Talk:0.999...

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Updated the introduction

I've updated the introduction to make it clearer that in conventional mathematical usage "0.999..." is simply a shorthand abbreviation for the limit of a specific convergent series. This should hopefully head off the common misconception that 0.999... can have some other significant meaning other than as a limit of a series; or if the reader does not believe in this interpretation, make it clear to them at this early point that this is where they depart from the standard framework of mathematics. -- The Anome (talk) 12:21, 28 March 2008 (UTC)

I think we need to be careful here. 0.999... is not a shorthand notation to the limit of (0.9,0.99,...) or to the series $\sum _{k=1}^{\infty }9\cdot 10^{-k}$. It is a shorthand notation to the real number represented by the decimal expansion with zero integer part and all fractional digits 9, which is simplest to formalize as $a_{n}={\begin{cases}9&n\in \mathbb {Z} ^{+}\\0&n\in \mathbb {Z} \backslash \mathbb {Z} ^{+}\end{cases}}$. It can then be trivial, depending on how we have defined the whole thing, to prove that this is equal to the limit, or to the sum - but this is still a distinct step. -- Meni Rosenfeld (talk) 20:53, 29 March 2008 (UTC)

I've now changed the wording to:

“ In conventional mathematical usage, the value assigned to the notation "0.999…" is the real number which is the limit of the sequence (0.9,0.99,0.999,0.9999,…), ... ”

Is this any better?-- The Anome (talk) 09:18, 30 March 2008 (UTC)

I suppose so. -- Meni Rosenfeld (talk) 09:52, 30 March 2008 (UTC)

I've now removed the TeX formula, after realizing that it didn't improve the argument: it's confusing for the non-mathematician who does not yet know the notation, yet annoying to mathematicians striving for a rigorous treatment, since stating this apparently obvous formula actually jumps over several logical steps by making numerous hidden assumptions. -- The Anome (talk) 07:56, 31 March 2008 (UTC)

The section is certainly reading better! While you're at it, could you have a look at the phrase "this can be formally demonstrated using a closed interval defined by the above sequence and the triangle inequality"? With apologies to whoever wrote it, it's a pretty ambitious claim given the context, and it could use either a clearer explanation or scaling back. Melchoir (talk) 08:19, 31 March 2008 (UTC)

I've chopped it out, and instead just appealed directly to the Archimedean property of the reals, and provided a link to that article (which uses least upper bounds to prove it) for those seeking more detail. -- The Anome (talk) 09:29, 31 March 2008 (UTC)

Okay. I don't understand this recent revert though. It's true that no real is smaller than all reals, but this is trivial and hard to exploit. What's interesting is that no real is smaller than all rationals, which is almost exactly the Archimedean property. Melchoir (talk) 17:04, 31 March 2008 (UTC)

I don't understand it either. Can someone explain it or un-revert it? Tlepp (talk) 16:00, 2 April 2008 (UTC)

You're probably right, but I think introducing rational numbers is unnecessary. They aren't mentioned in the definition of the Archimedean property as far as I remember (although you could reformulate it in terms of them), so let's not confuse the matter by introducing them here. Perhaps it needs to be phrased differently, but I think mentioning a completely different field is unnecessary. --Tango (talk) 16:13, 2 April 2008 (UTC)

I gave it shot by introducing some vagueness. It's not ideal, but it's still better than saying that Archimedes is necessary to show that the only nonnegative real smaller than all positive reals is zero. Melchoir (talk) 21:27, 6 April 2008 (UTC)

Dubious caveats

I have a big problem with the following recent addition to the Real analysis section:

All proofs given above have certain problems and aren't really rigorous mathematical proofs. Let's take a closer look.

1. The proof on fractions assumes that $1/3=0.333...$, how do we know it's true? If any decimal is equal to 1/3 it is $0.333...$ but perhaps no decimal is equal to 1/3, in which case that proof fails.
2. The proof on digit manipulation assumes that the obvious digit manipulations are valid. The result $1=0.999...$ might cause us to reject this assumption. The manipulations can be justified by more fundamental considerations, but these also establish $1=0.999...$ so the proof is unneeded.
3. The proof on infinite series says: "The last step — that lim ¹/_10ⁿ = 0 — is often justified by the axiom that the real numbers have the Archimedean property." Here we have some axiom, that magically solves the infinitesimal problem.
4. The nested interval proof uses the nested intervals theorem, which is just another form of Archimedean property.

1. Use the division algorithm, duh!
2. Why should the number 0.999... behave differently from all other real numbers? If we accept that the real numbers form a field, as everyone normally does, then those digit manipulations are necessarily correct.
3. The wording "magically solves the infinitesimal problem" is suspicious in itself. Actually, the Archimedean property is not introduced into mathematics ad hoc to save the 1 = 0.999... equality. The Archimedean property is a fundamental property of the real numbers in their standard formalization. Many people would even add that it's intuitive.
4. "Just another form of Archimedean property"?! How glib! Are we really expected to brush aside the whole of the modern construction of the real numbers just like that? FilipeS (talk) 18:58, 30 March 2008 (UTC)

1. Division can only give you a terminating approximation - the "repeat ad infinitum" bit is a little dubious (it does work, but it's not trivial).
2. How do we know other non-terminating decimals work like that? It's non-trivial for all of them.
3. I agree, it's not an axiom, it's a fundamental property.
4. Once you've dismissed 3 as not being a problem, 4 is moot really.

--Tango (talk) 19:06, 30 March 2008 (UTC)

Division can only give a terminating approximation? I'm not sure I agree with that. (My new section on the Arguments page would be a great place to discuss this.) Gustave the Steel (talk) 19:23, 30 March 2008 (UTC)

With a finite number of steps, anyway. You can spot the pattern and add the "..." to the end, but that's not particularly rigorous (it can be made so, of course). --Tango (talk) 19:42, 30 March 2008 (UTC)

Back in the early days of this article, one of my personal goals was to get rid of all the "that's not a proof, this is a proof" sentiment. There are many possible logical foundations for the standard real numbers, all of them provably equivalent by design. Logically, there is absolutely nothing wrong with a proof that exploits known properties of the reals, whether they are proved to hold in some model or taken as axioms.

For educational purposes, there is a range of viewpoints on whether axioms or set-theoretic constructions are more enlightening for the student. This is why I wrote the extended descriptions in the References section of authors' differing approaches; compare for example Munkres vs. Pugh. A reader who cares may trace a given proof back to its author, and then to the author's school of thought.

The offending passage not only obliterates these subtleties, it doesn't even claim to represent a notable viewpoint in logic or education. It should be removed. Melchoir (talk) 20:57, 30 March 2008 (UTC)

Agreed on removal. –Pomte 21:09, 30 March 2008 (UTC)

It shall be so. Melchoir (talk) 22:00, 30 March 2008 (UTC)

I disagree. If you want to get rid of sentiments, this is probably the worst math article. This passage tries to motivate set theoretic definitions below it. Digit manipulation is just plain wrong. Axiomatic approach may be pedagogically good, most people might think Archimedean property is intuitive, but for a non-believer it's not convincing. (Stupid analogy: It's intuitive to think world is flat, so lets assume axiom of flatness.) If the choice of words is offending, change them, don't remove. This article maybe your baby, but it's irrelevant. Do you really have to enforce your orthodoxy? Change it back. Tlepp (talk) 22:57, 30 March 2008 (UTC)

Please explain how digit manipulation is just plain wrong. Melchoir (talk) 23:22, 30 March 2008 (UTC)

With hyperreal numbers sequences (9.9, 9.99, 9.999,...) and (9, 9.9, 9.99, ...) modulo ultrafilter are different numbers. I know this article is about standard reals, but digit manipulation proof doesn't directly use any property of standard reals, so if I (stupidly) reinterpret decimal expansion to hyperreal case, is it still a proof? No. "The validity of the digit manipulations ... follows from the fundamental relationship between decimals and the numbers they represent." The validity follows from Archimedean property, one way or other. I can accept Archimedean property, but this proof hides it. It's a fraud acceptable at high school level. Tlepp (talk) 00:03, 31 March 2008 (UTC)

A fraud at high school level is exactly what this proof is supposed to be. Another point: For our approach to the real numbers, the Archimedean property is not an axiom: You can prove they have this property. The article constructs the reals from the rationals via Dedekind cuts or Cauchy sequences. The real number article doesn't mention the property in its axiomatic approach, and neither does Tarski's axiomatization of the reals. There might be other axiomatizations actually including the property as an axiom, but that doesn't mean we have to care about them here. -- Huon (talk) 00:11, 31 March 2008 (UTC)

The basic complaint I'm hearing is that proofs in the article are not explicit about their assumptions. If that's the case, we can certainly work on it.

