Talk:0.999.../Arguments/Archive 4

message: :Moved from main talk page. --Brad Beattie (talk) 02:22, 26 October 2006 (UTC)
replies: text: Yes you can. These operations are justified most easily by the infinite series formalism. Vague statements like "little bit more" and "never be reached" don't fly in mathematics. Melchoir 22:07, 25 October 2006 (UTC)

text: 0.999...=1 becuse each 9 brings the number closer to 1 each time, until it brings it infinitely close to 1, since the 9's go for infinity. Infinity is the largest possible number tht goes on forever. To put it this way, if you could count from 1 to infinity, it would take you one eternity to do so. It's the same concept with 0.00000...1=0 and 0.19999999999999...=0.2. I guess you could say there is an infinite number of numbers like this. AstroHurricane001 22:19, 25 October 2006 (UTC)

text: And there is no last digit. Read the article or get off the talk page. Melchoir 22:46, 25 October 2006 (UTC)

text: Yeah, that material is cited to articles in Mathematics Education Research Journal and Mathematics Teaching. The later treatment of last digits is due to books published by Dover and Oxford University Press. I trust your theories about who doesn't understand what are reflected in similar literature? Melchoir 23:10, 25 October 2006 (UTC)

text: Oh, if you wanted a debate, just enter the relevant search term into Google and you'll get all kinds of incoherent shouting. This is an encyclopedia article. Melchoir 23:42, 25 October 2006 (UTC)

text: Yes, you can go find a reliable source that disputes a claim made in the article. When you bring that source to our attention, it will be integrated into the article in a manner consistent with WP:NPOV. If you have no sources, then you have no power here. This is an encyclopedia, not a court of truth. Melchoir 23:54, 25 October 2006 (UTC)

text: Of course, even if this was a court of truth, the article would stay, since it is true. It's inappropriate of you to project your own misunderstanding of mathematics in general, and decimal expansions in particular, onto others who happen to actually know what they are talking about. Read the article and this arguments page. All the answers are found within. -- Meni Rosenfeld (talk) 09:18, 26 October 2006 (UTC)
thread_title:  your loophole 

message: :Moved from main talk page. --Brad Beattie (talk) 02:23, 26 October 2006 (UTC)
replies: text: "Always"? Melchoir 01:58, 26 October 2006 (UTC)

text: Um, I'm no math expert, but if I start with 0.3 and only add digits to the right of the three, I'm not approaching infinity; I'm approaching 0.4  - aren't I? --Badger151 02:05, 26 October 2006 (UTC)

text: "Every value in the infinite set"? "Number of approaches tends towards infinity"? What are you talking about? Melchoir 02:08, 26 October 2006 (UTC)

text: And where, in the definition of the addition of two decimals, do those concepts arise? Melchoir 02:19, 26 October 2006 (UTC)
thread_title:  0.333...+0.666... isn't 0.999... 

message: 0.333... is less than 1/3 because every time you add another 3, the decimal only gets 10 times closer, but never really hits the fraction of 1/3. The difference between the two numbers gets infinitely small, but the numbers are not the same.
:You're thinking of a finite number of threes, with a final three at the end. The value 0.333... does not have an end, so it is exactly 1/3. Read the article; this misconception is explained. -- Schapel 03:01, 26 October 2006 (UTC)
thread_title:  0.333... < 1/3 

message: :Moved from Talk:0.999... Supadawg (talk • contribs) 03:18, 26 October 2006 (UTC)
replies: text: Oy. 0.333… is easily proved equal to 1/3 by summing the implied geometric series. There are no last digits to multiply to zero. And all of the cited references support the article; that's why they're cited. There are plenty of textbooks among them. Melchoir 00:44, 26 October 2006 (UTC)
thread_title:  Failed Logic 

message: :Moved from Talk:0.999... Supadawg (talk • contribs) 03:18, 26 October 2006 (UTC)
replies: text: .9999-> of an apple is the same as 1 apple. Melchoir 22:01, 25 October 2006 (UTC)
thread_title:  Step 2??? 

message: :Moved from Talk:0.999... Supadawg (talk • contribs) 03:18, 26 October 2006 (UTC)
replies: text: I do this everyday. Everytime I take a step, there's a point where I take 90% of my step, 99% of my step, 99.9% and so on. Eventually I complete my step, and I've covered every point in between (there's infinite points in my path). "Infinitely small" for most purposes means 0. A point is infinitely small, and occupies no space. Falsedef 05:10, 26 October 2006 (UTC)

text: No. Argyrios 03:46, 26 October 2006 (UTC)

text: "...researchers have studied the reception of this equation among students, who often reject the equality. The students' reasoning is often based on an expectation... that 0.999… should have a last 9." 153.104.208.67 04:42, 26 October 2006 (UTC)
thread_title:  No!! 

