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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)

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)

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)

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 X = –1 (late fixing of a typo, 09:51, 28 July 2017 (UTC))

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. Furthermore, it is not difficult to generate other paradoxes of a very similar nature (Dmcq and Lazard) where it is indeed the other axioms that need to be questioned. I point out that the notation $...999$ does make sense, as a series or supremum, in the extend real number system, and so the Dmcq-Lazard argument is an equally flawed algebraic "proof" whose result turns out to be false in that interpretation —because we cannot subtract by infinity, but of course may be a valid proof in other situations such as the p-adic numbers. It is clearly the Archimedean property that distinguished the proof offered by Dmcq-Lazard, from the subject of the article, and so cannot be argued away by an apeeal to a "reasonable" set of axioms that does not include the axiom of Archimedes. Sławomir Biały (talk) 15:19, 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)

In my eyes that's more of a pseudo paradox than real paradox for kids, at least I see no reason why kids would accept 10 X = ...9990, as they learn multiplication means adding zero at at the end only for integers or left to the decimal point and not right of it.--Kmhkmh (talk) 16:20, 27 July 2017 (UTC)

I don't understand your point. In ...999 there is no decimal point; thus the decimal point is supposed to be implicit at the right, and multiplication has to be done as for integers. D.Lazard (talk) 18:02, 27 July 2017 (UTC)

I think I misread your notation and I'm probably still not quite clear what it is supposed to stand for, in particular for kids. If ...999 has no decimal point than we're talking about an arbitrary large integer only consisting of 9 as digits, i.e. a weird way of notating $\infty$?--Kmhkmh (talk) 21:04, 27 July 2017 (UTC)

Please, note that a good deal of reason for all the above quandaries of readability and correctness is based on this "weird way of notating" anything by "..." As long as there is no rigid definition of the object under scrutiny, everyone from kid to emerited non-matematician is endangered to perceive different objects wrt each others, and thus emerge conundrums. Imho, it's about "lying to children" by leaving them in unawareness vs. presenting "hard facts", with "hard" also pertaining partly to readability. Definitions are there to provide "unique meaning" to "weird ways of notation" (vaguely speaking).

For examples of even more embarassing dot collections may I point you to "continued fractions" (right and down) or "power tower" (right and up), or still worse, to "infinite matrices" for functions, employing both, all in the intention of furthering understanding in newbies? Purgy (talk) 07:50, 28 July 2017 (UTC)

• Also, we can take a number that has a 9 at *every* decimal position, positive or negative (Y = ...999.999...). Obviously, 10*Y = Y (by moving the decimal point we get a number that still has a 9 at every decimal position), so it should hold that Y = 0. Just as obviously, if we add 1 to X = ...999.0, we get a number that has 0 at every decimal position, so X must be -1. (I was wondering why the opening sentence said that X = -9, before realizing that it was a typo: 10*X+9 = X, so 9*X = -9 and X = -1.) - Mike Rosoft (talk) 18:06, 27 July 2017 (UTC)

Elementary proof

I have replaced section "Motivation" (which I have added myself to the article) by a section "Elementary proof". Here is the rationale.

I have been confused by the old version the article, and by the whole discussion, which suggested that the article was about infinite objects (infinite decimal representation). This confusion was supported by the algebraic pseudo proofs, which consist of manipulating infinite objects, without defining them neither the operations that are available for them. I have thus introduced the motivation for trying to explain why it is relevant to consider these object.

Presently, we have an accurate definition of 0.999..., which is conform with the history of the notation, and it appears that one does not need to manipulate infinite objects for proving 0.999... = 1. Therefore, this section "Motivation" seems unneeded.

On the other hand, the article deserves to have an elementary proof of this equality. By elementary, I mean using only properties of ≤ and the operations of (finite) decimal numbers. This is such a proof, which is the object of this new section. This proof is very simple, and is also simple to explain, even to young kids, by a drawing on the number line. Sławomir Biały, could you adapt to this the file that I have kept.

I apologise that I have no source for this proof. It is so easy that I am pretty sure that such a source exists. In any case the algebraic pseudo-proofs are not sourced either.