The article is sequenced in such a way that proof A makes certain intuitive assumptions; proof B makes more technical assumptions which justify the assumptions of proof A; proof C makes still more foundational assumptions which justify the assumptions of proof B. The chain ends with making certain assumptions about the ordered field of rational numbers, just enough so that we don't have to assume that it's Archimedean either. In principle you could continue the pattern ad nauseum, but we make the (logically arbitrary) choice to end the discussion there, trusting that the reader is comfortable with the rationals.

Hopefully, this means that a skeptical and curious reader is motivated by A to read B, and by B to read C. If not, we could do a better job with the intro to "Real numbers". First, I want us to agree that proof C doesn't makes proofs A and B obsolete or intrinsically fraudulent. They are simply not self-contained; neither is C or any other proof on Wikipedia. Melchoir (talk) 01:18, 31 March 2008 (UTC)

Well put, Melchior. This is what the editor who added these remarks clearly failed to understand. FilipeS (talk) 17:48, 1 April 2008 (UTC)

Why this article is so hard to get right

I believe that the difficulty in getting this article right hinges on a single point. Not only is 0.999... = 1 not -- for any number of reasons -- intuitively obvious, but more importantly the properties of the reals that make it true are not intuitively obvious even to most people with some mathematical training, and need to be proved in order to be believed. It is therefore perfectly reasonable for intelligent people to be skeptical about its validity and to refuse to believe some of the less rigorous proofs.

For example, most of the serious non-troll dissenters end up asserting the same thing, that 0.999... = 1 minus an infinitesimal quantity. As I understand it, this is actually not at all an unreasonable thing to say, since not only is it actually strictly correct, but also, when combined with the Archimedean property, which proves that the only infinitesimal real is 0, actually leads to a valid proof of 0.999... = 1. (While arguing this in a messageboard, I found a theory that said that if a certain progression is true in most cases, it is probably true in all cases i.e. .9+.1=1, .99+.01=1, .999+.001=1 and so on. Its right about then a finite being says I do not understand infinity. Heh. Will find name of theory)

However, the validity of the Archimedean property (or any of the other similar results necessary for a rigorous proof) is not obvious at all, and requires a quite sophisticated understanding of the construction of the reals in order to understand why it is true. All the handwavey proofs such as the digit manipulation proofs also have similar hard ideas buried within them, such as proof by infinite induction or the existence of limits.

This is why 0.999... = 1 is such a stumbling block, and also why this article is so important; rather than being stupid, I believe that many of the serious non-troll dissenters are actually nascent mathematicians, on their own personal mathematical journey that recapitulates the history of mathematics, logically picking away at the underlying structures of mathematics trying to understand why this non-obvious proposition is universally held to be true; and the only way forward for the serious dissenter is to start to get to grips with real analysis. And this is also why this article is so hard to get right; to satisfy the serious non-believer, it in effect has to be a standalone mini-course in real analysis that can be understood by a serious auto-didact with only high school mathematics and basic logical reasoning as tools. -- The Anome (talk) 10:33, 31 March 2008 (UTC)

To do it properly, yeah, we basically need to teach the reader a fairly large amount of real analysis - I don't think that's in the scope of this article, though. Wikipedia is an encyclopaedia, not a textbook, we should be giving a basic introduction to the topic. Someone wanting all the details should go and read an analysis text. Is it worth looking at Monty Hall problem and seeing how it's done there? That's an equally counter-intuitive mathematical result, but they don't have need of an /Arguments page. Is there something better about that article, or is the proof just easier to understand? (It certainly is easier to understand, but is it just that.) --Tango (talk) 14:19, 31 March 2008 (UTC)

OK, I've rewritten the "Introduction" section to emphasize the argument above as the basis of a "gold standard" proof of 0.999... = 1. It does -- very explicitly -- pull the Archimedean property out of a hat to do it, but apart from that, it should be entirely comprehensible by anyone with a high school education. The Archimedean property article -- to which the reader is directed -- then contains a paragraph-long demonstration of the Archimedean property for the reals based on least upper bounds. Hopefully, by segregating the hard stuff and the easy stuff, by making the arithmetic part very simple indeed, and making the hardness of the hard stuff very clear, this may make things easier for some skeptics, who we may be able to move on from arguing from obviousness, to engaging in more technical arguments about the Archimedean property, which should in turn pull them into learning about real analysis. -- The Anome (talk) 15:24, 31 March 2008 (UTC)

Let's not just fixate on the Archimedean property of the reals, though. From the education literature, and from our own experience on /Arguments, it seems like at least as many students are also hung up on the structure of the natural numbers. They would probably benefit from achieving a solid appreciation of Hilbert's hotel before starting to think about infinitesimals. Yet another reason why this article is so hard to get right: there are so many possible gaps in mathematics education. Melchoir (talk) 16:56, 31 March 2008 (UTC)

You mean the whole "the last 9" thing? Yeah, that's a definite stumbling block for some. I'm not sure the best way to correct that misunderstanding - it's such a fundamental misunderstanding of what "infinity" means that I don't know where to start. --Tango (talk) 17:11, 31 March 2008 (UTC)

Yeah, mostly I mean "the last 9", although I'm sure it's but one of many confusions around the naturals. Melchoir (talk) 17:28, 31 March 2008 (UTC)

The problem seems obvious to me. The article currently spends half its length furiously obsessing over the real number set before grudgingly admitting that the actual subject is to be found elsewhere. (in Alternative number systems) The imaginary unit and the color blue do not exist in the real number set either, but have escaped this treatment. The failure to even define the topic before attacking it's existence undermines the credibility of the article, and makes people want to defend the subject from a perceived unfair attack. How often do people use .999... outside the concept of suggesting "the highest value less then one"?" An encyclopedia is a place for words, and aren't words defined by how they are understood? Algr (talk) 22:50, 31 March 2008 (UTC)

So you're suggesting 0.999... should be given a different meaning to that it would have under the standard definition of decimal notation? --Tango (talk) 23:03, 31 March 2008 (UTC)

*sigh*

1. This article is about the real number 0.999..., as it should, since the real numbers are the most commonly used number system.
2. The "actual subject" is not found elsewhere. If you actually looked at the "Alternative number systems" section, you would see that in most suggested systems, the object called 0.999... is not at all the highest value less than one. Richman's decimals, a system in which it is, is not a number system at all - you can't even subtract in it. It's just a lexicographic ordering of sequences of digits.
3. This article is not about the imaginary unit or the color blue, but about the real number 0.999...
4. 0.999... is a particular case of decimal expansions of real numbers. If you feel those are not covered well enough in Wikipedia, take your case to the talk page of that article.
5. The existence of what has been attacked? 0.999... certainly exists. It can also be proven to be equal to 1.
6. If people use 0.999... to mean the highest value less than one, they are wrong. 0.999... is a decimal representation of a real number.
7. Aren't you confusing an encyclopedia with a dictionary?
8. Slang is defined by how it is understood. Words are defined by what they truly mean.
9. 0.999... is not a word. It is a mathematical symbol for a mathematical entity. The "word on the street" regarding what it means is irrelevant to a mathematical article.
10. You have admitted to having little knowledge of the concepts upon which this article is based, so why do you choose to keep poisoning this talk page with your nonsense, instead of studying up on it? Are you afraid that by learning math, you will lose your soul and join the dark side of people who think that 0.999...=1? -- Meni Rosenfeld (talk) 23:26, 31 March 2008 (UTC)

Tango, I am suggesting that you need to respect the reader's intelligence before advocating other ideas. If you can't do that, you will never convince anyone of anything, no matter how "right" you may be. The article grudgingly acknowledges that the only really important thing about .999... is how people understand it, but it is so disrespectful of those who don't agree with it that it causes hostility.

