Talk:0.999...

Induction proofs and 0.999.....

This is primarily a humerous arguement against those that think that because for any finite amount of 9s then 0.999...<1.

Start with the empty set, it's cardinality is clearly 0. Now the induction step is assume that for a set with a cardinality of n, finitely such, then attatching another element gives anotehr finite set with n+1 in cardinality. As this is always true, therefore natural numbers has finite cardinality.

Yes I know this is faulty and it is meant to be to show the errors of this arguement. TheZelos (talk) 13:02, 14 February 2017 (UTC)

I see the point that you're trying to show here. Obviously, the flaw of the argument as a whole is that no single finite set/sequence represents the complete infinite set/sequence represented by 0.999... It's equivalent to the argument that the limit of the sequence {0.9, 0.99, 0.999, ...} (which are all finite strings of digits) is 0.999..., which is not itself a member of the sequence. But more to the point, I'm not sure if using such a (purposely flawed) example argument for cardinality would be sufficiently instructive for readers of the article. Specifically, I'm not sure naive readers who are trying to grasp the equivalence of 0.999... and 1 are going to understand the concept of cardinality to begin with. — Loadmaster (talk) 18:30, 14 February 2017 (UTC)

They do not need to know very advanced cardinality such as bijections and stuff, all they need to know is that natural numbers are not finite, their arguement means they are, the contradiction is reached. TheZelos (talk) 07:49, 17 February 2017 (UTC)

A number system S is countable set if there exists an injective function f from S to the natural numbers N = {0, 1, 2, 3, ...}. If you say that some numbers can be written in more than one way then the function is surjective. "0.999...=1" is a declaration that there are now two ways to write the same number "one" in the otherwise countable decimal positional numeral system. In general, recurrence is an attempt to represent a rational number that is not completely divisible within the finite precision of the base of the decimal positional numeral system in which it is written. Recurrence denies the clear limitation in precision of decimal representation. Recurrence has the additional effect of voiding ordinality because it is not possible to say what is the previous/next sequential number before/after a number with digits of infinite recurrence. For example the next number after 0.333 is 0.334 but what is the next number after "0.333..." ? Thus the infinitely recurring digit voids the countable number system through loss of injectivity with the set of natural numbers, and loss of ordinality due to having no terminating positional numeral.

If, now, after all that, you still think "0.999...=1", then I suggest you find yourself a good psychiatrist. Because you clearly don't understand the cardinal rules of set theory, and logic and reason isn't helping you any more. You need real help, professional help. In the meantime, just what you think you're doing purporting to lecture to the world about a clear violation of basic set theory is anybody's guess.

Alexander Bunyip (talk)

This should really go on the Arguments page, and perhaps someone will move it there. Anyway, you say "recurrence is an attempt to represent a rational number that is not completely divisible within the finite precision of the base of the decimal positional numeral system in which it is written." That's like saying that complex numbers are an attempt to represent two dimensions. They're useful for that, but they're a construction that we can get from first principles and use however we want. After all, there are many other ways to represent a rational number; in fact, in most places you wouldn't see the "..." convention; you'd just see a fraction and/or round off at a point where people could surmise the precise value. And of course it's wrong that "the next number after 0.333 is 0.334." Unlike whole numbers, reals (and rationals, for that matter), have no "next number." 0.3335 is between the two you gave, as are uncountably many others. You seem to understand the basics of set theory, but this isn't set theory; it's arithmetic (and, I'm afraid, not the basics).

In any event, please stop vandalizing various Wikipedia articles with your misconceptions about mathematics. They violate policy, and, even if they make sense to you, I'm not sure they make sense to anyone else; with apologies to Tolstoy, everyone who understands that 0.999... is 1 understands the same thing, but everyone who does not misunderstands it in his or her own way. The purpose of Wikipedia is to help people learn, not to confuse them. Calbaer (talk) 16:25, 24 June 2017 (UTC)

@Abunyip: The property that there is no "next" rational number after any given rational number applies to any ordered field, and there is nothing wrong with it whatsoever. And this does not violate injectivity in any way, because for every number with finite decimal expansion, there is one and only one other way to write it, namely decrementing the last digit and appending an infinite string of 9's afterwards. You will then agree with me that there are countably many decimal expansions ending in .999999999999999999... (as they are a subset of the rationals, which are a countable set), and the union of countable sets is countable. The set of rational numbers is countable and, importantly, that does include rational numbers with repeating decimal expansion. Hence, while the rational numbers are a dense subset of the reals, they are also a meager subset and almost all real numbers are irrational.

By the way, regarding "then I suggest you find yourself a good psychiatrist": your comments from this point forward are at the minimum bordering on WP:NPA, please refrain from remarks of that sort (another editor's mental state is none of your business).--Jasper Deng (talk) 17:17, 24 June 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)

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Definition

It seems to me that one reason people tend to argue about the subject of this article 0.999... is the lack of a definition of the string "0.999..." Most people feel like it refers to a number, behaving in some ways as a familiar schoolboy type decimal expression, but are unable to define what real number actually is referred to by this sequence of glyphs on the page. There is a confusion here between being able to calculate something manually, using marks on a page, and identifying the thing itself as belonging to the real numbers.

I note that this misapprehension is apparently present in the article as well, since at no point is the sequence of symbols "0.999..." actually defined. Instead, the reader is referred elsewhere to the article on decimal expansion. I feel that the article should make more of an effort to indicate from very early on that a sequence of symbols $h_{0}.h_{1}h_{2}h_{3}\cdots$ is merely a special notation for referring to the sum of the infinite series $\sum _{n=0}^{\infty }h_{n}10^{-n}$, a limit by definition.

This fallacy in particular seems to be at the heart of the algebraic proofs. For this reason I feel that those "proofs" should be presented with greater scepticism then they are currently afforded. Perhaps these "proofs" should merely be presented to motivate the much more satisfactory analytic treatment.

In any case, I think that the definition of the real number denoted by the string of symbols "0.999..." should be presented with significantly more fanfare. Sławomir Biały (talk) 15:36, 19 July 2017 (UTC)

In the article, the string "0.999..." is defined in the first equality of the sequence of equalities in section 0.999...#Infinite series and sequences:

$0.999\ldots =\lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}=\lim _{n\to \infty }\sum _{k=1}^{n}{\frac {9}{10^{k}}}=\lim _{n\to \infty }\left(1-{\frac {1}{10^{n}}}\right)=1-\lim _{n\to \infty }{\frac {1}{10^{n}}}=1\,-\,0=1.$

The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven.

Perhaps we should emphasise this with equality overstrikes, and move that entire part to way up in the article:

$0.999\ldots \ {\overset {\underset {\mathrm {def} }{}}{=}}\ \lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \lim _{n\to \infty }\sum _{k=1}^{n}{\frac {9}{10^{k}}}\ {\overset {\underset {\mathrm {induction} }{}}{=}}\ \lim _{n\to \infty }\left(1-{\frac {1}{10^{n}}}\right)\ {\overset {\underset {\mathrm {limit\ property} }{}}{=}}\ 1-\lim _{n\to \infty }{\frac {1}{10^{n}}}\ {\overset {\underset {\mathrm {standard\ limit} }{}}{=}}\ 1\,-\,0\ {\overset {\underset {\mathrm {algebra} }{}}{=}}\ 1.\,$

or simply:

$0.999\ldots \ {\overset {\underset {\mathrm {def} }{}}{=}}\ \lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \lim _{n\to \infty }\sum _{k=1}^{n}{\frac {9}{10^{k}}}\ =\lim _{n\to \infty }\left(1-{\frac {1}{10^{n}}}\right)=1-\lim _{n\to \infty }{\frac {1}{10^{n}}}=1\,-\,0=1.$

This would show what is actually defined; and what is provable. - DVdm (talk) 15:52, 19 July 2017 (UTC)

Yes, I agree with this suggestion.

