Talk:0.999.../Archive 16

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MfD for Arguments/FAQ

FYI someone has nominated /Arguments/FAQ for deletion here. 28bytes (talk) 15:38, 22 November 2010 (UTC)

Congratulations

Among article talk pages listed in this database report, this talk page has the largest archive of any talk page that isn't about religion or politics. --Damian Yerrick (talk | stalk) 14:01, 21 November 2010 (UTC)

SHOCK AND HORROR. --COVIZAPIBETEFOKY (talk) 21:23, 21 November 2010 (UTC)

Who says this isn't a page about religion? — Carl (CBM · talk) 00:38, 22 November 2010 (UTC)

Carl, I am shocked. I was supposed to say that. Are politicians' archives long because they have something to hide? Tkuvho (talk) 03:15, 22 November 2010 (UTC)

No, I'm pretty sure that Talk:9 is ahead of Talk:0.999.... —mc10 (u|t|c) 06:03, 1 December 2010 (UTC)

On redirecting

How about redirecting this article to the one dealing with the number one? —Preceding unsigned comment added by 189.74.242.116 (talk) 20:05, 25 December 2010 (UTC)

No, as they are completely different. 1 (number) is about the number 1 in general. This is about the particular representation 0.999... They have little in common, and neither has so little content it would be best merged.--JohnBlackburne^words_deeds 20:15, 25 December 2010 (UTC)

Uncommon misconception?

Is there a common misconception that 0.999...<1? You can comment at Talk:List of common misconceptions#0.999.... Tkuvho (talk) 21:01, 15 January 2011 (UTC)

Yep, and you can see how great those misconceptions are simply by looking at Talk:0.999.../Arguments. It's rife witrh people who believe in that misconception, and others who try to persuade them in the error of their belief. --JB Adder | Talk 04:42, 23 January 2011 (UTC)

Long division and hyperreals

I've reverted Tkuvho's claim that long division yields a hyperreal. I was taught long division in primary school, and I'm pretty sure hyperreals didn't occur back then. Besides, I believe Tkuvho's claim that

a simple division of integers like 1⁄9 produces the sequence (0.1, 0.11, 0.111, ...). What one does with such a sequence has nothing to do with long division. Thus, if one takes its equivalence class [0.1, 0.11, 0.111, ...] in the ultrapower construction, one obtains a hyperreal that falls infinitesimally short of 1⁄9, and adequal to the latter, typically represented by the recurring decimal, 0.111...

is wrong on several levels. First of all, long division indeed produces a recurring decimal, not a sequence. Secondly, the result of a division of integers in the hyperreals should not depend on the choice of an ultrafilter, but the equivalence class of the sequence (0.1, 0.11, 0.111, ...) does. Even more obviously, the result of the division 1⁄9 in the hyperreals is still 1⁄9, not infinitesimally short of 1⁄9. I'd also be interested in what hyperreal, in Lightstone's system of representation, 1⁄9 is supposed to be according to Tkuvho. What's the first digit where 1⁄9 deviates from 0.111... (in the hyperreal sense)? Huon (talk) 10:28, 27 March 2011 (UTC)

It was added to the FAQ too, so I removed it for the same reason, along with a further addition to another question which contradicted the rest of the answer so would only cause confusion. The FAQ is best left to straightforward (as possible) answers to basic questions. Digressions and examinations of the topic in terms of non-standard number systems should be confined to the relevant part of the article.--JohnBlackburne^words_deeds 13:43, 27 March 2011 (UTC)

Again the answer to the question 'Can't "1 - 0.999..." be expressed as "0.000...1"?' is 'No', as it now says with a detailed explanation. Adding a sentence that flatly contradicts that will confuse readers, defeating the point of the FAQ. I'm not even sure what it meant: "infinitesimal enriched" is hardly a mathematical term.--JohnBlackburne^words_deeds 17:44, 27 March 2011 (UTC)

Welcome to a pluralistic society. Sometimes it can be confusing to have more than one point of view represented. However, that's no reason to suppress all points of view other than your own. Rather, we should work toward expressing properly sourced viewpoints in such a way as not to cause confusion. An "infinitesimal-enriched" continuum is a continuum containing infinitesimals. You may not like this, but this page is not your private back yard. Tkuvho (talk) 18:29, 27 March 2011 (UTC)

One simple error

To make things easier, I would like to point just one simple thing shortly:

The fact that this topic never really points out that .333... isn't really a third. The correct form of describing a third is 1/3. —Preceding unsigned comment added by 83.248.233.85 (talk) 03:07, 13 February 2011 (UTC)

Actually, both are correct ways of describing a third. One is the decimal representation (illustration here), the other is a fractional representation. 28bytes (talk) 03:33, 13 February 2011 (UTC)

I'd bet this misconception comes from fixed precision computing (and the usage of calculators) where 1/3 becomes 0.3333... but with a fixed finite number of digits. 75.18.188.35 (talk) 01:01, 13 March 2011 (UTC)

To be a little more precise, 0.333... is an approximation of a third, but it's "as close as you like" to the actual value of 1/3; kind of like how differentiation takes the smallest interval on a certain scale to measure change. Teamabby (talk) 05:05, 24 March 2011 (UTC)

That's a very intuitive way of approaching both infinite decimals and the derivative. To formalize it as correct mathematics, one needs an infinitesimal-enriched number system such as the hyperreals. Some of the editors in this space are hostile to this approach, however; don't be surprised to get condescending comments sooner or later. Tkuvho (talk) 06:06, 24 March 2011 (UTC)

No, 0.333... is 1/3 exactly. Decimal expansions were invented to represent real numbers, and are unsuited to the hyperreals because you can't represent all hyperreals unambiguously. Your characterization of differentiation is incorrect because there is no "smallest interval" -- even in the hyperreals. You can use hyperreals to define differentiation, however, in a way that turns out to be equivalent to the usual one. Eric119 (talk) 16:40, 24 March 2011 (UTC)

Why can't you represent hyperreals unambiguously? Tkuvho (talk) 17:04, 24 March 2011 (UTC)

Sorry, I meant to say that you can't represent all hyperreals unambiguously,using decimal expansions. For example, which hyperreal does 0.333... represent? Whichever one you pick, there will be many other hyperreals infinitesimally close to it, which have just as much reason to be called 0.333... . Unless there's a natural method of picking a preferred one, which I suppose, there could be.

But, really, the idea of understanding decimal expansions in terms of hyperreals is trying to answer the "wrong" question, because it treats decimal expansions as the primary objects. For that matter, a lot of the discussion of 0.999... seems to be based on treating decimal expansions as primary, and viewing the reals, hyperreals, etc., as potential ways of understanding them. But, both historically and philosophically, the situation is the complete reverse: it is the real numbers that are the main topic of interest, and the purpose of decimals is to help understand the reals. We can also invent the hyperreals to try to understand the notion of infinitesimal, and then we can wonder if we can contrive some kind of decimal expansions to understand the hyperreals. If we can, they probably wouldn't be exactly the same as those for the reals. But using hyperreals to understand decimal expansions (which are fine-tuned for the real numbers) is backwards. Eric119 (talk) 18:05, 25 March 2011 (UTC)

A quick historical correction: it is altogether untrue that historically speaking, the reals came first and "the purpose of the decimals is to help understand the reals". The opposite is true. The decimals were introduced by Simon Stevin a century before Newton and Leibniz (or Isaac Barrow, if you prefer) invented calculus.

As far as the hyperreals are concerned, the point is not so much to use them so as to understand decimal expansions, but rather to find a mathematical expression for persistent student intuitions that "zero, dot, followed by an infinity of 9s" can be a meaningful entity falling short of 1. This is precisely the case in the context of the hyperreals, but can be accomplished in other number systems, as well. Thus, such student intuitions are not "erroneous" but rather "non-standard", as argued in a recent article by Robert Ely in a leading (perhaps the leading) mathematics education journal. Tkuvho (talk) 16:46, 29 March 2011 (UTC)

There is a way to represent hyperreal numbers by a variant of decimal expansions; the article mentions it under 0.999...#Infinitesimals. Of course, the number 0.999... where all digits are nines is still equal to 1. Huon (talk) 19:29, 25 March 2011 (UTC)

Suggested addition

I suggest the following material be added to either the main page or the FAQ: "Using long division, a simple division of integers like 1⁄9 produces the sequence (0.1, 0.11, 0.111, ...). What one does with such a sequence afterwards has nothing to do with long division 'per se'. Thus, if one takes the equivalence class [0.1, 0.11, 0.111, ...] of the above sequence in the ultrapower construction, one obtains a hyperreal that falls infinitesimally short of 1⁄9, and is adequal to the latter. The identification of 0.999... with 1 results from a commitment to the real number system, and is not a consequence of long division." Tkuvho (talk) 15:12, 27 March 2011 (UTC)

I disagree. I assume you accept that a simple division of integers like 1⁄8 produces 0.125, not the sequence (0.1, 0.12, 0.125, 0.125, ...). In that case, all such divisions should result in the same type of object, not sometimes in a number and sometimes in a sequence.

Furthermore, you probably agree that a division of integeres like 1⁄9 should result, no matter whether we are in the real numbers or the hyperreals, in a number which, when multiplied by 9, gives 1, and not a little less. Otherwise our division algorithm would produce patently wrong results. I have detailed further objections to that line of reasoning above; I don't think we need two sections for the same proposed addition. Huon (talk) 15:35, 27 March 2011 (UTC)

No, I don't accept your claim about 1/8. In the case of 1/8, the algorithm terminates. In the case of 1/9, the algorithm does not terminate. By no stretch of imagination can one claim that long division already "knows" about the real number system. You can continue long division indefinitely, but what you construct is a sequence of digits. Tkuvho (talk) 16:11, 27 March 2011 (UTC)

Technically, the algorithm does not terminate with 1/8 either, it just gives nothing but zeroes after a certain point. Or you could argue that it terminates in both cases, with the termination event being the second occurrence of the same remainder, that is, once we have ascertained what precisely will be repeating.

Anyway, you mean long division produces an infinite string of digits, ie a repeating decimal? I agree. It happens to be the repeating decimal which represents the correct real number. I still don't see the use of having an algorithm of long division which produces wrong results. Could you please clarify? Huon (talk) 17:34, 27 March 2011 (UTC)

The algorithm of long division does not produce wrong results. It produces a string of digits. At the next stage, this string of digits is interpreted in a certain way. We are in agreement that one useful way of interpreting the outcome is in such a way that when you multiply back by 9, you get the same fraction you started with. But relying on this as evidence for the fact that long division must give such an outcome, amounts to circular reasoning. There is an infinitesimal ambiguity in interpreting the result of the long division. Afficionadoes of the real number system are free to suppress all infinitesimal differences and proclaim there is no life after real numbers. However, there are other points of view, which should not be suppressed in a pluralistic society. Tkuvho (talk) 18:24, 27 March 2011 (UTC)

We agree that long division gives a string of digits in which every digit has a finite distance from the first. This string of digits obviously is the decimal representation of a real number, and a meaningful real number at that. Of course you can interpret it otherwise, but I would like to see a reliable source for someone actually doing that. I'm pretty sure that any school textbook explaining long division will not mention hyperreals, and it will present the result as one number, not an equivalence class of infinitesimally close numbers. Anything else is a fringe position. Huon (talk) 18:53, 27 March 2011 (UTC)

You are absolutely right that we could define the result of long division to be a hyperreal. The fact of the matter is, though, that we don't. We define it to be a real number. Real numbers are a very useful way of modelling all sorts of things in our lives. Hyperreals aren't (they do have their uses, but not anywhere near as many as reals do). --Tango (talk) 19:16, 27 March 2011 (UTC)

Tango, this is exactly the problem I was pointing out. We define the result of long division to be a real number. Therefore purported "proofs" of .9=1 based on long division are merely brow-beating the students and their useful intuitions. I was certainly not proposing that we should redefine long division. Huon's "fringe" claims are WP:OR and should not be allowed to dominate this page. Tkuvho (talk) 13:45, 28 March 2011 (UTC)

Long division, as it is universally taught and understood, is about numbers. You divide one number with another number and get a third number which can be a terminating decimal or a recurring one. If can be extended to other things, polynomials for example, or other bases, but the common sense is the one about decimal numbers. Your proposed digressions into number systems that are little known even among mathematicians are fringe, and the consensus is clearly against their suggested inclusion.--JohnBlackburne^words_deeds 14:05, 28 March 2011 (UTC)

For example, our article on long division cites this paper, which in turn on page 4 cites a report published in the Notices of the AMS: "To understand that rational numbers correspond to repeating decimals essentially means understanding the structure of division of decimals as embodied in the division algorithm." So it's the official position of the AMS that long division of integers yields repeating decimals. The paper itself discusses this in greater detail, including a variant of this very proof. Huon (talk) 14:10, 28 March 2011 (UTC)

The paper you cited is a fine paper, and I think everybody here is in agreement that repeating decimals correspond to rational numbers. If you read the article carefully you will also notice that it deals with pre-calculus highschool education. At the pre-calculus level, there is no need for either infinitesimals or a nonzero entity "1-.999...". At the calculus level, there is a rich education literature about the advantages of the infinitesimal approach that you cannot write off with the stroke of your "fringe" pen. The point is not whether infinitesimals are useful or not, but rather if there is a legitimate literature to that effect. Supression of such literature is what is causing student frustrations which you seem interested in perpetuating. Tkuvho (talk) 16:52, 28 March 2011 (UTC)

suggested addition to FAQ

I suggest we add the following material to the FAQ: "The string '0.000...1' can be assigned definite meaning in the context of an infinitesimal-enriched number system, see the discussion of A. H. Lightstone's extended decimal notation in the main article." This is properly sourced in R. Ely's article referenced on the main page. Tkuvho (talk) 18:21, 27 March 2011 (UTC)

See my comments two sections above.--JohnBlackburne^words_deeds 18:27, 27 March 2011 (UTC)

To repeat, this page is not your private back yard where you can impose your opinion on everybody. Tkuvho (talk) 18:30, 27 March 2011 (UTC)

I agree with JohnBlackburne. There is no need to cover hyperreals in that question. Besides, we already discuss h

yperreals in another question, where we argue (rightly so, imo) that the hyperreal number with the best claim to being called 0.999... is still 1, which would imply that while a hyperreal number 0.000...;...0001 exists, it's not 1-0.999... Huon (talk) 18:57, 27 March 2011 (UTC)

I have to agree with JohnBlackburne and Huon that this proposed addition is outside the scope of the article. 28bytes (talk) 19:04, 27 March 2011 (UTC)

Apart from the issue of the relation of 0.000...01 to "1-.999...", there is a separate issue that the current version of the FAQ makes in incorrect claim to the effect that "0.000...01" cannot be meaningfully interpreted. This is inaccurate, as documented in the literature. The suppression of this point of view is not helpful to the students. Tkuvho (talk) 16:55, 28 March 2011 (UTC)

The FAQ says "[t]he string "0.000...1" is not a meaningful real decimal", which is correct. That it may be interpreted a hyperreal with no connection to 0.999... does not help the reader. Huon (talk) 17:03, 28 March 2011 (UTC)

Technically speaking the claim is correct but most readers have interpreted it and will interpret it as saying "[t]he string "0.000...1" is not a meaningful decimal". By the way, the way the question is formulated does not imply a commitment to the real number system on the part of the questioner. Tkuvho (talk) 17:08, 28 March 2011 (UTC)

It isn't a meaningful decimal – which digit is supposed to be 1? There is no "next" digit just after the standard naturals in any nonstandard model of PA (since 0 is the only non-successor natural number), so it isn't clear from the notation 0.000...1 which power of 10 is supposed to be multiplied by that 1. — Carl (CBM · talk) 20:33, 31 March 2011 (UTC)

There are two separate items here: (a) a pre-mathematical student's intuition of a tiny number with an infinity of zeros before 1; (b) a hyperreal infinitesimal 10^{-H} for an infinite H. What you seem to be pointing out is that there is no relation between (a) and (b), and that the student intuition is merely erroneous. This is precisely the claim contested by Robert Ely's paper. Ely argues that the student intuition is not erroneous, but rather nonstandard. What he means by that is that the intuition is robust, withstands challenges from the traditional interpretations, and provides a fruitful way of learning calculus. I happen to agree with Ely, but this is irrelevant. What is relevant is that this viewpoint is properly sourced. Tkuvho (talk) 20:47, 31 March 2011 (UTC)

All I am saying is that 0.0001 has a clear meaning of $10^{-4}$ but 0.000...1 has no meaning, even in the hyperreals, because there is no canonical candidate for which nonstandard power of 10 it is supposed to represent. — Carl (CBM · talk) 20:54, 31 March 2011 (UTC)

Again, 0.000...1 is not a precise mathematical notion, but a pre-mathematical intuition. Compare it to naive concepts students have of the continuity of functions. A teacher's role is not to discourage their naive concepts, but help them connect those concepts with precise mathematical structures. Note that there is no canonical candidate for 10^{-H} in the hyperreals, either. But it has an infinite number of zeros nontheless! The student in Ely's study naturally developed an arithmetic of infinite numbers to accomodate her infinitesimal intuitions. This does not mean she developed nonstandard analysis, or that she cannot benefit from her insights until she learns the details of the ultrapower construction. Tkuvho (talk) 21:06, 31 March 2011 (UTC)

There is a canonicial 10^{-H} for every nonstandard integer H. Peano arithmetic proves that for 10^{H} exists and is unique, by induction, and the nonstandard integers model PA because the standard ones do, via transfer of all the first-order axioms of PA. Then 10^{-H} is the reciprocal of 10^{H}. Alternatively, you can just adjoin the exponential function 10^x to the structure before you take the ultrapower, and then you can use the nonstandard extension of this function to give a unique value 10^x for every nonstandard x. — Carl (CBM · talk) 21:15, 31 March 2011 (UTC)

You pointed out earlier that there is no canonical choice for "0.000...1". I was merely pointing out that there is no canonical choice of an infinite hypernatural, either, in the sense that it depends on the choice of the ultrafilter (one can choose the nearly canonical sequence (1, 2, 3, ...), but the properties of the hypernatural it generates will depend on the ultrafilter). Of course transfer implies that exponentiation is well-defined. We are talking about the transition from student intuition to formal mathematics, not about the technicalities of the ultrapower construction. The transition from "0.000...1" to a formal infinitesimal is a very interesting one, and one that is documented in the recent education literature, in a leading math education journal. I fail to see why such a hullabaloo is being made about including a short sentence about this at the FAQ. Are we maintaining an ideological purity here? Tkuvho (talk) 04:37, 1 April 2011 (UTC)

(od) There is simply no need to mention hyperreals in this context at all. You may call that "ideological purity", but I'd call it "staying on topic". Besides, I don't think a field study involving a single student can be generalized to suggest changing mathematics education for all students, no matter where it has been published. Huon (talk) 11:59, 1 April 2011 (UTC)

(←) I was really responding to Huon's comment of 17:03, 28 March 2011 which suggested that the notation "0.000...1" did have a well defined value in each model of the hyperreals. I wanted to point out that that notation doesn't have a well-defined value in any model of the hyperreals.

