366:
this process never terminates." The typical reader neither uses nor feels comfortable with this language. (Nor with zooming in by an infinite amount!) Also, it gets us into dangerous territory. For example, a natural question it raises is, "How long do we have to wait to decide equality?" And the answer would seem to be, "Forever!" Not quite what we were going for, eh? In fact, one of the core mental errors people have in this area is a conception of limits as a process rather than a static fact. They want a definite answer sometime before they die, so they artificially terminate the process — and reach the wrong conclusion.
1574:". Possibly the 10-adic expression ...999 could make an appearance somewhere. "Applications" could use a second application to justify the plural: perhaps the role of the converse in the decimal proof that the reals are uncountable? "In popular culture" needs a bit of expansion, and finally-- this is the hard part, but doable-- the introductions of the three major proof sections should start explaining their contexts: the desirability of theorems with weak hypotheses, and the diversity of pedagogical treatments of real numbers. Any takers for those tasks?
3836:(As for half-open -type subdivisions, 1.000… becomes undefined. Either style breaks equivalency with the supremum definition and prevents one from applying the Nested Intervals Theorem. I can probably dig up a source for either convention, and Apostol considers both in parallel. We could put a note, possibly towards the top of "Nested intervals and least upper bounds" since that's where it becomes most relevant, that some authors refuse to define or even consider 0.999…, but nonetheless whenever it
346:"All numbers can be placed on the number line. If two points are in the same place on the number line they are the same number, if they are in different places they are different numbers. If you place 0.999... and 1 on the number line they appear to be in the same place. If you zoom in on the number line these two points still appear to be in the same place. If you zoom in by an infinite amount one still appears to be on top of the other. Therefore they must be the same number."
2679:. In my opinion, the article should be more general, giving historical context and analysis of skepticism over the equality to 1. This is already reflected in the body of the article, and the title should follow suit. If there is enough material, somewhere down the line the article could be split in two, with one devoted to proofs of the equality and the other devoted to everything else. This would follow the pattern of the articles you've cited.
31:
2630:, the title makes sense. But Melchoir has hated both the title and the content since first encountering the article, and has come back again and again and again to try to bend it to something different. Just recently, lacking supervision, Melchoir launched a large-scale assault. So please read what the article was before that when deciding what the title should be. Also, this is not the first time a move has been proposed, with Melchoir
1992:
at classical and established processes and not "news." Do you see what i mean? Calling it "Proof that..." led me to expect the article to be establishing something new, just as a newspaper headline might read, "Proof That There Are
Planets Outside Our Solar System." I could be totally offbase or maybe just nit-picky, but I feel like it would give the article the informational we're looking for instead of an argumentative feel.
2939:, and leaving behind a lighter summary here. This move might become necessary, as the article is getting long. In fact, I can even envision two more sub-articles: one on connections with other mathematics (generalizations, applications, other number systems) and one on conceptual difficulties alone. But these are big decisions, and I'd rather postpone them until all the content is laid out and we can do a Peer Review.
3577:"The goal of this article is to enliven Abraham Robinson's concept of an infinitesimal by exhibiting infinitesimals in a simple and direct manner. ...To this end we shall exploit the familiar notion of the decimal expansion of a real number. After all, this notion improves our grasp of real numbers; just so, it will help us appreciate infinitesimals, indeed all numbers of our extended number system."
3877:
3787:
3734:
3540:(Okay, I wanted to make sure that you didn't think I was aggravating the situation.) Maybe we should err on the side of reversible, light-handed solutions for the article. If you cancel the request, the worst that could happen is a few more reverts. Meanwhile, we'll have the opportunity to capitalize on recent participation and take the article down the home stretch to GA.
3771:
rectangle depicts the part of the number line covered by ; the cyan ; the yellow ; and the gray . The upper segments and their rectangles are , , , and . The green line shows how the number 1 is contained in all of the intervals, and therefore can be written either as 0.222… or 1.000…. The extra gray marks are unused subdivision points. The vertical gray line is 0.
1769:...Okay, in addition to the reference I used just now, I also found "A sheaf of R-algebras on the set of Dual Numbers", whose author uses an "obvious and useful" metric topology and notes that other authors use different topologies. Of course, nobody is saying anything about 0.999…. Maybe we should just leave it the way it is and avoid drawing any conclusions.
1936:
flow more naturally into the following section on rational numbers. One of them will take the sequence limit within the real numbers; I realize that we were trying to avoid such a limit and the associated infinity sign, but it's an important idea that gets referenced again and again. Also, some subsections will have to be renamed more precisely.