Something continues to bother me about the algebraic proofs. What they are intended to show is: To accept that $0.999...\not =1$, one must also accept the less palatable $0.111..\not =1/9$. But this is under the implicit assumption that the naive elementary school manipulation of digit expressions (multiplication by nine in this case) continues to work as stated for infinite expressions.

However, it seems possible that this argument might have the unintended effect of convincing one not that $0.999...=1$, but rather that multiplication by nine is not actually permitted for infinite expressions like $0.111...$.

For the schoolboy, the identity $1/9=0.111...$ is not the equality of two "numbers" (!, since the school child does not yet have a satisfactory concept of "number"), but rather refers to the outcome of a process of long division: "one divided by nine produces a decimal expansion of all 1's". The meaning of "=" in this scenario is also contextual, and indeed this multiplication by nine is not actually permitted, because the decimal obtained from $9/9$ by long division is $1.000...$ rather than $0.999...$.

And indeed here is where it seems that most of the confusion arises: the contention that we can multiply in the obvious way actually does work, as we know, provided all infinite expressions are interpreted in the conventional way as real numbers (which requires the use of limits). But we cannot multiply the schoolboy's "numbers" in this way, because the meaning of the sign "=" has changed (it is read as: the outcome of this computation is this expression). Sławomir Biały (talk) 17:24, 19 July 2017 (UTC)

I don’t see the problem with the article as it is. It starts with the a definition of 0.999... that is the best for a general audience, that it is a repeating decimal, and the first section builds on that, treating it as just another fraction like 0.111..., then as a number. It only moves on to more formal definitions after that. And that is the correct order, as the article should be as much as possible accessible to a general audience. This is especially important for an article like this, one of the few mathematical featured articles.--JohnBlackburne^words_deeds 20:22, 19 July 2017 (UTC)

Yes, that's a fair point. And indeed there is sufficient emphasis on the lack of rigor in the section 0.999...#Algebraic proofs and again in 0.999...#Discussion. Meanwhile I have added ([1]) the two "def"-overstrikes in that sequence of equalities—I think that will be sufficient. - DVdm (talk) 20:55, 19 July 2017 (UTC)

My problem, I suppose, is that it does not start out with "a definition of 0.999...". The actual definition of the real number represented by the sequence of digits 0.999... is not given until the eleventh paragraph of the article, in the section on "Analytic proofs", in not a very auspicious location, after a set of misleading non-proofs using repeating decimals. The definition should be in the first or second paragraph of the article. The subject of this article, in fact, is a standard cautionary example against the naive view of numbers as decimals, and it should not reinforce this idea by opening with a set of deeply misleading proofs. We should not tell lies to children. Incidentally, I would prefer the leading algebraic "proof" to conclude what it actually shows, namely that either $0.999...=1$ or $9\times 0.111...\not =0.999...$. Sławomir Biały (talk) 21:00, 19 July 2017 (UTC)

The article is structured from a pedagogical point of view. That's why it starts with what it concedes is simple and not mathematically rigorous because that actually convinces many people who don't grok it immediately. The reference to limits is deliberately further down because any mention of limits confuses people who mistakenly think that a limit is close to rather than exactly a number. The format of the mathematical article always has the simple up the top and the advanced down the bottom. While the reverse might be more logical, it allows the reader to stop at their level of expertise and interest, providing a better article for everyone. Hawkeye7 (talk) 21:32, 19 July 2017 (UTC)

So what is 0.999...? It is apparently not a real number. What is the article actually about? Sławomir Biały (talk) 22:02, 19 July 2017 (UTC)

It is a real number. It is called "one". Hawkeye7 (talk) 00:22, 20 July 2017 (UTC)

This is the second time you have expressed the belief that "0.999..." is a real number that by definition is equal to one. If that is so, why does the article bother presenting not one but five different "proofs"? If 0.999...=1 is your definition of the sequence of symbols 0.999..., then there is nothing to prove.

I also question this edit. By definition the number 0.999... is a limit. Surely this is at the heart of the matter of an article whose subject is that number. Sławomir Biały (talk) 00:32, 20 July 2017 (UTC)

You have three editors disagreeing with you. Seek consensus for your changes. Hawkeye7 (talk) 01:02, 20 July 2017 (UTC)

Err... what? I have you writing 'It is a real number. It is called "one".' That's not disagreement, it's not even wrong. You've made similar such pronouncements elsewhere in the discussion archives, being corrected on this point by User:Trovatore in 2016, and pressed on the matter by myself, twice now (once in the archives, and once here, which you also failed to respond to).

If you want to discuss things substantively, you are welcome to do so. But what you're doing here is obstructionist, and arguably trolling. As to the other changes, no one claimed that the algebra proofs are proofs. They are not. In fact, they perpetuate the very misconceptions about the real numbers that the example 0.999..=1 is supposed to dispel. Sławomir Biały (talk) 01:06, 20 July 2017 (UTC)

I'm not sure quite what comment of mine Sławomir is referring to here, but that may be because I don't really understand what the dispute is all about. In isolation, I would not disagree with the claim that 0.999... is a real number, and that it is called "one". But I might disagree with some argument that included the claim, if, again, I understood the dispute in the first place.

There seem to be at least a couple of possible levels of use–mention confusion, or Hesperus is Phosphorus-type paradoxes. We all agree that 0.999... equals one. By the principle of substitution, you could claim therefore that this article should be entitled "one", but of course in that case its current content would make no sense, so we have to find more careful ways of expressing what exactly we're talking about, at least for this meta-level discussion.

I would say there are at least two levels of denotation here. The literal eight-byte string 0.999... is a symbol for the infinitely long numeral consisting of a zero, a decimal point, and then infinitely many nines. That latter string, in turn, denotes the real number 1.

Then you can ask why we give it that interpretation. I think Sławomir's statement that the interpretation is defined to be a specific limit is ... possibly a little too specific. That's a very natural, direct way of specifying the interpretation, but not necessarily the only one. What the reader needs to be convinced of is that it (or any other way of specifying the interpretation) does not yield an arbitrary interpretation, and therein lies the difficulty. --Trovatore (talk) 03:42, 20 July 2017 (UTC)

See this. Bubba73 ^{You talkin' to me?} 04:14, 20 July 2017 (UTC)

So I went ahead and watched that, and having watched it, I'm not sure what point you're trying to make as regards the current discussion. --Trovatore (talk) 04:33, 20 July 2017 (UTC)

It shows why 0.999... = 1. Bubba73 ^{You talkin' to me?} 05:06, 20 July 2017 (UTC)

But no one is arguing that point. The question under discussion is how to convince the reader. --Trovatore (talk) 05:15, 20 July 2017 (UTC) Or, I should say, at least I think that's the question under discussion. As I mentioned, I'm not entirely sure I understand the dispute, so maybe I shouldn't be too confident in saying what it's about. --Trovatore (talk) 05:16, 20 July 2017 (UTC)

Perhaps we should ask the following question: Is there anyone who reads this article, comes away confused/disbelieving, but then — after rereading or online/offline discussion — "gets it"? If so, there's room for improvement; if not, maybe not. People who will never understand this aren't the audience here.