It is amazing that so many people have discussed on 0.999... = 1 and the way to present this equality to kids, without realizing that it is easy to prove it without any reference to any calculus concept. D.Lazard (talk) 16:39, 31 July 2017 (UTC)

That's much better than what it replaces, which I had a lot of problems with. It might be somewhat daunting to some people - kicking off the article with a rigorous proof relying on a analytical property you don't prove, but instead link to - but it's a significant improvement nonetheless. You are correct to fear an objection on the grounds of original research, but I wouldn't challenge it on those grounds. I disagree that it doesn't rely on real analysis, since it assumes that no number is smaller than the inverse of an integer. That's something that anyone accustomed to such proofs would take as a given, but doubters might not. Nonetheless, I changed the portion about what's required for formal proof to say, "such as rigorous proofs relying on non-elementary techniques, properties, and/or disciplines" rather than mentioning real analysis specifically. Calbaer (talk) 17:49, 31 July 2017 (UTC)

The article now is in a far worse state than a month ago. Previously it started with the most elementary of proofs, ones you might encounter in high school and which require only a knowledge of recurring decimals. But now the first proof is a far more advanced one, using notation and a degree of formalism not encountered until university. This makes the article far less accessible than it was. As I noted a week ago, it’s important this article is accessible as possible, as one of the few mathematical FA, on a topic easily understood by someone with high-school mathematics.--JohnBlackburne^words_deeds 09:25, 1 August 2017 (UTC)

Surely the section on "Algebraic proofs" cannot possibly count as proofs without some version of the Archimedean property. I'm all for making the article as accessible as possible, but not at the expense of obscuring the most important aspects of the subject. Sławomir Biały (talk) 10:41, 1 August 2017 (UTC)

This "elementary proof" is unsourced, so it should be removed per wp:NOR. With a source, it is welcome. I also think that all the unsourced algebraic pseudo-proofs should either be backed by a solid source, or removed. Failing that, this discussion will never end. - DVdm (talk) 09:56, 1 August 2017 (UTC)

The algebraic pseudoproofs appear in the sources cited in the discussion, by Richman, Byers, and Peressini and Peressini. However, as I have said in the RfC, the section does not currently neutrally summarize those sources. Sławomir Biały (talk) 10:04, 1 August 2017 (UTC)

Ah yes, indeed, sorry. Could you try to create a few inline citations and stick them somewhere? TIA.

Of course, that other new section still really needs a solid source. - DVdm (talk) 10:09, 1 August 2017 (UTC)

The paper Does 0.999… Really Equal 1? is fairly close in structure to this article. It gives a proof but only mentions that it used the Archimedean property a bit later. I agree that using the Archimedean property is a better route rather than going through the completeness property of the reals even though some people derive the Archimedean property from the completeness. Dmcq (talk) 10:51, 1 August 2017 (UTC)

Yes, that seems closer to the original structure. However, I note that here (as in other such sources) the elementary proofs are not presented as altogether convincing. ("maybe 0.333... doesn't equal 1/3", "skeptics might reject the equality by claiming that not all numbers can be subtracted from one another!") He clearly presents the resolution of such apparent paradoxes by giving 0.999... a meaning as a limit. Also, it seems to me that completeness cannot be totally omitted from the discussion, since it is completeness that ensures that the notation "0.999..." is meaningful as a number. I note that some editors have been waving the "repeating decimals" magic wand (notably JohnBlackburne and Trovatore), but I am uncertain what repeating decimals means to them if not via limits and completeness. I have concerns that completeness is also foregone in the new "Elementary proof" section. There are algorithms for constructing rational numbers from repeating decimals, but the "Algebraic arguments" make no reference to these algorithms. (And indeed, there is actually nothing to prove in that case: $0.999...=1$ is simply true by definition.) Sławomir Biały (talk) 11:20, 1 August 2017 (UTC)

I don't think any proof can be skeptic-proof; as I noted, the new proof requires one to buy that there's no positive number less than 1/x for all integers. I'm reminded of skeptics talking about the mythical 0.000...1, being convinced that 1-0.999.. is that number, an infinite number of 0s followed by a 1, which presumably would be less than any 1/x. Most other people don't seem to have a problem 0.333... = 1/3, so that demonstration helps bridge a gap for some people, even as if leaves other gaps open (i.e., assumed non-axiomatic properties). In spite of the improvements of the past few days, I agree with JohnBlackburne that the article is in a worse state than a month ago. I'm hoping that this is just a matter of transition, though.

