Simple Questions - April 19, 2019

It's actually easier to generalize and talk about arbitrary roots of 1, i.e. complex numbers z such that zn = 1.

Now if z is a complex number, it's length (or "modulus") is the length of the line from the origin to z in the complex plane, and we write it like |z|. If z = x + iy, then we have |z| = sqrt(x² + y²). Do you see how this is related to the pythagorean theorem, or the distance formula?

We also define the angle (or "argument") of z to be the angle between the line we drew above and the x-axis. You should try and draw some pictures to visualize it.

Now if you know the length and angle of a complex number, you know it completely. In fact, z = |z|*(cos(Arg(z)) + i*sin(Arg(z))). This is basically just polar coordinates in the plane. This is useful, becuase it let's us look at what happens when we mutiply two complex numbers. We see z*w = |z|*(cos(Arg(z)) + i*sin(Arg(z))) * |w|*(cos(Arg(w)) + i*sin(Arg(w))) = (|z|*|w|)*(cos(Arg(z))*cos(Arg(w)) + i*cos(Arg(z))*sin(Arg(w)) + i*sin(Arg(z))*cos(Arg(w)) - sin(Arg(z))*sin(Arg(w))). All I've done here is foiled out (cos(Arg(z)) + i*sin(Arg(z))) * (cos(Arg(w)) + i*sin(Arg(w))). It's annoying but not super hard to show |z*w| = |z|*|w| by

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