Talk:0.999.../Archive 17

Request
184.56.245.136 (talk) 21:29, 3 October 2012 (UTC)
 * ❌ – You need to give details of the request. Without details it's not actionable: nothing can be done.-- JohnBlackburne wordsdeeds 21:34, 3 October 2012 (UTC)

Wikipedia is wrong to say "the symbols "0.999..." and "1" represent the same number"
Mathematical discussion moved to the arguments page. Huon (talk) 02:02, 5 October 2012 (UTC)

Missing information
Where is the history section of .999? Who is the first one came up with the proof? Who is the first one who mentioned .999 = 1? 174.20.15.246 (talk) 23:40, 2 January 2013 (UTC)


 * Math textbooks usually don't provide that kind of historical information, and I wouldn't know where to look for it. The article does mention that Euler had a proof by 1770 (the infinite series and sequences proof). Huon (talk) 00:38, 3 January 2013 (UTC)

"Simple visual verification"
I've removed a section on "simple visual verification" added by Tdadamemd. Firstly, that approach seems to be original research; it cited no sources and I cannot remember ever seeing it before. Secondly, it's basically a variant of the fractions and long division proof and presupposes that 0.333... = 1/3 and that 0.999... = 3 * 0.333... - people accepting those statements probably don't need a visual aid for accepting 0.999... = 1. Thirdly, as I point out in the image caption, the sequence of squares actually does a bad job at illustrating 0.333... For these reasons I don't think the section was an improvement. Huon (talk) 16:49, 5 January 2013 (UTC)
 * For many people, this image does a great job for illustrating how an infinite series can have a convergent sum of 1/3 (and 3/3). The biggest problem that people have with 0.999... = 1 is that they cannot grasp how the infinite series of 0.9 + 0.09 + 0.009 + ... can converge to exactly 1.000...  When people are presented with this visually compelling proof that the area of these squares form an infinite series that converges to exactly 1/3, then it becomes very easy to take the step that 0.999... converges to exactly 1.  This covers your third objection.


 * As for your first two objections, this image is found in elementary school math textbooks going back decades. And the reason why I placed it after the section algebraic proofs is for exactly the reason you point out: it uses the same approach. But it also gives a very powerful element that the strictly algebraic version does not and cannot.  It shows how an infinite series that does not intuitively seem like it should add up to 1/3 does.  How can an infinite series of 1/4's add up to exactly 1/3?  The proof of symmetry is compelling.  Huon, by eliminating this Simple Visual Verification, you are alienating the group of people who are perfectly ok with understanding that 0.333... equals a third, yet fall in the gap of how three times that, when written as 0.999... does not leave out the tiniest morsel of one.  Wikipedia has an image that has the potential to reach each and every one of those "non-believers".  The question facing us editors at large is whether to banish this approach to the Talk page, where most of those people will never see it, or move it back into the article where it can reach out and connect with everyone who balks at the more complicated and less intuitive proofs.


 * If the primary objection is that I did not cite a reference for this image, I am certain that anyone who has a textbook with this image can provide the name of the book and the page it is located. I don't own it.  I first saw it in my nephew's text book back in the 80's when he was in middle school.  And we can simply add it back with a "citation needed" tag.  No harm at all.


 * For those who missed it, here is the original caption that was posted for the image:
 * "One-third (or 0.333...) of this unit square is highlighted (as seen by the largest three squares forming an 'L'-shape with one-third highlighted, and this shape repeats throughout the rest of the square). 0.999... is when the other two-thirds of the square are highlighted, which results in the entire unit square being filled, with absolutely nothing left unfilled."


 * The caption that you changed it to completely clouds the crystalline clarity that had been provided above. And that is a general criticism of the way a lot of math is taught.  Something that is very simple is presented in a way that becomes very difficult to comprehend.  This trend loses a significant portion of the audience.  And they just walk away with a belief that math is hard, when all along it was their friend.--Tdadamemd (talk) 19:41, 5 January 2013 (UTC)


 * I've restored the section. The visual description is so simple that in a stretch it could be covered by WP:CALC (the Greeks made their proofs with geometry, didn't they?) and is quite similar to File:999_Intervals_C.svg, though knowing that the explanation comes from textbooks is enough to make it easily verifiable. I'm in favor of providing simple explanations before approaching the heavy math stuff, and WP:TECHNICAL agrees with this. Diego (talk) 20:14, 5 January 2013 (UTC)
 * I don't see how that image helps at all. It's not a picture of 0.33333 the series. The simplest mathematical representation is


 * $$\frac{1}{4} + \frac{1}{4^2} + + \frac{1}{4^3} + ...$$


 * I.e. an entirely different series. Of course this sums to 1/3 as does 0.3 + 0.03 + 0.003 + ... but the text does not explain this or even how they are related, and general geometric series is beyond the scope of this section. They could also be related by interpreting the picture as 0.111... in base 4, with 0.333... equalling one in that base. But again that is not explained and again is far too advanced an approach for this section.-- JohnBlackburne wordsdeeds 20:38, 5 January 2013 (UTC)
 * See, the problem with knowing too much about the subject is that you lose perspective. The image embodies the idea that an infinite collection of mathematical objects can reach a limit or fill a complete covering and/or without leaving a hole - exactly what students have problem with. You don't need the reader to understand what particular series is being represented - only the core insight that guides the mathematical proofs. For someone that hasn't got a solid grasp on formal methods, that insight is more valuable than an exact procedure. Diego (talk) 21:27, 5 January 2013 (UTC)
 * Yes, an infinite collection. Just not the one 0.3 + 0.03 + 0.003... which is the decimal mentioned. In fact the text relates the two as follows: "so the summation of the infinite series is 1/3, or in decimal notation it is 0.333... ". I.e. it just asserts that 0.333... equals 1/3, and does not explain how the diagram justifies it. Anyone who tries relating to the diagram to 0.333... will come away confused.-- JohnBlackburne wordsdeeds 21:42, 5 January 2013 (UTC)
 * Why? I got the gist of the diagram quite fast, and I didn't even read the text in much detail. There's no need to explain the equivalency of the diagram to the decimal notation; as you said, that's too advanced at this point, but it's not the essence of the explanation. The idea is that you get a third of an object (the L-shaped area), and then add a third of the remaining area, and so on until you get a third of the original square. By the analogy explained in the text, you can do the same with the three squares in the L-shape and get the unity of the whole square. Of course analogies are not exact mathematical descriptions, but that's precisely what makes them useful. Maybe you don't see it, but that doesn't make it less useful - you're just not the target audience. Diego (talk) 22:39, 5 January 2013 (UTC)


 * For people who accept 0.333...=1/3 we already have the fractions and long division proof which seems fine to me without an image. The image above does indeed show how a sequence may converge, and it may be a good addition to our articles on sequences or limits, but like JohnBlackburne I don't see how it shows anything about this sequence, 0.333... - and convergence of sequences in general is beyond the scope of this article.
 * As an aside, I'm not at all sure the image will prove as convincing as we might hope. People doubting 0.999...=1 are quick to resort to infinitesimals, and I don't see why they shouldn't claim that there's an infinitesimally small uncovered area in the top right corner. After all, they're not convinced by the one-dimensional equivalent either. Huon (talk) 01:09, 6 January 2013 (UTC)

I would take this out. It is a really nice picture and idea but it does not belong here. I resisted the urge to edit it as it stands. Actually I started but decided to cancel. First of all, it says very emphatically in two places that the entire unit square is covered with absolutely nothing left uncovered. But that is not the case. The upper righthand corner is not covered. So, a wag could claim, it is less. Or maybe more, after all, the horizontal and vertical midlines are covered twice (as are lots of other places) so that more than over balances the missing corner. Also, at a crucial point we pass from the picture to the point that in decimal notation 1/3=0.333... If that is OK then (as someone else already noted) the "opponent" has already given up. If there was to be a diagram like this (which I think a bad idea) then I would think it better to have at the bottom nine horizontal 1/10 by 1 strips colored in 9 colors. Then in the upper left nine 1/10 by 1/10 squares in the same colors in the same order. In the final 1/10 by 1/10 square at the upper right, repeat the same picture etc.etc. etc. Actually one could go in one dimension and color the unit interval from right to left with a point colored color Ci if the first non-zero digit is i. Then everything is colored and each color class has obvious area 1/10+1/100+1/1000+... You probably would not like that picture (good for you!) and then you should no longer be satisfied with this one. Gentlemath (talk) 06:12, 9 January 2013 (UTC)


 * I agree, this is not really the right article for this picture. It is not really a demonstration of 0.9...=1. I wanted to suggest it belongs in Geometric series, but I see it is already there with different colours. --Qetuth (talk) 06:29, 9 January 2013 (UTC)


 * OK having thought about it, I am going to remove the section.


 * It is an awesome picture and might have a happy home at geometric progression in the section on infinite progressions. But not here. In a nutshell it says these things:
 * 1) The yellow region is by its decomposition 1/4+1/16+1/64+... and is also 1/3 since it with, a vertical translate and horizontal one, covers the "entire" square (with nothing missing and nothing covered twice.. don't think about that too hard.) OK now you have just agreed that an infinite sum "like this" can be a simple fraction that you did not expect.
 * 2) You already accepted in your past that 3/10+3/100+3/1000+...=0.33333...=1/3.
 * 3) Now multiply by 3. DONE!


 * So 1/3 is used in two places, but there is no real connection between them. Gentlemath (talk) 06:35, 9 January 2013 (UTC)


 * Thanks for finding that other image, Qetuth.


 * Ok, here's why I see this "visual verification" image to belong in this article perfectly. The crux of the issue is convergence of an infinite series.  What this image brings to the explanation for people reading this article is the feeling of complete understanding that an infinite series can converge to something that you did not expect it to.  People can leave this article with a complete sense of mastery in comprehending how 0.999... = 1.  That is a confidence that is lacking in all of the other efforts toward proving the equality.


 * I totally agree that the series illustrated (1/4+1/16+1/64+...) has no direct connection with the series: 3/10+3/100+3/1000+...=0.333...
 * Yet there is a very strong connection: equality.  Both series converge to the exact same value.
 * Most people have no problem with the decimal notation of 1/3=0.333...
 * Almost no one has a problem with 3 x 1/3 = 1
 * The common point of incredulity is that the infinite series: 0.9 + 0.09 + 0.009 + 0.0009 + ... converges to exactly 1.000...


 * What this image adds is the fact that you never need to explore whatever may or may not happen at the end of those ...dots. All you have to look at is the first three squares, understand that exactly 1/3 of them is highlighted, and then observe a symmetry that continues unbroken.  The reader ends up with a sense of being totally at ease with the notion of infinite convergence simply from analyzing the finite and noting the symmetry.


 * This is the very same reason why people are ok with 1/3 = 0.333... They prove it to themselves by doing the long division, not out to infinite places but just a few places and then noting the symmetry of the progression.


 * And this leads to the potency of the image: In highlighting all three squares of the pattern, you can visually verify that 100% was covered in this finite realm.  And then you can extend that 100% out by symmetry.  Perfectly comprehensible without needing to pull out any electron microscopes.  The verification is DONE.  QED.


 * This is a feeling of understanding and satisfaction that cannot be achieved by these other methods for many people. Delete the image if you want, but in doing so know that you are depriving them of this:  a perfect way for the finite mind to gain comprehension of the infinite.


 * ...and if the consensus here does decide to leave it out, you all can sleep soundly knowing that you're in good company. Why do I say this?  Well just look over at that other image.  That, and its non-svg predecessor were both posted way back in 2008.  Neither of those images nor the article on geometric progression offer any explanation as to how this series adds up to 1/3.  Think of all the thousands of people who have looked at that article and looked at that image.  All of those people can be placed into one of two groups:
 * - Those who understood it, but decided not to add the explanation, and
 * - Those who didn't understand it, but decided not to ask for the explanation (the Talk page for both of those images have yet to be created).


 * I myself belong to the first category. I saw the image, understood it, yet decided not to add the explanation.  But my reasons are VERY different.  It is not because I don't care.  Perhaps my problem is that I care too much.  I am not going to add the explanation to either of those images, nor to the article on geometric progression.  And I am not going to readd this Simple Visual Verification to this article we've been focusing on.  I am not going to do any of these things myself because I maintain the belief that there are plenty of other editors beside me who care to provide simple explanations to very complex things.


 * And I even believe that most of the editors who oppose this particular addition care as well. The problem becomes one of seeing the benefit where others do not.  I've given this forum my best persuasive effort.  I will leave it to others from here.  And if the basic objection is that readers will fail to connect, you can always decide to keep it in the article as an experiment.  Try it out, see what people think, and if it works for some then great.--Tdadamemd (talk) 06:03, 10 January 2013 (UTC)
 * I tend to agree with your position, but then people above show a strong opposition based on their personal opinions of not getting it. They seem to be the people that don't need images to understand mathematical concepts. Diego (talk) 07:20, 15 January 2013 (UTC)


 * I indeed don't get how the image illustrates what it's meant to illustrate. The image-based proof presupposes that we accept 0.333...=1/3, and it also presupposes that we accept 0.333...*3=0.999... If we accept both of that, what do we need the image for? All it shows is that some series with no immediate relation to either 0.333... or 0.999... may indeed converge, which is certainly nice but beyond this article's scope. What it does illustrate is that in base 4, 0.111...=1/3, but that's probably once again too complicated for beginners. Huon (talk) 13:54, 15 January 2013 (UTC)
 * It also shows that in base 4, 0.333...=1 which is a simplification with respect to 0.999...=1 in base 10, and thus easier to understand - thanks to the geometrical, instead of decimal representation. Diego (talk) 14:52, 15 January 2013 (UTC)


 * Diego's new text makes quite a few interesting presuppositions. It's not at all obvious that 0.999...10 = 0.333...4 - when we operate in a number system where 0.999... doesn't equal 1, it probably won't equal 0.333...4 either. (For fun I just constructed a number system where 0.999... equals 1, but 0.333...4 doesn't.) And I somehow doubt "easy to verify by simply looking" will convince anybody - I expect claims that there is an infinitesimal un-covered square in the top right of the image.
 * On an unrelated note, the image caption seems highly redundant to the section's text. Can we pare down one or the other? Huon (talk) 20:04, 19 January 2013 (UTC)


 * It is also unsourced. This is a featured article and so all content should be adequately sourced. Given the problems with the text it really needs sourcing so other editors can see where it is from and try and improve it. Otherwise it is original research and so cannot be used. The image still does not illustrate 0.999... or any of the existing proofs, and as it requires a long and yes presumptuous argument in two different bases to establish how it relates it doe not belong either. I have therefore removed the section and image.-- JohnBlackburne wordsdeeds 21:26, 19 January 2013 (UTC)
 * There is a misunderstanding somewhere in your comment - the image is not illustrating 0.999..., just like this image immediately above is not illustrating 0.999... either. They're both illustrating 0.3334 as a geometric series, one in linear and the other in bidimensional form. The justification is exactly the same for both images - either both should be included or both deleted. Diego (talk) 00:34, 20 January 2013 (UTC)


 * The image immediately above just illustrates the idea of a sequence, in a section that mostly generally discusses sequences. And it has been in the article for quite a while, including back to the last FA review; no-one until now has raised any concerns over it. If as you say they are both illustrating 0.333... in base 4 then there is no need for two of them.-- JohnBlackburne wordsdeeds 04:10, 20 January 2013 (UTC)
 * The need has been expressed in detail in this whole thread - the illustrate the same concept with alternate forms, that are useful to different audiences. Several editors here agree that the examples previously available in the article weren't enough to explain the visual intuition besides the geometric series. If an image without direct sources can be part of a featured article when editors decide to do so, then two similar images with different explanatory power can be too. If only one can be kept, I'd use the new one as is clearer than the previous one. Diego (talk) 10:45, 20 January 2013 (UTC)

Excellent job here, Diego! I intuitively felt the connection, but my brain did not take that simple step into Base 4 as you've shown us all. I'm gladly eating my "no direct connection" words from my previous post. What Diego has done is simply connect two pieces that have long been established in Wikipedia: the Base 4 one-dimensional number line convergence and the quartered-square convergence (from Geometric Series). Easy as pi! (Oh WAIT, that's another debate...) It's clear to me that Diego's re-addition of the "Visual representation" section needs to be put back in this article permanently. For the section as I had added it, there was one seemingly valid objection and that was having no direct connection. Diego showed us that we were all mistaken on that one point, and the proof was right there in the article all along.--Tdadamemd (talk) 06:11, 22 January 2013 (UTC)


 * Look, it seems to me that nothing has changed from this observation (which I haven't checked, but will accept is correct). The average reader is not thinking about base-4 when he sees the image, so if he thinks, "Aha!  With this picture, the identity 0.999... = 1 is clear!" then he's deluding himself.  Tdadamemd seems to be more interested in achieving this "Aha!" moment than in ensuring that the reader achieves real, not imagined, understanding (pardon me if this is harsh, Tdada, but that really is how it seems to me and your comment that you "intuitively felt" a connection that you could not understand supports this view).
 * I do not think that the image conveys a real explanation (accessible to the average reader without a ream of text) of why 0.999... = 1 and hence it should not be included here -- as pretty as it is. Phiwum (talk) 13:03, 22 January 2013 (UTC)
 * Do you realize how all editors opposing the inclusion of the image do so based on personal feelings? ("I dont't hink it helps", "I think it's not needed...") -except for Huon, who rightly requires an identified reliable source-, while those of us who get it are providing an exact mathematical description of the represented object? The image works and provides a real understanding because the intuition of "full coverage" is independent of the base provided, but is not independent of the linear / planar distinction. People that don't get the idea of the limit over a line can grasp it over the unit square - the human mind plays such tricks; the exact logical reasoning is not enough, and often also not required to understand the concept once you really get the insight that generates it (different, alternate theories based on the insight can provide equivalent explanations, as illustrated by the numerous examples in this page). The only requirement to include such image should be a school-level textbook that used this example - Tdadamemd assures us that there the example came from one, but we really need to identify it. With that source, the example could be used based on the editorial judgement of those editors that understand the explanation. Diego (talk) 13:33, 22 January 2013 (UTC)
 * I should think that, if we include a pictorial "argument" in the article on the identity 0.999... = 1, then the picture ought to represent 0.9 + 0.09 + ... and show that it equals 1. This picture is a very nice representation of a wholly different identity and as such seems utterly irrelevant.
 * But by all means, give a reference to some text which uses this picture to show that 0.999... = 1. Then, of course, the balance will be on your side. Phiwum (talk) 17:10, 22 January 2013 (UTC)