1. The article is not Real number 0.999... or Real number infinity but 0.999.... Since when does "most commonly used" mean "appropriate for this task"? The only reason you want to use real numbers here is that it makes you "right". But this treatment looks a lot more like politics then math, which is why so many people reject it. The only alternative explanation proposed for why people reject the equality is basically "because only us chosen few possess the intelligence to understand the truth." No matter how nicely you rephrase this, history shows that people who think like this tend to do far more harm then good.
2. No matter how many real numbers aren't 5, and no matter how sucky you think 5 is, that doesn't change the fact that 5 is a real number. Richman's decimals exist.
3. furiously obsessing over the real number set before grudgingly admitting that the actual subject is to be found elsewhere.
4. "
5. "
6. "
7. Both are based on people's understanding of words.
8. What does "Gay" mean? Who changed it from it's use in the 1950s? (Remember The Flinstones theme song.)
9. "
10. But the entire article right now is about refuting "the word on the street" - it doesn't say anything else.
11. I have plenty of knowledge about reasonable and fallacious arguments, (some learned from the Moon Hoaxers!) and it was upsetting to see such tactics used here. I have no fear of ideas. I could rewrite this article and make it much more convincing that .999...=1, but it still would know that I was misinforming people.

Algr (talk) 04:14, 1 April 2008 (UTC)

The article 0.999... has a section, Skepticism in education, which is chiefly concerned with "explanation[s] proposed for why people reject the equality". By some terrible oversight, we seem to have neglected to include the statement that "only us chosen few possess the intelligence". How strange that none of the education research journals would have published this theory.

For that matter, your question "How often do people use .999... outside the concept …" was probably intended to be rhetorical, but there is another section, Applications, that answers just that question. Melchoir (talk) 04:52, 1 April 2008 (UTC)

It's true that slang sometimes becomes language if it lasts long enough. I have yet to see 0.9... used as a slang term, however, and I'm not aware of any mathematical errors which became regarded as mathematical facts due to simple overuse. Gustave the Steel (talk) 05:21, 1 April 2008 (UTC)

Melchoir, how would you summarize the way that non-believers are described here? (I don't just mean your posts, but in general.) Algr (talk) 06:20, 1 April 2008 (UTC)

Described where, the article, this talk page, or /Arguments? Melchoir (talk) 06:53, 1 April 2008 (UTC)

Algr, could you cite some references for your assertion that 0.999... is most commonly used to refer to something other than a real number? I'm sure you'll agree that 0.5, 0.333..., 0.125, 3.14159..., etc. are all used to refer to real numbers, so why would people use 0.999... to refer to something else? I'm quite certain that people usually use it to refer to "0 units, 9 tenths, 9 hundreds, 9 thousands, etc.", since that's what the notation means in every other context. --Tango (talk) 08:58, 1 April 2008 (UTC)

Gustave the Steel, how about mathematical 'error' called Axiom of Choice, which has became regarded as mathematical fact due to simple overuse. There are many proofs why it is an 'error' (=contradicts intuition), see Banach–Tarski paradox. It is now 'fact' because those who reject it are minority. Fair amount of skepticism isn't bad thing, it is essential for any science. Tlepp (talk) 10:06, 1 April 2008 (UTC)

Tlepp, this is a warped description. AC is not, and is not considered by anyone to be, a "fact". It is an axiom which is independent of ZF, and can be included or excluded (or its negation can be included) in an axiomatic system we are investigating. Most people find that the overwhelming reasons to consider it intuitive far outweigh the few reasons to the contrary, and thus find it more interesting to add it to the list of axioms.

Skepticism towards unfounded claims is important, but it is not the same as taking up arms against proven mathematical theorems. -- Meni Rosenfeld (talk) 12:25, 1 April 2008 (UTC)

Here is more warped descriptions. Do you agree that axiom of Dedekind-completeness is not a fact and it can be excluded (or it's negation included) in the axiomatization of real numbers? As a consequence "0.999... = 1" (or it's negation) is not a fact, but a hypothesis that can or can't be proven.

"Only us chosen few possess the intelligence to understand ... that truth is not god given but declared by man, and the reasons why this is the only right declaration are so profoundly difficult and complicated that any attempt to explain it in a wikipedia article is doomed to fail." Tlepp (talk) 09:40, 2 April 2008 (UTC)

I agree with your observation - this is just as warped. The choice of whether we should work with ZF, ZFC, ZF+AD or something else, is a matter of personal preference. The fact that the word "ZFC" refers to an axiom system with AC included, is fact. So is the fact that from ZFC we can prove Zorn's lemma. So is the fact that the word "real numbers" refers to a system where Dedekind-completeness is provable\included as an axiom, and the fact that from this we can prove that 0.999... = 1. -- Meni Rosenfeld (talk) 10:03, 2 April 2008 (UTC)

Because this is the talk page and not /Arguments: Does anybody (except Algr) claim that 0.999... usually denotes anything but a real number? Does anybody (including Algr) have reliable sources for such a claim? Does anybody have a source for something without Dedekind-completeness (or something equivalent) included in its axioms still being called "real numbers", and if so, should this article discuss that case? -- Huon (talk) 11:49, 2 April 2008 (UTC)

If those conditions are met, then I am not opposed to including a mention to that alternative definition of "0.999..." — in a small section on "non-standard mathematics" at the bottom of the article. The thrust of the article, though, should be that, while many people doubt this result at first glance, 0.999...=1, and this can be proven in various ways (some "deeper" than others, but all of them mathematically sound). FilipeS (talk) 17:43, 2 April 2008 (UTC)

I suggest some research on how the 0.999... confuses mathematicians

I have unsuccesfully suggested a POV tag more than one year ago Talk:0.999.../Archive_11, others may have before me and I see that the issue still comes up at times. All arguments are quickly and abruptly settled by a group of people who actually do know what they're saying, but, i suspect, don't understand what they're being told. My view is that this article is POV and systemically biased. Not because there may be disagreement over the validity of the claim. Demonstration and sources prove that 0.999... indeed equals 1. But it's not enough to tell the truth, in an encyclopedia you have to tell it to everyone. Why I think there should be a POV tag is because this is a story about how frustrated some mathematicians and how wrong some students are. Well, this may be fun in a scientific journal and does not serve the purpose of wikipedia. The specific origin of the debate is that the conclusions and asking the general public to think like mathematicians. Maybe mathematicians should try to think like the general public in order to really make this NPOV. Here's an example of a similar topic done right: Monty_Hall_problem Luciand (talk) 01:20, 4 April 2008 (UTC)

The problem with your suggestion is that the editors of this article are not aware of a reliable source that discusses how 0.999... confuses mathematicians. Therefore, for Wikipedia's purposes, there is no reason to assume that it does confuse mathematicians, and there is nothing to say in the article.

It is, of course, conceivable that there is some small minority of mathematicians who have studied calculus and real analysis and are confused about 0.999.... And I think it is likely that when infinite decimals were first introduced, the pioneers in the field disagreed with each other over what they should mean. Possibly 0.999... was a part of that discussion. But we have no evidence of such confusions or disputes, and not for a lack of searching.

