Geometric Interpretation of Roots

Complex Analysis · Axiom Academy

LESSON Geometric Interpretation of Roots Taking the n th roots of a complex number is the same thing as drawing a regular n -gon. 1. The Roots Live on One Circle Every n th root of a complex number z has the same modulus , so they all lie on a single circle centered at the origin. If z has modulus r , that circle has radius: Every root sits on the circle of radius r^ 1/n Take z = 8 . Its cube roots all have modulus , so they ride a circle of radius 2 — and connecting them traces an equilateral triangle. Press play to watch the three roots land and close into the polygon. Because the roots are evenly spaced by an angle of , they always form a regular n -gon inscribed in their circle. Each root is the previous one turned by one identical step around the origin. The angle between consecutive roots — always equal Below, watch a single root get stamped, then rotated by exactly ( ) again and again until the equilateral triangle is complete. Same hop every time is what guarantees a regular polygon. Hop each time — an equilateral triangle. Hop each time — a regular pentagon. Hop each time — a regular hexagon. Bigger n means a smaller hop and more vertices: the polygon has more sides but the rule never changes — divide the full turn into n equal slices. The spacing fixes the shape , but one root anchors the whole figure: the principal root . Its angle is the argument of z divided by n :

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