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)

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)

I assume you'll dismiss my comments since you have pigeonholed me as "not a mathematician" (based on what evidence?), but talking of a systematic attempt to downplay the role of the real number system is ridiculous. I don't remember seeing you around when people argued that we should give more weight to the hyperreals, or whatever number system would allow them to make the article say that after all, 0.999... isn't equal to 1. I don't remember seeing you around when we argued that Katz&Katz shouldn't be given undue weight (in fact I saw you add it to the lead). There's megabytes worth of archives (and /Arguments archives); did you take a look at them? When you've spent a couple of years debating the infenitesimals cranks, then you can accuse others of downplaying the importance of the real number system.

You also misunderstood my comment about the philosophical difference. What I meant was that being desecribed as "a real number that can be shown to be the number 1" was, up to subtle philosophical differences, functionally equivalent to saying "it is the number 1". You're welcome to help ill-informed non-mathematician me by explaining the difference between a real number that can be shown to be 1 and 1, from a mathematical point of view. I'll add that saying "0.999... is defined to be 1" is no less correct than saying "0.999... is defined to be the least equal bound of the set {0, 0.9, 0.999, ...}"; in fact you added the second to the article but argued here that instead 0.999... should be defined as the sum of a series, which (at least the way my calculus course, long ago, introduced it) is a different concept. I also rather doubt that you can show that the scholarly consensus favours either of those two definitions over the other, or that there is a single definition that is overwhelmingly used in the literature at all. So if you want to talk about OR and NPOV, start with what you introduced. Huon (talk) 18:55, 22 July 2017 (UTC)

Yes, one could define "0.999...=1". But that's not how it's usually defined, so I don't see what your point is. We can probably easily list all of the usual definitions. One is the least upper bound of the sequence {0.9,0.99,0.999,...}. One is the sum of the series $\sum _{n=1}^{\infty }{\frac {9}{10^{n}}}$ (which, by definition, is the limit of the partial sums). I imagine that these two alone probably account for 90% of the definitions in published academic literature on the subject. There are probably more exotic ones too (there's a cute description of Dedekind cuts using lattice paths, for example). I don't mean to express a fundamental preference of one of these definitions over the other, but it seems to me that a straightforward application of the least upper bound principle recognizable to someone who has made it through the second chapter of (say) Bartle and Sherbert, is more pedagogically accessible to the definition using the infinite series, which is not covered until the very end of the third chapter. But if you feel really attached to the other definition, I don't see why both cannot be mentioned. Are there any other definitions you feel would be relevant to include? Sławomir Biały (talk) 19:38, 22 July 2017 (UTC)

"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 seems wrong: completeness property asserts the existence of limits, while the equality 0.999... = 1 relies on Archimedean property or, equivalently, to the non-existence of infinitesimals. D.Lazard (talk) 13:17, 23 July 2017 (UTC)

True, but: first, the Archimedean property is not an axiom; it is true by completeness. (Proof: By contradiction suppose 1/epsilon is an upper bound of the set of integers. Let N be the least upper bound. Then there is an integer n greater than N-1 and so n+1 is greater than N, a contradiction.) And secondly, which is what I was thinking of at the time, is that the equality $0.999...=1$ is ipso facto meaningless without the completeness axiom, because the LHS is a limit (or supremum) by definition. (Note, what I actually said above was that algebra alone is insufficient because "the existence of that representation requires the completeness property".) Sławomir Biały (talk) 14:07, 23 July 2017 (UTC)

Algebraic proof, clarified

I think I can clarify my objection to the algebraic proof. One can define the sequence of digits of a number. That is, there is a map $d:\mathbb {R} \to D=\mathbb {Z} \times \{0,1,\dots ,9\}^{\mathbb {N} }$ from the reals to the set D of allowed decimal sequences (including trailing nines). This mapping is easy to define just using "elementary algebra" (although we should be quick to point out that some concept of completeness is already required at this early stage, we shall attempt to make the algebraic proof rigorous relying on that fact as minimally as possible). In fact, the function d is actually one-to-one (which I believe requires the Archimedean property). However, this mapping is not surjective (as the subject of this article illustrates). Nevertheless, this map does have a left-inverse: that is, there is a function $s:\mathbb {Z} \times \{0,1,\dots ,9\}^{\mathbb {N} }\to \mathbb {R}$ with the property that $s\circ d$ is the identity automorphism of $\mathbb {R}$. There are, in fact, an infinity of such left-inverses. More on that later, but note that at this point the reader has no reason for preferring any of these left-inverses to any other.

Now, let us unpack the first algebraic proof. Here, I am assuming only naive algebra on the part of the reader, without a detailed knowledge of the completeness property. So, the equation $1/9=0.111...$ by definition means precisely the same thing as $d(1/9)=(0;1,1,1,1,...)$. At the next step of the proof, we have $9/9=9\times (0.111...)$. This is problematic, for the following reason: although we can multiply the left-hand side of the equation by 9 (it is a "real number", whatever that might mean to our algebra student), we cannot actually multiply the right-hand side by nine. Indeed, this operation admits no interpretation, because we haven't said how to "multiply by nine" an infinite decimal sequence. It has no intrinsic "numberhood", it's just an element of the set D.

Let's not quite give up so easily. Why not simply define "multiplication by nine" (or indeed any integer value by an element of D) so that the usual rules of multiply-with-carry decimal manipulation are correct? More generally, we can give D the structure of an abelian group. Then we certainly have $9\times (0.111...)=0.999...$.

So, let us try now to make sense of the proof. We have

$d(1/9)=(0;1,1,1,...)$

so

$9\times d(1/9)=9\times (0;1,1,1,...)=(0;9,9,9,...)$

so

$d(9/9)=(0;9,9,9,...)$

or

$1=0.999...$

Now the fallacy in the argument becomes quite clear. At the last two steps we used that $9\times d(1/9)=d(9\times 1/9)$. But at no point did we establish that d is a homomorphism of abelian groups. Indeed, it is not, as the very example of $9\times d(1/9)\not =d(1)=(1;0,0,0,...)$ shows!

In spite of this fallacy, the theorem remains true, for the following reason. Although d is not a homomorphism, there is a unique left-inverse that is a homomorphism. This left-inverse, s, is summation: $s(a;b_{1},b_{2},...)=a+\lim _{N\to \infty }\sum _{n=1}^{N}b_{n}10^{-n}$ (or any other equivalent definition). Notice that s, a necessary device to make the statement of the result correct, here explicitly requires calculus. That is, by the notation $0.999...=1$, we actually mean that $s(0;9,9,9,...)=1$. Then the steps of the argument follow, because s is a homomorphism.

One could argue that one does not really need the full force of the real number system to make sense of the arguments in this section, and so requiring calculus may seem a bit heavy-handed. If we confine attention solely to rational numbers, then the relevant subset of D is all of the ultimately repeating decimals. There exists a standard algorithm to construct a rational number from a repeating decimal. We could then define the value of s to be that algorithmically-constructed rational number. However, one of the rules in that algorithm is exactly that $0.999...=1$ (and likewise for any sequence that ends in trailing nines). This, then, becomes circular: we took, as one of the rules of the algebraic structure, what we wished to prove.

(I should add that there is an element of this same circularity in the opinion of some editors here who feel that because "0.999..." and "1" are both apparently names for the same real number, and therefore there is no difference between the two expressions. But then we should not claim to have proved anything. We have simply given the real number referred to by "1" a different name, and the equation is true simply by fiat, not because it expresses any mathematical content whatsoever! The article even invites this interpretation with the sentences "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 seems to say that it's the decimal representation scheme (d) that is relevant, and that the identity is "built into the definition", that is true by fiat.)

