There is a theorem that states that every valuation ring of a field $k$ comes from a valuation function on $k$. In the proof, you take a valuation ring $R$ and define the valuation $v:K^{\times}\longrightarrow K^{\times}/R^{\times}$ as $v(x)=[x]$, and also define an order in the codomain as follows: $[x]\geq [y] \Longleftrightarrow xy^{-1}\in R$.
I have trouble showing that this is a valuation, however. For this to be a valuation, we have to give some additive structure to the codomain, because we want $v(ab)=v(a)+v(b)$. Here is where my first problem arises. We have that $K,R$ are rings, so we are able to add their elements. But when we take $K^{\times}/R^{\times}$, we are seeing it as a quotient of abelian multiplicative groups, so I am not even sure how we would define addition there.
Secondly, I have trouble showing that $v(ab)=v(a)+v(b)$. With how we have defined $v$ we have $v(ab)=[ab]$ and I don't see why this should be equal to $[a]+[b]$.