One difference with the Monty Hall problem is that it is a recent invention, and at its birth it was communicated simultaneously to a global audience in a very non-technical forum. Therefore we have the opportunity to observe that "several hundred mathematics professors" disagreed. I understand that, if you are of a certain mindset, it is attractive to suppose that there was/is a similar outcry over 0.999.... But without evidence, Wikipedia will remain silent on the question. Melchoir (talk) 03:11, 4 April 2008 (UTC)

My comparison to Monty Hall problem reffers strictly to the neutrality of the structure. As for how the current matter confuses mathematicians, it is just a rethorical suggestion for further research :) Luciand (talk) 15:03, 7 April 2008 (UTC)

I'm afraid I don't know what you're talking about. What is this "neutrality of the structure"? Melchoir (talk) 20:18, 14 April 2008 (UTC)

Why would mathematicians want to think like the general public when the general public are provably wrong? When trying to understand a mathematical concept, you have to think like a mathematician. The true story is one of the general public not understanding an accepted mathematical fact, so that's what the article should discuss - that's not POV. I'm not sure why we don't have the same issues with the Monty Hall problem, but I suspect it's simply because the problem is easier to understand (it doesn't involve infinity, for a start), and is experimentally verifiable (while rigorous proofs mean more to mathematicians, an experiment is often more convincing for a layman). --Tango (talk) 14:47, 4 April 2008 (UTC)

Mathematicians may want to think like the general public when writing in an encyclopedia, which is meant for the general public. By the way, you will notice that, despite being very unobvious, few people dispute the validity for monty hall's solution. My opinion is that the neutral structure has a merrit in creating a neutral mindset, which in turm allows people to more easily accept fresh viewpoints. Luciand (talk) 15:03, 7 April 2008 (UTC)

My father died today, so I won't be able to participate here for a while, even though there are some things I hope to explain. I'll be back when I can, but I might want to start on a less combative article. Algr (talk) 05:00, 5 April 2008 (UTC)

Sincerest condolences. Be back when you feel like it.--Noe (talk) 15:26, 5 April 2008 (UTC)

I'm sorry to hear that, my thoughts are with you. We'll still be here when you're up to it - take as long as you need. --Tango (talk) 15:34, 5 April 2008 (UTC)

My deepest condoleances Luciand (talk) 15:03, 7 April 2008 (UTC)

Dedekind cut proof

From the page under the section Proofs->Real numbers->Dedekind cuts

Conversely, an element of 1 is a rational number ${\begin{aligned}{\tfrac {a}{b}}<1\end{aligned}}$, which implies ${\begin{aligned}{\tfrac {a}{b}}<1-({\tfrac {1}{10}})^{b}\end{aligned}}$.

Prove it. I'm not showing skepticism to the truth of the statement; just wondering how one would go about proving it. --69.91.95.139 (talk) 15:10, 6 April 2008 (UTC)

We have a any integer, b a positive integer. If ${\tfrac {a}{b}}<1$, then a<b. Thus, ${\tfrac {a}{b}}\leq {\tfrac {b-1}{b}}=1-{\tfrac {1}{b}}$. Now we can prove by induction that for any positive integer b, $b<10^{b}$, leading directly to the statement to be proven. Huon (talk) 16:12, 6 April 2008 (UTC)

Thanks. --69.91.95.139 (talk) 16:34, 6 April 2008 (UTC)

This is not a good way to test students intuïtive knowledge about mathematics.

0.999... is not equal to 1 without first establishing that 0.999... is defined as the limit of that notation if you say that n is the amount of digits you allow while cutting the rest of. With n going to infinity, which is done at some place too late in this article. But it was never taught that way to the students, who were never introduced to the formal concept of the limit. In fact, saying that a student is 'right' or 'passed the test' when he or she says 'It is equal to 1.' is nonsense I assert without having told the student that the value of the repeating decimal is defined as its limit as the number of digits goes to infinity. They just vaguely tell them 'It means that you keep on repeating the nine until infinity', 'Until infinity'? Infinity is somewhere one 'stops' or something? If it is put like that, it is mathematical nonsense. Children who say 'It is equal to 1.' fail the test if this is all they have been told is my opinion. It is not equal to one, it is not defined, it is mathematical nonsense. Students who say 'What you are saying here is not anything I can say mathematically exist.' pass cum laude as far as I think. Of course if they do it well and they tell students 'If I give you any positive value, you matter how small. Can you make the difference between 1 and 0.999.. where n is the amount of digits you allow before cutting of the rest smaller than that value if you can make n as large as you want?' they are of course going to answer 'Yes.', each and everyone of them. /rant Niarch (talk) 23:04, 5 May 2008 (UTC)

I'm not sure which test you are talking about here. If a teacher doesn't teach anything useful about 0.999... or decimal expansions in general, and then expects students to discuss 0.999... (and its equality to 1) in a test, then yes, the teacher isn't doing a very good job. However, 0.999... means what it does, and without any alternative context specification, the correct answer to the question "is 0.999... equal to 1?" is "yes", just as the correct answer to "is 2+2 equal to 4?" is "yes". Whether any given student understands the reasoning behind these assertions is a different matter. -- Meni Rosenfeld (talk) 23:13, 5 May 2008 (UTC)

Moreover... Where I went to high school, derivatives were introduced without first discussing limits and the definition of derivative. We were simply taught how to find basic derivatives. Nevertheless, without the definition, we were tested and held accountable for the results of what we were taught. Tparameter (talk) 23:21, 5 May 2008 (UTC)

I like the point Niarch is making, but I'm not sure if this implies any changes to be made to the article. By the way, Niarch, I think you will agree that students "know" that 1/3 = 0.333..., with no end of threes. Even without formal limits, this allows one of the informal proofs of 0.999... = 1. --Noe (talk) 23:45, 5 May 2008 (UTC)

It says in the article that students are tested to this question. Written in such a way to give the impression that they 'fail' if they do not answer 0./9/ = 1. Which I find ludicrous. I often see debates on the internet with people who have never heard of limits saying 'Why can't you just get that 0.999... = 1 exactly, idiot?' I can't blame anyone who has never been introduced to limits to not 'getting' this and I blame people who have not been introduced to limits to do 'get' this as without knowing that the value of the recurring decimal is defined as the limit you are talking nonsense. I never learnt that 0./3/ = 1/3. All I knew was that you it was an approximation to get many 3's to get the exact value of 1/3. The Dutch curriculum in secondary school requires children not to know that 0./3/ = 1. And we were actually explained the definition of the derivative via the intuïtive definition of the limit without knowing what the limit was. Simply by saying 'You see that if h becomes smaller you approach a certain value, that value you get closer to is your derivative.' Niarch (talk) 01:01, 6 May 2008 (UTC)

I searched for the word "test" in the article and didn't find it. Melchoir (talk) 03:17, 6 May 2008 (UTC)

{sound of crickets}... Tparameter (talk) 13:46, 6 May 2008 (UTC)

I think introducing the word "infinity" is unnecessary and also misleading, since, as you point out, it sounds like a destination rather than the direction it really is. It's easy to say "the 9's go on forever" or "the 9's go on without end". --Tango (talk) 00:28, 6 May 2008 (UTC)

Exactly, I am extremely opposed to the notation using the $\infty$ sign in limits and such. Either for the supposed 'value' the limit equals or the thing the variable 'approaches'. The first is simply saying that the limit doesn't exist and the function keeps on being monotonous after a certain point. The second says you increase the variable with no bound. It is not 'equal' to infinity and you are not 'approaching' it. Niarch (talk) 01:01, 6 May 2008 (UTC)

That's a rather narrow interpretation. Please see One-point compactification and pages 7-8 of [1]. Melchoir (talk) 03:25, 6 May 2008 (UTC)

Limits are normally only introduced formally in a fairly advanced setting where people should be capable of understanding what's going on. The symbol $\lim _{x\rightarrow \infty }$ has a vary precise meaning, it doesn't really matter what symbols it's made up of, it's the symbol as a whole which has meaning. --Tango (talk) 13:45, 6 May 2008 (UTC)

Why this equality is so hard to prove

I found a mathematical reason: Löwenheim–Skolem theorem, Skolem's paradox. Axiom of Dedekind-completeness is a statement in second order logic. The real numbers are not absolute, but categorical. Could this article mention the mathematical fact that equality "0.999... = 1" is not a theorem in first-order logic? Tlepp (talk) 06:05, 13 May 2008 (UTC)

I don't believe "0.999... = 1" is a theorem, period. "0.999..." has been defined as a matter of notation, just as the operator "+" has by convention. It is defined by the corresponding limit - which is why I don't like all of the "proof" rhetoric in the article. Tparameter (talk) 14:07, 13 May 2008 (UTC)

0.333...=1/3 is not a DEFINITION, but it FOLLOWS from some definitions, and hence can be PROVED as a THEOREM (a simple one!). So can 0.999...=1; it is a theorem too. Or perhaps a lemma or a corollary, if you prefer - the point is, it is something that FOLLOWS from the definition of decimal notation, but it is not something you'd mention specifically as a part of such a definition, a special case or something. I got to check out Löwenheim–Skolem theorem, Skolem's paradox, first-order logic and Absoluteness (mathematical logic); it sounds interesting!--Noe (talk) 14:42, 13 May 2008 (UTC)

Those topics are interesting, but they have nothing to do with the first-order theorem that 0.9999... = 1. And (to Tparameter) it is true that the theorem depends on the convention regarding what 0.999..., = and 1 mean, but so what? In exactly the same way, the Pythagorean theorem depends on conventions regarding the meanings of "right triangle" and so on.