2167:
started it, but my objection was taken as a signal to do even more. Previously there were exactly four proofs, each brief and each serving a different purpose. The history of
Melchoir's opinions and "contributions" to this article has been one unmitigated disaster after another. It no longer benefits the audience that needs it. --
677:, Chapter 1, "Construction of the System of Real Numbers" G. Pickert and L. Görke provide several approaches. They first demonstrate an approach using decimal expansions, but note "addition can only be defined in a very lengthy way, and the rules for calculation are not very convenient to prove." They then note the fact that "
2654:
was never the only one to dislike both the content and title. The nature of the topic of this article (the fact that so many people seem to think that 0.999... ≠ 1) means that an article with just a proof (or proofs) won't really benefit anyone (I have the impression that anyone with enough knowledge
2166:
Hell no! Melchoir has trashed the article quite enough. We have a number of "Proof…" articles. This was a fine article, and will be again once I have the stomach to correct the damage that has been done to it. The "extensive proving" and out-of-control citing is absurd, as I pointed out when
Melchoir
1935:
I've just made a trip through Google Book Search, and it appears that the treatment of 0.999… in real analysis textbooks is even more diverse than I thought. I'll be expanding the section, possibly adding a couple more subsections to clarify the differences; these will have the benefit that they will
1639:
I'm not sure that there's really a relevant difference between not recognizing the characters and not being able to execute the task. The whole danger with calculators as a thinking aid is that you don't have to interpret for them: they're supposed to be able to take an input as it looks on your page
658:
operator, and induction. And it turns out that he actually did this not just on his personal homepage, but in an MAA pedagogical publication! After I hit "Save page" on this comment, I'll hit "Save page" on the "Order proof" section of the article. It will be different, but hopefully not so different
365:
This is a version of the argument I have nicknamed the "squeeze proof". Many variations of this can be found on the web, though a lengthy
Socratic dialog is unusual. Having read far too many debates in researching this article, I noticed that I balked early on here at the sentence, "They are equal if
472:
into a
Socratic dialog! Nor would I say it is "obvious", for two reasons. One reason is that I've seen too many nay-sayers who want to insist on the existence of infinitesimals. The other is that we are perfectly capable of constructing number systems that somewhat resemble the rationals (or reals),
449:
Picture two kids playing a game of "name the largest number". The first says "one hundred", the second responds "one hundred and one", the first goes for "one bazillion", and the second counters with "one bazillion and one!" The game gets boring quickly; the kids needn't play forever to realize that
3635:
Well, that's reasonable. I could go either way on the sentence listing the methods; they're an important part of the article but maybe the reader doesn't care so early on. If you feel strongly about it, okay. But what about the former third paragraph on skepticism, popular discussion, and alternate
3172:
state it can be done, it can only apply to "numbers" which can be equated to a rational number, i.e., if both sequences are periodic. This pretty much amounts to equating a rational p-adic with the rational real, which seems arbitrary. (Furthermore, if we go to the p-adics, we have to mention the
2025:
I also sympathize about the opening sentence. The idea of approximate equality belongs in the "Skepticism" section, properly sourced of course, and in the lead section it should appear only alongside a statement about the frequent confusion surrounding 0.999…. You're welcome, of course, to edit the
1991:
I would also suggest one other small but significant change that would help someone like me understand what the point of the article is: Could we change the title to "Proofs that 999... equals 1"? I feel that simply making the title plural would immediately let a reader know that they are looking
3770:
Ideally an illustration should be understandable without a long caption, but just so we're on the same page.... the horizontal line is the real number line, and visual elements are displaced above or below it only so that they can be told apart; the vertical scale is meaningless. The lower magenta
3386:
Let's agree that the wholesale revert, without reaching consensus, was inappropriate. Let us also agree that it doesn't really matter - the article is adequate in its current form, and reverting it back and forth won't help anything. Our energies should be focused on discussing the desired form of
1950:
This article is quite interesting, but I'm not sure if it belongs in an encyclopedia instead of a journal of mathematics. I feel like an encyclopedia isn't really where I would look for a mathematical proof. It may just be the fact that I'm not a mathematics expert that makes me wonder. Perhaps
1304:
You reverted me because the definitions of "Dedekind cut" and "Archimedean" I found in the literature are different from the ones you personally prefer? One of them already has a citation to two references. I will add a citation for the other. If you have further questions, please direct them here
767:
