What points in the complex plane satisfy $|z|=arg(z)$. Question: What points in the complex plane satisfies $|z|=arg(z)$.
Work thus far: Let $z=a+bi$ and $arg(z)$ is the angle between the vector $z$ and the axis. So $|z|=\sqrt{a^2+b^2}$ and $a=|z|\cos(\theta),b=|z|\sin(\theta)$. I would expect from intuition that the shape traced out by the points is a spiral. From here I don't how to proceed. Any hints would be appreciated. 

Try thinking the other way:
Fix some $\theta_0$ and imagine a line starting at the origin and making an angle $\theta_0$ with the real axis. How many points $z $, in that line, have modulus $|z|$ equal to $\theta_0$? Can you write an expression for it/them?