As for adding the sentence to the FAQ, my personal opinion is that it's out of place. But even if it was added, we can't claim that the notation "0.000...1" has a value in the hyperreals. The strings "0.1", "0.01", "0.001" can be assigned definite meaning, but this doesn't extend to strings like "0.000...1" and "0.000...01". — Carl (CBM · talk) 11:51, 1 April 2011 (UTC)

fuzzy rendering... (not the same old question)

In the section on non-standard systems, the N renders on my browser as fuzzy. not clear or sharp. Anybody know why this might be? Cliff (talk) 16:55, 22 April 2011 (UTC)

Shows up as a blackboard bold N on my browser. 28bytes (talk) 17:03, 22 April 2011 (UTC)

mine too, but it's not clear. It's fuzzy. Cliff (talk) 17:06, 22 April 2011 (UTC)

same person, different browser. It renders fuzzy in chrome and in firefox. 134.29.231.11 (talk) 17:11, 22 April 2011 (UTC)

Hmm. It shows up as hollow but clear on my machine on both IE and FF at standard resolution, but if I zoom in or out even slightly, it does fuzz up a bit. 28bytes (talk) 17:16, 22 April 2011 (UTC)

Ah, that's probably it. I think I had zoomed in a bit because my eyes were bothering me...time to visit the opthamologist again I think. Cliff (talk) 04:55, 23 April 2011 (UTC)

Nope, that's not it. The N in text renders nicely at default zoom. The N in the equation still seems out of focus. Where does it come from, does anybody know? Is it a font on my machine or is it a "picture" that is downloaded with the webpage? Cliff (talk) 21:03, 25 April 2011 (UTC)

The N in the text (... where $[\mathbb {N} ]$ is...) is significantly larger than those in the equation (e.g. ${\frac {1}{10^{[\mathbb {N} ]}}}$); maybe that's why it renders better at default zoom? Both should be images; the TeX code used to write them is, I believe, turned into PNG images. The large one can be seen here. Huon (talk) 21:16, 25 April 2011 (UTC)

Is the small one just a shrunken version of the large one or its own image? Cliff (talk) 00:02, 26 April 2011 (UTC)

There's more to this. It's not just the $\mathbb {N}$. All the text in the math equations are fuzzy. I guess they are PNG images, I don't know, but at 150% zoom, all the math symbols and digits in between the standard text appear with the same fuzziness as other images, like for example the Wikipedia logo at the upper left of all pages and also the Wikimedia and Mediawiki banners at the bottom of most all pages. The $\mathbb {N}$ seems to appear even more out of focus because of the double-barred diagonal. More of an optical illusion going on there. Zoom to 150 and you will see many examples of other text and symbols with the same fuzziness.

For an even more curious example, zoom on the following, the lead from this article:

In mathematics, the repeating decimal 0.999... which may also be written as 0.9, $\scriptstyle \mathbf {0} .\mathbf {\dot {9}}$ or 0.(9), denotes a real number that can be shown to be the number one.

The $\scriptstyle \mathbf {0} .\mathbf {\dot {9}}$ appears fuzzy when zoomed to 150%, but all the other numbers and letters in this sentence remain clear and sharp on my browser. Kinda weird eh? Racerx11 (talk) 04:05, 26 April 2011 (UTC)

I can explain that one, at least... The 0.9 with the dot over it is the only one of the three rendered as a PNG; the others are text. Previously the 0.9 with a dot over it was also rendered as text using either CSS or a Unicode combining dot character (I forget which), but the dot couldn't be positioned correctly (depending on the browser, it would be shifted to the left or right) so that was changed to use the PNG method. 28bytes (talk) 04:18, 26 April 2011 (UTC)

I think you are exactly right. The PNG characters all appear fuzzy when in zoom. The $\mathbb {N}$ is particularly troublesome because, like I said, it has that double-barred element, in addition to it apparently also being rendered as a PNG image (at least it looks the same as the PNG images). This double diagonal makes it appear more out of focus even at low and normal zoom. Racerx11 (talk) 04:33, 26 April 2011 (UTC)

And to follow up on the original point; when certain PNG characters are shrunk down such as in (e.g. ${\frac {1}{10^{[\mathbb {N} ]}}}$), the small $\mathbb {N}$ in particular looks even more out of focus. Racerx11 (talk) 04:54, 26 April 2011 (UTC)

Is there a way of "fixing" this? can the images be replaced with higher resolution ones that don't have this problem on zoom? Cliff (talk) 14:54, 28 April 2011 (UTC)

Probably not without modifying of the MediaWiki software. The images are automatically created from the TeX code - I don't think there's a way to manually change that creation process to increase image resolution. Huon (talk) 11:55, 2 May 2011 (UTC)

There is mathJax, which instead of PNG generates SVG, and arguably looks better if scaled. It's not built into MediaWiki, but can be added manually with a couple of steps described at User:Nageh/mathJax.--JohnBlackburne^words_deeds 12:05, 2 May 2011 (UTC)

1 = .999 . . . is technically a mistatement

A better statement is that the limit of 1 - 1/n as n approaches infinity is equal to 1.

The distinction is clear if you consider the limit of (1 - 1/n)^n as n approaches infinity. The value of this limit is 1/e or approximately .38787. . .

On the other hand the limit of (1)^n as n approaches infinity is 1. In this case 1 does not give the same limit as .999 . . ., which shows that the equivalence depends on the context.

Prokrop (talk) 01:18, 17 May 2011 (UTC)

Actually the statement is correct as long as you interpret 0.999... to be the real number whose decimal expansion consists entirely of 9's, for all negative powers of 10. (And 0 otherwise of course.) But if you read the history of this page, you'll find many discussions on the subject. Thenub314 (talk) 01:29, 17 May 2011 (UTC)

Especially, .999... is the limit of 1 - 1/n (or more precisely, the limit of 1-1/(10^n)) as n approaches infinity; it's not the sequence itself. The limit of $(\lim _{n\to \infty }1-10^{-n})^{m}$ as m approaches infinity is just as much 1 as the limit of (1)^m as m approaches infinity. Huon (talk) 02:48, 17 May 2011 (UTC)

Hi Huan, can you summarize my position for me here? Perhaps we can write a separate FAQ to address this type of comment, which as you know I know you know, is not completely misguided. Tkuvho (talk) 05:18, 17 May 2011 (UTC)

Sorry, but I don't think I know your position on this issue well enough to summarize it for you; if I had to speculate I'd assume that you'd want to say that modulo the choice of an ultrafilter, interpreting 0.999... as a sequence will lead to a hyperreal. But that ultrafilter step is essential and non-trivial, and I don't think it's what those who think about 0.999... as a sequence have in mind. A hyperreal isn't more of a sequence of reals than a real number is a Cauchy sequence of rationals. Anyway, an addition to the FAQ regarding the "sequence vs. number" question may be useful. Huon (talk) 12:16, 17 May 2011 (UTC)

Thanks :) However my starting point wouldn't be to jump to ultrafilters, but rather to relate to the fact that many students think of an infinite string of 9s as a terminating one, i.e. there is a "last" one. That particular intuition can be fruitfully implemented in an infinitesimal-enriched continuum. The particular construction you choose (such as ultrafilters, etc) is not so important, as such systems can be described axiomatically, as well. Tkuvho (talk) 12:58, 17 May 2011 (UTC)

A "last 9" doesn't seem to be the issue here. If I understood Prokrop correctly, he didn't argue that 0.999... is different from 1 because there is a "last nine at infinity" or something like that, but rather says that 0.999... is a sequence of reals which simply never reaches its limit (which he agrees would be 1, not something a little short of 1). Those are two different misinterpretations of 0.999... Huon (talk) 13:17, 17 May 2011 (UTC)

Nonzero or non-zero

I would be writing the term as "nonzero." Any opinions, or arguments against that? Twipley (talk) 01:14, 30 March 2011 (UTC)

My dictionaries show "nonzero" as the standard spelling, so go for it. 28bytes (talk) 03:29, 30 March 2011 (UTC)

While that may be the case with your dictionary, non-zero is also correct. It appears both ways all throughout WP, oftentimes in the same article, as this Google search and this Google search demonstrate. So it's not really clear at all which form is better. — Loadmaster (talk) 17:46, 27 April 2011 (UTC)

Be sure to check whichever article you are editing for existing spelling and remain consistent within each article. Cliff (talk) 13:02, 26 May 2011 (UTC)

New thread

1/9 does not equal .111.... It is an approximation. —Preceding unsigned comment added by 165.155.196.69 (talk) 15:26, 14 April 2011 (UTC)

More precisely, it is an adequality. Tkuvho (talk) 15:32, 14 April 2011 (UTC)

The article could devote more attention to the student intuition that "0.999..." is adequal, rather than equal, to 1. Before the number system is specified (e.g. as being the real numbers), such intuitions are not erroneous but rather nonstandard, see R. Ely's article. Tkuvho (talk) 18:34, 16 April 2011 (UTC)

That's precisely what the article already says. Giving more prominence to the Ely article would be undue weight. Regarding the re-addition of 165.155.196.69's comment, please have a look at WP:TALK: "Article talk pages should not be used by editors as platforms for their personal views on a subject", and "[i]rrelevant discussions are subject to removal." This clearly includes your n-th attempt to proclaim that students somehow intuitively think of hyperreals instead of reals. I'd say the arguments page is a nice case study to the contrary: Hardly anybody who disagrees with the equality instead thinks of the hyperreals - the hyperreals just violate their intuition in other places than the reals. Huon (talk) 18:44, 16 April 2011 (UTC)

OK, I agree with what you wrote. At any rate, it was a legitimate issue to be raised, and I see no reason to delete it summarily as some kind of an expletive. Tkuvho (talk) 19:48, 16 April 2011 (UTC)

The mention of adequality was just deleted from the lede, which is a pity. The comment deleted nicely summarized the infinitesimal section. I have mentioned numerous times that it is not hyperreals that the students intuit (that would be remarkable indeed!), but rather a number system containing infinitesimals, as envisioned by Fermat and Leibniz. The latter certainly were not thinking in terms of hyperreals. Tkuvho (talk) 12:42, 17 April 2011 (UTC)

Well, in the decades in which I taught mathematics, I never once came across a student who appeared to "intuit" either hypereals or a number system containing infinitesimals. What I saw year after year was students who perceived decimals as strings of figures, rather than as an abstract concept which has strings of numbers as a convenient concrete representation. If your concept of what a decimal number actually is is a string of figures, then clearly 0.99999... is not the same as 1. And if you think of the order relation on decimal numbers being defined in terms of that string in some such tems as "Compare the numbers digit by digit from the left until you find the first difference: then the number which has the larger digit in that place is the larger number" then clearly 0.99999... < 1. Since these students have learnt the properties of decimals by rote from a very early age, they have internalised some process along those lines. From the point of view of a person with a high level of mathematical understanding it is clear that logically the difference in that case would be infinitesimal, but in my experience that is not how it is perceived by the vast majority of people who cannot accept that 0.999... = 1. They simply are not thinking in such terms. Yes, you can push them in the diection of thinkinbg in such terms by, for example, asking them what 1 - 0.999... is, but I have never seen any evidence that they think in such terms spontaneously. However, in my opinion this is not the main point. The main point is that the article essentially is about what 0.999... represents in the real number system, about the popular miconception that it does not represent the number 1 in that system, and about what reasons there are for accepting that in fact does. To insert stuff about hyperreals or infinitesimals into such an article confuses and muddies the issue for the average reader, and detracts from the clarity of the essential point that the article seeks to convey. Mathematicians are not the principal readership. JamesBWatson (talk) 18:11, 17 April 2011 (UTC)

Speaking directly about the article, I'm not sure that adequality is worth mentioning in the lede. We do link the term lower in the article, in the section on infinitesimals. Because infinitesimals are already not directly the subject of this article, I personally prefer to keep the lede sentences about them very tight. — Carl (CBM · talk) 18:12, 17 April 2011 (UTC)

Yes, that is very much in keeping with my own thoughts. It is mentioned, but it is not germane to the central point of the article, and should be resricted to the place where it is most relevant to the context. Certainly not in the lead. JamesBWatson (talk) 18:20, 17 April 2011 (UTC)

OK, whatever consensus emerges here is fine. Tkuvho (talk) 07:32, 18 April 2011 (UTC)

To respond briefly to JamesBWatson's detailed remarks above: Huon pointed out correctly that R. Ely's paper should not be given undue weight. On the other hand, it cannot be ignored altogether, either. JamesBWatson argues that student misconceptions about .999... result from their thinking of a number as being a string of digits. This is one possible interpretation. However, his claim that there is no evidence for any other interpretation is not correct, as Ely documents in his field study. The claim that some students think of 1-0.999... as an infinitesimal is documented in the education literature and is no longer in the realm of pure speculation. Tkuvho (talk) 10:17, 20 April 2011 (UTC)

I did not say that "there is no evidence for any other interpretation". I said that I had not seen such evidence. JamesBWatson (talk) 22:42, 20 April 2011 (UTC)

If you want students to intuitively grasp .999... = 1 there's an easy way imo, tell them a story about a King that wishes his cooks to make him a pie but to pre-slice it in diminishing divisions 1/10th the size of the last because the King wishes the option to eat any sized slice he wants without having to cut the pie again. The 1 single pie would come out and be presented to the king on a plate in an infinite amount of smaller slices starting with 9 slices of 1/10th size each, 9 slices 1/100th sized each, 9 slices 1/1000th sized each and so on. If you looked at the pie with your eyes squinted you wouldn't see the slices but the whole of what they form - 1 single pie. And there is no infinitesimal bit of pie missing because the cooks did nothing but take a knife to the existing pie and were careful to not let any crumbs fall out :P 76.103.47.66 (talk) 23:23, 25 April 2011 (UTC)

I like it. Cliff (talk) 00:00, 26 April 2011 (UTC)

Each slice the cook makes does not even come close to the "infinitesimal bit" at the end. At what stage do you claim that your "infinitesimal bit" disappears, if not when you apply the limit to sweep all infinitesimals under the rug? Tkuvho (talk) 04:59, 28 April 2011 (UTC)

I find this is a particularly smart metaphor. There are not infinitesimal pieces of cake that disappear, there are infinitesimally closer and closer cuts - each cut represented by a digit in the secuence. When you apply the limit, the final cut is performed at exactly 1.0 distance from the origin. Wait, this is now the Zeno's paradox!! Diego Moya (talk) 08:51, 28 April 2011 (UTC)

Please keep in mind that it was the previous editor, rather than myself, who pointed out that "there is no infinitesimal bit of pie missing". If you assume from the start that there are no infinitesimals, of course there is going to be none left at the end of the process. What editors keep trying to do all over again is prove "from first principles" the non-existence of infinitesimals. This is circular reasoning. Tkuvho (talk) 08:56, 28 April 2011 (UTC)

I understand what each of you said. But this is this what I find clever about the metaphor. There 'are' infinitesimals in it, but they are not used to describe the pie (the real number 1) but the cut actions (each of the "0.9...9" fractions). With this intuition there can't be an infinitesimal missing part of the pie which makes the series distinct from a whole pie; all the "payload" -the pieces of pie are always part of the cake. Diego Moya (talk) 10:16, 28 April 2011 (UTC)

It is unlikely we will get a culinary degree this way, but if you do assume that there are infinitesimals in the pie, it is difficult to claim that "there is nothing left". As you say, the pieces of the pie are always part of the cake, but at each cut there always remains a last uncut slice. If one assumes that there are infinitesimals there, then an infinitesimal "at the end of the pie" is always in the remaining slice. Therefore it is never eliminated. Bon appetit. Tkuvho (talk) 10:23, 28 April 2011 (UTC)

And therefore movement is impossible? :-) Of course you always have infinitesimals - before you reach the infinite, while you have "arbitrarily small but bigger than zero" uncut slices. But when you 'jump' to the infinite you arrive at exactly 0 distance from the origin to make the 'last' cut, so there can't be any uncut pie left. This jump is more difficult to understand when the limit is explained as 'what you have when you add an infinite amount of increasingly small fractions'. Is there a referenced demonstration that uses sectors of a circle? If so it could be used in the article to illustrate this insight. Diego Moya (talk) 11:31, 28 April 2011 (UTC)

On the contrary, if you do have a last cut, then you have a precisely measurable infinitesimal leftover, see 0.999...#infinitesimals. Tkuvho (talk) 12:28, 28 April 2011 (UTC)

I thought we were talking about real numbers (physical cake and all), where there are no non-zero infinitesimals? I don't think the pie metaphor is a good fit for alternate numeral systems. Diego Moya (talk) 12:56, 28 April 2011 (UTC)

Tkuvho, the cake analogy is not offered as a proof, but as a way of helping ones students overcome their misunderstanding when it comes to the real numbers, and the 0.999...=1 statment. Diego Moya, yes. We are working from the assumption that our number set is the reals, this is why Tkuvho's arguments make no sense here. Underlying the statment 0.999... = 1 is an assumption that our set is the real numbers, (it works in the rationals too). Cliff (talk) 14:52, 28 April 2011 (UTC)

To clarify my original pie story, I am not starting with the assumption that there is no infinitesimal missing pie. I started with the assumption that there is 1 pie and you can cut it up in to any number of slices you wish. If you cut it into 10 slices you have 10/10 = 1. If you cut one of those slices into 10 you have 9/10 + 10/100 = 1. Cut one of those and get 9/10 + 9/100 + 10/1000 = 1. Next is 9/10 + 9/100 + 9/1000 + 10/10000...If instead of this linear geometric approach you simply think of every possible division existing at once without having to cut down to it you can see that any number of slices in any the particular division would be 9. So another way to think of 1 whole pie is at 9 slices each in infinite descending place values in base 10. Or, .999....76.103.47.66 (talk) 00:37, 2 May 2011 (UTC)

What Tkuhvo is saying, anon 76, is that according to some number systems, your story leaves an infinitely small slice out of the sum. Such number systems are usually doctoral level study and not widely known. Look for information on "non-standard analysis" to see what he's saying. Cliff (talk) 03:22, 2 May 2011 (UTC)

Thanks, a fair summary except for the word "doctoral". Recent education studies report successful undergraduate calculus teaching using infinitesimals. Tkuvho (talk) 04:14, 2 May 2011 (UTC)

Yeah most of that goes over my head. Like a lot of people I found the concept of .999... = 1 to be counter-intuitive and was never completely convinced by any of the proofs (even the ones i understood thoroughly) until I approached the problem in a way that I could visualize the equality rather than simply acknowledge it, and that's how I came up with the pie idea. I thought it might help others as it did me in solidifying the equality intuitively without having to resort to complex theories. As for the non-standard analysis stuff, I cannot comment on that and leave it for smarter people than me :) 76.103.47.66 (talk) 06:07, 2 May 2011 (UTC)

Another way of seeing it geometrically is by flipping the problem to the other side of 1, namely by considering 2-0.999... What kind of number could it be? Here the decimal digit 1 at rank n gets pushed off further and further to infinity. Since there are no infinite ranks in the ral number system, in the limit we get 1.000... on the nose. Tkuvho (talk) 12:03, 2 May 2011 (UTC)

Oh good grief. 1/9 is equal to 0.111..., in the sense of being simply different textual representations of the same real number (just like 0.5 + 0.5 is equal to 1). It is not an adequality, as defined by that article, except in the trivial sense of being an identity William M. Connolley (talk) 16:09, 3 May 2011 (UTC)

Tkuvho, can you point me in the direction of the articles you mentioned above? I'd like to read them. Thanks, Cliff (talk) 13:25, 26 May 2011 (UTC)

A good place to start is the article by Robert Ely entitled "Nonstandard student conceptions about infinitesimals", published in Journal for Research in Mathematics Education. It is referenced in the bibliography of this .9 page. Do you have access to the journal from your server? Tkuvho (talk) 13:30, 26 May 2011 (UTC)

I believe so, and if not it doesn't matter. I'm joining NCTM later this month next month and will get access to JRME then. Cliff (talk) 13:34, 26 May 2011 (UTC)

will 0.999... always equate the same as 1 in all instances?

