I'm stuck with this equation:

> 
> Find all the $z\in\mathbb{C}$ such that $z-1=z^{-1}$, with $z\neq{0}$
> 
> 
> 

So far, I've tried writing $z=a+bi$ and $z^{-1}=\frac{\bar z}{|z|^{2}}$ but got nowhere.

Then, I wrote: 

${re^{i\theta}+e^{\pi i}}=r^{-1}e^{-\theta i}$, where $r=|z|$

Also, I thought of considering the argument on both the LHS and the RHS of the equation, but that's not a good idea when you have a sum.

For some reason I am not being able to continue and/or realize what am I missing.

Thanks for the help!