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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 diffi

• 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.