As for representation versus represented, I'm not sure that semantic difference is tripping anyone up but those arguing over the semantic difference itself. Making too big a deal of it initially might confuse more people than it helps. Calbaer (talk) 05:33, 20 July 2017 (UTC)

@Trovatore: Sorry, I just looked at part of it and thought that it was yet another arguement about it. Bubba73 ^{You talkin' to me?} 05:40, 20 July 2017 (UTC)

Sorry, I just don't get how to indent this. I'm here to balance the statement by Hawkeye7 of "Sławomir Biały having three editors disagreeing with him" by explicitly supporting Sławomir Biały's view on this topic and contesting the opinion that pedagocical reasons could ever justify calling mathematical rubbish a proof. This is not to say that I oppose to depicting heuristics as sculpting reasons to select this and not that definition. So I see the indestructible desire to have infinitesimal small numbers as the source for axiomatizing hyperreals or similar, which turned out to be less useful than the standard reals in average math. Evidently, multiplying through even infinitely long strings is very seductive to beginners, and so requires also a very intense caveat. Generally, math education suffers from perceived, but bad head starts, imho. Purgy (talk) 08:23, 20 July 2017 (UTC)

Well, you can formally add and multiply infinitely long decimal strings. For any n, the n^th digit of the sum/product depends on only finitely many digits of the addends/multiplicands, so you can define the sum/product by saying, for each n, the n^th digit is the eventual value.

If you then identify strings that end in ...999... with different strings that end in ...000..., meaning you take the quotient by the obvious equivalence relation, you wind up with a structure that is isomorphic to the reals.

So that is one way of defining the real numbers, and using that methodology, it's actually true that 0.999... is equal to 1.000... by definition.

It's not the standard construction of the real numbers, and not for my taste a very good one (its biggest flaw is its apparent radix-dependence; it's true but not obvious that you get the same structure if you use a different base). But it is a construction of the real numbers, and to me it makes it problematic to claim that the denotation of an infinite string is defined specifically as a limit. --Trovatore (talk) 08:45, 20 July 2017 (UTC)

@Trovatore, I just do not want to miss to reply to your comment. I do know about the formal introduction of reals via decimals (or in other bases: 2 being far less clerical in treating the ripple), but I am convinced that hiding deep difficulties like suprema or the inherently(?) necessary equivalence classes for the sake of lying to children makes things worse.

I am not convinced that those nitpicks are necessary in the first line, but I'd rather confess to the readers that there's more difficulty than meets the eye, than present those numberphile wisdoms, easily going viral, but just detracting from any of the possible rigorous views.

For the time being I do not object to the suggestion by DVdm above, and I share the reservations of Sławomir Biały, but I am not d'accord with Calbaer and Hawkeye7. I am afraid they satisfy the property of not fully discriminating decimals from numbers. Purgy (talk) 09:26, 21 July 2017 (UTC)

To me, this is the problem with the algebraic "proofs". Ultimately, they wind up begging the question by simply defining 0.999... to be 1, at least at some level. But no one is going to be convinced that it's something true of the "real" numbers if it's simply true by fiat. And I think the algebraic proofs have cunningly concealed this in a fallacy, which is another reason the more clever readers continue to fail to be convinced by our article. Sławomir Biały (talk) 09:38, 20 July 2017 (UTC)

I don't see how they "define 0.999... to be 1." They merely illustrate in an intuitive fashion - without full rigor - why the two terms represent the same number. When people on this talk page say that 0.999... is 1 "by definition," they just mean that the definition of repeated decimals has the logical result that they're the same, not that they're taking 0.999... and "defining" it as 1. Again, I ask, is anyone put off by this who might otherwise "get it"? I'm not sure if there's a way to convince people who are "clever" enough to see that the intuitive demonstrations aren't rigorous but ignorant enough to not be able to follow any rigorous proofs. Calbaer (talk) 14:31, 20 July 2017 (UTC)

Actually, they do beg the question. The recipe for getting a rational number from a repeating decimal does have 0.999..=1 as a rule. To make sense of a repeating decimal as a number (that is as an object in its own right) require the use of some properties of the real number system that are not present in naive arithmetic. Indeed, the equation may not be true in non-archimedean fields. So there is a fallacy that needs exposing. Sławomir Biały (talk) 15:07, 20 July 2017 (UTC)

You seem to be confusing "definitions" with outcomes that result from those definitions and the axioms of mathematics. That's the type of confusion we'd like to avoid in the article. Calbaer (talk) 15:18, 20 July 2017 (UTC)

What "axiom of mathematics" is being used when we write $1/9=0.111...$ or $9\times 0.111...=0.999...$? Students without a knowledge of calculus do not have the axioms of the real numbers at their disposal. That's the whole problem with the supposed "proofs". Before a student has a concept of a real number, the very concept of a repeating decimal as a number-object is contingent upon the rules of conversion to a rational normal form. And one has, as an axiom, that 0.999...=1. It didn't have to be this way, as non-archimedean arithmetic shows. Sławomir Biały (talk) 15:32, 20 July 2017 (UTC)

Because of that, I don't think the skepticism to this article inherently reflects any flaws, but instead the tendency of the very subject to attract skeptics and cranks. It was, after all, a featured article. The changes Sławomir Biały keeps attempting to make to the article result in a more confused article that would never get that distinction. They might make it seem clearer to one person, but I doubt many more would agree with that. This isn't anything I especially have for the text as it is or against Sławomir, but when I looked at the article - unaware of Sławomir's unilateral changes - I thought, "Wow, this is pretty bad. What gives?" What gives is an editor who's changing an article contrary to any consensus of an ongoing discussion. Calbaer (talk) 14:43, 20 July 2017 (UTC)

The current article does not actually say what is meant by the notation "0.999...", but it refers to the fact that it is a theorem that this notation is equal to the number one. Is this really an acceptable state of affairs?? Sławomir Biały (talk) 15:07, 20 July 2017 (UTC)

I agree with your removal of the additions. Although well meant they seem to be trying to solve a problem that is not there, and doing so in a way that was stylistically rather jarring with the introductory text there. The article already addresses this appropriately in my view, starting off with less formal and more elementary definitions and proofs, before moving on to consider it with more mathematical rigour and formality. That is the best approach for mathematical articles, wherever possible, and is especially appropriate here, in a featured article on a topic of wide and general interest.--JohnBlackburne^words_deeds 15:11, 20 July 2017 (UTC)

The added clarification much better summarizes the sources, which approach the supposed "algebraic proofs" with far more scepticism than the current article. As currently written, the article makes it seem as though these proofs merely suffer from lack of rigor. But in fact they actually rely on the same problematic fallacy that makes the subject of this article so difficult.

The current revision of the article commits the act of telling lies to children. I hope we can arrive at a consensus that, although this revision may not be easier to read, it does a much better job of explaining the central issue of why the properties of the real numbers are essential to understanding the subject of the article, and is fully supported by direct quotes from sources. If so, I motion that the revision should be restored under the WP:NPOV policy. The current article assigns undue weight to the class that the algebraic proofs are actual proofs, and fails to summarize appropriately the central issues of the topic. Indeed, the proofs have been cribbed from the literature either without mentioning the corresponding take-away lessons from those sources, or in some cases relegating them to footnotes (possibly without understanding, given that they have been pared down to the point of meaninglessness, as well as given the nature of objections here to making the text more policy-compliant). If there are no policy based objections, I will be restoring the content. (So far, I count one objection from an editor who apparently doesn't understand what is being talked about or the subject if the article, one editor who apparent objects on the non-policy reason "What gives is an editor...", one based on the belief that ease of reading apparently trumps neutrally presenting reliable sources and defining the subject of the article.)