 * After Qetuth gave the pointer to the similar image from 2008, I updated the Source info on the image I had uploaded.
 * And I need to clarify exactly what I had meant when I said "I intuitively felt the connection". The specific issue that statement refers to was whether or not there was a direct connection between the infinite series from the quartered square and the infinite series of repeated 9's.  My reasons for adding the section in the first place are all thoroughly explained above, and have nothing to do with any hunch. Even without the direct Base 4 connection, I saw everything I had added to belong in this article because the image gives a compelling explanation of how such an infinite series can converge to what you might not expect, and that it provided the missing element for people who found the algebraic proof to be just shy of perfectly convincing.
 * As I've explained above, the quartered-square image gives the same sense of understanding and acceptance as people have with accepting that 0.333... converges to exactly 1/3. They recognize the symmetry, and it is that aspect that gives compelling proof. This symmetry reveals itself in the finite realm, so there is no need to continue examining it toward the infinite realm.
 * That, in a nutshell, is the rationale for adding the section to the article.
 * The key step that Diego made was to eliminate the primary objection for removing the section. Your own position is that after all of this explanation you still maintain that the image is "utterly irrelevant".  If you want to be consistent, then you would be pushing to eliminate the 1-D Base4 image, as Diego stated earlier.  But that image has been a long established part of this article because the community recognizes it to be an expression of the exact same issue when formulated in Base 4.  And with this understanding, we know that the quartered-square image addresses this exact same issue directly.
 * I never thought I'd quote Kelly LeBrock in a math debate, but when it comes to the emotion I see in people wanting to eliminate the quartered-square image, we can imagine this image talking back to us to say... "Don't hate me because I'm beautiful!"--Tdadamemd (talk) 21:25, 22 January 2013 (UTC)


 * Ok, I will now present the direct connection between Kelly LeBrock and Math...
 * There is this strange phenomenon when it comes to beauty products, where women actually prefer to pay more. The psychology behind this is that the more they pay, then the more self-worth they feel.  A HUGE empire has been created around this effect.
 * Well there is a similar phenomenon in Academia, where people actually prefer the complex and obscure proofs over one that is plain and simple. It is like preservation of caste society that separates the "smart people" from the ones who just don't get it.
 * There's a natural beauty to the quartered-square. I see no need for anyone to freak out over it.  Wikipedia is about democratizing knowledge.  This "simple visual verification" has the potential to connect with absolutely everyone - including all of the non-believer holdouts who remain unconvinced by all of the various other explanations provided in the article.--Tdadamemd (talk) 21:47, 22 January 2013 (UTC)
 * No, it's not that editors prefer obscure and complex proofs. You should always assume good faith, and not try to assign to other editors motives which you think lessen their arguments. Nor is your pop-psychology relevant, and whoever Kelly LeBrock is he or she doesn't sound like an authority on maths. And none of this addresses the objection I raised; that the content was unsourced and so original research, and so not suitable for any article, especially one adhering to the featured article criteria.-- JohnBlackburne wordsdeeds 22:59, 22 January 2013 (UTC)


 * The very first sentence in my previous reply stated how the image now has a source.
 * The image that is still in need of a reference is the 1-D Base4 image.
 * I patiently sat on my hands for an entire week after Diego had established the Base4 meaning of the quartered-square. I was amazed to see how this community first gave him no response, and then hit with such a negative response.  I turned to pop-psych because I was at a loss for explaining what I had observed here.  Yes, I reached all the way out to Kelly LeBrock.  No, she's not a math authority.  (The link takes you to a hair shampoo commercial.)  And I was not saying that anyone was not acting in good faith.  I was examining the mechanisms that drive our good faith efforts.  Wikipedia is an amazing website.  And that's because of people like you and everyone here.  It all works itself out with a net result that gives great results far more often than not.  Our particular point in question here will get worked out as well.  Not everyone will be happy with how this resolves, and that's ok.  What you see as best doesn't have to match what I see as best.  Our aggregate effort produces an optimization that is "best".  My frustration here is that I saw all valid objections to have been answered, yet still this community resisted what I see to be a major improvement to the article.  And that's where Kelly LeBrock comes in...  Ok, maybe she's a lame explanation.  If anyone would like to offer a better one, I'd be glad to hear it.  Or better yet, we can incorporate the change and sit back for a bit and see how it goes from there.--Tdadamemd (talk) 02:54, 23 January 2013 (UTC)


 * I don't believe the have been addressed in all of this back and forth (other than being sourced now). It is still a good visual demonstration of Geometric series in general, and I wholeheartedly support its inclusion in such articles. It is a poor demonstration of 0.999...=1, except in explaining other related concepts which could be better explained with links to appropriate articles (where this picture WOULD fit). This is a Wikipedia article, not a textbook, something which is oft forgotten in the struggle to make this article enough on its own to single-handedly convince all doubters of the real number system.
 * To specifically address the above: in Base 4, it seems to me to be only an argument that 0.111...= 1/3rd, which is a single step in an unnecessarily roundabout way of proving the main point. You argue it fits well with the other base 4 image, which is not really explained in the text and has its caption marked as citation needed. You argue we don't like pictures, or elegant proofs, or certain shampoo brands? Well, I'd like to put my hand up and say I don't think this proof is remotely elegant, I often choose which textbooks and study guides to recommend to students based on having simple explanations with lots of pictures which aid the explanations, and I have no idea who Kelly LeBrock is. --Qetuth (talk) 03:32, 23 January 2013 (UTC)
 * No one is opposed to elegance, but I am at least opposed to the illusion of understanding that elegant but irrelevant pictures can yield. (For the record, I do know who Kelly Lebrock is.)  I happen to love the understanding the right picture can give.  I heartily recommend the book, "Proofs without Words", where (by the way) this very image appears -- as a proof that 1/4 + 1/16 + ... = 1/3.  No one disputes that this is a pretty picture for discovering that fact.  The question is whether it really helps one understand 0.999... = 1. Phiwum (talk) 04:21, 23 January 2013 (UTC)


 * If someone could create a similar image for depicting the equivalence of $9/10$+$9/100$+$9/1000$+··· = 1, then I think most of us would agree that such an image would be quite appropriate for this article. But I'm not exactly sure what that would look like; perhaps 9 out of 10 boxes colored, and the remaining box further subdivided, ad infinitum? Or perhaps 90% of a square shaded, and the remaining 10% further subdivided, ad infinitum? In such an image, all parts of the total image would be shaded/colored. — Loadmaster (talk) 18:59, 23 January 2013 (UTC)


 * This is the kind of image you are asking for:
 * --Tdadamemd (talk) 20:46, 23 January 2013 (UTC)
 * The caption should probably be 36°, not 36', but apart from that, is there any reason to make that image a circle as opposed to a line? Huon (talk) 21:00, 23 January 2013 (UTC)
 * The image is now fixed. And I also added a full explanation in the description for anyone who may not see the proof immediately.--Tdadamemd (talk) 04:48, 24 January 2013 (UTC)

I had a think about it and think it's hard to do anything for ten. To do something similar with a square you need a square number; so 4, 9 but not 10. To do at all alike in 2D the pieces need to be self-similar with 1/10 the area so 1/√10 smaller. The only think I can think of is a rectangle 1 x √10, cut up into 2 × 5 pieces but that's far from obvious.

A circle's no good as it's not really 2D. It's just a line wrapped around. So it has the same problem as a line, that the pieces vanish too quickly – at normal size only the first three pieces are visible – and the pieces aren't self-similar like the line so it's even less clear how/why it repeats.-- JohnBlackburne wordsdeeds 21:17, 23 January 2013 (UTC)


 * You all are hitting on the exact reason why the quartered-square has so much potency. As explained at length previously, it uses symmetry so that no "electron microscope" is needed.  As to me using that prime symbol instead of a degree, it was a rush job.--Tdadamemd (talk) 21:25, 23 January 2013 (UTC)


 * ... and John, you might want to use more care in you words. The plain meaning of what you just said is that this circle is not really 2D. That's just silly.  But I expect most of us here can infer what you meant .--Tdadamemd (talk) 21:53, 23 January 2013 (UTC)


 * Well, if we're sticklers for precision, then a circle is indeed one-dimensional. What you mean is a disk... Huon (talk) 04:29, 24 January 2013 (UTC)


 * Au contraire. You cannot loop a line back onto itself in only one dimension.--Tdadamemd (talk) 04:48, 24 January 2013 (UTC)
 * ...and I just deleted (strikedthrough) my previous reply, as it now seems clear to me that the plain meaning of the words was actually what was intended. Let's all be clear here:  a circle is a 2-d object, and a disc is a 2-d object.  The blue circle/disc image is not 1-d.  It may correlate directly to the 1-d representation, but it is not itself 1-d.  You can say that you made the circle with a 1-d curved line, but the end result is not 1-d.  The moment the line is connected back onto itself, the topology has changed.--Tdadamemd (talk) 05:13, 24 January 2013 (UTC)

I'd like to go back and directly answer Huon's question: "...is there any reason to make that image a circle as opposed to a line?"

This 2-D version can give a more substantial sense of unity of the whole. We do not sense 1-D. Chopping up a 1-D number line is a purely mental exercise, and this can result in a feeling that the mind has played some kind of trick. But in the 2-D realm, we sense this directly. The world projects onto the 2-D retinas of our eyes. When we see the unit area of the circle, we see it as a tangible unity. When the process starts for chopping up slices, we can relate that directly to cutting into a pie (which is again far more tangible than a 1-D line). Of course chopping toward the infinite realm becomes a mental exercise, but at least we started in a realm of substance, and the end result is understood back in this realm of substance. We started with a unit area, and we ended with a unit area. All we did was put a bunch of cuts in it.

An alternative way to do this geometrical proof in 2-dimensions is to start with a unit square, and then take vertical slices of it at the 0.9 point, the 0.09 point, the 0.009 point, etc. This approach, of course, is what you get by expanding the 1-D number line from 0 to 1 out orthogonally into a plane by 1 in the 'y' direction. For the same reasons as with the circle above, chopping up a 2-D unit square gives a proof that is substantial and tangible. So this too has the same advantage over the number line.

And if anyone wanted to be even more tangible than the 2-D realm, we could take a unit cube and slice that up in a similar way as the unit square was chopped up after expanding it by 1 up in the 'z' direction. Of course this realm of substance is not the physical realm of matter that we live in. It is an imaginary realm where the substance is infinitely divisible, totally unlike the limits we encounter in dividing matter down to the atomic level, or finer still to the subatomic level, or finer still to a Planck scale where "space atoms" are encountered (or strings, or whatever it is that lives down there). The point for this forum is that the circle area substance, or the square area substance, or the cube volume substance may be an imaginary realm, but it is a realm that is far more closer to our real world experience than a 1-D line. For all three of these cases, we start in this tangible realm and we end in this same realm. The process starts with a whole unit, and ends with that same unit, albeit chopped up a whole bunch.--Tdadamemd (talk) 08:47, 24 January 2013 (UTC)

Call for closing remarks

All valid objections to re-adding this 'Simple visual verification' section have been answered. We have placed our arguments on the scales and it is clear to me which direction those scales have tipped. After all of this lengthy discussion, the strongest objection remaining has been succinctly put by Phiwum: "The question is whether it really helps one understand 0.999... = 1", and we have the answer for that too, because people here have stated emphatically that it does help. If some of us here see it as a big help, it is reasonable to assume that it will be a big help to a significant fraction of readers.

There have been strong objections raised, but there is an emotional component to reasons given behind those objections. Our job as editors here is to set those emotions aside and make our best collective decision as to what is best for this article in regards to the readers' experience with it, with the totality of Wikipedia standards and our collective best judgement as our guide in making that decision.

The crux of this issue here came down to a matter of taste. For some, we thought it gave a great explanation. For others, they saw it as unhelpful. The endgame strategy was presented by Loadmaster putting out a call for a 2-D image that depicted the 0.9... geometric progression. One was provided and it was seen by everyone who voiced an opinion to not have any significant advantage over the quartered square.

Based on this, and everything covered in this Talk section at length, I plan to re-add the 'Svv' section.

I will suggest that the best place to do this is right after the 'Algebraic proofs' section, as this is the proof that the image directly supports. And I will also suggest that the best way to do this is to have the blue sliced circle image right below the quartered square image to help make the connection between the Base 4 representation to the Base 10, for the benefit of anyone who does not immediately see the connection. With the inclusion of this blue circle, it now becomes a visual proof - so the proposed change to the title of the section is "Simple visual proof".

I will plan to make this change on Sunday, unless someone would like to re-add it themselves before then, or if someone would like to re-iterate an objection that they believe has not been sufficiently answered. And of course, if anyone can think of a new objection that none of us have thought of throughout all of this, that would be helpful too.--Tdadamemd (talk) 19:10, 25 January 2013 (UTC)


 * Honestly, I don't care enough to keep arguing one way or the other, but I want to put a big citation needed tag on your entire closing remarks. Since my last entry, please point to where any of the issues have been addressed since they were last raised, where any of the objections have had an emotional component, and where anyone has claimed the picture helps them understand the equality in question, as opposed to helping them understand a the general concept of geometric series in general. To be clear - I still do not see the direct relevance of this very nice picture to this article, and the only discussion since I posted my objections has been Phiwum seeming to agree with me, and people discussing that the circle picture doesn't help. --Qetuth (talk) 03:16, 26 January 2013 (UTC)


 * Besides, this is synthesis - Nelsen doesn't mention decimals or base 4, and while he has a geometrical proof a few pages earlier (p. 118 if I'm not mistaken) that might actually be relevant to 9/10+9/100+9/1000+..., it's not this one. Huon (talk) 04:04, 26 January 2013 (UTC)


 * Nelsen says it is a visual proof of the summation equaling 1/3. Yes, it would be far too bold for us to extend that here in this article by extrapolating that 3 x 1/3 = 1.  Shakey ground indeed.  We really should wait for someone to publish a PhD dissertation and then we can safely cite that three times one-third does in fact equal one.


 * As to the Base 4, I've expressed above that the biggest benefit I saw from Diego's connection with that was removing anyone's grounds for objecting to the image on the basis that it had no direct connection. If it were re-added with no mention of Base 4, I'd see it to be just as useful.


 * Regarding this huge concern about Featured Articles needing everything sourced, I saw that as totally understandable. Particularly when I saw that only a small handful of math articles were FA-status.  Well I just happened to click on the very next one beside this one.  It was 1 − 2 + 3 − 4 + · · ·.  I looked it over and saw that it has 7 images.  And looking closer, I found that the first 6 images had absolutely no reference.  STOP THE PRESSES!  Let's all go and delete those images from all the articles.  Oh wait.  That's a Featured Article, so it means that editors were totally ok with that.  And guess what the situation is that we have here on this article we have been discussing?  It was quite some time ago that Diego highlighted the fact that the 1-D Base4 image has no reference.  And we all were ok with that - as evidenced by the fact that no one deleted it.  [Edit:  Whoa, I'm feeling foolish for having gone to another article to make that observation, because I just went and looked at the images of the article we have been discussing this whole time and saw that only one of its five images has a reference!  One image goes so far as to mix Base10 and Base3 in the same image without even indicating that it is doing this!--Tdadamemd (talk) 11:54, 26 January 2013 (UTC)]


 * So I suggest we get real on this hangup that we are breaching Wikipedia policy by captioning this square with the conclusion that 3 x 1/3 = 1. And if you really are hung up on it, I've long been suggesting that we examine our true reasons for being so resistant to this proposed improvement to the article.


 * Does anyone here really want to maintain that 3 x 1/3 = 1 constitutes original research? Does anyone here really want to maintain that looking at a unit square divided into 4 quadrants with one of those highlighted represents 0.1 in Base 4 is original research?  Do you even know what the definition of OR is?  From the very first line of the explanation of the policy:  "The term "original research" (OR) is used on Wikipedia to refer to material—such as facts, allegations, and ideas—for which no reliable, published sources exist."


 * For everyone here who has dug their heels in on an NOR-based objection, let's be perfectly clear that the foundation of your position is that absolutely no reliable published sources exist that can lead us to conclude that 3 x 1/3 = 1. And that absolutely no reliable sources exist that 1/4th of a unit square is the representation of 0.1 in Base 4.


 * The BS flag has been raised as high as I can raise it. Because obviously my previous rebuttals were not sufficient enough for several people here.--Tdadamemd (talk) 10:59, 26 January 2013 (UTC)


 * Oh, wait. I see now that your objection was "Nelsen doesn't mention decimals".  So the foundation you are standing on is that it constitutes OR by anyone here being so bold as to conclude on their own, without citation, that 1/3 = 0.333...--Tdadamemd (talk) 11:04, 26 January 2013 (UTC)


 * Qetuth, if the basis of our discussion here is the limit of how much of our lives we wish to invest in this debate due to the limits on how much we care, then I too can just walk away from this. I can think of far more productive things I can be doing with my time.  I know that 0.99... = 1.  I have no need to prove it to anyone.  What we are all motivated by here is our desire to help others gain what they are seeking when they visit this article.  Some editors here see that to be one way.  Other editors here see that to be another way.  If you want to know exactly who, it was Diego and me as a minimum who found the quartered-square to be helpful.


 * And if you see valid objections to be upheld, please let me know what that is, because I saw all of them to have been answered substantially.


 * Regarding the blue circle, I don't recall anyone saying it didn't help. What I saw was people pointing out that it didn't help much more than a 1-D image would.  Well this article doesn't even have the 1-D image.  And I offered a wealth of explanation as to why people could find the blue circle to be better than a 1-D image, and no one responded.


 * This is exactly the kind of breach in logic that leads me to a conclusion that people have dug their heels in to staunchly object while putting their reasoning abilities on emotional override. How do you explain that more than one person here doesn't recognize a circle as 2-dimensional?  I'm baffled.--Tdadamemd (talk) 10:59, 26 January 2013 (UTC)


 * Sorry, I mis-summarised, I should have said the circle discussion was on how much the circle helped, but I was trying to express that opinions on the circle seemed to be generally negative (microscopes were mentioned) and worded it poorly. Note that a reason I personally would say the circle is not a very helpful picture to readers (translated to avoid further misunderstanding) is that it is 1D in the sense that its use can be easily parameterised by a single parameter, and hence for visual demonstration purposes its 2D-ness is little more use than the technical 2D-ness of a line with measurable thickness. And having to specify the difference between topological dimension, algebraic dimension, fractal dimension etc should not be necessary to make such a point. In any case, it would surprise me if a student claimed this picture aided their understanding when others failed, but I've been surprised before.


 * I don't strenuously object to the 1/3 picture being added with appropriate explanation and logical link to the surrounding text, certainly not to the point of 'digging my heels in'. I personally think the article could do with a lot of improvement, and am glad you seem interested in that. One of the issues I have is its attempt to explain in detail connected topics which I think defeats the purpose of this being a wiki, and duplicating articles on the background maths used in a proof is a step in the wrong direciton. Hence I don't think demonstrating convergeance of geometric series, or base conversions, or anything like that, should be here, it shold be linked to. And I don't think this picture is directly relevant. I stated my position, others disagree, I have no real problem with that and would be happy to be overruled by consensus.


 * I had a much bigger problem with statements that seemed to repeatedly misrepresent others arguments, make assumptions about the reasons behind them, and misrepresent the discussion in general. I see that you disagree with my objection, fair enough, but I do not see it being "answered substantially" or why you clearly don't consider it "valid". I asked 3 questions and the only answer you gave to any did not even address the question - You just said you and Diego found the picture useful. Yes, reading the posts where you each said so is what caused me to pose that question in the first place. Are you saying that this picture demonstrated to you that 0.9...=1, not just that 0.3...=1(four) after it was pointed out to you, or that the infinite sum of (3/4)^n=1, and you extrapolated from there?


