The Humble Roots of Unity Quietly Build the Fourier Matrix

Notes on the Complex Roots of Unity

Starting from a middle-school question — what's the square root of 1? — this walkthrough extends roots of unity into the complex plane using polar coordinates. It shows how the nth roots sit evenly on the unit circle, form regular polygons, and pair with their conjugates. The set of nth roots turns out to be a cyclic Abelian group whose generators are the primitive roots, and their powers form an orthogonal basis — the Fourier matrix that underpins the DFT.