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I've updated the introduction to make it clearer that in conventional mathematical usage "0.999..." is simply a shorthand abbreviation for the limit of a specific convergent series. This should hopefully head off the common misconception that 0.999... can have some other significant meaning other than as a limit of a series; or if the reader does not believe in this interpretation, make it clear to them at this early point that this is where they depart from the standard framework of mathematics. -- The Anome ( talk) 12:21, 28 March 2008 (UTC)
“ | In conventional mathematical usage, the value assigned to the notation "0.999…" is the real number which is the limit of the sequence (0.9,0.99,0.999,0.9999,…), ... | ” |
I have a big problem with the following recent addition to the Real analysis section:
All proofs given above have certain problems and aren't really rigorous mathematical proofs. Let's take a closer look.
- The proof on fractions assumes that , how do we know it's true? If any decimal is equal to 1/3 it is but perhaps no decimal is equal to 1/3, in which case that proof fails.
- The proof on digit manipulation assumes that the obvious digit manipulations are valid. The result might cause us to reject this assumption. The manipulations can be justified by more fundamental considerations, but these also establish so the proof is unneeded.
- The proof on infinite series says: "The last step — that lim 1/10n = 0 — is often justified by the axiom that the real numbers have the Archimedean property." Here we have some axiom, that magically solves the infinitesimal problem.
- The nested interval proof uses the nested intervals theorem, which is just another form of Archimedean property.
With hyperreal numbers sequences (9.9, 9.99, 9.999,...) and (9, 9.9, 9.99, ...) modulo ultrafilter are different numbers. I know this article is about standard reals, but digit manipulation proof doesn't directly use any property of standard reals, so if I (stupidly) reinterpret decimal expansion to hyperreal case, is it still a proof? No. "The validity of the digit manipulations ... follows from the fundamental relationship between decimals and the numbers they represent." The validity follows from Archimedean property, one way or other. I can accept Archimedean property, but this proof hides it. It's a fraud acceptable at high school level. Tlepp ( talk) 00:03, 31 March 2008 (UTC)
I believe that the difficulty in getting this article right hinges on a single point. Not only is 0.999... = 1 not -- for any number of reasons -- intuitively obvious, but more importantly the properties of the reals that make it true are not intuitively obvious even to most people with some mathematical training, and need to be proved in order to be believed. It is therefore perfectly reasonable for intelligent people to be skeptical about its validity and to refuse to believe some of the less rigorous proofs.
For example, most of the serious non-troll dissenters end up asserting the same thing, that 0.999... = 1 minus an infinitesimal quantity. As I understand it, this is actually not at all an unreasonable thing to say, since not only is it actually strictly correct, but also, when combined with the Archimedean property, which proves that the only infinitesimal real is 0, actually leads to a valid proof of 0.999... = 1. (While arguing this in a messageboard, I found a theory that said that if a certain progression is true in most cases, it is probably true in all cases i.e. .9+.1=1, .99+.01=1, .999+.001=1 and so on. Its right about then a finite being says I do not understand infinity. Heh. Will find name of theory)
However, the validity of the Archimedean property (or any of the other similar results necessary for a rigorous proof) is not obvious at all, and requires a quite sophisticated understanding of the construction of the reals in order to understand why it is true. All the handwavey proofs such as the digit manipulation proofs also have similar hard ideas buried within them, such as proof by infinite induction or the existence of limits.
This is why 0.999... = 1 is such a stumbling block, and also why this article is so important; rather than being stupid, I believe that many of the serious non-troll dissenters are actually nascent mathematicians, on their own personal mathematical journey that recapitulates the history of mathematics, logically picking away at the underlying structures of mathematics trying to understand why this non-obvious proposition is universally held to be true; and the only way forward for the serious dissenter is to start to get to grips with real analysis. And this is also why this article is so hard to get right; to satisfy the serious non-believer, it in effect has to be a standalone mini-course in real analysis that can be understood by a serious auto-didact with only high school mathematics and basic logical reasoning as tools. -- The Anome ( talk) 10:33, 31 March 2008 (UTC)
The problem seems obvious to me. The article currently spends half its length furiously obsessing over the real number set before grudgingly admitting that the actual subject is to be found elsewhere. (in Alternative number systems) The imaginary unit and the color blue do not exist in the real number set either, but have escaped this treatment. The failure to even define the topic before attacking it's existence undermines the credibility of the article, and makes people want to defend the subject from a perceived unfair attack. How often do people use .999... outside the concept of suggesting "the highest value less then one"?" An encyclopedia is a place for words, and aren't words defined by how they are understood? Algr ( talk) 22:50, 31 March 2008 (UTC)
Algr ( talk) 04:14, 1 April 2008 (UTC)
Algr, could you cite some references for your assertion that 0.999... is most commonly used to refer to something other than a real number? I'm sure you'll agree that 0.5, 0.333..., 0.125, 3.14159..., etc. are all used to refer to real numbers, so why would people use 0.999... to refer to something else? I'm quite certain that people usually use it to refer to "0 units, 9 tenths, 9 hundreds, 9 thousands, etc.", since that's what the notation means in every other context. -- Tango ( talk) 08:58, 1 April 2008 (UTC)
Because this is the talk page and not /Arguments: Does anybody (except Algr) claim that 0.999... usually denotes anything but a real number? Does anybody (including Algr) have reliable sources for such a claim? Does anybody have a source for something without Dedekind-completeness (or something equivalent) included in its axioms still being called "real numbers", and if so, should this article discuss that case? -- Huon ( talk) 11:49, 2 April 2008 (UTC)
I have unsuccesfully suggested a POV tag more than one year ago Talk:0.999.../Archive_11, others may have before me and I see that the issue still comes up at times. All arguments are quickly and abruptly settled by a group of people who actually do know what they're saying, but, i suspect, don't understand what they're being told. My view is that this article is POV and systemically biased. Not because there may be disagreement over the validity of the claim. Demonstration and sources prove that 0.999... indeed equals 1. But it's not enough to tell the truth, in an encyclopedia you have to tell it to everyone. Why I think there should be a POV tag is because this is a story about how frustrated some mathematicians and how wrong some students are. Well, this may be fun in a scientific journal and does not serve the purpose of wikipedia. The specific origin of the debate is that the conclusions and asking the general public to think like mathematicians. Maybe mathematicians should try to think like the general public in order to really make this NPOV. Here's an example of a similar topic done right: Monty_Hall_problem Luciand ( talk) 01:20, 4 April 2008 (UTC)
My father died today, so I won't be able to participate here for a while, even though there are some things I hope to explain. I'll be back when I can, but I might want to start on a less combative article. Algr ( talk) 05:00, 5 April 2008 (UTC)
From the page under the section Proofs->Real numbers->Dedekind cuts
Conversely, an element of 1 is a rational number , which implies .