rationals; and it's clearly not just a typo. It is also puzzling to see the name "cut" applied to a single set, rather than a pair of sets partitioning the rationals. Logically, the second set is merely the complement of the first in the reals and so would seem to convey no additional information;
3442:
Thanks for your ideas, everyone (especially those who complimented and started cleaning up my work)! My immediate concern, besides the desire to continue editing the article, is that with Chuayw2000's recent edits to KSmrq's version, a content fork has in some sense developed. We don't want to be
3040:
Third: Two editors with similar edit summaries do not make a consensus. Using a revert as an opportunity to vent frustration through the edit summary box is, in fact, actively harmful to the process of building consensus. There are many editors watching this article who would rather stay out of a
2682:
Attacking
Melchoir, instead of the issue itself, does nothing to help anyone's cause. I dislike the phrase "large-scale assault" and your implication that an editor in good standing requires supervision, especially when none have voiced opposition to his edits except you. Yes, some of his edits
1987:
Thanks for your help. I think you're right, and a lead section would help immensely. Now that you mention it, I think it was the opening sentence that set me up to think of the whole article the wrong way. As it stands, it sounds a bit polemic, especially to a layman such as myself, as if the
3339:
I am well aware that double-decimals don't work in the p-adics. I did not claim that they did, nor did I write that they did in the article. Now, if I recall correctly, the reference requires only that the left end of the sequence is periodic. Either way, you are welcome to add a caveat to that
1380:
You distort the history. The chorus of answers to your old questions was "Stop calling standard mathematics
Original Research." It was clear from your attitude at the time that you were extremely hostile and that such a citation would accomplish nothing. Then, too, I expended considerably words
585:
Thoroughly excellent! I was about to ask about this, whether the 'find a number between 0.999... and 1' is a proof or just a sneaky way to explain this to people. There is no number between 0.9999.... and 1 (because you can't just add another '9' in the trailing 9's), thus the two numbers are
1619:
The point is that the calculator doesn't recognize that character, not that the calculator can't do it (it can't anyway, but that's not what the picture is saying). It's like typing in 3×5 into a calculator that's looking for 3*5 and then saying that calculators struggle with multiplication.
2009:
Awesome! On the naming issue, my personal preference is simply "0.999…". The current title comes from early versions of the article that contained only a proof or proofs; it now contains much more material of general relevance to 0.999… from diverse approaches. (See, for example, the article
2091:
number between zero and one with an infinite number of nines after the decimal point. It can be proven that this number is, in fact, equal to one, a result which some/many people find counter-intuitive for a number of reasons." with, of course, appropriate sourcing for the various bits.
755:
authors, as a partition of the rationals satisfying certain properties. And so on. I took the liberty of cutting through the voluminous discussion to distill the parts relevant to this article, but the arguments — including the pivotal role of the
Archimedean property — are essentially
1704:
Is that saying, possibly, that one applies the topology of the plane and not the natural order topology? If so, wouldn't 0.999… still be well-defined and equal to 1, just as in the regular complex numbers? Is there a source that addresses decimal expansions in these sets, anyway?
653:
Okay, since I was last seen here, I've lined up some half-dozen references. First up, and the most exciting, is a fresh look at one of the first sources ever to be mentioned on this talk page: Fred
Richman. He uses Dedekind cuts in a way that avoids partitions, upper bounds, the
3090:). The other recent edits don't seem to agree with either group. 2 to 1 qualifies as a super-majority, if not a consensus. But, thinking about it, your proofs do seem better than the other version, for the most part. Perhaps I'll edit in some of the simpler versions, later.
445:
plays a decisive role. You might say it tells us we don't actually have to compare forever, so long as we are prepared to compare as much as needed on demand. For those who have studied calculus, the same kind of subtle but effective termination appears in the "epsilon-delta"
3741:
While we wait for things to settle down, I'll work on some images. If anyone has feedback on the illustration to the right or suggestions for other illustrations, fire away. I have the Adobe
Illustrator sources of course, so even the most minor changes are easy to implement.
3432:
Without commenting on which version is better, I also agree that the wholesale reverting isn't helpful. There are multiple editors with 2 or 3 reverts on this article in the past couple of days. Given the long history here, perhaps mediation by a third party would be
2585:
about the number written "0.999…", but about proofs of an equality. Nor is the nature of the equality limited to this specific example, though the article has become so cluttered that the generality may no longer be clear. I draw your attention to the following list:
1640:
and give you the answer. I'm not aware of any calculator that inputs either three dots or an ellipsis and doesn't choke. As for syntax, I don't think we have to worry about a reader who thinks, "Well, if they'd put spaces between the dots, maybe it would have worked"!
