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 Posted: Tue Dec 05, 2006 10:58 pm
I recently came across this image.  There has to be something wrong with it! I can't figure out what, though. sweatdrop
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Posted: Wed Dec 06, 2006 8:17 am
There isn't. At worst, the proof is incomplete, since we should assure that "0.999..." represents a valid real number. But that's easy because the reals are complete and from the definition of infinite decimal representation as limit of partial decimals, in this case {0.9, 0.99, 0.999, ...}.
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Posted: Sun Dec 10, 2006 10:02 am
Indeed. However you look at it, as a real number, a fraction or a limited process, 0,999999... = 1
Another (correct) proof:
0,11111111... = 1/9
So: 0,9999999... = 9*0,1111111... = 9*1/9 = 1
Another proof
n -> 1 - (0,1)^n
lim (n->infinity) 1 - (0,1)^n = 1
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Posted: Wed Jan 03, 2007 7:59 pm
Nothing wrong with it. It's all legit.
It's one of my favourites. I always show the kids I tutor on the first day. xd
It's using limits like Bartner said.
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Posted: Fri Jan 19, 2007 9:43 am
Ah!! You meanie! lol.. I do it too. =p
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Posted: Tue Jan 23, 2007 11:28 pm
wait a moment, it states quite plainly that c = 0.999...
yet says that 9c=9.
9c, using the definition above, should be equall to 8.999...
...
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Posted: Thu Jan 25, 2007 2:21 am
BloodlvsTxBvtterflies wait a moment, it states quite plainly that c = 0.999... yet says that 9c=9. 9c, using the definition above, should be equall to 8.999... ... You're right, in that 8.999... is what you get when you multiply 0.999... using the standard method for multiplying numbers. However, 10c - c= 9.999... - 0.999... seems to directly imply that 9c= 9. This comes from using the standard method for subtracting numbers. From this you get 0.999... = 1. Note that if we take that as true (which it is) then 8.999... = 8 + 0.999... = 8 + 1 =9, so really you're saying the same thing. Which isn't to say that the argument is circular, just that its conclusion is consistent when you apply it to arithmetic, as one would hope.
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Posted: Sun Jan 28, 2007 11:12 pm
I was actually just reading about this in a book and yes, it is true that 0.999... is equivalent to 1. It is just the sum of an infinite series. Much like the flaw in the common story that is used to explain asymptotes where (there are many different versions of course) an arrow is shot at a target. The target is one unit away and the arrow is traveling at 1 unit per second. To get to the target, the arrow must travel half of the distance to the target. Then it must travel half of the remaining distance to the target (which is half of a half or a fourth). Then the arrow must travel half of that distance, so on and so forth. The point is that if the arrow keeps traveling half of the remaining distance over and over again, it would never actually get there, it would only keep getting closer and closer to the target but never actually hit it. Yet we know from common experience and common sense that the arrow does indeed reach the target. The problem arises from the assumption that the infinite series just keeps continuing while in point-of-fact it does make it there, hence the sum of infinite series. So yes, 0.999...does equal 1 which means you can make all of the rest of the assumptions that are made in this problem which makes it correct.
-first real post here, go easy... smile
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Posted: Thu Mar 01, 2007 8:22 am
__penguin__ I recently came across this image. There has to be something wrong with it! I can't figure out what, though. sweatdrop Using series you can go into more detail, with geometric sums (and power sums?) or something. I could find the packet and try to scan it and upload it ... did it near the end of Calc. II (sadly).
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Posted: Wed Mar 07, 2007 5:24 am
I've always found the 'series' proof to be the clearest for me. .999 can be written as .9+.09+.009+.0009+etc. This is an infinite gemoetric series where each term in the series is equal to (.9)(1/10)^n: .9 = .9(1/10)^0 .09 = .9(1/10)^1 .009 = .9(1/10)^2 etc. The formula for the sum of such a series is: In this case a = .9 and r = 1/10 a/(1-r)=.9/(1-1/10) = .9/.9=1 There was a thread a while back on gaia and the debating just kept going and going and going. bleh
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Posted: Wed Mar 07, 2007 7:35 pm
I've got a slightly related question that will sound dumb to mathematicians and philosophers alike even though they apparently come up with different answers:
Why is it that, when you look at philosophical refutations of the "refer to calculus" answer to Zeno's Achilles/Tortoise paradox, or his equivalent dichotomy paradox, the main argument seems to be "one cannot finish the act of sequentially going through an infinite sequence," to quote Wikipedia?
Maybe it's just because I've been immersed in mathematics rather than philosophy, but I don't see any problem with finishing a countable sequence.
It mentions that any resolution to this problem that involves time makes the argument into a strawman, and that the real problem involves finishing a countable sequence, which is "logically impossible." However, to say that, to me, changes the entire problem, in that time is integral to the idea of "finishing." If one disregards restrictions on time, then one can say that the sequence finishes at t = infinity. After all, any open interval of the real numbers is homeomorphic to the real numbers in the standard topology.
