Talk:0.999... - Knowledge (XXG)

875:: Yes, although such systems are neither as used nor as useful as the real numbers, lacking properties such as the ability to take limits (which defines the real numbers), to divide (which defines the rational numbers, and thus applies to real numbers), or to add and subtract (which defines the integers, and thus applies to real numbers). Furthermore, we must define what we mean by "an infinitesimal amount." There is no nonzero constant infinitesimal in the real numbers; quantities generally thought of informally as "infinitesimal" include 1660: 1639: 1547: 421: 1537: 1512: 1268: 1411: 1301: 2784:

representations occur, ideally independent of the axiom of choice and it would be nice if it gave a sense of the lengths one might have to go to to avoid these multiples. The class of standard positional systems suffices, but clearly the class for which this is true is larger, and this gives room for possibly finding a class that is simpler to define. Improving on just copying the second sentence in

1194:, it shouldn't be surprising that you find difficulty understanding some of the proofs, and, indeed, might have some skepticism that 0.999... = 1; this isn't a sign of inferior intelligence. Hopefully the informal arguments can give you a flavor of why 0.999... = 1. If you want to formally understand 0.999..., however, you'd be best to study real analysis. If you're getting a college degree in 351: 1460: 1442: 381: 32: 1609: 1523: 2752:(presumably the same) set of real values, but now a pair of equal representations differs in one digit instead of infinitely many. This leads me to wonder whether a representation composed of an infinite sequence of discrete symbols could avoid doubles, once we have the the restriction that every real number must have a representation. — 2826:

know what you mean by 'finitely many digits': we are dealing with a countably infinite sequence of digits, each of which is an element of a finite (or possibly infinite) set of symbols. I'm afraid educating me on this is, for the moment, a lost cause, so I suggest that we just focus on language to include in the article. —

3119:

I think you're a bit off on that point, Prof. Lazard. My impression is that the rejection of the actual infinite was more in theory than in practice, and its systematic use considerably predated the 20th century, since real analysis was developing in the mid-19th century and used the actual infinite

2446:

I rewrote the first paragraph of the lead that used many terms (technical or jargon) that are useless for people that know infinite decimals and are confusing for others. Also, I added that the equality can be proved, and therefore that is is not a convention that may be rejected by people who do not

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tags. I propose wrapping the second kind in {{efn}} instead, which has what I consider the advantage of distinguishing between the two types of notes (efn get labeled , , etc. instead of , ). One disadvantage is that there are clearly some judgement calls to be made. How do other people feel about

2695:

While I agree that "terminating representations" are a little peripheral, including that to answer the immediate question "How easy is it to find such values?" seems reasonable, although extrapolating from the example would seem obvious to us, and the phrase is adequately defined as linked. I don't

2208:

Agree that the simpler get-to-the-point jargon-free lede is better. The intended audience here is not mathematicians, it's lay people who likely are not familiar with the idea that the decimal representation of a real number is not unique in all cases (ie a "terminating" decimal that repeats zeros

2825:

I'm not of the opinion that anything more complicated than a sentence defining 'positional numeral system' is called for here to address the concerns that have been expressed, and it seems that my hope that a broader class could be defined more easily was too much. I have to confess that I do not

2783:

I was having difficulty framing the issue properly, and I thought about the cardinality argument after my post. I this context, all I would care about in this context is finding a simple constructive definition of a class of surjective maps from sequences of symbols to the reals for which multiple

2241:

The problem is this is like telling lies to children. An unprepared reader has no idea what the notation 0.999... refers to. The current lede makes clear what that is. The proposed lede is actively misleading, in the name of being more accessible. The problem is that the subject of this article is

1967:

I realize the mathematicians love precision and thus those special words that have meaning in math, but this article has an important point for a broader audience. I change the intro yesterday to concentrate the ideas that "It is the number one!" into the first paragraph and move the two (or is it

2799:

In this context, a nice way to think of continuity is that, if you want finitely much information about the answer, you need only finitely much information about the input. On the "real" side, finitely much information means an open interval. On the "representation" side, it means finitely many

2761:

If you put no restrictions at all on what you mean by a "representation", the answer is clearly yes, you can avoid duplicates. For example the set of all countably infinite decimal strings and the set of all reals have the same cardinality, so there's a bijection between them. You can even make

1166:

in which the string of symbols "0.999..." is different than the number 1. This object would represent a different number than the topic of this article, and this notation has no use in applied mathematics. Moreover, it does not change the fact that 0.999... = 1 in the real number system. The fact

2479:

Glad to see the work done to make this article more readable to a general audience. I hope "The utilitarian preference for the terminating decimal representation..." (last para of lede) can also be simplified, as I see what it means but as written it's well above most of the population's reading