Not using the word "proofs" for the series of equations at 0.999...#Algebraic proofs is one thing. But leaving them until after a formal proof (deceptively called "elementary," as though it's easier to understand than those in 0.999...#Algebraic proofs) or a discussion of a tangentially related philosophical problem of Achilles and the tortoise is going to lose people. (I call the now-removed tortoise section "tangentially related" because of its many differences: twos instead of tens, a moving target instead of a stationary target, and physics/philosophy versus pure math.) The only question is how many people we'll lose. It might be worthwhile to leave up the section to satisfy those who believe it's important and to see what type of reaction we get for it. I think it's interesting, but the OR and readability concerns remain. (And "original research" isn't just an arbitrary concern; if no one else has used this to bridge understanding, there might be a reason. After all, we're coming from the other side of understanding, trying to put ourselves in the shoes of someone who doesn't buy 0.999...=1, which is a different position than a math educator who has to deal with this problem all the time.) Calbaer (talk) 14:34, 1 August 2017 (UTC)

Except the algebraic proofs don't actually prove the theorem, which is that one is the smallest number greater than all of the finite truncations 0.9, 0.99, 0.999, etc. In this respect, they are manifestly deficient. If anyone wants to clarify what actually is proven by the algebraic proofs, I would be happy to comment at greater length. Sławomir Biały (talk) 15:20, 1 August 2017 (UTC)

For those who find that the elementary proof is too difficult, I have split this section into two subsections "intuitive explanation" and "formal proof". This allows the reader, who does not really care of proofs, skipping the formal proof. Although this is not the objective of this article, this may allow also helping some reader to better understand the difference between a proof and an explanation. D.Lazard (talk) 16:29, 4 August 2017 (UTC)

Introduction

I greatly shortened the introduction, which was too long and had a number of unnecessary, confusing, and even wrong content. For example, it implied that 0.999... was a number other than 1 in the hyperreals. (I'd also argue that the current body doesn't make this clear either, but that's another matter.) It also gave a definition of decimal representation different than that in the decimal representation article. (It's actually technically incorrect as a definition, since "smallest number that is greater than all of the finite truncations" may not exist, e.g., for a decimal like 0.900....) However, it is useful to give a correct but short definition. I'd suggest this as a short post-introduction section:

Natural numbers can be represented by the sum of each digit multiplied by the specified non-negative power of 10 for its position, e.g., $409=4\times 10^{2}+0\times 10^{1}+9\times 10^{0}$. A decimal representation adds a fractional part to extend this to real numbers. Each fractional part corresponds to the unique real number which is the (potentially) infinite sum of each digit multiplied by the specified negative power of 10. This can be shown to correspond to (and is occasionally instead defined as) the smallest number no less than the finite decimal representation of all truncated versions. In the case of 0.999... (with truncated versions 0.9, 0.99, 0.999, etc.), this smallest number is 1. This is rigorously proved below.

Calbaer (talk) 18:21, 29 July 2017 (UTC)

This fixed some problems with the earlier lead, but others still remain. A proper definition of the subject should appear. Sławomir Biały (talk) 19:28, 29 July 2017 (UTC)

Something like the first equation at 0.999...#Infinite series and sequences could be added with appropriate text to the above and put under a "Definition" section just below the introduction. One problem is how far you want to go with the definition. You could define it as an infinite sum, but you'd have a question of notation and you then might want to define infinite sums. Assuming that's defined in terms of limits, you then might want to define limits. If you properly define limits, then you'd have a fair bit of text and probably lose some folks, people who'd be left wondering why the definition of a simple decimal is going into college-level math. It's probably best to leave off at infinite sums, then use the properties infinite sums in the aforementioned section. I would certainly avoid using sigma notation in the first section. Calbaer (talk) 03:46, 30 July 2017 (UTC)

Re Introduction I want to emphasize the sole justification of the existence of this article:

it's about the ed's problem in teaching the kids the right way of thinking about math notions!

