Finding Cube Roots of Unity

Complex Analysis · Axiom Academy

EXAMPLE Finding the Cube Roots of Unity Solving z^3 = 1 with the exponential form and the nth-root formula. Find every complex number z that satisfies z^3 = 1 . These three values are called the cube roots of unity . The roots are the vertices of an equilateral triangle inscribed in the unit circle, spaced 120° apart. Nice work. You found all three solutions of z^3 = 1 by converting to exponential form and stepping k through 0, 1, 2 . Geometry: they sit on the unit circle as the vertices of an equilateral triangle, exactly apart. They sum to zero: , because z^3 - 1 = (z-1)(z^2 + z + 1) and the quadratic factor's roots are and . Conjugate pair: , and the product . The same recipe finds the n roots of any z^n = 1 : they always land equally spaced around the unit circle.

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