2751:

I intended that a digit incrementing reverses the interpretation of all subsequent digits – in my quest to get rid of the infinite number of digits 'rolling over' (e.g. 0.999... to 1.000...). Looking more closely, I see that my supposition was incorrect: you still get two representations for a

2661:

Well, the trouble is that "balanced ternary" is not a "usual" positional numeral system. Perhaps it might be better to write, "and this is true of all bases, not just decimal". In the end it depends whether you think Knowledge (XXG) is here for the benefit of lawyers, or just to help people

2415:

Independently from Trovatore's post, I have rewritten the lead for removing jargon (in particular "denotes" is less jargonny/pedantic than "is a notation for") and unneeded technicalities from the beginning. This required a complete restructuration. By the way, I have removed some editorial

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for any values, this is not an axiom of decimal representation, but rather a property for terminating decimals that can be derived from the definition of decimals and the axioms of the real numbers. Systems of numbers have axioms; representations of numbers do not. To emphasize: Decimal

2907:

The representations are totally disconnected whereas the reals are connected, so intuitively, to map the representations to the reals, you have to "connect something", which is where the duplicates come from. The p-adics are already totally disconnected, so the problem doesn't come up.

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In particular, it's too early in the article to assume that the reader knows anything about infinite sequences and convergence. Statement like "The notation refers to a real number, namely the least real number greater than every truncation of the decimal." will be lost on the average

1154:: If it is possible to construct number systems in which 0.999... is less than 1, shouldn't we be talking about those instead of focusing so much on the real numbers? Aren't people justified in believing that 0.999... is less than one when other number systems can show this explicitly? 2676:

A better viewpoint is that such systems have no "terminating" representations at all, but only ones that eventually repeat the digit 0. That's beyond the scope of this article. Still, I think we should de-emphasize the notion of "terminating" representations. We don't really

1912:{{efn|{{harvtxt|Bunch|1982}}, p. 119; {{harvtxt|Tall|Schwarzenberger|1978}}, p. 6. The last suggestion is due to {{harvtxt|Burrell|1998}}, p. 28: "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."}} 1008:, there are systems for which 0.999... and 1 are distinct, systems that have both alternative means of notation and alternative properties, and systems for which subtraction no longer holds. These, however, are rarely used and possess little to no practical application. 3062:(essentially a weighted sum) is precisely correct for the statement as it now stands (i.e. including all nonstandard positional systems that meet this definition, with the proviso that they can represent all real numbers), and as supported by the text of the article. — 491:: No. The string "0.000...1" is not a meaningful real decimal because, although a decimal representation of a real number has a potentially infinite number of decimal places, each of the decimal places is a finite distance from the decimal point; the meaning of digit 503:· 10 toward the value of the number represented. It may help to ask yourself how many places past the decimal point the "1" is. It cannot be an infinite number of real decimal places, because all real places must be finite. Also ask yourself what the value of 2706:

That aside, it is interesting that having multiple representations depends on the definition of a positional numeral system as having position weights and values associated with symbols that do not depend on the value of other digits; I say this, because

3102:, a concept that is so counterintuitive that, before the 20th century, it was refused by most mathematicians. It seems that some teachers hope that kids could understand easily concept that were refused by mathematicians and philosophers a century ago. 1245:. Regardless of how confident you may be, at least one published, reliable source is needed to warrant space in the article. Until such a document is provided, including such material would violate Knowledge (XXG) policy. Arguments posted on the 2498:: authors in mathematical education have different explanation on the difficulties of the students with the equality; this is not to Knowledge (XXG) to select one amongst several. I did several other edits that are explained in the edit summaries. 1968:

three or maybe one) definitions to a separate section. The waffle-worded, footnoted definition will be completely opaque to naive readers. They will stop reading and never discover "This number is equal to 1.". Unfortunately my change was

3181:

So we have two people saying that a clear definition of a 'positional number system' is needed in the article, and I tend to agree in the context of this claim. I imagine that this can be omitted from the lead, but it might make sense in

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just as the first sentence, then we can continue on with elaborations. This way we can (as Johnjbarton put it) "get to the point" in the first sentence, and we haven't told any lies-to-children. By not insisting on including the phrase

3162:

Interesting point about p-adic numbers. I think the lead should mention infinity somewhere. I think the issues are resolved if the article is clear on what a "positional number system" is. I am unclear exactly what is meant.