Prove it. I'm not showing skepticism to the truth of the statement; just wondering how one would go about proving it. -- 69.91.95.139 ( talk) 15:10, 6 April 2008 (UTC)
0.999... is not equal to 1 without first establishing that 0.999... is defined as the limit of that notation if you say that n is the amount of digits you allow while cutting the rest of. With n going to infinity, which is done at some place too late in this article. But it was never taught that way to the students, who were never introduced to the formal concept of the limit. In fact, saying that a student is 'right' or 'passed the test' when he or she says 'It is equal to 1.' is nonsense I assert without having told the student that the value of the repeating decimal is defined as its limit as the number of digits goes to infinity. They just vaguely tell them 'It means that you keep on repeating the nine until infinity', 'Until infinity'? Infinity is somewhere one 'stops' or something? If it is put like that, it is mathematical nonsense. Children who say 'It is equal to 1.' fail the test if this is all they have been told is my opinion. It is not equal to one, it is not defined, it is mathematical nonsense. Students who say 'What you are saying here is not anything I can say mathematically exist.' pass cum laude as far as I think. Of course if they do it well and they tell students 'If I give you any positive value, you matter how small. Can you make the difference between 1 and 0.999.. where n is the amount of digits you allow before cutting of the rest smaller than that value if you can make n as large as you want?' they are of course going to answer 'Yes.', each and everyone of them. /rant Niarch ( talk) 23:04, 5 May 2008 (UTC)
I found a mathematical reason: Löwenheim–Skolem theorem, Skolem's paradox. Axiom of Dedekind-completeness is a statement in second order logic. The real numbers are not absolute, but categorical. Could this article mention the mathematical fact that equality "0.999... = 1" is not a theorem in first-order logic? Tlepp ( talk) 06:05, 13 May 2008 (UTC)
This has a lot to do with Lowenheim-Skolem. The article on second-order logic has a section "Why second-order logic is not reducible to first-order logic" that tell us how we can interprep LUB in first-order logic. Relevant part of that section is:
The article on archimedean property contains proof that full least upper bound property implies archimedean property. Relevant part of the proof:
If we interprep LUB in first-order logic, the problem is that Z is not necessarily an internal set, all we can say is:
Archimedean property, least upper bound property and (non-)existence of infinitesimals is relevant to this article and so is Lowenheim-Skolem. Tlepp ( talk) 21:43, 13 May 2008 (UTC)
It is an undefined term in ZFC and hence occurs only informally. Some of those informal uses can be explained in terms of ZFC. I translated what I meant in the following two cases: When I say, " In ZFC, is a set," I mean "ZFC proves ." Of course, someone might speak of sets in some context other than ZFC. If so, they mean something else. Perhaps we should take this to email, since it has strayed from the point of this page. I can be reached at jesse@phiwumbda.org. Phiwum ( talk) 22:24, 14 May 2008 (UTC)
Can anyone find a number divided by a prime number that equals to 0.33333333333333333... ?
I tested and things like 1000000000000000000000000000 / 3000000000000000000000000001 and 1000000000000000000000000000 / 2999999999999999999999999999 never repeat forever, but they eventually stop. William Ortiz ( talk) 07:52, 13 May 2008 (UTC)
It is not true that 1000000000000000000000000000 / 3000000000000000000000000001 does not repeat forever; in fact it does. See rational number. Michael Hardy ( talk) 09:11, 14 May 2008 (UTC)
I was thinking about what the smallest possible number would be, i.e "0.00...1", and then I realized if you're trying to do that, in reality it's "0.00...". You never get to the point where you can affix the "1", because the zeros never stop. Something "infinitely small" is the same as "zero", then. Likewise, saying something is "infinitely smaller" than "1" is equivalent saying it's not at all smaller than "1". That's how I finally came to accept this. —Preceding unsigned comment added by 69.62.140.50 ( talk • contribs) 17:01, 16 May 2008
Wow! Thanks for the insight..I can finally understand why this is true. Its so silly that they choose to prove 0.999.. = 1 by first stating 1/3 = 0.333.. and multiplying by 3, when they are BOTH displaying the exact same fallacy. But of course, infinity doesn't end when I kept imagining that it MUST end somehow, and when ti does it won't be equal to 1. I guess its hard when people try to imagine what infinity is - being unimaginably large/small!! 128.100.25.53 ( talk) 17:34, 22 August 2008 (UTC)
Count me in the camp of evil conspiracists who accept that . But, it seems to me that the entire dispute may have to do with the meaning of the ellipsis in the expression. There is no reason to think that three dots, in themselves, represent any particular thing, just as there is no reason to think that the superscript 2 in inherently represents the concept of x multiplied by itself. These symbols become meaningful only because they are widely held to represent those concepts. They become standard, and communication becomes possible among people who are using standard definitions of symbols.
If you were to privately define the ellipsis in to represent some (finite) large number of 9s, then of course, for you, the equality would not hold. The fact that the ellipsis--by standard definition--does not represent a finite number of 9s is meaningful only to those who accept the standard definition of the ellipsis.
It's akin to the dispute over stealth creationism--if one accepts the widely accepted definition of science as a rigorous method, then so-called "intelligent design" fails, but if one considers science to embody the opinion of people who call themselves scientists, then one can disregard the rigors of the scientific process and call just about anything into dispute as "just a theory."
The question, then, is: Do you accept the widely accepted definition, or not? If not, you will be using the the same words to describe entirely different things. The possibility of consensus among people using incompatible terminology becomes remote. These days there are many people who think that, if you believe something strongly enough, it automatically becomes true. Rangergordon ( talk) 05:02, 25 May 2008 (UTC)
I'm wondering whether or not it would make sense to introduce the controversy in its own section; as the introductory section stands, it barely has time to establish the existence of the identity before it begins refuting counterarguments.
This happens before any explanation that there is a controversy to begin with.
Many readers will refer to this page in order to help them understand fine points of their teacher's explanation of the identity; the existence of alternative theories will only serve to confuse them. Only after they understand the basic theory can they begin to make up their minds as to alternatives.
I suggest a structure such as this:
Rangergordon ( talk) 05:16, 30 May 2008 (UTC)
I'm confused by this proposal on a couple levels. First, I don't think the Introduction section refutes counterarguments. It's essentially positive and descriptive. It takes a few opportunities to distinguish between things that need to be distinguished, but I don't think that makes it argumentative.
