I am wondering about how to compute the degree of r(z) := \frac{z}{| z |}. I know that $deg f_n = n$ where $f_n (z)= z^n$ i.e. the degree is how many times it goes around $0$. I also know that $deg(f \circ g) = deg (f) \cdot deg (g)$. Now with $f(z)=z$, $deg f = 1$, but what about $f(z) = \frac{z}{|z|}$?
Edit:
Where $r: \mathbb{C} \backslash \{ 0 \} \rightarrow S^1$. The degree is defined for continuous functions $f:S^1 \rightarrow S^1$. $ \mathbb{C} \backslash \{ 0 \}$ is homotopy equivalent to $S^1$ so I was thinking $r$ can be viewed as function $r: S^1 \rightarrow S^1$. I'm asking this question because I think $deg (r)$ is used in a proof of the fundamental theorem of algebra.
Many thanks for your help!