Roots of Complex Numbers

Complex Analysis · Axiom Academy

LESSON Roots of Complex Numbers Every nonzero complex number has exactly n distinct n th roots — and they sit at the corners of a regular polygon. 1. Where the Formula Comes From We want every w with w^n = z . Write the unknown root as . By De Moivre, raising it to the n th power gives . Matching that to forces the length and the angle to agree, so and . That is the principal root — but it is not the only one. The catch: an angle is only defined up to full turns. The angle and the angle point to the same number z for every integer k . So matching the angle really means , which gives a whole family of solutions. one root for each whole turn k we add before dividing Letting run through one full set of turns produces every distinct root. Past k = n-1 they repeat, because adding n turns puts you back where you started. This is the master formula: Every root has length |w| = r^ 1/n — so they all sit on one circle. The first (principal) root sits at angle . Each next root is further around — i.e. apart. k = n would repeat the principal root, so exactly n roots exist. Reading the formula off the picture The animation finds the cube roots of . The principal root lands at on the radius- 2 circle; stepping by twice drops the other two at and . 3. A Circle of Equally Spaced Roots

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