In fact, as the footnote says: William Byers suggests that a student who agrees that 0.999… = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation (Byers pp. 39–41). We have here done exactly what Byers warns us against: we have presented the proofs without any effort to resolve the ambiguity, and so conveyed a non-understanding of the equation. Worse than that, we have packaged this non-understanding as a "pedagogical" device, to make readers feel like they understand, and presented the algebraic arguments as if they are merely lacking one or two formal details. This is wrong. Period. Sławomir Biały (talk) 17:27, 20 July 2017 (UTC)

Summary

So, just to summarize: the present article does not actually define the subject, and it presents examples of "proofs" from the literature, intending them to be "pedagogically" convincing, when the literature explicitly presents them as fallacious, in gross violation of the neutral point of view and original research policies. Sławomir Biały (talk) 16:18, 20 July 2017 (UTC)

No. It actually defines the subject in the very first sentence: 0.999... "denotes a real number that can be shown to be the number one." Now there may be subtle philosophical differences between defining 0.999... to be 1 or defining it to be something that can be shown to be 1, but it ends up being 1 either way, and the article says so right away. I also disagree that the indicated revision of the article is inherently better than the current one. Our audience, especially for this topic, is greater than people doing math at calculus level. Requiring that level of understanding from readers does them a disservice. Instead, we correctly note that the algebraic proofs are not fully formal, and provide the more formal proofs later on. Huon (talk) 23:49, 20 July 2017 (UTC)

Firstly, no, the subject is not actually defined. We say that the notation "0.999... denotes a real number that can be shown to be the one." We do not actually define what that real number is. I am very alarmed at the presence of so many editors on this discussion page who do not seem to think this is a problem, and who have either expressed a belief that the notation "0.999..." intrinsically refers to 1, or otherwise seem to think that the details of what the notation "0.999..." actually means are irrelevant to the article on 0.999..., or ultimately want to minimize the meaning of the expression "0.999..." because it's equal to one anyway, regardless of why that happens to be the case. Furthermore, I reject that a reader will hope to understand the equation "0.999... = 1" based on the current text: indeed, we have sources that say precisely this.

Secondly, it is the real number system, and in particular its completeness, that is essential to understanding the subject of the article. In fact, we already note this in a somewhat muddled and confused footnote, that a student who accepts the algebraic justifications, but still has not resolved the difference between the potential infinity of an infinite process and the actual infinity of the completeness axiom, does not understand the equation at all. This is what the source says, on which the "Algebraic proof" section is largely based, and therefore which has significant WP:WEIGHT in how we should present things in the article. But every effort instead appears to have been made to minimize the role of the real number system and especially the completeness axiom, even thought this is obviously against black-letter non-negotiable Wikipedia policies, because it is thought better to spare our poor readers' feelings. I am astonished to see so many experienced Wikipedians, and one who usually knows what they are talking about mathematically, express this wrong belief.

Finally, I continue to await actual policy-based rationale that justifies the minimization of the properties of the real number system (despite obvious WP:WEIGHT in sources), and the failure of the article to define the subject until the eleventh paragraph (after the undefined thing denoted by "0.999..." has been "proven" to be equal to one in two different ways (!)). So far, sparing our readers' feelings has been presented by several different editors, but we are under no policy obligation to place our readers' feelings above the neutral point of view policy. Indeed, WP:NOTCENSORED. Sławomir Biały (talk) 01:53, 21 July 2017 (UTC)

As far as the question of the first sentence goes, I agree with the not-Sławomir Biały consensus. Articles about mathematical objects need not contain a complete, rigorous definition of the object in their first sentence (or even necessarily in their introductory section). In the case of this particular article, that level of detail makes it worse, not better. The current first sentence conveys the essential facts at a broadly accessible level of technicality, and that's good. --JBL (talk) 03:47, 21 July 2017 (UTC)

For the record, here's the current article versus the version when it was first deemed "featured": https://en.wikipedia.org/w/index.php?title=0.999...&diff=791499454&oldid=80638011

And here's the current article versus the version when its featured status was reviewed: https://en.wikipedia.org/w/index.php?title=0.999...&diff=791499454&oldid=382076790

It might be fruitful to see whether there's anything in the older versions of the article we think should be restored in the interest of clarity, rather than going back and forth on a single editor's suggested changes. And, of course, clarity should be valued over and above any particular ordering of this information. Calbaer (talk) 04:35, 21 July 2017 (UTC)

"going back and forth on a single editor's suggested changes": This is the second time that you have made this about the editor rather than the changes. Clearly not including the information is not an option under policy. This therefore justifies making it better by editing, rather than by reverting. I will restore the information, after the first sentence. Sławomir Biały (talk) 11:00, 21 July 2017 (UTC)

Here. Feel free to improve. But note that these edits were made in order to comply with policy. That has so far not been challenged by a single editor in this discussion. Sławomir Biały (talk) 11:08, 21 July 2017 (UTC)

I have not followed this discussion.Thus I'll not comment the various opinions, and will focus only on the comparison between the two disputed versions of the article. I agree with JBL that the present version of the lead is better than Sławomir Biały one, and for the same reasons. However, the present version of section "Discussion" is mathematically wrong, and, IMO, must be replaced by Sławomir Biały version. Here are the main issues:

• Although these proofs demonstrate that 0.999… = 1: The discussion consists essentially of explaining that these proofs are not really proofs, and that they "demonstrate" nothing.
• The extent to which they explain the equation depends on the audience: A proof never explains anything, it proves (if it is correct), or it is not a proof. Moreover the correctness of a proof cannot depend on the audience.

Because of these issues, the remaindier of the section is highly misleading: it tries to explain common misunderstanding by introducing confusion about proofs and mathematical correctness. It is exactly the contrary of what has to be done; we must, here, explain that mathematical correctness may be, sometimes, counter-intuitive, and explain also why this occurs here. For these reasons, I suggest to restore Sławomir Biały version of the section, and to rename section "Algebraic proofs" as "Algebraic explanations". D.Lazard (talk) 13:30, 21 July 2017 (UTC)

Thank you for the comment. I am curious what you think now of the current placement of the definition of the real number $0.999\dots$ in the third paragraph of the lead. My feeling is that the old version of the lead already mentioned infinitesimals, which seems like undue weight if we are not also permitted to include the mainstream view. If this is not suitable, my objection still remains that a satisfactory definition of the subject of the article is not actually given suitable prominence in the article, if it appears at all. Sławomir Biały (talk) 14:07, 21 July 2017 (UTC)

Modification of the article page is not the best way of resolving disagreements. No editor has supported all of Sławomir's proposed alterations (though many, such as D.Lazard, have sympathy with Sławomir's arguments, see potential in some changes, and - if Sławomir had the patience to do so - might work to come up with improvements; the "equals def" change is a start). It's true that no one has rebutted Sławomir point by point, likely because no one wants to invest time in a Gish gallop with someone who's ignoring everyone else anyway, just modifying the article unilaterally and making it far less readable. It also might be that it's not at all clear how the policies cited have anything to do with the article in the first place. What does WP:WEIGHT and WP:NPOV have to do with using intuitive explanations before formal proofs? Those policies are about making sure that viewpoints are properly represented, but method of explanation is not viewpoint, so the policy is not applicable to what is being discussed here. The concern of the other editors is not about "sparing people's feelings." And Sławomir's concern should not be "representation" of any "viewpoint." Instead, the overall concern should be explaining the matter without sacrificing either accuracy and comprehensibility.

