How to plot the 5th roots of a complex number

Question

I want to draw the roots of z^5=1-i.

I am totally novice to latex and really have no idea about drawing. Thanks for helping!

Answer 1

Hmmh, nobody seems to want to answer this. I agree that this problem is very similar to the linked questions, but personally would not consider it a duplicate since you do not ask to take the nth root of unity, but of a different complex number with nontrivial radius and phase.

\documentclass[fleqn]{article} 
\usepackage{tikz}
\usepackage{amsmath}
\begin{document}
You wish to plot the 5th roots of 
\[ 1-\mathrm{i}~=~\sqrt{2}\,\mathrm{e}^{-\mathrm{i}\pi/4}\]
for some complex number $z=r\cdot\mathrm{e}^{\mathrm{i}\,\varphi}$.
This means that you need to solve the equation
\[ z^5~=~r^5\,\mathrm{e}^{5\cdot\mathrm{i}\cdot\varphi}
~=~\sqrt{2}\,\mathrm{e}^{-\mathrm{i}\pi/4}\;,\]
which translates to 
\[r~=~2^{1/10}\quad\text{and}\quad \varphi~=~\frac{\pi}{20}\left(-1+8\,n\right)
~=~9^\circ\left(-1+8\,n\right)
\]
with $n\in\{0,...,4\}$.

\begin{tikzpicture}[scale=3]
\pgfmathsetmacro{\ticklength}{0.06}
\draw [-latex] (-2,0) -- (2,0) node[below left]{Re$\,z$};
\draw [-latex] (0,-2) -- (0,2) node[below left]{Im$\,z$};
\draw (1,\ticklength) -- (1,-\ticklength) node[below] {1};
\draw (\ticklength,1) -- (-\ticklength,1) node[left] {i};
\draw (0,0) circle({pow(2,1/10)});
\foreach \X in {0,...,4}
{\node[scale=0.4,circle,fill,label={{9*(-1+8*\X)}:$n=\X$}] at ({9*(-1+8*\X)}:{pow(2,1/10)}) {}; }
\end{tikzpicture}

\end{document}