message: What is the last number before 1.0 that is not equal to 1.0?
Is 11:59pm and 59.999... seconds: today or tomorrow?
If 0.999...=1 then why don't you do the same for other repeating decimals?
:Do you mean write a wiki article on each of the infinite other equalities?  "We" do do the same for all other decimals ending in a repeating 9. e.g. 1.99999... = 2.  This is covered in the article under generalisations.
I already know the answers, but go ahead and amuse me.
Why don't you sign your comment so we know who you are? The Wiki team will respond that there is no previous or next number because as they claim the real number system contains no infinitesimals (not only do these not exist but they are not even well-defined. For a good laugh, see Wiki's definition of infinitesimal). Thus the real number system contains holes. Hmmm, this is kind of strange because in Calculus we talk about continuous functions. We further define continuity in terms of limits. I wonder what happens to continuity when we reach these holes? Does a function experience what is called a bump? See, the problem lies with the subject of real analysis - it is mostly illogical nonsense that contains serious contradictions. 198.54.202.254 06:15, 27 October 2006 (UTC)
replies: text: Hard to know from this if you're a troll or genuine, since you seem to be claiming as evidence something that proves the exact opposite of what you're arguing.  It's exactly because there are no holes that there are no infinitesimals.  If there were infinitesimals there would be holes, for example half way between the infinitesimal and the next number.  Since the real numbers are continuous and have no holes, there can be no infinitesimals.  You seem to be saying that if the real numbers do not have infinitesimals, they would have holes.  Can you point to any of them?
Anonymous: It would help if you read my comment carefully - I do not believe in 'infinitesimals'. Furthermore, I agree that reals do not have holes. I am saying that Wiki's arguments imply that holes exist and they use this as an excuse to support their argument which goes something to the effect that the underlying structure of the reals must have 0.999... = 1. This is false. And can I point to an example? Why of course: 0.999... - if this were a rational number, one would be able to express it as a ratio of two integers. This is impossible unless you start out by assuming the two are equal in which case you are defeating your argument through faulty logic. 198.54.202.254 11:40, 27 October 2006 (UTC)
thread_title:  Questions about the article 

message: the argumentation used has a fault in from line 2 to line 3.
"c  = 0.999… 
10c  = 9.999… 
10c − c  = 9.999… − 0.999… 
9c  = 9 
c  = 1"
Because this is an equasion, in this step the same value has to be subtracted form both sides of the equasion for it to continue beeing true. The author does not consider this. Actually c would equal 0.999…, (which is the inital statement) if the equasion continued valid.
c               =0.999… 
10c             =9.999…
10*0.999…-0.999…=9.999…-0.999…
9=9
this is the mathematically correct version of the equasion, which actually has no meaning at all.
:In the third step, c is subtracted from both sides. On the left side, it is written as c, and on the right side it is written as 0.999... The first step says these are the same. -- Schapel 12:50, 29 October 2006 (UTC)
replies: text: I don't understand what's being argued in this section... and I don't think it's because of the usual reason (i.e. my poor maths). Could you please restate the objection as clearly as possible. --Dweller 11:40, 2 November 2006 (UTC)
thread_title:  The algebraic proof is wrong 

message: Our mathematical system defines a numberline as a one-dimensional string of infinite points. Perhaps to look at the concept of dimensions would help people understand this debate.
0-dimensional = a point. (space-type)
1-dimensional = line: an infinite series of points. (line-type)
2-dimensional = plane: an infinite series of lines. (plane-type)
3-dimensional = space: an infinite series of planes. (space-type)
4-dimensional = time: an infinite series of spaces. (line-type)
5-dimensional = ?: an infinite series of times? we cannot comprehend this (plane-type)
-1-dimensional = that which, when strung together, makes a point. ditto. (plane-type)
A numberline is a 1-dimensional string of infinite points. There are an infinite number of points on the whole line and even between any two points. 0.999..., 1.0, and 1.000...1 represent three consecutive 0-dimensional points on a numberline such that if you say, "gotcha, there are an infinite number of points even between two points," you are only just beginning to understand the value of 0.999... Gotta run, discuss-- --JohnLattier 12:37, 30 October 2006 (UTC)
replies: text: Points on the real number line are not consecutive; they form a continuum. Between any two distinct points, there are infinitely many points between them. Between any two distinct points, there is a non-zero distance between them. -- Schapel 12:50, 30 October 2006 (UTC)
replies: text: This is exactly why Metric space theory cannot be applied to real numbers. The main idea in a real space which is metric is that two numbers/points are the same if the distance between them is zero. In real life, this is untrue and is the real reason behind all misunderstandings regarding this very issue. You can define a point as a location but it is not very helpful assigning real numbers to points on a number line. Real numbers are finite and have finite representations.  This is how arithmetic was invented - with finite quantities. This characteristic gives real numbers their realness. The limit concept is derived fro the upper bound principle. The reason we talk about 1 being the limit of the series 9/10+9/100+9/1000+...  is because we don't know what the sum of this series is nor can we ever hope to find out. To blatantly define the limit of an infinite sum as the number illustrates a human tendency to resolve the uncertainty. 198.54.202.254 05:59, 31 October 2006 (UTC)
replies: text: Ahhh, finally, the core misunderstanding surfaces. Firstly, the real numbers ARE a metric space. They are that, by their very construction, and the metric is defined by d(x,y) = |x - y|. Raising issues of "real life" reveals a failure to appreciate the nature of mathematics, and is characteristic of folk mathematics. In maths, we have created the real numbers as a model for quantities we observe in real life - but that model is a wholly-developed mathematical number system, all the same. It is formally defined, and it is subject to all the rigourous methods of axiomatic proof, just as any other mathematical number system is. If you want a number system in which every number is "finitely representable", then you want the rational numbers, because there are uncountably many numbers in the reals that are NOT finitely representable. Maelin (Talk | Contribs) 06:11, 31 October 2006 (UTC)