As Noe says, the fact is that 0.999... = 1 is a mathematical theorem (or lemma or corollary). It is noteworthy only because some people seem to have trouble understanding it. Phiwum (talk) 14:53, 13 May 2008 (UTC)

According to this very article, 'in conventional mathematical usage, the value assigned to the notation "0.999…" is the real number which is the limit of the convergent sequence (0.9, 0.99, 0.999, 0.9999, …)', i.e. it is defined as the limit - which happens to have the value 1. However, I would be interested to see a published source that refers to "0.999... = 1" as a "theorem". That's not how I was taught. Tparameter (talk) 16:10, 13 May 2008 (UTC)

The fact that that limit equals one is a theorem. A rather trivial one, really, but a theorem none the less. --Tango (talk) 18:25, 13 May 2008 (UTC)

That's obvious, given the definition of limits, then we can calculate that particular limit. However, the symbol "0.999..." was defined as that limit - so, that it equals one is by definition, and by subsequent consequence of the value of that limit. "0.999..." was not calculated in the sense of the trivial theorems that you are speaking of. Tparameter (talk) 21:57, 13 May 2008 (UTC)

Just as Tango says. I have never seen any text that stated "17+21 = 38" is a theorem, but it is (in the mathematical logic sense, but not in the informal terms found in math texts). Phiwum (talk) 19:48, 13 May 2008 (UTC)

"0.999... = 1" is a theorem in second-order logic. That article says verbatim: "one needs second-order logic to assert the least-upper-bound property for sets of real numbers". When I say it's not a theorem in first-order logic, i don't claim it's false in first-order logic. I don't claim it's provably unprovable in first-order logic. I claim there is no such theorem in first-order logic. It is not an argument against the equality. It's a statement that first-order logic has it's limitations. Take it as an opportunity to say a few words about mathematical logic in this article. Tlepp (talk) 16:18, 13 May 2008 (UTC)

That surely has nothing to do with Lowenheim-Skolem! In any case, there's a conventional proof of such facts by modeling the real numbers in ZFC, which is a first-order theorem. Whether the LUB can be stated in a first-order theory of real numbers is a different matter. But we can interpret reals as sets in ZFC in such a way that the LUB principle is a theorem and hence so is the topic of this article. Phiwum (talk) 19:48, 13 May 2008 (UTC)

This has a lot to do with Lowenheim-Skolem. The article on second-order logic has a section "Why second-order logic is not reducible to first-order logic" that tell us how we can interprep LUB in first-order logic. Relevant part of that section is:

Every nonempty internal set that has an internal upper bound has a least internal upper bound.

The article on archimedean property contains proof that full least upper bound property implies archimedean property. Relevant part of the proof:

Denote by Z the set consisting of all positive infinitesimals, together with zero. ... Therefore, Z has a least upper bound c.

If we interprep LUB in first-order logic, the problem is that Z is not necessarily an internal set, all we can say is:

If Z is an internal set, then Z has an internal LUB.

Archimedean property, least upper bound property and (non-)existence of infinitesimals is relevant to this article and so is Lowenheim-Skolem. Tlepp (talk) 21:43, 13 May 2008 (UTC)

I agree with what Phiwum said above. It's not helpful to talk about first-order logic in such generality. You have to talk about specific theories. If, for example, you intend the theory of real closed fields with addition and multiplication, it's not obvious to me that something like 0.999... is even expressible in the language of the theory, so what is there to say about it?

I'm not really asking you this question, though; this is the sort of background context for which I'd certainly want to see a reference work before adding a new direction to the article. Melchoir (talk) 05:20, 14 May 2008 (UTC)

Bruno Poizat's book "A Course in Model Theory" discuss limitations of axiomatic systems. An excerpt from section "7.12 A Little Mathematical Fiction":

"It is hardly surprising to see the author of a textbook yield to the ease of the axiomatic method, prefixing the work with a Chapter 0, a so-called "set theory", fixing the rules of the game for the rest of the book; this is yielding to the ease of an a priori, dogmatic, and noncritical exposition of the contents of a discipline; it reassures students, who love to be hit with absolute truths, even if they understand none of them. What is surprising, however, is to see confirmed mathematicians very rigidly stick to this viewpoint..."

Book obviously doesn't say anything about "0.999...". Why is it related to this article? Because some people seem to dogmatically insist that equality "0.999... = 1" is a universal truth that follows from ZFC. First-order theory "ZFC + axioms of real closed fields" can't prove it. This is a rationale and justification why a new rule/axiom needs to introduced into our axiomatic system. The new rule is axiom of Dedekind-compleness. Some people don't want to believe that no axiomatic system is ever going to be perfect and sometimes new axioms are required. Tlepp (talk) 07:50, 14 May 2008 (UTC)

You're mixing meanings of "axiom". Once the axioms of ZF are admitted, no further axioms are necessary to bestow special privileges to certain kinds of fields or ordered sets. "Real closed" and "Dedekind complete" become definitions that a given set can either satisfy or not. And it becomes possible to construct the real numbers, which are explicitly the topic of most of this article. Now I am perfectly aware that real analysis contains questions that are undecidable in ZF, or ZFC, or what have you; and this suggests to some mathematicians that new axioms of set theory are "required". However, 0.999... = 1 is certainly not one of these questions. Melchoir (talk) 08:39, 14 May 2008 (UTC)

I'm just arguing semantics here, but I would say that "axioms" and "definitions" are pretty much synonymous. A set is defined to be something with satisfies the ZF axioms, a group is defined to be something with satisfies the group axioms, etc. If you add a new definition, that's pretty much synonymous to adding one or more new axioms. We define a "complete field" to be something which satisfies the axioms for a field and the new "completeness axiom". --Tango (talk) 14:20, 14 May 2008 (UTC)

Your view regarding axioms and definitions is odd indeed. After all, it makes no sense to say that a set is something satisfying the ZF axioms. Those axioms stipulate relations between sets, not a set. More specifically, they serve to "define" (in scare-quotes) the element relation.

In any case, Melchoir is quite right. We construct a set that we call R, using the axioms of ZFC. We show that R satisfies certain properties, including the LUB property, which certainly *can* be interpreted in ZFC. We can also show, using only FOL and the axioms of ZFC, that 0.999... = 1. It is in this sense that 0.999... = 1 is provable using FOL.

Whether it is provable in some other theory, say a first-order theory of reals, is another matter entirely. Phiwum (talk) 14:39, 14 May 2008 (UTC)

Then what is the definition of a set? A collection of sets? That's a little circular... You need the ZF axioms to work out what is and isn't a set. For example, the class of all sets isn't a set precisely because it fails to satisfy the axiom of regularity. --Tango (talk) 16:02, 14 May 2008 (UTC)

Tango, I think you lack some background in formal logic, which leads you to confusion whenever something like this comes along. Being no expert myself, I will not be able to explain anything clearly (or even correctly, probably), but ZFC set theory describes a world of sets. Every object in that world is a set. The axioms don't pose restrictions that an object in the world needs in order to be a set, but rather restrictions on the world in order to be a model of the theory. There is no such thing as "class" in that world - the latter is a meta-entity we use in our descriptions of the world. Some classes (which again, are meta-objects) have corresponding sets (objects in the world). The class of all sets does not correspond to any set, because that set would have to contain all sets, and a theorem of ZFC is that no set can contain all sets.

