I hesitate to reply to a thread that should have been over last night, but there does appear to have been some "piling on" since. Rapidcoder's mention of 1/3 was equivalent to my 73.000555 and his mention of pi had already been mentioned by cire. I replied, not to prove L B wrong (he already knew he was wrong), but rather to answer his question:
|So where is my logic flawed? What invalid assumption have I made?|
His error arose from not realizing that a real number is rational if and only if its decimal representation either terminates or ultimately contains a pattern that repeats ad infinitum and hence none of the irrationals can be represented in his scheme.
I would hope, L B, that you would not be discouraged from posting such questions in the future because of any "piling on" which I don't think was intentional.
I for one, cannot help but admire the thought processes that led to the question, that having learned about countably infinite sets and seen a proof that the rationals are countable, you would try to extend or modify the argument to see why it doesn't work for the reals. It is precisely those types of thought processes, even when they lead to dead-ends, that result in a better understanding of new concepts and how to use them. I believe (just an amateur programmer's opinion here) those same types of thought processes are what a programmer uses in solving programming problems. Haviing thought about these things places you are far ahead of your fellow students who learned the definition of countable and uncountable, saw a couple examples and haven't thought about the ideas since.
I must warn you however, that if you persist in thinking such thoughts and your math professor finds out about it, one day he may ask you to see him after class and say to you:
"L B, have you ever thought about changing your major from computer science to math?"
Best of luck with your studies.