Second, I'm not sure what you mean by controversy or alternative theories. The closest we can get to that direction, without doing Original Research, is to describe observed behavior patterns in students, and to describe educators' theories about that behavior. If you want to invent a point/counterpoint, we'll just wind up with straw man arguments that probably have little to do with the real concerns students have. Melchoir ( talk) 19:43, 1 June 2008 (UTC)
The article is internally consistent about the number of ways of representing the same number as a decimal expansion. In the article summary, it states that there are many ways:
"..all positional numeral systems contain an infinite number of alternative representations of numbers. For example, 28.3287 is the same number as 28.3286999…, 28.3287000, or many other representations."
(It's not even totally clear to me what this is saying - I guess it's referring to: "0.99...", "1", "1.0", "1.00", "1.000", ..., "1.000...").
Whatever it means, this statement is contradicted by a statement in the Introduction section stating that there are only two possible representations:
"Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on)."
I'd be inclined to delete the statement in the summary: it doesn't add anything to the article. AndrewBolt ( talk) 08:40, 16 June 2008 (UTC)
Moved to /Arguments. -- Tango ( talk) 20:17, 13 June 2008 (UTC)
I like some of FilipeS's recent edits, but not so much these two. I think "has a limit if" is more natural than "has the limit if". And the section in question doesn't concern a construction of the real numbers. Rather it concerns a proof that 0.999... = 1 based on an axiom of the real numbers, namely the nested intervals property. I'll boldly revert... Melchoir ( talk) 03:36, 18 June 2008 (UTC)
I don't understand a sentence in the Applications section: "A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines."
What is the "simple method", and what is the "opposite method"?-- 78.145.239.113 ( talk) 17:57, 22 June 2008 (UTC)
I'm not going to lie. This is a largely heuristic proof, whose explicit purpose is much the same as the algebraic proofs are: to "give you a flavor of why 0.999...=1". I feel that after those other heuristics fail, this one may have a tad more success.
Looking at increasingly accurate decimal expansions of the square root of 2, we see that:
We notice that the number normally notated squares to, not 2, but rather 1.999... Noting this, we have two options: accept that the number normally notated is falsely attributed as such, and that number is actually , or that the two are one and the same.
In the first case, there is demonstrably no decimal expansion that would correspond to , since increasing any arbitrary digit would bump the square up above 2. Thus, assuming that every real number has a decimal expansion to correspond (a fact that is drilled into our heads from day 1 in school), does not exist. Since this goes against the Greeks' historical purpose in extending their number set from the rationals to the reals, we conclude that this cannot be the case. Therefore, the alternative plays out as follows:
Sorry for the horrible display; I don't know how (or if it's possible) to force Wikipedia to display a PNG no matter what (or at least be consistent across lines). Anyway, that's my 'proof'. It's a sort of historically based proof by contradiction. I think it should follow the two algebraic proofs already put up, as one last filter to try and convince the skeptics. This proof drives home the original historical purpose of extending the rational set of numbers to the reals: to allow for a number whose square is 2. I feel that that may make this proof a bit more convincing than the other algebraic proofs, despite being perhaps a bit of a backwards step in rigor.
What do you think? -- 69.91.95.139 ( talk) 13:42, 13 July 2008 (UTC)
Both the and the .333... proofs fail because they both make unjustified assumptions about what happens when a process is repeated infinitely. 1/3 x 3 = 1 by definition, but assuming that .333...=1/3 requires the Archimedean property, which is precisely what is being questioned in the first place. The definition of "real numbers" includes an inherent contradiction, in that infinity is required to make certain decimal expansions mean what we say they do, but infinity itself is not a real number. The Greeks were not happy about the need for irrational numbers. (They threw Hippasus overboard a ship!) I think what is happening with .999... and hyperreals is basically the same thing. Algr ( talk) 21:58, 13 July 2008 (UTC)
Moved to /Arguments.
"When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which many write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999…."
This sentence seems to be based around the word novices. I'd rewrite it myself, but i'm not familiar with this subject. If anyone would like to tackle it, that would be much appreciated. Thanks, -- MattWTadded on 07:59, 12 August 2008 (UTC)
I made the following edit for clarity: [2]. The wording is obviously improved, in my opinion. However, some editors seem to feel that this suggests that this wording lends legitimacy to the representation 1-0.999... = 0.000...01. I really strongly disagree that there is any possibility for such a misapprehension. First of all, the entire article up to this point is basically devoted to dispelling such a notion. Secondly, the part of the sentence in question is a subordinate clause which serves as the direct object of the verb "believe", so that by the rules of English grammar, such a misinterpretation would be completely unfounded. Thirdly, the very next sentence begins "Whether or not this makes sense..." which obviously reinforces the (grammatically correct) interpretation that the fictional equation 1-0.999... = 0.000...01 exists only as part of the novices' belief. siℓℓy rabbit ( talk) 16:48, 12 August 2008 (UTC)
My concern with the sentence, "When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which can be written as "0.000…1"." is that it's not immeadiately clear whether the "believe" refers to just 1-0.999... being a positive number or whether it refers to the way of writing it as well. From context, you can probably work out what it means, but why bother when "many write" is much clearer and perfectly accurate? What's wrong with how it was? -- Tango ( talk) 16:49, 12 August 2008 (UTC)
I've merged your two sections; hope you don't mind. Anyway, I think the best compromise here would be "they write": the novices write the number. I'm being bold and editing accordingly, but feel free to discuss further. -- 69.91.95.139 ( talk) 20:01, 12 August 2008 (UTC)
The most simple way I know of to prove 2/3 + 1/3 = 1 in only decimal form is this.
Change the base system to a base_12 system. Thus you have 1 2 3 4 5 6 7 8 9 I O
I represents 10, O represents 11
2/3 (base_10) = 8/12 8/12 = .8 base_12
1/3 (base_10) = 4/12 4/12 = .4 base_12
.4 + .8 = 1 [All in Base_12]
Does anyone have a disagreement with this addition of this?