Now, it seems to me that a proper algebraic proof could be made rigorous and non-circular (even over the rationals) if we were to show that there is a unique homomorphism $s$ that is left-inverse to $d$. But it's extremely unlikely that a reader would infer this from what is written in the article now, since it seems to offer no indication that this is what in fact is required to make the algebraic "proof" an algebraic proof. It is the first, fallacious, interpretation of the argument that seems much more likely to me. So what is wrong about these proofs is that students almost universally think they are about the object d (which they do not realize is not a homomorphism), when in fact they are about the object s, which is a homomorphism, but is not something that they necessarily have any reason to think about (indeed, they may not even be capable of thinking about it).

This is borne out by the discussion in all three of the sources that we currently cite: Richards notes the "indoctrinat[ion]" to accept that $1/9=0.111...$, which is an equation wanting an interpretation. Byers distinguishes between number-as-process (d) and number-as-object (s). Peressini and Peressini discuss the need for a discussion of completeness, so that the mapping s can be constructed. Accordingly, all of these algebraic proofs are actually perfectly valid in the s-interpretation, but are subtly fallacious in the d-interpretation. To be sure, the s-interpretation is the one that is standard in mathematics, so the indeed the proofs are not quite wrong. But they are deceptive, because they have the great potential to mislead the reader into thinking that we have used a property that d lacks (homomorphismhood), when the argument promptly establishes that the opposite is true. Sławomir Biały (talk) 22:22, 23 July 2017 (UTC)

I (and at least some sources) look at it differently:

For any natural number n, write $S_{n}=\mathbb {Z} \times n^{\mathbb {N} }$. Then there is a function $s_{n}:S_{n}\to \mathbb {R}$ defined as the sum of the power series, using the completeness of the reals. (Whether we define S as the union of all the S_n and note all the s's are compatible is a pedagogical choice.) In either case, we can define "+" and "10 ×" in the obvious manner, and show that $s(A)+s(B)=s(A+B)$ and $s(10\times A)=10\times s(A)$ where defined, by using properties of power series. To show s₁₀ is onto (by constucting a right-inverse d) is a little tricky, but $9\times s_{2}(0;1,1,1,\ldots )=s_{10}(0;9,9,9,\ldots )=s(1;0,0,0,\ldots )=1$ follows immediately from the properties above. — Arthur Rubin (talk) 14:42, 24 July 2017 (UTC)

Yes, I agree with everything you have just said. This is compatible with the first s-interpretation; detailed above, before the purely algebraic one that does not rely on completeness of the reals. The trouble, however, is that algebra students often believe that "real numbers" and "decimals" are effectively the same thing, which they are not. (Such a student is not aware that there is an "s" at all!) Rather than discuss this difficulty, the current algebraic proofs section misleadingly seems to reinforce this same presumption. Of course, the result is then paradoxical: one cannot have numbers that are the same and yet different. If I had to speculate, this is one overriding reason that the subject continues to engender so much "skepticism" from those lacking mathematical literacy. We do a great disservice by failing first to validate this paradox, and then to explain how it is resolved. Sławomir Biały (talk) 15:08, 24 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, and it takes considerable effort to make these arguments rigorous: that requires, in particular, a proper definition of the real number system and a derivation of its basic properties. According to Peressini and Peressini (p.186), simple arguments like these fail to "explain why this equality holds." They note that such an explanation involves the distinction between numbers and their decimal representations, the concept of 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 had not noticed that you changed one of the versions. Doing that causes problems with same !votes being cast for the old version and some for the new.

Here is the best way to do that: let's say someone writes

"It is a little known fact that automobiles are the masters and humans exist on to serve their needs" --Example (talk) 01:11, 01 January 2000 (UTC)"

A few people !vote on it, then they realize that they meant to say something else. Rather than just changing it, they should do it like this:

"It is a little known fact that automobiles cats are the masters and humans exist on to serve their needs" --Example (talk) 01:11, 01 January 2000 (UTC) Modified 02:22, 01 January 2000 "'

(Five "~" characters instead of four gets you just the date). That way it is clear what was changed and when it was changed.

The removal of the scare quotes and the "purportedly" does indeed settle my specific objection about promoting a fringe view. I neither support nor oppose the other proposed charges at this time. --Guy Macon (talk) 12:58, 27 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)

I don't see why being an expert in mathematics is mutually exclusive with being an expert in math pedagogy. On the contrary, I would say that the two attributes exhibit a strong positive correlation. But that's beside the point: it seems like pedagogy they're writing to me. I mean, it seems like our criterion for someone being an "expert in pedagogy" here, would be someone who writes peer-reviewed pedagogy research. And we have that. So I don't see what the problem is. Sławomir Biały (talk) 19:05, 22 July 2017 (UTC)

Please show me where it says the book is "peer reviewed." It's edited. That's a completely different process than peer review. Don't get me wrong, non-peer-reviewed sources are fine for Wikipedia. But don't claim that you're a mathematician struggling against us less qualified "non-mathematicians" when you don't even know what peer review is. Calbaer (talk) 19:48, 22 July 2017 (UTC)

This is, sadly, becoming rather less about improving the encyclopedia. Does it matter if the content in the reliable sources meets some criterion for peer review? Both are serious academic sources. If you think that the sources are unreliable, then this is a fruitful line of inquiry. If I used "peer-review" in an off-the-cuff way to refer to sources about which there is no real dispute, then I don't see that resolving this irrelevant ambiguity is constructive. If you wish to discuss your own qualifications, showing a knowledge of the subject of the article would be a good start, or at least an eagerness to acquire the necessary background. But attacking those editors in this process who are the most qualified to improve the article, seems like a waste of everyone's time, and not beneficial to the encyclopedia. Sławomir Biały (talk) 20:07, 22 July 2017 (UTC)

You are the one who appointed yourself most qualified to improve this article; no one else did. I am merely questioning your qualifications, given that you don't know one of the most basic aspects of mathematical and scientific research. I had hoped this observation might convince you to stop throwing stones from your glass house (i.e., writing ad hominem attacks against the qualifications of others). If this results in less elitism and hostility on your end, then, yes, it was indeed constructive, since these are hampering efforts to improve the article. Calbaer (talk) 03:08, 23 July 2017 (UTC)

From academic publishing: "many academic and scholarly books, though not all, are based on some form of peer review or editorial refereeing to qualify texts for publication." I have little doubt that both sources were subject to some form of peer review. But that is not relevant to improving the article, and the matter is not worth further discussion. Regarding glass houses, let me remind you that (from Wikipedia:Credentials matter): "Amateurs may not have the experience or education necessary to evaluate sources adequately, or may not understand the material well enough to organize it into a coherent whole. And they may not be aware of how poor their understanding might be (the Dunning–Kruger effect). Experts are not perfect, but amateurs are on the whole less perfect, and especially in their judgement of the work of experts." Sławomir Biały (talk) 10:37, 23 July 2017 (UTC)

The first words of Wikipedia:Credentials matter are: "This page is an essay. It contains the advice or opinions of one or more Wikipedia contributors. Essays are not Wikipedia policies or guidelines. Some essays represent widespread norms; others only represent minority viewpoints." Not a policy, not a guideline, just an editor's opinion. As for Dunning–Kruger, I'd point you to the transcript this interview with Dunning: "We all have our specific pockets of incompetence, and we know some of them. But there are a lot of them we simply don't know." When challenged on teaching expertise - given that the point of the article is to educate - you replied that expertise in math should be sufficient. It's clearly not (assuming you have credentials in the first place; for all your talk of their importance and how they make your arguments superior, you curiously never state yours). Anyway, please don't cite opinion as though it were policy, and please don't misrepresent sources. Calbaer (talk) 13:40, 23 July 2017 (UTC)

