Find the cube roots of unity.

Unity just means 1, so we need to find all z such that z³=1. However, although we can immediately see that z=1 is a solution, when dealing with complex numbers some solution may not lie on the real axis. To find all the soltuions, it is necessary to re-write the equation in polar form, that is z³=e^2ipi, since 1 has argument 0 on the complex plane, and argument 2pi is the same as argument 0, as both will lie on the positive real axis. Now, we recall that if we add 2pi to the argument, the value is unchanged, since spinning a point 2pi around the origin in the complex plane puts the point back where it started, as 2pi is a full turn. Also, recall that cubic equations have 3 roots, so we can write z³=e²ⁱpi, e^4ipi or e⁶ⁱpi. Now, we can take cube roots on both sides, and recall that this is equivalent to raising to the power third, and that when raising a power to a power we multiply the powers. So we obtain z=e^2i/3pi, e^4I/3pi or e^6i/3pi. Recall that we can subtract 2pi from the argument to ensure that the argument is between minus pi and pi, and that 6/3pi=2pi which is the same as an argument of 0, i.e. this value is one. So the cube roots of unity are 1, e^-1i/3pi and e^2i/3pi.