All without recurring decimal places. —Preceding unsigned comment added by CheskiChips ( talk • contribs) 14:31, 22 August 2008 (UTC)
The proof as written doesn't even mention repeating decimals, so it has nothing to do with the article. What you have discovered, CheskiChips, is that (IMHO) a fraction cannot be expressed in decimal unless it's denominator is a factor of the base. Base 10 = 2x5, so 1/2 and 1/5 terminate, but 1/3 does not. Base 12 = 3x2x2, so 1/3 in base 12 DOES terminate, but you will discover that 1/5 does not. My position is that infinitely repeating decimals do not truly solve this problem because all sorts of mathematical principles break down when infinity is treated as an accomplished fact. I don't know how to express the value of 1/3 - .333... except to assign it to some variable, but that is exactly how the square root of negative one was handled, so I don't see why this concept would create such a crisis. Algr ( talk) 07:42, 14 September 2008 (UTC)
A proof which I find quite powerful relies on the density of the real numbers: for any two distinct real numbers, there is a third real number that lies between them on the number line. In other words, if then there exists a such that
Assuming that a student accepts this property (which is quite intuitive), then the proof of 0.999... = 1 becomes simple. If 0.999... is less than 1, then there must be some real number (in fact infinitely many real numbers) that is greater than 0.999... and less than 1. It is clear that there can be no such numbers, therefore 0.999... = 1. Grover cleveland ( talk) 17:04, 13 September 2008 (UTC)
Note: the paper referred to above can be found at Computational Construction as a Means to Coordinate Representations of Infinity doi:10.1007/s10758-008-9127-5 -- The Anome ( talk) 10:32, 12 October 2008 (UTC)
Moved to /Arguments. -- Tango ( talk) 22:31, 20 September 2008 (UTC)
I intend to move this page to 0.999… (the difference: the current title uses three seperate dots, while the target article has a Unicode ellipsis in it). Are there any objections? -- Church of emacs ( Talk | Stalk) 17:31, 11 October 2008 (UTC)
"Although <snip>, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in Mathematics: A Very Short Introduction that the resulting identity 0.999… = 1 is a convention as well:"
"Although <snip>, the decision to interpret the phrase "Venus" as naming a planet is ultimately a convention, and Timothy Showers argues in Space: A Very Short Introduction that the resulting statement "Venus has mean surface temperature 461.85 degrees C" is a convention as well:"
See what I did there?
What one needs to say is that the phrase "0.999...=1" is intended to be interpreted as "the real number 0.999... is equal to the real number 1". There is no need for talking about p-adic numbers and Richman decimal expansions [personally I don't find "topos theory" coherent, and have yet to see a satisfactory developement of a number system that can serve the purpose of "calculus with infinitesimals", but that's by the by], because they have no relevance, and serve no purpose except to possibly confuse the student. It's like teaching a child arithmetic and giving them some answers to some basic sums as follows:
Answers: 1+1=2 (BUT 1+1=0 IN ARITHMETIC MODULO 2 !!) 3x5=15 (BUT 3x5=0 IN ARITHMETIC MODULO 5 !!) 5+9=14 (BUT 5+9=4 IN ARITHMETIC MODULO 10 !!) ...
It's pedagological madness! And the people who might gain something by reading this article are, in terms of mathematical maturity at least, children!
If nothing else, it is *nonsensical* to write "in another number system, it might not be the case that 0.999...=1". What you really mean is that "if the meanings of the symbols 0,.,9,=, or 1 are changed, then the reinterpreted statement, using the same symbols, may have a different truth value".
If I was trying to replace this nonsense with sense, I would say something like "what 0.999... really means is [lim n->+infinity](<sum k=1 to n> 9x(10^-k))" and then refer the reader to the meanings of these terms in real analysis. Oh, wait, that already happened earlier in the article. Therefore there is no need for this entire section, the charge being two-fold: (1) irrelevant, and (2) nonsensical. —Preceding unsigned comment added by 212.183.134.64 ( talk) 13:04, 23 October 2008 (UTC)
Is there any reason for JackyCheung's recent page move, replacing three dots with six? Phiwum ( talk) 19:57, 9 November 2008 (UTC)
The sources for this article seem to have gotten out of hand a little. The "notes" operate in the way that most articles handle "references", but the "references" section here could do with being properly integrated into the text. It's more like a disconnected "further reading" section at the moment (albeit a very long one). I'll try to have a look myself, but this isn't exactly my forte and I'm liable to make mistakes. Cheers, -- PLUMBAGO 16:37, 10 November 2008 (UTC)
(Decrease indent) Thanks for the discussion above. That's been helpful. Looking again at the article, I've not been able to find any "references" that do not also occur as "notes". So that's a good start. I guess that I'd like some way of retaining both the full citation ("references") and the specific location ("notes"), but doing so in a more transparent way than we have at present.
Poking around WP:CITE reveals that this issue has got a template to "solve" it. The example below tries to illustrate this:
How would that sit with people? I personally think that it's a bit ugly, but it would certainly "resolve" my original problem with this article's citations. In passing, and to save anyone from trying the same thing, I tried putting references inside references (e.g. "<ref>pg. X of Source 1<ref name=s1/></ref>"), with predictably dire results.
Anyway, does this help at all, or should I just get over it and shut up? ;-) Cheers, -- PLUMBAGO 13:00, 12 November 2008 (UTC)
You are talking about "recurring decimals". Would you so kind to present how we cuold get it by deviding two rational numbers? (Like 1/3). (Sorry I'm Hungarian my English isn't at the top. If there where any questions please contact me [cerna@richter.hu]). -- 80.99.184.247 ( talk) 20:07, 16 November 2008 (UTC)
This article is very good and informative (if not a little bit prolix), but one thing that might make this article better are pictures, with a few colours even. The first two pages of this article are black and white and even though it doesn't diminish the educational value of it, a few colours would liven it up. Only problem is that I can't for the life of me think of a suitable picture. Maybe if somebody has a textbook which covers how 0.999... equals 1 they could take a picture of it up-close? Any ideas? -- BiT ( talk) 01:14, 20 November 2008 (UTC)
I am wondering if the following comment in the article may be somewhat misleading: as one of the "erroneous intuitions about the real numbers" that students typically have, the article lists the intuition "that nonzero infinitesimal real numbers should exist". Now technically speaking it is correct to describe such an intuition about real numbers as "erroneous". However, a student grappling with this issue is certainly not sophisticated enough to mean "standard real number" when he formulates an intuition about "numbers". Would it constitute a mathematical error to experience an intuition that infinitesimal numbers should exist? Katzmik ( talk) 14:57, 30 November 2008 (UTC)
Copied from my talk page:
Now the edits in question took a paragraph from the lead, giving a summary of some sections further down, and made it the first section aftre the lead instead, preceeding the section Introduction. This was clealy unacceptable. Editing the lead to make it a slightly shorter summary of the whole article might be a good idea.-- Noe ( talk) 17:25, 30 November 2008 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 10 | Archive 11 | Archive 12 | Archive 13 | Archive 14 | Archive 15 | → | Archive 20 |
I've updated the introduction to make it clearer that in conventional mathematical usage "0.999..." is simply a shorthand abbreviation for the limit of a specific convergent series. This should hopefully head off the common misconception that 0.999... can have some other significant meaning other than as a limit of a series; or if the reader does not believe in this interpretation, make it clear to them at this early point that this is where they depart from the standard framework of mathematics. -- The Anome ( talk) 12:21, 28 March 2008 (UTC)
“ | In conventional mathematical usage, the value assigned to the notation "0.999…" is the real number which is the limit of the sequence (0.9,0.99,0.999,0.9999,…), ... | ” |
I have a big problem with the following recent addition to the Real analysis section:
All proofs given above have certain problems and aren't really rigorous mathematical proofs. Let's take a closer look.