The argument that I feel that expertise in math is sufficient for math pedagogy is incorrect, and I would point out that the sources under discussion clearly concern math prdagogy, not research mathematics. But we could focus the discussion on mathematics teaching expersience. How many mathematics courses have you taught, in which the subject of this article was discussed, for example? I heartily agree the we should not misrepresent sources, and am grateful that you finally acknowledge this. It seems like we should be discussing how to do that. Do you agree that the best way not to misrepresent dources is by summarizing what those sources Ave to say, supported by in-text attribution and direct quotation? If not, how to you propose this non-misrepresentation be achieved? Sławomir Biały (talk) 13:52, 23 July 2017 (UTC)

Appeal to probability (many do, ergo this one does) is a fallacy. And it's silly to expect me to answer questions you refuse to answer yourself, such as those on qualifications. My point is that you should drop the "you're unqualified" attacks, because they're counterproductive and you don't have a leg to stand on there anyway. Calbaer (talk) 14:23, 23 July 2017 (UTC)

Not sure what "appeal to probability" you're referring to: Experts in mathematics are "probably" experts in pedagpgy, and we're discussing sources on pedagogy. You seem to be saying that if something is "probably true", then it must be false, like your earlier apparent belief that because authorities say something, we should (apparently) disregard it, regardless of what other reasons are given in support. This same logic pervades your denunciation of the supposedly "loaded question" that you object to: simply saying that "the idea that due weight should be the only criteria in judging which text is better for the article is false" does not absolve one of the responsibility to address what is and is not due weight. Nor does dismissing the credentials of other "experts" or "supposed experts" or whatever, absolve you of the responsibility of responding to what they write in a substantive manner (if you respond at all: no one can oblige you to have an opinion on the matter one way or the other).

And no one besides you seems to be keeping the issue of credentialism alive. Nothing you have said in this discussion appears to relate in a substantive way either to the specific policy points (which have nothing to do with qualifications), or the specific sources themselves, aside from question the extent to which "peer-review" is applicable. You don't have to have formal qualifications to edit Wikipedia, but to continue productively to a discussion about mathematics does require a certain competence that you're not demonstrating. The only mathematical thing you mentioned was an apparently mistaken belief that the proofs under discussion follow from the axioms of mathematics, which is just wrong, while insinuating that it is my own understanding of the subject that is flawed [2]. I hope you can see how this comes across, and how by doing this you did make qualifications relevant to the discussion. If you do not wish qualifications to be relevant, you should stop discussing them. And, in particular, stop trying to attack the qualifications of others who, plainly, have far more substantive things to say about the article than you do. That is not constructive, unless your goal is to prevent those who know what they're talking about from contributing. Is it? I'm beginning to wonder. Sławomir Biały (talk) 15:00, 23 July 2017 (UTC)

The appeal to probability ("some therefore all") is your conclusion that because some books are peer reviewed, the Springer one is, in spite of there being there being no evidence that this is true. Note that the link above shows you confusing language, not mathematics. Language, not math, is your primary difficulty in making a positive contribution here. A secondary difficulty is hostility. On that point, you are the one keeping credentialism alive. Editors such as Huon, MjolnirPants, and myself are trying to dissuade you from doing so, to consider ideas on their own merits rather than attacking the qualifications of those presenting them. If you believe this line of argument is useless, then just stop starting such arguments. Calbaer (talk) 19:46, 24 July 2017 (UTC)

Calbaer, I have been doing nothing but discussing "ideas on their own merits" with other editors. I don't care what your (or anyone's) credentials are, in truth, and dismissing others as "non-mathematicians" was worded in a less-than-ideal manner. Indeed, I feel strongly that contributing productively to a discussion about a mathematical topic does require a willingness to admit that there is a great deal that one does not know. But your behavior throughout this discussion, beginning with this post, your dismissal of what I have to say as a Gish gallop (while perversely at the same time saying that I am ignoring you), your reverts to the edits of the article, and finally this current inquest, seem to be intended to shut down discussion about ideas and their merits. They are just meta-discussions rather than proper discussions, observations of form rather than substance. If you want me to discuss with you "ideas on their own merits", give me some ideas with merit to discuss with you and I would be happy to do so. Or you could discuss my ideas on their merits. It's up to you. Sławomir Biały (talk) 09:27, 25 July 2017 (UTC)

I gave alternatives to your suggestions several times, so you can't credibly accuse me of not producing any ideas or not discussing yours. And your attacks and mistakes might be matters of unfortunate wording to you, but all we have are your words, so that's what we have to go by. If you have difficulty conveying your thoughts in print, then presenting your ideas on talk pages for refinement before inclusion in articles might be preferable to editing first, only to find your contributions reverted by multiple editors. Calbaer (talk) 18:58, 25 July 2017 (UTC)

I responded to your suggestions. If I did not, please tell me which suggestions you would like my input on. You are certainly welcome to respond substantively to any of my posts as well (or not). But please don't make excuses like that I "have difficulty conveying [my] thoughts in print". If something is unclear to you, try reading it. If it's still unclear, try reading it a second time. If it's still unclear, you can always ask for clarification. There are people here from whom you can actually learn something, if you're not so busy trying to shoot them down. Sławomir Biały (talk) 19:46, 25 July 2017 (UTC)

"[D]ifficulty conveying your thoughts in print," is indicated by your statements that it was your wording to blame for the appearances that you were lodging ad hominem attacks ("worded in a less-than-ideal manner") and didn't know the definition of "peer review" (which you characterized as "an off-the-cuff way to refer to sources about which there is no real dispute"), rather than credentialism and ignorance, respectively. Unfortunately, an encyclopedia article is not something from which a reader will be able to "ask for clarification," so we need to be careful about wording there. Calbaer (talk) 20:11, 25 July 2017 (UTC)

Fine. Point taken. Sławomir Biały (talk) 20:14, 25 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)

Calbaer, thank you for responding in very calm words (quite rare these times) to my rather harsh accusations. Nevertheless, I disagree with holding all three variants equal wrt their distance to the non-proofs. While I perceive version A as, blatantly and fully intentionally, distracting from the problems at the heart of "infinitely long" division and "infinite" multiply through, version B constantly appeals to the higher ideas in the respective math. I agree that these approaches reflect themselves in the current readabilities, but I rather take the degraded readability of B, hoping for achievable improvement, than the dishonest soothing of everything is easy in the first section of A, suggesting that the nitpickers, best to ignore, may articulate their troubles and pipe dreams in the rest of the article, best ignored by the average reader. Imho: A (and C) blissfully fail accuracy, B is improvable, especially on readability. Purgy (talk) 12:32, 23 July 2017 (UTC)

I was judging based on which is better for inclusion, not which is improvable. I appreciate the concern here, but I'd personally prefer an incomplete (and thus inaccurate) but readable text to complete and accurate but unreadable text. Thus my desire to wait for a "version C" (not necessarily mine, since I just quickly typed it out as an example). I'd like something that is both accurate and not so opaque and meandering as to lose everyone who might actually gain knowledge from the article. Calbaer (talk) 19:52, 24 July 2017 (UTC)

