I am reconsidering, but I still believe the algebra.
And I have seen quite a few misconceptions about decimals/rounding on this thread.
We aren't talking about .999. Yes, we all know that doesn't equal 1.
Also, there is a way to display 1/3 (1/6, 1/9, etc) in decimal form.
Anyway, I have seen several proofs (most of which have already been posted). If you can show my why the algebra doesn't work. I will change my mind.
Even intuitively, this is true:
This is one of the most complete explanations I've seen. Go here: http://polymathematics.typepad.com/photos/uncategorized/9s_thirds.
Oh, x9, you mentioned that calculus holds more information about this. I have taken calculus and I was wondering if you could be more specific.