1 raised to the power of any number = 1 as in 1ⁿ as n→∞ = 1. Does that hold true for 0.999...ⁿ as n→∞ = 1. I have tried to work the math behind this but have as yet been unable to determine how to fit an infite series into an infite series. Any help here would be great. —Preceding unsigned comment added by 68.170.209.249 (talk) 08:21, 17 May 2011 (UTC)

Short answer: Yes. If you have an expression like $\lim _{n\to \infty }(\lim _{m\to \infty }1-10^{-m})^{n},$ you should evaluate the limits from the inside out. And since $\lim _{m\to \infty }1-10^{-m}=1,$ you end up with $\lim _{n\to \infty }(\lim _{m\to \infty }1-10^{-m})^{n}=\lim _{n\to \infty }(1)^{n}=1.$

Note that while $\lim _{n\to \infty }(\lim _{m\to \infty }1-10^{-m})^{n}=\lim _{n\to \infty }\lim _{m\to \infty }((1-10^{-m})^{n}),$ in general $\lim _{n\to \infty }\lim _{m\to \infty }a_{n,m}$ and $\lim _{m\to \infty }\lim _{n\to \infty }a_{n,m}$ will differ, and neither need be equal to $\lim _{n\to \infty }a_{n,n}.$ Huon (talk) 12:37, 17 May 2011 (UTC)

0.999_ does not equal 1

Discussion moved to Talk:0.999.../Arguments#0.999_does_not_equal_1. Tkuvho (talk) 03:58, 20 May 2011 (UTC)

Dead links

links all work (all checked), bot is having a bad day

During several automated bot runs the following external link was found to be unavailable. Please check if the link is in fact down and fix or remove it in that case! http://www.math.umt.edu/TMME/vol7no1/ In 0.999... on 2011-05-20 22:34:14, 404 Not Found In 0.999... on 2011-05-31 15:18:57, 404 Not Found --JeffGBot (talk) 15:19, 31 May 2011 (UTC) You seem to have copied the link incorrectly, as there were two curly braces at the end that are not needed. I removed the braces four lines above, and now the link cited in this section of talk should work, just as the link in the article itself. Tkuvho (talk) 03:59, 1 June 2011 (UTC) Bot malfunction. The "}}" are a remnant of the citation template. I have notified the bot owner. Huon (talk) 04:07, 1 June 2011 (UTC) During several automated bot runs the following external link was found to be unavailable. Please check if the link is in fact down and fix or remove it in that case! http://elib.mi.sanu.ac.rs/files/journals/tm/20/tm1114.pdf}} In 0.999... on 2011-05-20 22:34:23, 404 Not Found In 0.999... on 2011-05-31 15:19:06, 404 Not Found --JeffGBot (talk) 15:19, 31 May 2011 (UTC) During several automated bot runs the following external link was found to be unavailable. Please check if the link is in fact down and fix or remove it in that case! http://elib.mi.sanu.ac.rs/files/journals/tm/24/tm1312.pdf}} In 0.999... on 2011-05-20 22:34:23, 404 Not Found In 0.999... on 2011-05-31 15:19:20, 404 Not Found --JeffGBot (talk) 15:19, 31 May 2011 (UTC)

Dead link

more bot failure

During several automated bot runs the following external link was found to be unavailable. Please check if the link is in fact down and fix or remove it in that case! http://www.math.umt.edu/TMME/vol7no1/}} In 0.999... on 2011-05-20 22:34:14, 404 Not Found In 0.999... on 2011-05-31 15:18:57, 404 Not Found In 0.999... on 2011-06-20 03:32:11, 404 Not Found --JeffGBot (talk) 03:32, 20 June 2011 (UTC) The bot still does not recognize the "}}" as part of the citation template. Huon (talk) 11:19, 20 June 2011 (UTC) So maybe message to bot operator is good idea? (I did this) Bulwersator (talk) 11:30, 29 June 2011 (UTC)

Image for 0.999...

Why did the "perspective" image get removed? I don't see any discussion or consensus about the removal. — Loadmaster (talk) 17:37, 5 July 2011 (UTC)

Because it is a kitsch decoration that contributes nothing whatsoever to the reader's understanding of the article. Hesperian 05:20, 9 July 2011 (UTC)

This is purely a matter aesthetics though. Which means you might wanna ask other folks what they think first before removing it. I sort of like the image...Volunteer Marek (talk) 06:38, 9 July 2011 (UTC)

The image does contain information and contributes to understanding. It shows the unfolding of the 9s beyond the first three digits used in the text, with an emphasis on the infinite procedure. It portrays at a glance the definition of the number as a repeating decimal, which otherwise would have to be understood through careful study of the math formulas. Diego Moya (talk) 09:32, 9 July 2011 (UTC)

There's also a nice artistic subtlety in the image. It makes sense in this particular context as soon as you realize that the 9's are not "getting smaller" but rather appear smaller because they are "moving further away". Put it back in! Keep it!Volunteer Marek (talk) 09:48, 9 July 2011 (UTC)

Keep it. It neatly illustrates the subject of the article. (I have looked at numerous definitions of "kitsch", and not found one anywhere that could conceivably apply to this. Apart from anything else, this does not claim to be art.) JamesBWatson (talk) 20:33, 10 July 2011 (UTC)

Keep it, per Diego Moya's comment about the number of nines. While three are standard notation, there are in truth infinitely many, and the image clarifies that point better than if we tried to express it in words. Huon (talk) 20:42, 10 July 2011 (UTC)

Keep — I'm certainly all for keeping it, kitsch and all. (And you might want to compare it to the image for pi → PI.svg.) — Loadmaster (talk) 19:00, 11 July 2011 (UTC)

Ugh, that's even worse. Hesperian 00:28, 14 July 2011 (UTC)

Hesperian, you will not win this battle if you fail to suggest a reasonable alternative. As of now, the only alternative you've offered is removing the image altogether, leaving a huge wall of text, which will only serve to drive laymen away from the article. If you think there's a better representative image that can be placed there, then by all means, make a suggestion! --COVIZAPIBETEFOKY (talk) 21:59, 15 July 2011 (UTC)

Keep it. It's an interesting yet economical visual hook for this article. SkyMachine (talk) 00:00, 16 July 2011 (UTC)

• IS this a reasonable summary of the discussion so far? Hesperian hates it. Nobody else has any problem with it, and several people think it should stay. If so, is it reasonable to consider the discussion settled, with consensus very clear? JamesBWatson (talk) 10:45, 16 July 2011 (UTC)
  • Yes. Hesperian 10:59, 16 July 2011 (UTC)

How many nines in a repeating decimal?

The title in this article and ALL appearances of the repeating decimal use the same ZERO-DOT-NINE-NINE-NINE-DOT-DOT-DOT symbol. This extremely consistent usage of exactly three '9' digits makes it look like the number of digits in the repeating decimal is significant, when it's not; any number of appearances would represent the same number through a slightly different symbol, but now it seems as if 0.999... is the topic of this article, but 0.9999999... is a different thing for which the properties described may or may not apply. I think that the article must not give that false impression and made an edit to address the problem, but it has been reverted as not clarifying. What else can be done to the article to avoid this problem? Maybe if the first sentence explained it? ("...which may also be written as 0.9, $\scriptstyle \mathbf {0} .\mathbf {\dot {9}}$, 0.(9), or with any number of 9s in the repeating decimal") Diego Moya (talk) 14:43, 9 July 2011 (UTC)

I prefer to think that the consistent use of exactly three "9" digits makes it look like the most common accepted mathematical notational convention for repeating trailing digits. I'm no expert, but I think that is far more common to see three trailing repeated digits used than any other specific number of digits. — Loadmaster (talk) 19:05, 11 July 2011 (UTC)

(1) Yes, three nines is usual. (2) is there any serious risk of people being seriously misled into thinking that 0.9999... would not mean the same? It would require someone with a remarkably literal and pedantic mind. I am for keeping it as it is. JamesBWatson (talk) 20:08, 11 July 2011 (UTC)

If 0.9999... did appear somewhere in the examples it wouldn't be a problem, but in fact it doesn't. The problem lies not with a literal mind, but with someone without full comprehension of repeating decimals; someone with little knowledge on the subject is more prone to literal thinking. The fact that all the same-digit repeating decimals in this article use exactly three digits may prompt someone with little knowledge to think that this is the only valid representation (whether by a mathematical property or a historical reason), when that isn't true. An article in which the used notation is part of the main topic should be more careful to not imply invalid inferences; and fixing it is as easy as including the clarifying sentence in the description of valid representations, or showing a different number of 9s in one of the salient examples in the lede. Diego (talk) 20:57, 11 July 2011 (UTC)

I reverted your first edit. Simply writing a larger number of nines did not seem to convey the flexibility of notation that you intended. The sentence now in place does a much better job, though I too question its necessity. Cliff (talk) 09:14, 12 July 2011 (UTC)

Averaging / continuum proof

I just removed the following as unnecessary and unclear

Extended content

Averaging / Continuum The average, A, of two unique numbers, b and c, is a value that is in between the two values. $b<A<c$ This fact can be proven separately, but it should also be intuitively obvious that a larger value will bring up an average, and a lower number will bring down an average. If we assume that b = 0.9999... is a unique number separate from c=1, we may find the average of the two. ${\begin{aligned}b&=0.9999\ldots \\c&=1\\\\A&={\frac {b+c}{2}}\\A&={\frac {0.9999\ldots +1}{2}}\\A&={\frac {1.9999\ldots }{2}}\\A&=0.9999\ldots \end{aligned}}$ We may conclude that the average is equal to the smaller value, A=b. This implies that the average is also equal to the larger value, A=c and therefore that the two are equal, b=c. This proof stems from the fact that the real numbers exist on a continuum. The argument states that there is no value larger than 0.999... that is smaller than 1. Because there is no distinct value between these two, they are the same.

Apart from minor formatting issues my main concern is the assertion that 1.999... / 2 is 0.999.... . Of course this is true because it's just 2/2 = 1, but it cannot be just asserted as we try and establish 0.999..... == 1.

If this is fixed then it is equivalent to the digit manipulation proof, but instead of multiplying by 10 we multiply 0.999... by 2. As that is shorter and clearer this third proof is not needed.--JohnBlackburne^words_deeds 21:19, 3 September 2011 (UTC)

Alternate representation

Does anybody know of any source that expresses the equality this way?

0.9 = 1.0

IMO this representation is less likely to induce the incorrect intuitions about infinitesimals that are so common in education, since now the span can be seen as an operator over both the 0 and 9 digits, and it highlights that the number 1 is also expressed by a repeating decimal with infinite trailing 0s. (Is there any distance at the first decimal, even though one has a 9 and the other a 0? No. And at the second? No! And at the third?...)

Maybe this form could be added to the FAQ. Diego (talk) 15:52, 27 October 2011 (UTC)

Wikibooks

The last time I was here, I recall having an interesting discussion about how the rigor of a proof depends on the background knowledge one can assume. Well, I've been thinking about it, and I've come to the conclusion that, for this reason and other reasons, we need more than just a Wikipedia article on 0.999.... Therefore I present:

• Wikibooks:0.999...

It has no content yet, but it does describe my vision on the top-level page. Eventually I think it can be a great supplement to this article! Melchoir (talk) 01:10, 27 November 2011 (UTC)

At the level you are aiming at, it would be sufficient to say that even to those who feel that .999... falls short of 1, the discrepancy is infinitesimal, and all infininitesimal differences are suppressed in the received approach, by definition. This would clarify that the "proofs" you are aiming at are actually "definitions". Tkuvho (talk) 09:08, 27 November 2011 (UTC)

I disagree, but let's leave it there. :-) Whatever the reader's prejudices are concerning the received approach, I believe there is value in understanding what the received approach is and how it works. (Only then can one hope to reach an informed opinion about why it works, including the extent to which it's merely a matter of definition.) And I believe that Wikibooks offers a unique opportunity for communicating this information! Melchoir (talk) 10:28, 27 November 2011 (UTC)

Benardete quote

I'm not sure what to make of the Benardete quote. From a mathematical point of view, Benardete says nothing useful. He would have to do more than say we "include infinitesimals" to specify his system of continuum numbers, and he would have to do something to say which number in his enriched system is supposed to be 0.999... Our own article gives various different models of infinitesimal-enriched number systems, and I cannot tell which of them, if any, is supposed to be the domain of continuum numbers. I'll try to get access to Benardete's book, but that may take some time. Thoughts? Huon (talk) 15:30, 27 November 2011 (UTC)

Benardete does not provide any construction of such a system. He leaves such work to mathematicians. His book came out after Robinson's seminal 1961 paper but before Robinson's 1966 book. What Benardete seems to be pointing out is that some natural pre-mathematical intuitions cannot be expressed if one is limited to a number system that's too restrictive. R. Ely's recent field research confirms the sentiment. Tkuvho (talk) 15:58, 27 November 2011 (UTC)

I think we may be giving this idea undo weight. We have already sourced the idea that 0.999... is less then 1 but infinitesimally close to it by means of Lightstone, Stewart, and Katz&Katz. Do we need another lengthy quote to further describe it? Thenub314 (talk) 20:52, 27 November 2011 (UTC)

It does seem unnecessary, not only per undue weight but also quotations should not be used unless pertinent, i.e. unless the quotation is being commented on or the point cannot be made without using a quotation. If the same information could be presented by paraphrasing then it should be paraphrased, though not too closely.--JohnBlackburne^words_deeds 22:28, 27 November 2011 (UTC)

Archimedean property

In the Infinite Series and Sequences part of the Analytical proofs section the article claims that the Archimedean property is an axiom of the real numbers. This is incorrect. The Archimedean property can be proven from the Completeness axiom of the real numbers. I am editing the article to reflect this. NereusAJ (talk) 07:28, 20 December 2011 (UTC)

Standard real number system?

In the fourth paragraph of the Introduction the article refers to the "standard real number system". This is misleading. There is only one real number system. Alternative number systems which can be constructed are never referred to as real numbers. There is therefore no ambiguity in just saying "the real number system" instead of "the standard real number system". NereusAJ (talk) 20:55, 20 December 2011 (UTC)

I think what was meant is that the real numbers are the standard number system. I can't think of a way to clarify that without making it sound awkward in other ways. Huon (talk) 22:35, 20 December 2011 (UTC)

For the sake of accuracy note that the real line of Edward Nelson discussed in Internal Set Theory does contain infinitesimals. The editor who introduced the term "standard" may have had this in mind. Tkuvho (talk) 12:14, 21 December 2011 (UTC)

or as "0."

why 0. means 0.(9) rather than 0.(8)? Bulwersator (talk) 14:59, 21 December 2011 (UTC)

That actually says "or as "0." followed by any number of 9s in the repeating decimal" - while not the best wording, it's pretty obvious that we don't talk about 0.8888... Huon (talk) 15:14, 21 December 2011 (UTC)

But, but... zero is not a number!!!! ;-P (Thanks Bulwersator for noticing and fixing the degenerate case). Diego (talk) 18:16, 21 December 2011 (UTC)

Meaning

What I don't like about this article is that it takes for granted that 0.999... exists and has a meaning, and all we have to do is show that it =1. It isn't until we get down into the analytical proof section that we start saying what 0.999... means. "0.999...." is just a text string, there is no a-priori right to assume that it represents a number William M. Connolley (talk) 08:40, 20 December 2011 (UTC)

The first sentence in the first few words says it is not a text string but a repeating decimal. If readers are not sure what that means (it is something most high school educated mathematicians should know) they can follow the link to repeating decimal or read on and hope to make sense of it from e.g. the proofs which initially only use basic arithmetic.--JohnBlackburne^words_deeds 09:27, 20 December 2011 (UTC)

I agree with William M. Connolley's sentiment. The text string "0.999..." evokes certain definite pre-mathematical intuitions, but its mathematical interpretation/implementation depends on the needs of the resercher, teacher, or student. The fact of the existence of multiple such implementations is documented in the body of the article, but is not mentioned in the introduction. We have the authority of no less a philosopher than C. S. Peirce concerning the fruitfulness of alternative interpretations. Certainly, the traditional interpretation was and remains the dominant one, and this needs to be made clear. But perhaps it need not be made clear at the expense of other interpretations. Tkuvho (talk) 10:35, 20 December 2011 (UTC)

I would also suggest that the following material be placed early on in the article, perhaps even in the introduction: the sequence .9, .99, .999, ... gets closer and closer to 1 in such a way as to get arbitrarily close to it. This is all that is meant by the claim that 0.999... is equal to 1, by definition. Tkuvho (talk) 10:39, 20 December 2011 (UTC)

But it's not about the sequence it's about the number. The number is perfectly well defined: if you can define 0.333... or 0.0909... then 0.999... is just as valid, and by common sense or by any one of the proofs, some of which just require basic arithmetic, equals 1. Certainly the series is one way of proving it but it's not fundamental to the definition.--JohnBlackburne^words_deeds 11:05, 20 December 2011 (UTC)

Thanks for your comment. I agree with every word you said except for the last sentence which I can't follow very well. It seems to me that the number is defined by the series. Translating this into the terminology of sequences by taking partial sums results precisely in the statement I made above. Tkuvho (talk) 11:10, 20 December 2011 (UTC)

It is defined as a real number represented by a recurring decimal. It can be shown to be equal to 1 using an infinite sequence but that is not the easiest or most elementary proof.--JohnBlackburne^words_deeds 11:30, 20 December 2011 (UTC)

I also agree with the sentiment. The lead section should mention the role of "0.999..." as a text string that represents a real number, and elaborate on the nature of the number as a repeating decimal. Linking to the article about recurring decimals is not enough, since the relation between the representation and the number is integral to the contents of this article (if it was only about the number, 0.999... could just be a redirect to 1). Diego (talk) 11:39, 20 December 2011 (UTC)

True. A recurrent decimal is defined by means of an infinite series, or equivalently an infinite sequence. I am not familiar with a definition of an unending decimal other than by means of series. Tkuvho (talk) 11:42, 20 December 2011 (UTC)

I like your description above "the sequence .9, .99, .999, ... gets closer and closer to 1 in such a way as to get arbitrarily close to it"; it gets the point through in a concise way. As JohnBlackburne said, the series demonstration may not be the easiest one, but is the one that more closely matches the intuition about the recurring decimal. Diego (talk) 11:46, 20 December 2011 (UTC)

I really don't find the current wording helpful: "The equality 0.999...=1 highlights the fact that the sequence 0.9, 0.99, 0.999, ... gets arbitrarily close to 1." And I don't see how this is going to clarify anything for anyone, especially not someone (the students who have problems with the subject of this article) whose intuition is that "arbitrarily close" is not the same as "equal". As for "the equality...highlights the fact that..." -- I am not sure what that is supposed to mean at all. Forgetting the awkward wording, those who have problems with the definition of limits of sequences may deny that 0.999... has anything to do with the sequence 0.9, 0.99, etc. I find that this new sentence just adds muddle, not clarity. --Macrakis (talk) 16:01, 21 December 2011 (UTC)

As I see it, there are two central ideas justifying the existence of this article:

1. The strings "0.999..." and "1" represent the same real number.
2. The string "0.999..." represents the recurring decimal defined as the infinite remainders of a division, which also are the decimal terms added in an infinite series.