Regarding "the idea that due weight should be the only criteria [sic] in judging which text is better for the article is false." That is true. But do you agree that: compliance with WP:WEIGHT is a mandatory requirement of all article content? On the assumption that the answer to this question is "yes", how is it that Version A is due weight, when it spends just a third of the time on the discussion, with two-thirds dedicated to the proof; whereas, in the Byers source, only 10% of the text is actually devoted to the proof itself? Also, does version A accurately and neutrally summarize the viewpoints expressed in Byers, Richman, and Peressini? Is relegating the opinions of these cited authors to a footnote, rather than the article text, consistent with the "prominence of placement" expressed in WP:WEIGHT? Finally, it does not have to be a binary decision, as I noted in my comment to Guy. You are certainly free to make suggestions on how Version A or B might be improved, if you feel that it can present the viewpoints expressed in the reliable cited sources in a more satisfactory manner. Your suggested Version C, however, does not attempt to express those views at all, and so it also fails the mandatory policy requirement of NPOV. Thus it is not an acceptable alternative. Sławomir Biały (talk) 17:16, 23 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)

@Arthur Rubin: Arthur, the Peressini source mentions "(potentially) infinite series", which I read as a reference to potential infinity, which is referred to elsewhere as well in the research literature (Katz and Katz). But if you are more comfortable, would simply changing the phrase "the relation between potential infinity and completed infinity" to "the nature of infinity" be sufficient to sway your opinion? Sławomir Biały (talk) 19:12, 22 July 2017 (UTC)

Also, on a slightly more nuanced note, I do not think it is correct to dismiss the idea of "0.999..." as a process. When we mathematicians use it, we do mean a process (or rather its result, in the limit). We write 0.999... as a limit of partial sums. The sequence of partial sums is the "process", which still plays a fundamental role. Students understand this process aspect, but fail to grasp the limit aspect that also makes it an object because that is much subtler. Sławomir Biały (talk) 19:21, 22 July 2017 (UTC)

• Support a shorter Version B; the last paragraph seems incorrect. There are ways to give "0.999..." and "1.000..." meaning in other number systems. If sources argue that's impossible, I'd like to know what they are. That said, much of B is indeed an improvement over A. Huon (talk) 19:38, 22 July 2017 (UTC)
• I agree with Guy, and therefore argue for no change based on the current proposals until something better is found William M. Connolley (talk) 11:33, 23 July 2017 (UTC)

William, could you please clarify how you think that the material can better summarize, in a proportionate way, the cited sources? Do you feel that "Version B" does not properly summarize the views expressed in the cited sources, while "Version A" does summarize those views in a more proportionate way? If so, could you please explain why? Version B, in particular, is supported by more page references, in-text attribution, and quotations to the sources, than is version A, and so at least superficially seems to be the more policy-compliant of the two. Accordingly, I feel that a view that the less detailed version lacking such attribution in fact summarizes those sources "better" requires some justification. This is not a vote, and you will note that the phrasing of the RfC specifically concerns this question. Sławomir Biały (talk) 12:38, 23 July 2017 (UTC)

• No change per Guy & William until a better alternative is presented. Keira1996 23:53, 23 July 2017 (UTC)
• Reminder to all participants (and the closing administrator). This is not a vote on which version we happen to like better. The parameters of the RfC are, specifically, which version properly gives the opinions expressed by reliable sources WP:DUE weight. It is true that Wikipedia is based on consensus, but the consensus must be based on valid policy-based arguments. Not WP:IDONTLIKEIT non-arguments that fail to respond to requests for clarification. (Much less now the apparent piling on to Guy's suggestion which seems to me to have been amicably resolved with a minor edit to the first sentence.) None of the most recent posts have addressed the question set forth in the RfC, so these will be of limited use in consensus building. The only compelling policy argument advanced thus far has been this one, and it has not been challenged by any of the participants here. I note that the Wikipedia:Arbitration committee has repeatedly held that articles must comply with the neutral point of view policy. For example [3]: "Wikipedia articles are to be written from a neutral point of view and without bias... To comply with the verifiability policy, assertions of fact, particularly controversial ones, should be supported by an inline citation to a reliable source." [4]: "All Wikipedia articles must be written from a neutral point of view. Where different scholarly viewpoints exist on a topic, those views enjoying a reasonable degree of support should be reflected in article content. An article should fairly represent the weight of authority for each such view, and should not give undue weight to views held by a relatively small minority of commentators or scholars. The neutral point of view is the guiding editorial principle of Wikipedia, and is not optional." (And so on...) I wish to emphasize that the closing administrator should not be counting votes, but read the reasoned policy-based arguments, knowing that the Arbitration Committee may review their decision if our policies fail to prevail. Sławomir Biały (talk) 00:23, 24 July 2017 (UTC)

You're quick to cite WP:NPOV to support your argument, but you blatantly ignore WP:UNDUE and WP:FRINGE which state that we're not required and in fact 'discouraged' from providing a platform for promoting fringe views (like the view that 0.999... isn't equal to 1. It is.). 74.70.146.1 (talk) 01:52, 24 July 2017 (UTC)

How about actually looking at the sources? Have you better? They like to promote mathematics as based on reason rather than faith. I hope we're not into the territory of the Australian PM who recently said 'The laws of mathematics are very commendable, but the only law that applies in Australia is the law of Australia'. Nobody is saying that as real numbers they are unequal - it is just that the algebraic proofs don't state their assumptions, as it says in the lead of the section 'However, these proofs are incomplete or not rigorous, as they do not include a clear definition of 0.999… and of the operations that are allowed on such a notation.' Dmcq (talk) 07:19, 24 July 2017 (UTC)

My objective is not to support the view that 0.999... and 1 are anything but equal (which would be fringe). But presenting as a proof a subtly misleading argument of a true fact is not justifiable, when the cited literature presents those proofs actually as examples of misleading arguments. Indeed, it would be better to give no such proof at all: our obligation to "the truth" stops at saying that the real numbers $0.999...$ and $1$ are equal, supported by a reliable source. The article is under no policy obligation to convince the reader of this standard fact, just like we aren't under any obligation to convince the reader that the statement "An electron is a fundamental particle" is true by presenting them with scattering data from particle accelerators. This having been said, if we include these proofs at all, then policy does require us firstly to cite sources appropriately, and secondly to summarize what those sources actually have to say on the matter in a proportionate way. As I noted above, the Byers source spends 10% of the text on the "proof" (with Byers' scarequotes), and 90% of the text on the discussion. So it's hard to see how spending more text on the proofs than the discussion is any way proportionate (and indeed, no effort is made to summarize his views in a coherent way at all). If you can think of a way to summarize the sources better, then please suggest it. That's why there is a request for comment. I'm rather alarmed though that so many editors do not seem to care what sources actually say. Needless to say, views that are not actually based on any analysis of sources should be disregarded by the closing administrator. If they are not, then ArbCom remains a likely outcome. Sławomir Biały (talk) 11:15, 24 July 2017 (UTC)

If you argue this way, it seems to me neither version A nor version B are a solution, but strictly speaking you'd need to ask for the deletion of the article.