 * No-one asked for every picture to be referenced that I saw, although pictures should of course be sourced as necessary. Explanations and analogies, even if they are squeezed into the captions of pictures, are another matter. Nor is the only synthesis being done (particularly in the original caption) multiplication by 3 - this picture is still only linked to the article by the base 4 version of the equality. So its use should be restricted to a demonstration of such, or else needs to be referenced to a reliable source using base 4 as an intermediate step in proving the base 10 result, or at the very least talking about the equivalency of such statements in different bases (Note this is not a statement about the quality of any current or proposed references, just an attempt to explain others' statements which your response makes it sound like you have misunderstood). --Qetuth (talk) 16:21, 26 January 2013 (UTC)


 * The circle "is 1D in the sense that..."? I cringe when I see anyone saying that the circle is 1D in any sense.
 * I hope we could all agree that card carrying members of the Flat Earth Society are missing a key part of the picture. We know the Earth is not flat.  Your one local part of the earth in a very limited sense might be flat.  But that is not the geometry of the Earth.  And to say so would be to totally ignore the vast majority of the picture.


 * On that note, I will re-assert: "It may correlate directly to the 1-d representation, but it is not itself 1-d."
 * To go further than that is to venture into the realm of error.


 * And then follow that up with these statements:
 * "...hence for visual demonstration purposes its 2D-ness is little more use than the technical 2D-ness of a line with measurable thickness."
 * "...it would surprise me if a student claimed this picture aided their understanding when others failed..."


 * I am now going to reiterate in bold what I had previously stated, because for whatever reason the message has repeatedly been ignored:
 * This article, as it stands, has no image at all that presents the very problem of 0.999...=1. Nothing.  Nada.  Zilch.  Not 1-D.  Not 2-D.  Not 3-D.


 * I can't state that more clearly. I post a blue circle and there's this huge pushback.  This forum is arguing about throwing out the bath water when I am pointing out that you have no baby.  Base4 is not the baby.  Nested intervals is not the baby.  Cantor dust is not the baby.  Whatever that last thing is, it is not the baby.


 * My Twilight Zone experience here continues. I really don't get you folk.  What I am facing now is a pile of evidence pointing toward a conclusion that, for whatever reason, some people here just want an inferior article.  Can that be true?  I sure hope not.  Ok, let's say we all agree that the blue circle is no better than a 1-D image.  Then why aren't we adding a 1-D image?  There is no foundation for an objection to say that the blue circle is no better than a 1-D image when that 1-D image is not in the article in the first place.


 * Now Qetuth, you are telling me that you have questions/objections that were not sufficiently covered. You're saying you asked three questions that went unanswered.  I've looked back through this entire section and I am not clear on what those are.  I will ask again for anyone to reiterate any question or objection they have that they see to not be covered.  If I don't know what they are, I cannot address them.


 * In your most recent post you've reiterated:
 * "...duplicating articles on the background maths used in a proof is a step in the wrong direciton. Hence I don't think demonstrating convergeance of geometric series, or base conversions, or anything like that, should be here, it shold be linked to. And I don't think this picture is directly relevant."


 * Perhaps this is one of the 3 points you were referring to. I saw this to have been fully addressed.  Here was my answer to this:
 * I totally agree that base conversions are a totally unnecessary diversion, and this article could communicate far more clearly by never mentioning that topic. What I remember saying is that it would be best to bury the Base4 interpretation of the quartered square as a footnote.
 * But I totally disagree that convergence is not something to be covered. This is the very essence of the question at hand!  I've repeatedly stated that the very potency of the quartered square is in how it gives direct insight into the convergence issue.  This is not background maths.  This is the crux of the article.  This is why the quartered square is directly relevant.  It goes hand-in-hand with the algebraic proof.  It communicates this visually.  It communicates this powerfully.  And it is a perfect stepping stone to the next image of the blue circle that presents the visual proof of the exact problem.


 * (Maybe you were totally clear on my position on that. I've reiterated it just on the chance that it was one of the questions you had that you feel went unanswered.)


 * You say: "I had a much bigger problem with statements that seemed to repeatedly misrepresent others arguments, make assumptions about the reasons behind them, and misrepresent the discussion in general."
 * Are these your three questions? This is far too vague for me to give a direct answer.  And all three of these points are critiques on the style of discussion.  I don't see how these constitute a valid objection to the question of whether or not we will include the section being proposed.


 * "Are you saying that this picture demonstrated to you that 0.9...=1, not just that 0.3...=1(four) after it was pointed out to you, or that the infinite sum of (3/4)^n=1, and you extrapolated from there?"
 * I don't care for Base4. I hope that's clear.  Once again, I see the article to be much better off without any mention of it within the body.
 * And yes, the quartered square does demonstrate that 0.999...=1. This has been explained at length.  The series converges to 1/3.  Color in the other 2 squares of the 'L'-shape and you have 3 x 1/3 = 1.  Crystal clear to me.  I don't even know why you are bringing any of this up.  This is not an extrapolation.  It is simple arithmetic that an average 5th Grader can see the truth of.  Mentioning anything about Base4 just clouds the issue.
 * (I should explain here my reasons for going back to the caption on the image on the Talk page here to include the Base4 stuff. I did not do that as a suggestion that Base4 should be mentioned when transferring this into the article.  I did that for the benefit of anyone who might peruse this Talk section as a concise summary of how it was established that this image has a direct connection to the original problem.  If I saw a good way to do that with a footnote, I would have.)


 * You also say: "...this picture is still only linked to the article by the base 4 version of the equality...".
 * You repeatedly maintain this position while ignoring the fact that I've repeatedly covered how the image relates directly to the algebraic proof. I've done that several times in previous posts, and I've done it once more in this post.  It is totally fine for you to disagree.  But throughout this entire discussion, people are just simply choosing to ignore rebuttals they are presented with.  These are signs of emotions overriding rationality.


 * I maintain that these two images, presented with a short explanation, will be an excellent improvement to this article.


 * But I have no plan, now, to re-add this tomorrow as originally intended. It is very clear that major objections are persisting.  I am not totally clear as to the what or why.  Something very weird has been happening here.  Certainly none of us have to agree.  But when person after person after person dig in on saying things like a circle is one-dimensional, well that is just an E-ticket ride into some twilight dimension.  Freaky.


 * Maybe the healthiest thing for me to do is walk out of the amusement park, because frankly I don't find any of this to be amusing. Weird, yes.  Amusing, no.--Tdadamemd (talk) 22:27, 26 January 2013 (UTC)

When this call for closing remarks was placed, I had stated "if someone would like to re-iterate an objection that they believe has not been sufficiently answered". I'll invite feedback once again, this time with an emphasis on valid objection. I am hopeful that we can wrap this up here without diving back into the most fundamental concepts of mathematics that are widely accepted by college and high school students without anyone crying out for any need for source references. The end game of this discussion has devolved into topics that we all learned, or were supposed to have learned, back when we were in middle school or earlier.--Tdadamemd talk) 11:12, 26 January 2013 (UTC) (reason for strikethrough given below--Tdadamemd (talk) 22:41, 26 January 2013 (UTC))

I just thought of a solution that if I had thought of this three weeks ago, it would have saved everyone here a lot of time and energy. I will have that implemented within the next couple of hours. Hopefully everyone will see this as an acceptable fix, and a beneficial fix.--Tdadamemd (talk) 22:35, 26 January 2013 (UTC)


 * I won't respond to the above as it is struck, except to say: The 1d circle thing is not the basis for objection and should probably be dropped as a distraction from the main conversation, but since it is bothering you you might want to read Dimension_(mathematics_and_physics) and Lebesgue covering dimension where a circle is the specific example used of a 1d object. Otherwise, I look forward to seeing what you come up with. --Qetuth (talk) 23:30, 26 January 2013 (UTC)


 * I'll have a look at that! Thanks for the pointer.  And my fix is completed.  I hope everyone sees it to be a satisfactory resolution to all of this.--Tdadamemd (talk) 00:18, 27 January 2013 (UTC)
 * Your link to your user page? Such links are not allowed in articles as they are not part of the encyclopaedia, with very few exceptions. See e.g. WP:LINKSTYLE.-- JohnBlackburne wordsdeeds 00:37, 27 January 2013 (UTC)


 * If the problem is that UserSpace is considered to be external to the encyclopedia, then the simple, productive change is to move the link I had added down to the External Links section. Done.--Tdadamemd (talk) 03:59, 30 January 2013 (UTC)


 * Very curious developments here of late...
 * As some of you have seen, in my best effort toward a solution that everyone might find as acceptable, I went and built this page: Visual proof of 0.999... = 1


 * I added it in what I saw to be a totally harmless way as a 'See also' link. After all of 14 minutes it got clobbered by JohnBlackburne, with his comment above.  He's saying that UserSpace is not part of the encyclopedia.  Ok, then.  So I re-added it as an External Link today.  After all of 16 minutes this time, it got clobbered by Hawkeye7, citing the somewhat non-descript "Cannot link to a user page".  I reverted with an appeal to what might be the most important WikiPolicy of all, that if our change is for the purpose of improving Wikipedia then we are directed to W:Ignore all rules.  Hawk took all of 16 minutes once again to revert, citing W:User where it says: "encyclopedia articles should never link to any userspace pages".


 * Really? Never?  The first thing I note is that it does not specify "Not even as an External Link."  And above any of that, there is as I had said W:IAR.
 * And even if someone were to argue for some reason that this is one rule in which W:IAR does not apply, then I find the logic very curious in that I can host the exact same content with the only change being that the URL does not start with "en.wikipedia.org" and all of a sudden it would be perfectly legal to have it as an External Link.


 * I see no problem at all with having this as an External Link. If the Wikipedia policy really means that absolutely nothing on UserSpace should be linked to anywhere on the entire page of a MainSpace article, then W:User should be clear in that.  As it stands, the policy is not clear about that.


 * Furthermore, in order to make such a clarification stick, it would be necessary to change W:IAR to say that "No External Links to a User Page is the one exception to Ignore All Rules".


 * ...and even if we were to get that far in revising time-honored Wikipedia policies, I would still say "fine". Because the best solution I saw here in the first place was to include these images in the article itself.  The only reason why I moved it to UserSpace was because a vocal faction here had a cow.


 * It is totally possible that the actual consensus here is to INCLUDE. And that is the fix I would vote for at the top of my list of options.--Tdadamemd (talk) 09:47, 30 January 2013 (UTC)


 * Thank you again, Qetuth, for those pointers. I clearly owe an apology to people who were using the accurate definition of a math dimension.


 * I expect that someone long ago had tried to teach me that definition, and even today it strikes me as a very odd way to define a dimension. For instance, if a circle is perfectly round throughout its circumference except for, say, 270-to-271 degrees where instead of a circular arc it has a straight line connecting those two points, it still looks a lot like a circle, but suddenly it's math dimension has jumped from 1 to 2.  Very strange meaning of dimension.


 * Also, I could define a coordinate system of concentric squares so that a square is now identified by just one parameter. Sure, it is no longer an orthonormal set of coordinates, but it still forces a conclusion, by that definition, that the square is an object of one math dimension.  Again, a very strange behavior of the concept of dimension.


 * What I am seeing is that what these math people are calling dimension is really something else entirely. Maybe a better word is "parameterization".  But I'm sure many smart people have put a lot of thought behind their need to arrive at this definition that is so totally different from the physical notion of dimension, and if anyone knows of a good article that might explain that to me, I would appreciate it.--Tdadamemd (talk) 04:42, 30 January 2013 (UTC)


 * Why is it that you think that the almost circle has dimension 2? The circle, almost circle, and square are all homeomorphic to each other and so topologically they should have the same dimension. Also, we can deform the circle into a closed curve that does not lie in a plane but does lie in 3-dimensional space. This curve will still have dimension 1 as it is homeomorphic to the circle. Eric119 (talk) 21:28, 30 January 2013 (UTC)


 * You are now introducing a third definition of dimension, separate from the ordinary one I had been using (plain and simple spatial dimension), and separate from the alternate math definition that was pointed out to me above (Dimension_(mathematics_and_physics)). And this "homeomorphic" concept of dimension, if that's what it's called, is exactly the point I was making with the almost circle.  By the math definition, putting the smallest of dents into the circle changes its dimension, when our intuition might point us toward a completely different answer, as topology never changed.  My original assessment using the ordinary spatial meaning of dimension was that it was 2-D before and 2-D after, under that math definition it was 1-D before and 2-D after, and now you are pointing out this other definition where it was 1-D before and it is 1-D after.  (And we could probably conjure up some other definition that would give some other answer.)


 * As a lesson learned for the future, I will suggest that people assume a common definition for a term unless an alternate definition gets specified. Someone above had tried to do that by saying that the circle is defined by only one parameter, and if the math definition had been pointed to at that time then all of this would probably have been avoided.  I'll chalk this up to a lesson learned.


 * I'd also like to post a reminder that the only connection this whole circle dimension tangent has to the article is in regards to the consideration being given to having the blue circle included in the article, as the only direct image representing 0.999... = 1. I am now favoring direct inclusion, as the most negative comments were to the effect that it offered little advantage over a 1-D image.  Since there is no 1-D image, it would be logical to conclude that this blue circle remains the best image to add.--Tdadamemd (talk) 04:30, 31 January 2013 (UTC)


 * My intended point was that you were not understanding what you call the "math definition". (We could easily parametrize the dented circle with one variable, and indeed, if two spaces are homeomorphic, then any parametrization of one space transfers via the homeomorphism to the other.) Also, note that the "number of coordinates" notion isn't a formal definition of dimension, just the intuitive idea. A lot more could be said, but since, as you point out, this dimension thing is rather off topic, I will stop here. Eric119 (talk) 21:31, 31 January 2013 (UTC)


 * I have one final comment on that topic for this forum here...
 * If what you are saying is an accurate description of the math definition of dimension, then I would suggest that you go over to that article and make some big changes, because my understanding of what you're saying and my understanding of what that article says are two very different things.--Tdadamemd (talk) 06:23, 1 February 2013 (UTC)

Isn't this article assuming that the decimal system is absolute?
If we use a trinary system for this, this discussion becomes irrelevant. .3333.... is just a representation of 1/3 in a numerical system that doesn't support that number in a "normal" way. .9999.... is just an error caused by that respresentation.

In trinary 1/3 would be represented by .1 If we were to multiply that by 3 we would get 1.

This discussion is senseless and it makes me sick to think that proper mathematicians actually consider this to be correct.

-Renato Fontes Tapia, 2013-01-28 14:17 UTC


 * In trinary (base 3), there is no digit '3'; however, 0.222... is equal to 1. Similarly, in a base 4 system, 0.333... is also equal to 1. — Loadmaster (talk) 21:05, 28 January 2013 (UTC)

1/10 = .1 in trinary then... the thing is... you are trying to represent numbers in numerical system that don't support them. — Preceding unsigned comment added by 63.192.82.30 (talk) 20:50, 30 January 2013 (UTC)


 * This article is about an artifact of our place value system (often ambiguously called the decimal system) which actually occurs in any base we apply, but of course looks slightly different in each base, as most things do. The article is named for the base 10 version because base 10 is overwhelmingly the most commonly used base by the average reader. However, the phenomenon exists in any base using that place system: 0.999...=1 in base 10, 0.222...=1 in base 3, 0.FFF=1 in base 16, and in general 0.nnn...=1 in base (n+1) for any integer n>1 (it is a little more complicate for non-integer bases). --Qetuth (talk) 21:18, 30 January 2013 (UTC)

Misrepresentation of the hyperreals
Hi, From the article:

"The equality of 0.999... and 1 is closely related to the absence of nonzero infinitesimals in the real number system, the most commonly used system in mathematical analysis. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals. In most such number systems, the standard interpretation of the expression 0.999... makes it equal to 1, but in some of these number systems, the symbol "0.999..." admits other interpretations that contain infinitely many 9s while falling infinitesimally short of 1."

This paragraph is poorly written and misleading.

The hyperreal numbers are not an "alternative number system." They are a field extension to the real numbers just as a the reals are an extension of the rationals. This also has nothing to do with the topic of the article because this field extension has no effect on the .9~=1 equality whatsoever since these are both real numbers and not hyperreal.

It seems as though people are interpreting this paragraph to mean that "in the hyperreal system, there are infinitesimals, and thus .9~=/=1," but this is false. The hyperreal numbers is an extension that does contain infinitesimals, but that does not change anything about the real numbers, as a field extension only gives us more ideas to work with, it does not change the old ones. To claim that considering the hyperreals, somehow "changes" the real number .9~ is no different from saying that considering the irrational numbers somehow "changes" the number 5. OF course, since this is false, interpreting this paragraph correctly makes it entirely irrelevant.