- The proof on fractions assumes that , how do we know it's true? If any decimal is equal to 1/3 it is but perhaps no decimal is equal to 1/3, in which case that proof fails.
- The proof on digit manipulation assumes that the obvious digit manipulations are valid. The result might cause us to reject this assumption. The manipulations can be justified by more fundamental considerations, but these also establish so the proof is unneeded.
- The proof on infinite series says: "The last step — that lim 1/10n = 0 — is often justified by the axiom that the real numbers have the Archimedean property." Here we have some axiom, that magically solves the infinitesimal problem.
- The nested interval proof uses the nested intervals theorem, which is just another form of Archimedean property.
With hyperreal numbers sequences (9.9, 9.99, 9.999,...) and (9, 9.9, 9.99, ...) modulo ultrafilter are different numbers. I know this article is about standard reals, but digit manipulation proof doesn't directly use any property of standard reals, so if I (stupidly) reinterpret decimal expansion to hyperreal case, is it still a proof? No. "The validity of the digit manipulations ... follows from the fundamental relationship between decimals and the numbers they represent." The validity follows from Archimedean property, one way or other. I can accept Archimedean property, but this proof hides it. It's a fraud acceptable at high school level. Tlepp ( talk) 00:03, 31 March 2008 (UTC)
I believe that the difficulty in getting this article right hinges on a single point. Not only is 0.999... = 1 not -- for any number of reasons -- intuitively obvious, but more importantly the properties of the reals that make it true are not intuitively obvious even to most people with some mathematical training, and need to be proved in order to be believed. It is therefore perfectly reasonable for intelligent people to be skeptical about its validity and to refuse to believe some of the less rigorous proofs.
For example, most of the serious non-troll dissenters end up asserting the same thing, that 0.999... = 1 minus an infinitesimal quantity. As I understand it, this is actually not at all an unreasonable thing to say, since not only is it actually strictly correct, but also, when combined with the Archimedean property, which proves that the only infinitesimal real is 0, actually leads to a valid proof of 0.999... = 1. (While arguing this in a messageboard, I found a theory that said that if a certain progression is true in most cases, it is probably true in all cases i.e. .9+.1=1, .99+.01=1, .999+.001=1 and so on. Its right about then a finite being says I do not understand infinity. Heh. Will find name of theory)
However, the validity of the Archimedean property (or any of the other similar results necessary for a rigorous proof) is not obvious at all, and requires a quite sophisticated understanding of the construction of the reals in order to understand why it is true. All the handwavey proofs such as the digit manipulation proofs also have similar hard ideas buried within them, such as proof by infinite induction or the existence of limits.
This is why 0.999... = 1 is such a stumbling block, and also why this article is so important; rather than being stupid, I believe that many of the serious non-troll dissenters are actually nascent mathematicians, on their own personal mathematical journey that recapitulates the history of mathematics, logically picking away at the underlying structures of mathematics trying to understand why this non-obvious proposition is universally held to be true; and the only way forward for the serious dissenter is to start to get to grips with real analysis. And this is also why this article is so hard to get right; to satisfy the serious non-believer, it in effect has to be a standalone mini-course in real analysis that can be understood by a serious auto-didact with only high school mathematics and basic logical reasoning as tools. -- The Anome ( talk) 10:33, 31 March 2008 (UTC)
The problem seems obvious to me. The article currently spends half its length furiously obsessing over the real number set before grudgingly admitting that the actual subject is to be found elsewhere. (in Alternative number systems) The imaginary unit and the color blue do not exist in the real number set either, but have escaped this treatment. The failure to even define the topic before attacking it's existence undermines the credibility of the article, and makes people want to defend the subject from a perceived unfair attack. How often do people use .999... outside the concept of suggesting "the highest value less then one"?" An encyclopedia is a place for words, and aren't words defined by how they are understood? Algr ( talk) 22:50, 31 March 2008 (UTC)
Algr ( talk) 04:14, 1 April 2008 (UTC)
Algr, could you cite some references for your assertion that 0.999... is most commonly used to refer to something other than a real number? I'm sure you'll agree that 0.5, 0.333..., 0.125, 3.14159..., etc. are all used to refer to real numbers, so why would people use 0.999... to refer to something else? I'm quite certain that people usually use it to refer to "0 units, 9 tenths, 9 hundreds, 9 thousands, etc.", since that's what the notation means in every other context. -- Tango ( talk) 08:58, 1 April 2008 (UTC)
Because this is the talk page and not /Arguments: Does anybody (except Algr) claim that 0.999... usually denotes anything but a real number? Does anybody (including Algr) have reliable sources for such a claim? Does anybody have a source for something without Dedekind-completeness (or something equivalent) included in its axioms still being called "real numbers", and if so, should this article discuss that case? -- Huon ( talk) 11:49, 2 April 2008 (UTC)
I have unsuccesfully suggested a POV tag more than one year ago Talk:0.999.../Archive_11, others may have before me and I see that the issue still comes up at times. All arguments are quickly and abruptly settled by a group of people who actually do know what they're saying, but, i suspect, don't understand what they're being told. My view is that this article is POV and systemically biased. Not because there may be disagreement over the validity of the claim. Demonstration and sources prove that 0.999... indeed equals 1. But it's not enough to tell the truth, in an encyclopedia you have to tell it to everyone. Why I think there should be a POV tag is because this is a story about how frustrated some mathematicians and how wrong some students are. Well, this may be fun in a scientific journal and does not serve the purpose of wikipedia. The specific origin of the debate is that the conclusions and asking the general public to think like mathematicians. Maybe mathematicians should try to think like the general public in order to really make this NPOV. Here's an example of a similar topic done right: Monty_Hall_problem Luciand ( talk) 01:20, 4 April 2008 (UTC)
My father died today, so I won't be able to participate here for a while, even though there are some things I hope to explain. I'll be back when I can, but I might want to start on a less combative article. Algr ( talk) 05:00, 5 April 2008 (UTC)
From the page under the section Proofs->Real numbers->Dedekind cuts
Conversely, an element of 1 is a rational number , which implies .