Sławomir, take a look at the featured versions of the article. If you have the same objections to them, then I'd hope you'd be convinced that your position is an aberrant one not just among editors who've looked at this page in the past week, but in general, including among those who review for featured articles.

If you do not have the same objections to the featured versions, then perhaps you should restore content from the featured versions, rather than adding your own over the objections of others. Even better would be proposing such changes on the talk page, since - at this point - some editors might come to the conclusion that all your additions need to be reverted, that being the pattern so far. Yes, this should be about the content rather than the editor, but an editor who stubbornly, knowingly, and repeatedly introduces anti-consensus content requiring reversion has already made the issue about himself or herself rather than the content of the article.

Otherwise, we can just discuss things point by point. For example, the objection over the validity of "algebraic proofs": Perhaps we should just avoid the word "proofs" rather than adding a bunch of explanations and other apologia? For example, we could say, "Showing via algebra: Algebra can be used to show that 0.999… represents the number 1, using concepts such as fractions, long division, and digit manipulation to build transformations preserving equality from 0.999… to 1. However, these intuitive explanations are not rigorous proofs as they do not include a careful analytic definition of 0.999…." Sławomir's current version seems a lot more awkward (using scare quotes, warning for the need for "sophistication") than such a minor change would be. Calbaer (talk) 14:15, 21 July 2017 (UTC)

I have simply made the article compliant with the neutral point of view policy, and brought the text of the "Algebraic proofs" section into line with the sources that are actually cited there. Perhaps editors mistakenly believe that the subject of the article can be understood through elementary algebra alone, without any knowledge of the real number system, and this is why those who I rate as non-mathematicians (User:Calbaer, User:Huon, and User:Hawkeye7) do not apparently realize that presenting the subject as if it were something that could be understood independently of the real number system actually strongly violates the neutral point of view policy. In particular, such editors have expressed at various points of view obvious falsities like that 0.999... equals one by definition, that the decimal manipulations in the algebraic proof are somehow direct consequences of "the axioms of mathematics". Such ill-informed arguments are easily demolished, and carry no weight whatsoever.

Fortunately, the opinions of ill-informed editors are actually irrelevant in this matter: Wikipedia is based first and foremost on sources, and the sources that we have cited for this content are absolutely crystal clear. Peressini and Peressini (p. 186) indicate that the supposed proof "offer[s] nothing to explain why this inequality holds. Such an explanation would probably involve something considerably more, e.g., explaining the distinction between the rational numbers themselves and a decimal representation of them, how the decimal representation is related to a (potentially) infinite series, and also the Cauchy-Weierstrass property (or an equivalent one)" (Peressini and Peressini, p. 186). Byers (p. 41), discusses the distinction between process and object at great length in the context of these arguments, and in particular concludes with: "That is, understanding involves the realization that there is 'one single idea' that can be expressed as 1 or as .999..., that can be understood as the process of summing an infinite series or an endless process of successive approximation as well as a concrete object, a number."

The arguments against policy enforcement are actually very weak. These are, essentially, that the article should be made accessible to all readers, with the implied subtext even if it makes the article wrong or inadvertently misleads readers into thinking the subject is an algebraic triviality instead of a highly non-trivial property of the real number system. That's not right. Since no one seems willing or able to address the actual policy concerns, it would be inappropriate to revert the edit. The neutral point of view noticeboard is available for anyone wishing to bring in outside input.

I note, finally, that the latest argument is an appeal to past consensus. I have examined the three featured article reviews, and found no attempt to address the neutrality of presenting the subject of the article as divorced from the properties of the real number system. Instead, the FAR process appeared to be more concerned with accessibility than with neutrality and accuracy, perhaps because most or all of the featured article reviewers are totally unaware of the problem. So, since WP:FACR#1d was never adequately addressed, I'm afraid that appeal to the previous featured article reviews cannot override the present policy enforcement. Sławomir Biały (talk) 15:33, 21 July 2017 (UTC)rs is not about "sparing people's feelings." And Sławomir's concern

I agree with Calbaer. There is obvious room for consensus-based editing here, but it is probably not compatible with widespread unilateral changes. --JBL (talk) 15:37, 21 July 2017 (UTC)

On to the specifics: "Showing via algebra: Algebra can be used to show that 0.999… represents the number 1, using concepts such as fractions, long division, and digit manipulation to build transformations preserving equality from 0.999… to 1. However, these intuitive explanations are not rigorous proofs as they do not include a careful analytic definition of 0.999…." This is actually wrong. Algebra cannot be used to show that the notation "0.999…" represents the number 1, because the existence of that representation requires the completeness property. This is in fact why it is necessary to point this out, referring to the greater mathematical sophistication that you would like to banish from the article. Sławomir Biały (talk) 15:33, 21 July 2017 (UTC)

You are twisting people's words (on "sophistication"), inventing quotes out of whole cloth (on "definition"), engaging in multiple ad hominem attacks (e.g., about whom you "rate as non-mathematicians" and who's "ill-informed"), making various assumptions about motives and presentation (multiple in the "non-mathematicians" statement alone), and not adequately explaining how the policies you cite support the things you say they support. And unilaterally making changes most editors are asking you not to make. And attacking alternative suggestions. (I made suggestions in order to suggest a middle ground that would both address your concerns and not degrade readability or accuracy. Feel free to reject them, but don't make false personal attacks about "sophistication that [I] would like to banish.") Those are not consensus-building actions. If you feel the article should change, I believe you'll be disappointed so long as you retain the tactics we've seen from you so far. The patience of others is a limited resource. Calbaer (talk) 16:34, 21 July 2017 (UTC)

You've still not addressed the NPOV rationale, Calbaer. I have no idea what "inventing quotes out of whole cloth" means. This latest post seems to be increasingly divorced from my concerns with the article. But in any case, let me summarize some of the views that editors have expressed here, that I find very worrying:

1. Your rational in defence of the status quo "Algebraic proofs" section is that they "explain... the matter." However, this is directly and explicitly refuted by the sources we cite in that section.

2. Here you baldly suggest that the neutral point of view policy is "not applicable to what is being discussed here". Presenting arguments that sources present explicitly as fallacious arguments as if they were proofs is not neutral. And the neutral point of view affects presentation as well, including prominence of placement, and faithfully discussing the context of sources. In particular, the status quo revision fails both.

3. A number of views have been expressed by editors on this page, suggesting a failure to understand the subject and the sources. For example, here you apparently expressed a belief that the "Algebraic proofs" section follows from the "axioms of mathematics". That is false. Here an editor apparently dismisses the difference between the definition of the number 0.999... (which is the subject of this article) and the number 1 as a "philosophical difference", when in fact it is clearly at the heart of the matter. An editor here who seems to believe that 0.999... is just "called" one, as if by fiat. These posts reflect a grave failure to grasp the subject of the article.

I remain very concerned at what I see as a systematic attempt here to downplay the role of the real number system in the equality $0.999...=1$. This is too a grave failure to adhere to the neutral point of view. Sławomir Biały (talk) 17:08, 21 July 2017 (UTC)

Request for comment: Which version neutrally summarizes the cited sources with appropriate weight?

For the discussion section of the "Algebraic proofs" given in the article, which of the following pieces of text more accurately reflects the opinions expressed by the cited sources, and represents established scholarship with appropriately due weight: Sławomir Biały (talk) 16:20, 21 July 2017 (UTC)

Version A:

Although these proofs demonstrate that 0.999… = 1, the extent to which they explain the equation depends on the audience. In introductory arithmetic, such proofs help explain why 0.999… = 1 but 0.333… < 0.34. In introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all. Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999… and 1.000… both represent the same real number; it is built into the definition. This is done below.

