Any complex number can be written as a sum of complex numbers of modulus 1? I found this problem in a text-book, no solution offered. I'm curious because it seems like a very interesting result. Full statement is: Let $M \subseteq \mathbb{C}$, a set with the following properties: 1\. if $x\in{\mathbb{C}}$ with $|x|=1$, then $x \in{M}$ 2\. if $x=a_1+a_2$ and $a_1,a_2 \in{M}$, then $x \in{M}$ Show that $M=\mathbb{C}$. Any suggestions are welcome, thanks in advance :)

An analytic proof using Old John's hint.

Let $z=r\,e^{i\phi}$ with $r\ge0$ and $0\le\phi<2\,\pi$. Let $[r]$ be the integer part of $r$ and $\\{r\\}$ its fractional part. Then $$ z=e^{i\phi}+\dots([r]\text{ times})\dots+e^{i\phi}+\\{r\\}\,e^{i\phi}. $$ It is enough to show that the las term in the sum is the sum of complex numbers of modulus $1$. Let $\theta=\arccos(\\{r\\}/2)$. Then $$ \\{r\\}\,e^{i\phi}=(e^{i\theta}+e^{-i\theta})\,e^{i\phi}=e^{i(\phi+\theta)}+e^{i(\phi-\theta)}. $$