The real reals are hidden behind the rationals, and low level, dubious articles about "repeating decimals" with all their idiosyncrasies dominate even the rationals. One can easily see this problem also in how rules about irrational exponents are taught: almost no one dares to mention that the extension to real exponents is beyond this level of rigor.

Therefore, I miss the mentioning of the lack of those deep formal notions, which are indispensably necessary for a strict treatment of this notation "0.999..." At High school level and even beyond, this lack is responsible for the extensively researched trouble, which many, and not the silliest, students have, to take this equality for granted.

Trying to sweep this lack under the rug (again!), by dealing at a just cursory level with indispensable necessary, but higher/deeper facts, misses the raison d'etre for this article. This article's justification is NOT in pure math, but in a flaw in math education, and therefore, is bound to point to these difficult notions, otherwise best avoided with the uninitiated. Purgy (talk) 08:03, 30 July 2017 (UTC)

(edit conflict) I appreciate the shortening of the lead. However the new lead suffers of several issues. Firstly, it is (implicitly) too technical, as repeating decimal is defined in terms of decimal representation, which is itself defined in terms of a series, a concept which is clearly outside of the knowledge of the intended audience. Secondly, the phrase "the real numbers, the system over which 0.999... is defined" is wrong, as 0.999... is well defined over the rational numbers, without any need to refer to real numbers. Thirdly (IMO, it is the main issue), the new version reintroduces the confusion between a notation (0.999...), a way for representing a number (repeating decimal), and the number itself. If the phrasing is not clear about this, there is no hope of not confusing the layman.

I'll try to fix these issues. For this, I'll reintroduce in the lead the the definition as "smallest number no less than". By the way, I disagree with the comment "occasionally instead defined as", as infinite decimal expansions have not originally be introduced through series, but as successive lower approximations of the number to be represented. This view of the decimal representation, is clear when considering long division (continued after the decimal point), and the common way of truncating when allowed accuracy is sufficient. It is true that the definition in terms of series is also widely used, as being easier for experimented mathematicians, but people, who can understand it, can also understand easily that both definitions are equivalent. D.Lazard (talk) 08:21, 30 July 2017 (UTC)

Regarding Purgy's comment, I don't think that shortening the introduction and adding a brief definition section can be regarded as sweeping anything under the rug. It's just, as D.Lazard notes, we don't want the introduction to be too technical.

Regarding D.Lazard's comment, I'm not sure about the best way to distinguish representation versus what's represented. On the point of the definition being tied to infinite series and that being an advanced topic, there's really no way around infinite sequences; it's going to be defined via infinite sum (which is defined via infinite sequence) or infinite sequence. (I must say that, personally, I had infinite sums in high school - perhaps before then informally - but I didn't know what "infimum" meant until grad school. That's likely because limits are necessary to mention before elementary calculus, while inf/sup/lim inf/lim sup are not. While "unique lowest number not less than an infinite sequence" might seem like a simpler concept than limit, I'd suspect that it is a concept less familiar to a general audience.)

As for which definition is used, I relied on the Wikipedia articles for repeating decimal and decimal representation for that. If you insist that they're both wrong and it should be defined in terms of sequence, not sum, then that's a larger Wikipedia problem. If you like, "occasionally defined" can be replaced with "equivalently defined." But I don't think you can get around mentioning repeating decimals, whether or not you want to talk about rationals (which all repeating decimals are) rather than reals (which allow you to talk about limits and infimums). Calbaer (talk) 14:20, 30 July 2017 (UTC)

I had concerns about the old lead that are adequately molified in the present revision. I think it is not a problem to use whatever definition is more likely to be understood by the average reader. Of course, the article should discuss both such definitions and their equivalence. But if we assume no familiarity with limits in a like reader of the lead, then a definition as an infinite series does not sufficiently clarify the meaning, I think. Sławomir Biały (talk) 15:33, 30 July 2017 (UTC)

