Finding nth Roots

Complex Analysis · Axiom Academy

In the complex plane, a single number hands you a whole symmetric set of roots — not just one. How many solutions does x^3 = 8 really have? Over the real numbers, gives a single answer: x = 2 . But a degree- n equation is supposed to have n roots — so where are the other two? They are there; they are just complex . And once you plot them, a strikingly regular picture appears. Watch the three cube roots of 8 drop onto the complex plane. They land on a circle of radius and split it into three equal slices — a perfect triangle. That even spacing is the whole story of nth roots. Three roots, 120° apart, on a circle of radius . The reals only ever showed you the one sitting on the positive axis. Turn the dial: n roots, always evenly spread Slide n from 2 up to 12 and watch the nth roots of the same number rearrange themselves. However many you ask for, they spill out equally spaced — n points, 360°/n apart, sitting on a circle of radius . The roots are always a regular n -gon. As n grows, the points crowd closer — but they never bunch up. The gap 360°/n keeps them perfectly balanced around the circle. Move the number z , and its roots move as one These are the roots of one specific z . Drag z around — change its size |z| and its angle — and watch the whole bundle respond. Make z bigger and the root-circle grows like ; spin z by and every root turns by just . The set never breaks its symmetry; it only scales and rotates.

This is the written version of the interactive lesson above. See the full Complex Analysis course.