The last statement in particular is unacceptably bad. While reusing the incorrect notion that the hyperreals are an "alternate number system," it talks about how in "some" (which?) "other alternate number systems" the symbol means something different. Why are we bringing up the meaning of symbols and unnamed "alternate number systems" on a page about a simply mathematical equality? This just doesn't belong in the article. — Preceding unsigned comment added by 184.190.201.114 (talk) 21:15, 11 February 2013 (UTC)


 * I don't see these problems. We explicitly say that in most non-standard number systems 0.999... will still equal 1. This includes the hyperreals, unless we follow Katz' approach and denote some numbers a little short of 1 as 0.999... The connection between hyperreals and decimal representations, however, isn't quite the same as for the reals (see the article's infinitesimals section), and it's not obvious that 0.999... in a hyperreal context still denotes a real number. Again, see Katz for an approach that disagrees.
 * The other number systems are all discussed in the article; they include, for example, Richman's decimal numbers and arguably the Hackenbush numbers. The lead is supposed to be a short summary; we cannot provide that much detail, and discussing, say, Richman in the lead would give him undue weight. The fact that the hyperreals are a field extension of the reals does not mean that they aren't an alternative number system; they behave quite differently (topologically, for example). Huon (talk) 23:40, 11 February 2013 (UTC)


 * How does Katz's approach to using the notation 0.999... to mean something different than the standard (real or hyperreal) interpretation square with the mathematical meaning of the notation, vis., $9/10$+$9/100$+$9/1000$+···? If he takes it to be something different, then this does seem confusing, since the sentence in question is introducing  a different meaning for the notation discussed in the article. If Katz's notation truly has a different value, then perhaps it should be discussed as such in a separate section, e.g., "Other meanings of 0.999...". — Loadmaster (talk) 01:16, 12 February 2013 (UTC)


 * I'm not really the one to defend Katz' approach, but he denotes as 0.999... a sum of the type $$\sum_{k=1}^\mathbf{N}\frac{9}{10^k}$$, where N is some infinite hyperinteger. This sum doesn't equal 1, but it is a sum of just the type we're talking about - except the fact that it ends somewhere among the infinitesimal summands. It even has a few additional summands compared to the "real" 0.999...=$9/10$+$9/100$+$9/1000$+···. I believe $$\sum_{k\in\mathbb{N}}\frac{9}{10^k}$$ isn't well-defined within the hyperreals. To reach 1, we have to sum over all integers and hyperintegers. Huon (talk) 02:24, 12 February 2013 (UTC)
 * Just a quick technical comment: User:Huon's observation that the sum $$\sum_{k\in\mathbb{N}}\frac{9}{10^k}$$ isn't well-defined is correct as far as I understand. The reason is that the set $$\mathbb{N}$$ isn't internal, and therefore one can't have a hyperfinite sum over it.  Also, note that an $$\mathbb{N}$$'s worth of decimal digits is not enough to specify a hyperreal uniquely; that's why there will be infinitely many hyperreals whose decimal expansion starts with .999...  The canonical choice is of course the number 1 itself.  The claim seems to be not that other choices are preferable on purely mathematical grounds, but rather that they are more consistent with student intuitions which can then be fruitfully exploited to introduce the student to calculus, as illustrated in the fine study by R. Ely.  Tkuvho (talk) 14:02, 13 February 2013 (UTC)

Article title
Hello. I think this article needs to be renamed to something like Equality of 0.999... and 1, because it is what the article is about. Regards, Freewol (talk) 19:24, 16 March 2013 (UTC)
 * I think not. Hawkeye7 (talk) 10:46, 25 March 2013 (UTC)
 * I wouldn't be so quick to dismiss the suggestion without discussion. It is true that virtually the entirety of the article has to do with the issue of whether 0.999... is equal to one (with all the proofs showing that it is).  I would compare it to Parity of zero.  Ultimately, though, I am against the move for two reasons:
 * Awkwardness of the new title (although if this was the only issue, it can probably be remedied by the redirect created by the move)
 * Unlike the number zero, there are no notable properties of 0.999... other than its equality with one and the seemingly endless number of people refusing to accept the proofs. Singularity42 (talk) 11:25, 25 March 2013 (UTC)
 * If we want to discuss this further I think it would need a proper Wikipedia:Requested move.--Salix (talk): 11:27, 25 March 2013 (UTC)
 * Thank you for your remarks. I am opposed to the current article title since if it was really about the number 0.999..., then it would need to be merged with 1 (number). You don't have a different article for each possible writing of a thing. Do we really have to discuss in WP:RM ? It seems to me that this talk page is the most appropriate ? Freewol (talk) 12:10, 25 March 2013 (UTC)
 * PS : I am very puzzled by the move comment of : "page moved in error. There is no equality". There is no equality ??

Looking at WP:NAMINGCRITERIA, I prefer the shorter name. There is no other article on 0.999 that we need to distinguish this one from, and the current title is both recognizable and brief. In general we try not to make article titles longer than they need to be, even if they could be made slightly more "precise" by making them longer. The current title is sufficiently descriptive to identify the article. &mdash; Carl (CBM · talk) 12:24, 25 March 2013 (UTC)
 * I understand your point, along with 's one seen in a new light. Anyone seeing the current title can probably guess that the article will be about the equivalence between the writing "1" and lots of other ones looking like "0.999...".
 * Rigorously, it is disturbing to have an article named "0.999..." that is not about the number "0.999..." (which is already described in article 1), but I can understand that having a simple name is more important than having a rigorous name here, since WP is not a specialized encyclopedia. Thank you for your feedback. Freewol (talk) 13:02, 25 March 2013 (UTC)


 * I too prefer the short name. It's not so much about the number 0.999... but about the representation 0.999..., which is discussed even in contexts where the number it represents does not equal 1. Huon (talk) 14:13, 25 March 2013 (UTC)
 * 0.999... is another way of writing 1. I agree with Huon and Singularity42. The short name is preferred; that is where the readers will expect to find it. Hawkeye7 (talk) 19:18, 25 March 2013 (UTC)
 * Against (renaming). This article is indeed about the number (or representation) 0.999...; since it is by definition equal to 1, most of the discussion is about that equality. Most of the rest of the discussion is about the representation form of 0.999... and its implications. The remaining text is about possible alternative meanings and infinitesimals, which are generally not directly related to the number 1, and therefore belong here instead. — Loadmaster (talk) 20:03, 25 March 2013 (UTC)

Another example
I don't know if this is useful for the article, but in addition to the example of using the decimal expansion of $1/9$ × 9 = 1, you can also use $1/11$ × 11 = 1:
 * $$\begin{align}

\frac{1}{11}              & = 0.090909... \\ \frac{1}{11} \times 11    & = 0.090909... \times 11 \\ \frac{1}{11} \times (10+1) & = 0.090909... \times 10 + 0.090909... \\ \frac{(10+1)}{11}         & = 0.909090... + 0.090909... \\ 1                         & = 0.999999... \end{align} $$ — Loadmaster (talk) 20:22, 25 March 2013 (UTC)


 * A single version of this proof should be enough for the article. It works with various numbers, and I think 1/9 is slightly easier than 1/11. Huon (talk) 20:53, 25 March 2013 (UTC)
 * It works using any number with a reciprocal with a repeating decimal representation in base 10. Double sharp (talk) 16:14, 6 April 2013 (UTC)


 * In theory, yes. In practice, not all numbers are suitable for numeric manipulations that work out to 0.999... . For example, the reciprocal of 7 in base 10:
 * $$\begin{align}

\frac{1}{7} & = 0.142857142857... \\ \frac{10}{7} - \frac{3}{7} & = 1.428571428571... \\                           & - 0.428571428571... \\ \frac{7}{7} = 1            & = 1.000000000000... \end{align}$$
 * — Loadmaster (talk) 16:58, 15 April 2013 (UTC)
 * Similarly:
 * $$\begin{align}

\frac{1}{11} & = 0.90909090... \\ \frac{12}{11} - \frac{1}{11} & = 1.90909090... \\                           & - 0.90909090... \\ \frac{11}{11} = 1            & = 1.000000000000... \end{align}$$
 * Which is the equivalent to what you did, (n+a)/n - a/n will give 1.abcd... - 0.abcd = 1.0000 (for n, a positive). A better example for sevens would be:
 * $$\begin{align}

\frac{1}{7} & = 0.142857142857... \\ \frac{6}{7} + \frac{1}{7} & = 0.142857142857... \\                           & + 0.857142857142... \\ \frac{7}{7} = 1            & = 0.999999999999... \end{align}$$
 * or in general (n-a)/n + a/n (a, n real, 1/a has a repeating decimal representation in base 10) MChesterMC (talk) 12:17, 1 May 2013 (UTC)


 * So in general, prefer addition examples over subtraction. — Loadmaster (talk) 18:27, 4 May 2013 (UTC)

Consistency across-the-board for all the reals
[copied from Talk:Infinitesimal ]

Tkuvho, I am fine with the Wikipedia's verifiability not truth policy and I intend to keep my editor's hat on when it comes to deciding appropriate placement of content. Essentially, Katz & Katz is proposing to subtract terms in a way that with the hyperreal notation, defines some sets of numbers that according to their rank, gives numbers such that 1 - infinitesimal. But, as my demonstration above shows, I don't see how this allows for consistency across-the-board for all the reals, thus its a pedagogically nightmare for me to understand, and I wouldn't want to try to teach such a major conceptual revision without learning it for myself. But putting my opinion of this wiki's coverage of these papers (which I will read once I get a chance to visit the library) aside, there shouldn't be any need for me to discuss the content's veracity further. Anyway, this article is not about teaching students Katz & Katz's work on hyperreals! Since its nonstandard, its not a notable "introduction"! Further, without any tertiary sources (which cover well-seasoned research) actually discussing the K&K's recent proposal, it does not yet come close to meeting wp:DUE to warrant being placed in any article's lede. -Modocc (talk) 20:43, 30 April 2013 (UTC)
 * Your objection to the material on infinitesimal [1-"0.999..."] seems to be based on your concern for consistency for the reals. I appreciate your acknowledgment at Talk:Infinitesimal that you are not a mathematician.  Editors at this page should be able to help dispell your concerns about consistency.  My impression is that your misconception is based on your interpretation of the expression "infinite sum" that you interpret too literally.  Perhaps we can take it from here.  Tkuvho (talk) 08:37, 1 May 2013 (UTC)
 * Thanks Tkuvho, and I am presently asking the guys at Reference_desk/Mathematics to help correct any misconception(s) I might have of the Katzes' proposal. --Modocc (talk) 12:56, 1 May 2013 (UTC)

Adding clarification in the beginning on the nature of the "proof" of .999... = 1
I think it would greatly help to clarify the common misconceptions of the proofs that .999... = 1 if there was a discussion added at the very beginning of the article that highlighted the fact that because the decimals representation is not and can not be made unique, that all nonzero numbers with a finite decimal notation are placed into a equivalence class with the number with the last digit decremented and with trailing 9s when they are treated as the real numbers. Therefor it could be said that .999... = 1 is by definition and that all the proofs provided show only the consistency of the standard operations on the real numbers under this equivalent relation. Linket (talk) 06:47, 20 June 2013 (UTC)


 * This point is mentioned (in a less technical form) in the second paragraph of the article. Gandalf61 (talk) 07:55, 20 June 2013 (UTC)


 * My point is that saying they are equal isn't logically sound since if you treat decimals as just strings or sequences of digits then clearly they are not equal. The 4th paragraph on the Rational numbers does a very concise and actuate job of defining rationals by not only as ordered pairs but as ordered pairs under an equivalence relation setting pairs like (1,2) and (2,4) equal, otherwise (1,2) = (2,4) is nonsensical outside of that context. So adding a few words to the second paragraph highlighting that they are equal under an imposed equivalence relation by definition would then justify writing that they are all equal. In that light trying to prove .999... = 1 is just as silly as trying to prove 1 = 1 in the natural numbers or 1/2 = 2/4 in the rational numbers... Linket (talk) 19:00, 20 June 2013 (UTC)
 * But 0.999... is not a string or sequence. It's a number. And it's a perfectly valid and unique number under our understanding of decimals and recurring decimals. If 0.111... is a number so is 0.999..., just a different one. Further both are rational numbers, as are all recurring decimals. If we work out their values then 0.111... is 1/9 while 0.999... is 9/9 or 1. It can't be anything else. The article demonstrates this in various ways of varying rigour.-- JohnBlackburne wordsdeeds 20:47, 20 June 2013 (UTC)
 * But calling it a number isn't well defined. Any set of objects could be called numbers and for 2 different elements to be shown to be the same is clearly a logical contradiction. All the "proofs" provided (except for the proof by constructions) aren't rigorous at all since they are appealing to the semantic meaning of the numbers and not their syntactic meaning. Linket (talk) 23:12, 20 June 2013 (UTC)
 * It's perfectly well defined as a number. I would suggest you review the article as it does in my view explain it very well. You could also look at Decimal representation, especially Decimal representation, which considers such numbers more generally. Or any of the references as this is only one place you can learn about it, and Wikipedia's encyclopaedic nature means it's not always the easiest thing to learn from.-- JohnBlackburne wordsdeeds 23:44, 20 June 2013 (UTC)
 * "But 0.999... is not a string or sequence. " It sure does look like a string, the database of Wikipedia stores it like a string different from "1", and we all know that repeating digits forever is known as a sequence. Saying that 0.999... is the same number as 1 probably means that mathematically you can replace one for the other in any formula without getting a different result, but they are not "the same" in every sense possible. 1 (number) is a completely different article, not a redirect to this one or vice versa. Joepnl (talk) 00:22, 21 June 2013 (UTC)


 * Firstly, "repeating digits forever" is not a sequence. Secondly, while this article is mainly concerned with the real numbers, where 0.999... is indeed the same number as 1, it also discusses number systems where 0.999... is a different number, so redirecting this article to 1 would not be helpful. There's also a size issue: Merging this article into 1 (number) would make that article too large.
 * Now technically you can argue that "0.999..." is not a number but only the representation of a number, but then you'd by analogy also have to claim that "1" is just the representation of a number, too, and I don't see people doing that. Huon (talk) 01:02, 21 June 2013 (UTC)
 * Firstly, no, each digit of a number is a well defined sequence and often treated as such when proving that certain real is a Normal number or the like. On your last point, what I'm trying to say is that if you do treat ".999..." as a number in the reals and not as a label for some element coming from some other way of construction the reals then this is a logical contradiction for 2 different objects are shown to be the same which means both x=y & x≠y are true. Hence to make this view of the reals logically sound, we must define them to be equal by an equivalence relation. Linket (talk) 05:44, 21 June 2013 (UTC)
 * The sequence behind 0.999... is the sequence (0, 0.9, 0.99, 0.999, ...) - that is a sequence, 0.999... itself is not. On the "number or label" issue, what about, say, 1, 2, e or π? Are those real numbers, or "labels for some elements coming from some other way of constructing the reals"? Do you really want to argue that 1 (number) is a misnomer and should instead be 1 (label for some number)? If you accept that 1 is a number, why should 0.999... be something other than a number?
 * Since we were discussing sequences anyway, I like the Cauchy sequence approach to reals, and I'd say 0.999... is the real number given by the equivalence class of Cauchy sequences of rational numbers represented by (0, 0.9, 0.99, 0.999, ...). Similarly, 1 is the real number given by the equivalence class of Cauchy sequences of rational numbers represented by (1, 1, 1, 1, ...) (where, by an abuse of notation, "1" denotes both the real number and the rational number used to construct it - I could use subscripts, but it should be clear from context whether I'm talking about "real 1" or "rational 1"). 0.999... and 1 are then well-defined real numbers, just as 1/2 and 2/4 are well-defined rational numbers. 0.999... and 1 then are equal (and I never claim that they are not equal, so I don't see where the "x≠y" part of the contradiction is supposed to come from), but not by definition - I'll have to prove the equality. Huon (talk) 07:06, 21 June 2013 (UTC)
 * No, .999.... is itself a sequence (0,9,9,9,9, ...) and so are all reals in decimal form. The real number 1 has no other intrinsic meaning other than it is the element of the reals that satisfies the multiplicative identity axiom. Nothing more nothing less.
 * If you read the article on rational numbers, they make it very clear that 1/2 and 2/4 are not rational numbers but labels for the equivalence classes that contain 1/2 and 2/4. Most people do not consider 1 and .999... to be labels for Cauchy sequences or Dedekind cuts but that the very sequence of digits to be the underlying object that makes up the real numbers as in Stevin's construction. Using what people typically refer to as the real numbers, then one does not show that 1=.999... but it is defined in the definition of those real numbers. Even with the Cauchy sequence construction, the real number are not Cauchy sequences themselves but equivalence classes over the set of all Cauchy sequences that also makes the equality more or less definitional. Thus the beginning of the article should at least say that it is defined or shown to be equal depending on how the reals are construction. Linket (talk) 18:22, 21 June 2013 (UTC)

Citation needed. I'm well aware that Katz&Katz have some interesting ideas on how numbers should be constructed and on what 0.999... should be, but for all I can tell that's rather far from the mathematical mainstream which uses either Dedekind or Cauchy if they want to construct the reals. The Cauchy sequence approach is more relevant here because, as I pointed out, a decimal representation gives you a Cauchy sequence of rationals which in turn (indeed via an equivalence relation) gives a real number. That different Cauchy sequences are equivalent requires proof. This is the approach taken by classics such as Dieudonné, and I could easily cite lots of other introductory analysis texts taking this route. This article should not present comparatively uncommon approachses as if they were the mainstream; see WP:WEIGHT. Huon (talk) 20:01, 21 June 2013 (UTC)
 * "then you'd by analogy also have to claim that "1" is just the representation of a number, too, and I don't see people doing that." Actually some editor that contributed to Number did make a very similar distinction: "In common usage, the word number can mean the abstract object, the symbol, or the word for the number." Joepnl (talk) 02:02, 22 June 2013 (UTC)


 * 0.999... is not a sequence (which implies more than one element), but a single value, being the sum: 9×10$−1$ + 9×10$−2$ + 9×10$−3$ + ... . A true sequence would be something like: 0.9, 0.99, 0.999, ... . — Loadmaster (talk) 18:10, 15 August 2013 (UTC)

1 - (.99999...)^2 = x
So if 1 - (0.99999...)^2 = x, what is the value of x? Is it 0 or 0.00000...19999...? — Preceding unsigned comment added by Reddwarf2956 (talk • contribs) 15:42, 22 August 2013 (UTC)
 * Zero. The second thing you've written isn't even a number.-- JohnBlackburne wordsdeeds 16:57, 22 August 2013 (UTC)


 * The fact that 0.999...2 = 0.999... = 1 is somewhat interesting, but almost certainly not interesting enough to mention in the article, since 12 = 1 (trivially). — Loadmaster (talk) 18:26, 3 October 2013 (UTC)

.999... = 1/1 by long division
It seems like some people are getting tripped up on .999...=1 because they believe that .999... is an irrational number. But you can directly show that 1/1 = .999... by doing intentionally inefficient long division.

1/1 = .9 R .1

.1/1 = .09 R .01

.01/1 = .009 R .001

Carrying this out ad infinitum gives 1/1 = .999... because the remainder goes to zero.

