t we be doing absolute value squared?

  • @iAvicenna
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    610 hours ago

    you are imagining things

  • @owenfromcanada
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    2113 hours ago

    This is pretty much the basis behind all math around electromagnetics (and probably other areas).

      • @[email protected]
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        179 hours ago

        Circles are good at math, but what to do if you not have circle shape? Easy, redefine problem until you have numbers that look like the numbers the circle shape uses. Now we can use circle math on and solve problems about non-circles!

      • @owenfromcanada
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        2513 hours ago

        The short version is: we use some weird abstractions (i.e., ways of representing complex things) to do math and make sense of things.

        The longer version:

        Electromagnetic signals are how we transmit data wirelessly. Everything from radio, to wifi, to xrays, to visible light are all made up of electromagnetic signals.

        Electromagnetic waves are made up of two components: the electrical part, and the magnetic part. We model them mathematically by multiplying one part (the magnetic part, I think) by the constant i, which is defined as sqrt(-1). These are called “complex numbers”, which means there is a “real” part and a “complex” (or “imaginary”) part. They are often modeled as the diagram OP posted, in that they operate at “right angles” to each other, and this makes a lot of the math make sense. In reality, the way the waves propegate through the air doesn’t look like that exactly, but it’s how we do the math.

        It’s a bit like reading a description of a place, rather than seeing a photograph. Both can give you a mental image that approximates the real thing, but the description is more “abstract” in that the words themselves (i.e., squiggles on a page) don’t resemble the real thing.

        • @A_Union_of_Kobolds
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          39 hours ago

          Makes sense, thanks. More of a data transmission than an electrical power thing.

      • Dr. Bob
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        413 hours ago

        Use the Pythagorean theorem: the length of the hypotenuse is equal to the sum of the squares for the other two sides. 1x1=1. ixi=-1. 1+(-1)=0.

        • @A_Union_of_Kobolds
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          313 hours ago

          Yeah I get that, but what’s the application to electromagnetism? I’m an electrician, it’s been a few years since I had to think about induction and capacitance calculations, but I do recall them being based mostly on trigonometry. Where does i come into play, I guess is what I’m asking.

          • @marcos
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            312 hours ago

            I think the GP is going with complex numbers representing a magnitude and a rotation angle, so that side with length “i” is rotated, and A is an angle of 0°.

            But this image is out of order for that. This one would lead to A = 180°. Either way, you can’t use Pythagoras theorem anyway.

          • Dr. Bob
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            211 hours ago

            Sorry I insta-deleted because I realized I wasn’t answering the question but it looks like it still slipped through.

            I wasn’t answering the question because I don’t know. I’m aware that imaginary numbers play a major role in circuit math, but I also need an expert to ELI5.

          • @owenfromcanada
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            212 hours ago

            The “i” terms represent the induction and capacitance of a system, while the real component represents the resistance. You can think of “i” terms as the characteristics that hold energy in some way (in mechanical terms, something springy or something with inertia).

    • @Bassman1805
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      3013 hours ago

      The reason it doesn’t work is that 1 is a scalar while i is a vector (with magnitude 1). The Pythagoras theorem works with scalars, not vectors, so you’d get 1^2 +1^2 = 2.

      • @someacnt_
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        13 hours ago

        I am sorry, but… to be pedantic, pythagorean theorem works on real-valued length. Complex numbers can be scalars, but one does not use it for length for some reason I forgor.

      • @[email protected]OP
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        8 hours ago

        Far as I understand it (which is not very far), i is a scalar even if you take it to be the complex number 0 + i. Just by itself i is the imaginary unit that’s defined as i = sqrt(-1) (edit: or, well, the solution to x² + 1 = 0, but same difference), and nothing in that definition says it’s a vector quantity.

        Even though complex numbers do extend real numbers into a 2D plane doesn’t mean they’re automatically vectors, and – again, as far as I’ve understood things – they’re still treated as single entities, ie. scalars. i by itself isn’t a complex number I think, though.

        The joke is that i² = -1 by definition, so i² + 1² = 0²

        Edit: eg. nothing on the imaginary number wiki page implies that the imaginary unit is not a scalar value

        • @affiliate
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          66 hours ago

          whether or not i is a scalar depends entirely on the context.

          every vector space has an associated field of coefficients. in practice, this field is typically the real numbers. but you can have lots of other kinds of vector spaces as well, and they can be useful for certain things.

          anyways, if you have a vector space over the complex numbers, then i is a scalar, because it is a complex number. if you have a vector space over the real numbers, then i is not a scalar, because it’s not a real number.

          its worth mentioning that you can view the complex numbers as a vector space over itself. this is just a fancy way of saying that you can add complex numbers together, and you can multiply a complex number by a complex number. (one of those numbers is playing the role of scalar, and the other is playing the role of vector.) but you can also view the complex numbers as a vector space over the real numbers. and this is just a fancy way of saying that you can add complex numbers, and you can multiply a complex number by a real number.

          • @[email protected]OP
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            15 hours ago

            Right, sort of my vague understanding as well although it’s been 15 years since my university math courses. My point was more that “1 is a scalar while i is a vector” just didn’t seem correct to me, at least on a general level

    • @owenfromcanada
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      1112 hours ago

      If AB = i and BC = 0, then B would be in the same 2D space as C, but one of them would be “above” the other in 3D space (which doesn’t exist in this context, just as sqrt(-1) doesn’t exist in the traditional sense).

      So this triangle represents a 2D object that is “standing up” on the page.

      • @rtxn
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        12 hours ago

        It makes sense if you represent complex numbers as (a, b) pairs, where a is the real part and b is the imaginary part (just like the popular a + bi representation that can be expanded to a * (1, 0) + b * (0, 1)). AB’s length is (1, 0), AC’s length is (0, 1), and BC’s length will also be a complex number.

        I think.

        • @[email protected]
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          19 hours ago

          Yes. Also if you think of i as a 90° rotation (with a length of the scalar coefficient infront of i, in this case 1) . Thus one rotates you outwards away from the 2D plane, and two of those gets you back to the 2D plane, just going the other direction.