Version B:

Although these arguments demonstrate that 0.999… = 1, they fall short of being valid mathematical proofs. Furthermore, these arguments fail to supply a satisfactory explanation of why the equation should hold. Peressini and Peressini (p. 186) note that such an explanation involves considerably more sophistication, such as the distinction between numbers and their decimal representations, the relation between potential infinity and completed infinity, and the Cauchy completeness property.

For someone with no knowledge of the detailed properties of the real number system, a plausible reading of the first equation in the first proof $1/9=0.111\ldots$ is "the division of one into nine leaves one-tenth, with a remainder leaving one-hundredth, and a remainder leaving one-thousandth, and so forth". Based on this reading, the equation is not an equality of numbers, but reporting the result of a computation that can be carried out indefinitely: what Byers (p. 40) identifies as a process rather than an object. In order to make sense of $1/9=0.111\ldots$ as an equation of numbers, it is necessary to have a conception of the decimal $0.111\ldots$ itself as an object rather than a process.

According to Fred Richman (p. 396), the first argument "gets its force from the fact that most people have been indoctrinated to accept the first equation [$1/9=0.111\ldots$] without thinking." However, as Byers notes, for someone without knowledge of the real number system, the number $0.999\ldots$ may make sense only as process rather than an object, and so the equation $0.999\ldots =1$ is difficult to resolve, because it appears to be a category error: one cannot have a process (a verb) equal to an object (a noun). He suggests that a student who agrees that 0.999… = 1 because of the above proofs, but hasn't resolved this ambiguity, doesn't really understand the equation (Byers pp. 39–41).

The completeness axiom of the real number system is what allows infinite decimals like $0.111\ldots$ and $0.999\ldots$ to be regarded as objects (real numbers) in their own right, independently of their realization as common fractions. Once the real number system has been formally defined, its properties can be used to establish the decimal representation of real numbers. The properties of the decimal representation can then be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999… and 1.000… both represent the same real number. This is done below.

References

• Version B. The cited sources do not present the algebraic arguments as convincing demonstrations (as evidenced, for example, by Byers placing scarequotes around "proof"). On the contrary, in the Byers source, these arguments are presented as convincing but fallacious arguments to illustrate that students can become convinced of the identity of 0.999... and 1 without understanding that identity, and he goes to great lengths to distinguish between process and object. This context therefore carries a significant caveat that is completely lost in version A. Byers' view is suppressed, being relegated to a meaningless tweet in a footnote. Furthermore, version A also fails to capture the full context of the view of Peressini and Peressini, who say "Such an explanation would probably involve considerably more, e.g., explaining the distinction between rational numbers themselves and a decimal representation of them, how the decimal representation is related too a (potentially) infinite series, and also the Cauchy-Weierstrass property (or an equivalent one)." They do add that "This simple proof may actually, in certain less obvious contexts, have explanatory power", but these "less obvious contexts" (such as why $0.33...<0.34$) are unexplained and have little bearing on the subject of this article. Finally, it is also a fact that significantly more is required to prove that 0.999...=1 than what has been offered in the "Algebraic proofs" section. The identity requires the completeness property of the real number system. Version A (as well as the earlier version of the lead-in to the section on the proofs) attempts to minimize this aspect of the issue, when in fact it is at the very heart of the matter. A reader could easily walk away from the article believing that the identity follows from some trivial algebra, apart from one or two finer points of rigor. This is directly undercut by the quote to Peressini and Peressini given above, and also by Byers', for instance (p.41): "understanding involves the realization that there is 'one single idea' that can be expressed as 1 or as .999..., that can be understood as the process of summing an infinite series or an endless process of successive approximation as well as a concrete object, a number." However, fruitless discussions on the talk page have lead me to the conclusion that actually some routine editors of the article and its talk page seem not to appreciate the importance of these nuances that are amply evidenced in reliable sources, and wish to place the article's accessibility ahead of the need to represent sources in accordance with the demands of due weight and accuracy. (This is evidenced in particular by User:Calbaer's remarkable assertions that "Sławomir's concern should not be 'representation' of any 'viewpoint.' Instead, the overall concern should be explaining the matter without sacrificing either accuracy and comprehensibility." and "It also might be that it's not at all clear how the [NPOV and WEIGHT] policies cited have anything to do with the article in the first place... Those policies are about making sure that viewpoints are properly represented, but method of explanation is not viewpoint, so the policy is not applicable to what is being discussed here.) Sławomir Biały (talk) 16:20, 21 July 2017 (UTC)

• • Subcomment: Proportionate representation. I see that certain editors here are entertaining the possibility of shortening the added material. I do not believe that is consistent with the WP:WEIGHT policy, which requires that we cover topics proportionate to their coverage in reliable sources. The Byers source, in particular, spends less than 10% of the text on the proof, and more than 90% of the text discussing the issues that have been condensed into several short paragraphs. Less than a quarter of the total character count in the coverage in the Peressini and Peressini is the actual proof itself. My argument is, and always has been, that we include the full context of each source that we use. I submit that to do otherwise would specifically violate the proportionate treatment aspect of the neutral point of view policy. I have no objection to editors cleaning up the treatment to make it more palatable, but the whole summary of the sources must be there if we are to include the proofs at all. Policy is absolutely crystal clear in the matter. Content doesn't get a pass if it's just for pedagogical or educational purposes. All content is subject to the neutral point of view policy, no exceptions. Sławomir Biały (talk) 18:38, 22 July 2017 (UTC)

• Version B.It would be nice if the problem could be dismissed easily but the example of Hackenbush game theoretic values shows it is not altogether straightforward. Version A would contradict the lead and the lead is correct - it is true within the standard real number system but can be false in other systems. There should be citations for the algebraic proofs. Citation 1 in the discussion is rather cryptic and should be expanded to reference the actual publications - it might help if Harvard citation templates were used as the artcle does tend to that style. Dmcq (talk) 17:03, 21 July 2017 (UTC)
• Version B. seems to reflect the situation best.Slatersteven (talk) 17:19, 21 July 2017 (UTC)
• Version B. Version A comes across to me as too facile, too sloppy about the distinction between the definition of real numbers vs the definition of decimal notation and whether there is even a single universal definition for either, too condescending, and also problematic from the point of view of egg submarines. —David Eppstein (talk) 17:44, 21 July 2017 (UTC)
• Version B, provided the section title Algebraic proofs is amended to Algebraic motivations. - DVdm (talk) 18:21, 21 July 2017 (UTC)

• Neither: What is being suggested here is replacing "Although these proofs demonstrate that 0.999… = 1" with "Although these 'proofs' purport to demonstrate that 0.999… = 1". But the lead of the article clearly says "In mathematics, the repeating decimal 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. Proofs of this equality have been formulated with varying degrees of mathematical rigor..." Version B looks like a blatant attempt to give the fringe theory that 0.999… does not equal 1" undue weight. This is not to say that version A cannot be improved -- those who have pointed out the flaws in it have a point -- but we should not "fix" the problem by supporting fringe science. --Guy Macon (talk) 19:16, 21 July 2017 (UTC)

@Guy Macon:: It is not the intention of the section to advance a fringe position. In fact, these supposed "proofs" are presented in the very sources that we cite as examples of fallacious arguments. A fallacious argument is a fallacious argument regardless of the truth value of its conclusion, and it is not fringe to point that out. The article does contain several more rigorous proofs. These still need work, because the present article fails to define clearly what is meant by the notation "0.999..." But if we are going to present flawed proofs, then it is very important that the article point out that these proofs are flawed. A failure to do so, apart from violating the neutral point of view, is just fodder for the usual bunch of fringe theorists who will attempt to undercut those supposed "proofs"; ironically, in this case, they would have a point. Perhaps you are leaning towards a third option, namely: why should the article present false or misleading proofs at all? That might be worth discussing. There seems to be something inherently dishonest about using these arguments as "explanations" of the concept, when they are explicitly denounced by reliable sources as having little explanatory value. On the other hand, mathematical explanations often benefit by having both examples and non-examples. If we present them as non-examples, then we must be upfront that this is what they are. This is the reason version B is written in the way that it is. Sławomir Biały (talk) 19:39, 21 July 2017 (UTC)

Please don't ping me. When I make a comment I watch for replies. I have no problem if the article presents false or misleading proofs, but they should be clearly labeled as such. Changing proofs to "proofs" and changing ...demonstrate that.. with ...purport to demonstrate that... does not make it clear that the argument is false or misleading.