If the purpose of this article is simply to state the well known fact 0.999...= 1, then we don't need an article for that all. That fact can simply be mentioned in the article on Real numbers and/or Decimal representation and that's it (maybe instead of deletion turns this article into a redirect). Having a separate article only makes sense, if it deals explanations and backgrounds infos on that fact, which loosely speaking is kinda like "convincing the reader of that fact". Also having a (minimal) encyclopedic obligation does not block us from doing more. So while we are not obligated to do more, we very well may choose to nevertheless.--Kmhkmh (talk) 16:40, 24 July 2017 (UTC)

I did not mean to suggest that we do not include a proof at all. By all means, we should include a really convincing proof; several even! What I object to is presenting a proof that absolutely should not be convincing to anyone that actually needs to be convinced, as if it were convincing. Indeed, our sources say that students who find these proofs convincing do not understand the identity. Yes, to someone with prior knowledge of the completeness axiom, they can be made precise and rigorous; but I've been repeatedly told that it's not that group for whim these proofs are intended. On the contrary, the algebraic proofs are also being presented as easy to understand, which they are not. Indeed, the algebraic proofs are every bit as hard to understand as the analytic proofs, possibly moreso. If we aren't committed to conveying a proper understanding of these proofs, supported by reliable sources, then I do not believe that they should be included at all. Does this make sense? Sławomir Biały (talk) 17:17, 24 July 2017 (UTC)

I'm somewhat neutral on the A versus B and I'm not going to state a preference. I can live with the inclusion of the problematic/iffy proofs as long as they are found in (reputable) literature and their potential issues are pointed out in the article. In particular if they tend to show up math books for general audiences or (high) school books, it might be a good idea to have a Mathematically proper treatment of them in WP. Here i just wanted to point out that the logical consequence of strict "just state mathematical fact"-argument would imho be a request for a deletion or redirect, as that bit of information doesn't warrant an article.--Kmhkmh (talk) 17:40, 24 July 2017 (UTC)

• Version B: Version A is too handwavey. I'll note here that several people read the second version as questioning the fact that 0.999... = 1. That is not what it's doing: indeed it notes the fact in the first sentence. What it says is that a rigorous proof is tricky. I agree with the sentiment, therefore prefer Version B. It could perhaps be shortened (not sure how). Even though it's longer, it's a bit easier to read and has less jargon than the shorter version. Kingsindian ♝ ♚ 16:26, 24 July 2017 (UTC)
• Version B: Is it perfect? No. Is it too long? No, because either the reader is equipped to follow it or not. If not s/he will pass on, skipping all of say, 200 words. Readers capable of following are condemned to read the extra 200! Biiig deal... Could B be improved on? Sure, and feel welcome, but until then B should be used because it is far more helpful than A. C did not cut the mustard. Pardon me for not participating in the wall-of-text choir, but feel welcome to rattle my cage when the new all-singing-all-dancing Version D arrives; meanwhile, sorry, things to do... JonRichfield (talk) 05:58, 25 July 2017 (UTC)
• version B. I don't know who the intended audience of the paragraph is – but version B makes more sense to me. Maproom (talk) 08:05, 25 July 2017 (UTC)
• Something else. Look, I'm on board with avoiding lies-to-children; Sławomir is correct that those have no place in this sort of project. But I don't see the point of the article in the first place, if readers have to understand the real numbers rigorously. Anyone who understands the real numbers rigorously doesn't need this article.
  It seems to me that the "algebraic" proofs can be saved with some caveats. Emphasize that if there is a way of getting from infinite decimal strings to numbers, and if that way has certain properties that will seem reasonable (the behavior of multiplication or subtraction, and maybe the Archimedean property), then it follows that 0.999... equals 1.
  I think the Archimedean property is much easier to follow for the target audience than completeness, and we should emphasize it over completeness in the early going.
  Then, further down into the article, we can segue into the motivation for the reals. The Zeno stuff goes well there — I've complained a couple of times that the real number article doesn't mention Zeno, which I think is a major oversight, but I haven't gotten around to doing anything about it. But I think we can start with the "algebraic" arguments, provided we stress that they make assumptions that can't be justified at the level of that exposition. --Trovatore (talk) 09:37, 25 July 2017 (UTC)

Doesn't the definition of 0.999... already rely on completeness? Sławomir Biały (talk) 10:33, 25 July 2017 (UTC)

My proposal is that, for the introductory part of the article, we not try to prove that 0.999... denotes a number, but that we rather show that, given that it denotes a number, and given that the interpretation has certain properties that will seem reasonable to the reader, then the number it denotes must be 1. Then we don't need to mention completeness.

That will be enough for the naive reader, and we haven't told him/her any lies. Then the more advanced, or more curious, reader can read on to see how we justify the "given" parts of the above. --Trovatore (talk) 10:46, 25 July 2017 (UTC)

Yes, that could work. That discussion would ideally go before the proofs I think. But don't the proofs still obscure the main point, which is either completeness or the Archimedean property? I do not think that making the algebra explicit axioms that are assumed to be valid eliminates the need for a discussion such as "Version B". Sławomir Biały (talk) 10:59, 25 July 2017 (UTC)

Yep we have to mention either completeness or the Archimedean property if we're not indulging in flimflam just trying to convince people who don't know better. Probably the Archimedean property is best. To some extent it just gets rid of the problem by making an assumption but we can't get rid of it just using algebra - that would be like showing parallel lines exist without assuming Euclid's fifth postulate. Just saying real numbers doesn't cut it as people will just assume they know what is meant by that. Dmcq (talk) 11:19, 25 July 2017 (UTC)

I imagine we could infer that some version of the Archimedean property is a consequence of taking the algebraic properties for decimals as axioms, which may not be so bad. But I think that would probably stray into original research, and I imagine that it would make a Platonist uncomfortable. Sławomir Biały (talk) 12:19, 25 July 2017 (UTC)

I can't see how one could do that without assuming the result is true. For instance the very first line of the first algebraic proof goes wrong when it says 1/9 = 0.111... Yes one gets a sequence of 1's using the algorithm but one can't say they are equal just using algebra without assuming the Archimedean property. Dmcq (talk) 13:07, 25 July 2017 (UTC)

I see your point. Perhaps all sides would be more satisfied if, instead of presenting these arguments as proofs of anything, they are presented as seeming paradoxes that, instead of definitively demonstrating something specific, suggest a re-examination of the number concept? If real numbers and decimals meant the same thing, then these arguments lead to a paradox, because the decimal "1.000..." is not the same as "0.999...".

As we know, algebra students nowadays are entirely reliant on their calculators for even the most basic of tasks. An algebra student perhaps would agree that $1/9=0.111...$ because he checks his calculator, which reports that $1/9=0.11111111$, and perhaps infers that the 1s actually go one forever, but are simply rounded to the number of digits of precision. Then multiplying this through by nine, he would get $1=0.99999999$. This equation is, of course, not actually true any more than the equation $1/9=0.11111111$ was true. But the question is, how to interpret it?

I think we start with the equation $1/9=0.111...$ (with infinite 1s), multiply through by nine to give $9*(1/9)=0.999...$. By the associative law for multiplication, the left-hand side simplifies to unity. Now, we started with a "true" equation, "multiplied" by nine, then applied a valid algebraic rule for multiplication, and arrived at an equation. The student knows that multiplying both sides preserves the equation of the two sides. But here we have an equation that his calculator tells him is false: he enters "1", and the calculator tells him "1". So how can this apparent paradox be resolved? Sławomir Biały (talk) 13:34, 25 July 2017 (UTC)

The same algebraic manipulation that gives 1 + 1/9 = 1/(1-1/10) = 1.111... also gives -1/9 = 1/(1-10) = 1+10+100+.. = ...11111.0 so 0 = 1/9-1/9 = ...1111.1111... ;-) It's really got to be based on the sources. Dmcq (talk) 14:04, 25 July 2017 (UTC)

"It's really got to be based on the sources." I don't think anything else really needs to be said ;-) Sławomir Biały (talk) 14:12, 25 July 2017 (UTC)

• Questions I see in the article an explanation of the "Archimedian Property" but no explanation of the "completeness axiom" in the proposal or in the article and how, if at all, they relate or compare. Also, why is not the "completeness axiom", what it is and what it means upfront in the proposal (and in the lede), since the "completeness axiom of the real number system is what allows" the equality? Alanscottwalker (talk) 16:07, 25 July 2017 (UTC)

I must confess that I too am puzzled by the article's aversion to the completeness axiom. Several knowledgeable editors have expressed the opinion that it is the Archimedean property that should be emphasized, as opposed the completeness axiom. I might be missing something, but it seems to me that completeness is required anyway, regardless of whether the Archimedean principle is emphasized (at least, without modifying things in an exotic way). Sławomir Biały (talk) 16:18, 25 July 2017 (UTC)