Perhaps I should mention that there are alternative theories which describe a world of classes, and then "set" is a special kind of object in the world. But that's not ZFC anymore.

I tend to agree that all definitions can be reformulated as axioms, but not the other way around. -- Meni Rosenfeld (talk) 17:08, 14 May 2008 (UTC)

You don't need the axioms to work out what is a set. A set is a fundamental thing, like a point. All you need are two mathematicians that agree what it is, then they can use it accordingly. Tparameter (talk) 17:51, 14 May 2008 (UTC)

Meni, you're right, I do lack the background for this kind of discussion - I really don't understand why my degree course doesn't cover logic beyond a very vague description of set theory, and it continuously annoys me. I tried reading up on it, but failed somewhat - that was in my first year, though, perhaps I should try again now. That said, I really don't get what you're saying. I can say "$\{\emptyset ,\{\emptyset \}\}$" and I can say "All sets", how do I know that the first is a set and the second isn't? --Tango (talk) 18:46, 14 May 2008 (UTC)

I agree - you didn't get what I was saying. The objects in ZFC set theory are sets and nothing but sets. It makes no sense to say that something is not a set. It is provable from ZFC that there is a set whose elements are φ and {φ}, and it is provable from ZFC that there is no set containing all sets. -- Meni Rosenfeld (talk) 19:01, 14 May 2008 (UTC)

I don't care about "the language of ZFC set theory", I'm speaking English. In English, I can say "the collection of all sets is not a set", and that makes sense and is true. There must be a definition of the word "set", since all English language words have definitions. --Tango (talk) 19:56, 14 May 2008 (UTC)

Sure, there's an English definition of the word "set". In fact, lots of them and some of them are relevant to you. But set doesn't mean "something satisfying the ZFC axioms". Nonetheless, it is sensible to say something like, "In ZFC, $\{0,\{0\}\}$ is a set, but $\{x|x{\text{ is a set}}\}$ is not a set."

What would those statements mean? Something like this: by $\{0,\{0\}\}$, I mean a set $y$ such that $(\forall x)(x\in y\leftrightarrow x=0\vee (\forall z)(z\in x\leftrightarrow z=0))$. Let that big formula be $\varphi (y)$. Then ZFC proves $\exists !y\,.\,\varphi (y)$.

Similarly, let $\psi$ be the formula $x\in y\leftrightarrow x=x$. Then any set satisfying $\psi$ would be the set of all sets, but ZFC proves $\neg \exists y\,.\,\psi (y)$. Now, that's a way in which ZFC tells you what counts as a set or not, but it's not the same as saying "a set is something satisfying the axioms of ZFC".

Make sense? Sorry if I was brusque earlier. Phiwum (talk) 21:48, 14 May 2008 (UTC)

Yes, it makes sense, and don't worry about the brusqueness. However, you haven't actually answered my question. You used the word "set" repeatedly in that explanation - what did you mean by it? --Tango (talk) 21:56, 14 May 2008 (UTC)

It is an undefined term in ZFC and hence occurs only informally. Some of those informal uses can be explained in terms of ZFC. I translated what I meant in the following two cases: When I say, " In ZFC, $\{0,\{0\}\}$ is a set," I mean "ZFC proves $\exists !y\,.\,\varphi (y)$." Of course, someone might speak of sets in some context other than ZFC. If so, they mean something else. Perhaps we should take this to email, since it has strayed from the point of this page. I can be reached at jesse@phiwumbda.org. Phiwum (talk) 22:24, 14 May 2008 (UTC)

It is a consequnce of Gödel's completeness and incompleteness theorems that any axiomatic system has the good, the bad, and the ugly. In ZFC there are non-measurable sets, and there are sets that slice a ball into five pieces and can be glued together to get two balls identical to the original. Any reasonable person would agree that in a perfect set theory there shouldn't be such sets.

Informally a set is a set, and any attempt to rigorously and formally define it is a swamp or wet sand. In this article the most important axiom is the axiom of Dekekind-completeness, which is equivalent to least upper bound property, which implies archimedean property, which implies non-existence of infinitesimals. ZFC is pure technicality, we could replace ZFC with any other reasonable set theory and not a single word had to changed in any proof written in english. Tlepp (talk) 06:35, 15 May 2008 (UTC)

When you drive a car you don't have to know the technical details of the engine. Car needs an engine, but any reasonable engine will do. You have to know whether to use left-side traffic or right-side traffic, and no matter how well you know the engine details, it doesn't imply that right side is the right side. We can construct a car that is better for right side traffic. (=wheel and pedals are on the left side, and the lights won't dazzle someone on front left) We can choose one such car and call it the car. It is in this sense that technical details of the car imply the right side.

And my point here is not that choice between left and right is arbitrary. That's the biggest weakness in my analogy. Real numbers are not arbitrary. My point is that if someone asks what is the justification for right side traffic, then only a complete idiot would answer: well there's is this car, the car, and it's a logical consequence of the car that right is right. Some platonists among us may believe that there is the car, but that kind of religious arguments prove nothing. Tlepp (talk) 10:11, 15 May 2008 (UTC)

Question to people

Can anyone find a number divided by a prime number that equals to 0.33333333333333333...  ?

I tested and things like 1000000000000000000000000000 / 3000000000000000000000000001 and 1000000000000000000000000000 / 2999999999999999999999999999 never repeat forever, but they eventually stop. William Ortiz (talk) 07:52, 13 May 2008 (UTC)

Not sure what you mean. The fraction "1/3" is a number divided by a prime and equals 0.333... . And: Assuming the fraction is written to lowest terms, only division by a power of 2 times a power of 5 will terminate. All other rationals repeat. hence, dividing by 3000000000000000000000000001 or 2999999999999999999999999999 will produce infinitely repeating decimals (though the repeating period will not be just the digit "3").--Noe (talk) 08:30, 13 May 2008 (UTC)

And -- how did you "test" those divisions? --jpgordon^∇∆∇∆ 15:25, 13 May 2008 (UTC)

Well I'm thinking 0.999... = 1 seems also to have some relationship with the decimal value of 1/3 = 0.333.. But I've been looking around and it appears only 1/3 = .333.. as long is the divisor is reduced to a prime number. I tested some close values in Windows Calculator. 1000000000000000000000000000 / 3000000000000000000000000001 = 0.33333333333333333333333333322222. 1000000000000000000000000000 / 2999999999999999999999999999 = 0.33333333333333333333333333344444. 1000000000000000000000000001 / 3000000000000000000000000000 = 0.33333333333333333333333333366667. 999999999999999999999999999 / 3000000000000000000000000000 = 0.333333333333333333333333333. None of these numbers keep going after the last decimal I gave. It appears 1/3 is unique in being 0.333... and 2/3 is unique in being 0.6666... as long as either the top or bottom numbers in the equation is a prime number. William Ortiz (talk) 01:15, 14 May 2008 (UTC)

I don't really follow you here. Of course 0.999... = 1 is related to 1/3 = 0.333..., because 1 = (1/3) * 3 = (0.333...) * 3 = 0.999...