In other words, this RfC is an example of "A is wrong. Something must be done. B is something. Therefore, B must be done" See False dilemma. --Guy Macon (talk) 19:53, 21 July 2017 (UTC)

I object strongly to the characterization that version B comes from version A by simply "changing proofs to 'proofs' and changing ...demonstrate that... with ...purport to demonstrate that...". In any case, a full context can be seen at this revision (diff), where the first paragraph of the section includes an explicit indication that they fall short of being valid mathematical demonstrations and why, and also the lead section of the article contains a definition of the subject (which remains absent from the status quo revision). Perhaps your concerns are assuaged by that revision, seen in full?

I am aware that option A/option B RfC's can often miss nuances, but I wanted to avoid going out into the weeds regarding revisions to other parts of the section. I do offer my apologies for making you decide in a binary fashion like this, if you feel that both versions have serious shortcomings. Since you seem to have identified just the first sentence of Version B as problematic, I'd like to invited you to attempt to rewrite it so that both of our concerns are satisfactorily addressed. Also, sorry for the ping. I won't do it again. Thanks, Sławomir Biały (talk) 20:11, 21 July 2017 (UTC)

Use

"Although these proofs demonstrate that 0.999… = 1, they fall short of being valid mathematical proofs."

instead of

"Although these "proofs" purport to demonstrate that 0.999… = 1, because they do not actually rely on the relationship between decimals and the numbers they represent, they fall short of being valid mathematical proofs."

The rest seems fine. --Guy Macon (talk) 21:49, 21 July 2017 (UTC)

I heartily endorse this suggestion, with one small emendation: "Although these arguments demonstrate that 0.999… = 1, they fall short of being valid mathematical proofs." Something that is not a mathematically valid proof should not be called a proof. I will make the change. I do not anticipate any objections to making this change in "Version B", so I have gone ahead and done it. Anyone may feel free to revert me if they object. Thanks, Sławomir Biały (talk) 21:57, 21 July 2017 (UTC)

• Version B is the better of the two, but this of course does not rule out the possibility that a still-better version could be written. XOR'easter (talk) 20:06, 21 July 2017 (UTC)

• Version B, for the reasons that I have given in my previous post in #Summary, and with changing the heading Algebraic proofs to Algebraic motivations. D.Lazard (talk) 20:11, 21 July 2017 (UTC)

• Version A: What Guy said. Neither is perfect, but B is verbose and the fact that a reader could come away from it thinking that it supports 0.999... not equal to 1 means that it is deceptive. Better to be retain the current material than to replace it with something that is deceptive and more difficult to read, and gets us further away from the desired state. Of course, ideally, someone would be present an alternative that is both clear and precise. But for now, Guy nailed it: classic false dilemma. I'd also add loaded question; the idea that due weight should be the only criteria in judging which text is better for the article is false. Calbaer (talk) 22:48, 21 July 2017 (UTC)

Version C: "Although these arguments demonstrate that 0.999… = 1, they are not rigorous proofs. They are useful for the sake of pedagogy, as rigorous proofs might be inaccessible to those without knowledge of higher math. However, they do not prove why the intuitive mathematical steps within them work on these repeating decimal representations[1]. As such, formal definition of the decimal representation scheme and use of real analysis are necessary for formal proofs of 0.999… = 1." Short, sweet, and to the point. Don't get ahead of yourself by introducing advanced math that we don't even use, let alone expect the reader to know. The important thing is that the initial "proofs" are not rigorous, but can be used to give intuition and think about how a formal proof might work. Calbaer (talk) 00:08, 22 July 2017 (UTC)

I am not sure what part of it you feel would lead a reader to "could come away from it thinking that it supports 0.999... not equal to 1". Not misleading the reader on the matter of whether $0.999...$ is equal to $1$ is actually very easy to accomplish. Apart from being telegraphed in the very first sentence of the article and the paragraph preceding the arguments in question, the very last paragraph of Version B says, quite explicitly: "Once the real number system has been formally defined, its properties can be used to establish the decimal representation of real numbers. The properties of the decimal representation can then be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999… and 1.000… both represent the same real number. This is done below." This is, actually, considerably more detailed in its description of how the proofs can be made correct, than version A. The only caveat is that they indeed require a detailed analysis of decimal representations. Without the added indication of what is wrong with these proofs, however, the reader is far more likely to be convinced that the proofs are actually correct without modifying in a deep way their understanding of concept of "decimal representation". This is the chief danger for readers of the section in question. Furthermore, we don't even need to be hypothetical about this being the problem. The research literature, including the sources that we cite, tell us explicitly that this is the problem with reader understanding, in an extremely detailed way. I happen to think the published, peer reviewed, assessments of what readers do and do not find confusing about the subject of the article should be given considerable weight in our assessments of what we think readers will and will not find confusing. If indeed the true aim is pedagogy, as you've repeatedly raised, then the opinions of highly qualified published experts on mathematics education should at the very least inform our own approach to the question of presentation, and certainly not be tossed out just because we think we know better than they do what will help readers understand the problem. Sławomir Biały (talk) 00:17, 22 July 2017 (UTC)

Regarding "I am not sure what part of it you feel would lead a reader to 'could come away from it thinking that it supports 0.999... not equal to 1,'" I am referring to Guy's statement that, "Version B looks like a blatant attempt to give the fringe theory that 0.999… does not equal 1 undue weight." Regarding "This is, actually, considerably more detailed in its description of how the proofs can be made correct, than version A," first of all, it doesn't describe how they can be made correct. Moreover, something that includes unnecessary details will only confuse the reader, which is a point I've repeated made. So you shouldn't argue "considerably more detailed in its description" as a point in that description's favor. I'll gladly toss out any details that fail to improve the article. Calbaer (talk) 00:44, 22 July 2017 (UTC)

Firstly, "Guy's statement" is not an answer to "what part of it you feel would lead a reader to..." (etc). In any case, Guy's objection has now been settled, and so this is a moot point entirely. Secondly, from Version B: "Once the real number system has been formally defined, its properties can be used to establish the decimal representation of real numbers. The properties of the decimal representation can then be used to justify the rules of decimal arithmetic used in the above proofs." This says exactly how the proofs can be made rigorous. Sławomir Biały (talk) 01:10, 22 July 2017 (UTC)

I strongly object to the claim that "Guy's objection has now been settled" without any attempt to ask Guy whether he considers it to be settled. I haven't even read the comment that supposedly settled this (busy with real life) and probably won't find time until Monday or Tuesday. --Guy Macon (talk) 09:25, 22 July 2017 (UTC)

