Roots of Unity

What this is?

An Argand diagram of the n-th roots of unity: points zk = cos(2πk/n) + i sin(2πk/n) for k = 0,…,n−1. They lie on the unit circle, equally spaced, forming a regular n-gon. Multiplying by a unit complex number rotates; raising to the n-th power maps all points to 1.

Answer — drawing

Reason

  1. Solutions of zn = 1 are zk = cos(2πk/n) + i sin(2πk/n), k = 0,…,n−1 (De Moivre).
  2. All have |zk| = 1, arguments 2πk/n, hence equal spacing by 2π/n and a regular polygon.
  3. Sum of all roots is 0 and product is (−1)n−1. The polygon (circumradius 1) has area A = (n/2) · sin(2π/n).

Check

    We verify zn=1, unit modulus, equal angle steps, sum/product identities, chord length 2·sin(π/n), area formula, and DFT sums Σ zkm = 0 for 1 ≤ m ≤ n−1.