I repeat my stance on this whole article, that its existence is solely justified by educational problems with the abstract foundations of the reals, as used predominantly in contemporary math. There is no math problem within this whole topic. The widespread troubles, giving the article its name, cannot be soundly remedied by calming down the burdened readers with "repeating decimals" (in their current state) and "algebraic proofs". Neither the disbelievers for ignorance of higher math (e.g., limits), nor the disbelievers for reasoned opposition (e.g., finitists, or N.J. Wildberger) will accept these "arguments". I do believe that the lead should contain a statement, making explicit that just being "acquainted" to repeating decimals, and cursorily brushing over "infinite" algebra, does not suffice easily to deal with the gap from rationals to reals, readying for the treatment of infinity. It's not that there should be some "Abandon hope, all ye who enter here.", but I'd prefer something that unambiguously states the necessary technical requirements to resiliently deal with this matter. This is not to say that I wouldn't prefer a short lede, but, yes, I do consider the "algebraic arguments" as blatant "sweeping under the rug", as I do with the use of "repeating decimals", even when I also regard these as nice heuristics. Purgy (talk) 09:49, 31 July 2017 (UTC)

The article is here because the topic satisfies WP:Notability. It is not our job to preach to people or cure the ills of the world, the aim is to produce an reasonably neutral and reliable encyclopaedia. Yes we should emphasize mainstream views, but we're not supposed to bowdlerize and censor things in order to to eliminate erroneous thought and avoid the misdirection of the weak minded like many on the fringe theories noticeboard seem to think. What we should do is follow the sources and try to make articles readable especially their introductions. Dmcq (talk) 11:02, 31 July 2017 (UTC)

Most humbly I beg pardon, especially from those, in whose perception I caused the impression of preaching, ills-curing, bowdlerizing, of censorship, or of elimination of whatever misdirection. My pure intention, so I -also most humbly- assert, is to have this highly desirable "reasonably neutral and reliable encyclopaedia", readably following the sources, especially in its introductions. Sooo sorry! Purgatorio (talk) 11:46, 31 July 2017 (UTC)

It doesn't seem like the text above is getting much notice (except for the comment objecting to "occasionally instead defined as") compared to the new intro section. I added a comment to the intro that one of the reasons for doubt was the need for real analysis in formal proofs, and changed "defined over the reals" to "defined within the reals," since my point here was not to assert that real numbers were required to define 0.999..., but to say that 99.9+% of the time you'd see 0.999..., it would be the number within the real number system, not within some other system with different axioms. If there's a clearer way to say this (without the unverifiable 99.9+% figure, of course), then the text should be altered accordingly. I also took off the header tags since the main discussion died down and any revitalized discussion should see the tags added to the specific section(s) they apply to rather than implying that the entire article is shot through with wrong information from start to end. Calbaer (talk) 15:59, 31 July 2017 (UTC)

The new section that I have just added shows clearly that there is no "need for real analysis in formal proofs" of 0.999... = 1. Thus a large part of the lead deserves to be rewritten.D.Lazard (talk) 16:52, 31 July 2017 (UTC)

I voice my opposition here to the proposed addition. Once more, the problem is not "decimal representations", it is "real numbers". This section merely serves to perpetuate the wrong belief that understanding real numbers (or even understanding rational numbers) boils down to magic with decimal representations. Sławomir Biały (talk) 10:26, 5 August 2017 (UTC)

It is unclear if your opposition is against section "Elementary proof" (which has been recently added, and does not really rely on decimal representation) or section "Decimal representation" that you have just removed. I agree with this removal, but not exactly for the same reason. The main reason is that it does not make clearly the distinction between finite and infinite decimal representations. If this section would be exclusively about finite decimal representation, it could be fine at this place, as the properties of these finite decimal representation are used in the definition of 0.999..., and in the elementary proof of its equality with one. However I cannot say wether recalling here properties of finite decimal representation is useful or not. But this section is not about finite decimal representation, or, more precisely, it does not make a clear distinction between finite and infinite decimals, which are very different objects. A section about infinite decimal representation and their relation with real numbers could be useful, but not as the first section of the article. Moreover, as all the numbers that are considered in this article are rational, I am not sure that this article is the place for discussing real numbers. D.Lazard (talk) 11:04, 5 August 2017 (UTC)

