In the study of algebraic curves,we define intersection number of $F$ and $G$ at a point $p$ to be $I(F\cap G,p)=\dim_K(\mathcal O_p(\mathbb A^2)/\langle F,G\rangle$.But it is a rather unintuitive definition to me.In fact, I am cofused between intersection multiplicity and intersection number at $p$.Also it is not clear to me why the intersection number should have the following properties:
$1.$ $I(F\cap G,p)\geq m_p(F)\cdot m_p(G)$ with equality exactly when they have no tangent lines in common at $p$.
$2.$The intersection numbers should add when we take union of curves i.e.if $F=\prod F_i^{r_i}$ and $G=\prod G_j^{s_j}$, then $I(F\cap G,p)=\sum_{i,j}r_is_j I(F_i\cap G_j,p)$.
$3.$ $I(p,F\cap G)=I(p,F\cap (G+AF))$ for any $A\in K[X,Y]$.
(Note: There are seven properties that determine a unique intersection number but I understand the motivation behind the other ones.Just having problem with these ones.)
I am looking for a little help.