I think it might help people who say, "But there is no whole number ratio that gives .999..." Anyhow, if people like it, maybe it should go in. 68.12.180.246 (talk) 04:29, 8 October 2013 (UTC)


 * I gave essentially the same argument a few years ago, and I also provided a diagram of the long division involved. See this archive page from 2009-02-13. — Loadmaster (talk) 18:10, 10 October 2013 (UTC)

Three reference lists?
Why are there three reference lists in the article? Can't the sections just be merged? Epicgenius (give him tirade • check out damage) 15:53, 21 October 2013 (UTC)


 * Not really, because they serve different purposes. The numbered "Notes" section is footnotes for citations and other notes from the article text. For citations it uses the author-page referencing style, so to find the actual work cited you look up the author in the "References" section - for example, a citation "Grattan-Guinness p. 69" is referring to page 69 of Grattan-Guinness, Ivor (1970), The development of the foundations of mathematical analysis from Euler to Riemann. The third section, "Further Reading", is a list of works that are not specifically referenced in the article text, but which a reader may find useful or interesting. Gandalf61 (talk) 16:12, 21 October 2013 (UTC)

Merge with 1 (number)
If this article needs to exist at all, and isn't just a slightly hysterical response to some highly successful trolling, it should be merged with 1 (number). There are no other integers that have two different articles about them when expressed in different notation: there aren't separate articles on 2 (number) and 10 (binary number), and there's no reason why there should be. I don't see any difference between that situation and this one.GideonF (talk) 11:46, 16 October 2013 (UTC)
 * Definitely not. This is not about the number 1 but about the decimal representation of it 0.999... . Yes this equals one but this is non-obvious to many, has much theory underlying it, as well as pedagogical and other aspects to it, which easily justify a standalone article.-- JohnBlackburne wordsdeeds 12:12, 16 October 2013 (UTC)
 * That the article is about a particular representation of 1 is exactly my point. No other numbers have multiple articles about multiple notations for them.GideonF (talk) 12:26, 16 October 2013 (UTC)
 * That's not entirely true. Diego (talk) 12:05, 18 October 2013 (UTC)
 * This article is a dozen pages with fifty academic references - even if it's the "trolling" you suspect, an act of trolling with a 240 year history and a huge weight of sources would deserve its own article . If we could fill twelve pages with how 10 (binary number) has been historically considered to be interestingly distinct from 2 (number) (or how wily Leonhard Euler trolled everyone about it), then that would have an article too. --McGeddon (talk) 12:43, 16 October 2013 (UTC)
 * You seem to be arguing with a point I'm not making. I'm not saying the content shouldn't be included (although it could use some substantial pruning), it should just be included elsewhere.GideonF (talk) 12:58, 16 October 2013 (UTC)
 * The article includes some well-sourced alternative interpretations of the symbol "0.999..." that differ from 1. Merging it with 1 (number) makes as much sense as merging infinitesimal with zero. Tkuvho (talk) 13:50, 16 October 2013 (UTC)
 * Perhaps a disambiguation page?GideonF (talk) 13:55, 16 October 2013 (UTC)
 * On second thoughts, that's probably unnecessary. If we had a disambig for every number symbol that has different meanings in different unusual number systems, we'd have to have a disambig for every number, e.g., 10 (number) would need a disambig for whether it was ten in decimal, two in binary, sixteen in hex, et cetera ad infinitum.  The nonidentity of 0.(9) with 1 in various exotic number systems can be dealt with perfectly well in the various articles about those number systems.GideonF (talk) 14:08, 16 October 2013 (UTC)
 * You would first have to convince the folks here to do this. Tkuvho (talk) 14:31, 16 October 2013 (UTC)
 * I'm not sure I take your meaning, I'm afraid.GideonF (talk) 14:54, 16 October 2013 (UTC)

I see no reason to break up this article merely for the purpose of not having it. It is long and detailed, covering all the aspects of 0.999... - we would do our readers a disservice by splitting that up into multiple parts. Huon (talk) 21:17, 16 October 2013 (UTC)
 * That seems to me to be an argument for my proposal rather than against it. The article on the number 1 is currently split into 2 articles: on for when it is notated as 1, and another for when it is notated as 0.(9).  If you think, as I do, that splitting articles on the same subject into multiple parts is a bad thing, you ought to support merging the two articles about the number 1.GideonF (talk) 08:32, 17 October 2013 (UTC)
 * This article is too large to be merged with 1 (number), especially considering its format which is basically a list of facts. One of those facts is a link to this article if the reader is intrigued enough to follow it, so it's not as if the information is unreachable via 1 (number). Being two separate articles is an arbitrary "problem" that I feel might be due in part to disrespect of the subject. JaeDyWolf ~ Baka-San (talk) 08:50, 17 October 2013 (UTC)
 * I really don't see how you can think having two articles about the same number isn't a problem. I don't even begin to know what you could mean by "disrespect."  The length of the article is an issue, but the length of the article is already excessive for its subject. What about a compromise where we redirect 0.999... to 1 (number), and move most of the content currently in this article to a new, more appropriately titled article called List of proofs that 0.999... equals 1?GideonF (talk) 13:50, 17 October 2013 (UTC)

Are you still banging on about this? Give it over per WP:DEADHORSE William M. Connolley (talk) 14:09, 17 October 2013 (UTC)
 * What a strange thing to say at such an early stage in the discussion.GideonF (talk) 14:16, 17 October 2013 (UTC)
 * There is a vast education literature on the subject of student perceptions of 0.999... and their resistance to the idea of identifying it with 1. This does not necessarily mean that the students are right but it does mean that the subject is notable. There is also considerable literature devoted to developing "proofs" of various degree of rigor of the equality 0.999...=1, so that subject is notable as well, and is naturally related to the first one. The reactions of your fellow editors here may indeed indicate that we are in a WP:STICK situation. Tkuvho (talk) 14:50, 17 October 2013 (UTC)
 * I think the reactions of many of my fellow editors are based on their labouring under the same misapprehension that you appear to be, that I have somehow suggested removing the content. The non-strawman portion of the conversation has barely begun.GideonF (talk) 15:19, 17 October 2013 (UTC)


 * I am opposed to this merge proposal for the same reasons already explained at length by other editors - this article is about a separate and distinct subject; it is notable in its own right; and it is too large to be practically merged into 1 (number). Gandalf61 (talk) 15:31, 17 October 2013 (UTC)
 * I accept that the merge is not going to happen. What do you think of renaming the article as proposed above to more accurately reflect its content?GideonF (talk) 15:52, 17 October 2013 (UTC)
 * I think the name "0.999..." is a perfect title, since that's what this article is about. Attempting to rename it to something like "Equality of 0.999... and 1" only muddies the waters for the reader. Besides, even if we did rename it, 0.999... would still redirect to this same page. — Loadmaster (talk) 16:28, 17 October 2013 (UTC)
 * But 0.999... isn't what the article is about. It is about only one specific aspect of the number 0.999..., namely its identity with 1. And what it consists of is a list of proofs of that  identity.  All other aspects of the number 0.99... are covered in a different article: 1 (number).  And if the title is changed, 0.999... shouldn't redirect here, it should redirect to 1 (number).GideonF (talk) 11:04, 18 October 2013 (UTC)
 * Please see Use–mention distinction. Diego (talk) 12:00, 18 October 2013 (UTC)
 * So I take it you support renaming on that basis?GideonF (talk) 13:05, 18 October 2013 (UTC)
 * No. I meant to say that 0.999... is a notation for the number 1. As this article is about the notation as a repeating decimal (a "mention" of the number), and not the number itself (which would be a "use" of the number 1 as the topic), it is OK to name it with the string "0.999..." and contain only information about the notation. (See the introduction of the article defining its actual topic: "In mathematics, the repeating decimal 0.999...")
 * As I said above with the link to Representations of e, this arrangement is not unique to this number; and it's according to the WP:SPINOUT guideline to have separate articles for different aspects of the same topic. You have not explained why do you believe that all the information related to the number 1 should be held in one single article, but certainly there are reasons why there should be a separate article. Diego (talk) 15:15, 18 October 2013 (UTC)
 * I wouldn't have thought it required much in the way why there being two articles ostensibly about the same number is undesirable. There aren't, to use an analogy, two different articles: Canberra; and Capital of Australia.  The latter is a redirect to the former.  It happens that there are some people in the world who mistakenly believe that the capital of Australia is Sydney.  If this fact were deemed sufficiently noteworthy to merit an article, the article wouldn't be at Capital of Australia; it would have a name referring to the misconception because it would be the misconception, not the city, that the article was about.  This article isn't about the number 0.999... (a.k.a. the number 1), it is about a misconception and a list of proofs (a list which I happen to think strays into indiscriminate collection of information territory and mostly serves as a troll magnet, but I can see that I am in a minority on that point and am not going to convince anyone).  I think it should have a title other than 0.999... because articles that are about names, and not about the objects that the names refer to, should reflect this in their titles.  0.999... (repeating decimal) would be an improvement over 0.999... if List of proofs that 0.999... equals 1 is not acceptable.GideonF (talk) 17:28, 18 October 2013 (UTC)

Oppose. This article is not about a number at all. It is about the characters "0.999..." and how to interpret them. Algr (talk) 13:16, 19 October 2013 (UTC)

Oppose merge: the article is about the notation "0.999...", and the fact that that notation denotes the number 1. As the article shows, this is not obvious to the average non-mathematician. Putting this in the "1" article would give disproportionate influence to this particular issue. -- The Anome (talk) 23:35, 20 October 2013 (UTC)

Oppose merge: the article is about the notation "0.999...", not the number. Hawkeye7 (talk) 08:06, 30 October 2013 (UTC)

Question (by an amateur)
Ok, I'll just try to pose this question here

(1)*(0.5)= (0.5)

therefore we can deduce

1*(0.99...)= (0.99...)

If it is true that 1=0.99... that would mean that

2*(0.99...)= (1.99...)= 2 1*(0.99...)=(0.99....)=1

AND

(0.99...)*(0.99...)= (0.99...)

If we applied this logic consistently, then we would also have to say that (0.99...)*(0.99...)=(0.99...) which obviously poses a problem since

(0.99...)*(0.99...)=(0.99...)=1

(0.99...)*(0.99...)*(0.99...)*(0.99...) ........... = 1

I hope that I have been able of getting my point across. Basically what I'm proposing is that if 0.99... truly equals 1 then one could theoretically infinitally reduce a given number by multiplying it an infinite number of times by 0.99..., and that number would also equal 1 by this logic, since it would automatically have to equal 0.99... which again equals 1.

Maybe I'm just incredibly stupid and haven't really looked into this debate enough, but if 0.99... equals 1 then what would stop me from claiming that 0.2 also equals 1 by the reasoning just outlined? --88.73.3.251 (talk) —Preceding undated comment added 19:41, 27 October 2013 (UTC)


 * Everything you say above is correct until the word "reduce". Multiplying by 0.999.. does not "reduce" a number, since 0.999... is not less than 1 -- it is equal to 1. --Macrakis (talk) 21:35, 27 October 2013 (UTC)


 * To rephrase what you wrote,    (x)(0.999...)(0.999...)(0.999...)···(0.999...) = (x)(1)(1)(1)···(1) = (x)(1) = x. Multiplying x by 0.999... an infinite number of times is exactly the same as multiplying x by 1 an infinite number times, which is of course equal to x. — Loadmaster (talk) 02:34, 30 October 2013 (UTC)


 * A good way to think about it is that while everyone can agree that 0.999... = 1-&epsilon; where &epsilon; is at most an infinitesimal number, the big twist here is that the real numbers have the Archimedean property, and the only infinitesimal in the real number system is exactly zero, so that 0.999... exactly equals 1. This is certainly not an intuitive fact, and is not obvious at all. If you read the Archimedean property article. it's technical, and difficult to explain in simple terms, and that's the difficulty that's at the root of the difficulty people have in accepting 0.999... = 1, particularly among lay people who have thought deeply about the matter, but haven't gone quite far enough into the mathematics to get all the way there. But although it's difficult to explain without a lot of work, it's true, and that's why mathematicians all agree that 0.999... = 1, providing you're talking about the real numbers, and not some non-standard decimal number system that does not obey the conventional laws of arithmetic. -- The Anome (talk) 09:16, 30 October 2013 (UTC)

Digit Manipulation
If

\begin{align} x          &= 0.999\ldots \\ 10 x      &= 9.999\ldots \\ 10 x - x   &= 9.999\ldots - 0.999\ldots \\ 9 x        &= 9 \\ x          &= 1 \end{align} $$

then \\ When a number in decimal notation is multiplied by 10, the digits do not change but each digit moves one place to the left. Thus 10 × 0.333... equals 3.333..., which is 3 greater than the original number. To see this, consider that in subtracting 0.333... from 3.333..., each of the digits after the decimal separator cancels, i.e. the result is 3 − 3 = 0 for each such digit. The final step uses algebra:

\begin{align} y          &= 0.333\ldots \\ 10 y      &= 3.3333\ldots \\ 10 y - y   &= 3.333\ldots - 0.333\ldots \\ 9 y        &= 3 \\ y          &= 3\\ Then\\ 3          &= 0.333\ldots \\:) \end{align} $$

why the first digit manipulation is true for 9 and not true for 3? — Preceding unsigned comment added by CristianChirita (talk • contribs) 21:50, 26 November 2013


 * Your calculations for 0.333... are wrong. This is maybe what you meant, or is at least correct:



\begin{align} y          &= 0.333\ldots \\ 10 y      &= 3.3333\ldots \\ 10 y - y   &= 3.333\ldots - 0.333\ldots \\ 9 y        &= 3 \\ y          &= \frac{3}{9} = \frac{1}{3}\\ Then\\ \frac{1}{3}           &= 0.333\ldots \end{align} $$
 * -- JohnBlackburne wordsdeeds 22:00, 26 November 2013 (UTC)


 * @Cristian Where the hell did you come up with y=3??? Since 9y=3, then 3y=1 and y= 3/9 (1/3) which is 0.333... But instead you wrote 9y=3, y=3. And then you ask us why it doesn't work? Are you trolling or what? GreyWinterOwl (talk) 22:03, 26 November 2013 (UTC)
 * first it was that kind of mistake(when do you want to prove something and see what you want to see) now it is an experiment because a lot of people have seen what they desire to see and not the obvious truth.CristianChirita (talk)
 * You're not making any sense. What's "an experiment?" What "obvious truth" are you referring to? JaeDyWolf ~ Baka-San (talk) 09:22, 28 November 2013 (UTC)

Repeating decimal
Given the fact that any number with a repeating decimal representation is always a rational (e.g., 0.142857142857... = $1/7$), would it be useful to mention that since 0.999... is a repeating decimal it therefore must be a rational, and that rational is 1? Or to phrase it another way, 0.999... must be some rational, and there are no other rationals that it could possibly be other than 1. Repeating decimals are mentioned in the "Applications" section, but I don't think this rather obvious fact about 0.999... is mentioned explicitly there. — Loadmaster (talk) 19:26, 4 December 2013 (UTC)
 * I don't see it illuminates things. All whole numbers are rational numbers but you don't generally make this point when referring to whole number properties, for example in number theory. Yes, it's "some" rational, it's also "some" real number. That number is 1 so it can't be any other rational or real number. But that's just a statement, and a fairly obvious one if you accept 0.999... equals one.-- JohnBlackburne wordsdeeds 19:45, 4 December 2013 (UTC)


 * My point was more specific, that since 0.999... is a repeating decimal, it must therefore be a rational number and not something else. This is meant to be another one of the justifications to present to those who have problems accepting that 0.999... is 1, and a rather obvious justification at that. For those people, they may have in mind that 0.999... is somehow a different kind of number than whole numbers, rationals, etc. Which of course it isn't, it's just an ordinary rational. — Loadmaster (talk) 17:13, 6 December 2013 (UTC)

Is 0.999... a rational number?
1 certainly seems to be, but how can they be the same if one is irrational and the other is not? 85.252.131.79 (talk) 13:48, 6 January 2014 (UTC)


 * Not here please—see wp:talk page guidelines. You can try at the wp:Reference desk/Mathematics. Good luck. - DVdm (talk) 13:52, 6 January 2014 (UTC)


 * You should also take a look at the Repeating decimal article before going any further here. — Loadmaster (talk) 22:01, 6 January 2014 (UTC)

0.999... is indeed a rational number; it is equal to 9/9, which happens to be equal to 1. (Just like 0.888... equals 8/9.) - Mike Rosoft (talk) 18:26, 9 January 2014 (UTC)

0.999... = 1 is just a trivial truth about the convergence of a sequence
What do you think of this:

0.999... = 1 is just a trivial truth about the convergence of a sequence.

For the decimal expansion 0.999... is defined as the infinite series 0.9+0.09+0.009+...

And the sum of the series 0.9+0.09+0.009+... is defined as the limit of the associated sequence of partial sums (0.9, 0.99, 0.999, ...)

So 0.999... = 1 just means that the sequence (0.9, 0.99, 0.999, ...) converges to 1. And whereas this is undoubtedly true, there is nothing counterintuitive about it. For even though this sequence converges to 1, it, of course, never reaches 1.

(Note that this "proof" of 0.999... = 1 also holds if we only consider the rational numbers. So it also holds for the rational numbers that 0.999... = 1).

Pulvertaft (talk) 22:08, 7 November 2013 (UTC)
 * This page is for discussions of proposed suggestions for possible improvements of the page. Do you have a proposed suggestion? Tkuvho (talk) 09:11, 8 November 2013 (UTC)

Yes: https://en.wikipedia.org/w/index.php?title=0.999...&diff=580649687&oldid=580647913 Pulvertaft (talk) 15:29, 8 November 2013 (UTC)


 * That is an undesirable change. Common problems are thinking that 0.999... is a sequence. It isn't. That it converges. It doesn't. That it is close to but somehow not equal to 1. It isn't. That it has something to do with sequences. It doesn't. 0.999... is just another way of writing 1 and has nothing to do with the convergence of a sequence. Hawkeye7 (talk) 17:33, 8 November 2013 (UTC)


 * Agreed, it's unnecessary. If we're trying to show that the infinite sum 0.9+0.09+0.009+··· is the limit of the partial sums, or that it is the sum of an infinite geometric series, that's already covered in the "Infinite series and sequences" section. — Loadmaster (talk) 19:27, 8 November 2013 (UTC)

It's all a question about definition. If 0.999... is defined as just another way of writing 1, then no wonder that 0.999=1 ;-). Similarly, if 0.999... is defined as a series, and the sum of a series is defined as the limit of its partial sums, then again it is no wonder that 0.999...=1. I was just trying to spell out that the mystery about how 0.999... can be equal to 1 disappears in this "proof" - which is given in technical terms just above my proposed suggestion: "According to this proof 0.999... = 1 just means that the associated sequence (0.9, 0.99, 0.999, ...) converges to 1. And whereas this is undoubtedly true, there is nothing counterintuitive about it. For even though this sequence converges to 1, it, of course, never reaches 1". As so many people are mystified about how 0.999... can be equal to 1, it seems important to get rid of the mystery where it's possible. — Preceding unsigned comment added by Pulvertaft (talk • contribs) 21:33, 8 November 2013 (UTC)


 * From my point of view, the "mystery" happens to disappear with every proof which is already in the article and I don't see why what you're describing should be held in distinction. You simply strike me as somebody who is incredibly good at visualising mathematical series :) JaeDyWolf ~ Baka-San (talk) 23:27, 8 November 2013 (UTC)
 * The last time this was discussed as I recall someone pointed out that the fact that "0.999... = 1 just means that the associated sequence (0.9, 0.99, 0.999, ...) converges to 1" is already mentioned in the page. If it isn't we should certainly add it.  This fact can easily be sourced.  Tkuvho (talk) 20:21, 9 November 2013 (UTC)
 * It is mentioned on the page, under "Cauchy sequences". Hawkeye7 (talk) 21:19, 9 November 2013 (UTC)

It is indeed correct that the number 0.999... can be thought as a limit of the sequence (0.9, 0.99, 0.999, ...), just like the number 1.000... (which, by convention, is usually written simply as "1") can be thought as a limit of the (constant) sequence (1.0, 1.00, 1.000, ...). But the key point is that the two sequences have the same limit; in other words, "0.999..." and "1.000..." represent the same real number. - Mike Rosoft (talk) 21:10, 8 January 2014 (UTC)
 * No. The inverse sequence is 1.000...1 (i.e. an infinite sequence of zeros followed by a one). Because 1 - 0.1 = 0.9, 1 - 0.01 = 0.99 (and so on ad infinitum), so the inverse is 1 + 0.1 = 1.1, 1 + 0.01 = 1.01 (and so on ad infinitum). And 1.0000... != 1.0000....1. Alexander Gras (talk) 19:23, 27 February 2014 (UTC)


 * 1 - 0.1 = 0.9
 * 1 - 0.01 = 0.99
 * 1 - 0.001 = 0.999
 * and so on ad infinitum
 * 1 - 0.000...1 = 0.999...