Prove it. I'm not showing skepticism to the truth of the statement; just wondering how one would go about proving it. -- 69.91.95.139 ( talk) 15:10, 6 April 2008 (UTC)
0.999... is not equal to 1 without first establishing that 0.999... is defined as the limit of that notation if you say that n is the amount of digits you allow while cutting the rest of. With n going to infinity, which is done at some place too late in this article. But it was never taught that way to the students, who were never introduced to the formal concept of the limit. In fact, saying that a student is 'right' or 'passed the test' when he or she says 'It is equal to 1.' is nonsense I assert without having told the student that the value of the repeating decimal is defined as its limit as the number of digits goes to infinity. They just vaguely tell them 'It means that you keep on repeating the nine until infinity', 'Until infinity'? Infinity is somewhere one 'stops' or something? If it is put like that, it is mathematical nonsense. Children who say 'It is equal to 1.' fail the test if this is all they have been told is my opinion. It is not equal to one, it is not defined, it is mathematical nonsense. Students who say 'What you are saying here is not anything I can say mathematically exist.' pass cum laude as far as I think. Of course if they do it well and they tell students 'If I give you any positive value, you matter how small. Can you make the difference between 1 and 0.999.. where n is the amount of digits you allow before cutting of the rest smaller than that value if you can make n as large as you want?' they are of course going to answer 'Yes.', each and everyone of them. /rant Niarch ( talk) 23:04, 5 May 2008 (UTC)
I found a mathematical reason: Löwenheim–Skolem theorem, Skolem's paradox. Axiom of Dedekind-completeness is a statement in second order logic. The real numbers are not absolute, but categorical. Could this article mention the mathematical fact that equality "0.999... = 1" is not a theorem in first-order logic? Tlepp ( talk) 06:05, 13 May 2008 (UTC)
This has a lot to do with Lowenheim-Skolem. The article on second-order logic has a section "Why second-order logic is not reducible to first-order logic" that tell us how we can interprep LUB in first-order logic. Relevant part of that section is:
The article on archimedean property contains proof that full least upper bound property implies archimedean property. Relevant part of the proof:
If we interprep LUB in first-order logic, the problem is that Z is not necessarily an internal set, all we can say is:
Archimedean property, least upper bound property and (non-)existence of infinitesimals is relevant to this article and so is Lowenheim-Skolem. Tlepp ( talk) 21:43, 13 May 2008 (UTC)
It is an undefined term in ZFC and hence occurs only informally. Some of those informal uses can be explained in terms of ZFC. I translated what I meant in the following two cases: When I say, " In ZFC, is a set," I mean "ZFC proves ." Of course, someone might speak of sets in some context other than ZFC. If so, they mean something else. Perhaps we should take this to email, since it has strayed from the point of this page. I can be reached at jesse@phiwumbda.org. Phiwum ( talk) 22:24, 14 May 2008 (UTC)
Can anyone find a number divided by a prime number that equals to 0.33333333333333333... ?
I tested and things like 1000000000000000000000000000 / 3000000000000000000000000001 and 1000000000000000000000000000 / 2999999999999999999999999999 never repeat forever, but they eventually stop. William Ortiz ( talk) 07:52, 13 May 2008 (UTC)
It is not true that 1000000000000000000000000000 / 3000000000000000000000000001 does not repeat forever; in fact it does. See rational number. Michael Hardy ( talk) 09:11, 14 May 2008 (UTC)
I was thinking about what the smallest possible number would be, i.e "0.00...1", and then I realized if you're trying to do that, in reality it's "0.00...". You never get to the point where you can affix the "1", because the zeros never stop. Something "infinitely small" is the same as "zero", then. Likewise, saying something is "infinitely smaller" than "1" is equivalent saying it's not at all smaller than "1". That's how I finally came to accept this. —Preceding unsigned comment added by 69.62.140.50 ( talk • contribs) 17:01, 16 May 2008
Wow! Thanks for the insight..I can finally understand why this is true. Its so silly that they choose to prove 0.999.. = 1 by first stating 1/3 = 0.333.. and multiplying by 3, when they are BOTH displaying the exact same fallacy. But of course, infinity doesn't end when I kept imagining that it MUST end somehow, and when ti does it won't be equal to 1. I guess its hard when people try to imagine what infinity is - being unimaginably large/small!! 128.100.25.53 ( talk) 17:34, 22 August 2008 (UTC)
Count me in the camp of evil conspiracists who accept that . But, it seems to me that the entire dispute may have to do with the meaning of the ellipsis in the expression. There is no reason to think that three dots, in themselves, represent any particular thing, just as there is no reason to think that the superscript 2 in inherently represents the concept of x multiplied by itself. These symbols become meaningful only because they are widely held to represent those concepts. They become standard, and communication becomes possible among people who are using standard definitions of symbols.
If you were to privately define the ellipsis in to represent some (finite) large number of 9s, then of course, for you, the equality would not hold. The fact that the ellipsis--by standard definition--does not represent a finite number of 9s is meaningful only to those who accept the standard definition of the ellipsis.
It's akin to the dispute over stealth creationism--if one accepts the widely accepted definition of science as a rigorous method, then so-called "intelligent design" fails, but if one considers science to embody the opinion of people who call themselves scientists, then one can disregard the rigors of the scientific process and call just about anything into dispute as "just a theory."
The question, then, is: Do you accept the widely accepted definition, or not? If not, you will be using the the same words to describe entirely different things. The possibility of consensus among people using incompatible terminology becomes remote. These days there are many people who think that, if you believe something strongly enough, it automatically becomes true. Rangergordon ( talk) 05:02, 25 May 2008 (UTC)
I'm wondering whether or not it would make sense to introduce the controversy in its own section; as the introductory section stands, it barely has time to establish the existence of the identity before it begins refuting counterarguments.
This happens before any explanation that there is a controversy to begin with.
Many readers will refer to this page in order to help them understand fine points of their teacher's explanation of the identity; the existence of alternative theories will only serve to confuse them. Only after they understand the basic theory can they begin to make up their minds as to alternatives.
I suggest a structure such as this:
Rangergordon ( talk) 05:16, 30 May 2008 (UTC)
I'm confused by this proposal on a couple levels. First, I don't think the Introduction section refutes counterarguments. It's essentially positive and descriptive. It takes a few opportunities to distinguish between things that need to be distinguished, but I don't think that makes it argumentative.
