the roots of unity
The n-th roots of unity are the solutions of the simplest possible n-th-root problem: z^n = 1. They are just the n-th roots of the number 1, and since 1 has modulus 1, all of them lie on the unit circle. Applying the n-th-root recipe to w = 1 (which has angle 0), the roots are z_k = e^(i 2 pi k/n) for k = 0, 1, ..., n - 1 — n points spaced evenly around the unit circle, starting at 1 and stepping counterclockwise by 2 pi / n each time. They form a perfect regular n-gon inscribed in the circle, with one vertex always at 1.
There is a beautiful structure hidden here. Write omega = e^(i 2 pi/n), the first root after 1. Then every root is a power of this single one: the roots are 1, omega, omega^2, ..., omega^(n-1). Multiplying two roots gives another root, and 1/omega is a root too — so the roots of unity form a self-contained group under multiplication, a cyclic group generated by omega. They are the algebraic skeleton of rotational symmetry.
Roots of unity are everywhere once you look. Their sum is always zero for n at least 2 (the vertices of a regular polygon balance at the center), which makes them the foundation of the discrete Fourier transform and the fast Fourier transform that powers digital signal processing. They classify the symmetries of regular polygons, appear throughout number theory (cyclotomic fields), and are the cleanest possible illustration that the complex numbers turn 'solve z^n = 1' into a single, elegant picture.
The fourth roots of unity solve z^4 = 1: they are 1, i, -1, -i (that is, e^(i 0), e^(i pi/2), e^(i pi), e^(i 3 pi/2)) — four points at the compass corners of the unit circle, forming a square. Their sum is 1 + i - 1 - i = 0.
The n-th roots of unity are equally spaced on the unit circle; for n >= 2 they sum to zero.
Not every n-th root of unity 'generates' all the others — only the primitive ones (those with the angle 2 pi k/n where k shares no factor with n) cycle through the full set. For instance among the fourth roots, i and -i are primitive, but -1 only generates 1 and -1.