The sentence adds clarity by expressing the second idea in the lead section, where it's otherwise not explained at all. In my view, excluding this idea from the introduction creates a biased article. As I said above, if only the first of the two ideas were relevant we wouldn't have this article at all, a simple redirect from "0.999..." to the article about "1" would be all the information that Wikipedia needs about the topic. Maybe the same idea could be expressed as 'the addition of "0.9", "0.09", "0.009"...', but I think the current wording about "the sequence 0.9, 0.99, 0.999,..." is easier to understand and conveys the same meaning. Diego (talk) 16:17, 21 December 2011 (UTC)

P.S. As this second fact is essential to the topic, exposing it as soon as possible is the best way to get readers in the track to understand that, in this case, "arbitrarily close" is in fact the same as "equal". Much better in any case than burying it in the article's body under a difficult mathematical proof.Diego (talk) 16:23, 21 December 2011 (UTC)

It seems to me that the history of arguments on Talk shows that readers go wrong in many ways about understanding 1=.999.... One fallacious argument, for instance, is that 1 - .999... = .00...01, and that .00...01 is (obviously! :-) ) different from 0. I do like the current wording better than the original ("the mathematical content...highlights the fact"), but I think it simply complicates the lead without helping convince the skeptic. --Macrakis (talk) 19:26, 21 December 2011 (UTC)

The lead is not there to "convince the skeptic", it's intended to give a summarized version of all the relevant facts. You don't think that the sequence 0.9, 0.99, 0.999,... is relevant to the topic? More than the infinitesimals and the Cantor set?Diego (talk) 08:14, 22 December 2011 (UTC)

Structure of the "0.999..." representation in the lead

Rather that arguing on the particular wording of the sentence that Tkuvho introduced, I'll broaden the scope of the discussion: I feel that the lead should elaborate on the structure of the "0.999..." representation, and that not doing it is hurting both the clarity and the usefulness of the article.

Above I have presented two alternative ways to address this need that don't involve limits (the reminders of fractions and the 1/10^n terms) and that are closer to the subject than the "limit" definition, which is the actual source of disagreement. So I'll ask you: what is for you an acceptable way to include the above idea in the lead (that the structure of repeating 9s is important)? Or if unacceptable, what are the reasons why this shortcoming should not be addressed? Note that I will not accept as valid arguments the too common "I don't think it's needed", "The lead has been this way for a long time" or "We are more than you". Diego (talk) 09:55, 22 December 2011 (UTC)

Firstly, could you state exactly what your alternative way is. I'm not sure what you mean with "remainder of fractions an the 1/10^n terms". If you are referring to expressing 0.999... as 1-1/10^n you should note that this is incorrect.

Secondly, 0.999... is an infinite series, which means that the limit is central to its definition. There can't therefore me a definition "closer to the subject" as you put it. NereusAJ (talk) 07:38, 23 December 2011 (UTC)

This proposal is a follow-up of an ongoing conversation above at #Meaning and #0.999... = 1, sequences, etc, in the lead., I'll clarify below what the alternatives are. The definition "closer to the subject" refers to what user JohnBlackburne said about 0.999... being introduced as repeating fractions and with limits not being the most elementary proof; for some reason there are some editors in this debate that strongly oppose including any mention to limits in the lead section.

My proposal is based on a feeling that the actual content of the lead conveys a meaning that "we know that 0.999... is the number 1, but where not telling you why because it will blow your mind". I find this stance patronizing and contrary to what a proper explanation should be; at least it should include the idea that "it has something to do with being infinitely close to 1", which is a solid insight to understand the topic. I'm trying to include this explaining idea in a way that editors in the conversation above will deem acceptable. Diego (talk) 09:53, 23 December 2011 (UTC)

Detailed proposal

The current article introduction does not have a definition of "0.999...". This proposal intends to address the Wikipedia:Jargon guideline that "the first section [...] be accessible to a broad readership" by "explaining technical terms [...] when they are first used" (this goes for "repeating decimal"), so that "those who are only looking for a summary or general definition may stop reading at the end of the lead".

To clarify it, these are the approaches I think ought to be explored in order to have an explanatory lead section, that in some way defines what the symbol means:

1. Mention that the meaning of the repeating decimal representation of 0.999... is composed by an infinite collection of decimals:

$b_{0}.b_{1}b_{2}b_{3}b_{4}\ldots =b_{0}+b_{1}\left({\tfrac {1}{10}}\right)+b_{2}\left({\tfrac {1}{10}}\right)^{2}+b_{3}\left({\tfrac {1}{10}}\right)^{3}+b_{4}\left({\tfrac {1}{10}}\right)^{4}+\cdots .$

$0.999\ldots =9\left({\tfrac {1}{10}}\right)+9\left({\tfrac {1}{10}}\right)^{2}+9\left({\tfrac {1}{10}}\right)^{3}+\cdots ={\frac {9\left({\tfrac {1}{10}}\right)}{1-{\tfrac {1}{10}}}}=1.\,$

2. Mention that digits in repeating decimals are usually generated as the remainders of a long division, and that the corresponding division for 0.9... is the fraction 9/9.

3. Explain that there's a formal definition of 0.999... that has to do with the limit of an infinite series.

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

At least one of these three approaches should be included in as to explain the mathematical meaning of the "0.999..." string (though any other equivalent approach is welcome). While the actual equations are not needed and should not be placed in the lead, the idea that they express should definitely be in there - otherwise, "0.999..." remains undefined. Diego (talk) 09:53, 23 December 2011 (UTC)

No, it is defined, simply as the number 0.999... . If a reader accepts that 0.111..., 0.333..., 0.142657..., etc. can be understood as the decimal representations of fractions then the similar 0.999... can be understood the same way. This may seem an informal definition but it's much preferable to what you're proposing because of the requirement that "the first section [...] be accessible to a broad readership". At least in the UK students learn decimal fractions long before they learn infinite series, so such a definition in terms of a repeating decimal is far more more accessible. It's also enough to prove that 0.999... equals 1, so is a good enough definition for that purpose.--JohnBlackburne^words_deeds 10:09, 23 December 2011 (UTC)

Your 1st and 3rd points are the same. Both just point out that 0.999... represents an infinite series. This concept is developed in the Analytic proofs section. I don't see including this representation in the introduction as improving the clarity of the article.

I don't think that relating repeating decimals to long division would make them easier to conceptualize. NereusAJ (talk) 10:44, 23 December 2011 (UTC)

@JohnBlackburne I believe we should follow the advise given in the guideline Write one level down "to consider the typical level where the topic is studied [...] and write the article for readers who are at the previous level". If this concept is for the level of UK students, it should be written for students in other countries or people without formal education; this means explaining how the words "repeating decimal" relate to the symbol "0.999..." instead of assuming it as given knowledge.

If the only explicit definition you include is that "0.999.. is a number, the number one" then it's not clear why this article is different than the article "1 (number)". Also I don't get why you can be so strongly opposed to introducing the limit in the lead (which as NereusAJ rightly stated, is "central to the definition" of the symbol's meaning), specially when you don't seem to have a problem about including Cantor sets and infinitesimals; I think that's a fairly obvious contradiction in your position.

@NereusAJ I also don't think that the long division is the best way to define the symbol, but it at least is an example of an infinite recursion, and is one idea that JohnBlackburne seems to think is less advanced than the limit. I don't think that the lead can be complete without expressing in some way that the symbol is connected to the infinite process of generating all the 9 digits in it. In summary, I'm trying to find a way to answer "why does 0.999... represent the number one?" in the lead. For some reason, it seems that I'm the only one thinking this question is what all readers are trying to learn when reading the article's introduction. Diego (talk) 11:33, 23 December 2011 (UTC)

P.S. @NereusAJ The first and third are have the same mathematical definition, but intuitively are not the same: the first one is "you keep placing 9s to the right of the decimal, which never ends", the third one is "you keep adding smaller and smaller terms in a series and find the limit". Maybe the distinction would be made clearer by treating the first case "programatically" as a string, not an addition. Diego (talk) 11:45, 23 December 2011 (UTC)

My reasoning is given above so I'm not going to repeat it, but I will expand on one point. Decimals are part of basic mathematical understanding. You need to know them to pay for anything in most countries, to measure a quantity precisely, to use a spreadsheet, etc.. Many non-mathematicians use and understand general decimals, including recurring ones, so they are taught at an early age. Infinite series are required only by mathematicians, or at least for those using much more advanced mathematics: hence they are taught much later. In this country that's only to students following an advanced course in mathematics, usually in preparation for university. I don't know where compulsory mathematics education ends in other countries but I would think it is true elsewhere that people understand decimals long before they understand formal infinite series.--JohnBlackburne^words_deeds 14:03, 23 December 2011 (UTC)

Agreed. It is pointless to address this article to people who have mastered infinite series and limits, let alone non-standard analysis. --Macrakis (talk) 15:21, 23 December 2011 (UTC)

So you agree that the bits about fractals, hyperreals and uncountability of the real numbers should be removed from the lead too? They are out of reach to people with an entry level to mathematics. Diego (talk) 08:52, 26 December 2011 (UTC)

No, that would be disrupting Wikipedia to illustrate a point, so I hope no-one here would consider doing it or even using it as an argument about some totally unrelated material.--JohnBlackburne^words_deeds 10:03, 26 December 2011 (UTC)

Nobody's here trying to make a WP:POINT by disrupting Wikipedia, I'm trying to get both of you to discuss the argument you made because it contains self-contradictions, but you're still using it to revert other Wikipedians edits. You know, to get you into the process of consensus building. I don't really want to discuss about Wikipedia policy now, I know it's on my side in this, so please stick to discussing the given arguments or let the edits pass. Diego (talk)

no my argument doesn't "contain self-contradictions". As for consensus there is a clear consensus against your changes, with the reasons given above addressing all your arguments. There has been for days. Suggesting other content should be deleted because consensus is against adding yours is disruptive, and if that is now your main argument I suggest you accept the consensus is against you and move on.--JohnBlackburne^words_deeds 18:04, 26 December 2011 (UTC)

Your definition of consensus, where only one side of the discussion says that there is consensus, is not how it is defined in Wikipedia. Repeated calls for other editors to stop editing, wikilawyering and accusations of misbehavior is not how it's done either. I can make a summary of all the arguments I have made that you have not addressed but I'm afraid you would keep ignoring them and saying that your unrelated comments solved them.

Why yes your insistence that limits are a too advanced topic while the lead paragraph (P.S. section, thanks Macrakis) contains advanced topics is particularly contradictory, but that's not the "horse carcass I'm beating"; it's the other three alternative ways to change the article that I proposed and that you didn't respond to, but that sadly I'm sure you will keep reverting without discussion if I, alone or with others, try to improve the article in that direction.

So if I don't go away only because you wish that I'd go away, will you start to be civil and begin a discussion centered on ideas, or should we begin to try the other ways in which real consensus can be achieved? Diego (talk)

The current lead paragraph does not mention fractals, hyperreals, and the uncountability of the reals. The last paragraph of the introduction says "The equality of 0.999... and 1 is closely related to the absence of nonzero infinitesimals in the real number system" and then elaborates on alternative number systems which do have non-zero infinitesimals. That seems fine, to delimit the scope of the discussion. Beyond that, the discussion of p-adic numbers seems out of place in this article, because it really doesn't contribute to the discussion of 1=0.999....

As for the discussion of the construction of the Cantor set and handling the non-uniqueness of the representation in Cantor's diagonal argument, yes, I do not believe those belong in the introduction. --Macrakis (talk) 18:32, 26 December 2011 (UTC)

The first approach Diego proposes to mention seems overly detailed to me; we would be recapitulating significant parts of the decimal representation and/or repeating decimal articles. The second approach is not taken up in the main article at all, and the lead should summarize the article. JohnBlackburne has already adressed the third. As an aside, I agree with Macrakis that the introduction need not mention the Cantor set or the diagonal argument. Huon (talk) 21:14, 26 December 2011 (UTC)

The summary has gotten filled with many excursuses which don't contribute to the main point. Here is my proposed summary text, which removes or modifies material which goes far beyond the topic of 0.999...=1, is overly wordy, or is pedantic: (I tried to do this with ins/del,but found it became difficult to read)

In mathematics, the repeating decimal 0.999... (which may also be written as 0.9, , 0.(9), or as "0.9" followed by any number of 9s in the repeating decimal) denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

In fact, every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is almost always preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other bases.

The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks. Nonetheless, some students question or reject it. Some can be persuaded by an appeal to authority from textbooks and teachers, or by arithmetic reasoning, to accept that the two are equal. However, some remain unconvinced, and seek further justification.

The equality of 0.999... and 1 is closely related to the absence of nonzero infinitesimals in the real number system. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals. In these systems, unlike in the reals, there can be numbers whose difference from 1 is less than any positive rational number but greater than 0. Although the real numbers are the most common object of study in the field of mathematical analysis, the hyperreals and other mathematical structures also have applications.

Thoughts? --Macrakis (talk) 01:11, 27 December 2011 (UTC)

Decidedly an improvement. It provides a clearer narrative over the current one, which fails to differentiate what's important to the topic and what's accessory. I would move the removed bits about applicability to to the corresponding "applications" section. It still doesn't address my concerns that 1) it doesn't include any mathematical definition of 0.999... (neither formal nor informal) and 2) it doesn't offer any explanation as to why 0.999... = 1, relying only on appeals to authority to assert it. A simple mention that 0.999... is a representation of a fraction would solve the first, and an enumeration of the kinds of proofs used to prove the fact would solve the second. Diego (talk) 08:41, 27 December 2011 (UTC)

A comment on the technical side of the matter: there are two ways in which unending decimals can be exploited: (A) one can define a real number as an unending decimal, modulo the identification of every decimal ending with 999..., with the appropriate finite decimal; (B) one can define the real numbers by Cauchy sequences or Dedekind cuts. In approach (A), one has 0.999...=1 by definition, so there is nothing to argue about. In approach (B), one needs to explain how to associate to an unending string of digits, a real number. This can be done by taking the limit of the sequence of finite decimals obtained by truncating the infinite one. Note the the sequence is monotone increasing, so the notion of limit in this case is more elementary than the general case, and can be replaced for instance by the least upper bound. This page does put us in a somewhat paradoxical situation of either talking about an undefined object, or else providing definitions that may be beyond the typical reader of this page. Tkuvho (talk) 11:52, 27 December 2011 (UTC)

Good call. That problem could be solved by having a new "Definition" section; with that section there wouldn't be any worries about limited space or target audience. Or a more general "Introduction" section could also have the excursuses (loving that word!) that are now in the lead about other positional number systems, hyperreals and p-adic, and applications of the decimal representation to the Cantor set and the uncountability of reals, plus a summarized introduction to the proofs. Diego (talk) 12:46, 27 December 2011 (UTC)

I think the current intro has enough about hyperreals and there is more detail in the body. As for p-adic numbers and p-adic analysis and the Cantor set, I doubt we need them in this article at all. I suppose the non-uniqueness issue in the diagonalization argument is somewhat relevant, though you can avoid it entirely by restricting to reals representable with only digits 0..8!

PS Re excursuses, since I was complaining about pedantry, I figured I shouldn't say excursūs (like status, the Latin plural is not in -i!). ☺ --Macrakis (talk) 00:21, 28 December 2011 (UTC)

0.999... = 1, sequences, etc, in the lead.

I am not particular fond of the newly added statement "The equality 0.999...=1 expresses the fact that the sequence 0.9, 0.99, 0.999, ... gets arbitrarily close to 1, which is therefore its limit." in the lead. For example one often encounters this equality with an algebraic proof long before one encounters limits.

