I have been readingA Long Way From Euclida book by Constance Reid. In the chapter on complex numbers she gives an equation, and the solutions to the equation, but does not show how the solutions are arrived at. I am stumped, and (finally) realize I need help. Just as important as knowing how to solve this equation where i equals "1" i, I would like to know how to solve it if the i equals 2i, 3i, 4i. . .ni, where n is any real number. Can the equation be solved algebraically by converting the x and the i into a + bi form? Or, does DeMoivre's theorem for finding complex roots have to be used? Both? Or, somthing else?

The equation is

$\displaystyle x^2-i=0$

The solutions that Reid gives are --

$\displaystyle x=\pm\left(\frac{1+i}{\sqrt{2}}\right) \;=\; \sqrt{i}$