Going through some of the applications of the proof that $\pi_{1}(S^{1}) \cong \mathbb{Z}$ in Hatcher, I have noticed that almost all of these arguments rely on a contradiction whose assumption involves the absence of some division by zero. For example, for his proof of the fundamental theorem of Algebra, his assumption of the non - existence of the roots of a nonconstant polynomial $p(z): \mathbb{C} \rightarrow \mathbb{C}$ leads him to claim that the following function: $$f_{r}(s) = \frac{p(re^{2\pi is})}{\lVert p(re^{2\pi is}) \rVert}$$ Is continuous everywhere, which he uses then to prove that $[f_{r}(s)]$ is trivial (by restriction of $r$ to zero).
As such, my question is this: why is it that so many of these applications involve the division by zero in such a way? Is there any deeper connection between division by zero and the fundamental group of a circle? In particular, could you please explain to me the intuitive connection between the construction of $f_{r}(s)$ and being nullhomotopic? For some reason, I cannot satisfyingly explain to myself why this construction should reveal something as deep as the triviality of a homotopy class in spite of it's simplicity. Why is it that the function must first "break down" when $f_{r}(s)$ has a root before it can "re - establish" itself, this time no longer being nullhomotopic?
Here is the intuitive deeper reason: The division by zero is coming in from the way that complex numbers multiply. That is multiplication by complex numbers is rotation (adding up the angles) and scaling. That is, $(re^{i\theta})(se^{i\phi}) = rse^{i(\theta+\phi)}$. In these formulas where you take $\frac{p(z)}{\|p(z)\|}$ you are trying to ignore the scaling, and so it is the rotation, which is precisely what the fundamental group measures.
Each factor of $z$ or $\frac{1}{z}$ is a full rotation in either this or that direction. Each one of those gets you a "division by zero" in some argument. The argument principle or Rochè's theorem from complex analysis are another good example of this kind of reasoning at work.
Hopefully I am understanding what you are asking correctly ...