Did I misunderstand your assertion "The rest seems fine"? Sławomir Biały (talk) 11:03, 22 July 2017 (UTC)

I strongly oppose to the argumentation for Version C. Either something is a (sketch of a) proof, lacking details, necessary for the not fully initiated to follow, maybe even accessible to strict formalization, or it is detrimental -and not useful- to achieving a sound mathematical education, when promulgated as mathematical reasoning (I do not want to deny occasional inspirational potential). All these numberphile-isms, even when viral, striving for acceptance by as many as possible, disregarding their mathematical competence, should be refuted within a serious encyclopedia, rather than constituting "content". The provided line of thoughts just demonstrates wishful thinking -wouldn't it be nice if it worked like this?-, but provides no fruitful intuition. Purgy (talk) 08:30, 22 July 2017 (UTC)

Also, the only reliable peer-reviewed sources that we have for these proofs even tell us that they provide no fruitful intuition. Sławomir Biały (talk) 11:12, 22 July 2017 (UTC)

What peer-reviewed sources? Both references are books (here and here), which are not peer reviewed. (How can you try to discredit other editors on an appeal-to-authority basis — "those who I rate as non-mathematicians" — and not know that?) Calbaer (talk) 15:33, 22 July 2017 (UTC)

Peressini and Peressini is published in the peer-reviewed Springer series Perspectives on Mathematical Practices. Byers is published by the academic publisher Princeton University Press, which certainly does conduct peer review. But a bigger question is, why are you trying to question obviously reliable sources, and also dismissing adherence to sources as "appeal authority"? You aren't new to Wikipedia, and should know better. In any case, citing expert sources in support of a carefully argued position is not an appeal to authority. It is the hallmark of all scholastic discouse. But I suppose you wouldn't know that, would you? Sławomir Biały (talk) 16:14, 22 July 2017 (UTC)

Actually, it can be questioned whether the authors and publishers are experts in mathematical pedagogy. They are experts in mathematics, but all the arguments can be made rigorous, and they may not be experts in which arguments are most questionable. — Arthur Rubin (talk) 18:23, 22 July 2017 (UTC)

Purgy, that's a problem with A, B, and C, not just C. All present the above material as not complete enough to constitute sufficient formal proof, not as "bad" material that the reader should avoid being fooled by. Even B's proponent has claimed that B showed "how the proofs can be made rigorous." Although I don't buy that, that indicates a desire to keep the material and present it as incomplete, not as a cautionary tale. B would just muddy the waters, trying to have it both ways. If you dislike C, that's fine; I just want something that resolves the objection to accuracy while retaining readability, criteria B fails to satisfy. Calbaer (talk) 15:33, 22 July 2017 (UTC)

• Version A: I am going out on a limb here, I know, but I have multiple concerns about Version B. First of all, we say you have to understand the relationship between compressed infinity, completed infinity and the Cauchy completeness property. None of which are heard from again. And the last-mentioned points to the article on Cauchy sequences where the completeness property is hard to find; perhaps the Construction of the real numbers would be a better place to send them? Then we mention "the completeness axiom of the real numbers". What is that? An an axiom? The reader might think that we've just produced a new concept, which we are asking to be accepted without proof. And it begs some more questions. First of all, the idea of the equality of 0.999... and 1 arising from the construction of the reals is a good one, but as we just pointed out in the preceding paragraphs, we haven't established that 0.999... is a real number yet. Secondly, the reals can be constructed in multiple ways, and we go on to use others below, so are we saying that we must use Cantor's construction, using the Cauchy sequence? If not, then doesn't that invalidate our argument? Is it true that 0.9999... = 1 under any construction of the real numbers, or just some? Hawkeye7 (talk) 00:00, 22 July 2017 (UTC)

  I am happy with the change from "proofs" to "arguments" in both versions. Hawkeye7 (talk) 00:02, 22 July 2017 (UTC)

I'm not. They have been presented as proofs not arguments even if they are not halfway rigorous proofs. And we don't need handwavey arguments here. We should say what they are which is purported or incomplete proofs. Dmcq (talk) 12:16, 22 July 2017 (UTC)

The real numbers are axiomatically defined as the unique complete ordered field up to isomorphism. There are many different ways to construct this field, but there is only one of them (at least in the standard foundations of mathematics), and the model does not affect the truth of the equality $0.999...=1$. This is another thing that (in my opinion), the present article gets wrong: it appears to hang the question of equality on the specific model of the reals. This also needs fixing. But that is a different discussion. The further objection seems to be that many unfamiliar mathematical concepts are required to understand the subject of the article, and these unfamiliar concepts seem very sophisticated. There is a reason for that: the subject of the article is a genuinely difficult thing to understand, even for students who have a thorough understanding of university calculus. For many individuals, it requires a radical restructuring of the very concept of "number". We should not present proofs that secretly rely on implicit assumptions that do not correspond to the assumptions that the target readers will have going into the proof (it would be like saying "triangle" in a proof, but really secretly meaning hyperbolic triangle). Those implicit assumptions should be made explicit, and we should use the mathematically correct vocabulary for them. Sławomir Biały (talk) 01:27, 22 July 2017 (UTC)

• Version B: Obviously, this version is amenable to improvement, too. However, it is by far mathematically better reasoned, and therefore, up to now, also prevails in the !voting by several mathematically educated editors. Attempting to achieve a fallacious understanding of a "deep" fact, not easily accesible, by pretending accessibility via simple mechanisms does not serve well unwary readers. As for the original question in the RfC, to me the obvious concerns in the sources are swept under the rug by Version A. Purgy (talk) 08:53, 22 July 2017 (UTC)

• Weak support for Version B. Once I saw Sławomir Biały involved, I expected to side with him. And to large extent I do. However Version B is simply too long, and reads as a bit axe-grindy. I think this part is very important, and should stay: The completeness axiom of the real number system is what allows infinite decimals like $0.111\ldots$ and $0.999\ldots$ to be regarded as objects (real numbers) in their own right ...establish the decimal representation of real numbers." I think the other added bits specifically critiquing algebraic proofs should be reduced. I appreciate the subtlety of Sławomir's objections to version A, but I think he's responded with overkill. What constitutes a rigorous proof is context-dependent, and it's not clear that this is the best place to dive in to a level of detail that most readers will find to be some mix of baffling and unnecessary. I would like to Support a shorter Version B. I realize now this is bad form for an RfC, but I can't support either as-is. SemanticMantis (talk) 13:36, 22 July 2017 (UTC)

That is perfectly in order. An RfC is supposed to be a request for comments - not a vote. I'd be quite happy with a slightly expanded version of the top of the section on algebraic proofs and remove the discussion altogether, the recent change by D.Lazard is a good basis for that. Version A is just wrong and jangles. The sooner the article passes over the algebraic section the better. And really a lot of the interest is just in how people try proving something without really understanding what they're working from or what they're actually doing. In fact a lot of the next section on analytic proofs is the same trying to do it without referencing the definition of real numbers. I wouldn't mind the whole 'analytic proofs' section disappearing too and just go on to proofs from the construction of the real numbers. Dmcq (talk) 17:19, 22 July 2017 (UTC)

• Version A is the better of the two. Most of the differences are wrong (irrelevant) and fringe (relevant). The completeness axiom should be emphasized more, but potential and completed infinities should not, because potential infinity is a fringe concept, at best. The difference between seeing 0.999... as an object and incorrectly seeing it as a process might be emphasized, but not using fringe terminology. — Arthur Rubin (talk) 18:23, 22 July 2017 (UTC)