We illustrate with examples a suitable method to compute square roots and roots of order 4 of a given com[plex number.
The extra equation is equivalent to . Solving this system for and , we have or . This means that has two square roots, namely and .
By iterations with this method more, we can solve compute roots of order 4,8,16,etc.
First we compute the square roots of . Denote , where . The following holds:
By the same method we compute the square roots of .
We leave to the reader the task of computing the square roots of . Finally, we found the four roots of order 4 of ; they are the complex numbers , , and .
Noah Dana-Picard 2007-12-24