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> sqrt(-1) occurs in "real" voltages / currents all the time today in Electrical Engineering I'd say sqrt(-1) describes "real" voltages / currents; but we can

> sqrt(-1) occurs in "real" voltages / currents all the time today in Electrical Engineering I'd say sqrt(-1) describes "real" voltages / currents; but we can find other models which don't involve sqrt(-1). Those models may be more complicated, and have fewer rich connections to pre-existing work, but they're just as valid. This is usually where appeals to Occam's Razor come in, and physicists start appealing to "elegance", "beauty", etc.

dragontamer 4y ago
You're not going to be able to describe how voltages/currents work in an AC circuit unless you reach into 2-vectors / matrices. In Complex Arithmetic, we call it the "imaginary axis". But reach into Linear Algebra and you'd just call it a "2nd dimension". When things oscillate (and AC current is oscillating), there's just certain mathematical properties that occur. It doesn't matter if its electricity, a spring/mass system, a bridge, sine-waves, music theory, Fourier Series or anything else, its all going to be represented by complex-math extremely elegantly. Or, have a 2-vector linear-algebra representation, since its equivalent. EDIT: I guess there's the differential-equations representation, but I feel that reaches into linear-algebra very, very quickly... or complex numbers. (E^(pi * i) keeps coming up...)
- DontchaKnowit 4y ago
  I have never seen complex numbers represented via linear algebra... would love a crash course on this as I find reasoning about complex numbers to be very difficult and find linear algebra to be imminently logical in all cases.
  - dragontamer 4y ago
    A complex number is "just" a 2-vector. Instead of calling the two dimensions "x and y", consider them "(unnamed) and i". For example: the vector <1, 1> is equivalent to 1 + i. The complex conjugate is a reflection across the x axis: <1,1> into <1, -1>, or in the Complex representation from 1+i into 1-i. Addition of numbers progresses just like addition of vectors: <1,1> + <2,2> == <3,3>. Similarly: (1+i) + (2+2i) == (3+3i). -------- IIRC, Multiplication of Complex numbers is a rotation across the two axis. The overall concept is pretty difficult to describe succinctly, but it is "obvious" in a simple case. Multiplication by "i" is just a 90-degree rotation counter-clockwise. That is: <1, 1> rotated 90-degrees is <-1, 1>. Or 1+i rotated 90-degrees is (i * (1+i)) == (-1 + i). Rotating again is another multiplication of i, or i * (-1 + i) == (-1 - i) == <-1, -1> in vector form. Then <-1, 1> and finally <1, 1> again (4x 90-degree rotations == 360 degrees). -------- Etc. etc. etc. Everything you do in complex arithmetic has a good interpretation in the vector / geometric space. A lot of the intuition needs to be built from the polar form of complex numbers: r * e^(theta * i), since the use of complex numbers is largely about rotations about the axis.
- chriswarbo 4y ago
  > It doesn't matter if its electricity, a spring/mass system, a bridge, sine-waves, music theory, Fourier Series or anything else, its all going to be represented by complex-math extremely elegantly. This is an appeal to "elegance", which I mentioned in my last sentence. I don't disagree; but it's not quite the same as proving "existence".

dragontamer 4y ago

You're not going to be able to describe how voltages/currents work in an AC circuit unless you reach into 2-vectors / matrices. In Complex Arithmetic, we call it the "imaginary axis". But reach into Linear Algebra and you'd just call it a "2nd dimension". When things oscillate (and AC current is oscillating), there's just certain mathematical properties that occur. It doesn't matter if its electricity, a spring/mass system, a bridge, sine-waves, music theory, Fourier Series or anything else, its all going to be represented by complex-math extremely elegantly. Or, have a 2-vector linear-algebra representation, since its equivalent. EDIT: I guess there's the differential-equations representation, but I feel that reaches into linear-algebra very, very quickly... or complex numbers. (E^(pi * i) keeps coming up...)

DontchaKnowit 4y ago
I have never seen complex numbers represented via linear algebra... would love a crash course on this as I find reasoning about complex numbers to be very difficult and find linear algebra to be imminently logical in all cases.
- dragontamer 4y ago
  A complex number is "just" a 2-vector. Instead of calling the two dimensions "x and y", consider them "(unnamed) and i". For example: the vector <1, 1> is equivalent to 1 + i. The complex conjugate is a reflection across the x axis: <1,1> into <1, -1>, or in the Complex representation from 1+i into 1-i. Addition of numbers progresses just like addition of vectors: <1,1> + <2,2> == <3,3>. Similarly: (1+i) + (2+2i) == (3+3i). -------- IIRC, Multiplication of Complex numbers is a rotation across the two axis. The overall concept is pretty difficult to describe succinctly, but it is "obvious" in a simple case. Multiplication by "i" is just a 90-degree rotation counter-clockwise. That is: <1, 1> rotated 90-degrees is <-1, 1>. Or 1+i rotated 90-degrees is (i * (1+i)) == (-1 + i). Rotating again is another multiplication of i, or i * (-1 + i) == (-1 - i) == <-1, -1> in vector form. Then <-1, 1> and finally <1, 1> again (4x 90-degree rotations == 360 degrees). -------- Etc. etc. etc. Everything you do in complex arithmetic has a good interpretation in the vector / geometric space. A lot of the intuition needs to be built from the polar form of complex numbers: r * e^(theta * i), since the use of complex numbers is largely about rotations about the axis.
chriswarbo 4y ago
> It doesn't matter if its electricity, a spring/mass system, a bridge, sine-waves, music theory, Fourier Series or anything else, its all going to be represented by complex-math extremely elegantly. This is an appeal to "elegance", which I mentioned in my last sentence. I don't disagree; but it's not quite the same as proving "existence".