There are infinitely many complex numbers $z$ such that $|z|= 1$. Can anybody just explain this to me so I can get the picture.
There are not only infinitely many complex numbers on the unit circle, there are infinitely many Gaussian numbers (both real and imaginary parts being rational) there. For, take a Gaussian integer $z=m+ni$, both $m$ and $n$ being ordinary integers, and calculate $z/\bar z$, something that you should have done many times in high school. You see that the resulting complex number $x+iy$ has both $x$ and $y$ in $\mathbb Q$, and of course its absolute value is $1$.