Let $V$ be a projective variety over a field $k$. For any affine patch we can define the function field of $V$ to be $K(V)=K(V\cap\mathbb{A}^n)$, and these are all canonically isomorphic.
In Silverman's "The Arithmetic of Elliptic Curves", the author states that we may also view $K(V)$ as the field of rational functions $F(X)=f(x)/g(x)$ such that 1)$f$ and $g$ are homogeneous and of the same degree; 2) $g\not\in I(V)$; and 3) two functions $f_1/g_1$ and $f_2/g_2$ are identified is $f_1g_2-f_2g_1\in I(V)$.
My question is: why does the first definition of $K(V)$ imply that the $f$ and $g$ in the first condition of the second definition above are of the same degree?