Completeness axiom is the existence of least upper bounds for upper bounded sets. This is the axiom, which implies that ${\sqrt {2}}$ is a real number, as being the least upper bound of the rational numbers whose square is less than two. Here, as 1 is a rational number, the Archimedean property is sufficient for proving that 1 is the least upper bound of all 0.999...9. It is what was shown in the first version of section "Motivation". I am not sure that it was a good idea of removing this from the end of the section, as the only possible definition of the notation 0.999... is to denote the least upper bound of all 0.999...9. D.Lazard (talk) 17:07, 25 July 2017 (UTC)

Thanks. "Completeness axiom is the existence of least upper bounds for upper bounded sets" . . . bear with me: The completeness axiom holds that for upper bounded sets there is [a] least upper bounds. All real numbers, such as 1 are upper bounded sets and 0.99 is its least upper bound. Is that close? Alanscottwalker (talk) 17:58, 25 July 2017 (UTC)

I've tried to explain it here. Any good? Sławomir Biały (talk) 21:14, 25 July 2017 (UTC)

Thanks. Two things: 1)"The meaning of the notation $0.999...$ is the first point after the sequence of finite truncations $0.9,0.99,0.999,\ldots$ (and similarly for the meaning of any infinite decimal)." Why cannot that thought be completed explicitly with reference to the statement one equals 0.999 . . . (thus, therefore, and, etc.) . . . close the hanging issue for the reader. 2) Avoid reference to it as a "property" and an "axiom", unless your directly explain somewhere something in the article, like, 'this property is an axiom in real number theory.' Alanscottwalker (talk) 12:29, 26 July 2017 (UTC)

How about this? Sławomir Biały (talk) 12:54, 26 July 2017 (UTC)

It was not my intention to remove it, but rather to rephrase it in an intuitive way suitable for motivation (thus, $P_{\infty }$ is the first point following all $P_{n}$). But you're right that it should be made explicit that this is the completeness property that one is referring to. Sławomir Biały (talk) 18:53, 25 July 2017 (UTC)

I do appreciate you being responsive, and I do hope these questions/comments have improved the article. On the overall issue of this RFC, I am persuaded of one thing, with respect to A and B - B is sourced with RS and I am therefore inclined to it, although I do wish it could be shorter. Alanscottwalker (talk) 22:24, 26 July 2017 (UTC)

I take that "shorter" part a bit back, I now think in particular the introduction to thinking about a "process" and an "object" is important for better understanding the article's later sections. -Alanscottwalker (talk) 12:42, 27 July 2017 (UTC)

• Monkey wrench — I think the first paragraph of Option B is justified by the sources, although "these arguments fail to supply a satisfactory explanation of why the equation should hold" begs the question of satisfactory to whom? But, the framing seems to imply that the equality was false, or impossible to adequately explain before the formalization of real numbers. Yet all repeating decimals are rational numbers and the mathematics of their values and the process of conversion from repeating decimals to rationals was well established before. Hence all the (scare-quote) proofs. So, I think if we're to say that the arithmetic explanation is "unsatisfactory" now, we need to indicate that it was satisfactory to (probably the vast majority) of mathematicians before the elaboration of real numbers and theorizing about the meaning of decimal representations.--Carwil (talk) 17:40, 25 July 2017 (UTC)

That's an interesting perspective, and I wonder if there are sources. I have serious doubts that the algebraic arguments would have been regarded as satisfactory before the introduction of the real number system. Generally, mathematics dealing with infinity were viewed with heavy suspicion prior to the 19th century. Archimedes famously found the area enclosed by a parabolic segment by what would now be recognized as the infinite sum of a geometric series, but he did so with the method of exhaustion: proving that the presumptive "sum" would need to be neither less than nor greater than some given value. So I think the idea that least upper bounds were important to proving statements like this were recognized well before the real number system was properly organized in the 19th century. Another issue with these historical kinds of questions, too, is that it is in some sense meaningless to impose our standards of what a proof meant, for someone in the past like Euler (or Archimedes). It may have been a rigorous proof to Euler, but it would have meant something different to Euler, and so the comparison to the proof that is understood today, by an algebra or calculus student, is a very different "proof-idea". I fear that makes such comparisons meaningless without very good historical sources.

In any case, I think it is simplest to forgo these questions by rewriting the sentence. An inline attribution, like "According to Peressini and Peressini, ..." or possibly even a direct quotation, would then not be something asserted in Wikipedia's voice. I have gone ahead and done this. Sławomir Biały (talk) 21:00, 25 July 2017 (UTC)

• Note: It is difficult to conduct an evaluation such as this if Version B changes in-place, as observations might no longer be applicable. (In general, anything on the talk page should be added to, rather than modified in place, though I'll admit to occasionally - but rarely - modifying something a few minutes after writing it if I find an error or ambiguity in what I wrote.) It might be more useful to add the new version to the bottom. Calbaer (talk) 21:34, 25 July 2017 (UTC)

I like the modification to attribute "unsatsifying." Encyclopedia Brittanica (1796) offers an explanation that references limited difference (1-0.9999… = 1/10, no 1/100, no 1/1000 etc.) but not infinitesimals or real numbers. See here.--Carwil (talk) 23:28, 25 July 2017 (UTC)

Thank you, that is most interesting. They say that 9/9=0.999... "signifies" 1. Thus it seems to be a matter of defining it do be something, rather than a statement about numbers, to be proved. Sławomir Biały (talk) 23:35, 25 July 2017 (UTC)

• Version B: This discussion (and article) stopped being ridiculous from reading B. TVGarfield (talk) 02:04, 26 July 2017 (UTC)
• Version B as more rigorous, albeit could be shortened a bit. Let's work on that after the RfC closes. — JFG ^talk 22:12, 26 July 2017 (UTC)

Dicussion brought to WP:FTN

Sorry for originally wrong name of noticeboard

As I noticed, just by chance, the above discussion was brought to WP:FTN under the accusation of giving undue weight to a fringe view of 0.999... not equaling 1.

Personally, I do not see this as a fringe view within mathematics, but as a realistic consequence in some rarely employed number constructs, and as an everyday assumption, in many a physicists' views in appealing to intuition, in spite of the equality being undisputed within the real number system. Additionally, I did not perceive the above attempts of improving the article as giving undue weight to the possibility.

Just for your information. Purgy (talk) 08:14, 24 July 2017 (UTC)

Indeed, no fringe is being pushed here. On the contrary. That ANI incident should be closed. I made a comment there. - DVdm (talk) 08:49, 24 July 2017 (UTC)

Motivation for section "Motivation: Achilles and the tortoise"

When thinking about the lengthy discussion/dispute about the algebraic pseudo-proofs, I got the conclusion that it is about a false problem, or more exactly a wrong way for considering the problem. In fact the challenge of this article is to explain to people who don't know real numbers why 0.999... = 1. Section "Algebraic proof" presents some pseudo-proofs without explaining why there are not proofs, because its authors thought that it is the only way of explaining the equality to people who don't know of real numbers. On the other hand other editors want to preserve mathematical accuracy, which may be too technical for some readers. Also this way of presenting things may be anti-pedagogical, as it could be understood as "things are like this because mathematicians are decided so".

My experience in such a situation, is that the solution comes from the history of the subject, which may only explain mathematicians choices. This lead me to explain the problem in term of Zeno's paradox, and this explanation showed me that the true problem is "what is the best number system for measurement?". Rewording Zeno's paradox in terms of 0.999... = 1, it appeared to me that the properties of a number system that are required for resolving the paradox are exactly the axioms of the reals. This is what I have tried to explain in terms that are understandable to people who do not know reals.