As to what your calculations are showing you, it's that Windows Calculator can calculate that many decimal places. Any fraction p/q will have a repeating, non-terminating decimal representation, although the period of repetition could be anywhere from 1 (as in the case 1/3 = 0.333...) to q-1 (as in the case 1/7 = 0.142857142857...). Since $10^{n}/(3*10^{n}\pm 1)$ is very close to $\,10^{n}/(3*10^{n})$ which reduces to 1/3, of course its decimal representation is going to be close, for about the first n decimal places. It's worth noting that the only time that p/q has a terminating decimal representation is when, after you've reduced it to simplest terms, q can be written in the form $2^{a}5^{b}$. Confusing Manifestation(Say hi!) 02:40, 14 May 2008 (UTC)

At one point I didn't think 1/3 = 0.333... exactly but was slightly more. However I noticed recently that first it seems nothing else done with primes gets 0.333 and also when you divide 1 by 3, your remainder is 1 each time, so it's a unique thing and it then should be exactly 0.333... I don't know quite what this means, but I think it relates to the article somehow. William Ortiz (talk) 05:26, 14 May 2008 (UTC)

Well, the "nothing else" part is certainly straightforward. Let a/b = 0.333...; then b = 3a. The only prime multiple of 3 is 3 itself, so a=1, b=3 is the only solution. Melchoir (talk) 05:31, 14 May 2008 (UTC)

While not all real numbers have unique decimal expansions (one is both 1 and 0.999..., for example), all decimal expansions refer to a unique real number. Having two different numbers having the same decimal expansion would make the whole decimal system completely pointless... --Tango (talk) 14:14, 14 May 2008 (UTC)

It is not true that 1000000000000000000000000000 / 3000000000000000000000000001 does not repeat forever; in fact it does. See rational number. Michael Hardy (talk) 09:11, 14 May 2008 (UTC)

How I finally got a grasp on this.

I was thinking about what the smallest possible number would be, i.e "0.00...1", and then I realized if you're trying to do that, in reality it's "0.00...". You never get to the point where you can affix the "1", because the zeros never stop. Something "infinitely small" is the same as "zero", then. Likewise, saying something is "infinitely smaller" than "1" is equivalent saying it's not at all smaller than "1". That's how I finally came to accept this. —Preceding unsigned comment added by 69.62.140.50 (talk • contribs) 17:01, 16 May 2008

That's well spotted, and is basically a case of the Archimedean property of the real numbers - that there is no number "just slightly bigger than zero", so to speak. Of course, as this article points out, you can develop a number system where you do have such things, but that number system is not the real numbers, and working in it would presumably be a bit of a pain. Confusing Manifestation(Say hi!) 07:02, 16 May 2008 (UTC)

I agree, this was also one of the reasons that I accepted this. (Another one being the fact that this is on wikipedia, and wikipedia is a relaible source *not saracasm*).--Sunny910910 ^{(talk|Contributions|Guest)} 00:31, 21 May 2008 (UTC)

Perhaps it's all a matter of notation

Count me in the camp of evil conspiracists who accept that $0.999\ldots =1$. But, it seems to me that the entire dispute may have to do with the meaning of the ellipsis in the expression. There is no reason to think that three dots, in themselves, represent any particular thing, just as there is no reason to think that the superscript 2 in $x^{2}$ inherently represents the concept of x multiplied by itself. These symbols become meaningful only because they are widely held to represent those concepts. They become standard, and communication becomes possible among people who are using standard definitions of symbols.

If you were to privately define the ellipsis in $0.999\ldots$ to represent some (finite) large number of 9s, then of course, for you, the equality would not hold. The fact that the ellipsis--by standard definition--does not represent a finite number of 9s is meaningful only to those who accept the standard definition of the ellipsis.

It's akin to the dispute over stealth creationism--if one accepts the widely accepted definition of science as a rigorous method, then so-called "intelligent design" fails, but if one considers science to embody the opinion of people who call themselves scientists, then one can disregard the rigors of the scientific process and call just about anything into dispute as "just a theory."

The question, then, is: Do you accept the widely accepted definition, or not? If not, you will be using the the same words to describe entirely different things. The possibility of consensus among people using incompatible terminology becomes remote. These days there are many people who think that, if you believe something strongly enough, it automatically becomes true. Rangergordon (talk) 05:02, 25 May 2008 (UTC)

Sorry, but the scientific method is based on empirical data and inductive reasoning. Rigor is deductive. Tparameter (talk) 05:51, 25 May 2008 (UTC)

Sorry I think the creationism debate is COMPLETELY unrelated to this, and Rangergordon is messing up his own valid point about notation by comparing the two issues. Notation is fundamentally arbitrary (i.e. a matter of definition). The scientific method as such is not at all arbitrary, and the element af arbitrarity in its results is of a very different nature from the arbitrarity of notation. But could we just ignore the creationism thing and stick to the meaning of "..."?

The standard meaning of "..." in maths seems to depend a bit on where it's used. In the sequence "1, 2, 3, ..." , it represents infinitely many elements. In "0.999..." or "1 + 1/2 + 1/4 + 1/8 + ..." , it does NOT represent infinitely many terms (I think); it represents the LIMIT of the sum as the number of terms grows beyond all limits. (In "0.999...", the sum in question is of course "9/10 + 9/100 + 9/1000 + ..." .) If anyone can find ONE way of expressing the meaning of both uses of "...", that would be nice.--Noe (talk) 09:12, 25 May 2008 (UTC)

They're pretty much the same meaning - limits are the way we rigorously define what we mean by "going on to infinity". You can express your first example as a limit: $(1,2,3,\dots )=\lim _{n\rightarrow \infty }(1,\dots ,n)$. --Tango (talk) 12:53, 25 May 2008 (UTC)

I apologize to Noe for bringing creationism into this. It was an analogy that made sense to me (that is, that the firmness of one's belief cannot be used to prove or disprove a proposition). However, I recognize that those who have a different outlook might not find it to be apt.

Still, I don't see that it's difficult to understand that "..." means one thing in "1, 2, 3, ..., n" and another in "1, 2, 3, ..." They're two different notations. Rangergordon (talk) 05:28, 30 May 2008 (UTC)

You're absolutely right - if you define "0.999..." to be something other than the standard meaning, the equality won't hold. That's true of anything in maths. 1 + 1 = 2 only because of how we define 1, +, = and 2. You define them some other way, you get a different answer. However, there is only one definition of "..." in this context that is ever used by anyone other than people disputing this Wikipedia article (your example of it being a large finite number of 9's isn't well-defined, different large numbers of 9's give different real numbers, you would have to pick one and that would be even more arbitrary). If anyone can find a published use of some other definition, then we can include it. --Tango (talk) 12:53, 25 May 2008 (UTC)

Ah, I've never laughed so hard at a discussion page. Tango, in your very last comment you've got two different uses for ... in the same equation; on the left hand side they represent infinitely many terms. On the right hand side they represent finitely many terms: exactly n-2 terms to be precise. With a straight face, ellipses for mathematicians generally mean "and the rest of the sequence which continues in such an obvious way I'm not going to describe it further." The sequence can be anything, be they infinite repeating decimal digits as in 0.999... or non-repeating decimal digits as in 3.141... or real numbers in a series, elements in a matrix, or arbitrary sets in a sequence, itterations in a convergent algorithm, anything really. Ellipses are what mathematicians use when they're not investing the effort in being completely formal, so it's kind of wierd to want a deffinition of what ellipses mean.

One general rule is that ellipses represent finitely many somethings if it's terminated at both sides as in 1,2...n and infinitely many somethings if it's not, such as 1,2,3... or ...-3,-2,-1,0. Yes, the ellipses in 0.999... do represent infinitely many 9's. This "rule" doesn't always hold, for instance in 0,1,2...$\aleph _{0}$,$\aleph _{0}$+1,$\aleph _{0}$+2...$\aleph _{\aleph _{0}}$... each of those ellipses represents infinitely many terms even though some are terminated at both sides. I guess the best general rule is that ellipses tell you that a mathematician is just assuming you know what's going on, and if you don't you should stop them and get them to explain it better. If they can't then they don't really understand it either. Endomorphic (talk) 16:25, 28 May 2008 (UTC)

As I explicitly said, what an ellipses means is dependant on context. In some contexts, it may even be ambiguous, but in the case we're talking about it's very clear what it means. --Tango (talk) 17:38, 28 May 2008 (UTC)

Exactly. It's clear in this case, because of the way the "unrestricted ellipsis" (if I may call it that) is defined, that "0.999..." does not involve any finite number on 9s; therefore, arguments that $0.999\neq 1$ based on some finite number of 9s are invalid, and the "controversy" can be chalked up to an incorrect interpretation of notation. Rangergordon (talk) 06:58, 1 June 2008 (UTC)

Refutation of Counterarguments in the Introduction: Confusing?