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No matter. This isn't really a math article, or shouldn't be. Mathematicians are unlikely to care about 0.999... per se. We should keep that in mind when thinking about how to present the material. I'm totally against

2903:

Or more importantly, the p-adic integers are zero-dimensional, or it might be easier to think of it in terms of the p-adic integers are totally disconnected (not quite equivalent but it gets at the same point for our

3097:

By the way it is astonishing that nobody mention what is, in my opinion, the main reason for which there is so much confusion with the subject of the article: it is that "infinite decimals" make a systematic use of

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One thing to consider is that opening sentences don't always have to have the form "foo is a bar"; when that's awkward, it's fine to pick a different structure. In this case, maybe something along the lines of

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It's misleading to say that "0.999..." is notation referring to the number 1. The notation refers to a real number, namely the least real number greater than every truncation of the decimal. The fact is that

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I tend to agree that this is getting off-scope for the talk page. I'll drop a note on your talk page. I don't think it's a lost cause; I probably just haven't found the right way of explaining myself.

1190:: Yes. The initial proofs are necessarily somewhat informal so as to be understandable by novices. The later proofs are formal, but more difficult to understand. If you haven't completed a course on 1343: 3080:. But this supposes a precise definition of a positional numeral system, and of a positional numeral system that accepts infinite strings. Without such a definition, everything is original research. 3219: 676:≠ 0). However, the limit as x goes to 0 is 0, which is not positive. This is an important consideration in proving inequalities based on limits. Moreover, although you may have been taught that 1599: 2839:

But as long as we're here, I do want to correct the record for the benefit of any lurkers. Turns out my maunderings about the continuity of the inverse mapping were unnecessary. As long as

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I think it's fine to add efn. Additionally, maybe both the notes and references sections should be merged into one section, containing three different lists (notes, footnotes, works cited)?

2973: 2730:, which seems to be the natural one to put on them, whereas the reals are one-dimensional. So a continuous surjection from the representations to the real numbers cannot be an injection 538:

would be. Those proposing this argument generally believe the answer to be 0.000...1, but, basic algebra shows that, if a real number divided by 10 is itself, then that number must be 0.

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don't have to say that it's a notation for a different (infinitely long) notation, which is true if you're super literal-minded, but is extremely confusing in the first sentence, and

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I think this small tweak could open up a lot of possibilities for making the opening sentence (at least) more understandable to non-mathematicians, without saying anything false. --

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is not defined as the highest number in the sequence, but as the smallest number that is higher than any number in the sequence. In the reals, that smallest number is the number 1.

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feel strongly about keeping "terminating representations" or any other specific description of the class, though. I have clarified the statement in a way that fits the section

2140:

Sorry I mis-edited. I know you disagreed with "notation" as it means definitional equivalence, but I accidentally left the word. Here is the alternative I should have written:

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with an edit summary, "Restored old lede. It is important that the lede refer to an actual number, not merely some notation.", which I do not understand. Note that my lede was

1027:: If notation '0.999...' means anything useful in hyperreals, it still means number 1. There are several ways to define hyperreal numbers, but if we use the construction given 1182:: The initial proofs don't seem formal and the later proofs don't seem understandable. Are you sure you proved this? I'm an intelligent person, but this doesn't seem right. 2635:

Every positional numeral system has two representations for certain numbers, but is this necessarily true of terminating representations? A counterexample would seem to be

2553:

I have provisionally trimmed the passage I couldn't find support for. It wasn't technically wrong, as far as I could tell, but we aren't a repository for everything that

557:: If you have a number like 0.999...9, it is not the last number in the sequence (0.9, 0.99, ...); you can always create 0.999...99, which is a higher number. The limit 2734:, because otherwise it would be a homeomorphism, contradicting the previous observation about dimension. Therefore there must be reals with non-unique representations. 1941:

tags are ugly enough when used in isolation. Stacking three in a row and then trying to fit in a quote as well would be a mess. We handled the concerns in this section

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This still boils the subject of the article down to a tautology, which it is not. 0.999... definitionally means something. It is not the same thing as the numeral 1.

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level. I'm confused by the use of the {{spaces}} template before the 1 in the first paragraph: it looks like a formatting error. What's the point of that big space?

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this? (Obviously this is not urgent, am happy to have "I'm busy trying to preserve featured status and don't want to think about/deal with this" as an answer.) --

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article? I doubt it. This article ought to be pitched considerably lower. Anyone who understands the above argument isn't looking to understand 0.999.... --

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considerations that do not belong here. By doing this, I deleted the last Johnjbarton's edit, but I think that my version is better for the intended audience.

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If the article has been moved from its initial review period to the Featured Article Removal Candidate (FARC) section, you may support or contest its removal.

2267:

I agree that the concept of infinity is core to the article. How about a sentence in the first paragraph that explicitly calls out the concept of infinity?