774:, in section 14.8 discusses the fact that in many topoi not only are the reals defined by a "Cauchy" approach different from those defined by a "Dedekind" approach, but slight variations in the demands on a cut are also of interest. I'll quote Goldblatt's definition.
1961:
Um... that's a lot of questions! I'd say that, yes, this is a classical mathematical truth, even more classical than some of the proofs! It isn't strictly necessary to prove it more than once, but it's one of those popular problems that has attracted lots of proofs
2892:
You may be right about that, but in cases like this one, it is not enough to aim at a correct and sensible exposition; we should also aim at something that will not be corrupted all the time - if it can be done without compromising the integrity of the articles.
2279:
Simpler titles are more likely to be found either by search engines or by humans. I'd oppose it if I thought more material properly belongs under the shorter title than under this one. Why object to argumentatitiveness? All mathematical proofs are arguments.
3572:
Apparently A. H. Lightstone, a student and coauthor of Robinson, addressed decimals in NSA not long after the field was created. Lightstone's POV, contrary to this article's suggestion, is that decimals in NSA are (1) easy and (2) a useful educational device:
500:
Well, you're spot on about the problem of non-termination. I did realize that little hitch when I wrote the dialog. But, honestly, pretend you aren't a mathematician for a moment. You don't know a lick of calculus: you are the "boy" in the Socratic dialog
303:
At the outset, I assumed that two real numbers are comparable according to their sequences of digits. Then I assumed that between any two different numbers, there lies a third which is also different. These two assumptions must be incompatible in some way.
2914:
I'm not sure I understand how having 2 articles will prevent them (or one of them) from being corrupted. Are you saying that people are less likely to insert nonsense into a proof article? That doesn't seem to agree with the history of this article. --
2134:
Haven't contributed anything in a while, but I also support a move. Give it a day or so, and if there aren't a plurality of objections, then I say Melchoir does the move. Also, since '…' is a Unicode character, there will have to be a redirect from
2507:
And just to make sure that the move proposal isn't misunderstood, this article should not back down from that position. While we describe popular doubts over "0.999… = 1" per NPOV, we must not assert those doubts, since no reliable source ever does.
3929:
Rudin p.20 or Richman p.399. To be precise, Rudin and Richman would call this cut "1" and "1", respectively; both identify these with the traditional real number 1. Note that what Rudin calls a Dedekind cut, Richman calls a "nonprincipal Dedekind
3236:
That was the other guy, but I can understand why you think we're the same person. Checking "breaking subtraction" references. For what it's worth, as I reverted one move and two of your edits/reversions, so I'm out of this for a while, under
1148:
I'm intending a minor rewrite pass over the whole article to fix some of the mess that has accumulated, but I've been postponing it while I work on other things. I'll hang on to a copy of your variation and see what of it I can profitably use.
1831:
For this article, perhaps we'll change "orderings" to "metrics" and segregate the p-adics into a new third section of "Other number systems" dedicated to systems where decimal expansions don't make sense, but which are still somehow relevant?
1911:
The new archive's date isn't a typo; I went from the absolute earliest post to the absolute latest post. By a freak coincidence, someone replied to a two-month old discussion while I was moving the text, so I had to move that section back.
170:
Compare each expansion, number by number in sequence. When we encounter a number in the expansion of one which is different from the other, then we may say that the first real number is greater or less according as the number is greater or
1969:, well, it's currently undergoing several changes. I suspect that its broad structure isn't going to shift too much more, but one problem is that it doesn't have a lead section that provides an overview of the article. We'll write one soon.
3832:
Well, it's hard to prove anything with an illustration. I tried to indicate closed intervals by capping them with square brackets and reducing the transparency of the rectangles' edges; perhaps the brackets and edges should be thicker and
3520:
Sorry, I should have been clearer. I prefer your version personally, so that's why I didn't protect it myself (not supposed to protect pages I'm involved with). Should I cancel the request now though, as KSmrq hasn't reverted again?
3358:
I agree about game theory. That wasn't my writing anyway, although it's a good suggestion for a topic, as Hackenstrings are the only known example of a number system where something like 0.999… is less than 1 due to infinitesimals.
