Expanding on @Stefan's answer for the article document class
When you are using the book document class, you have chapters, but not when using article. For that use case, you redefine \thesection as above, without redefining \thechapter, and then follow that with redefining \thesubsection instead. That gives you the same result as sought in the OP without having to use the book document class:
\renewcommand*{\thesection}{\arabic{section}.\arabic{subsection}}
\renewcommand*{\thesubsection}{\arabic{section}.\arabic{subsection}}
\subsectionetc. picks up that "definition without\thechapter" from\thesection... sweet. Why didn't I see this? :-D – DevSolar Aug 08 '11 at 16:07