Provide the exact list of steps needed to find, with ruler and compass, the two square roots of a given complex number. (The points $0$ and $1$ are given)
I don't really understand what I have to do in this question. We start off with the two numbers $0$ and $1$ and then we can choose our complex number? So the simplest one would be $i$.
But then what. I am very lost.
$$ Z = \sqrt{ r e^ {i \theta} } = \sqrt{ r}\cdot e^ {i \theta/2} $$
For the argument bisect the angle of given complex number using ruler/compass.
A unit length should be available on the ruler. Else there is another method.
Construct two line segments $PA,PB=r,1$ in a line along diameter of a circle. Draw the perpendicular at $P$ to find $ PN= \sqrt{ r}$. Set off this length along bisected ray to find point representing sqrt of $Z$
EDIT 1:
The real and imginary parts of $x +i y $ are $$ \frac{\sqrt{x-\sqrt{x^2+y^2}}}{\sqrt{2}},$$
$$ \frac{-\sqrt{2} x \sqrt{x-\sqrt{x^2+y^2}}+\frac{\left(x-\sqrt{x^2+y^2}\right)^{3/2}}{\sqrt{2}}}{y}$$
and variants by sign change of radicals. But it cannot probably used as a method for such construction.