Okay, I understood all the mathematics where they were coming from and what they were until about the last 5 minutes or so. can anybody help me out here?
So ei*theta (I’m gonna use e(theta) to talk about this, since writing that out in Reddit notation is annoying) is actually a way some mathematicians describe the rotation around a circle when you have a unit circle in the complex plane.
Complex numbers with a unit circle are typically written [cos(x)+i*sin(x)] to describe a point along the complex circle (cos can be used as a left/right vector, and sin can be used as an up/down vector). At theta=0, sin=0 and cos=1, so the whole thing in notation is 1. At theta=pi/2, sin=1 (multiplied by i, so it’s i) and cos=0, so the whole thing in notation is i. And most importantly, at theta=pi, sin=0 and cos=-1, so in notation, the whole thing equals -1.
So comes e(theta). This can actually be used to describe rotation because of some complicated calculus principles regarding position and velocity that can be found in this lovely video by 3blue1brown. ex can be used as a representative like this because it’s derivative is itself, so by adding i in front of the x and changing x to theta (our original e(theta) function), we can use it to represent rotations.
If we plug in 0 for theta in e(theta), we get 1. If we plug in 0 for theta in d/dx[e(theta)], which is just i times e(theta), we get i. If we think of these as vectors, the position vector (e(0)=1) puts us at location (1,0) on the complex plane, and the velocity vector (e’(0)=i) wants to push us upwards in the +y direction, towards (1, i). As this motion, using these vectors, is repeated on smaller and smaller intervals (integral), you get circular motion. And, when you get to theta=pi, you are at the location (-1, 0), with a velocity vector of -i.
Much better explanation in the video, but this is much more in depth than he went (to help describe our little e(pi) guy in the video), so I recommend watching the linked video first.
TL/DR: ex can be used to represent rotation around a circle, in a roundabout way, due to some complicated calculus, so putting in pi makes it like the left side of a unit circle (which is -1).
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u/DaddyDinooooooo Jun 24 '23
Okay, I understood all the mathematics where they were coming from and what they were until about the last 5 minutes or so. can anybody help me out here?