Overall, the changes to the lead today seems much less user friendly for those who do not have much mathematical background. I am personally in favor of going back to previous version. Thenub314 (talk) 01:16, 22 December 2011 (UTC)

Absolutely agree. The equation primarily expresses the fact that two terms denote the same number. This is the fundamental meaning of the equation. As it happens, the LHS is defined as a limit of a sequence, but it seems distracting to start off in this way. Phiwum (talk) 01:28, 22 December 2011 (UTC)

Yes. As I said before the way most people are introduced to this, as a particular repeating fraction, has nothing to do with formal limits and is learned long before those studying maths encounter sequences and limits. Certainly it's one way of proving that 0.999... equals 1, but it's not the most elementary proof.--JohnBlackburne^words_deeds 02:17, 22 December 2011 (UTC)

Since there is some initial agreement, I am going to be bold and go back to the previous version. Thenub314 (talk) 02:40, 22 December 2011 (UTC)

There's no initial agreement. If you read the above discussion, at least two editors agreed and have been working on the current that you so boldly removed completely. Please read the above motivations to have it and comment about it before removing the topic that the sentence tries to illustrate. Diego (talk) 08:06, 22 December 2011 (UTC)

I see only one other editor maybe agreeing with you, while three here and one there (not counting myself twice) disagree that it should be added. It's not a matter of how it's presented, but whether it should be included at all, per the reasons given. I've therefore removed it again, and would ask it not be added again unless there's consensus for it.--JohnBlackburne^words_deeds 08:54, 22 December 2011 (UTC)

I'm OK with discussing it here before making changes to the article; you'll see that I only reintroduced it because Thenub314 had reverted some other unrelated edits in the hurry. All I ask is having a rational debate based on the arguments I have presented, not just on emotional arguments ("I don't feel it's needed") nor the number of editors supporting or opposing it -like "there is some initial agreement" to remove (there wasn't) and "there's consensus" (there are some new arguments on the table). Diego (talk) 09:55, 22 December 2011 (UTC)

I suggest you reply to other peoples arguments and address their concerns, not simply dismiss them as 'emotional'. As for consensus that's how Wikipedia works; again there are four editors disagreeing with the change and only two supporting it, so there is no consensus for adding it.--JohnBlackburne^words_deeds 10:18, 22 December 2011 (UTC)

This is what I'm honestly trying to do. The only argument so far not based on personal preferences or statu quo is the one about the limit, which I have already answered; my answer to it is still unadressed by anybody. It's hard to stay cool when the only answers I get are about secondary concerns and the central ideas are ignored, but I'll try my best. Diego (talk) 10:25, 22 December 2011 (UTC)

I agree with Thenub314 et al. that mentioning the sequence in the lead is not helpful, at least not in this way. Talking about sequences and limits implies not only a topological space, but a Hausdorff space. For example, in the hyperreals, the sequence (0.9, 0.99, 0.999, ...) still comes arbitrarily close to 1, which thus is still the sequence's limit - but that limit is no longer unique. Conversely, in the more exotic number systems we discuss, 0.999... is no longer necessarily given by the sequence (0.9, 0.99, 0.999, ...) - Richman's decimal numbers do not care about sequences, and while arguably the various hyperreals that could be called "0.999..." do have some connection to the sequence, not all those hyperreals which Katz calls "0.999..." can be given by the same sequence, and there's also that pesky ultrafilter that needs to be given for any sequence of reals to become a hyperreal in the first place. Claiming that "the number represented by the sequence (0.9, 0.99, 0.999, ...) falls infinitesimally short of 1" in exotic number systems is thus arguably wrong either because such a number need not exist in the first place, or because the sequence may represent entire sets of numbers which need not contain 0.999... at all. Huon (talk) 10:49, 22 December 2011 (UTC)

(Thanks). What do you think could be a precise but accesible way to mentioning the infinite 9s in the lead? (either as a sequence or in any other way). Note that the possible alternate interpretations of the sequence in the hyperreals and other number systems is not relevant to its first mention, the same way that the current "the symbols 0.999... and 1 represent the same number". When introducing the idea for the first time its safe to assume that we're discussing the real number interpretation; the article is already doing this. Diego (talk) 11:11, 22 December 2011 (UTC)

Huon: Your sequence does not come arbitrarily close to 1, since infinitesimal distances count in that setting, and nor does it converge at all in the topology of Hyperreal numbers (presuming the natural order topology, which is, incidentally, Hausdorff, as all order topologies are); it would have to converge to a smallest number larger than each of 0.9, 0.99, ..., which doesn't exist in the Hyperreals.

Diego: you should acknowledge that the consensus is clearly against including sequences in the lead. I'm not sure why you think it's insufficient to refer to the existence of proofs of varying degrees of rigor in the lead. --COVIZAPIBETEFOKY (talk) 18:33, 27 December 2011 (UTC)

I only should acknowledge consensus against sequences if the rules for building consensus had been followed. Since rules have not been followed, I don't need to acknowledge consensus on that point.

I very much prefer the consensus formed here that the introduction should be kept simple and which is now implemented in the article. This still lefts open the question as whether to define 0.999... in some simple way (maybe using limits, as Tkuvho suggested and NereusAJ explained); current consensus is not built on it. Diego (talk) 11:13, 28 December 2011 (UTC)

As I said elsewhere, referring to the existence of proofs doesn't explain the equality. I.e. it doesn't answer the question "why does 0.999...=1?". It's only justified from authority ("some mathematicians think so"), not by identifying the mathematical concepts involved. This forces readers to study the proofs provided in the article, which is against the Lead guidelines. Diego (talk) 11:19, 28 December 2011 (UTC)

You do need to follow the consensus here, as has now been pointed out to you by more than one editor, especially as you have been using purely disruptive arguments to make your point, along with a seriously flawed interpretation of guidelines (e.g. which 'Lead guidelines' indicate proofs should not be in the body of the article?).--JohnBlackburne^words_deeds 11:40, 28 December 2011 (UTC)

I'm following consensus since consensus exist, which is since Macrakis edited the lead tonight. I simply don't have to "acknowledge that the consensus is clearly against including sequences in the lead" like COVIZAPIBETEFOKY said. You should check for the beam that is in thine own eye; how about your need to assume good faith? I already stated that my question about removing difficult content was not there to prove a point; your accusation that I meant to be disruptive even after I denied it is plainly insulting. And your assertion that it was against consensus is disproved by the fact that this very same idea was used to build a new consensus, since other editors agreed to remove that content. So will you stop misconstruing my claims, step out of this harassment and learn to respect my opinion even if you don't agree with it? Diego (talk) 13:31, 28 December 2011 (UTC)

Then please stick to discussing the change in question. So again, which 'Lead guidelines' indicate proofs should not be in the body of the article?--JohnBlackburne^words_deeds 13:48, 28 December 2011 (UTC)

As long as I get you to only discuss changes and not editors' behavior, this discussion was worth it. I'll clarify: I never said that WP:LEAD indicates that proofs should not be in the body of the article. Forcing readers to read the body is what is against WP:MOSINTRO, where it says that the introduction must "stand on its own as a concise version of the article". If the lead section doesn't say anything about the nature of proofs other than they exist, it's the same as if the article's body didn't say anything about the proofs other than they exist. This is why I still think that the question "why does 0.999...=1?" is not answered in the introduction (and why it should). Diego (talk) 16:49, 28 December 2011 (UTC)

That makes no sense: the one to four paragraphs of the lead simply cannot contain everything that's in the body of the article. And on proofs in particular there is no point having them, or details of them, in the lead (I can think of no examples). They are by the nature best used in the body of the article to expand on or support assertions made in the lead or early in the article.--JohnBlackburne^words_deeds 07:49, 29 December 2011 (UTC)

Certainly proofs should be described in detail in the body of the article; that's what the body is for. But it makes sense to choose one representative proof and tell in the lead the basic idea that makes it work, without including the whole of it; that's what the lead section is for. The lead should not hint at startling facts without describing them. The current "Proofs of this equality have been formulated..." sentence does not describe any proof; one of them should be mentioned as a concrete example of this abstract assertion. Diego (talk) 11:08, 29 December 2011 (UTC)

0.999...=1 and hyper-real numbers

Hello. I'm new to Wikipedia, so please feel free to move this around if I've put it in the wrong place, or to correct me on matters concerning style as well as substance.

In nonstandard analysis, the transfer principle states that any definition or theorem involving the standard universe has a corresponding definition or theorem within the nonstandard universe.

Interpreting 0.999...= 1 within the standard universe is a little complex, because first we imagine the string "0.999..." designates a decimal expression, an unending sequence of decimal digits. That unending decimal expression in turn designates the real number 1 (using a definition involving a limit).

So we have three things going on.

1) On the paper, I write a finite string, "0.999...".

2) This designates a "decimal expression". A decimal expression for a real number between 0 and 1 can be modeled in turn as a sequence of decimal digits, that is, a function from the natural numbers to the set of decimal digits {1,2, . . ., 9}. The particular decimal expression represented by "0.999..." is the function that maps every natural number to the digit 9.

3) By definition, the decimal expression above designates the real number that is the limit of the sequence it defines 0.9, 0.99, 0.999, . . . That is, the real number 1.

If we now interpret this in the nonstandard universe, the transfer principle tells us the following.

1)Just as every real number can be designated by a decimal expression, every hyper-real number can be designated by a hyper-decimal expression.

2) Just as every real number between 0 and 1 can be designated by a decimal expression, which in turn can be thought of as a function from the natural numbers to the set of decimal digits, every hyper-real number between 0 and 1 can be designated by a hyper-decimal expression which in turn can be thought of as an internal function from the hyper-natural numbers to the set of decimal digits. (The transfer principle introduces the term "internal" when doing the translating).

3) According to the transfer principle, when working in the non-standard universe, the expression "0.999. . ." designates the hyper-decimal expression which we can think of as an unending hyper-sequence of 9's which in turn through a limit process designates the hyper-real number 1.

So 0.999... = 1 for the hyper-real numbers just as it does for the real numbers; it's just the way we interpret the terms is different (as guided by the transfer principle).

Lightstone uses a notation for hyper-decimal expressions 0.999 = 0.999...;...999... the digits before the semicolon are all indexed by standard natural numbers, those after by nonstandard ("large") natural numbers. In particular, this is the hyper-decimal expression of endless 9's, and designates the hyper-real number 1.

Using this notation, one can also speak of 0.999...;...999. We imagine this to be a terminating hyper-decimal expression that terminates after a "large" number of 9's, and it designates a hyper-real number infinitesimally less than 1. The notation is ambiguous, as we haven't specified how many 9's are in the expansion, but except for the requirement this be a nonstandard natural number, this generally doesn't matter.

Karin and Mikhail Katz define 0.999... to be the decimal expansion Lightstone designated as 0.999...;...999. Of course one can redefine terms however one wishes, and if one does this then 0.999... < 1 is true. But in this case I don't see why one would reject the definition given by the transfer principle. In particular, it seems natural to me to interpret 0.999... to be designating a non-terminating expression, in particular one in which every decimal digit 9 has a 9 to the right of it.

Finally, the expression 0.999...;000... does not give an internal function from the hyper-natural numbers to the set of decimal digits, and therefore isn't really a hyper-decimal expression; it does not designate any hyper-real number. — Preceding unsigned comment added by RodericT (talk • contribs) 16:38, 28 December 2011 (UTC)

Thanks for your stimulating comment. The current version of the article clearly states that "0.999... = 1 for the hyper-real numbers just as it does for the real numbers", as you write. It also mentions the fact that the expression "0.999...;000..." does not correspond to any hyperreal number. Tkuvho (talk) 16:44, 28 December 2011 (UTC)

The question discussed in the literature is whether student intuitions of ".999..." falling infinitesimally short of 1 are necessarily erroneous, or whether they can be formalized mathematically in a way that's useful in learning calculus. I recommend Robert Ely's suggestive field study referenced in our page. After all, there is more than one way of interpreting "an infinite string of 9s". Tkuvho (talk) 16:47, 28 December 2011 (UTC)

To comment specifically on your point "3) By definition, the decimal expression above designates the real number that is the limit of the sequence it defines 0.9, 0.99, 0.999, . . . That is, the real number 1" : as you are probably aware, the limit operaton on sequences decomposes into two stages: (1) evaluating the sequence an an infinite rank H; and (2) taking the standard part. Stopping before the second stage gives a mathematical implementation of student intuitions about infinite strings of nines. This does not conflict with the fact that the equality 0.999...=1 must remain true in the hyperreals by transfer. Tkuvho (talk) 16:52, 28 December 2011 (UTC)

New 'Definition' section

The 'Definition' section added in this changes has a few issues, re the style and writing, but the biggest problem is it's entirely misplaced, giving a very advanced definition before the much more elementary proofs. One point of those proofs is they don't depend on anything more than basic arithmetic. I would include in that even the series proofs as they are more elementary than a formal definition of the reals. The definition of the reals appears in the text later, in e.g. the section 0.999...#Dedekind cuts, so it seems unnecessary to add it again. I have therefore removed it, as unnecessary and entirely out of place.--JohnBlackburne^words_deeds 13:36, 28 December 2011 (UTC)

I agree with the removal. The concept that mathematics ever "develops" a number system is already beyond a large portion of the people who will be interested in reading some of the proofs.

Also, I disagree with the content of the paragraph itself, there would certainly be more to say about why one makes this definition (A), and the list of possible definitions is not complete. If these were the only two issues I would just fix it, but as comment above this is unnecessary so early in the article, and much of it is mentioned later anyways. Thenub314 (talk) 15:27, 28 December 2011 (UTC)

It is an error to think that the proofs given here depend on nothing but "basic arithmetic", because the manipulations are being applied to infinite strings. Basic arithmetic is not defined for infinite strings. Arithmetic operations on the reals are not performed on infinite strings directly in most approaches; rather, they are performed on truncated finite strings. The finite operations are later proved to "extend" to infinite strings, typically by a non-constructive method. The basic problem is infinite carry-over. Tkuvho (talk) 15:34, 28 December 2011 (UTC)

Well that is not precisely true. The arithmetic operations can be defined for infinite strings of repeating decimals, and students frequently discover how to do addition, subtraction, and multiplication by an integer for themselves. Which is enough to carry out all the algebraic/digit manipulation proofs here. One place I know of to cite the actual operations is: "Fractions without Quotients: Arithmetic of Repeating Decimals" by Richard Plagge The College Mathematics Journal, Vol. 10, No. 1, (1978), pp. 11-15. Infinite carry over is actually fairly easy to deal with within the rationals. Thenub314 (talk) 16:11, 28 December 2011 (UTC)

Those definitions of arithmetic operations on infinite strings would be a great addition to the article; they would provide a sound mathematical basis for the algebraic proofs. Do you have access to the source, and would you copy here the relevant parts so that they can be incorporated? Diego (talk) 16:57, 28 December 2011 (UTC)

As the paper won a writing award I believe it is publicly available here: [1] just follow the "Read the article" link. Mostly the operations can be defined as follows. To add

$a_{0}.a_{1}\cdots a_{k}{\overline {r_{1}r_{2}\cdots r_{l}}}+b_{0}.b_{1}\cdots b_{n}{\overline {p_{1}p_{2}\cdots p_{m}}}$

you first: align the non repeating parts to have the same length. Then by repeating the repeating parts as many times as necessary you extend both repeating parts to have length $\mathrm {lcm} (n,m)$. In this way you ensure k=n, and l=m.

Then you simply add the repeating parts. If the repeating parts are less then 10^n, then you result gives you the new repeating part, and you add the non-repeating parts and your done. If the sum is not less than$10^{n}$ then you ignore the leading $1\cdot 10^{n}$ in the decimal expansion of the sum and add 1. The resulting number gives the periodic portion of the resulting number. You then add the non-periodic parts and add $1\cdot 10^{-n}$ to the result. My description is quick and sloppy and libel to contain some error, but the moral is about right. The paper has a lot of nice examples, but since addition/subtraction are easy he spends most of his time developing an algorithm for multiplication.

I think it may be a bit distracting in this article to give the operations, and he never discusses 0.999... directly in the article which is why I never saw a reason t cite it here. Thenub314 (talk) 18:29, 28 December 2011 (UTC)

To respond to the comment above "Infinite carry over is actually fairly easy to deal with within the rationals": if one assumes rationality there is a trivial proof that 0.999...=1. Therefore the rational framework is not the appropriate one for this page. Any page dealing with mathematics should have a definition of the objects it is dealing with. Otherwise we are not explaining but rather urging acceptance of traditional authority. This may not be so bad but in this case it is a source of unnecessary student frustrations. Tkuvho (talk) 09:07, 30 December 2011 (UTC)

As I said above, I think there is nothing really to add, as it would be distracting to the article. So I suppose we agree that "the rational framework is not appropriate for this page". The ideas discussed in the definition section were already discussed elsewhere in the article, and it made the beginning of the article too technical. Thenub314 (talk) 21:32, 30 December 2011 (UTC)

Definitions

The following material was recently removed from the page: "Definitions."

"There are two ways in which unending decimals can be exploited in developing a number system: (A) one can define a real number as an unending decimal, modulo the identification of every decimal ending with 999..., with the appropriate finite decimal; (B) one can define the real numbers by Cauchy sequences or Dedekind cuts. In approach (A), one has 0.999...=1 by definition. In approach (B), one associates to an unending string of digits, a real number by taking the limit of the sequence of finite decimals obtained by truncating the infinite one. Since such a sequence is monotone increasing, the notion of limit in this case is more elementary than the general case, and can be replaced for instance by the least upper bound."

Tkuvho (talk) 13:38, 28 December 2011 (UTC)

It IS the case

The current version of the lede states that "In these alternate number systems it may or may not be the case that 0.999... and 1 represent the same number". As mentioned numerous times in this talk (and specifically in an earlier section), the equality 0.999...=1 remains true in most such "alternate" systems, with the exception of one article by Richman. Emphasizing this little-noticed pearl in the lede is giving it undue weight. Tkuvho (talk) 08:19, 29 December 2011 (UTC)

More specifically, any mention of a strict inequality ".999...<1" has to be accompanied by its description as an alternative to the traditional interpretation. While I am in favor of briefly mentioning such alternatives in the lede, I am not in favor of emphasizing rare (typically constructivist) frameworks where the main interpretation of .999... is different from 1. Tkuvho (talk) 09:46, 29 December 2011 (UTC)

The comment that all bets are off if your working outside the real numbers wasn't really in reference to the article by Richman. Mostly it was just a generic statement in other number systems it will depend on the particular number system in question. There are plenty of fields which the article here doesn't address, and addressing them would probably be inappropriate. I am all for emphasizing that the standard interpretation is that 0.999...=1. The way the lead was phrased before it seemed to imply that if infinitesimals were present that you would have 0.999... < 1. Finally, I am not sure what constructivism has to do with this conversation, but I will comment that 0.999...=1 within the constructive framework as well. Thenub314 (talk) 07:11, 30 December 2011 (UTC)

Richman explicitly states in his article that 0.999... falls short of 1 by "a kind of an infinitesimal" which he denotes 0_. The old version was preferable since it did not appear to assert that 0.999...<1 in the hyperreals. Tkuvho (talk) 09:04, 30 December 2011 (UTC)

I think we fundamentally agree here. I never intended to make it sound as if 0.999... < 1 in the hyperreals. I actually think the current version as of this morning does this more the the previous version so I have attempted to clarify the main interpretation is that 0.999... is 1. Thenub314 (talk) 17:15, 30 December 2011 (UTC)

The algebraic "proofs" are not proofs at this level of discourse

To multiply 10 by .999... and get 9.999... you need the following machinery:

• A rigorous definition of the real numbers;

• A rigorous definition of the limit of a sequence of real numbers;

• A definition of an infinite series as a sequence of partial sums.

