I am trying to understand what the intuition behind the intersection multiplicity of two plane algebraic curves is/should be. I know that there are different ways to define this value, but the book I am working with right now uses the following: Let $g,f \in k[x,y]$ s.t. $f$ and $g$ have no factor in common. Assume that $f(0,0)=g(0,0)=0$. Denote the associated curves with $F,G$. The intersection multiplicity of $F,G$ at $P=(0,0)$ is the number:
$$ I(P,F\cap G):=\dim_k B $$
where $B=(k[x,y]/(f,g))/(x,y)^n$ and $n$ is the number s.t. $(x,y)^N=(x,y)^{N+1}$ for all $N\geq n$. Such a number exists because we know that $k[x,y]/(f,g)$ is a finite dimensional vector space.
So far so good. I know what all of this means and I can work with it i.e. I can compute the dimension for examples. My question is:
What is the intuition behind this definition? Or in other words: How does one come up with this definition?
Here are my thougths so far: Let's assume $k=\mathbb{C}$ from here on. We can write the Taylor expansion for $f$ and $g$ at $P=(0,0)$ using multi indices.
\begin{eqnarray} f(x,y) &=& f(0,0) + \sum_{\vert\alpha \vert\geq 1}\dfrac{D^\alpha f(0,0)}{\alpha!}x^\alpha\\ g(x,y) &=& g(0,0) + \sum_{\vert\alpha \vert\geq 1}\dfrac{D^\alpha g(0,0)}{\alpha!}x^\alpha \end{eqnarray}
My intuition now tells me that $I(P,F\cap G)$ should agree with $\vert \alpha \vert$ s.t. $$f(x,y)-g(x,y)=\sum_{\vert \alpha \vert=I(P,F\cap G)}\dfrac{D^\alpha f(0,0)-D^\alpha g(0,0)}{\alpha!}x^\alpha+\text{higher order terms},$$
where of course the $\sum_{\vert \alpha \vert=I(P,F\cap G)}\frac{D^\alpha f(0,0)-D^\alpha g(0,0)}{\alpha!}x^\alpha \neq 0$. The intuition is coming from the geometric picture that the multiplicity at $P$ should somehow measure how "tangent" the two curves are to each other. I computed some examples and for those it is indeed true but I have a hard time proving it in full generality. And maybe it's wrong all together. I would appreciate if somebody could give me a proof or point to one or in general can give me more insight to my question above.