Roots of Complex Number/Examples/Square of Cube Roots of i

Example of Roots of Complex Number: Corollary
The squares of the complex cube roots of $i$ are given by:
 * $\paren i^{2/3} = \set {\map \cis {60 + 120 k} \degrees}$

for $k = 0, 1, 2$.

That is:

Proof

 * Square of Complex Cube Roots of i.png

Let $z^3 = i$.

We have that:
 * $z^3 = \map \cis {90 + 2 k \pi}$

Let $z = r \cis \theta$.

Then:

Hence: