Proving the square root of $z\in \mathbb{C}$ geometrically I hope somebody could help me to prove that the squareroot exists for all numbers in $\mathbb{C}$. A complex number is Always a stretch and a Rotation at the same time. For example (3,2) Projects (1,0) to (3,2), if we trat (3,2) like we did treat (3,2), we do the same stratch and Rotation again. Visually it would look like this: ![enter image description here]( Can one derive an explicit Formula with that idea or with other words is it possible to argue that every Point on a plane can be described as the composition of two equal streteches and Rotation? Many Thanks for your Input and time. Here is also another idea of mine to find such a Point but not sure whether it is Right and how to prove it ![enter image description here](

From that point of view, consider half of the rotation and the square root of the strecht. If you multiply this by itself, you will get the original number.