1119:
Here we're in the context of topos logic, and we need to keep our wits about us. But classical logic is a special case, so for present purposes we can interpret simply. Anyway, the thing to realize is that we can't really define
1722:, by including a new element ε defined to combine with other reals in the usual way, but such that its product with itself is zero. Every dual number then consists of a standard real component and an "infinitesimal" component,
1566:
Well, I think "Order proof" and "Limit proof" are looking good. Next, that geometric series is getting lonely, so I'll be adding a complementary method in "Real analysis" using subdivided intervals, due to Bartle and Sherbert.
3677:
Possibly "the non-terminating decimal 0.999… (where the 9s recur)", or "the infinitely repeating decimal 0.999… (where the 9s recur)". Or maybe the "(where the 9s recur)" part isn't necessary; it's obvious by the notation.
1609:] What's wrong with using this image to illustrate the point-- all but made by Mazur, especially if you read the book-- that a student who relies on a calculator is screwed when it comes to reasoning about 0.999...?
559:
Anyway, the offending passage of the dialog can easily be swept away. The student could answer something along the lines of: "They are equal if the one is neither less than nor greater than the other." Of course a
1747:ε lies between 0.9999...9 and 1 (so the limit is undefined). This order has the property that all sums of squares are positive, but of course it is not a field. We need merely leave the bit about displacement out.
342:"Between any two distinct real numbers there must be another number which is larger than one of them, but not as large as the other. There can be no numbers between 0.999... and 1, so they must have the same value."
1750:
Right, I'm on board with you mathematically. But does anyone actually work with the dual numbers in the order topology? It seems to me that physicists would be much happier taking limits in the plane instead.
3267:
were omitted from the definition of a cut. Keep the same definitions of order and addition. Show that the resulting ordered set has the least-upper-bound property, that addition satisfies axioms (A1) to (A4)
2814:), the vast majority of the community seems to agree with him. Wholesale reverts without even a scrap of discussion is not the way to go. In my book, the unilateral removal of sourced content even borders on
1193:
Every positive decimal expansion then easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers
1995:
But now that I better understand the purpose of the article, the content is exactly what I would expect. Someone curious about established proofs of this truth should have this resource available to them.
348:(I expect both of these ideas have been suggested before and probably rejected because they are not rigorous enough. They are however good for trying to convince people who do not believe that 0.999...=1.)
2687:
it and working together to reach a compromise, instead of doing blanket reverts? Again, KSmrq, you stand alone in opposition. Please do not assume you have authority over the rest of the editors here.
1594:
Okay, nested intervals is up. I'm not real comfortable with having two long-yet-dense paragraphs that don't address 0.999..., so perhaps they should be exported to another article and allowed to expand.
1951:
this is a well-established, classical mathematical truth? If so, why is the extensive proving necessary? If it is an exploration, perhaps there is a better venue in which it ought to be published?
1491:
Another absurd twist. I will again revert to my decision to ignore you, and my plan to rewrite the article at some point in the future to clean out the mess you have almost single-handedly created. --
251:
It cannot be less than 9, since this would make it less than 0.999... by the method of comparison. It cannot be greater than 9, because each digit is an integer less than 10. So it must equal nine.
2655:
to understand the proofs originally presented, would also be able to derive them himself anyway). Any sensible article must discuss 0.999... within a broader context, and with a title to match. --
1221:
The equation "0.999... = 1" means that these two Dedekind cuts are the same set, each containing the same rational numbers. If the equation were untrue, there would have to be some rational number
3034:
me. The div headings are only for backwards compatibility with incoming section links. They invisibly demarcate areas that are currently too short to necessitate or merit subsections of thier own.
1403:
This section did not need to be rewritten just to satisfy your obstinacy, and the rewrite has several problems which I will not waste more time discussing if you have not changed your attitude. --
255:
So here is the number you have so far produced for me: 0.9????... where the digits represented by a "?" are yet to be determined. What is now the second digit to the right of the decimal place?
3581:
And that, kids, is why following references is better than doing original research. I was going to briefly expand on Hackenstrings next, but I think I'd better fix the overall readability first.
3306:
Yeah, and there's more that Richman has to say about the relationship between the two structures, especially in relation to negatives. I figured that material would stray too far from the topic.