With those things in hand you can then prove that term-by-term multiplication of an infinite series by a constant is valid.

But if you already understand those things then you would already understand that .999... = 1.

If someone is brand new to this topic and sees the 10 * .999... type proofs (same with 9 * .111... etc.) it is important to point out that these are not proofs, but rather heuristic arguments indicating a possible mathematical truth requiring rigorous proof.

The reason I think this is important is that in online discussion forums, one commonly sees the 10 * .999... type proof quoted as definitive; when in fact at this level of naive discourse, it is NOT a proof at all, as it depends on term-by-term multiplication of a series by a constant.

It's very misleading to students to give them the idea that there's an inherent infinite distributive law; when in fact there is not. Rather, the distributive law is given by the field axioms only for finite sums (or rather, it's given only for the sum of two numbers, and easily extended to finite sums). To apply it to an infinite sum requires a definition of the meaning of an infinite sum, followed by a proof of the infinite distributive law.

I see the 10 * .999... argument repeated uncritically all over the Internet; and it's wrong for Wikipedia to promote this misunderstanding.

I do understand that it's difficult to present mathematical subtleties like this on a Wiki page intended for beginners; but still, it's not mathematically correct to uncritically invoke the infinite distributive law without mentioning that at this level of discourse it's only an informal heuristic; and not yet a logically solid proof.

The purpose of a Wiki entry should be to educate students; not to flimflam them and leave them with the impression that the validity of term-by-term multiplication of an infinite series requires no proof.

76.102.69.21 (talk) 05:36, 31 December 2011 (UTC)

See the end of that section, 0.999...#Discussion, which does precisely what you suggest, points out that the previous subsections are not the final word on whether 0.999... equals 1. They are useful and valid though, useful as they are easily understood and valid as they help convince many of the equality. They might not stand up to formal analysis but then nor might the many geometric proofs of the Pythagorean theorem, many of which come down to rearranging shapes and asserting from looking at resulting pictures that e.g. a² + b² = c².--JohnBlackburne^words_deeds 06:08, 31 December 2011 (UTC)

I did read that section initially, and I was moved to comment here because that Discussion section makes the problem much worse. It begins: "Although these proofs demonstrate that 0.999... = 1 ..." This statement is flat-out wrong. At this level of discourse, those are not proofs, merely plausibility arguments. The Internet is now full of copies of this argument. Wiki didn't start it, but Wiki is now perpetuating it. To understand .999... one need to increase one's intuition about the real numbers and limits. Magically invoking the unjustified multiplication of an infinite series doesn't do that. At this level it's pure trickery. It makes people think they've understood something when in fact they haven't. Your remarks about bad proofs of the Pythagorean theorem are not relevant. You can't justify bad exposition and pedagogy here by saying, "Well there's some other bad stuff on the Internet too!" I thought we were trying to incrementally improve things. I can't fix everything that's wrong on the Internet. Tonight I'm just trying to fix this one little thing. It's true that these arguments help convince people, as you say, but that's the problem. They convince people by flim-flammery, not by elucidating the subtleties of the subject.

I would be in favor of adding something along these lines, though I haven't put much thought yet into the exact wording. "It should be noted that these are plausibility arguments, not proofs. In fact the notation .999... actually stands for the infinite series 9/10 + 9/100 + ... and in order to make this argument rigorous, we would need to carefully define what it means for such an infinite sum to have a finite value; and then we'd have to prove that we can multiply each term of the infinite sum by 10."

I realize that such a sentence would no doubt confuse people. So I'm not committed to that wording. But I don't think it's right to leave people with the impression that they've seen a proof, when in fact they haven't. A mathematical article has to increase readers' actual understanding; not just fake them out to make them feel better. 76.102.69.21 (talk) 08:02, 31 December 2011 (UTC) stevelimages@your-mailbox.com

Hi Steve, it would be helpful if you created a login name. There are several editors who have commented here in support of a proper section on definitions of 0.999... rather than relegating the issue to a vague "discussion" section. Such a section does not have necessarily have to precede the proofs, but it should be there in my opinion. So far I count 4 editors in favor of such a section. Tkuvho (talk) 20:43, 31 December 2011 (UTC)

Whether or not 0.999... actually stands for the infinite series depends a bit on the definitions your working with. For example sever sources define the real numbers to be formal decimal expansions, in which case the expression 0.999... is a real number by definition, and not directly refer to a infinite series. The final paragraph in the section JohnBlackburne points to sums it up well, once a definition of the real numbers is chosen the the validity of the proofs above can be investigated. Thenub314 (talk) 18:14, 31 December 2011 (UTC)

There is room for many kinds of "proof" in mathematics. There is rigorous proof, the kind you need when you're presenting a new result to an audience of skeptical professional mathematicians. There is formal proof (a la Principia Mathematica), which tries to use strictly formal manipulation -- but in practice no one ever does this. There are informal proofs, which demonstrate the plausibility of some statement. And there's everything in between. Different levels of proof can be useful for different audiences and in different settings. After all, the goal of a proof is to convince someone that something is indubitably true (or false) based on what they believe before the proof. See Imre Lakatos's brilliant Proofs and Refutations for a discussion of the role of proof in mathematics.... --Macrakis (talk) 01:07, 1 January 2012 (UTC)

I disagree. The goal of a proof is to conclusively demonstrate the veracity of a statement. As I understand the use of the term rigorous, a proof which isn't rigorous would have logical holes in it and would therefore not really prove anything. A proof that isn't rigorous wouldn't be very useful.NereusAJ (talk) 01:19, 1 January 2012 (UTC)

I think Steve is right. I've looked at the Argument archives. Many readers reject the equality because they have a problem with the proofs presented in the Algebra section. As they are presented as proofs, many people think that if they can find a hole in one of the two proofs, they have disproved the equality. I think it should be empathized that the analytical proofs and the proofs from the construction of the real numbers are the true justification for the equality and that the algebraic "proofs" are just plausible arguments or intuitive explanations or something along those lines. NereusAJ (talk) 01:39, 1 January 2012 (UTC)

I doubt those people on the Arguments page would stop rejecting the equality if we omitted the algebraic proofs; they just wouldn't understand the analytic ones, and not understanding a supposed proof is unlikely to be convincing. We already state that the proofs' level of mathematical rigor varies. (And non-rigorous proofs may still be useful for providing ideas on why something should be true even if there are some holes which would require work to fill but which one might take "on faith".) Huon (talk) 01:53, 1 January 2012 (UTC)

I don't think we should omit the algebraic proofs. Just relabel them to something that doesn't include the term "proof". When I stated earlier that non-rigorous proofs wouldn't be very useful, I admit I spoke rashly. I was trying to say that non-rigourous "proofs" are not useful in as far as they actually prove anything (which of course is the main objective). But you are right. They are useful for facilitating understanding. NereusAJ (talk) 02:11, 1 January 2012 (UTC)

You say "non-rigourous "proofs" are not useful in as far as they actually prove anything (which of course is the main objective)". If that were the objective, one simple rigorous proof would suffice and we could throw out most of this article.... --Macrakis (talk) 03:01, 1 January 2012 (UTC)

I didn't mean that the main objective of this Wikipedia article is to prove something. And I am not suggesting we delete anything. The algebraic "proofs" aren't proofs, but should still be included because they aid understanding of the subject matter. They help get an intuitive grasp of why the equality is true. All I am suggesting is that we are more clear in emphasizing that the analytical and real number construction proofs are the real justification for the equality. As the Algebra proofs are mentioned first they are given a prominence over the other proofs. This creates the misimpression that they are the primary justification for the equality. NereusAJ (talk) 03:15, 1 January 2012 (UTC)

The Discussion subsection tries to indicate this, but contributes to the misapprehension as Steve noted earlier. Saying that these proofs demonstrate the equality implies that they are sufficient in themselves. We could help clarify this my just renaming the section to something like Intuitive Explanations or Intuitive Arguments. The Discussion should also be altered to state that these intuitive explanations are in fact not sufficient to prove the equality, however the equality can be proved with the following rigorous proofs. Thoughts? NereusAJ (talk) 03:26, 1 January 2012 (UTC)

I disagree that these proofs are not rigorous, or that relabeling the section something other then proofs would help. This is one of the most frequently debated topics in the history of the internet. One must choose an appropriate place to rely on known results otherwise you end up with a book.

Yes they rely on other results that state that arithmetic works out the way we expect it too. But to explain why we would need to involve the definition of the real numbers. We are still not done, because which ever for what ever construction one chooses, its still necessary to show will need to be show real numbers so constructed will satisfy the axioms of real numbers. Of course all of that only makes sense if you show that any two objects which satisfy the axioms are order isomorphic. But I strongly suspect even with all of this there would still be arguments, as people then say that the geometric number line is the "true" definition, so you need to appeal to Hilbert's axiomatizations of geometry to show this works out as one hopes, but some wills still not be happy because Hilbert's axioms don't correspond to Euclid's and it goes on and on.

In the end these are in fact proofs, and I think they should be labeled as such. Thenub314 (talk) 06:01, 1 January 2012 (UTC)

It appears as if you are saying that because the real numbers are hard to define, the algebraic proofs are in fact real proofs. I don't understand your argument. Furthermore, there is no ambiguity or difficulty in defining the real numbers. The real numbers are a complete ordered field.

Whether or not you regard the algebraic proofs as real proofs, I still think a shift in emphasis is needed. NereusAJ (talk) 06:22, 1 January 2012 (UTC)

Hi Nereus, There is good reason for your not understanding the argument, as the argument is in fact absurd. This article needs a "definitions" section. I have great admiration for Imre Lakatos and his book, and there is room in mathematics for proofs at varying levels of rigor, but we can't prove anything about a concept without defining it. Tkuvho (talk) 07:58, 1 January 2012 (UTC)

But what would be in this definition section? The definition of the real numbers? This article clearly states we are discussing real numbers and it links to the Real number article in which the real numbers are formally defined. Including such a definition in this article would simply be duplicating content.

If you are talking about the "..." notation, this article uses the notation in the context of repeating decimals. Again, this article links to the Repeating decimal article, so defining the "..." notation in this article would also be duplicating content. NereusAJ (talk) 08:15, 1 January 2012 (UTC)

See reply in the next section. Tkuvho (talk) 08:26, 1 January 2012 (UTC)

Proofs and sketch proofs

Many of the proofs in this article are really "sketch proofs" -- that is, they give the basic idea of a proof which could in principle be made rigorous. This is in fact the nature of most proofs in education, and for that matter in professional mathematics. When a proof says e.g. "2a-1=0 therefore a=1/2", no sane mathematician would actually write out the details needed to derive this from the axioms of arithmetic. If it were the first time a high school algebra teacher were showing it to the class, he or she would probably break it into smaller steps, e.g. 2a-1=0 => ("add one to both sides") 2a=1 => ("divide both sides by 2") a=1/2, but would still not justify each step in detail (e.g. "since we know 2 <> 0, we can divide both sides by 2"), but would still not explicitly reference the axioms of arithmetic (or prove the appropriate lemma from them) to justify steps like "add one to both sides". The only case where a mathematician or mathematics teacher might go into greater detail is when working in an unfamiliar algebraic structure.

The proof 10*.999... = .999... - 9 is similarly a sketch proof. Of course making it rigorous it depends on interpreting .999... as sum(9*10^-i,i,1,inf) and invoking various relevant theorems about convergent infinite series (e.g. that sum(f(i),i,1,inf) = f(1) + sum(f(i),i,2,inf), that a*sum(f(i),i,1,inf) = sum(a*f(i),i,1,inf), etc. -- you don't even need uniform convergence). It is not a pseudo-proof which turns out to be invalid when you try to make it rigorous but rather a valid sketch proof, which I submit is just what is needed in this article. --Macrakis (talk) 22:38, 2 January 2012 (UTC)

I agree that the Algebra proofs should be in the article. Except, the long division proof. It cannot possibly be made rigorous. As for the proof assuming rationality, it is not a question of rigor. I am saying that proving 0.999... = 1 assuming 0.999.... is rational is absolutely useless as 0.999... can only be shown to be rational by proving 0.999... = 1. Therefore proving 0.999... = 1 assuming 0.999... is rational doesn't prove anything.

This article has excellent rigorous proofs of the equality. It also has excellent sketches of proofs for the equality (the digit manipulation and fractions proof). So I do not see why we have to add dubious non-proofs like the long division and rationality ones.NereusAJ (talk) 23:00, 2 January 2012 (UTC)

Proof assuming rationality

I was curious for the reference for this proof, as it is not one I do not remember seeing before. Thenub314 (talk) 17:19, 30 December 2011 (UTC)

What rational number could 0.999... possibly be? QED, Bitches! --COVIZAPIBETEFOKY (talk) 17:52, 30 December 2011 (UTC)

Yes, it doesn't prove they are the same; it just shows 0.999.... is irrational if different from 1. Even then it seems very incomplete. If unsourced it should be removed.--JohnBlackburne^words_deeds 18:39, 30 December 2011 (UTC)

The following material was recently removed from the page: "Proof assuming rationality. Assuming that 0.999... is a rational number p/q, one can immediately deduce the equality 0.999...=1 as follows. Let n be the number of decimal digits of q. If p<q, then ${\frac {p}{q}}<0.999\ldots 9000\ldots$ where the last 9 appears at rank n. Therefore p=q and 0.999...=1, as required." The proof is so trivial as not to require sourcing. It indicates clearly that an assumption of rationality trivializes the issue. This is relevant to the discussion of repeated decimals. Tkuvho (talk) 11:08, 3 January 2012 (UTC)

See WP:CALC for what mathematics is allowed without sourcing: "routine mathematical calculations, such as adding numbers, converting units, or calculating a person's age". This really needs sourcing for context, such as to justify the assumption of rationality. Otherwise it doesn't really prove the equality.--JohnBlackburne^words_deeds 11:31, 3 January 2012 (UTC)

Unless you can find a source that uses this proof, it fails notability. Effectively it becomes your personal proof, FRINGE, etc, and doesn't belong. Correctness becomes irrelevant William M. Connolley (talk) 12:59, 3 January 2012 (UTC)

A section with definitions

Here is a rough draft of a "definitions" section:

"There are two ways in which unending decimals can be exploited in developing a number system: (A) one can define a real number as an unending decimal, modulo the identification of every decimal ending with 999..., with the appropriate finite decimal; (B) one can define the real numbers by Cauchy sequences or Dedekind cuts. In approach (A), one has 0.999...=1 by definition. In approach (B), one associates to an unending string of digits, a real number by taking the limit of the sequence of finite decimals obtained by truncating the infinite one. Then the mathematical content of the assertion that "0.999...=1" is the fact that the sequence 0.9, 0.99, 0.999, ... gets closer and closer to 1, no more and no less. Since such a sequence is monotone increasing, the notion of limit in this case is more elementary than the general case, and can be replaced for instance by the least upper bound."

This is very informal and needs to be polished up, but it attains several objectives: (1) the reader is clearly and honestly told that in one of the traditional approaches, one has 0.999...=1 by definition, and specifies which approach; (2) gets to the gist of the mathematics without veiling it. Tkuvho (talk) 08:26, 1 January 2012 (UTC)

We already discussed a version of this, but to re-iterate as it does seem a little different. (A) is frankly useless: if you include 0.999... equals 1 in the definition then yes there's nothing to prove. Would that all proofs were this easy! (B) is already covered in detail in the article, appropriately ordered after more elementary explanations.--JohnBlackburne^words_deeds 08:37, 1 January 2012 (UTC)

I disagree with your claim that it is useless to frankly tell the students that in one of the traditional approaches, 0.999...=1 by definition. That might reduce some of the frustration. The definition (B) is camouflaged in one of the proofs. This is indeed an odd way to proceed on a math page. Tkuvho (talk) 08:44, 1 January 2012 (UTC)

I agree with JohnBlackburne. In (A) all you are saying is: if we define 0.999... to be equal to 1 then 0.999 would be equal to 1, which isn't very useful. Furthermore, you omit the most natural conception of unending decimals as infinite sums (most natural because all decimals represent sums). NereusAJ (talk) 08:52, 1 January 2012 (UTC)

The idea that this page should be dedicated uniquely to proving that 0.999... equals 1 seems a bit narrow. Why shouldn't we make it clear that sometimes they are equal by definition? Also, keep in mind that the expression "infinite sum" is a figure of speech. The mathematics behind the metaphor is, as mentioned above, that the sequence 0.9, 0.99, 0.999, ... gets closer and closer to 1, no more and no less. Tkuvho (talk) 12:30, 1 January 2012 (UTC)

I agree with NereusAJ and JohnBlackburne, adding a section like this will not improve the article. Thenub314 (talk) 15:48, 1 January 2012 (UTC)

I agree with Tkuvho, a definition section will improve the article.

