I am struggling with an example from my lecture notes. Let $K$ be a non-Archimedean field with valuation ring $R=\{x:|x|\le 1\}$, maximal ideal $\mathcal{M}=\{x:|x|<1\}$ and residue field $k=R/\mathcal{M}$. Then the reduction map $R\to R/\mathcal{M}$ given by $r\mapsto r+\mathcal{M}$ is a surjection denoted by $a\mapsto \tilde{a}$.
Here are two examples, working in $\mathbb{Z}_5$. Firstly, if $a=3+2\cdot 5^1+\cdots$ then $\tilde{a}=3$. Secondly, $\tilde{(\frac{17}{3})}=2/3=2\cdot 2=4$.
With the definitions given above, I don't understand how we get these results in the examples. Any ideas?