2619:
3443:
working on two versions simultaneously. I guess the most neutral thing to do would be to protect the article, but I hope there's enough of a consensus here that we can agree to simply revert to
1743:
In fact, this is plainly a confusion; the dual numbers can be viewed as a plane, but they can also be ordered lexicographically, in which case they clearly have non-Archimedian elements and 1 -
2738:
Um... the whole second half of the article consists not of proofs that 0.999... = 1 but of Generalizations, Other number systems, Applications, and Skepticism. What do you think it's missing?
1458:. The number of footnotes is only two less. But now, every cite appears at the end of a sentence, and each paragraph has at least one and no more than two. I suggest that this is not absurd.
2901:
have an article with stuff that you say is without enough merit to have an article of its won. Having both articles, everyone (well, more of us, at least) can be reasonably happy, I think.--
2968:
I thought we were over this. I'm anxious to get back to work on the article, and I'm confident that I can fix whatever problems are brought to my attention. Can I get a consensus here that
2350:
I gather from that link that p-adic expansions are sometimes written to the right? It would be confusing to adopt that notation, but I guess we could make a note in the relevant section.
1786:
Why are the p-adic numbers offered as an example of an ordered algebra that is non-Archimedean? Isn't it true that the p-adics can't be ordered, and no p-adic has an infinitesimal norm?
1853:
I've tried moving them to a new subsection and motivating them through Gower's discussion of 0.000…1. I don't have a reference for ...999 = −1, but given the number of books on the
3349:...or at least you'd be welcome if we were doing this the civilized way. KSmrq, could you please either join the discussion or undo your revert? I find it personally offensive.
3413:
I agree with Meni Rosenfeld, the revert was wrong in my opinion and I think Melchoir's version is much better (even if it does have some issues which need to be discussed ). —
3166:
3139:
3613:
says the summary should be concise, and I interpret that as telling the reader what will be done, not how; that's for the body of the article. The less fluff, the better.
1446:
Still, the number of footnotes is easily reduced by combining some of them, particularly those that cite the same author within a given paragraph. I'll work on that now...
369:
A less conversational, though more rigorous, use of the same idea is embodied in the "order proof". Here's its last paragraph, looking for a rational between 0.999… and 1:
2599:
259:
Again, it must equal 9, since it cannot be less as that would make the number less than 0.999..., and it cannot be greater for there is no individual digit greater than 9.
1570:
Then "Other number systems" could use more explicit ties to 0.999..., including a subsection on systems where "0.999..." doesn't mean 1: hackenstrings and Richman's "Cut
1384:
The current article is worse than when I left it, as predicted. The number of footnotes is absurd, especially when we have separate articles on the cited topics, such as
2874:
Possible, but I think the proofs are neither hard nor interesting enough to merit their own article. With the proper setup, they're no more than trivial one-liners. --
3111:
Could you check the "Breaking subtraction" section again; I don't think that works correctly. But I don't have the reference available to check, unles it's online.
2018:
contains just the one proof.) So I also see what you mean about "proofs", but plurals are a tricky issue on Knowledge, and usually they're omitted from titles. See
1866:
This is what I get for not owning a history book... I could swear that the construction of the reals by Cauchy sequences was due to Cantor. Are you sure it isn't?
564:
is that ultimately one must face the possibility of non-termination in the case of equality. It also fails to highlight the student's "error" in stark terms.
3013:
and I have reverted your changes, pretty much for the same reasons. I don't see how you can say there is consensus that the reversions are inappropriate. —
2631:
134:
1443:. When there are multiple ways of formulating a concept, or an accuracy dispute between editors, it is especially important to seek external verification.
215:
If two different points on a line are given, then there is a third point between them which is not equal to either of the other two. Is this reasonable?
1696:
It seems pretty clear to me that you can make the dual numbers into an ordered ring, such that epsilon is a true infinitesimal. So what's the deal with:
1585:
In local news, I don't have a history book on hand. Can anyone fill in the details on the origin of the geometric series formula and its early proofs?
1355:, and I made sure of that before taking any action. If you extend me the same courtesy by exercising a little caution, we can be civilized about this.
768:
however, things are not so simple. Also, standard reals are rather forgiving, but in advanced treatments there are subtleties. For example, Goldblatt,
263:
Here is the number so far: 0.99???... Do you see that the argument you have just offered must apply to each digit, taken in turn, of this expansion?
3814:
intuitive to someone who does not accept the theorem. Is there anything, in and of the illustration, that forces it to be closed on the right side?