text:  To 198.54.202.254: You claim that real numbers have finite representations. I guess you had a finite alphabet in mind. All possible finite strings over a finite alphabet (and in fact, also over a countably infinite alphabet) form a countable set, while the set of real numbers is uncountable. So the reals cannot have a finite representation over an at most countable alphabet. --Kprateek88(Talk | Contribs) 11:47, 31 October 2006 (UTC)

text: 1.000...1 is without meaning; it requires a lack of understanding of the concept of "infinite". 
--jpgordon∇∆∇∆ 14:50, 30 October 2006 (UTC)
replies: text: While 1.000...1 is not a formally defined term, I think I have a mathematical definition of the concept it probably is meant to convey.  Namely, define 1.000...1 as the limit of the sequence {1.1, 1.01, 1.001, 1.0001, ...}.  Or, arithmetically, it would be the limit as x approaches infinity of f(x)=1 + 10^-x.  You'll find when you try and calculate this number that, not surprisingly, the limit equals 1.  The proofs are similar to the proofs that the limit of {0.9, 0.99, 0.999, ...} equals 1.  Dugwiki 18:29, 30 October 2006 (UTC)
replies: text: Corrected your equation.  mdf 19:02, 30 October 2006 (UTC)

text: Thanks. :)  Also, another interesting thing I just noticed is that another poster had tried to ask about what 1.000...2 would represent.  Extending this limit idea, 1.000...2 would be the limit of the sequence {1.2, 1.02, 1.002 ...}, which is the limit as x approaches infinity of f(x)=1 + 2*10^-x. Again, that limit equals 1.  In fact, you can show that for all postive finite integers k, "1.000...k" equals 1 when 1.000...k is defined as the limit of {1.k, 1.0k, 1.00k, 1.000k, ...}. In other words, 1.000...1 = 1.000...2 = 1.000...k = 1 Dugwiki 20:21, 30 October 2006 (UTC)
replies: text: Yes, this is obvious. But it's beside the point. JohnLattier was obviously not defining the meaning of these strings according to their limits in the usual sense. I was asking him what the reciprocal of 1.000...2 would be, according to his reasoning. --Trovatore 20:24, 30 October 2006 (UTC)
replies: text: The reason I posted in this thread and not the thread you're talking about is that it is a footnote for my post.  I never said it was directly related to the other JohnLattier thread.  So it's not "beside the point" - it's just an observation that in fact all such uses of that notation will end up having the same value. And judging by some of the earlier posts in this talk page, that fact apparently is not obvious to all the readers here; it's only obvious to people who are used to dealing with limits. Dugwiki 20:41, 30 October 2006 (UTC)
thread_title:  Numberline concept 

message: Is 0.999... contained in the interval [1,2]? Discuss. --JohnLattier 02:04, 5 November 2006 (UTC)
replies: text: Of course it is; 0.999... is strictly equal to 1. What is the point of this question? --jpgordon∇∆∇∆ 02:48, 5 November 2006 (UTC)

text: Also note that 0.999… is not contained in the interval [0,1), because that interval does not contain 1. Likewise, the set [0,1) ∪ (1,2] excludes exactly one real, which is 0.999…, a.k.a. 1.
— Loadmaster 22:41, 6 November 2006 (UTC)

text: Anyone who thinks a set need contain its upper bound should read the supremum article.  It seems some people (and/or sockpuppets) stick to critiquing this article in particular when so many mathematical concepts have identical underlying axioms. Calbaer 18:52, 8 November 2006 (UTC)

text: You're wrong, and if you're just going to make things up, we can't discuss this any more. Go get published, then come back here. Maelin (Talk | Contribs) 13:14, 9 November 2006 (UTC)
thread_title:  [1,2]