What's the magnitude of a real number? As a student of mathematics (first year master degree) I have to admit that I'm somewhat ashamed to ask this.
We know that if $z=x+iy$ is a complex number then we can identify it as $z=r\cdot\exp(i\theta)$. But what if $z$ is real - in other words its $y$ equals 0? Then $z=r\exp(i\cdot0)=r$ and this means that $z$ would be equal to its magnitude $r$ if $z$ is positive. But what if $z$ is negative? We know that the magnitude is always positive and so we'll get $z$(negative) = $r$(positive)?
I'm sure there's something I'm missing here.

Since it has not been pointed out so far, it is crucial to realize that while cartesian coordinates are unique, polar coordinates are not unique, because the pair $(r\cos(t),r\sin(t))$ is exactly the same as $(r\cos(t+2πn),r\sin(t+2πn))$ for any integer $n$. Furthermore, $(0\cos(t),0\sin(t)) = (0,0)$ for any real $t$. So if we want to represent points by polar coordinates $(r,t)$, we usually stipulate a range for $t$. Commonly $0 \le t < 2π$ or alternatively $-π < t \le π$. In either case, if we require $r>0$ then $(r,t)$ is unique for any point except the origin, because the distance from origin and the angle of the ray from the origin are uniquely determined. There is still no unique polar coordinates for the origin.
One could choose to allow $r$ to be negative and restrict $t$ further to $0 \le t < π$, so that the point $(-1,0)$ (cartesian coordinates) can have 'polar' coordinates $(-1,0)$ rather than $(1,π)$. However, this is not as elegant or convenient as the usual definition of polar coordinates.
