FranticMad –
The sensation you are experiencing is called “algorasm”. It’s why pure mathematics is so much fun.
I agree with the conclusion here, but not the statement. Clearly, it’s fairly easy to prove that a decimal converges. However, there are NOT an UNcountable number of digits to the right. It is, in fact, a very countable infinite sequence. Unless I’m missing something. Am I?
No, it’s countable. Pleonast may have been using “uncountable” in the sense of “not countable in finite time”, which is bad but understandable.
That’s what I assumed, but in an argument as full of mathematical pedantry as this one, I figured that shouldn’t go uncaught.
I always hated that term. Why on earth would something with cardinality Aleph-Null be called “countable”? Couldn’t they come up with a Latin word for it instead?
Because, given infinite time, every item in the sequence/set can be counted. If it’s uncountable, then infinite time isn’t enough to count every item in the set.
I guess everyone’s intuition is different. Maybe my “intuition” has been influenced by my education, but I’ve always thought that it’s intuitive that .999… = 1. I can’t comprehend it being anything else. Any other identification makes no sense.
And, Achernar, it seems as though introducing Latin terms causes mass confusion. 
I also find it intuitive that 0.999… = 1, but ultrafilter’s point still stands. Read about the Banach-Tarski Paradox and then say that intuition is a good guide to mathematical truth.
Doh! I meant “uncountable” in the sense of “countable, but my fingers typed the wrong word”! Sorry about that, I really do know the difference.
I agree it dangerous relying on intuition. But how explicit must we be? Even Phage (I think?) accepts that 0.999… converges (he just disagrees what it converges to). And does anyone else find the theorem “a==b && c==d => a-c == b-d” unintuitive? I do, yet no one has complained about the failure to mention it in step 3 of RealityChuck’s proof.
Once we begin to leave off the justification for any step in a proof, we’ve opened the door for “stylistic” differences in our choices of what to leave implicit, and what to make explicit.
1/3 is expressed as .333… because of a limitation of the base 10 number system not because there is anything magic about the number. In the base 9 number system …
1/3 = .3
1/6 = .6
So 1/3 + 1/6 = .3 + .6 = 1.0
The best we can do in base ten is say that 1/3 is the limit that .333… is approaching. So the limit that .999… is approaching
just happens to be a number that can be expressed in base 10 as 1.0.
Mods,
Can I start a book on when this thread will end?
(I’ll split with ya!)
I’ll put 1/infinity dollars on July 32nd at 24:03 hours!
I’ll take $i-1 on June 10th at 10:56 AM
I’ll take sqrt(-peanut_butter) on the hrung collapse of Betelgeuse 7.
typo, didn’t notice until I tried to log in again. I’ve considered changing my name but means losing my post count.
No, the administrators can change your name, and you won’t lose any stats. Email TubaDiva, and she’ll take care of it for you.
thanx, it’s been bugging me for quite a while 
Quoth Jabba:
[hijack, though more on topic than Latin terminology] Does the B-T theorem rely on the Axiom of Choice? Because the pieces the ball is cut into are obviously unmeasureable, and the last time I saw an unmeasureable set constructed (on this board, in fact), the Axiom of Choice was required.[/hijack, though more on topic than Latin terminology]
And while I’ll certainly agree that one shouldn’t rely on or trust one’s intuition in mathematics, I wouldn’t go so far as to say that it’s useless. Intuition can be a very good guide for suggesting what problems to attempt, and how to go about attempting them.
(i) Yes, it does
(ii) I agree, intuition is vital for success in mathematics, but it must be the sort of intuition one develops through years of rigorous training. Man-on-the-Clapham-omnibus intuition is useless.
I suppose the answer would be in solving the infinite series of sums:
limit when n tends to infinite of:
sum(from i=1 to i=n) of 9/10(to i power)
I’m not too good with infinite sums but if someone could solve it he would end this dilemma.