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Frequently asked questions (FAQ)
Q: Are you positive that 0.999... equals 1 exactly, not approximately?
A: In the set of real numbers, yes. This is covered in the article. If you still have doubts, you can discuss it at Talk:0.999.../Arguments. However, please note that original research should never be added to a Wikipedia article, and original arguments and research in the talk pages will not change the content of the article—only reputable secondary and tertiary sources can do so.

Q: Can't "1 - 0.999..." be expressed as "0.000...1"?
A: No. The string "0.000...1" is not a meaningful real decimal because, although a decimal representation of a real number has a potentially infinite number of decimal places, each of the decimal places is a finite distance from the decimal point; the meaning of digit d being k places past the decimal point is that the digit contributes d · 10-k toward the value of the number represented. It may help to ask yourself how many places past the decimal point the "1" is. It cannot be an infinite number of real decimal places, because all real places must be finite. Also ask yourself what the value of would be. Those proposing this argument generally believe the answer to be 0.000...1, but, basic algebra shows that, if a real number divided by 10 is itself, then that number must be 0.

Q: The highest number in 0.999... is 0.999...9, with a last '9' after an infinite number of 9s, so isn't it smaller than 1?
A: If you have a number like 0.999...9, it is not the last number in the sequence (0.9, 0.99, ...); you can always create 0.999...99, which is a higher number. The limit is not defined as the highest number in the sequence, but as the smallest number that is higher than any number in the sequence. In the reals, that smallest number is the number 1.

Q: 0.9 < 1, 0.99 < 1, and so forth. Therefore it's obvious that 0.999... < 1.
A: No. By this logic, 0.9<0.999...; 0.99<0.999... and so forth. Therefore 0.999...<0.999..., which is absurd.
Something that holds for various values need not hold for the limit of those values. For example, f (x)=x 3/x is positive (>0) for all values in its implied domain (x ≠ 0). However, the limit as x goes to 0 is 0, which is not positive. This is an important consideration in proving inequalities based on limits. Moreover, although you may have been taught that must be less than for any values, this is not an axiom of decimal representation, but rather a property for terminating decimals that can be derived from the definition of decimals and the axioms of the real numbers. Systems of numbers have axioms; representations of numbers do not. To emphasize: Decimal representation, being only a representation, has no associated axioms or other special significance over any other numerical representation.

Q: 0.999... is written differently from 1, so it can't be equal.
A: 1 can be written many ways: 1/1, 2/2, cos 0, ln e, i 4, 2 - 1, 1e0, 12, and so forth. Another way of writing it is 0.999...; contrary to the intuition of many people, decimal notation is not a bijection from decimal representations to real numbers.

Q: Is it possible to create a new number system other than the reals in which 0.999... < 1, the difference being an infinitesimal amount?
A: Yes, although such systems are neither as used nor as useful as the real numbers, lacking properties such as the ability to take limits (which defines the real numbers), to divide (which defines the rational numbers, and thus applies to real numbers), or to add and subtract (which defines the integers, and thus applies to real numbers). Furthermore, we must define what we mean by "an infinitesimal amount." There is no nonzero constant infinitesimal in the real numbers; quantities generally thought of informally as "infinitesimal" include ε, which is not a fixed constant; differentials, which are not numbers at all; differential forms, which are not real numbers and have anticommutativity; 0+, which is not a number, but rather part of the expression , the right limit of x (which can also be expressed without the "+" as ); and values in number systems such as dual numbers and hyperreals. In these systems, 0.999... = 1 still holds due to real numbers being a subfield. As detailed in the main article, there are systems for which 0.999... and 1 are distinct, systems that have both alternative means of notation and alternative properties, and systems for which subtraction no longer holds. These, however, are rarely used and possess little to no practical application.

Q: Are you sure 0.999... equals 1 in hyperreals?
A: If notation '0.999...' means anything useful in hyperreals, it still means number 1. There are several ways to define hyperreal numbers, but if we use the construction given here, the problem is that almost same sequences give different hyperreal numbers, , and even the '()' notation doesn't represent all hyperreals. The correct notation is (0.9; 0.99; 0,999; ...).

Q: If it is possible to construct number systems in which 0.999... is less than 1, shouldn't we be talking about those instead of focusing so much on the real numbers? Aren't people justified in believing that 0.999... is less than one when other number systems can show this explicitly?
A: At the expense of abandoning many familiar features of mathematics, it is possible to construct a system of notation in which the string of symbols "0.999..." is different than the number 1. This object would represent a different number than the topic of this article, and this notation has no use in applied mathematics. Moreover, it does not change the fact that 0.999... = 1 in the real number system. The fact that 0.999... = 1 is not a "problem" with the real number system and is not something that other number systems "fix". Absent a WP:POV desire to cling to intuitive misconceptions about real numbers, there is little incentive to use a different system.
Q: The initial proofs don't seem formal and the later proofs don't seem understandable. Are you sure you proved this? I'm an intelligent person, but this doesn't seem right.
A: Yes. The initial proofs are necessarily somewhat informal so as to be understandable by novices. The later proofs are formal, but more difficult to understand. If you haven't completed a course on real analysis, it shouldn't be surprising that you find difficulty understanding some of the proofs, and, indeed, might have some skepticism that 0.999... = 1; this isn't a sign of inferior intelligence. Hopefully the informal arguments can give you a flavor of why 0.999... = 1. If you want to formally understand 0.999..., however, you'd be best to study real analysis. If you're getting a college degree in engineering, mathematics, statistics, computer science, or a natural science, it would probably help you in the future anyway.

