Roots of Unity

Complex Analysis · Axiom Academy

The n solutions of z^n = 1 — equally spaced points on the unit circle that form a perfect regular polygon. 1. The n Roots Form a Regular Polygon An nth root of unity is any complex number with z^n = 1 . Writing (modulus 1 , angle 0 ) and matching modulus and angle, every solution has modulus 1 and an angle that is a multiple of : the n roots, for k = 0, 1, …, n−1 each lands on the unit circle at angle 2πk⁄n 2. One Primitive Root Generates Them All Let , the primitive nth root of unity — the first root past 1 , with the smallest positive angle . Multiplying by rotates a point by exactly , so the powers of step around the circle and hit every root in turn before returning to 1 . has angle . Multiplying by it adds that angle — one step around the polygon. are the n roots; sits at angle . , so after n steps you are back at the start — the angles are read mod . is again a root: multiply two roots, stay in the set. Here . The three roots are , and indeed brings the third step back home to 1 . For , the nth roots of unity always add up to zero . Lay the n root-vectors head-to-tail: because they point in perfectly symmetric directions, the chain closes back onto the origin — the total displacement is 0 . Why, algebraically: the roots are exactly the zeros of . By Vieta's formulas the sum of the roots is -1 times the coefficient of , and that coefficient is 0 — so the sum is 0 . Since but , the numerator is 0 while the denominator is not — the sum collapses to 0 .

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