From near the beginning of baby Rudin there is the following passage.
Let $A$ be the set of all positive rationals $p$ such that $p^2 < 2$ and let $B$ consist of all positive rationals $p$ such that $p^2 > 2$. We shall show that $A$ contains no largest number and $B$ contains no smallest.
More explicitly, for every $p$ in $A$ we can find a rational $q$ in $A$ such that $p < q$, and for every $p$ in $B$ we can find a rational $q$ in $B$ such that $q < p$.
To do this, we associate with each rational $p > 0$ the number$$q = p - {{p^2 - 2}\over{p + 2}} = {{2p + 2}\over{p + 2}}.\tag*{(1)}$$Then$$q^2 - 2 = {{2(p^2 - 2)}\over{(p + 2)^2}}.\tag*{(2)}$$If $p$ is in $A$ then $p^2 - 2 < 0$, $(1)$ shows that $q > p$, and $(2)$ shows that $q^2 < 2$. Thus $q$ is in $A$.
If $p$ is in $B$ then $p^2 - 2 > 0$, $(1)$ shows that $0 < q < p$, and $(4)$ shows that $q^2 > 2$. Thus $q$ is in $B$.
Given that there is something which looks like polynomial division that shows up here in the proof, I can't help up but wonder two related things.
- Translated into the language of algebraic geometry (https://en.wikipedia.org/wiki/Algebraic_geometry), what would this statement or salient features of the proof offered by Rudin look like?
- What is the significance of this particular simple phenomena in algebraic geometry? As in, when does something like this show up, and why is it important?
Thank you, I know this is a vague question, but in the past plenty of vague questions like this on MSE have gotten very excellent responses, so here's to hoping that a seasoned algebraic geometry student or practitioner might be able to provide their perspective.