Q: But I still think I'm right! Shouldn't both sides of the debate be discussed in the article?
A: The criteria for inclusion in Wikipedia is for information to be attributable to a reliable published source, not an editor's opinion. Regardless of how confident you may be, at least one published, reliable source is needed to warrant space in the article. Until such a document is provided, including such material would violate Wikipedia policy. Arguments posted on the Talk:0.999.../Arguments page are disqualified, as their inclusion would violate Wikipedia policy on original research.
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Simple Explanation for a Pedestrian Mind?[edit]

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 (talkcontribs) 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...#Infinite series and sequences, 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, 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 Face-smile.svg. - 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 Face-smile.svg - 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. Face-smile.svg 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[edit]

[Non-editorial remarks moved to arguments subpage] --Trovatore (talk) 18:14, 4 April 2016 (UTC)

On secondary sources[edit]

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 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.--JohnBlackburnewordsdeeds 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. Sławomir
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! Sławomir
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. Sławomir
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. Sławomir
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. Sławomir
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. Sławomir
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[edit]

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.--JohnBlackburnewordsdeeds 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 is wrong William M. Connolley (talk) 13:44, 15 July 2016 (UTC)
Thanks, but the notation 0.999... by definition is , 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 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 implicit in the notation 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. KSFTC 21:37, 20 July 2016 (UTC)

Article is overtly biased toward the veracity of 0.999... = 1[edit]

This article is propagandist and does not adopt an unbiased view of the subject "0.999...". 1. It offensively belittles "students" as the group mostly holding to the "wrong" view that "0.999... < 1", with complete disregard to the possibility that the intuitive result may be right.

2. It overtly treats proofs supporting the "right" result more favorably than proofs supporting the "skeptical" result. The very word "skeptical" is used in the overall tone of the page as a pejorative. The correct headings would be "Arguments supporting "0.999...=1" and "Arguments supporting "0.999...<1" with equal treatment of both.

3. I have attempted to add a reference to a blog post containing a robust (and I might add, formidable) proof that very clearly demonstrates (possibly rigourously) in elementary school math that "0.999... < 1". This reference has been excluded on the basis that (in the excluder's opinion) the poster "does not understand limits". That may well be the case, however, the proof makes no reference to limits and has not dependency upon them. My reading of the blog post is that the discussion regarding Limits" is merely an opinion piece to promote debate, not offered as any part of the proof. Consequently, the reason for exclusion is both spurious and irrelevant, and simply reinforces my feeling that this page is far from objective. Alex Alexander Bunyip (talk) 15:07, 24 July 2016 (UTC)

It's a theorem that the real number represented by the infinite decimal expansion 0.999... is identical with the real number 1. There are high quality sources that have proofs of this, beginning with the axioms of the real number system. As a mathematical theorem, a disproof would essentially imply that all of mathwmatics involving real numbers is inconsistent. One of the pillars of Wikipedia is WP:NPOV, which in particular implies that subjects like this are discussed according to the weight of different viewpoints in reliable sources. There are various sectioning of the article that discuss septicism, alternative number systems in which 0.999... is different from 1. Sławomir Biały (talk) 15:21, 24 July 2016 (UTC)
@Abunyip: We rely on reliable sources as our primary basis for weighing claims in articles. The claimed proof you cite is on a self-publishing website, and thus does not count as being a reliable source. This is to be contrasted with the many proofs given in reliable sources which demonstrate that 0.999... = 1, which provide the basis for the article's presentation of 0.999... = 1 as established mathematical fact. Please do not re-add the material without providing a reliable source that supports it. (Also: have you see this proof, which demonstrates 0.999... = 1 from first principles?) -- The Anome (talk) 15:41, 24 July 2016 (UTC)
I think we should include a sentence about the formal proof, citing the metamath source. Also, a standard challenge to anyone claiming to have discovered a "watertight" proof that 0.999... ≠ 1 should be "ok, well formalize your proof in metamath" (or Coq or HoLight, etc) Sławomir Biały (talk) 16:09, 24 July 2016 (UTC)
I'm not sure we can cite Metamath directly, as it's not a WP:RS of itself, which is why it's in the external links section rather than the body of the article itself. Are there papers on Metamath that we cite in its stead? -- The Anome (talk) 16:22, 24 July 2016 (UTC)
But yes, inviting people to formalize their argument would go a long way to helping clarify things. Not least for them themselves. -- The Anome (talk) 16:37, 24 July 2016 (UTC)
I would consider Metamath to be a reliable source. It does not seem like a theorem proven in Metamath is likely to be challenged. Indeed, Metamath is probably much more reliable than many textbooks, etc. Sławomir Biały (talk) 17:47, 24 July 2016 (UTC)
The article should be biased in favor of the viewpoint that 0.999.. = 1, because that is the viewpoint of essentially every mathematics reference. The idea of neutral point of view does not mean that we are neutral between all viewpoints; it means that we are neutral between viewpoints to the extent that they are represented in high-quality sources. The viewpoint that 0.999... is the same real number as 1 is so overwhelmingly dominant in the mathematics literature that, even if some other viewpoint might be possible, this article should reflect the viewpoint that the numbers are equal. A better question to ask might be: why do so many sources say that 0.999... is equal to 1? What do they mean by "equal"? That will help clarify what is going on in the literature. — Carl (CBM · talk) 16:11, 24 July 2016 (UTC)