If there is a consensus for accepting this new section, my opinion is that the whole article must rewritten and restructured in function of it. In particular most pseudo-proofs and proofs become useless, as, in fact, 0.999... = 1 is directly proved from the axioms of the reals. The algebraic pseudo-proofs could be replaced by a section "Arithmetic of infinite decimal expansions", where it could be shown that the definition of 0.999... is compatible with this arithmetic (this is the only thing which is really proved in the present state of the article. Similarly, the analytic proofs could be replaced by sections "Interpretation in terms of series, sequences, ... But all of this requires more discussion.

By the way, I know that my English is not very good. Thus any improvement of my wording is welcome. D.Lazard (talk) 14:59, 24 July 2017 (UTC)

I am against it. The article is written to start with the algebraic proofs that are sufficient for most readers. We point out that these are not sufficiently rigorous and provide proofs using the Dedekind Cut and Cauchy Sequences. The kids don't encounter these until Analysis I at uni, so high schoolers are accustomed to thinking of real numbers as being points on the real number line, identified by their distance from the origin. They probably encountered decimals before they encountered the reals. Now, the discussion of Zeno's paradox resembles a proof by gesticulation of the more rigorous arguments below, and therefore I don't think it can add anything except confusion. Hawkeye7 (talk) 00:19, 25 July 2017 (UTC)

The algebraic proofs cannot be understood by someone who encountered decimals but not the reals, and they should not be presented as if they can. Sławomir Biały (talk) 09:40, 25 July 2017 (UTC)

I agree; although Zeno's paradox has a similar flavor (add another X% infinitely), most discussions on it come to completely different conclusions, and thus it will only confuse. Not every interesting idea is pedagogically useful.

And I think multiple authors angling for the article to be completely restructured in mutually exclusive ways will only add to the current headaches where it's clear there's an utter lack of consensus.

(I, for one, would restructure it to remove the "discussion" section and instead make it clear at the outset that the initial equations are meant for intuition and assume properties which themselves need to be proved via real analysis. I'd also emphasize that the real number system is the one upon which almost all science and engineering is based and that the systems where 0.999... is not 1 have little use. But we already have too many cooks in the kitchen for me to advocate for my ideas at this point.) Calbaer (talk) 01:17, 25 July 2017 (UTC)

So, would it help if I considered myself as silenced and banned from the kitchen? Purgy (talk) 09:53, 25 July 2017 (UTC)

I think I should be able to notice that the number of contradictory desires are working at cross-purposes without being accused of "silencing" anyone (especially since you're neither of the two editors who've unilaterally decided how the article should be). Calbaer (talk) 13:46, 25 July 2017 (UTC)

... and shouldn't I be able to offer me being quiet to ease your perception of heat in the kitchen? I'd never admit you being able to silence me in an honorable way. Purgy (talk) 07:52, 26 July 2017 (UTC)

We cannot say that these proofs are "meant for intuition" without a source. Since the proofs seem to provide no useful intuition about the subject, since they obscure the main point rather than shed light on it, I for one am opposed to presenting them in this way without strong sourcing, and the current sources appear to support this view. Sławomir Biały (talk) 10:49, 25 July 2017 (UTC)

By the logic, the Zeno section should be deleted, since there's no source saying it's useful. I think that's too high a bar - we should be allowed to explain without every explanation having a source that it's a great explanation! - but I still maintain the section does more harm than good. Calbaer (talk) 13:46, 25 July 2017 (UTC)

Under the WP:V policy, any statement that is challenged requires a source. I challenge the statement that these proofs are "meant for intuition". I am not required or expected to present sources of my own in support of this challenge, although I would happily discuss my findings that this assertion is apparently in contradiction with the sources that I have consulted. The suitability of discussing Zeno's paradox is another matter. You are free under policy to challenge it if you so wish, but I would recommend seeking sources first. Sławomir Biały (talk) 14:08, 25 July 2017 (UTC)

The new section itself concludes with, "There are less common number systems, such a[s] hyperreal numbers, which are perfectly valid, and do not have Archimedean property. In such a number system, the number 0,999..., as defined above, is not equal to 1." However, the part of the article that addresses hyperreals leaves the impression that 0.999... is not well-defined there; it could represent 0.999...;...999... (which is one) or 0.999...;...999000... (which is not a valid number). So nothing in this section indicates that 0.999... in hyperreals represents a valid number different from 1, which your statement contradicts. Calbaer (talk) 01:27, 25 July 2017 (UTC)

While I am currently undecided on the usefulness of this notable effort on explanation, I object with utmost intensity to the insinuation that "The algebraic proofs are sufficient for most readers." based on the saying about the impossibility of fooling all the people all the time, and via the impossibility of reducing the readers of an encyclopedia to some.

I am firmly convinced that exactly this sloppy treatment, popularized by numberphile and their likes, defended here by a group of editors, is to be blamed for the current weight of the wrong view of "inequality within the reals", verbalized by those being denied an honest explanation (not necessarily to the formal level, but making the difficulties visible). Any attempt of presenting the current lines of thought as a "proof", even of calling this "giving an intuition" founds profound misunderstanding of the heart of the problem and is no solid defense against the refuted claims of "inequality within the reals". I also do believe that frankly stating that there are other, rarely used, and troublesome(!) number systems, in which a similar construction leaves space for an inequality between the somehow embedded entity of 0.999... and 1 is rather helpful than detrimental for defending the established properties of reals against non-truths. "Inequality within the reals" is not fringe, it is simply wrong. Purgy (talk) 08:42, 25 July 2017 (UTC)

This summarizes my own feeling as well. By presenting a misleading argument in the hopes of tricking the reader into believing, we only give ammunition to the deniers. Sławomir Biały (talk) 09:42, 25 July 2017 (UTC)

Purgy, no one here has argued for equality within reals, just whether and how to present number systems in which 0.999... is undefined or unequal to 1. Some of the most passionate editors here are using straw man argumentation, refuting points that were never made by anyone. Mischaracterizing those who disagree with you, however, generally moves matters further away from consensus, not toward it. Calbaer (talk) 13:59, 25 July 2017 (UTC)

Even in this largely revised and redacted version I fail to correlate your comment to me having mischaracterized someone, especially, since I am not even aware who of the debaters disagrees or agrees to what extent on which item I uttered. Just recently I noticed one comment, which I interpreted as consent. Should I feel myself attacked by your comment as a straw man? Purgy (talk) 07:52, 26 July 2017 (UTC)

Speaking from experience, Version B in the above RfC was denounced as a "a blatant attempt to give the fringe theory that 0.999… does not equal 1" undue weight" (and other commentators have rallied to that misguided cause). The same points were raised at the WP:FTN discussion linked above. I have argued consistently that tricking the reader into believing something they do not understand (even if it happens to be true) is not a good basis for discouraging deniers. Sławomir Biały (talk) 13:53, 25 July 2017 (UTC)

A person arguing that something is a "fringe theory" is saying it's widely held to be wrong. You might object that that's the wrong language, because it is wrong in the reals, not just "widely held to be wrong," but the distinction there is not worth arguing about (especially since the argument in question was about giving undue weight to alternative number systems, in which it is not merely wrong). The point is we should make sure that it's clear that (1) 0.999...=1 in the reals, and (2) other number systems, some of which have 0.999.. undefined or not equal to 1, are not commonly used. Calbaer (talk) 14:10, 25 July 2017 (UTC)

