Computation of roots in algebraic form.

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:

Thus, the square roots of the complex number are and .

By the same method we compute the square roots of .

Hence, the square roots of are and . These are two of the four roots of order 4 of the number .

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 .

- Compute the square roots of -1; these are and ;
- Compute the square roots of ; these are . Then compute the square roots of ; they are equal to . The reader can have a confirmation of the validity of these results by computing the roots using the polar method.

Noah Dana-Picard 2007-12-24