Second, I'm not sure what you mean by controversy or alternative theories. The closest we can get to that direction, without doing Original Research, is to describe observed behavior patterns in students, and to describe educators' theories about that behavior. If you want to invent a point/counterpoint, we'll just wind up with straw man arguments that probably have little to do with the real concerns students have. Melchoir ( talk) 19:43, 1 June 2008 (UTC)
The article is internally consistent about the number of ways of representing the same number as a decimal expansion. In the article summary, it states that there are many ways:
"..all positional numeral systems contain an infinite number of alternative representations of numbers. For example, 28.3287 is the same number as 28.3286999…, 28.3287000, or many other representations."
(It's not even totally clear to me what this is saying - I guess it's referring to: "0.99...", "1", "1.0", "1.00", "1.000", ..., "1.000...").
Whatever it means, this statement is contradicted by a statement in the Introduction section stating that there are only two possible representations:
"Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on)."
I'd be inclined to delete the statement in the summary: it doesn't add anything to the article. AndrewBolt ( talk) 08:40, 16 June 2008 (UTC)
Moved to /Arguments. -- Tango ( talk) 20:17, 13 June 2008 (UTC)
I like some of FilipeS's recent edits, but not so much these two. I think "has a limit if" is more natural than "has the limit if". And the section in question doesn't concern a construction of the real numbers. Rather it concerns a proof that 0.999... = 1 based on an axiom of the real numbers, namely the nested intervals property. I'll boldly revert... Melchoir ( talk) 03:36, 18 June 2008 (UTC)
I don't understand a sentence in the Applications section: "A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines."
What is the "simple method", and what is the "opposite method"?-- 78.145.239.113 ( talk) 17:57, 22 June 2008 (UTC)
I'm not going to lie. This is a largely heuristic proof, whose explicit purpose is much the same as the algebraic proofs are: to "give you a flavor of why 0.999...=1". I feel that after those other heuristics fail, this one may have a tad more success.
Looking at increasingly accurate decimal expansions of the square root of 2, we see that:
We notice that the number normally notated squares to, not 2, but rather 1.999... Noting this, we have two options: accept that the number normally notated is falsely attributed as such, and that number is actually , or that the two are one and the same.
In the first case, there is demonstrably no decimal expansion that would correspond to , since increasing any arbitrary digit would bump the square up above 2. Thus, assuming that every real number has a decimal expansion to correspond (a fact that is drilled into our heads from day 1 in school), does not exist. Since this goes against the Greeks' historical purpose in extending their number set from the rationals to the reals, we conclude that this cannot be the case. Therefore, the alternative plays out as follows:
Sorry for the horrible display; I don't know how (or if it's possible) to force Wikipedia to display a PNG no matter what (or at least be consistent across lines). Anyway, that's my 'proof'. It's a sort of historically based proof by contradiction. I think it should follow the two algebraic proofs already put up, as one last filter to try and convince the skeptics. This proof drives home the original historical purpose of extending the rational set of numbers to the reals: to allow for a number whose square is 2. I feel that that may make this proof a bit more convincing than the other algebraic proofs, despite being perhaps a bit of a backwards step in rigor.
What do you think? -- 69.91.95.139 ( talk) 13:42, 13 July 2008 (UTC)
Both the and the .333... proofs fail because they both make unjustified assumptions about what happens when a process is repeated infinitely. 1/3 x 3 = 1 by definition, but assuming that .333...=1/3 requires the Archimedean property, which is precisely what is being questioned in the first place. The definition of "real numbers" includes an inherent contradiction, in that infinity is required to make certain decimal expansions mean what we say they do, but infinity itself is not a real number. The Greeks were not happy about the need for irrational numbers. (They threw Hippasus overboard a ship!) I think what is happening with .999... and hyperreals is basically the same thing. Algr ( talk) 21:58, 13 July 2008 (UTC)
Moved to /Arguments.
"When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which many write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999…."
This sentence seems to be based around the word novices. I'd rewrite it myself, but i'm not familiar with this subject. If anyone would like to tackle it, that would be much appreciated. Thanks, -- MattWTadded on 07:59, 12 August 2008 (UTC)
I made the following edit for clarity: [2]. The wording is obviously improved, in my opinion. However, some editors seem to feel that this suggests that this wording lends legitimacy to the representation 1-0.999... = 0.000...01. I really strongly disagree that there is any possibility for such a misapprehension. First of all, the entire article up to this point is basically devoted to dispelling such a notion. Secondly, the part of the sentence in question is a subordinate clause which serves as the direct object of the verb "believe", so that by the rules of English grammar, such a misinterpretation would be completely unfounded. Thirdly, the very next sentence begins "Whether or not this makes sense..." which obviously reinforces the (grammatically correct) interpretation that the fictional equation 1-0.999... = 0.000...01 exists only as part of the novices' belief. siℓℓy rabbit ( talk) 16:48, 12 August 2008 (UTC)
My concern with the sentence, "When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which can be written as "0.000…1"." is that it's not immeadiately clear whether the "believe" refers to just 1-0.999... being a positive number or whether it refers to the way of writing it as well. From context, you can probably work out what it means, but why bother when "many write" is much clearer and perfectly accurate? What's wrong with how it was? -- Tango ( talk) 16:49, 12 August 2008 (UTC)
I've merged your two sections; hope you don't mind. Anyway, I think the best compromise here would be "they write": the novices write the number. I'm being bold and editing accordingly, but feel free to discuss further. -- 69.91.95.139 ( talk) 20:01, 12 August 2008 (UTC)
The most simple way I know of to prove 2/3 + 1/3 = 1 in only decimal form is this.
Change the base system to a base_12 system. Thus you have 1 2 3 4 5 6 7 8 9 I O
I represents 10, O represents 11
2/3 (base_10) = 8/12 8/12 = .8 base_12
1/3 (base_10) = 4/12 4/12 = .4 base_12
.4 + .8 = 1 [All in Base_12]
Does anyone have a disagreement with this addition of this?