 * 1 + 0.1 = 1.1
 * 1 + 0.01 = 1.01
 * 1 + 0.001 = 1.001
 * an so on ad infinitum
 * 1 + 0.000...1 = 1.000...1


 * 1 + 0.0 = 1.0
 * 1 + 0.00 = 1.00
 * 1 + 0.000 = 1.000
 * and so on ad infinitum
 * 1 + 0.000... = 1


 * 1 - 0.0 = 1
 * 1 - 0.00 = 1
 * 1 - 0.000 = 1
 * and so on ad infinitum
 * 1 - 0.000... = 1


 * 0 != 1
 * 0.0 != 0.1
 * 0.00 != 0.01
 * and so on ad infinitum
 * 0.000... != 0.000...1


 * therefore
 * 0.999... != 1.000...


 * 0.000...1 is not well-defined; such a string does not represent a real number. See Talk:0.999.../Arguments for some recent discussions about other number systems and their shortcomings. In particular, what is 1/1.000...1? Anyway, this is getting rather off-topic; if you want to discuss other number systems that may have a number 1.000...1, please do so on the arguments page. Huon (talk) 19:27, 27 February 2014 (UTC)

Article should start with explaining real numbers (and not start with proofs that 0.9999... = 1)
Confusion about 0.9999... shows that students have difficulties understanding what real numbers are. Instead of trying to convince them of something they do not understand, it would be better to start with an explanation about the nature of limits and real numbers. Start with other examples where we have two different sequences that nevertheless have the same limit. When students understands that, they may be more receptive to 0.999... = 1. The real issue is: what are real numbers? The 0.9999.... issue is only a symptom. MvH (talk) 14:58, 18 February 2014 (UTC)MvH


 * Actually, this is easier said than done. The 0.999... = 1 issue is generally discussed before the construction of the reals. So when students doubt that the teacher really proved 0.999... = 1, they are right to be suspicious, because a complete proof can not be given until the student understands the construction of the reals. The trouble is that this construction, though standard knowledge for any mathematician, is hard to grasp for many students. MvH (talk) 14:43, 5 March 2014 (UTC)MvH


 * It should also justify why the real set is the appropriate tool, given that reals are defined as ignoring the central reason for the disagreement, ie. infinitesimals. As I've said before, it is like trying to use whole numbers to prove that fractions don't exist.  Algr (talk) 20:11, 18 March 2014 (UTC)


 * That would be rather off-topic for this article. Here it suffices that the reals are the number system most commonly associated with decimal representations in general and 0.999... in particular in reliable sources. Huon (talk) 01:24, 19 March 2014 (UTC)

"Digit Manipultion" is in error
It switches a stated constant for a variable. It starts with the statement "x = .999..." That statement establishes x as constant .999... The following statements evaluate to true when x = .999... but the statement 9x = 9 evaluates to false when x =.999... IE 9 times .999... = 8.999... x only equals 1 in 9x = 9 when solving for x as a variable.

Simply put when x = .999... then 9x = 8.999... not 9.

when x = 1 then 9x = 9 not 8.999...

98.164.101.116 (talk) 10:33, 29 March 2014 (UTC)


 * Please put new talk page messages at the bottom of talk pages.
 * Note that the section has a misleading header. "== Algebraic proofs ==" should be replaced with "== Algebraic musings ==". There is no proof there, just undefined symbol and string manipulation. - DVdm (talk) 11:37, 29 March 2014 (UTC)


 * 98.164.101.116, so you agree that for x=0.999... we have $$10 x - x = 9.999\ldots - 0.999\ldots$$? Then you must disagree either with $$10 x - x = 9 x$$ (which holds true for any x) or with $$9.999\ldots - 0.999\ldots = 9,$$ which also seems rather obvious. What exactly do you disagree with?
 * DVdm, I have to disagree with that assessment. The only point where the algebraic proof may be hiding complicated issues is the 10*0.999...=9.999... line, which seems easy enough a consequence of the definition of decimals and, say, Hilbert's Hotel or, more basically, the rule that "multiplication by 10 shifts all digits one place to the left". Huon (talk) 13:38, 29 March 2014 (UTC)


 * Hi Huon, the rule that "multiplication by 10 shifts all digits one place to the left", applies to numbers, but at this stage, things like "0.999..." and "9.999..." are not numbers, so the string "10*0.999...=9.999..." is just an informal, non-commital musing, or formally, strictly nonsense. I.m.o. the only rigorous thing one can do, is define the string "0.999..." as the abbreviation of $$\lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k}$$, and then prove that this limit exists and equals one. - DVdm (talk) 14:09, 29 March 2014 (UTC)
 * Define your expression as x in the equation above and solve as before. This proves that it is equal to 1. Hawkeye7 (talk) 20:41, 29 March 2014 (UTC)

How can 0.999 be 1?
(I hope I got the decimal notation for infinitely small but not zero correct... a guess after not being able to find it. 0.000...1 (infinitely zeroes before the 1)

true or false? 0.999... = 1

true or false? 0.000...1 = 0

true or false? 1 - 0.000...1 = 0.999...

true or false? 0.999... - 0.000...1 = 0.99...8

true or false? 0.999... = 0.99...8 = 0.99...7 = 0.99...6

true or false? the difference between every neighbouring floating point number would be 0.000...1?

what's the total sum of all differences between every neighbouring floating point number? infinity or zero? - ZhuLien

(edited as I had actually meant floating point not real numbers. if I have the term incorrect, I meant a number with a decimal point - there is no preferred or intended 'numbering system' - just the existence of the numbers - I've have moved to the suggested Arguments Talk thread below. ZhuLien 27.32.141.11 (talk) 20:20, 7 March 2014 (UTC))

— Preceding unsigned comment added by 202.168.6.200 (talk • contribs) 04:49, 17 January 2014‎ (UTC)


 * Please sign your talk page messages with four tildes ( ~ ), not with a type name as you did here above. Thanks.
 * Not here please—see wp:talk page guidelines. You can try at the wp:Reference desk/Mathematics. Good luck. - DVdm (talk) 07:50, 17 January 2014 (UTC)


 * I'll reply at Talk:0.999.../Arguments. Huon (talk) 21:51, 17 January 2014 (UTC)
 * The first statement is correct, the rest are meaningless (as 0.000...1, 0.99...8, etc. do not exist in real numbers). - Mike Rosoft (talk) 19:43, 7 February 2014 (UTC)
 * Well, they do if you regard then as limits. So
 * $$0.000\dots1 = \lim_{n \to \infty} 10^{-n} = 0$$
 * $$0.999\dots8 = \lim_{n \to \infty}(1-2\times 10^{-n}) = 1$$
 * So the first five statements are all true, but the sixth is meaningless without a definition of "neighbouring real number". Gandalf61 (talk) 15:42, 5 March 2014 (UTC)
 * There is no such thing as 0.000...1 or 0.999...8 in the real number system. Every real number is fully specified by the number before the decimal and the string of digits at all digits at every positive integer position and for every positive integer, there exists a positive integer one higher. According to Construction of the real numbers, one possible definition of the real number system says that a number system is not called the real number system unless it's a field with respect to addition and multiplication. Therefore, it can be proven from the fact that we're using the real number system that multiplication is associative and that 9 has a multiplicative inverse, and from that it can be proven that 0.999... = 1. The only incompleteness in the proof I see in § Digit manipulation is proving the existence of a set with relations +, × nad ≤ satisfying the axioms of real numbers, defining which real number each notation refers to as is done later in § Infinite series and sequences, and proving from that definition that notations can added, subtracted, and multiplied in the way they were done in § Digit manipulation. Although not done in this article, I'm sure as has been proven somewhere that any 2 number systems defined as the real number system in Construction of the real numbers are isomorphic with respect to the + and × operations and ≤ relation once addition and multiplication are defined on them after they have been constructed if not defined during the construction, and that justifies the other proofs given in this article.  — Preceding unsigned comment added by Blackbombchu (talk • contribs) 01:18, 12 May 2014 (UTC)

Hackenbush

 * This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111...2 = 0.11000...2, which are both equal to $3/4$, but the first representation corresponds to the binary tree path LRLRRR... while the second corresponds to the different path LRRLLL....

Should not it say “LRLRLLL...” and “LRLLRRR…”? -- Zygmunt Zzzyzzyzkoff (talk) 18:08, 5 July 2014 (UTC)


 * Why should it? The current paths are correct, with "L" and "R" in the path corresponding to 0 and 1 in the binary representation, respectively. Huon (talk) 18:17, 5 July 2014 (UTC)
 * I thought it would work like this:
 * Until a color change, each segment is worth +1 or -1 (depending on whether it is Blue or Red, respectively).
 * Once a color change occurs, each subsequent segment (regardless of color), is worth half of the previous segment, with a +/- corresponding to the color.
 * Thus, the string BBRB would be worth +1+1-1/2+1/4=7/4.
 * Otherwise, this statement is false:
 * For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1/3.
 * -- Zygmunt Zzzyzzyzkoff (talk) 19:25, 5 July 2014 (UTC)


 * You're right, I was wrong. The L and R don't correspond to 1 and 0, but to 1 and -1. Thanks for providing the more thorough explanation. I'll correct the article. Huon (talk) 00:38, 6 July 2014 (UTC)


 * I reasonably understand the surreal numbers, but I didn't know the hackenstring notation. In surreal numbers there exist distinct numbers 1 (generation 1), 1-ε, and 1+ε (both generation ω). The first is simply {0|}, the latter are {0, 1/2, 3/4, 7/8 ...|1} and {1|2, 1+1/2, 1+1/4, 1+1/8 ...}. I guess in hackenstring notation the numbers would be R, RLRRRR... and RRLLLL... - Mike Rosoft (talk) 00:01, 7 July 2014 (UTC)
 * And I think I now understand the relationship between the hackenstring and set notation: L means that the number is one step smaller, i.e. put the current value in the right set; R means that the number is one step larger, i.e. put the current value in the left set. - Mike Rosoft (talk) 04:44, 7 July 2014 (UTC)
 * On the second thought, I think I have swapped the L and R symbols; i.e. L means "put the current value in the left set" (the resulting value is more than the current one), rather than "go left from the current value" (the resulting value is less than the current one). In that case: 1, 1-ε, and 1+ε are L, LRLLLL..., and LLRRRR..., respectively. (Otherwise, numbers starting with L would have been negative.) - Mike Rosoft (talk) 19:04, 9 July 2014 (UTC)

Cauchy sequences
The current example in the "Cauchy sequences" section uses the sequence (1, $1/10$, $1/100$, $1/1000$, ...), with a limit of 0. Could we instead provide a slightly more intuitive example using the sequence ($9/10$, $99/100$, $999/1000$, ...), having a limit of 1, to more closely reflect the number 0.999...? — Loadmaster (talk) 22:20, 10 July 2014 (UTC)
 * The section constructs real numbers as equivalence classes of Cauchy sequences of rational numbers, with two sequences considered equal if the limit of their difference is zero. So what needs to be proven is that given the sequence 0, 0.9, 0.99, 0.999, ... (corresponding to the decimal representation 0.999...) and the constant sequence 1, 1.0, 1.00, 1.000, ... (corresponding to the decimal representation 1.000...), the limit of their difference is indeed zero - meaning that they both represent the same real number. - Mike Rosoft (talk) 03:33, 12 July 2014 (UTC)

Graphical representation of the limit
Section Infinite series and sequences includes a graphical representation of the limit, but it's base-4 and doesn't represent the $$\epsilon$$ measure of "arbitrarily closeness". I think we can add a second graph representing the distance |x − xn| as a segment on the real line, and showing how it becomes closer to 1 than 1-$$\epsilon$$ as the series increases. This would provide a redundant, visual representation of the concept already explained in that section, which could help us visual thinkers. Diego (talk) 10:33, 15 October 2014 (UTC)

Subpage nominated at MfD
I have begun a discussion of this talk page's "Arguments" subpage at MfD, because it is being used as a forum in violation of Wikipedia policy. Lagrange613 04:18, 16 October 2014 (UTC)


 * Update: the MfD was closed as keep, for the same reasons as before: the existence of that page is the lesser of two evils. There was, however, one very good argument there that it should really be part of the Reference Desk system: perhaps we should consider moving it to Reference desk/0.999... or Reference desk/Mathematics/0.999...? -- The Anome (talk) 12:54, 3 November 2014 (UTC)

Fractions and long division
it seems like asserting that .1111... = 1/9 is circular to proving that .999...=1. I'm not saying it's not true, but if you don't believe that .9999..=1 then why would you believe that .3333...=1/3 or .1111...=1/9 — Preceding unsigned comment added by 24.19.2.53 (talk) 05:12, 27 August 2014 (UTC)


 * Long division cranks out an endless sequence of threes when applied to 1/3, and an endless sequence of ones when applied to 1/9. Although neither of these are infinite series, since no-one has infinite time to carry out long division, they add plausibility to the idea of these as infinite series. Since long division is a simple procedure we learn in school, and gives the correct results for other problems, it's easier for people to believe in than more abstract procedures. -- The Anome (talk) 09:11, 27 August 2014 (UTC)


 * Those are absolutely infinite series, btw. $$\frac{1}{9} = \sum_{k=1}^{\infty}(\frac{1}{10})^k$$, multiply by 3 for 1/3.  No one may have "infinite time to carry out long division," but that doesn't mean the solution isn't provably infinite. Gnassar (talk) 09:44, 29 November 2014 (UTC)

No citations in the introduction
Hi, after reading the article, I saw no citations in the introduction, and only one in algebraic proof. It is a valid argument, but it is not verifiable. Please see what you can do to fix this. Thanks! The f18hornet (talk) 19:25, 20 March 2015 (UTC)
 * See WP:LEADCITE. The lead does not generally require citations. If there is a direct quotation or other element that requires it then yes, but otherwise citations generally can be found in the main sections of the article. Repeating them wholesale in the lead is unnecessary and can lead to excessive clutter – where for example a paragraph summarises a whole section and so is based on all the sources in that section. If there are particular problem statements then they should be checked and if necessary reworked or sourced, but tagging every paragraph as you did is excessive.-- JohnBlackburne wordsdeeds 19:32, 20 March 2015 (UTC)
 * Okay, Good to know! Thank you for helping. I will point out however, some things in the fourth paragraph of the introduction:

The equality 0.999... = 1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some students find it sufficiently counterintuitive that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of numerous studies in mathematics education. These are just some of the things that I saw that could have been written better. So please put these issues into consideration. Thanks! The f18hornet (talk) 20:10, 20 March 2015 (UTC)


 * Hi there. I've reverted your changes, because every one of these points is already addressed in the body of the article, with citations to reliable sources as and where appropriate. Which mathematicians? We name and cite a representative sample (eg. Leonhard Euler, Tom Apostol and Georg Cantor, to name just the first three I registered in a quick glance at the article). General mathematical education? See the section on mathematical education. Some students? See the section on mathematical education. Numerous studies? See the section on mathematical education, where we cite several. -- The Anome (talk) 17:32, 21 March 2015 (UTC)

Dedekind cuts
According the Dedekind cut section, a real number is a subset of the set of rational numbers, but a rational number is a kind of real number, which means that all rational numbers contain themselves which according to ZF is impossible. Blackbombchu (talk) 00:18, 12 May 2014 (UTC)
 * Technically speaking, a mathematician would say that there is a subset of the real numbers that is in one-to-one correspondence with the rational numbers, and that the real number addition and multiplication exactly correspond to the rational number addition and multiplication. In other words there is a set within the real numbers that is isomorphic to the rational numbers as ordered fields. So while the are different we choose not to distinguish them in most contexts. Thenub314 (talk) 00:44, 12 May 2014 (UTC)


 * More generally, given any set S, I can form the set $$S' =\{\{x\} | x\in S\}$$, where obviously S and are in bijection. So while there's an obvious mapping which sends x to {x},  neither x nor {x} contain themselves as an element. Something similar happens for the rational numbers. Huon (talk) 21:22, 12 May 2014 (UTC)


 * This seems to really belong in talk:Dedekind cut, as it's not a question somehow unique to 0.999.... But really, if I'm understanding the comment correctly, it has a simple answer: The Dedekind cut of a rational number is the set of all real numbers *less* than that rational number.  Rational numbers, therefore, do not "contain themselves." Gnassar (talk) 09:25, 29 November 2014 (UTC)
 * I'm a bit late to the party here, but while it may be true that the Dedekind cut corresponding to a rational number does not contain that rational number, that actually isn't the real point here.
 * The real point is the one Thenub314 made above. The real number corresponding to a particular rational number, in this approach to their construction at least, is not the same as that rational number.  So literally speaking, Blackbombchu's claim that "a rational number is a kind of real number" is wrong.
 * That said, practically no one ever speaks quite that literally, except in very peculiar contexts like this one. For almost all purposes, it's just fine to say that a rational number is a kind of real number, and in fact people would look at you funny if you said the opposite.  Just the same, in this context, it's not literally true.
 * To keep this on-topic, does this point need to be clarified in the article? I haven't checked. --Trovatore (talk) 20:30, 28 December 2015 (UTC)

Problem rendering formulas in this article
In the Fractions and long division section, the formulas that use the "math & /math" format don't seem to render correctly. For example, the first one simply appears as, "\begin{align} \frac{1}{9} & = 0.111\dots  \\  9 \times \frac{1}{9}  & = 9 \times 0.111\dots \\  1 & = 0.999\dots \end{align}". Is this a problem with this article or is it the browser's? I noticed that another article that used the "{{convert|}" template rendered correctly. Am I the only person with this problem? __209.179.0.121 (talk) 02:22, 28 December 2015 (UTC)


 * No problem here, using standard Firefox 43.0.1 and IE 11 on Win8.1 Pro. Formulas are perfect, both in PNG and MathML — see Preferences, Appearance, Math. - DVdm (talk) 14:53, 28 December 2015 (UTC)


 * Then it must be I. Time to upgrade, again. Thanks for your help. __209.179.0.121 (talk) 02:34, 29 December 2015 (UTC)
 * No problem. Are you still working with IE3 on Win 95? - DVdm (talk) 10:25, 29 December 2015 (UTC)


 * Actually, I don't use Windoze at all, unless I'm forced to, like when I was still working. I'm one of those people who uses a Mac, and have proudly since I got my first one in 1985. Why my browser is behaving oddly is a mystery, as it no longer displays pictures, either. Again, thanks. __209.179.0.121 (talk) 16:28, 2 January 2016 (UTC)
 * I’m on a Mac too and don’t have any problems. Your problem sounds like a couple I’ve seen although not exactly like either. One is make sure you have Javascript enabled and working. Some extensions can interfere with it so check them too. At worst some malware can interfere with Javascript though that is much less likely on a Mac. A more general image problem could be a connection problem to the WP servers that supply the images and are different from the ones with pages. That could be one of a variety of things: a server problem, an ISP problem, a DNS problem or a caching one. Mostly such problems go away of their own accord/once someone else notices them though it may need you to restart your computer, router or modem.-- JohnBlackburne wordsdeeds 16:57, 2 January 2016 (UTC)
 * Or turn off your ad blocker. --jpgordon:==( o ) 16:35, 3 January 2016 (UTC)