The opposition is to the added "Decimal representation" section. To think of $0.999...=1$ as an equality of rational numbers (as in the "Elementary proof" section), then part of that proof should consist of showing that the notation $0.999...$ actually refers to some specific number: that is, that the least upper bound of the sequence $0.9,0.99,0.999,\ldots$ exists. It seems like this is at the heart of the matter, and confusing peripheral discussions about magical infinite series do not clarify this essential point. Sławomir Biały (talk) 12:21, 5 August 2017 (UTC)

When I proposed the section, the response was neutral to positive, so I incorporated the feedback, and added the section. If you don't like the title, that can easily be changed, e.g., to "definition" or "definition of decimal representation." But we have to at least define our terms before jumping into proofs; in fact, that was your only reply to the text of this section in the first place!

The definition in the introduction is insufficient and artificial. Why is 0.999... = sup (1-0.1ⁿ)? That's not the definition of decimal representation I've seen anywhere, including the Wikipedia page on decimal representation. Yet it's the definition used in both the introduction and the first proof. Even though it can be shown to be equivalent to the standard (limit-based) definition, we should at least acknowledge the difference. Because people will notice if we choose an alternative definition out of convenience rather than the one used elsewhere, whether on Wikipedia, math textbooks, of elsewhere. An obvious gap in logic - one that looks like cheating - is what we want to avoid in this article. I'm not sure why you had no problem when you replied to the section a week ago and now you suddenly find it "misleading and unhelpful." It should at least be somewhere in the article, preferably at least linked to earlier in the article, so that people don't wonder why this article uses a different definition of decimals that that found most other places, including Wikipedia itself. Calbaer (talk) 14:44, 5 August 2017 (UTC)

The equivalence between the limit and the least upper bound for an increasing sequence is one of the first results which is proven when teaching limits. I have recalled this equivalence by a note in the article. In a textbook which introduces infinite decimal representation after having introduced limits, it is natural to use limits. On the other hand, if an elementary textbook use limits before defining them or without defining them for a public who is not supposed to know limits, then this textbook is certainly not a reliable source for mathematics. D.Lazard (talk) 16:01, 5 August 2017 (UTC)

You made an edit to the introduction, which was neutral, and also proposed text for discussion. Those were two different things. It was not clear what you intended to do with the above paragraph, so generated little input. If you would prefer the earlier sections of the article to focus on limits instead, including a detailed discussion of the completeness axiom and existence of limits, we can do that too. But that is contrary to the goal of presenting an elementary argumemt. Sławomir Biały (talk) 17:21, 5 August 2017 (UTC)

I don't think there's anything unclear about "I'd suggest this as a short post-introduction section." I'm not sure how I could have been even more direct. Calbaer (talk) 17:26, 5 August 2017 (UTC)

I'm fine with having a note, though I've altered it for language and accuracy. Limits are generally taught using delta-epsilon definitions. It's true that one can show the equivalence between the two definitions and that a course might opt to do so very soon after defining limits. However, that requires similar machinery to showing 0.999... = 1. So jumping from one definition to the other is assuming much of which we need to prove, and glossing over that fact is undesirable. As far as defining repeating decimals before defining limits, I've not seen formal definitions for them before that; if a pre-pre-calculus textbook discusses repeating decimals, they generally do it similarly to the Encyclopedia Britannica article linked to elsewhere on this page, i.e., informally. I haven't seen a formal definition in terms of upper bounds outside of this article. (I'm not saying they don't exist, but, if they do, it seems like that would be good to reference here.) Calbaer (talk) 17:26, 5 August 2017 (UTC)

True, that's not unclear, but the discussion was chiefly focused on the edits you made to the article. Now we're discussing the thing you just added, and so far two editors have objected to it. I think the question of whether the notation is defined as a limit or supreme is largely a red herring. But it has been suggested by several editors that a goal of the article should be as accessible as possible. Since the least upper bound property precedes limits in the treatment of the real number system, it seems likely to be more accessible. Furthermore, as already noted, one does not need limits or the completeness axiom. No argument or definition using limits is elementary, for the purposes of the intended audience, as these rely on completeness properties of R.

