If you're having math problems, I feel bad for you, son...

[Jim__]

4. A set of numbers is countable if it has the same cardinality as some subset of the natural numbers.

Whether or not a set of numbers is countable has nothing to do with the how members of that set may be represented. For instance, 1/3 is a rational number, and its infinite decimal expansion, 0.33333... is a rational number, even though its exact decimal expansion is infinitely long.

... PI is not a rational number, because integers with infinitely long representations aren't integers. ...

Pi is not a rational number because there do not exist 2 integers, say n and m, such that pi can be written as n/m. Pi is not an algebraic number because there is no non-zero, one variable polynomial with integer (or rational) coefficients that has pi as a root.

... If you believe the decimal number are uncountable, fine. But now subtract the set of countable numbers from the uncountable numbers. Which is 1 in? Which is .999... in? How can a number be both countable and uncountable?

Your statement doesn't actually make sense. It's sets that are countable or uncountable. If you have a set of numbers that contains pi, then pi counts as one member of that set. The set of real numbers are uncountable because they cannot be put into one-to-one correspondence with any subset of the natural numbers. Since 1 and .999... are the same number, any set that contains one of them contains the other; the only difference is representation.