1617: 1467: 1447: 3153:-adic numbers do not embed the reals. The ability to represent all reals is core to the statement that there are necessarily multiple representations. — 1710: 468:

should never be added to a Knowledge (XXG) article, and original arguments and research in the talk pages will not change the content of the article—only

3056:

I have changed that redirect to be consistent with the singular form, after verifying that there are no mainspace uses. In any event, the definition at

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are remarkably close to being a positional system and (extended to a fractional part) they evidently have a unique representation for each real number.

2442:

Sorry to have forgotten to sign. I did not use the "reply" button, because this is an answer to the opening post, and, as such, should not be indented.

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accessible to someone unwilling to grasp in some way with the concept of infinity. But this important aspect cannot be written out of the intro.

402: 2583:, but I also don't see the point in making this an article about real analysis. If you understand real analysis you don't need this article. -- 3229: 1815:

Right now, we have footnotes that are references and footnotes that are explanatory notes or asides, the former using {{sfnp}} and <ref: -->

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you get continuity of the inverse mapping, and therefore a contradiction, for free. That's because the representation space is compact (by

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representation, being only a representation, has no associated axioms or other special significance over any other numerical representation.

867:: Is it possible to create a new number system other than the reals in which 0.999... < 1, the difference being an infinitesimal amount? 98: 2769:

However, if you care at all about continuity, you're going to need to deal with the dimension issue I mentioned in my previous comment. --

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but I think it may have some things that are helpful. Unfortunately the comment by D.Lazard was added but not signed nor set as a Reply.

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Maybe the representation by Gray codes you're talking about isn't continuous (or its inverse is not); would need to see what you mean. --

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I gather your primary concern is the lede. My primary concern is the definition sentences. I think we should move that out of the intro.

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that 0.999... = 1 is not a "problem" with the real number system and is not something that other number systems "fix". Absent a

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Very nice. This also in some sense explains how p-adic integers can have unique expansions: the p-adic integers are compact.

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These could be just ref tags with rp templates for page numbers and quotes, but I don't know if that is the style you want.

1277: 852:, and so forth. Another way of writing it is 0.999...; contrary to the intuition of many people, decimal notation is not a 133: 646:: No. By this logic, 0.9 < 0.999...; 0.99 < 0.999... and so forth. Therefore 0.999... < 0.999..., which is absurd. 2630:... every nonzero terminating decimal has two equal representations ... all positional numeral systems have this property. 549:: The highest number in 0.999... is 0.999...9, with a last '9' after an infinite number of 9s, so isn't it smaller than 1? 236: 2869:

say that any continuous interpretation of the representations would have to have duplicates. Is that appropriate for

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desire to cling to intuitive misconceptions about real numbers, there is little incentive to use a different system.

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This is interesting, but does it apply? As I understand it (and admittedly this is outside my area of knowledge),

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Not sure quite what you mean about the Gray codes. The key point here is that representations of this sort are

1954: 361: 2925: 2612: 2566: 1410: 1267: 1143:, and even the '()' notation doesn't represent all hyperreals. The correct notation is (0.9; 0.99; 0,999; ...). 506: 338: 334: 330: 326: 322: 318: 314: 310: 306: 302: 298: 1886: 1842: 1428: 1287: 954: 2294:

In mathematics, the notation 0.999..., with the digit 9 repeating endlessly, represents exactly the number 1.

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The alphabet is finite (or at least there are only finitely many choices for a digit at any given position),

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We should never have a redirect and its plural linking to different places, so that is not a solution. —

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on Knowledge (XXG). If you would like to participate, please visit the project page, where you can join

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on Knowledge (XXG). If you would like to participate, please visit the project page, where you can join

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on Knowledge (XXG). If you would like to participate, please visit the project page, where you can join

1004:. In these systems, 0.999... = 1 still holds due to real numbers being a subfield. As detailed in the 846: 2533: 1659: 1638: 1546: 138: 3124:, but the ideas of Bolzano and Cauchy and Weierstrass and Dedekind were already laying the groundwork. 2858: 2544: 1896: 193: 43:. A featured article should exemplify Knowledge (XXG)'s very best work, and is therefore expected to 1291: 3138: 3058: 2913: 2878: 2816: 2786: 2774: 2742: 2686: 2588: 2324: 1868: 1823: 876: 842: 652: 229: 1162:: At the expense of abandoning many familiar features of mathematics, it is possible to construct 3107: 3014: 2663: 2503: 2481: 2452: 2427: 1853: 1621: 1552: 366: 17: 3094:

is an example of a positional numeral system such that there is always a unique representation.