All without recurring decimal places. —Preceding unsigned comment added by CheskiChips ( talk • contribs) 14:31, 22 August 2008 (UTC)
The proof as written doesn't even mention repeating decimals, so it has nothing to do with the article. What you have discovered, CheskiChips, is that (IMHO) a fraction cannot be expressed in decimal unless it's denominator is a factor of the base. Base 10 = 2x5, so 1/2 and 1/5 terminate, but 1/3 does not. Base 12 = 3x2x2, so 1/3 in base 12 DOES terminate, but you will discover that 1/5 does not. My position is that infinitely repeating decimals do not truly solve this problem because all sorts of mathematical principles break down when infinity is treated as an accomplished fact. I don't know how to express the value of 1/3 - .333... except to assign it to some variable, but that is exactly how the square root of negative one was handled, so I don't see why this concept would create such a crisis. Algr ( talk) 07:42, 14 September 2008 (UTC)
A proof which I find quite powerful relies on the density of the real numbers: for any two distinct real numbers, there is a third real number that lies between them on the number line. In other words, if then there exists a such that
Assuming that a student accepts this property (which is quite intuitive), then the proof of 0.999... = 1 becomes simple. If 0.999... is less than 1, then there must be some real number (in fact infinitely many real numbers) that is greater than 0.999... and less than 1. It is clear that there can be no such numbers, therefore 0.999... = 1. Grover cleveland ( talk) 17:04, 13 September 2008 (UTC)
Note: the paper referred to above can be found at Computational Construction as a Means to Coordinate Representations of Infinity doi:10.1007/s10758-008-9127-5 -- The Anome ( talk) 10:32, 12 October 2008 (UTC)
Moved to /Arguments. -- Tango ( talk) 22:31, 20 September 2008 (UTC)
I intend to move this page to 0.999… (the difference: the current title uses three seperate dots, while the target article has a Unicode ellipsis in it). Are there any objections? -- Church of emacs ( Talk | Stalk) 17:31, 11 October 2008 (UTC)
"Although <snip>, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in Mathematics: A Very Short Introduction that the resulting identity 0.999… = 1 is a convention as well:"
"Although <snip>, the decision to interpret the phrase "Venus" as naming a planet is ultimately a convention, and Timothy Showers argues in Space: A Very Short Introduction that the resulting statement "Venus has mean surface temperature 461.85 degrees C" is a convention as well:"
See what I did there?
What one needs to say is that the phrase "0.999...=1" is intended to be interpreted as "the real number 0.999... is equal to the real number 1". There is no need for talking about p-adic numbers and Richman decimal expansions [personally I don't find "topos theory" coherent, and have yet to see a satisfactory developement of a number system that can serve the purpose of "calculus with infinitesimals", but that's by the by], because they have no relevance, and serve no purpose except to possibly confuse the student. It's like teaching a child arithmetic and giving them some answers to some basic sums as follows:
Answers: 1+1=2 (BUT 1+1=0 IN ARITHMETIC MODULO 2 !!) 3x5=15 (BUT 3x5=0 IN ARITHMETIC MODULO 5 !!) 5+9=14 (BUT 5+9=4 IN ARITHMETIC MODULO 10 !!) ...
It's pedagological madness! And the people who might gain something by reading this article are, in terms of mathematical maturity at least, children!
If nothing else, it is *nonsensical* to write "in another number system, it might not be the case that 0.999...=1". What you really mean is that "if the meanings of the symbols 0,.,9,=, or 1 are changed, then the reinterpreted statement, using the same symbols, may have a different truth value".
If I was trying to replace this nonsense with sense, I would say something like "what 0.999... really means is [lim n->+infinity](<sum k=1 to n> 9x(10^-k))" and then refer the reader to the meanings of these terms in real analysis. Oh, wait, that already happened earlier in the article. Therefore there is no need for this entire section, the charge being two-fold: (1) irrelevant, and (2) nonsensical. —Preceding unsigned comment added by 212.183.134.64 ( talk) 13:04, 23 October 2008 (UTC)
Is there any reason for JackyCheung's recent page move, replacing three dots with six? Phiwum ( talk) 19:57, 9 November 2008 (UTC)
The sources for this article seem to have gotten out of hand a little. The "notes" operate in the way that most articles handle "references", but the "references" section here could do with being properly integrated into the text. It's more like a disconnected "further reading" section at the moment (albeit a very long one). I'll try to have a look myself, but this isn't exactly my forte and I'm liable to make mistakes. Cheers, -- PLUMBAGO 16:37, 10 November 2008 (UTC)
(Decrease indent) Thanks for the discussion above. That's been helpful. Looking again at the article, I've not been able to find any "references" that do not also occur as "notes". So that's a good start. I guess that I'd like some way of retaining both the full citation ("references") and the specific location ("notes"), but doing so in a more transparent way than we have at present.
Poking around WP:CITE reveals that this issue has got a template to "solve" it. The example below tries to illustrate this:
How would that sit with people? I personally think that it's a bit ugly, but it would certainly "resolve" my original problem with this article's citations. In passing, and to save anyone from trying the same thing, I tried putting references inside references (e.g. "<ref>pg. X of Source 1<ref name=s1/></ref>"), with predictably dire results.
Anyway, does this help at all, or should I just get over it and shut up? ;-) Cheers, -- PLUMBAGO 13:00, 12 November 2008 (UTC)
You are talking about "recurring decimals". Would you so kind to present how we cuold get it by deviding two rational numbers? (Like 1/3). (Sorry I'm Hungarian my English isn't at the top. If there where any questions please contact me [cerna@richter.hu]). -- 80.99.184.247 ( talk) 20:07, 16 November 2008 (UTC)
This article is very good and informative (if not a little bit prolix), but one thing that might make this article better are pictures, with a few colours even. The first two pages of this article are black and white and even though it doesn't diminish the educational value of it, a few colours would liven it up. Only problem is that I can't for the life of me think of a suitable picture. Maybe if somebody has a textbook which covers how 0.999... equals 1 they could take a picture of it up-close? Any ideas? -- BiT ( talk) 01:14, 20 November 2008 (UTC)
I am wondering if the following comment in the article may be somewhat misleading: as one of the "erroneous intuitions about the real numbers" that students typically have, the article lists the intuition "that nonzero infinitesimal real numbers should exist". Now technically speaking it is correct to describe such an intuition about real numbers as "erroneous". However, a student grappling with this issue is certainly not sophisticated enough to mean "standard real number" when he formulates an intuition about "numbers". Would it constitute a mathematical error to experience an intuition that infinitesimal numbers should exist? Katzmik ( talk) 14:57, 30 November 2008 (UTC)
Copied from my talk page:
Now the edits in question took a paragraph from the lead, giving a summary of some sections further down, and made it the first section aftre the lead instead, preceeding the section Introduction. This was clealy unacceptable. Editing the lead to make it a slightly shorter summary of the whole article might be a good idea.-- Noe ( talk) 17:25, 30 November 2008 (UTC)