This is a hard topic, judging from the amount of discussion about it, you can find more references at: Cut-the-knot Wikipedia Polymathematics
The first time I saw 0.99999 ... = 1 I felt unsettled, seeing again all the tension generated by that equality (see polymath above) I decided to write this post.
That equality might be less troublesome if one remembers that the "real number system" (the mathematical setting where this is demonstrated) is just 'a model' for some quantities and not the absolute truth of the universe. Meaning that only after you assume the axioms of that model you can show that 0.99999...=1. Those axioms imply that arbitrary small numbers exist, the problem is do they?
Let me try to explain myself better.
0.99999...=1 is a proven theorem in mathematics (in the real number system), you can also consider it an annoying characteristic of the "real" numbers, since the equality implies that there are two different decimal representations of the same real number "1".
When I say "real number system" I mean a specific mathematical model generally used to describe quantities. As a mathematical model it has well defined axioms, properties, and proved theorems like the one in the title.
First, the name "real number" doesn't mean those numbers are in the real world, of course the real number system is based on some intuitions on how quantities like length, charge, mass (and many others) should behave, and it is indeed a very successful model, but it is nether less just a model.
In measuring quantities (for example length) in the real world always implies some approximation or error. The "real number system" is a mathematical model which supposes that arbitrary good precision is possible, fixing a maximum precision at some small measure would appear strange and it is actually easier, for the models sake, to suppose "infinite" precision.
But can we achieve arbitrary precision? That is something meaningful?
Consider also that this equality has antecedents in Zeno's paradoxes some simple but also unsettling paradoxes.
The way the real number system solved that paradox, had some "bad" consequences like the one in the title.
Some might say that what I wrote here is obvious but I think part of the reason for so much confusion is that people don't say explicitly that stuff. Now for the more arguable part.
Quantum mechanics says that really small quantities give unusual kinds of problems, and many physical theories put at doubt the arbitrarily small in the usual sense. Maybe we should not worry too much with that equality since many modern physicists don't know how to make sense of it too.
