n-th roots and rotations

One last contribution, the most visual: in C, extracting an n-th root holds no mystery. There are always exactly n solutions, and they draw a regular polygon.

The n-th roots of unity

We look for the z with z^n = 1. Setting z = r e^(iθ):

r^n e^(inθ) = 1 = 1 × e^(i·0)

  ->  r^n = 1 so r = 1          (r is a positive real)
  ->  nθ = 0 [2π] so θ = 2kπ/n

Hence the n solutions:

z(k) = e^(2ikπ/n)        for k = 0, 1, 2, ..., n-1

Beyond k = n - 1 the same points come back.

The picture

These n numbers all have modulus 1 and are evenly spaced by an angle 2π/n: their images form a regular n-gon inscribed in the unit circle, always with a vertex at 1.

    n = 3                        n = 4                    n = 6
        •                          •                    •     •
       / \                       / | \                 /       \
      /   \                     /  |  \               •         •
     •-----•---• 1              •---+---• 1             \       /
      \   /                      \  |  /                 •     •
       \ /                        \ | /
        •                           •
  1, j, j²  (angles 120°)     1, i, -1, -i          six vertices, 60°

For n = 3 we recover 1 and the two solutions of z² + z + 1 = 0 met in the previous course: the loop is closed.

A useful property: the sum of the n roots of unity is zero as soon as n ≥ 2 (evenly spread vectors cancel out).

The n-th roots of an arbitrary complex number

Same method for z^n = a, with a = ρ e^(iφ):

            n___              φ + 2kπ
z(k) =    ⁿ√ρ   ×  e^( i × --------- )        k = 0, ..., n-1
                                n

Take the real n-th root of the modulus, and split the argument into n, shifted by full turns. Again n solutions forming a regular polygon — simply rotated and resized.

Multiplying is rotating

Let us keep the geometric reading, the most fruitful one:

multiplication by e^(iθ)          ->  rotation by θ (modulus unchanged)
multiplication by i = e^(iπ/2)    ->  quarter-turn anticlockwise
multiplication by -1 = e^(iπ)     ->  half-turn

This gives a way to compose transformations: instead of multiplying rotation matrices, multiply complex numbers. A rotation about ω by angle θ is written in a single formula:

z' - ω = e^(iθ) (z - ω)

This is the standard tool in plane geometry exercises, and the principle behind quaternions in dimension 3, used in computer graphics and robotics to compose rotations without gimbal lock.

Where they really turn up

Electricity        : a sinusoidal current is a "phasor" A e^(iωt);
                     phase-shifting means multiplying by e^(iφ)
Signal processing  : the discrete Fourier transform is built on the n-th
                     roots of unity — that is what makes the FFT possible
Wave physics       : interference is explained by adding rotating vectors

The FFT — the most used algorithm in numerical computing — rests entirely on the symmetries of the regular polygon described above.

Summary

• z^n = 1 has exactly n solutions: e^(2ikπ/n), k = 0 … n-1.
• They form a regular polygon inscribed in the unit circle, with a vertex at 1.
• Their sum is zero as soon as n ≥ 2.
• For z^n = a: n-th root of the modulus, argument divided by n, shifted by 2kπ/n.
• Multiplying by e^(iθ) means rotating; a rotation about ω is z' - ω = e^(iθ)(z - ω).
• These roots underpin the Fourier transform and the FFT.