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
$$
\{r\}\,e^{i\phi}=(e^{i\theta}+e^{-i\theta})\,e^{i\phi}=e^{i(\phi+\theta)}+e^{i(\phi-\theta)}.
$$