I'm wondering whether or not it would make sense to introduce the controversy in its own section; as the introductory section stands, it barely has time to establish the existence of the $0.999...=1$ identity before it begins refuting counterarguments.

This happens before any explanation that there is a controversy to begin with.

Many readers will refer to this page in order to help them understand fine points of their teacher's explanation of the identity; the existence of alternative theories will only serve to confuse them. Only after they understand the basic theory can they begin to make up their minds as to alternatives.

I suggest a structure such as this:

Suggested structure

• Introduction
  • Repeating decimals, limits
  • $0.999...=1$
• Basic proofs
  • Algebraic
  • Real Analysis
  • Series
  • etc.
• Controversy/Alternative theories
  • $0.999...\neq 1$
    • $2\times 0.999=1.998$
    • How, in a sane world, can there ever be such a thing as infinity?
    • etc.
  • Counterargument A to basic proofs
    • Refutation of counterargument A
  • Counterargument B to basic proof
    • Refutation of counterargument B
  • etc.

Rangergordon (talk) 05:16, 30 May 2008 (UTC)

Re Suggested structure

I'm confused by this proposal on a couple levels. First, I don't think the Introduction section refutes counterarguments. It's essentially positive and descriptive. It takes a few opportunities to distinguish between things that need to be distinguished, but I don't think that makes it argumentative.

Second, I'm not sure what you mean by controversy or alternative theories. The closest we can get to that direction, without doing Original Research, is to describe observed behavior patterns in students, and to describe educators' theories about that behavior. If you want to invent a point/counterpoint, we'll just wind up with straw man arguments that probably have little to do with the real concerns students have. Melchoir (talk) 19:43, 1 June 2008 (UTC)

I wonder if I may have misunderstood the topic. I thought it was an article about a particular numeric property. If it is, I think the introductory section merely should contain a description of that property, and perhaps briefly introduce any mathematical concepts that will be necessary to understand later discussions of that property.

If there are to be discussions of behavior patterns among students as observed by educators, they seem to fall not under the rubric of number theory, but under that of educational theory. I am not advocating for an expanded treatment of misconceptions. I'm only saying that such discussions about educational theory probably merit their own sections--lower down in the article.

As it is, the introduction of these behavior patterns, right up at the top in the introduction, may mislead the reader to believe that those misconceptions are somehow central to the mathematical concept being discussed--that is, the article gives the misleading impression that there may be some sort of real "controversy" involved. I worry that this will confuse students who, after all, are probably referencing the page only in an attempt to clear up their own misconceptions. Rangergordon (talk) 11:10, 5 June 2008 (UTC)

I would propose an easier solution to the problem. At the moment, I have tried to swap the second and third paragraphs of the lead. This should help to disspell any misconception that somehow the opinions of students are relevant to the mathematical validity of the identity. A description of the "Skepticism in education" should be in the lead, since this is one reason why the identity is of such interest to begin with. But perhaps the bottom of the lead is a more logical place for it. I am neutral either way, so if anyone feels like changing it back, I will not oppose, as long as they hear me out.

I also oppose a major restructuring of the article along the lines you have suggested. I think I must share some of Melchoir's confusion with it. I would like to add to his reasons for opposition, which I agree with, that the proposal falsely conflates the "controversy" (i.e., skepticism by students and non-mathematicians) and the alternative number systems (which were not developed in order to make 0.999... = 1 false, but rather for other reasons entirely). siℓℓy rabbit (talk) 11:44, 5 June 2008 (UTC)

The new lead order is more logical, except that the "contributing to the misconception" phrase has to be moved. I do worry, however, that some readers will become frustrated with the technicality of the second paragraph and not make it to the third.

Rangergordon, I begin to suspect that we're talking about different sections. Are you talking about the section before the table of contents, or the section after it? Melchoir (talk) 22:29, 5 June 2008 (UTC)

Well, I am eager to agree that my suggested restructuring scheme is nonsensical. I'm very glad Melchior and Silly rabbit have pointed this out, though, because when I wrote it, I was under the impression that the authors of this article were taking the so-called "controversy" seriously and trying to give it equal time. In that context, the proposed restructuring scheme was meant to de-emphasize the misconception, place it in context, and provide ample opportunities for refuting it.

Since that (thankfully) is not the case--again, let me say how glad I am to have been mistaken--then, may I merely suggest that the experience of educators (and the incredulity of their students) is not necessarily a main feature of the properties of 0.999...?

I agree with Silly rabbit that this issue--that some students are reluctant to accept the identity--is an important and relevant one. (And it's interesting! I never knew how complex and sophisticated students' objections to it could get!) Also, the topic is valuable because it reveals something about the psychology of the learning process, and may even help educators come up with improved teaching methods.

But perhaps I'm hidebound; I'm looking for an introduction like something that might appear in an introductory analysis textbook--a sparse statement of the identity, followed by detailed justifications, and then rigorous proofs. After that, interesting associated topics such as the educational phenomenon, or implications for computational algorithms, might follow.

Melchior: I like the swapping of the paragraphs, and you're right: When I said "introduction," I was talking about the section below the TOC. (Well, it is headed "Introduction"!) Thanks, people.Rangergordon (talk) 08:10, 6 June 2008 (UTC)

How many representations of a number?

The article is internally consistent about the number of ways of representing the same number as a decimal expansion. In the article summary, it states that there are many ways:

"..all positional numeral systems contain an infinite number of alternative representations of numbers. For example, 28.3287 is the same number as 28.3286999…, 28.3287000, or many other representations."

(It's not even totally clear to me what this is saying - I guess it's referring to: "0.99...", "1", "1.0", "1.00", "1.000", ..., "1.000...").

Whatever it means, this statement is contradicted by a statement in the Introduction section stating that there are only two possible representations:

"Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on)."

I'd be inclined to delete the statement in the summary: it doesn't add anything to the article. AndrewBolt (talk) 08:40, 16 June 2008 (UTC)

I think the point must be made somewhere in the article that 1 can also be written 1.0, 1.00, 1.000 etc, but that this redundancy is not the subject of the article. How this point is best made I don't know, but it should not be removed. If it is not in the lead, it should be in the following section.--Noe (talk) 09:13, 16 June 2008 (UTC)

I agree, it's a little confusing. I'm not sure it's even true. 1/3=0.333..., that's the only representation (well, I guess you could have 00.333... if you really wanted to, but I don't think that's even worth mentioning). Only terminating decimals can be written in multiple ways, and only two of them are significant (the others are just trailing zeros). I think it's very misleading, and probably just plain wrong, to say there are an infinite number of alternatives. --Tango (talk) 14:57, 16 June 2008 (UTC)

I think it's saying that in any positional system, there exist an infinitude of numbers that each have multiple representations, not that every number has an infinitude of representations. Of course it then shoots itself somewhat in the foot with the somewhat contradiction (I think that if you you're strict about decimal representations, 28.3287 is just shorthand for 28.3287000...). Confusing Manifestation(Say hi!) 23:43, 16 June 2008 (UTC)

Yeah, this part has been misunderstood a couple of times; it changed meanings here and here. I don't think there was any discussion, so we're free to edit away. I'll try it out. Melchoir (talk) 05:52, 17 June 2008 (UTC)

I think it is useful to mention this point, in passing. The reader should be given a clear distinction between two different concepts: 1) in a given base, any number can be represented by at most two different digit sequences; 2) the conventional notation schemes permit numerous ways of transcribing a given digit sequence. So, there are two possible decimal sequences for 1. The following are all transcriptions of one of these sequences: '1', '1.0', '1,0', '1.000000', '1.000 000', '1.000...', '10.0E-1' etc.

1 does not equal .999 ... .

Moved to /Arguments. --Tango (talk) 20:17, 13 June 2008 (UTC)