Computing intersection multiplicity using the completion

Question

Let $X$ be a smooth projective surface over complex numbers. Let $C$ and $D$ be two curves and $p\in C\cap D$ be a smooth point of both curves. I want to calculate the intersection multiplicity $(C\cdot D)_p$ which is defined to be $$\text{dim}\frac{O_{X,p}}{(f,g)}$$ where $f,g$ are the defining equations of $C$ and $D$ in the stalk. Is this the same as $$\text{dim}\frac{\widehat{O_{X,p}}}{(f',g')}$$ where $f',g'$ are the local equations of $C$ and $D$ in the completion of the stalk?