Now, I've never put much faith in philosophers' abilities to use logic, since every philosophy text I've read involves unjustified and often false assumptions (false either due to empirical evidence, mathematical ignorance, or logical inconsistency) and more undefined terms than defined ones.
I put even less faith in their abilities to do mathematics (I hope you never read St. Augustine's attempt to justify the objective existence of mathematics).
So perhaps I cannot fault them for what, to me, seems like the whines of a distraught child. As far as I can tell, it all seems to be a vain attempt not to have to deal with the infinities, declaring the situation a paradox instead of giving it a perfectly consistent solution.
So I ask you, as fellow mathematicians and possibly philosophers: can one, logically (for whatever that may mean), finish (in whatever amount of time may be deemed necessary) a countable sequence, with complete disregard for the geometric nature of time?
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Posted: Thu Mar 08, 2007 3:28 am
Layra-chan I've got a slightly related question that will sound dumb to mathematicians and philosophers alike even though they apparently come up with different answers: Why is it that, when you look at philosophical refutations of the "refer to calculus" answer to Zeno's Achilles/Tortoise paradox, or his equivalent dichotomy paradox, the main argument seems to be "one cannot finish the act of sequentially going through an infinite sequence," to quote Wikipedia? Maybe it's just because I've been immersed in mathematics rather than philosophy, but I don't see any problem with finishing a countable sequence. It mentions that any resolution to this problem that involves time makes the argument into a strawman, and that the real problem involves finishing a countable sequence, which is "logically impossible." However, to say that, to me, changes the entire problem, in that time is integral to the idea of "finishing." If one disregards restrictions on time, then one can say that the sequence finishes at t = infinity. After all, any open interval of the real numbers is homeomorphic to the real numbers in the standard topology. Now, I've never put much faith in philosophers' abilities to use logic, since every philosophy text I've read involves unjustified and often false assumptions (false either due to empirical evidence, mathematical ignorance, or logical inconsistency) and more undefined terms than defined ones. I put even less faith in their abilities to do mathematics (I hope you never read St. Augustine's attempt to justify the objective existence of mathematics). So perhaps I cannot fault them for what, to me, seems like the whines of a distraught child. As far as I can tell, it all seems to be a vain attempt not to have to deal with the infinities, declaring the situation a paradox instead of giving it a perfectly consistent solution. So I ask you, as fellow mathematicians and possibly philosophers: can one, logically (for whatever that may mean), finish (in whatever amount of time may be deemed necessary) a countable sequence, with complete disregard for the geometric nature of time? Their major hangup stems from their inability to distinguish the reasoning behind mathematics and the manifested realitities that can be described by mathematics. Ironically, they're the ones setting up the straw man. By using the phrase 'finishing', they're placing infinity on the numberline. Assuming time stretches on indefinitely, and the person doing the counting can live forever, there would never be a point in time where the counter counts the infinit-th ball, or rock, or whatever he/she's counting, and thus stop counting any more. But mathematicians never claim this. Mathematicians are simply saying that, for example, given an infinite amount of seconds, and a set with an infinite amount of rocks, all the rocks in the set can be counted, or labelled with a corresponding second (assuming the person counts at the rate of one rock per second). Philosophers don't understand that 'at t=infinity' doesn't refer to an infinit-th second. Thus they don't understand the concepts of calculus or real analysis, or even predicate calculus.
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Posted: Mon Mar 12, 2007 4:50 pm
Morberticus Their major hangup stems from their inability to distinguish the reasoning behind mathematics and the manifested realitities that can be described by mathematics. Ironically, they're the ones setting up the straw man. By using the phrase 'finishing', they're placing infinity on the numberline. Assuming time stretches on indefinitely, and the person doing the counting can live forever, there would never be a point in time where the counter counts the infinit-th ball, or rock, or whatever he/she's counting, and thus stop counting any more. But mathematicians never claim this. Mathematicians are simply saying that, for example, given an infinite amount of seconds, and a set with an infinite amount of rocks, all the rocks in the set can be counted, or labelled with a corresponding second (assuming the person counts at the rate of one rock per second). Philosophers don't understand that 'at t=infinity' doesn't refer to an infinit-th second. Thus they don't understand the concepts of calculus or real analysis, or even predicate calculus. Huh. I never thought of it that way, perhaps because I'm too used to dealing with compact(ified) things.
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Posted: Fri Mar 16, 2007 1:13 am
Morberticus There was a thread a while back on gaia and the debating just kept going and going and going. bleh i see what you mean! i brought this up to my friend, and he refused to buy it. we argued for a long long time. this argument went all over the place. we even talked about singularities in the universe! we're actually still arguing about it lol. somehow... we just brought the argument into linguistics... this is a dangerous topic. haha
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Posted: Sat Mar 17, 2007 3:19 pm
Yes, .9999... is equal to one. That's why we round whee
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