Fundamental Theorem of Algebra


#1

From numberphile:

Key insight:

  • A complex number, when multiplied by itself (x^2, x^3, etc.) will scale the magnitude and add the angles
  • If we have a circle of possibilities (same magnitude), we could have many angles along that circle. For example, if 60 degrees is our angle, and we got there in 3 steps, then 20 + 20 + 20 would work (1/3 * 60), along with (60 + 360) / 3 = 420 / 3 = 140, since 140 + 140 + 140 = 420 = 60 degree angle. Along with (60 + 360 + 360) = 780 / 3 = 260. Three different angles that, when combined 3 times, give us 60 degrees.
  • So, we show that a root exists, and there are N points that (when multiplied) would get us to that angle.

#2

Reply to reader:

Basically, we have two things to show:

  1. There is a “circle of possibilities” in the blue graph that corresponds to a circle on the red graph

  2. The circle on the blue graph can be shrunk, also shrinking the corresponding red circle. Eventually this circle will hit the “zero” point.

  3. [most important] any point on the red graph can be reached by multiple points on the blue graph.

For example, let’s say the point on the red graph is length 10 at an angle of 60 degrees.

How can we get to it in 3 steps? (I.e., we’re trying to solve the polynomial x^3 = “10 at 60 degrees” or x^3 = 5 + 8.6i

Getting to 10 in 3 steps (|x| * |x| * |x| = |10|) means having the cube root of 10 as our magnitude.

Getting to 60 degrees in 3 steps means:

angle + angle + angle = 60

One solution is angle = 20. Another is angle = 140, and another is angle = 260. [140 + 140 + 140 = 420 which is equivalent to 60 degrees, it just loops around once].

The idea is to show that eventually some point on the red graph will hit the point we want, and there are several complex numbers [in the blue graph] that correspond to that red point. This wasn’t made very clear in the video!