Anyway gTownTrey.... you are likely correct in what you are talking about, but I think you are choosing your words incorrectly. I say thi because at first, everything you were saying just wasn't making sense to me. When you quoted you calc book, I could see exactly what you are talking about. So, I'll rep you for truce.


05202007, 11:53 PM #421Non nobis rerum modi iusto.

05202007, 11:55 PM #422

05202007, 11:59 PM #423
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I agree. What I disagree with are the fact that the "proofs" of .999... = 1 aren't pure mathematical proofs, they are more like conjectures with a little rounding here and there. Proving that there is a least upper bound to .999... when compared with 1 does not prove that it is 1, it just shows it is the closest thing to 1. I'm going to use whole numbers in a rebuttle to show that in [0,5) there exists a least upper bound "n" , namely n = 4 between [0,5). However just because 4 is the least upper bound to 5, it isn't 5. It never will be 5 just as .999... will never be 1. If it did reach 1 then you could use the same least upper bound argument to show that you can reach a whole in a plane. If you notice on wikipedia, in almost all the "proofs" they use "1/3 = .333..." or something similar when in fact .333... != 1/3, it is simply the decimal reprisentation of 1/3 just as the convergent summation s(n) = 2 does not equal 2, it is just the representation of convergence.
Last edited by gTownTrey; 05212007 at 12:03 AM.
Georgetown University
Biology and Math Double Major
Giardia lamblia researcher =P

05212007, 12:01 AM #424
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I will def. rep you back because I rarely hear intelligent people talk on these boards and I enjoyed having this discussion with you. It's always nice to have good discussions. I agree in truce. I am never one with words and I'm sure I used incorrect ones all over the place. I have a hard time writing what I'm thinking and it's a real problems when I'm trying to write abstracts and stuff hahaha. Sorry about all the confusion.
Georgetown University
Biology and Math Double Major
Giardia lamblia researcher =P


05212007, 12:02 AM #425
Well, given two numbers a and b that are real numbers that are not equal, yes, we can say that a > b or b > a. However, you make the assumption, whether you know it or not, that all the real numbers can be listed in some way. As the the mathematician Cantor has shown, one cannot do it. Here you can find a proof: http://www.classy.dk/log/archive/000906.html
Non nobis rerum modi iusto.

05212007, 12:06 AM #426
From your own quote:
"It is only in the sense that the sequence of partial sums s(n) converges to the limit 2 that we say 'the sum of the series is 2'."
Obviously, we can't add up an infinite number of terms because they are infinite.
Let me take it a little further:
0.999... = 0.9 + 0.09 + 0.009 + ...
0.999... = 9*10^1 + 9*10^2 + 9*10^3 + ...
0.999... = sum(9*10^n) where n=1 and n > infinity
0.999... = limit[sum(9*10^n) where n=1 and n > N], N > infinity
0.999... = limit[9*10^1 + 9*10^2 + ... + 9*10^N], N > infinity
0.999... = limit[0.9 + 0.09 + 0.009...], N > infinity
0.999... = limit[0.999...], N > infinity
0.999... = limit[1  10^N], N > infinity
0.999... = (limit[1], N > infinity)  (limit[10^N], N > infinity)
0.999... = 1  0
0.999... = 1

05212007, 12:07 AM #427
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The way I see it.. the seemingly nonexistant difference in time would eventually become noticable if you did this with a million computers. the very small lengths of time would eventually add up to the point where the first computer would have booted up already while say the millionth would not be done booting at the same time. and if there were an infinite number of computers that this occurred with the difference in booting times would just get larger and larger.
it reminds me of the idea they had in office space to steal money from the company by taking fractions of cents of transactions and putting it into an account. it wouldnt be much at first but over time that would add up to alot of money.Last edited by Joemama06; 05212007 at 12:09 AM.

05212007, 12:10 AM #428
http://www.tenthdimension.com/flash2.php
That have anything to do with what you guys are talking about?"Watch your thoughts, for they become words. Choose your words, for they become actions. Understand your actions, for they become habits. Study your habits, for they will become your character. Develop your character, for it becomes your destiny."

05212007, 12:13 AM #429
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From my own quote, it says "partial sums s(n) converges to the limit". Any sum you take converges to the limit. Converges is the key word here. They converge. They don't equal. You say equal because equal is the representation of convergence. It is not the same equal as in 1+1=2. 1+1 actually equals 2. 1+1 does not converge to 2, it equals it.
I honestly don't see the point in arguing with a calculus textbook on whether or not series actually yield a sum. They yield a number that converges to a sum. No matter what way you look at it that "=" sign everywhere you have typed is just a calculus representation of convergence when dealing with series. If you replaced the "=" with it's true definition here, convergence, the end of your argument would state:
0.999... converges to (limit[1], N > infinity)  (limit[10^N], N > infinity)
(limit[1], N > infinity) converges to 1
(limit[10^N], N > infinity) converges to 0
Therefore, .999... converges to 10
.999... converges to 1.
I literally took that right out of the textbook. Anyone can feel free to look up convergence in any textbook, and I can assure you (if it's a real textbook, not "Calculus 101 by Bb.com misc") that convergence approaches a number and does not reach it.Last edited by gTownTrey; 05212007 at 12:18 AM.
Georgetown University
Biology and Math Double Major
Giardia lamblia researcher =P


