Son Goku's explanation is the one that deals best with an actual mathematical understanding of what is going on.
When I got on to my degree course, one of the first things we were taught was a new definition of equality. It's not the same as Son Goku's point, but it is very closely related.
That is, two numbers a and b are equal if there is no number, e such that
0 < e <|a - b|
(Which is the same as saying that there is no number between them, if you think about it).
In fact we needed this definition to deal with the sums of infinite sequences. And if we couldn't do that, integration wouldn't work (and we'd need to find a physical solution to Zeno's paradox, too !) So it is important to mathematics that 1 and 0.999... are the same number. If they were different we'd have real problems.
Edited by PaulK, : Corrected after prompting from Dr. Adequate.