1536: 1511: 243: 1225:: But I still think I'm right! Shouldn't both sides of the debate be discussed in the article? 3189: 3154: 3063: 3048: 2827: 2791: 2753: 2714: 2652: 2348: 2228: 2181: 2100: 2014: 888: 884: 823: 425: 119: 2495: 3164: 2976: 2890: 2727: 2723: 2636: 2604: 2580: 2558: 2525: 2462: 2400: 2386: 2362: 2334: 2268: 2261: 2247: 2193: 2127: 2113: 2046: 2026: 1973: 1946: 1918: 1878: 1834: 1207: 1028: 1001: 363: 1909:

I assume that your goal is to eliminate footnotes that are in fact citations? For example,

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to talk about them. We can say, for example, that 3.4999... is the same as 3.5000.... --

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Similarly, it's not appropriate to assume that the reader knows the difference between a

1250: 1238: 460:, yes. This is covered in the article. If you still have doubts, you can discuss it at 2639:: the numbers that have two representations seem to be nonterminating, e.g. 1 = 1.000... 3134: 2909: 2874: 2812: 2770: 2738: 2682: 2584: 2320: 1864: 1819: 832: 1234: 638:: 0.9 < 1, 0.99 < 1, and so forth. Therefore it's obvious that 0.999... < 1. 469: 209: 3203: 3103: 3088: 2592: 2548: 2499: 2448: 2423: 2382: 2185: 2104: 2018: 1900: 1848: 1191: 3033:

on "Standard positional numeral systems", so I suggest we simply use that wikilink.

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don't have to say that it "is a notation"; we just put that part before the 0.999...

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Probably a correct removal, but sort of a pity, since it's the only bit of actual

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page are disqualified, as their inclusion would violate Knowledge (XXG) policy on

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is totally disconnected, and so is not the continuous image of a real interval.

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In my opinion we should change the content back towards the version I suggested.

1136:{\displaystyle 0.(9)<0.9(9)<0.99(9)<0.(99)<0.9(99)<0.(999)<1\;} 2177: 2144: 2096: 2063: 1981: 1935: 1565: 1199: 1195: 997: 838: 620:{\displaystyle 0.999\ldots =\lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}} 457: 3034: 1542: 1522: 1203: 1031:, the problem is that almost same sequences give different hyperreal numbers, 31: 2708: 2621:

Two representations in every positional numeral system with one terminating?

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digits. If that's true in both directions, then there must be duplicates.

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that bijection pretty explicit, by playing games with (the proof of) the

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don't have to talk about least upper bounds before we give the punch line

1233:: The criteria for inclusion in Knowledge (XXG) is for information to be 1005: 391: 92: 2539:

The first and the second, as well as the bullet list, remain unsourced.

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As far as I understand, this section discusses supposed properties of

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Please place discussions on the underlying mathematical issues on the

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Agree with removing the technical details to a definition section.

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Something that holds for various values need not hold for the

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come from somewhere? Other than that, the sourcing seems OK.

401:. For questions about the maths involved, try posting to the 1607: 499:

places past the decimal point is that the digit contributes

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Featured articles that have appeared on the main page once

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This article appeared on Knowledge (XXG)'s Main Page as

2651:(where T = −1) has two. Or maybe I need some coffee? — 1969: 1942: 1374: 1355: 1336: 431: 53: 2811:

article, but it would be satisfying if it could be. --

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always has another representation that repeats nines).

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Featured articles that have appeared on the main page