05212007, 12:21 AM #430

05212007, 12:28 AM #431

05212007, 12:31 AM #432

05212007, 12:43 AM #433
ah so when i prove you wrong you shall bail out, go for it.
i love when someone tries to be all high and mighty when they cant even prove their assumptions, ive proved my facts once and over in every way you kept asking and you still cant understand.
I didnt say "there is an infinite c", see this is where you begin crumbling under your weak argument and twisting my words again
I said "unless you claim that there is an infinite c between 0.999... and 1 "
so read the entire post intead of pulling things out of context
i love it when people think they are automatically right and they "give up" as in they think they are still 100% right

05212007, 12:48 AM #434


05212007, 12:51 AM #435

05212007, 12:58 AM #436

05212007, 01:08 AM #437

05212007, 01:10 AM #438
The quote that you gave doesn't tell the whole story. Even I misunderstood and foolishly requoted your quote (I apologize, I've been studying all day for finals tomorrow and my brain is almost dead) when I should have seen the root of the misunderstanding and point out the fallacy of you providing that quote (I am not saying that the quote is wrong, merely that you are using the wrong one). I have also been careless in using "convergence" so that is my bad.
Let me start over again. I am talking about the sum of the WHOLE geometric series. Not some, not partial. Consider a simple series: 1/2 + 1/4 + 1/8 + ...
As more and more terms are added, we get closer and closer to 1. I believe that is what you're trying to show me and I agree with that.
So we have established that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 is close to 1 and if we add 1/64 to the above we are even closer to 1, etc.
Now this is my point: I am NOT trying to get "closer and closer" to 1. Instead, I am trying to get the ENTIRE sum of the infinite series. 1/2 + 1/4 + 1/8 + 1/16 + ... = 1. If you have one blueberry pie and cut it in half, then cut each half another half, etc...you do NOT have the pie "converging" to one, do you? You still have one blueberry pie.
If we are talking about partial sums, then we can describe the series of partial sums as "converging." But if we are talking about taking the ENTIRE infinite series, much like gluing back the blueberry pie slices into a whole pie, we are using the symbol "=", not the term "converging."
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 1
Not "converging to 1." But EQUALS 1.
The same thing with 0.999...
It doesn't "converge to 1" but it is EQUAL to 1.
1 = 0.999...
0.999... = 1
They are one and the same.
Two different symbols representing ONE number.

05212007, 01:12 AM #439
er... didnt the OP say that ALL of the computer turn on at the EXACTLY same point in time?? well... then it doesnt matter if theres 2 of them or infinity of them. they just ALL turn on at the SAME time...therefore it has nothing to do with convergence theory... after ALL the comps get turned on at EXACTLY the SAME time... time continues on...OP needs to l2logic
from my understanding convergence theory is when two numbers are real close can be considered the same...this DOES NOT mean dat 2 numbers EXACTLY the SAME(as OP said) are slightly differentLast edited by stefanaustralia; 05212007 at 01:16 AM.
With each jacked inch I add to my biceps, the more beef I inherit on the streets.
Zyzz died for our gains, and when he returns he'll cast aside all your false prophets.
I live my life one set at a time. For those 10 reps or less, nothing else matters.


05212007, 01:15 AM #440
0.999... and 1 are not equal
/thread
However, you make the assumption, whether you know it or not, that all the real numbers can be listed in some way. As the the mathematician Cantor has shown, one cannot do it. Here you can find a proof: http://www.classy.dk/log/archive/000906.html

05212007, 01:17 AM #441

05212007, 01:18 AM #442

05212007, 01:27 AM #443
And in your case, your closeminded attitude is the problem. Read through the many links provided by Absane and several others (including me). From your posts, you seem to be a smart person but is just too stubborn.
http://en.wikipedia.org/wiki/.999
Start with that link, please. It is fairly simple and there are easy proofs in it.

05212007, 01:28 AM #444
I'll even link one just for you:
http://www.physicsforums.com/showthread.php?t=5513


05212007, 02:17 AM #445

05212007, 02:35 AM #446

05212007, 03:47 AM #447
ALRIGHT!!!! we've figured a way to stop ****in time!!!!
just get an infinite line of computers and turn them all on at the same time.
And i'm pretty ****ing sure the first computer will turn on. It takes 5 seconds to do, and after 5 seconds it will be on. The other computers can't sea what each other is doing and decide what they want to do
/thread

05212007, 05:23 AM #448

05212007, 05:28 AM #449


05212007, 10:25 AM #450
I DON'T vehemently believe there is a number beetwee .999... and 1. YOU do. I was asking you to provide one.
That would be true if there were "very small lengths of time" to add up, but these "very small lenghts of time" don't exist in my thought experiment. I'm saying the time difference between each event is 0.0 seconds. An infinitely small amount of time is zero, so nothing adds up to anything.
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