1322:
Like you did? You reverted once while I was typing extended material here, and a second time immediately after. In any event, as I said to you on my
3293:
under the appropriate definitions, and the cuts form an ordered groupoid, or something like that, and the reference seems to say that, as well. —
2581:. Melchoir has been warping the content from the intent of the article. The title of the article follows the convention of any proof article. It is
616:" doesn't return anything usable. Could someone make a link from the inverted titles to this article so that "=" may become commutative again? Thx.
2935:
I might support a variant of the idea: splitting off the sections detailing the mathematical development / proofs of 0.999… into a sub-article per
223:
Very well then. So according to your claim that 0.999... is different from 1, there must be a point between the two which is not equal to either.
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Now that you've finally convinced yourself that a direct Dedekind cut approach is not OR, it seems a good time to mention the three-volume series
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3391:'s version is better, and any new problems it may have introduced should be dealt with on a case-by-case basis (as has been initiated above by
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This approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in
2087:, including a fair/balanced section on Skepticism for NPOV, and a lead section along the lines of "0.999... is a common representation of the
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All of the examples you gave, with the exception of the last, have corresponding articles about the general concept. For example, there is
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This is certainly good, as well as comprehensive and understandable to the layperson, but I think its a little too long for the article. --
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Yes, that's what I meant. What about the first digit to the right of the decimal point? Is it less than 9, greater than 9, or equal to 9?
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point of the article is to argue against the notion of .999... equaling "approximately" 1. From what you've told me, this is not the case.
1284:, for which the proof is more obvious. Other modifications of the procedure can lead to structures other than the real numbers; see below.
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They are equal if this process never terminates. That is, if each digit in one sequence is equal to each corresponding digit in the other.
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I'm unable to understand the image, but that may be because I'm unfamiliar with the area. Perhaps an explanation of how the image works?
1730:ε, either of which may be zero. However, the infinitesimals are displaced off the real line, rather than ordered between standard reals.
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Good. Now I will demonstrate the flaw in your method of comparison. Are you aware that numbers may be represented as points on a line?
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We cannot rely upon the four articles you mentioned for verification because, for various reasons, Knowledge does not consider itself a
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For breaking subtraction, two links to the reference are in the References section, under "Richman". The free link is mostly the same.
1841:(By the way, I chose words poorly, but I secretly do know the difference between a field that's an ordered set, and an ordered field.)
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2569:. There could be a redirect from "improper decimal expansion" which is a commonly used term for decimal expansions with recuring 9's.
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Then applying this argument, we can conclude that the number between 0.999... and 1 which is not equal to either of them is 0.999...
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Maybe the "trouble" language is sending the wrong message. How about, "A typical calculator cannot help one reason with 0.999...?"
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And given two different real numbers as infinite decimal expansions, can you tell whether one is greater or less than the other.
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the article becomes more general on the number 0.999..., as opposed to its current state as merely a set of proofs, then I will
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As for first sentence's "the recurring decimal 0.999… (where the 9s recur)": surely there's a better, shorter way to say that?
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I've never seen it myself, but a reference was given (at least on the talk page, if not in the article — I haven't checked the
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If by first digit, you mean the digit to the left of the decimal point, then it must be a zero since it has to be less than 1.
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infinitesimals: they are non-Archimedean. (The article later mentions a few.) I doubt many nay-sayers would actually want to
2014:. It favors a specific proof under the spartan heading "Proof", hence the singular, describing some others as alternatives.
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claims that "This field cannot be turned into an ordered field." I'm no expert in this area, but I'm inclined to believe it.
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I'm winding down for the day. Further discussion or revert wars or whatever will just have to wait, at least on my end. --
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True, yet even a TI-89 requires its operator to do much of the interpretation for it. Okay, I'll try the second version.
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Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.
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Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.
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of two decimal expansions? Invariably, the answer you come up with is going to be inconsistent with the "geometry" of
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positive rational. The version of the Archimedean property used here tells us that if a non-negative rational number
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Typing "Proof that 1 equals 0.999..." etc returns this article on the search though, but you need to use "proof". —
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to either. So from this perspective 0.999... must be unequal to 0.999..., which is absurd. What is the problem?
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3104:-style games. (I don't remember where we put the article, although it's probably in my watch-list). Found it:
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Add "* Support" or "* Oppose" followed by an optional one-sentence explanation, then sign your opinion with ~~~~
1857:-adics and the obviousness of using negative one as an example, I'm sure there's a citation waiting to be found.
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You could explain it in a shorter (but less well laid out way) and then it would not be too long. You could say
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Okay. I was just wondering if it was impossible to use half-open intervals, and it looks like it is. Thanks.
