Finding Fourth Roots of 16

Complex Analysis · Axiom Academy

EXAMPLE Finding the Fourth Roots of 16 Using polar form to find all four complex solutions of a degree-4 equation Find all solutions to z^4 = 16 . Over the real numbers we'd only get z = 2 (and maybe z = -2 ), but over the complex numbers there are exactly four fourth roots. All four lie on a circle of radius 2, spaced 90° apart — the vertices of a square in the complex plane. Nice work — you found every fourth root of 16 by reading off a shared modulus and four equally-spaced angles. One modulus: every root has modulus , so they all sit on a circle of radius 2 . Equal spacing: the n roots are apart — here . The four roots: — two real, two purely imaginary — the vertices of a square. The same recipe — one modulus, n equally-spaced angles — finds the n th roots of any complex number.

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