If the definition is provided in the Analytic proofs section, it can be moved from there to the definition section, and the proof can refer to that definition to avoid having a duplicated definition. This way, the mathematical object that gives get a clear definition in a section of its own, instead of being hidden as a step in a proof. A section like this can also explain the implications of choosing between the different definitions, something that can't be done inside an ongoing proof. Diego (talk) 18:39, 1 January 2012 (UTC)

To address a possible source of concern: the idea of a definition section would not be to recycle the definition of the real numbers. The point I was trying to make that even assuming that the real numbers have been defined, one still needs to define unending decimals. The definition in question is short and is stated in terms of a limit of truncated decimals. Here the notion of limit is a rather intuitive one rather than the general epsilon, delta technique, since we are dealing with a monotone sequence. Tkuvho (talk) 18:46, 1 January 2012 (UTC)

An infinite series isn't a "metaphor" or "figure of speech" as you call it. It is a mathematical concepts. Now, you are right in saying that an infinite sum is defined as the limit of the sequence of partial sums. However, your approach in (B) makes it seem that 0.999.... is just arbitrarily defined to be the limit of the sequence (0.9, 0.99, 0.999, ...). This is in fact not arbitrary. 0.999.... is equal to the limit of the sequence (0.9, 0.99, 0.999, ...) because 0.999... is an infinite series and (0.9, 0.99, 0.999, ...) is a sequence of its partial sums. What I am trying to say is that the concept of infinite sums is crucial in conceptualizing unending decimals. NereusAJ (talk) 20:39, 1 January 2012 (UTC)

Furthermore, you say "Why shouldn't we make it clear that sometimes they are equal by definition?". Everything is always true by definition. Explicitly pointing out that if you define something to be true then it is true is just silly. NereusAJ (talk) 20:43, 1 January 2012 (UTC)

You say "What I am trying to say is that the concept of infinite sums is crucial in conceptualizing unending decimals". Aren't you and Tkuvho basically saying the same thing with different words? Both that (0.9, 0.99, 0.999, ...) is the sequence of partial sums and that it gets infinitely close to 1 are important facts for defining and understanding this particular unending decimal. Diego (talk) 21:49, 1 January 2012 (UTC)

The difference is that Tkuvho avoids mentioning infinite sums, which I believe is the key concept.68.146.78.154 (talk) 22:17, 1 January 2012 (UTC)

A further note on the proposed definition section. In (A) you say real numbers are defined in terms of unending decimals. Real numbers are never defined in terms of decimals. The decimal system is just a system to represent them. NereusAJ (talk) 23:37, 1 January 2012 (UTC)

You are in error. A number of references define a real number as an equivalence class of unending decimals. Here an equivalence class has either one member or two. While "infinite sum" is a metaphor, "infinite series" is not. Tkuvho (talk) 13:56, 2 January 2012 (UTC)

Real numbers are sometimes defined as equivalence classes of Cauchy sequences of rational numbers. That is not to say they are defined in terms of decimals. Furthermore, you say "While "infinite sum" is a metaphor, "infinite series" is not". Kindly explain the difference between an infinite sum and infinite series.NereusAJ (talk) 20:02, 2 January 2012 (UTC)

Our article constructions of the real numbers clearly indicates that construction from unending decimals is a different one from the construction out of Cauchy sequences. An infinite series is not really a "sum". In this sense, "infinite sum" is a figure of speech for an infinite series. In the hyperreals, it is possible to form literally an infinite sum of precisely a hypernatural's worth of summands, but I assume you did not have the hyperreals in mind. Tkuvho (talk) 11:12, 3 January 2012 (UTC)

No one ever claimed that real numbers are only defined in terms of decimals. Just that they are sometimes defined that way. --COVIZAPIBETEFOKY (talk) 14:56, 3 January 2012 (UTC)

To Tkuvho: See Series (mathematics), where in the first line it says "A series is the sum of the terms of a sequence". But this is besides the point, what I meant was obvious.

To Tkuvho and COVIZAPIBETEFOKY: I consulted the Construction of the real numbers article and it does talk about defining reals in terms of unending decimals (Stevin's construction). So you are both right. My mistake. NereusAJ^{T | C} 23:49, 3 January 2012 (UTC)

Rationality proof and proof by long division

As I have already indicated, I support relabeling the proofs offered in the Algebra section to something other than proofs (See discussion above). But I would now like to focus attention on two purported proofs which are particularly weak. Those are the long division "proof" and the "proof" assuming rationality. Long division is an algorithm, not a method of proof. In fact in this case the algorithm never terminates and therefore never has an final output. In the article this proof doesn't even have a final statement just an ellipsis indicating the algorithm goes on forever. Furthermore, the proof assuming rationality is also very weak. 0.999... is known to be rational only because it can be proven to be equal to 1 which is rational. Proving that 0.999... = 1 assuming that 0.999.... is rational is similar to proving 0.999... = 1 assuming 0.999... = 1. I strongly suggest the removal of these proofs.NereusAJ (talk) 21:26, 2 January 2012 (UTC)

I've been trying to tell them that for years. Algr (talk) 22:22, 2 January 2012 (UTC)

That's impossible. The two proofs I am referring to were only added last month.NereusAJ (talk) 22:51, 2 January 2012 (UTC)

Algr, I think you misunderstood me. I am not denying the equality. I think this article contains excellent rigorous proofs of the equality. I am just complaining about two specific proofs that were added very recently. The long division proof and the proof assuming rationality. NereusAJ (talk) 05:53, 3 January 2012 (UTC)

Seconded. The rationality proof was discussed already and it is seriously flawed, proving only that 0.999... cannot be rational, not that it's equal to 1. I am therefore removing it, based on the views expressed as and no-one has suggested a way to fix or improve it. The long division proof makes a bit more sense, is properly sourced, but it's using a very non-standard way of doing long division which really needs explaining, as it is in the source. The other proofs don't depend on such mental stretches, and so are I think much clearer, and the first of which uses similar arithmetic renders it unnecessary.--JohnBlackburne^words_deeds 08:23, 3 January 2012 (UTC)

See my comment above in Talk:0.999...#Proof assuming rationality. Tkuvho (talk) 11:14, 3 January 2012 (UTC)

I just noticed that the non-standard division proof was still in the article, even though we seem to agree it has too many issues, so per the objections above I've removed it.--JohnBlackburne^words_deeds 20:16, 22 January 2012 (UTC)

Cannot be written as stated.

From the article: "In mathematics, the repeating decimal 0.999... (which may also be written as ... "0.9" followed by any number of 9s." No it can't. That would represent a completely different number that is a terminating decimal as opposed to recurring decimal and would definately not equal 1. Zibart (talk) 19:57, 22 January 2012 (UTC)

It says up to the end of the bracket "or as "0.9" followed by any number of 9s in the repeating decimal", which means it could be

0.9... or 0.99... or 0.999... or 0.9999..., and so on.

It's simply stating that there's nothing special about 0.999... and its three nines. This is covered more fully at repeating decimal, which ends the sentence but is not linked as it's linked immediately before.

Does that make sense? I don't know if we want to add a detailed explanation, i.e. the above or even more from repeating decimal, as it would make the lead too long, and any reader who's not familiar with repeating decimals should start there before tackling this article. Or any suggestions on what could be added to clarify this?--JohnBlackburne^words_deeds 20:22, 22 January 2012 (UTC)

That was the intent of the sentence, but if Zibart has been confused about its meaning, maybe it could be replaced by the more literal: ...may also be written as "0.9" followed by any number of 9s and a final ellipsis. Diego (talk) 09:46, 23 January 2012 (UTC)

I was not confused I was simply worried that other people may write the number incorrectly. Zibart (talk) 14:33, 23 January 2012 (UTC)

The following edit was recently deleted: "(which may also be written as 0.9, $\scriptstyle \mathbf {0} .\mathbf {\dot {9}}$, 0.(9), indicating an infinite tail of 9s)". This captures the essence of the matter. Tkuvho (talk) 16:51, 23 January 2012 (UTC)

Yes, it removed Diego Moya's edit which I also agreed with (his edit was a good answer to my last question posed above). 'infinite tail' is much less clear: "tail" is not a mathematical term, and it is not necessarily infinite, just recurring i.e. repeating without end.--JohnBlackburne^words_deeds 17:11, 23 January 2012 (UTC)

I agree with JohnBlackburne and Diego Moya that we should mention "0.9999..." and similar notations as equally valid and that Diego Moya's edit does a good job. Huon (talk) 17:16, 23 January 2012 (UTC)

But the 0.999... notation already appeared before the parenthesis. Why do we need to do it again? On the other hand, the fact that there are infinitely many 9s should be mentioned. A recurring 9 produces infinitely many of them. Tkuvho (talk) 17:23, 23 January 2012 (UTC)

The sentence in parentheses is about the symbolic notation, which is neccesarily finite. The "any number of 9s" is to distinguish between "0.9...", "0.999..." or "0.999999..." which are all valid. Maybe its a good idea to add that all notations are meant to represent an infinite quantity of nines, but that's a separate idea (which applies to all notations, not only "0.999..."). If we agree to include it, it should be a separate sentence. What do you think of this version below? The new sentence is a way to write one level down by explaining how the technical term "repeating decimal" applies to this topic. Diego (talk) 18:40, 23 January 2012 (UTC)

In mathematics, the repeating decimal 0.999... denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigor, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience. The number may also be written as 0.9, $\scriptstyle \mathbf {0} .\mathbf {\dot {9}}$, 0.(9), or as "0.9" followed by any number of 9s and a final ellipsis; all these notations are ways to indicate that the dot is followed by an infinite number of 9s.

It's not necessarily infinite though. I mean recurring decimals are learned quite early, long before the formal notion of infinity is studied. So readers may have a clear concept of recurring decimals, that they repeat indefinitely/as many times as necessary. But that's not the same as an infinite number of 9's, and many readers won't understand it.--JohnBlackburne^words_deeds 19:02, 23 January 2012 (UTC)

I don't follow this; how would someone think that "indefinitely recurring decimal 9 digit" is different from an "infinite number of 9 digits"? And what does "as many times as necessary" mean when referring to an unending sequence of decimal digits? — Loadmaster (talk) 23:30, 23 January 2012 (UTC)

See e.g. the start of repeating decimal:

In arithmetic, repeating decimal is a way of representing a rational number. Thus, a decimal representation of a number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely.

It's possible to define repeating decimals, and so define 0.999..., without using 'infinity', which makes sense as repeating decimals are generally encountered by students long before infinity and infinite series. Again, in repeating decimal infinite series appears only towards the end, so the rest of the article does not depend on it.--JohnBlackburne^words_deeds 00:09, 24 January 2012 (UTC)

Excuse me, but nobody's talking of infinite series now. Recurring decimals is usually the first time students find infinite, in the form of an infinite string of characters (not a series). Repeating decimal just mentions infinite in the first paragraph and describes 0.99... as "an infinitely repeating sequence of nines". I agree with Tkuvho and LoadMaster that the ammount of repeating 9s in the number is best described as infinite even if any particular representation is not. I'm afraid the distinction you try to make between an infinite created with series and some kind of "not-infinite" created by an infinite repetition is too subtle and doesn't hold merit here. The problem you seem to have is the difficulty in understanding the idea of "an infinite number of 9s". That's one more reason to explain that infinite here means not having an ending condition that would produce a final '9' with nothing to the right; not to fully scrap the foundational idea of infinite. Diego (talk) 07:43, 24 January 2012 (UTC)

I think it is a good idea to mention the ellipsis, but the current sentence is a bit unwieldy. Why should the ellipsis be mentioned after all the alternative forms of denoting periodic decimals? It should be mentioned right after the notation 0.999... is first used. It seems to me that there is a consensus of scholars that there is a 9 for each of the natural numbers, which form an infinite set, which would make it an infinite tail of 9s. Students are reportedly confused in thinking that there are only finitely many 9s; shouldn't it be mentioned that there are infinitely many? Tkuvho (talk) 13:14, 24 January 2012 (UTC)

Any alternate wording is welcome anytime, of course. Diego (talk) 13:47, 24 January 2012 (UTC)

Except when repeating decimals are taught it is without referring to them as infinite. here e.g. is Wolfram (the second result after repeating decimal in this search) which describes repeating decimals fully and quite technically without using infinity. The concept of infinity is introduced somewhat later, after compulsory maths education ends, at least in the UK. So many if not most readers will understand 'repeating' but not 'infinite'. Per WP:MOSINTRO the lead should be as accessible as possible, so we should not use words in the definition which a large portion of the readership have not encountered, unless it is unavoidable.--JohnBlackburne^words_deeds 13:21, 24 January 2012 (UTC)

The wolfram page you linked to confronts terminating and repeating decimals, so repeating decimals are thus non-terminating (a term that is also used to describe non-repeating decimals for irrationals, that are explained at the same instruction level). I coincide with Loadmaster that I don't understand how you can make a distinction between "not terminating" and "infinite" (which is exactly the same, but in latin). Anyway, it didn't take much work to find several text books that describe "infinite repeating decimals" (see [2], [3], [4]) . This one in particular is targeted to Elementary School Mathematics and describes repeating decimals as infinite. Diego (talk) 13:59, 24 January 2012 (UTC)

I agree. User:JohnBlackburne's assumption that the article should be addressed to highschoolers is not necessarily endorsed by WP:MOSINTRO . Tkuvho (talk) 14:47, 24 January 2012 (UTC)

it should be as accessible as possible, which in this case means to highschoolers as the topic is not too advanced for them, and the first two proofs in particular are accessible to anyone that's encountered repeating decimals as they otherwise use only basic arithmetic. See e.g. MOS:MATH#Article introduction: "The lead should as far as possible be accessible to a general reader, so specialized terminology and symbols should be avoided as much as possible".--JohnBlackburne^words_deeds 16:41, 24 January 2012 (UTC)

The term 'infinite' is not specialized terminology, as in this case is used with the common meaning of "unbounded or unlimited" and not with any particular mathematical definition. If 'infinite' is not acceptable neither is 'repeating decimal' that does not have a meaning in common language (since "decimal" is always a mathematical term). Diego (talk) 17:48, 24 January 2012 (UTC)

See again my comments above and the Wolfram example: infinity may be common to me or you but there are many readers who will not know what it means because they have not learned it yet or finished their mathematics education before encountering it. They may know is as a common word but may not know what it means (it's used often just as a superlative, or for dramatic effect), and are unlikely to appreciate its mathematical meaning.--JohnBlackburne^words_deeds 01:21, 25 January 2012 (UTC)

You don't have evidence that using the "infinite" word will cause problems to readers, several editors agree that it's an accurate description, and we have reliable sources using that word in the context of repeating decimals as long division. The Wolfram page that you referenced uses "terminating"; we could explain infinite as "non-terminating" to differenciate it from the other common meanings you mentionted, which is in accordance to WP:Explain jargon (do not depend on wikilinking), Wikipedia:TECHNICAL#Don't oversimplify (not "tell lies to children") and WP:EXPLAINLEAD (provide an accessible overview). The only remaining question is the exact words by which this is to be described. I suggest asserting that having "infinite digits [created by a non-terminating long division] means that any '9' will have another '9' to the right, so there's no final digit". This would address one of the common confusions of students (the "final 9 at the infinite"). Diego (talk) 09:54, 25 January 2012 (UTC)

Teaching math

I've found this source discussing 0.999...=1 with respect to a "teacher's mathematics" category. It poses that students will not be convinced by one single proof or explanation but that showing a collection of varied explanations is what provides grounding for the concept. It has 30 citations, should we add it to the 0.999...#Skepticism_in_education section? Diego (talk) 10:29, 25 January 2012 (UTC)

"This last explanation does not help students who have not seen some of the other explanations, but it does show the consistency of mathematics and helps to give closure on the idea. This is teachers’ mathematics."

I was unable to access the reference, nor to reproduce the google scholar stats. Tkuvho (talk) 10:53, 25 January 2012 (UTC)

Sorry, it looks like I posted a link to a cache. Try the link now. The reference is for "Teachers' mathematics: A collection of content deserving to be a field" by Zalman Usiskin.Diego (talk) 11:26, 25 January 2012 (UTC)

Since it has 30 scholar cites, I see no reason why it shouldn't be mentioned. Tkuvho (talk) 11:34, 25 January 2012 (UTC)

Clear example why 0.999... != 1

moved to Talk:0.999.../Arguments#Clear_example_why_0.999..._.21.3D_1 Bulwersator (talk) 10:21, 2 February 2012 (UTC)

Why .999... does NOT equal 1

moved to Talk:0.999.../Arguments#Why .999... does NOT equal 1 Double sharp (talk) 03:20, 4 May 2012 (UTC)

By Definition

This article may benefit from the addition of a simple definition explanation. That is, there is a definition for this number, and that definition is a limit. From here, we can see that this limit converges. And thus, by a=b and b=c, we have a=c.

$0.999\dots =\lim _{n\to \infty }(1-{\frac {1}{10^{n}}})$ (by definition).

$\lim _{n\to \infty }(1-{\frac {1}{10^{n}}})=1$ (convergent).

Hence, $0.999\dots =1$ (a=c, because a=b and b=c from above).

This is irrefutable, true, simple, and easy to understand. Tparameter (talk) 03:08, 31 May 2012 (UTC)

Exactly. There is an even better definition. One of the ways of defining a real number, following Simon Stevin, is a string of digits modulo equivalence which declares .999... to be equal to 1 by definition. That's even more irrefutable, true, simple, and easy to understand. Tkuvho (talk) 07:53, 31 May 2012 (UTC)

Er, yes, but the definition you (Tkuvho) mention seems, oh, unpersuasive to those who doubt whether 0.999... = 1, no? Phiwum (talk) 10:57, 31 May 2012 (UTC)

The purpose of including a definition is not to persuade those who doubt, but to explain the topic to those wanting to learn. An encyclopedic article should at the very least define the subject of its topic, moreso for a mathematical topic. An mathematical definition will allow the reader to understand the meaning of the ellipsis in concrete unambiguous terms. Diego (talk) 12:53, 31 May 2012 (UTC)

@User:Phiwum: Field studies reveal that indeed the students remain overwhelmingly unpersuaded, whether by my definition or any other. We might as well tell them that .999... is 1 by definition, which is the exact truth as far as the most practical system of building the real numbers is concerned, namely in terms of unending decimals. Tkuvho (talk) 13:00, 31 May 2012 (UTC)

A definition is not a tool to persuade. It simply gives a factual meaning. 0.999... is BY DEFINITION the limit, the limit converges to 1 because of the nature of limits and mathematical theorems and whatnot. And consequently, 0.999...= 1. None of this is debatable. At this point, a skeptic can really only say they don't like it. Nevertheless, it is. I, for example, really want the defnition of "grape" to be a banana. However, it's not. It is what it is. I don't like it, but I must accept it. :) Tparameter (talk) 22:47, 31 May 2012 (UTC)

If fact, IMO it is ridiculous to "prove" such a thing. What do we mean when we say "0.999..." in mathematics? The mathematical definition follows, and thus the limit. If (only by the grace of God) this article was written in the most simplified way that any math prof would explain it to a student, there would be very little debate. The argument would evolve into why non-math people hate this definition. Simple. Tparameter (talk) 23:19, 31 May 2012 (UTC)

That's already in the article, at 0.999...#Infinite series and sequences, except properly explained. You can't just assert this:

$0.999\dots =\lim _{n\to \infty }(1-{\frac {1}{10^{n}}})$ (by definition)

it's undoubtably true but you need to say why, which is what the article does at the end of that section. The simpler proofs and explanations precede it.--JohnBlackburne^words_deeds 23:27, 31 May 2012 (UTC)

I agree with User:JohnBlackburne. The material is already there. Tkuvho (talk) 08:14, 1 June 2012 (UTC)

I think it should be highlighted at the very beginning that this is a notational definition, not something to prove. How do you write out the largest real less than 1? Is it possible in this notation? Or is it not even a real? This article is interesting on the matter, on infinitesimals and cognition: arxiv.org/pdf/1003.1501. /Vay --46.194.45.247 (talk) 23:14, 18 July 2012 (UTC)

Technically, 0.999... is a special case of the wider notational definition of decimals, and I'd say you indeed have to prove that this specific decimal equals 1 - the proofs closest to the standard definition of decimals are probably those under 0.999...#Infinite series and sequences.