2928: 1037: 957: 897: 741: 682: 563: 509: 2013:) is a notation for the number "1" represented as a 1671:, a collaborative effort to improve the coverage of 1564:, a collaborative effort to improve the coverage of 2807:skeptical that this can (or should) be worked into 2184:consisting of an unending sequence of 9s after the 2103:consisting of an unending sequence of 9s after the 2017:consisting of an unending sequence of 9s after the 483:: Can't "1 - 0.999..." be expressed as "0.000...1"? 242: 3210:Knowledge (XXG) featured article review candidates 2967: 2180:equal to "1". The real number is represented as a 2099:equal to "1". The real number is represented as a 1598:This article has not yet received a rating on the 1487:Knowledge (XXG):WikiProject Spoken Knowledge (XXG) 1214:, it would probably help you in the future anyway. 1135: 988: 935: 786: 727: 619: 530: 3127:That said, sure, it's a key psychological point. 2399:Yeah, I didn't like the new version as a whole. 2224:. We can explain all this later in the article. 1945:; nothing more in this regard needs to be done. 959: 951:(which can also be expressed without the "+" as 899: 574: 101:for general discussion of the article's subject. 2643:has no other representation, but 1/2 = 0.111... 2522:0.999...#Impossibility of unique representation 1019:: Are you sure 0.999... equals 1 in hyperreals? 936:{\displaystyle \lim _{x\rightarrow 0^{+}}f(x)} 856:from decimal representations to real numbers. 8: 2954: 2929: 1290:. Even so, if you can update or improve it, 1286:as one of the best articles produced by the 1280:; it (or a previous version of it) has been 3017:point out, the statement is valid in usual 1963:lede that gets to the point without jargon. 1490:Template:WikiProject Spoken Knowledge (XXG) 2968:{\displaystyle \{0,...,9\}^{\mathbb {N} }} 2417: 1877:OK, I've made a stab at dividing them up. 1633: 1506: 1436: 1315: 1262: 531:{\displaystyle {\frac {0.000\dots 1}{10}}} 444:: Are you positive that 0.999... equals 1 2959: 2958: 2957: 2927: 1036: 989:{\displaystyle \lim _{x\downarrow 0}f(x)} 962: 956: 913: 902: 896: 769: 759: 749: 740: 710: 700: 690: 681: 672:0) for all values in its implied domain ( 611: 593: 577: 562: 510: 508: 1408: 1131: 996:); and values in number systems such as 470:reputable secondary and tertiary sources 3240:Featured articles on Mathematics Portal 3120:implicitly. It took Cantor to make it 3023:non-standard positional numeral systems 1635: 1578:Knowledge (XXG):WikiProject Mathematics 1508: 1438: 3077: 2112:and restoring the Definition section? 887:, which are not real numbers and have 3235:Unknown-priority mathematics articles 7: 1665:This article is within the scope of 1558:This article is within the scope of 1465:This article is within the scope of 787:{\displaystyle 1.y_{1}y_{2}y_{3}...} 728:{\displaystyle 0.x_{1}x_{2}x_{3}...} 2625:The article contains the statement 1816:tags, the latter using <ref: --> 1685:Knowledge (XXG):WikiProject Numbers 91:for discussing improvements to the 1468:WikiProject Spoken Knowledge (XXG) 584: 25: 3215:Knowledge (XXG) featured articles 2494:I have removed this sentence per 879:, which is not a fixed constant; 118:New to Knowledge (XXG)? Welcome! 62:and improve the article directly. 18:Talk:Proof that 0.999... equals 1 1658: 1637: 1581:Template:WikiProject Mathematics 1545: 1535: 1521: 1510: 1458: 1440: 1409: 1299: 1266: 883:, which are not numbers at all; 419: 379: 349: 113:Click here to start a new topic. 3250:Mid-importance Numbers articles 2790:seems to me to be challenge. — 2599:Are any of the concerns in the 1705:This article has been rated as 1493:Spoken Knowledge (XXG) articles 655:of those values. For example, 3078:all positional numeral systems 2845:The mapping is continuous, and 1863:Strongly support using efn. -- 1427:It is of interest to multiple 1122: 1116: 1107: 1101: 1092: 1086: 1077: 1071: 1062: 1056: 1047: 1041: 983: 977: 966: 930: 924: 906: 814:: 1 can be written many ways: 581: 390:for discussing changes to the 1: 3230:FA-Class mathematics articles 1679:and see a list of open tasks. 