Is surjectivity superfluous for the definition of valuation?

Question

Let $F$ be a field and $\Gamma$ be an ordered abelian group and $v:F^*\rightarrow \Gamma$ be a function satisfying the following:

1. $v(xy)=v(x)+v(y)$

2. $v(x+y)\geq \min\{v(x),v(y)\}$ if $x+y\neq 0$

Then, we say $(v,\Gamma)$ is a valuation of $F$.

Some authors require $v$ to be surjective in addition, but some do not. Is the surjectivity condition for $v$ superfluous? If not, what is the standard definition for valuation of a field?