Simple Explanation for a Pedestrian Mind?
Mathematics has never been a strong point with me. To me, 0.9999... seems as if it will always APPROACH 1 but never REACH 1. I accept, obviously, that minds better suited to this have determined otherwise. Could someone possibly "dumb it down" for me? Accepting a truth, and understanding a truth, are two different things. — Preceding unsigned comment added by 100.11.128.55 (talk • contribs) 06:40, 27 January 2016‎ (UTC)


 * Please sign all your talk page messages with four tildes ( ~ ). Thanks.
 * It is dumbed down in the last part of section 0.999..., which i.m.o. is the only part of the article that you should read in this—again i.m.o., seriously overloaded—article. The thing labeled "0.9999..." is defined as the smallest number to which you can get as close as you like by adding more and more nines to the sequence ( 0.9, 0.99, 0.999, 0.9999, etc. ). That number can be proven to be 1. No matter how many elements you write down in that sequence, the number 1 will not be in it, but you can get as close to 1 as you like, and indeed not at 1 itself.
 * If this is not sufficient for you, per wp:talk page guidelines, the place to go is to our wp:Reference desk/science. Here we should discuss the article, not the subject. - DVdm (talk) 08:14, 27 January 2016 (UTC)
 * Or in other words, 0.999... does not approach anything; 0.999... is a limit of the sequence {0.9, 0.99, 0.999, ...}, and the sequence approaches the limit. (And, as User:DVdm has said, the limit happens to be equal to 1.) - Mike Rosoft (talk) 20:32, 9 February 2016 (UTC)


 * Another take on this is to realize that the trailing "..." of "0.999..." means that there is an unending (infinite) series of 9 digits. Which means that saying 0.999... approaches 1 is incorrect; it is already there, exactly at 1, because there are already an infinite number of 9 digits in place in 0.999...; all the 9s are already there. There is no process going on when you write a decimal number, whether it's 1 or 0.999... or 123.456; the number is right there in its entirety. It's not a "partial" numeric value waiting for someone to complete the digits on the right end. (If that were the case, then many written numbers would be inexact quantities that are always "moving around" their actual value. Which is nonsense.) Another way of approaching this is to realize that any decimal number is a written representation of a point on the real number line, and both "1" and "0.999..." are the same point on that line. There is no "approaching" to be "performed", those points are simply there on the line. — Loadmaster (talk) 18:12, 11 February 2016 (UTC)
 * This is covered in the section on pedagogy (poorly named "Skepticism in education"). So nI would suggest that section be read too. Hawkeye7 (talk) 21:08, 11 February 2016 (UTC)


 * I think that right now the article has the effect of confirming in the mathematical student-reader's mind that this equality is something mysterious and only true in some esoteric sense that only a mathematician could possibly understand. The introductory intuition should be in the lead, not buried in the places mentioned above, and that means starting out by defining what we mean definitionally by an expression of the form 0.999....
 * I like DVdm's wording from above in this thread:
 * The thing labeled "0.9999..." is defined as the smallest number to which you can get as close as you like by adding more and more nines to the sequence ( 0.9, 0.99, 0.999, 0.9999, etc. ). That number can be proven to be 1
 * I'm going to put this into the intro, with a couple tweaks.Loraof (talk) 16:01, 25 March 2016 (UTC)


 * Perhaps we should replace the emphasis, which I used in the explanation to 100.11.128.55, with wikilinks:
 * The thing labeled "0.999..." is defined as the smallest number to which we can get as close as we like by adding more and more nines to the sequence ( 0.9, 0.99, 0.999, 0.9999, etc. ). That number can be proven to be 1.
 * This way we'd have kind of encyclopedic emphasis. I'd also replace the "you" with "we". And close the door with a dot. - DVdm (talk) 16:49, 25 March 2016 (UTC)
 * 0.999.. is not a sequence. It is not a limit that is as close as we like to 1. 0.999... "denotes a real number that can be shown to be the number one. In other words, the symbols "0.999…" and "1" represent the same number." 'That is our simple explanation. Hawkeye7 (talk) 19:59, 25 March 2016 (UTC)
 * But that sentence does not say that it is "a sequence", and it does not say that it is "a limit that is as close as we like to 1." Not even close. I have no idea how anyone would be able to read that in the sentence . - DVdm (talk) 20:36, 25 March 2016 (UTC)
 * The very mention of sequences is to be avoided, as it has been known to cause misunderstandings that it is defined as a sequence. It is not. 0.999... is defined as 1. Hawkeye7 (talk) 20:56, 25 March 2016 (UTC)
 * Yes, quite indeed, it is not defined as a sequence, and indeed also, that often is part of the confusion. Excellent point.
 * There is way to avoid mentioning "sequence", replacing it with "list", which is not a standard math term. What about this?
 * The thing labeled "0.999..." is defined as the smallest number to which we can get as close as we like by adding more and more nines to the numbers in the list 0.9, 0.99, 0.999, 0.9999, etc. That number can be proven to be 1.
 * Note that there is no wikilink to list, even if we have a DAB on it. - DVdm (talk) 21:30, 25 March 2016 (UTC)
 * Wait, what? You want to avoid saying it's the limit of a sequence, because someone might skip over the words "the limit of"?  That can't be right.
 * Also, I would avoid any too-authoritative pronouncements on how something is "defined". There are lots of equivalent definitions and none of them has any special status as "the" definition. --Trovatore (talk) 21:35, 25 March 2016 (UTC)
 * Another good point. To avoid the authoritativity (I'm sure that is not a word) perhaps "can be defined", as in:
 * The thing labeled "0.999..." can be defined as the smallest number to which we can get as close as we like by adding more and more nines to the numbers in the list 0.9, 0.99, 0.999, 0.9999, etc. That number can be proven to be 1.
 * We're looking for something that is correct and simple. Yes, this is tricky: there are so many good points - DVdm (talk) 21:51, 25 March 2016 (UTC)

I'm rather unconvinced by the proposed definition. Is it one commonly used in the literature? Why "the smallest number"? It's also the largest - it's the only such number. Huon (talk) 22:04, 25 March 2016 (UTC)
 * I agree, "smallest" should not be there (and indeed I removed it in my (reverted) edit to the lead). Also there are lots of numbers that we can get closer and closer to by adding nines and then stopping, such as 0.99, so that needs to be worded differently. I recommend the following revised version of my reverted edit:
 * The expression "0.999..." can be defined as the number which we get ever closer to, and indeed closer than any difference however small, by adding more and more nines to the sequence ( 0.9, 0.99, 0.999, 0.9999, etc. ). That number can be proven to be 1.
 * I think that's both accurate and simple enough for non-mathematicians to understand. Loraof (talk) 00:42, 26 March 2016 (UTC)
 * It is not accurate because 0.999... is a number, not an expression.  It is true that 1 is the limit of the sequence 0.5, 0.75, 0.875,... but we want to avoid mentioning limits or sequences, because of misconceptions that 0.999... is a limit or a sequence. 0.999... denotes a real number that can be shown to be the number one. In other words, the symbols "0.999…" and "1" represent the same number.  Hawkeye7 (talk) 03:54, 26 March 2016 (UTC)
 * Hold on. Of course "0.999..." is an expression.  If it weren't, there would be nothing to "define".  I don't think we can "avoid mentioning limits or sequences".  Limits of sequences is how you interpret infinitely long decimal expressions (in general, not just for the case of 0.999...) and I don't think it's helpful to try to dance around that.  Sure, you can restate it in other words, but to what end? --Trovatore (talk) 09:21, 26 March 2016 (UTC)
 * Fair enough; One is indeed a mathematical constant, which can be considered a constant expression. But limits of sequences is not how I interpret numbers. Hawkeye7 (talk) 00:15, 27 March 2016 (UTC)
 * You're missing the point. No one is interpreting numbers.  What we're talking about is interpreting representations of numbers.
 * The string of digits 0.999... is not a number. The digit 1 is not a number either.  They are symbols for numbers, but they are not themselves numbers.
 * The number itself is a Platonic abstract object that is independent of any representation by symbols. One of these Platonic objects, the one we're talking about, is represented by two different infinite strings, the string 1.000... and the string 0.999..., but if you're given the strings instead of the number, how do you find the number?  You do it by a limit process (in both cases).  It happens to be the same limit.
 * But the really important thing to keep in mind is that the Platonic real number is the real thing. The symbols are less important.  --Trovatore (talk) 01:10, 27 March 2016 (UTC)

Ellipses denote approximations which ignore infinitesimally small remainders
[Non-editorial remarks moved to arguments subpage] --Trovatore (talk) 18:14, 4 April 2016 (UTC)

On secondary sources
There seems to be some confusion here about what a secondary source is in the context of the In popular culture section. One cited source is Straight Dope, a secondary source about several discussions that took place on a message board. Those discussions would be primary sources. Likewise, the Blizzard Entertainment press release was a secondary source concerning discussions that took place on the Battle.net discussion forum. Those discussions themselves are primary sources, but the press release is a third party secondary source.

I believe that the content here is important for establishing the wider cultural context for the subject of this article. It is not an indiscriminate collection at all, but indeed is very discriminate. It was suggested that editors here study the essay WP:IPC. I think that summarizes nicely why sections like this are an important part of Wikipedia. Sławomir Biały (talk) 12:02, 27 May 2016 (UTC)
 * Per this RFC, content in IPC sections requires reliable sourcing that indicates the significance of the reference to the topic. In this particular case, The Straight Dope and Blizzard Entertainment can't demonstrate their own significance. You would need much better sourcing to support an argument that a noticeboard post is significant enough to warrant discussion here. Nikkimaria (talk) 12:06, 27 May 2016 (UTC)


 * Well, we disagree about that. So does User:JohnBlackburne, apparently.  Perhaps some other editors watching this page, who aren't just drive-by taggers, might care to weigh in?  The consensus at the RfC is that secondary sources are required, pure and simple.  We have that.  If the secondary sources are adequate is a matter for local consensus, not administrative fiat.   Sławomir Biały  (talk) 12:31, 27 May 2016 (UTC)
 * To quote from the RFC close, "The source(s) cited should not only establish the verifiability of the pop culture reference, but also its significance". These sources don't do that. Nikkimaria (talk) 12:44, 27 May 2016 (UTC)


 * The sources do establish significance. They are secondary sources about message board discussions, indicating that the discussions had a broad impact upon that online community.  That is clear, and direct, "significance", even on a very strict reading of the term "significance" in the closer's rationale at the RfC you cited.  Also, typically on Wikipedia secondary sources are ipso facto evidence of significance.  See, for example, our general notability guideline.  I'm not seeing any evidence in the comments put forward at the RfC that sources like this would not be considered acceptable, the closer's comment notwithstanding.   Sławomir Biały  (talk) 12:52, 27 May 2016 (UTC)
 * A company remarking on its own noticeboards doesn't mean that the posts are significant to anyone but the company - it certainly doesn't mean that the posts are significant to an encyclopedic understanding of the topic. Nikkimaria (talk) 16:45, 27 May 2016 (UTC)
 * No, it doesn't necessarily mean this. That's why we have editors to decide whether something is noteworthy enough to go in the article.  Obviously not everything is necessarily worth including, but something that indicates the cultural impact of the topic outside of mathematics is definitely germane and important information worthy of inclusion in the article.   Sławomir Biały  (talk) 16:51, 27 May 2016 (UTC)
 * It would be if there were actually sourcing to demonstrate broad cultural impact and not just "someone on our noticeboard thought this was cool". Nikkimaria (talk) 17:05, 27 May 2016 (UTC)
 * Well, I happen to think that the Cecil Adams piece does establish broad cultural impact. In fact, he addresses precisely that question in a fairly direct manner.   Sławomir Biały  (talk) 17:31, 27 May 2016 (UTC)
 * To expand on my edit summary, WP:IPC is just an essay, but even looking at it I cannot see anything in the section that is problematic. It is not an indiscriminate unsourced list. It does not contain only meaningless irrelevant mentions. The RFC requires sources and that they establish the significance of the entries, but I think they do that – they all relate to how 0.999... is perceived in popular culture. It is a very unusual thing in mathematics, elementary enough to be widely 'understood', but also easily misunderstood because it involves subtleties which might not be apparent on first encountering it. So it has an outsize presence in popular culture, and this is worth documenting.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 12:55, 27 May 2016 (UTC)
 * It's worth documenting, but it isn't worth overemphasizing the impact of a couple of noticeboard posts on cultural understanding of the topic. Nikkimaria (talk) 16:45, 27 May 2016 (UTC)
 * If you have other sources to bring to bear, now would be a good time to bring them up. Here "overemphasizing" seems to mean "at the exclusion of other, better sources on the cultural understanding of the topic".  If you have such sources, I'm sure that the article would benefit from their inclusion.   Sławomir Biały  (talk) 16:51, 27 May 2016 (UTC)
 * "Overemphasizing" means presenting as "important information worthy of inclusion", something that really isn't absent better sourcing. I don't have sources to suggest these noticeboard posts are significant; unless you do, they shouldn't be here. Nikkimaria (talk) 17:05, 27 May 2016 (UTC)
 * It seems like you're alone in that view. WP:CON.   Sławomir Biały  (talk) 17:21, 27 May 2016 (UTC)
 * WP:LOCALCON. Nikkimaria (talk) 17:46, 27 May 2016 (UTC)
 * Right. But you're also the only editor who thinks that these sources aren't acceptable under that RfC and the essay you referred to.  As far as this article goes, the local consensus and community consensus apparently agree, unless you can make some convincing case, which so far you havent.  Indeed, your eagerness to disregard local consensus smacks of bureaucratic I-know-best cabalism.  I think we're done here, unless you have anything of substance to say. If all you can do is refer to alphabet soup that's not what Wikipedia is about. <span style="display:inline-block;vertical-align:-.3em;line-height:.8em;text-align:right;text-shadow:black 1pt 1pt 1pt"> S ławomir  Biały  18:05, 27 May 2016 (UTC)
 * Let's keep the discussion focused on content and not each other, shall we? Nikkimaria (talk) 00:49, 28 May 2016 (UTC)
 * Lovely. Let me know when you're ready to start!  <span style="display:inline-block;vertical-align:-.3em;line-height:.8em;text-align:right;text-shadow:black 1pt 1pt 1pt"> S ławomir  Biały  00:58, 28 May 2016 (UTC)
 * A video game reliable source search shows 3 foreign references (which does not mean they should be disincluded) to the Blizzard April Fools' day joke. I have not assessed the sources to see whether they are passing mentions or basically "echoing" the original press release. --Izno (talk) 17:17, 27 May 2016 (UTC)
 * It's great to look for additional sources, but let's not lose sight of the big picture. The subject of this article is one of perennial confusion in lots of online message boards.  The secondary sources that we have gathered amply show this I think.  While it might be reasonable to question, for instance, the Blizzard source (which is by far the weakest), it too is a secondary source attesting a similar kind of cultural impact that the others do.  So we have here a plurality of seconary sources showing a similar kind of cultural impact in different online communities.  They fit together. <span style="display:inline-block;vertical-align:-.3em;line-height:.8em;text-align:right;text-shadow:black 1pt 1pt 1pt"> S ławomir  Biały  18:27, 27 May 2016 (UTC)
 * Are there any sources that say anything beyond "people on noticeboards are confused"? We would expect "cultural impact" to extend past that. Nikkimaria (talk) 00:49, 28 May 2016 (UTC)
 * Go, find the sources about the kind of "cultural impact" that you feel is more momentous and significant. It seems like you have something very specific in mind, and if you can produce sources to match your high expectations, I'm sure we'd love to discuss them.  But I've already given reasons why the content is reasonable for the article.  So has another editor.  As far as I can tell, that point still stands.  <span style="display:inline-block;vertical-align:-.3em;line-height:.8em;text-align:right;text-shadow:black 1pt 1pt 1pt"> S ławomir  Biały  01:07, 28 May 2016 (UTC)
 * I don't think those sources exist, which is why I don't think this content should be included. Nothing you've said on this talk page explains why even minor changes are unacceptable. Nikkimaria (talk) 01:19, 28 May 2016 (UTC)
 * Those changes weren't minor. <span style="display:inline-block;vertical-align:-.3em;line-height:.8em;text-align:right;text-shadow:black 1pt 1pt 1pt"> S ławomir  Biały  01:20, 28 May 2016 (UTC)
 * Why is it important to say what else the FAQ covers, when those details have absolutely nothing to do with this article? Why must we go down the rabbit-hole that some unknown noticeboard about video games discussed something that was then discussed on another noticeboard which was then discussed in a column? Why do we have to have a quote that says that a proof was offered, and then say again that a proof was offered? It seems like all possible changes are being rejected. Nikkimaria (talk) 01:40, 28 May 2016 (UTC)
 * I've numbered your questions. 1. They clearly do have to do with this article.  This shows that you've not read the article, and one wonders if this is the best position from which to be making such "minor" edits.  2.  Those paragraphs in the section concern the impact of the identity 0.999...=1 in those online communities, so mentioning the online communities seems germane.  3.  There are various proofs covered in this article, and it is worthwhile pointing out that the proofs discussed in the press release are related to other parts of the article.  4.  I realize this is not a "question" per se, but if you do not make good edits, I do not know why you should expect those edits to stay.  You're certainly free to propose edits here that you think might be good.  That would certainly prevent bad edits from being reverted, and might actually lead somewhere constructive.  <span style="display:inline-block;vertical-align:-.3em;line-height:.8em;text-align:right;text-shadow:black 1pt 1pt 1pt"> S ławomir  Biały  01:50, 28 May 2016 (UTC)
 * You've stated that the point of overemphasizing these noticeboard posts is to demonstrate the cultural impact of the concept; we don't need these details to do that. Nikkimaria (talk) 14:41, 28 May 2016 (UTC)
 * I have replied in good faith to all of your questions. I think we are done here.  Come back when and if you have something constructive to say, like additional content-building sources.   Sławomir Biały  (talk) 14:47, 28 May 2016 (UTC)

Interpretation within the ultrafinitistic framework
''Exposition moved to Arguments page. A suggested change to the article related to this argument is possible, but this is not it.'' --Trovatore (talk) 00:49, 14 July 2016 (UTC)
 * I moved this back from the /Arguments page. I think this is a notable perspective that should be mentioned somewhere in the article.   Sławomir Biały  (talk) 23:58, 13 July 2016 (UTC)
 * I didn't say it wasn't. Just the same, the contribution itself didn't suggest anything like that; it just attempted an exposition of a POV.  I think those should get moved automatically to Arguments.  If Iblis wants to suggest putting something in the article, that request would take a considerably different form, and also a considerably different tone. --Trovatore (talk) 00:47, 14 July 2016 (UTC)
 * After Sławomir's edit and Hawkeye's reversion, I suppose we'd better talk about this.
 * I agree that it's a point of view that should probably be treated. But a few lines of get-off-my-lawn blogging from Zeilberger don't strike me as rising to the level of something that we should virtually copy into the article.  (By the way, I didn't notice that Iblis's post was essentially a direct quote of Zeilberger's proto-tweet.)
 * Surely there must be something better somewhere. Could even be from Zeilberger, if he's written about it more seriously somewhere. --Trovatore (talk) 23:23, 14 July 2016 (UTC)

