The nth roots of unity are the complex solutions of zⁿ = 1, equally spaced on the unit circle and forming a regular n-sided polygon when n ≥ 3. For eight roots, the spacing is 45°, and multiplying by the root at 45° rotates a point counterclockwise by that angle, returning it to its starting position after eight steps. For n > 1, all the roots sum to zero; in the eight-root example, opposite roots cancel in pairs.