Quote:

If you can't say that there is a difference between x and y if they are the same, then why bother with representing .999~ and 1.000~ at all if they are the same number? It seems to be an exercise in futility.

Because in math, we tend to define things generally. We didn't define decimal expansions number by number, we defined the decimal expansion in general. As it so happens, there is a decimal expansion for every single number, and because of repeating 9s (or repeating 1s in base 2, or in general, repeating n in base (n+1)), many numbers have two decimal expansions. It's similarly to how you can represent the same number as many different quotients. 1/1 = 2/2 = 3/3 = 4/4 etc.

Quote:

Yes, both numbers represent one whole "numerical unit". In essence, if it were a pie, you could say you have a whole pie (1/1 or 1.000~), or three thirds of a pie (3/3 or .999~). They all mean the same thing... Is this the point of the whole topic?

Well the point is whether or not this is in fact the case, but yes, those would all mean the same thing though it would be very odd to express an intact pie as anything other than "one pie".