1572:and see a list of open tasks. 1481:and see a list of open tasks. 464:. However, please note that 110:Put new text under old text. 39:This article is undergoing a 3255:WikiProject Numbers articles 3083:As an example, the standard 2557:be said about a math topic. 1688:Template:WikiProject Numbers 2865:So if we can source it, we 2361:Great, I made that change. 3271: 3185:0.999... § Generalizations 3029:redirects to a section of 3027:positional numeral systems 3019:positional numeral systems 2764:Schröder–Bernstein theorem 2699:0.999... § Generalizations 2381:I reverted the change by @ 2347:Also support this change. 1901:04:04, 19 April 2024 (UTC) 1887:23:31, 17 April 2024 (UTC) 1873:22:35, 17 April 2024 (UTC) 1859:22:29, 17 April 2024 (UTC) 1843:22:15, 17 April 2024 (UTC) 1828:21:54, 17 April 2024 (UTC) 1711:project's importance scale 1363:Featured article candidate 436:Frequently asked questions 3245:FA-Class Numbers articles 3193:18:09, 30 June 2024 (UTC) 3173:17:01, 30 June 2024 (UTC) 3158:16:57, 30 June 2024 (UTC) 3143:18:53, 30 June 2024 (UTC) 3112:16:44, 30 June 2024 (UTC) 3067:16:43, 30 June 2024 (UTC) 3052:15:54, 30 June 2024 (UTC) 3043:15:32, 30 June 2024 (UTC) 2831:23:35, 30 June 2024 (UTC) 2821:22:13, 30 June 2024 (UTC) 2795:21:40, 30 June 2024 (UTC) 2779:19:52, 30 June 2024 (UTC) 2757:19:12, 30 June 2024 (UTC) 2747:18:30, 30 June 2024 (UTC) 2732:with a continuous inverse 2718:15:11, 30 June 2024 (UTC) 2691:05:36, 30 June 2024 (UTC) 2672:04:07, 30 June 2024 (UTC) 2656:01:56, 30 June 2024 (UTC) 2613:02:30, 23 July 2024 (UTC) 2603:still outstanding, then? 2593:21:50, 29 June 2024 (UTC) 2567:17:47, 29 June 2024 (UTC) 2549:06:18, 29 June 2024 (UTC) 2534:05:10, 29 June 2024 (UTC) 2508:14:21, 28 June 2024 (UTC) 2490:11:34, 28 June 2024 (UTC) 2471:10:41, 28 June 2024 (UTC) 2457:10:37, 28 June 2024 (UTC) 2436:19:23, 27 June 2024 (UTC) 2409:17:44, 27 June 2024 (UTC) 2395:17:41, 27 June 2024 (UTC) 2371:16:35, 27 June 2024 (UTC) 2357:16:22, 27 June 2024 (UTC) 2343:16:18, 27 June 2024 (UTC) 2329:16:13, 27 June 2024 (UTC) 2277:15:59, 27 June 2024 (UTC) 2256:15:39, 27 June 2024 (UTC) 2237:15:31, 27 June 2024 (UTC) 2202:15:52, 27 June 2024 (UTC) 2136:15:26, 27 June 2024 (UTC) 2122:15:09, 27 June 2024 (UTC) 2055:14:58, 27 June 2024 (UTC) 2035:14:51, 27 June 2024 (UTC) 1955:21:16, 27 June 2024 (UTC) 1927:01:11, 27 June 2024 (UTC) 1704: 1653: 1615: 1597: 1530: 1453: 1435: 1395: 1318: 1314: 1288:Knowledge (XXG) community 1239:reliable published source 148:Be welcoming to newcomers 77:Skip to table of contents 3087:-adic representation of 2985:23:16, 1 July 2024 (UTC) 2922:Also, the product space 2918:20:05, 1 July 2024 (UTC) 2899:10:59, 1 July 2024 (UTC) 2883:03:06, 1 July 2024 (UTC) 2848:The mapping is injective 1600:project's priority scale 1308:Today's featured article 76: 3031:List of numeral systems 2333:I support this change. 1833:I'd be fine with that. 1561:WikiProject Mathematics 1382:Featured article review 1247:Talk:0.999.../Arguments 462:Talk:0.999.../Arguments 41:featured article review 2969: 1612: 1484:Spoken Knowledge (XXG) 1448:Spoken Knowledge (XXG) 1417:This article is rated 1137: 990: 937: 788: 729: 621: 532: 143:avoid personal attacks 35: 2970: 2520:Does the argument in 1611: 1421:on Knowledge (XXG)'s 1344:Articles for deletion 1138: 991: 938: 789: 730: 622: 533: 343:Auto-archiving period 168:Neutral point of view 34: 2926: 1584:mathematics articles 1310:on October 25, 2006. 1164:a system of notation 1035: 955: 895: 739: 680: 561: 507: 448:, not approximately? 173:No original research 49:Please feel free to 3059:Positional notation 2859:Tychonoff's theorem 2787:Positional notation 2662:understand things. 