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

[napoleon_in_rags]

2. I disagree. Is PI a rational number?

I mean, can't we write PI as 314159.../100000... A rational number who's denominator and numerator are INFINITELY long integers?

No, we can't. PI is not a rational number, because integers with infinitely long representations aren't integers. If they were they would not be countable, thus not integers. So rationals that are infinitely long, also uncountable, are a not a proper part of the set Q.

That brings us to the next question: is .999... really a decimal number? No. By the same logic that excludes pi from being a rational number, and the numerator of the "rational pi" above from being an integer, a decimal number with an infinitely long representation is not a number. (because I consider the decimal numbers countable.) 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?

You can resolve this by considering .999 repeating to be a computable number... Its a simple algorithm which prints 9's to infinity, taking finite code. So in this sense its countable. But within the context of the computable numbers, is it equal to the program which prints "1" and halts? No.

And my final example: If I gave you two choices: 1) I will give you 1 million dollars today, or 2) The amount of money I give you will converge to 1 million as time approaches infinity, which would you choose?

Good choice, and that's why you agree with me, not the OP.

PEace.