Regarding the "largest real less than 1": There is no largest real less than 1 (because between any two different reals there's another one), and number systems which do have a largest number less than one tend to be either the integers (zero is the largest number less than one in that system) or something non-standard with some pretty exotic and little-used properties; I believe you have to lose either subtraction or decimal representations. Huon (talk) 00:21, 19 July 2012 (UTC)

Only that for instance the open interval [0,1[ is not particularly exotic in math, so neither should its last included value. From a layman's view. /Vay — Preceding unsigned comment added by 46.194.38.174 (talk) 22:32, 19 July 2012 (UTC)

Considered as a subset of the rationals, the reals or the hyperreals, that interval doesn't have a last included value, nor is there any reason why it should. Huon (talk) 22:54, 19 July 2012 (UTC)

Let X denote 0.999... (written just like that). It does follow from certain standard definitions that X=1. It is possible to use other non-absurd definitions and have X<1 but it is hardly worth it, for example you have to either accept that X+X=1+X or say that the left hand side does not make sense. It is not an exaggeration that you can start with 0.999...=1 as a basic definition, combine that with reasonable rules of arithmetic and recover some of the standard definitions as consequences. This approach is rigorous and valid but not popular (with good reason) Two respectable texts using this approach are J. F. Ritt, Theory of Functions, 1946 (in full) and M. Rosenlicht, Introduction to Analysis (in outline). The first is hard to find, the second is reprinted by Dover. In short, it is true by definition or by easy consequence of the definitions. The distinctions are subtle and it would be a mistake to discuss them in the high school curriculum (or 99.993% of the undergraduate curriculum.) One could discuss them in Wikipedia but it would be highly upsetting to lots of people including students, former students, teachers. Gentlemath (talk) 18:56, 10 August 2012 (UTC)

POV in Skepticism in education

The "0.333... = ¹⁄₃" argument seems valid to me, but the paragraph about "Skepticism in education" reminds me of Lysenkoism. It is the very nature of science that it only gets better when people question widely accepted theorems. Sentences like "There are many common contributing factors to the confusion" imply that the students are simply wrong, something that would not be accepted at for instance any Global Warming article, and rightly so. Like I said, I do think 0.999... actually equals 1, but for the ¹⁄₃ argument I would like to see that ¹⁄₃ is actually 0.333... (apart from that it looks that way on a calculator) and that multiplying it by 3 it must yield 0.999... Maybe some briljant student will prove in 2045 that you cannot multiply repeating decimals at all by a syntactic trick (i.e. replace all the 3's with 9's), because there are an infinite number of 3's. Similar to not being able to divide by 0. The fact that students do doubt it is interesting, as well as the reasons why, but the paragraph needs other wording. Joepnl (talk) 00:12, 8 July 2012 (UTC)

This is Mathematics. Students simply are wrong. It is not like other sciences where theories can be overturned; everything has to be proven. But the fact that it is somehow easier for students to accept that 0.333... is ¹⁄₃ than 0.999... is 1 has implications for educators. Hawkeye7 (talk)

Ian Stewart mentioned in either Cabinet or Hoard that once shown that 0.999... = 1 is a direct consequence of 0.333... = 1⁄3, some people respond by beginning to doubt 0.333... = 1⁄3 as well, instead of accepting this as a proof of 0.999... = 1. Is this still in the article? I remember having seen it here before. Double sharp (talk) 11:37, 12 August 2012 (UTC)

Well that certainly is the path I took. It seems ridiculous and insulting to me to answer a question about the nature of repeating decimals by citing another repeating decimal as proof. Hawkeye, you are wrong. This isn't Mathematics, it's mathematicians. Humans who think they are beyond mistakes are often proven tragically wrong. Aristotle was right until Galileo came along. Algr (talk) 05:39, 24 August 2012 (UTC)

For the justification that ¹⁄₃ is 0.333..., check out the article on repeating decimal. FilipeS (talk) 14:40, 17 August 2012 (UTC)

SILLY redirects

From http://en.wiki.x.io/w/index.php?title=Special:WhatLinksHere/0.999...&hidetrans=1&hidelinks=1&limit=500

0.9999999 (redirect page) ‎ (links)
0.99999999 (redirect page) ‎ (links)
0.999999999 (redirect page) ‎ (links)
0.9999999999 (redirect page) ‎ (links)
0.99999999999 (redirect page) ‎ (links)
0.999999999999 (redirect page) ‎ (links)
0.9999999999999 (redirect page) ‎ (links)
0.99999999999999 (redirect page) ‎ (links)
0.999999999999999 (redirect page) ‎ (links)
0.9999999999999999 (redirect page) ‎ (links)
0.99999999999999999 (redirect page) ‎ (links)
0.999999999999999999 (redirect page) ‎ (links)
0.9999999999999999999 (redirect page) ‎ (links)
0.99999999999999999999 (redirect page) ‎ (links)

0.999999999999999999999 (redirect page) ‎ (links)
0.9999999999999999999999 (redirect page) ‎ (links)
0.99999999999999999999999 (redirect page) ‎ (links)
0.999999999999999999999999 (redirect page) ‎ (links)
0.9999999999999999999999999 (redirect page) ‎ (links)
0.99999999999999999999999999 (redirect page) ‎ (links)

0.9999999999999999999999999999 (redirect page) ‎ (links)
0.99999999999999999999999999999 (redirect page) ‎ (links)
0.999999999999999999999999999999 (redirect page) ‎ (links)

0.9999999999999999999999999999999 (redirect page) ‎ (links)
0.99999999999999999999999999999999 (redirect page) ‎ (links)

How many 9s should we have?

(See http://en.wiki.x.io/w/index.php?title=Special:Log/delete&page=3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068))

218.22.21.3 (talk) 13:55, 12 July 2012 (UTC)

There's maybe more that are needed but redirects are cheap and all of them make sense individually.--JohnBlackburne^words_deeds 14:09, 12 July 2012 (UTC)

Redirects may be cheap, but I don't see the point to these when they're wrong. Why should the number 0.9999999 redirect to the different number 0.999...? It'd be one thing if they all had ellipses. I can even see an argument for '0.999' to link here, as an obvious mistyping of '0.999...' but I'd like to hear why you think '0.9' for example should suffer the same fate. What about people interested in the properties or pupular culture uses of 0.9 as a number? --Qetuth (talk) 04:19, 19 July 2012 (UTC)

If you feel these redirects are misleading, WP:RFD is the place to go. Even if we wanted to, there's not much editors at this talk page can do about them. That said, while I don't expect they see much use, I doubt anyone is really interested in the properties of the number 0.99999999 and will be misled by the redirect. Huon (talk) 10:59, 19 July 2012 (UTC)

None of these make sense. WP:RFD is the way to go. These are silly redirects that are highly unlikely to be linked from anywhere else or have any significant traffic, and, as said above, make no mathematical sense whatsoever. -- The Anome (talk) 19:40, 10 August 2012 (UTC)

This series of silly redirects seems to stop at 32 nines. (Has nobody thought of making a redirect with an infinite number of nines? That would at least be mathematically correct. :-P) Double sharp (talk) 11:39, 12 August 2012 (UTC)

Seems to me it would make more sense to title these redirects with a trailing "..." in their names, if you're striving for technical correctness, to wit:

0.9...
0.99...
0.9999...
0.99999...
0.999999...
0.9999999...
0.99999999...
etc.

— Loadmaster (talk) 18:55, 12 August 2012 (UTC)

Agreed. (My previous comment was a joke.) Double sharp (talk) 14:26, 13 August 2012 (UTC)

Another point to make is that 0.99 could redirect to 99%, since it's exactly that and not equal to 0.999... (100%). — Loadmaster (talk) 00:34, 16 August 2012 (UTC)

Makes sense except 99% is a disambiguation page, with no link to the mathematical concept. --RacerX¹¹ Talk to me^{Stalk me} 01:23, 16 August 2012 (UTC)

True, but it's still a better target than 0.999... I have changed the redirect. Huon (talk) 01:25, 16 August 2012 (UTC)

Really intense

I am someone with a bachelors in mathematics who likes to read Wikipedia math articles for fun, and even I was disheartened and dissauded by the mountains of extremely intense mathematics filling up most of the middle of this article. Even the introduction is a bit dense. If the top .01% of your reader population is stymied by your article, you're setting an awfully high bar.

Why not write this popular math topic towards the majority of your population, for example a person with a high school diploma? (that may even be too high) All of the dense math can be in links to other pages for the fraction of a fraction of your readers interested. Instead, spend this article conveying in terms everyone can understand the meaning, importance, history, and relevance of the topic to the overwhelming majority of your readers who are interested. I feel like normal people will come to this article wanting to learn about this intriguing fact and be driven away instantly by overbearing scholarship it takes 10+ years of study to understand. This is probably more my issue with most math articles rather than this particular one, but I feel interesting, basic math articles that the general population might actually read should make an extra effort to be understandable and readable. Sorintheseeker (talk) 02:11, 14 June 2012 (UTC)

I have a bachelor's degree in mathematics too. I don't think that there is anything in the article that would not be comprehensible with high school mathematics (who of course, do have 10+ years of schooling), although in some countries they may not study mathematics in high school. With maths articles, there is a balance that needs to be struck between providing information for the layman and for the expert. The usual pattern is to work up to higher level mathematics as you go along. The article should also be a gateway to other topics. As you descend, these are likely to become more complex. I encourage you to work on other articles. WP:Wikiproject Mathematics have a long list of articles in need of attention. Believe it or not, this is one of only 24 top class maths articles. Hawkeye7 (talk) 04:42, 14 June 2012 (UTC)

Well the level changes as you read further through the article. The first proofs only use basic algebra and are very easy to understand. After that it get's harder but if your accustumed to limits and calculus you should find your way through the analytic stuff. At the point that reaches Dedekind cuts it does get hard. I think it's a goodway of building up an article. Starting with simple proofs and building up the rigor. — Preceding unsigned comment added by 2001:980:D0F5:1:3DA8:FD6E:38D3:3BE2 (talk) 22:55, 5 December 2012 (UTC)

Is there a simpler way?

I thought it's widely accepted that 1/3= 0.333... Were I had a problem with the 'simple' proof is that we need to accept that 0.3333... * 3 = 0.9999... I am not aware that this multiplication can be done this way, so I propose an extra step in the demonstration: We know that 1/3 = 0.3333... 2/3 = 0.6666... 1/3 * 2 = 2/3 thus 0.3333... * 2 = 0.6666... thus 0.3333... * 3 = 0.9999... — Preceding unsigned comment added by 99.226.188.218 (talk) 00:51, 23 August 2012 (UTC)

Then you have to show that 0.333... + 0.666... = 0.999..., and that addition "can be done this way". That's no less complicated than showing that the multiplication works just as well. Furthermore, I'm not quite sure people are as willing to accept 2/3=0.666... as they are to accept 1/3=0.333... - after all, shouldn't there be a 7 "in the end"? Huon (talk) 01:19, 23 August 2012 (UTC)

1/3= 0.333... is a elementary school simplification similar to "Columbus discovered America". If you get serious about history, you'll know that the Columbus statement is only true from a very narrow point of view, with specific definitions of what "discovery" means, and don't even try to defend this to a Native American. 1/3= 0.333... is similarly problematic in ways that would overwhelm kids who are just learning how fractions work. Algr (talk) 05:22, 24 August 2012 (UTC)

While that may be true, I, for one, learned about fractions long before I heard of decimal representations, and I don't see how this article can ever serve those "kids who are just learning how fractions work". This isn't middle school math, and we cannot simplify it to that extent. In short: Too bad for those kids, but I don't think we can do anything for them here and therefore need not be overly considerate of them. The 1/3=0.333... proof is meant to be a simplification to almost the lowest level at which talking about 0.999... makes any sense at all, and unlike the Columbus example, it remains true if you take a more nuanced view (unless you change the number system, and even then you have to arrive at something pretty exotic before it gets wrong). Huon (talk) 12:06, 24 August 2012 (UTC)

unless you change the number system - This is where you are wrong, I think. The real number system is not remotely as ubiquitous as you think. Ordinary people never encounter it in their daily lives. For example, I doubt any engineer has ever actually used more then 10 digits of pi in a blueprint. Algr (talk) 14:18, 24 August 2012 (UTC)

Whether they're aware of it or not, the vast majority of people who consider both fractions and decimal representations do so in the context of either the rationals or the reals, or possibly something in between (a field extension of the rationals that's a subset of the reals), and in those number systems the proof is basically true. You need quite some mathematical knowledge to construct a number system where that proof fails - the easiest example I could think of would be the Hackenstrings, and those technically don't really have a number "0.999..." but only the base-2 equivalent. I don't expect you want to claim those engineers use Hackenstrings for their blueprints? What number system do they use in your opinion? Huon (talk) 14:47, 24 August 2012 (UTC)

They use whatever their calculator or computer works under. It's a system where the largest possible number is 9.999999e+99. It's a number system with a limit of perhaps 9 significant digits for any number, so the idea of an infinitely repeating decimal is totally out of the question. Does that sound like the Real Set to you? Algr (talk) 17:36, 24 August 2012 (UTC)

This is getting rather far off-topic for this talk page; it's completely unrelated to 99.226.188.218's proposed changes. Debating whether people really use non-associative number systems seems far-fetched even for the arguments page. Huon (talk) 20:00, 24 August 2012 (UTC)

Real set is obscure and not how regular people use numbers.

New section via Div

On the contrary. The whole article is written with the assumption that the Real Set is the only game in town and everyone knows it. But I have just demonstrated that nothing could be further then the truth. You'll never convince anyone of anything if you can't be bothered to check what a word like "Number" means to the general public. Algr (talk) 08:17, 26 August 2012 (UTC)

If you want to change the article in so radical a way, how about a reliable source that discusses the public perception of "number" in the context of 0.999...? Preferably one that not just acknowledges that there are common misconceptions about 0.999..., but claims that people prefer to work with a non-associative number system which has a largest number? If you can't provide that, this looks like original research. Huon (talk) 12:51, 26 August 2012 (UTC)

Algr, you are committing a fundamental fallacy here. You made reference to the fact that people make their computations on a calculator or computer, which, you correctly point out, does not use the system of real numbers. This has nothing to do with any preference for an alternative number system, and everything to do with the limitations of the underlying electronics. When engineers built their various systems of arithmetic, they were not promoting theirs as superior, but rather theirs was intended to be used as a close approximation to a mainstream mathematical number system. The floating point and double precision floating point numbers are supposed to be approximately real numbers. The unsigned and signed integers are approximations for the actual integers. etc. --COVIZAPIBETEFOKY (talk) 14:55, 26 August 2012 (UTC)

That too, and of course the vast majority of users is aware of the limitations of the hard- and software they use - they don't believe that the largest number their calculator can display is the largest number there is. Since "calculator math" doesn't have a number "0.999..." in the first place, even if they truly did use the number system Algr describes, that would still be irrelevant to this article. Huon (talk) 16:07, 26 August 2012 (UTC)

What people understand about digital number systems is beside the point, the fact is they are using one and have little interest in the quirks the Real set. Today's CPUs could do a far better job of modeling the Real set if there was any interest in doing so, but their isn't. The Mathematica article doesn't even mention the Real set - although complex numbers are noted.Algr (talk) 03:55, 27 August 2012 (UTC)

Floating-point arithmetic is a useful number system, to be sure. However, it has all sorts of problematic properties that mean that it can't possibly function as the mathematical reals: basic identities like (a-b)+b = a don't hold in general; it is not true that for every number, a+1 > a; it is true that there is a largest number; it is not true in general that if a < b, there is a number c such that a < c < b; etc. etc. Without those properties, pretty much all of mathematics -- algebra, calculus, etc. -- goes POOF!

When engineers and scientists do calculations in floating-point, they need to be aware that floating-point is just an approximation to the reals. There is a whole field of applied mathematics called numerical analysis which is devoted precisely to using this approximation effectively, accurately, and efficiently in solving problems posed (and often solved) over the reals. Numerical analysis is not easy precisely because floating-point numbers don't have the important properties I mentioned above.

So, sure, 0.999999 in a six-digit decimal calculator is not equal to 1.00000. But so what? That is not 0.999999..., and calculator arithmetic is not real arithmetic. --Macrakis (talk) 05:02, 27 August 2012 (UTC)

Agreed. But more relevant is the lack of sources connecting 0.999... to floating-point or "digital" number systems. That's not really surprising because in those number systems 0.999... doesn't even exist. So why bring them up here? Huon (talk) 12:45, 27 August 2012 (UTC)

The reason to bring it up here is the first thing I wrote in this section. "floating-point is just an approximation to the reals" - The Rationals could be called an approximation of the reals, and floating-point better approximates the Rationals anyway. The Real set is obscure and not how regular people use numbers. If you want to be understood, you must first understand. Algr (talk) 01:50, 6 September 2012 (UTC)

Algr, as you've said above, in calculator arithmetic, "the idea of an infinitely repeating decimal is totally out of the question". But this article is explicitly about the infinitely repeating decimal 0.999.... It is not about calculator arithmetic. --Macrakis (talk) 03:20, 6 September 2012 (UTC)

I'd like to see a source for the claim that "the Real set is obscure and not how regular people use numbers" before I took it seriously. People -- or mathematicians and philosophers, at any rate, and this is after all an article on a mathematical notion -- have felt the need for real numbers ever since the Pythagoreans in ancient Greece. Throughout most of history, there were no electronic calculators around. FilipeS (talk) 11:07, 6 September 2012 (UTC)

If someone said "Good grammer helps you communicate." would you demand a source for that? I'm not proposing that "Real set is obscure" be stated in the article, but hoping that this idea helps put you in the mindset of your readers so that you may understand why so many people reject this article. The last time I brought up "y>1 ; y+x≥1 ; Solve for x" Huon's response basically amounted to denying the existence of the Real set and insisting that I invent some other number system for y. You guys keep contradicting the very things you expect your readers to know to accept that .999...=1 Algr (talk) 14:45, 10 September 2012 (UTC)

The problem is that you're arguing quite the opposite: Most people don't bother with good grammar, and we should be understanding of them and put ourselves in the mindset of those who use bad grammar. No, we shouldn't. We should explain good grammar and good math. I have yet to see another number system that's closer to popular conceptions of what a number system should do than the reals. It may be possible to invent number systems that's closer to what many people believe about 0.999..., but all attempts that I'm aware of have severe disagreements with common conceptions about number systems in other places. Algr, if you disagree please go ahead and provide an example.

Regarding the "y>1 ; y+x≥1 ; Solve for x" problem (in Archive 10 of the arguments page), I didn't deny the existence of the reals, but the existence of a real number x which solves the problem for all (real) y. So what? I expect every number system allows for systems of equations that aren't uniquely solvable, or not solvable at all. That's not a flaw of the reals. If you want to conntinue that discussion, we should once again take it to the argumets page. Huon (talk) 16:50, 10 September 2012 (UTC)