Ok. So you are saying that 0.333... is not equal to 0.3* where * is an infinite number of 3's?
You might be confusing 0.333... with the division process. Dividing, we cannot fully right out 1/3.
However, notation wise "..." means "continues forever." In mathematics, when you carry out the infinite, things happen that do not happen in the finite world. In the finite world, 0.333... doesn't "exist." And also "1/3" doesn't exist. There is not "1/3" of something. Everything exists on it's own. We just might call something "1/3" to mean "part of the whole in which it came," but in reality all "wholes" are made up of single objects.
I do not know if that makes sense, but that is how I think of it. Mathematics and reality do not run parallel.
Also, it is an axiom. I quote this from one of my many overpriced mathematics books:
The real numbers can be equated with the set of all base 10 decimal representations. That is, every real number can be written in a form like 338.1898..., where the decimal might or might not terminate, and might or might not fall into a pattern of repetition.
Furthermore, every decimal form you can construct represents a real number. Strangely, though, there might be more than one decimal representation for a certain real number.
So, if we assume that this axiom is true, then 0.333... = 1/3. However, if you want to assume this axiom is false, you can develop mathematics based on that fact. However, you will likely run into contradictions. If you really want to me try, I can try to come up with some.
No need for that. Maybe you do not understand what "..." means and what "=" means?
Do not give in just to ignore your misunderstanding.