1668:WikiProject Numbers 3015:user:Imaginatorium 2965: 2461:Looks fine to me. 2095:) is a notation a 1622:Mathematics Portal 1613: 1553:Mathematics portal 1423:content assessment 1319:Article milestones 1133: 1132: 986: 973: 933: 920: 885:differential forms 784: 735:must be less than 725: 671:is positive (: --> 617: 616: 609: 588: 528: 154:dispute resolution 115: 36: 3021:, but not in all 2862: 2854: 2851: 2516:Sourcing question 2443: 2438: 2422:comment added by 2182:repeating decimal 2151:(also written as 2101:repeating decimal 2070:(also written as 2045:is equal to one. 2015:repeating decimal 1988:(also written as 1792: 1791: 1725: 1724: 1721: 1720: 1717: 1716: 1632: 1631: 1628: 1627: 1505: 1504: 1501: 1500: 1403: 1402: 1391: 1390: 1261: 1260: 1251:original research 958: 898: 889:anticommutativity 594: 592: 573: 526: 466:original research 434: 409: 408: 374: 373: 134:Assume good faith 111: 82: 81: 69: 68: 45:meet the criteria 16:(Redirected from 3262: 3091: 3086: 2974: 2972: 2971: 2966: 2964: 2963: 2962: 2856: 2853: 2838: 2728:product topology 2724:zero-dimensional 2637:balanced ternary 2631: 2581:lies to children 2441: 2304: 2295: 2170: 2169: 2168: 2165: 2156: 2089: 2088: 2087: 2084: 2075: 2059:Ok so how about 2043:this real number 2007: 2006: 2005: 2002: 1993: 1940: 1934: 1856: 1851: 1797:Yet another anon 1727: 1693: 1692: 1691:Numbers articles 1689: 1686: 1683: 1662: 1655: 1654: 1649: 1641: 1634: 1618:selected article 1586: 1585: 1582: 1579: 1576: 1555: 1550: 1549: 1539: 1532: 1531: 1526: 1525: 1524: 1514: 1507: 1495: 1494: 1491: 1488: 1485: 1462: 1455: 1454: 1444: 1437: 1420: 1414: 1413: 1405: 1398:Featured article 1396:Current status: 1377: 1358: 1356:October 10, 2006 1339: 1316: 1303: 1278:featured article 1270: 1263: 1243:editor's opinion 1208:computer science 1142: 1140: 1139: 1134: 995: 993: 992: 987: 972: 942: 940: 939: 934: 919: 918: 917: 793: 791: 790: 785: 774: 773: 764: 763: 754: 753: 734: 732: 731: 726: 715: 714: 705: 704: 695: 694: 626: 624: 623: 618: 615: 610: 605: 587: 537: 535: 534: 529: 527: 522: 511: 456:: In the set of 424: 423: 411: 394:article itself. 383: 382: 376: 368: 354: 353: 344: 247: 246: 232: 163:Article policies 84: 71: 63: 27: 21: 3270: 3269: 3265: 3264: 3263: 3261: 3260: 3259: 3200: 3199: 3100:actual infinity 3089: 3084: 2953: 2924: 2923: 2712: 2650: 2646: 2642: 2623: 2518: 2166: 2163: 2162: 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2229:Mr. Swordfish 2226: 2223: 2219: 2215: 2211: 2207: 2203: 2199: 2195: 2192: 2187: 2186:decimal point 2183: 2179: 2175: 2171: 2157: 2150: 2146: 2142: 2141: 2139: 2138: 2137: 2133: 2129: 2125: 2124: 2123: 2119: 2115: 2111: 2106: 2105:decimal point 2102: 2098: 2094: 2090: 2076: 2069: 2065: 2061: 2060: 2058: 2057: 2056: 2052: 2048: 2044: 2039: 2038: 2037: 2036: 2032: 2028: 2020: 2019:decimal point 2016: 2012: 2008: 1994: 1987: 1983: 1979: 1978: 1977: 1975: 1971: 1962: 1956: 1952: 1948: 1944: 1943:back in April 1937: 1930: 1929: 1928: 1924: 1920: 1916: 1911: 1910: 1908: 1902: 1898: 1894: 1890: 1889: 1888: 1884: 1880: 1876: 1875: 1874: 1870: 1866: 1862: 1860: 1857: 1852: 1846: 1844: 1840: 1836: 1832: 1831: 1830: 1829: 1825: 1821: 1810: 1808: 1807: 1805: 1796: 1794: 1788: 1784: 1780: 1776: 1772: 1768: 1764: 1760: 1756: 1752: 1748: 1744: 1740: 1737: 1735: 1731: 1729: 1728: 1712: 1708: 1702: 1699: 1698: 1695: 1678: 1674: 1670: 1669: 1664: 1661: 1657: 1656: 1652: 1646: 1643: 1640: 1636: 1623: 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903: 890: 886: 882: 881:differentials 878: 874: 871: 870: 866: 863: 862: 861: 855: 851: 844: 840: 836: 835: 830: 829: 825: 821: 817: 813: 810: 809: 805: 802: 801: 800: 781: 778: 775: 770: 766: 760: 756: 750: 746: 742: 722: 719: 716: 711: 707: 701: 697: 691: 687: 683: 675: 670: 666: 662: 658: 654: 650: 649: 645: 642: 641: 637: 634: 633: 632: 612: 606: 602: 599: 596: 589: 578: 570: 567: 564: 556: 553: 552: 548: 545: 544: 543: 523: 519: 516: 513: 502: 498: 494: 490: 487: 486: 482: 479: 478: 477: 471: 467: 463: 459: 455: 452: 451: 447: 443: 440: 439: 433: 430: 427: 422: 415: 413: 412: 404: 400: 399: 393: 389: 385: 378: 377: 358: 357: 352: 348: 340: 336: 332: 328: 324: 320: 316: 312: 308: 304: 300: 296: 292: 288: 284: 280: 276: 272: 268: 264: 260: 257: 255: 251: 250: 245: 241: 238: 235: 231: 227: 223: 220: 217: 214: 211: 208: 205: 202: 199: 195: 192: 191:Find sources: 188: 187: 179: 178:Verifiability 176: 174: 171: 169: 166: 165: 164: 155: 151: 149: 146: 144: 140: 137: 135: 132: 131: 125: 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