I think there is some essential communication that is not happening here. Certain editors have argued that, by pointing out flaws in problematic proofs, we are supporting a fringe theory (see several of the statements in the above RfC, and the notice at WP:FTN and some of the comments there). We can quibble over what is fringe and what is simply wrong. But my understanding of Purgy's essential point is that by presenting flawed (/wrong/whatever) proofs as proofs (/intuitive arguments/whatever) in the first place, we simply invite readers who do subscribe to the wrong belief that the real numbers 0.999... and 1 are unequal, to note for themselves the flaws in the proofs (or rather of their understanding of those proofs, which cannot be corrected without some knowledge of the real number system). Instead of including language that communicates an incorrect understanding, we should strive to communicate a correct understanding, even if it means pointing out why not all reasons for believing that $0.999...=1$ are good. Sławomir Biały (talk) 14:26, 25 July 2017 (UTC)

I feel that the main point in this treatment of Zeno's paradox is somewhat obscured: why does Achilles reach the tortoise at $P_{\infty }$? I propose that a 1km finish line be included in the race, so that the reader will instantly agree that they cross the finish line at the same time. Then we can define the sequence of points, $P_{1},P_{2},\dots$. Then at $P_{\infty }$, Achilles will have crossed every part of the track to the finish line. We could then mention the Archimedean property, which is that if Achilles crosses every point of the track, then he must have reached the finish line (?). And so $P_{\infty }=1$. I do not think the explicit treatment of the Archimedean property is ideal in a motivation section. I would postpone it until a subsequent section. Sławomir Biały (talk) 10:10, 25 July 2017 (UTC)

Correction: The Archimedean property is that by crossing all of the $P_{n}$, he crosses every point of the track (and so reaches the finish line). Sławomir Biały (talk) 11:04, 25 July 2017 (UTC)

So, from what I can tell, Hawkeye7, Sławomir Biały, and I are all against having this section (at least in the introduction), and its creator is for having it. (Purgy Purgatorio has warned against being sloppy, but it's unclear to me whether that is opposition to this section, opposition to proposed alternatives, or just a general objection.) If I've either misinterpreted or omitted your opinion, let me know, but if that's the state - i.e., no one finds it useful except its author - then we should probably consider it a well-intentioned but failed experiment, end this particular discussion, and just remove it. Calbaer (talk) 13:57, 25 July 2017 (UTC)

I object to being included in the list of "oppose". I am "stay and improve". Furthermore, Trovatore expressed an opinion that Zeno is important and relevant in the RfC. Sławomir Biały (talk) 14:09, 25 July 2017 (UTC)

I (?)clearly(?) declared that I am undecided (yet) about the usefulness of a notable effort, and my warnings are quite unambiguously directed at numberphile-like attempts to make non-proofs popular talks of the town, clearly avoiding any contemporary rigorosity, and not against this effort. I oppose at this moment to remove this well-intentioned experiment. Purgy (talk) 07:52, 26 July 2017 (UTC)

Satisfaction by "proofs"

Obviously, a lot of witty students is nowadys hard to convince of the equality 1 = 0.999..., thereby establishing a problem within WP. In no way I want to insinuate that Euler and al. were fully satisfied by the then contemporary treatment of "repeating decimals" as ratinal numbers. Carwil pointed to this question in his comment Monkey wrench and Sławomir Biały expanded on it, and I like the thought, too.

I am absolutly not versed in the history of math, but I dare to ask, if the fancy algebraic manipulations, targeting to pretend the mentioned equality to be true, could perhaps be reported as historic attempts to take on this problem, sourced at a, b, and c, critisized by xyz and uvw, but superficially accepted by wishful thinking, according to d, in lack of the necessary rigorous wrenches, not yet available at this time (dates of Cauchy, Weierstraß ...).

I have heard of mathematicians, specifically about Euler, being very adventurous in successfully expanding methods to uncharted terrain, proving the admissibility only afterwards. Perhaps the witty students can take from this that they have to struggle a bit more to achieve Euler's level, but then will understand, why these blunt manipulations, introduced for historical reverence, are no proof, and why this hard to get claim of equality is nevertheless true.

Maybe, my suggestion allows to keep the "historic" algebra upfront, while giving it mercy on lack of rigor for being simply outdated, thereby calming those who shiver when smelling a limit, reserved for later sections, and pleasing also casual drive by readers.

Honestly, this article is mathematically BORING, it's for them eds. :p Purgy (talk) 13:51, 26 July 2017 (UTC)

I am also not an expert on history of mathematics, but I know the following. Before Georg Cantor (1845–1918), no mathematician accepted to manipulate infinite objects. The modern interpretation of 0.999... is that the ellipse represents infinitely many 9. I do not know if the notation 0.999... were used before Cantor, but, if it was, the ellipse represented an indefinite sequence of 9, that is a large number of 9, which can be enlarged as soon as this becomes useful. This implies that before the 20th century, nobody would write 1 = 0.999..., or if someone has written this equality, is was as an abbreviation for "1 is the limit of the sequence of the 0.999...9, when the number of 9 increases" (I may be wrong, but, if I am, a citation must be provided). At that time, the standard wording was something like "0.999... is infinitesimally close to 1" or "1 – 0.999... is infinitesimally small".

My impression is that the algebraic pseudo-proofs have been invented by the pedagogists, who have introduced the "modern mathematics" and set theory in elementary mathematical courses, during the second half of the 20th century. Maybe I am wrong, but, if not, sources must be provided. D.Lazard (talk) 18:14, 26 July 2017 (UTC)

I think there is very little if anything in this article that had to wait until the 20th century to be done. Michael Hardy (talk) 07:19, 27 July 2017 (UTC)

...999 = – 1

I found in :fr:Talk:0,999... a nice paradoxal consequence of the algebraic pseudo-proofs of this article:

Let X = ...999. Then 10 X = ...9990, and 10 X + 9 = X. Solving this easy linear equation gives X = –9.

How the proponents of the algebraic pseudo-proofs explain to kids why they have to accept that 1 = 0.999... when a very similar argument leads to a result that is blatantly wrong? D.Lazard (talk) 09:51, 27 July 2017 (UTC)

Sorry for invading, I just want to point to a related remark by Dmcq, above. Purgy (talk) 10:44, 27 July 2017 (UTC)

I wouldn't say it's "blatantly wrong" — it's true in the 10-adic numbers, not an extremely useful structure because 10 isn't prime, but the argument, adapted to that context, is correct, I think.

This is the thing — the algebraic arguments aren't really that bad. They just have some missing assumptions. If you assume without proof that every decimal string in fact denotes a number, and that certain manipulations on them work in the obvious way, then the rest of the argument is airtight. To put it classically, they're enthymemes; hope I'm using that word correctly. As long as we say that, I think it's appropriate to start with them, given the intended audience of this article. --Trovatore (talk) 10:23, 27 July 2017 (UTC)

Honestly, I do not want to see necessary premisses shoveled out of a math article, just to have some reason for placing seducing enthymems there, especially, when these are accused of vulgarizing intuition in students (sources!). Afaik, these mechanisms are not only employed in p-adics, but also served in special summations (Ramanujan?) in sketching a target, which still required serious work for formalization. Purgy (talk) 10:44, 27 July 2017 (UTC)

I mean, we could use it to show that some agreed upon set of axioms for manipulating decimal expressions, and including $0.999...\not =1$, is inconsistent. But it seems prejudicial to insist that it is alone the "axiom" $0.999...\not =1$ that deserves to be questioned. In other words, I would be happier if such an enthymeme were presented as a paradox rather than a proof. The resolution of that paradox into a convincing proof is not very conceptually easy. Sławomir Biały (talk) 10:55, 27 July 2017 (UTC)

I wouldn't mind an argument like that in the article. The problem is we'd need a reliable source before we said anything like that in the article and Wikipedia is not a reliable source. Dmcq (talk) 10:49, 27 July 2017 (UTC)