Most sources define the decimal expansion separately from the question of the meaning of the decimal as a number. Apostol's Calculus defines the real number represented by a given decimal using the least upper bound property. This is also implicit in his Mathematical analysis, where he writes "The fact that a real number might have two different representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum." Stillwell leaves the matter of supplying the interpretation as an exercise. Brannan "A first course in mathematical analysis" explicitly defines the real number associated with a given decimal as a supremum, and writes in particular: "Truncating each of these decimals and forming the sums, we obtain the set {0; 0:9; 0:99; 0:999; ...} The supremum of this set is, of course, the number 0.999...=1, which is the correct answer." Sławomir Biały (talk) 18:35, 5 August 2017 (UTC)

The desideratum of 0.999... = 1

Under the hypothetical assumption of "0.999... ≠ 1" all the algebraic efforts lead to a contradiction, thereby disavowing the generalized algebraic rules within "infinite" long division or "infinite" multiply through, which are highly plausible, and even "true", but which are neither trivial, nor proven to be de rigeur in this here context of purely algebraic steps. As the discussion shows, these generalisations are not provable in a simplistic way. For the sake of upholding a useful and correct(!) plausibility, it is therefore obviously desirable, that the equality "0.999... = 1" holds. Any correct plausibility is of high educational value for creating ease of understanding, but the more critically minded readers deserve a hint, in which steps of the argumentation nontrivial extensions are necessary. I hope this makes the spirit of my edit more clear and generally perceiveable.

The use of the word "show" -to me in math context synonymous for "prove"- is absolutely against my intended spirit at this place. Of course, I am convinced that my non-native wording is improvable. Purgy (talk) 13:01, 6 August 2017 (UTC)

The problem with "obviously desirable" is that it sounds like "We'd like it if this were true, but maybe it's not," rather than "This is true, and here's an argument that indicates, but not proves, that it could be." I'm not sure what is wrong with "attempting to show," since that language indicates that they're not proofs (due to the "attempting"), but also doesn't imply they're wrong (which would be improper, so each step can be proven correct). Perhaps you missed the "attempting" and just reverted when you saw "show." If not, perhaps there's a way to avoid both and be precise, e.g., "The following algebraic operations, though both intuitive and correct, contain implicit assumptions about the definitions and properties of the involved terms. In fact, properties such as 0.111… = 1/9 require proofs of similar form and complexity to that of 0.999… = 1. Thus, without formally establishing these definitions and proving these properties, such operations do not constitute complete proofs of 0.999… = 1. Nevertheless...." Calbaer (talk) 14:21, 6 August 2017 (UTC)

Sounds a bit like Saccheri trying and failing to prove the parallel postulate ""the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines". Dmcq (talk) 16:25, 6 August 2017 (UTC)

I think that the mindset of wishful thinking described above by Calbaer with "We'd like it if this were true, but maybe it's not." is nothing to be deprecated, but even prevails in many mathy situations, motivating to construct the proofs, necessary to bridge the gaps, left open by appealing to desirable plausibilities. The sequence of algebraic steps, presented here, imho, do not even resemble an attempt of proving the desired facts, since all of the serious difficulties in the pertinent proofs are not even hinted to, but casually, if not with aforethought, brushed over, for the lines to fit in the mentioned context of informal, elementary education, looking like a proof to the uninitiated and careless reader.

For the time being I stop commenting on this topic at my sole discretion, because of (traditionally?) perceiving repeated and unprovoked hostility in Dmcq's comments above. Purgy (talk) 06:27, 7 August 2017 (UTC)

The problem with the algebraic proofs is that they beg the question. They can be used to establish that certain hypothetical properties of infinite decimals are mutually inconsistent, but they do not show that anything in particular is "true" of infinite decimals, because they do not specify the meaning of infinite decimal expressions. I have attempted to clarify what axioms are required in order to make the proofs "valid". Chiefly, these are that multiplication of infinite decimals behaves as expected (which is inconsistent with the "simple division" definition given in the first proof) and that the expression 0.999... is a well-defined number. Sławomir Biały (talk) 12:42, 7 August 2017 (UTC)