I agree with the removal. as it was it was sourced only to a blog post which is far from a reliable source. It’s hard to extract any useful information from that reference, it certainly does not clearly reference ultrafinitistism. Absent proper sources it does not belong in the article.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 00:17, 15 July 2016 (UTC)
 * I also agree with Hawkeye7's action, but I think Sławomir is probably right that the POV deserves some mention. Needs a higher-quality source, and also better exposition (in particular, what Zeilberger means by "symbolically" is not well explained). --Trovatore (talk) 00:46, 15 July 2016 (UTC)
 * I think the solution is not removal, much less under WP:FRINGE, but to improve the content.  Sławomir Biały  (talk) 11:22, 15 July 2016 (UTC)
 * But perhaps you could get it right, before re-inserting.  such as the summation of infinitely many decimal digits implicit in the notation $$0.999\dots$$ is wrong William M. Connolley (talk) 13:44, 15 July 2016 (UTC)
 * Thanks, but the notation 0.999... by definition is $$\sum_{n=1}^\infty 9/10^n $$, which is an example of an infinite series. It is a theorem of mathematical analysis that this series is equal to one. And, since your reasoning that it's "wrong" is puely ad hominem ("It's wrong because I said so..."), perhaps you at least feel that your mathematical bona fides are relevant to the discussion?  For instance: How many post-graduate degrees in mathematics you have?  How many years of experience do you have teaching mathematical analysis at an undergraduate/graduate level?  Sławomir Biały  (talk) 14:46, 15 July 2016 (UTC)
 * I'm going to guess that WMC's point is that $$\frac{9}{10^n}$$ is not a "digit", which I think you could argue either way, but doesn't strike me as ideal wording in any case (precisely because you could argue it either way).
 * What does "some particular number (like 64)" mean? I'm afraid people will read this as something special about 64, which I don't think is what's intended.
 * By the way, people mix up "fringe" and "crackpot". Fringe views are not necessarily wrong.  Ultrafinitism is definitely fringe, but that doesn't mean it shouldn't be covered, just that it should be given due weight.  I'm still a bit uncomfortable that we have a reference to one of Zeilberger's belly grumbles. --Trovatore (talk) 18:04, 15 July 2016 (UTC)
 * Perhaps it would be best working to improve the ultrafinitism article first. Its been marked as no-decent-refs since 2008 William M. Connolley (talk) 14:26, 15 July 2016 (UTC)


 * Yes, and I think Zeilberger wrote his Ph.D. thesis on this subject, so there likely do exist some good sources for us to work with. But simply put, it's about replacing infinite sets by another formalism. Compare this to we not using use infinitesimals like Newton did and instead using a rigorous limit procedure. Or instead of using distributions non rigorously like is often done on physics where they are treated as if they are ordinary functions, one can set up a rigorous mathematical framework where they are functionals on a suitably defined space. Count Iblis (talk) 18:11, 15 July 2016 (UTC)

such as the summation of infinitely many decimal digits $$9/10+9/100+\cdots$$ implicit in the notation $$0.999\dots$$ is wrong William M. Connolley (talk) 21:55, 18 July 2016 (UTC)
 * You've already expressed that opinion, thanks. I assert that it is correct.  Your basis for making this proclamation is, apparently, "Wrong because I said so."  Obviously, if we are arguing purely ad hominem, then I win the argument.  (I sincerely doubt you boast similar qualifications in mathematical analysis to my own.)  If you wish to give reasons for your opinion, those are welcome.  Anyway, assuming you are no happier with this ad hominem reasoning than I am with yours, you are certainly welcome to look at the article decimal representation that discusses what the decimal representation of a number actually means.   Sławomir Biały  (talk) 22:06, 18 July 2016 (UTC)
 * While I wouldn't have noticed a problem if WMC hadn't pointed it out, our numerical digit article says that a digit is a numeric symbol (such as "2" or "5") used in combinations (such as "25") to represent numbers (such as the number 25) in positional numeral systems.
 * The "numeric symbols" here are all 9s (well, also some 0s, to the left of the decimal point). We aren't adding up infinitely many 9s (ignoring the question of whether you can even add 9s as numeric symbols as opposed to numbers).
 * So I don't know how much to make of this; I think the meaning is pretty clear, but the statement in the article is not literally correct. Could be "positional values of decimal digits" or some such, maybe. --Trovatore (talk) 07:22, 19 July 2016 (UTC)
 * I think this is splitting hairs well beyond the point of reasonableness. And Connolley's attitude of "It's wrong, delete it" and, when asked for an explanation merely repeats "It's wrong", does not strike me as constructive.  It seems like borderline trolling.   Sławomir Biały  (talk) 11:13, 19 July 2016 (UTC)

I'm moving this to the arguments subpage. KSF T C 21:37, 20 July 2016 (UTC)
 * @Sławomir Biały You are the one trolling here.  Sometimes you have to let someone else manage the problem. And the guy who wrote this "ultrafinitism section" is a troll not only on wikipedia but also on different forums. If you'd understand what it is about, you'd agree.  Please remove it, it has nothing to do on this article. 78.196.93.135 (talk) 03:04, 14 November 2016 (UTC)

Horizontal ellipsis (U+2026) vs. three periods
Considering that the title uses "..." as a single character that represents infinite nines, I think it would be more semantic if we called this article 0.999… instead and replaced all instances of "0.999..." in the article with "0.999…". Then a redirect could be set up. Please advise. lol md4 U&#124;T 00:32, 10 November 2016 (UTC)
 * Don't do it. See MOS:ELLIPSES for the rationale. Hawkeye7 (talk) 06:31, 14 November 2016 (UTC)

Ultrafinitism removal
Regarding this, I do not see how removing the ultrafinitist/finitist perspective from the "In alternative number systems" is constructive. Why is it that a group of things that includes "infinitesimals", "hackenbush", "revisiting subtraction", and "p-adic numbers", ultrafinitism is somehow ruled as the FRIGE perspective? I think it is just as relevant as these other strange mathematical objects, in the sense that I don't think the idea of a finite number system is any more "fringe mathematics" than branches of mathematics that include infinitesimals. (Consider Edward Nelson, for instance, who worked in fundamental questions regarding both such kinds of number systems.)

As an aside, looking at some of Trovatore's remarks earlier to Hawkeye, I think ultrafinitism rejects that the conventional interpretation of "0.999.." as an "infinite string" is meaningful. Sp one side of the equation "0.999... = 1" is meaningless under the conventional interpretation, in their view. In fact, I suspect that the ultrafinitist interpretation more accurately reflects Haweye's own perspective on this equality, when he writes "0.999... is defined as 1". The ultrafinitist would be perfectly happy adjoining the eight character string "0.999..." as a term, and adjoining the axiom "0.999... = 1". All conceptual problems, magically swept away ;-P  Sławomir Biały  (talk) 21:40, 16 February 2017 (UTC)


 * The paragraph was incompletely sourced, which is not permissible in a Featured Article. I thought it had been given a reasonable amount of time for this to be rectified. I've had a go at correcting it myself. I haven't changed the text, just the referencing. Assuming that the two of us have understood it correctly, the statement would be covered by Katz and Katz, who basically examine the notion that if the notation is what is confusing people, then maybe the notation could be changed to make the mathematics easier to teach. Hawkeye7 (talk) 22:18, 16 February 2017 (UTC)
 * I think we should get rid of the link to Zeilberger's poorly explained rant. It's a very low-quality source, possibly sufficient to establish that this is Zeilberger's opinion, but not remotely good enough to make it clear why the reader ought to care.  --Trovatore (talk) 23:21, 16 February 2017 (UTC)
 * What do you mean "completely unsourced"? It had two quality sources.  One of which is (apparently) an important one in ultrafinitist circles, which also explicitly discusses the subject of this article, and the other is on the subject of this article, which mentions ultrafinitism.  Both are in peer reviewed journals.  I think these sources easily establish due weight for the present context of "Alternative number systems".   Sławomir Biały  (talk) 12:15, 17 February 2017 (UTC)
 * It should also be added that ultrafinitism IS a fringe in mathematics, mathematics works on the axiomatic method which means you accept the axioms at hand to make claims about propositions within the system at hand. But ultrafinitists, like Zeilberger, rejects all axioms that does not fit his personal taste, that makes it a fringe and him a crank. TheZelos (talk) 07:53, 17 February 2017 (UTC)
 * Can't say I like it either.I took it out again William M. Connolley (talk) 08:05, 17 February 2017 (UTC)
 * We do say that it is not widely accepted mathematics. Ultrafinistists, like Zeilberger and Nelson are definitely not cranks.  Also, WMC's reason for removal is that he "doesn't like it".  This is not the first time that WMC has put his authoritarian and uninformed opinion ahead of actual reasonable discussion on this matter.  Sławomir Biały  (talk) 11:12, 17 February 2017 (UTC)
 * When Zeilberger deviate from the axiomatic method, the foundation of mathematics, to only play in his own prefered realm while rejecting findings in other axiomatic systems, then he most definately is a crank. TheZelos (talk) 14:01, 17 February 2017 (UTC)
 * Where did you get the idea that Doron Zeilberger chose to deviate from the axiomatic method? He is actually a considerable expert on formal systems, as a number of famous theorems bearing his name (and, most significantly, their proofs) show.  His work on formal systems in fact earned him an Euler Medal.  He earned a Leroy P. Steele Prize for a famous paper with Herbert Wilf that rational functions can computably verify certain combinatorial identities (see Wilf–Zeilberger pair).  He was recently made a fellow of the American Mathematical Society and, this past year, won the David P. Robbins Prize for a formalized proof of the q-TSPP conjecture.  The idea that he is a crank, or indeed is not a leading expert on formal systems, is simply laughable.  He does have ideas that are outside the mainstream, but we are here including ultrafinitism along with other alternatives to the standard interpretations, certainly not giving it equal validity or suggesting that it is the correct interpretation.  But it is notable, there are sources, and a number of high profile proponents such that I believe it deserves mention.   Sławomir Biały  (talk) 14:28, 17 February 2017 (UTC)
 * I don't get that either in my understanding (without being particularly familiar with it) ultrafinitism is legitimate math following an axiomatic method, just that its set of axioms is (unnecessarily) more restrictive than that what most mathematician would consider as appropriate. This might make it fringe as not that many mathematicians work with it, but it is still legitimate math and there is no "deviation from axiomatic method" I'm aware of, just different axioms. So as long as it is properly sourced (textbook or journal articles) I don't really see any problem with including it as a notable minority viewpoint.--Kmhkmh (talk) 14:46, 17 February 2017 (UTC)
 * Ultrafinitism is not a mathematical methodology, but rather a philosophical position. It is the position that methods that invoke completed infinite objects (or sometimes even arbitrarily large finite ones) are illegitimate or meaningless.
 * As such, it is a sharply minority viewpoint outside the mainstream. That's what "fringe" means.  "Fringe" does not mean "wrong"; plate tectonics, for example, was fringe, but is now considered proven. I think the problem is that some people use "fringe" as a euphemism, to avoid saying what they really mean, which is "crackpot".  That's unfortunate, but it shouldn't prevent us from using the word correctly.  Ultrafinitism is not crackpot.  But it is fringe.
 * I have no objection to covering an ultrafinitist take on 0.999.... I do have an objection to the link to Zeilberger's ill-explained opinion.  That is not a serious source; it's just venting.  Zeilberger is a serious mathematician and could have created a serious source if he wanted to.  Maybe he even has, for all I know.  But it's not that one. --Trovatore (talk) 18:39, 17 February 2017 (UTC)
 * I have removed the Zeilberger reference.  Sławomir Biały  (talk) 19:04, 17 February 2017 (UTC)
 * Fine with me.--Kmhkmh (talk) 19:24, 17 February 2017 (UTC)
 * And me too. Hawkeye7 (talk) 19:53, 17 February 2017 (UTC)
 * Does the article go too far in stating that "the philosophy lacks a generally agreed-upon formal mathematical foundation"? Hawkeye7 (talk) 21:25, 17 February 2017 (UTC)
 * I think it's not wrong, but it's misleading and poorly worded. There is no reason to expect a philosophical position to have a formal mathematical foundation.
 * What is probably meant is that (as is common with ontologically restrictive viewpoints) there is no clear agreement among its proponents about where to draw the line between the real stuff and the meaningless. For example, some accept arbitrarily large natural numbers and some do not (the former group are sometimes just called "finitists" without the "ultra-").  Among those that accept that every natural number has a successor, there is disagreement about whether we can justifiably assert that for every n there's a 2n.  That sort of thing. --Trovatore (talk) 22:01, 17 February 2017 (UTC)
 * Where does he deviate? The instant he rejects infinity due to personal taste and claim all propositions based on it is wrong/not-even-wrong due to including that one axiom he doesn't like it. TheZelos (talk) 06:43, 20 February 2017 (UTC)
 * One can reject the axiom of infinity, yet still employ the axiomatic method, just as there are mathematicians who reject the axiom of choice. For example, intuitionistic mathematics typically rejects the latter and sometimes the former, while finitism rejects both.  The mathematicians holding these perspectives are most definitely not cranks, but rather share a different idea about what constitutes a "reasonable" set of axioms.  Recall that hyperbolic geometry was discovered by rejecting Euclid's fifth postulate.  This was not a rejection of the axiomatic method as such, but a rejection of one of the postulates.   Sławomir Biały  (talk) 11:25, 20 February 2017 (UTC)
 * Absolutely. And if you believe that the total amount of computation possible in the lifespan of the universe is finite (see here for a paper that considers that hypothesis), so will be the number of theorems deducible from any finite axiom system. (Note however that if that axiom system allows infinite sets, it will allow construction of theorems about infinite sets: but only a finite number of such theorems. And of course Gödelization allows the generation of theorems about infinite sets of theorems, and so on... but the "top level" set of theorems must be finite. Or am I missing something Permutation City-like here?) -- The Anome (talk) 12:31, 20 February 2017 (UTC)
 * Not really relevant to this page, but the fact that there is an upper bound to the information density of the universe, a la Bekenstein, seems to be a strong reason against believing in the reasonableness of the axiom of infinity. (Bracing myself for the onslaught of Platonic idealists, who believe that infinite sets are more real than reality.  That's also a possibility, but to me requires something like faith in the infinite.)   Sławomir Biały  (talk) 13:19, 20 February 2017 (UTC)
 * Actually if you reject it and then proclaim the findings of it is false because you reject it, you've violated it. The axiomatic method means you accept the axioms at hand and then go where they lead, regardless of personal preferences. While one can question the utility, that is no relevance to axiomatic method. TheZelos (talk) 13:47, 20 February 2017 (UTC)
 * Once again, at least crudely, they are rejecting the axioms, not the method. There is a difference.  There is perhaps another sense, in which some mathematical theorems are regarded not as describing "reality", despite being derivable from the axioms.  It seems to me that this is a matter of interpretation: one is under no philosophical obligation to regard the outcome of logic games as describing reality any more than one is obliged to regard a chess game as describing reality.  Yet we can still agree that the moves proceeded according to the established rules.  Indeed, many chess players do not regard the game as descriptive of reality.  That would be silly.   Sławomir Biały  (talk) 14:12, 20 February 2017 (UTC)
 * I never claimed axioms have to do with reality. I know it doens't and don't care fori t really and when you reject axioms like that you are rejecting the methodology. By the methodology you accept the axiom at hand and move on so rejecting an axiom means rejecting the methodology. TheZelos (talk) 07:23, 21 February 2017 (UTC)
 * Hi TheZelos; we're getting off-track here. This is not the place to argue foundational philosophy.  Just very quickly, you are advocating a version of formalism, whereas at least some ultrafinitists (and in particular Zeilberger) are realists, albeit very ontologically restrictive realists.  Realism is a reasonably common position; it's the restrictive version of it that is not so common.  Whether it is correct or not is irrelevant.  However, if you think that it's the consensus position among mathematicians that axioms are arbitrary, then you're just wrong. --Trovatore (talk) 09:15, 21 February 2017 (UTC)
 * "By the methodology you accept the axiom at hand and move on so rejecting an axiom means rejecting the methodology." No, there is substantial disagreement about what constitutes a reasonable set of axioms.  Ultrafinitists, and indeed mathematicians of any stripe, are likely to agree on whether a proposition is derivable from a finite number of axioms, using a finite sequence of rules for inference (which also satisfy a finite set of axioms).  Now, if a theorem derived from such a set of axioms and rules has no "meaning" (as per your view that axioms have nothing to do with reality), possibly apart from some formal interpretation, then it does not matter if a mathematician views the result as "meaningless".  It is meaningless in the same way that a chess game is meaningless: it does not correspond to reality.  He does not claim that the theorem is not a theorem, or that the theorem is not "true" (i.e., its truth value that is determined in a given interpretation of the formal system), merely that it has no ontological meaning to him, in the sense of corresponding to a capital-T Truth of the "real world".  Just as he agrees that the rules of chess were observed throughout the chess game.  When asked "What is the meaning of the chess game?", I think most of us would probably agree that it has no meaning apart from the game itself.
 * For the ultrafinitist, in the real world there is no such thing as an infinite set, so of course conclusions about infinite sets are as meaningful as conclusions about invisible pink unicorns. We can agree from the "rules" of English predicates that invisible pink unicorns are invisible, that they are pink, and that they are unicorns.  But these are not actually meaningful statements about the world, because when we attempt to assign meanings to these predicates in the real world, we obtain a manifest inconsistency.  (Unicorns are entirely fictitious, and moreover cannot be both invisible and pink.)  So even though we are able to apply formal reasoning to the English compound noun "invisible pink unicorn", the results of the analysis are meaningless.   Sławomir Biały  (talk) 18:47, 21 February 2017 (UTC)

Infinitely many
Regarding this change: I am accustomed to thinking of transfinite numbers as numbers too, and would always say "countably many" rather than "infinitely many" in this context, but am willing to compromise on rigour if it is considered necessary to make the article more comprehensible. Hawkeye7 (talk) 21:12, 7 May 2017 (UTC)
 * I think "countably many" has two problems; first, some people don't know what it means at all; second, for those who do know, it can be slightly jarring on the grounds that finite sets are also countable. Pretty sure "countably infinitely many" is not on the table. --Trovatore (talk) 22:41, 7 May 2017 (UTC)
 * I think those readers who are aware of different sizes of infinity will also know that a decimal representation can only, by definition, contain a countable number of digits, and so "infinite" in this context must mean countably infinite. Gandalf61 (talk) 12:21, 8 May 2017 (UTC)
 * Infinitely many seems to be the correct idiom. Also as Gandalf notes, no reader would reasonably expect that we mean some uncountable infinity of nines in a decimal expansion, because decimal expansions are countable by definition.   Sławomir Biały  (talk) 21:33, 8 May 2017 (UTC)
 * I personally have no ideological objection to "an infinite number of", but it seems less graceful than "infinitely many", which also seems to upset fewer people for some reason. --Trovatore (talk) 21:23, 10 May 2017 (UTC)
 * I personally have no ideological objection to "an infinite number of", but it seems less graceful than "infinitely many", which also seems to upset fewer people for some reason. --Trovatore (talk) 21:23, 10 May 2017 (UTC)