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My stated goal is to make this a Featured Article. Other theories on my motivation and behavior are welcome at
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doesn't "equate" to 0, even if …999.999… can equate to 0. I've looked at that one, so, even if the reference
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Yes, it should summarize, but going into detail about how each proof is accomplished distracts the reader.
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1700:"However, the infinitesimals are displaced off the real line, rather than ordered between standard reals."?
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I've moved the tag to the sentence this seems to be objecting to. I'm not sure whether the p-adics are an
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Anyway, I hope this is a sign that we are converging, and I do think the Richman paper is a nice addition.
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it, as the intervals could just as easily be [.2, 1), [.22, 1), and [.222, 1), which might actually seem
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And I haven't installed popups. My reversions were both manual edits, although without much explanation.
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And so, using this method, would you then say that 0.999... is less than, greater than, or equal to 1?
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2022:. Another benefit of "0.999…", now that you mention it, is that it's the least argumentative of all.
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such numbers, but to realize that, they would have to confront some of the awkward implications. --
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And this third point must be greater than 0.999... and less than 1, without being equal to either?
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I didn't limit my explanation to one sentence, and I don't think we should. Nor need we have a
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1760:...ah, it looks like the computer science people are interested in our ordering. One moment...
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trying to talk with you before I walked away in frustration, leaving others to deal with you.
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Yes, I see now. It's a great illustration of .222... approaching 1, but I don't see how it
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1190:. In particular, the real number 1 is the set of all rational numbers that are less than 1.
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The step from rationals to reals is a huge extension, and order is an essential part of any
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question of the old material several times: "Are there sources for this"? The answer was
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2469:, as 0.999… does not equal one like the title "Proof that 0.999... equals 1" implies.
1259:, which is impossible. (There is no infinite rational number; the rational numbers are
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This is not a problem with the article but with the wiki search engine. Searching for "
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I don't see the problem. The notation "...999" is quite different from "0.999...". --
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And a pair of real numbers are equal exactly when the points they represent coincide?
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advises us that the lead section should summarize the entire article. Shouldn't it?
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In the p-adics, the double-decimals DOES NOT WORK. Adding one of the four 10-adic
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I've filed a request for protection (I can't do it myself because I'm involved). —
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revert war but who are often willing to provide their opinions on the talk page.
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3705:*checks edit history* Yup, guess I did. Sorry about that; re-removing it now.
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Clearly, it must be less, for 0.999... begins with a 0, while 1 begins with a 1.
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If you wish to start a new discussion or revive an old one, please do so on the
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Definitely better. Include "typical" though, because calculators such as the
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smaller than any positive rational we can name, we have only one choice: zero.
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But by your argument, this is the only number between 0.999... and 1 which is
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Given the lack of discussion after 60 hours, I have listed this article for
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of the rational numbers yields the same results; in particular, one can use
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So using your method of comparison, what is the first digit of this point?
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There, now the reader's head is guaranteed to explode, and it's all true!
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2488:... it does equal 1: that's what the article is supposed to be proving —
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Okay. While you check out Richman, I'll paste in the quote from Rudin:
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Wikifying the article. Then we can start discussing indivdual changes.
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ring to it. I can understand exactly how it made him uncomfortable. --
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Your proof incorrectly states the Archimedean property as the lack of
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Illustration that in base 3, 1 = 1.000… = 0.222… via nested intervals
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Illustration that in base 3, 1 = 1.000… = 0.222… via nested intervals
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Illustration that in base 3, 1 = 1.000… = 0.222… via nested intervals
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Is it true that every real number has an infinite decimal expansion?
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2227:– More accurate reflection of content; simpler; less argumentative.
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the reals. And for that we want pairs satisfying the five sentences.
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Now take reciprocals: The immediate implication is that there is no
3456:(By the way, Chuayw2000 says he wouldn't be offended by a revert.)
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DO NOT EDIT OR POST REPLIES TO THIS PAGE. THIS PAGE IS AN ARCHIVE.
396:. Continuing in this fashion through each decimal place in turn,
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The above discussion is preserved as an archive of the debate.
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The following discussion is an archived debate of the proposal.
2139:. On previewing, looks like there already is, made by Mets501.
1060:, ε denotes the standard membership relation, and the variables
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want to include some indication of these philosophical issues.
505:. Try to answer the question naïvely: how do you decide on the
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2292:. It would also have to include a note on the left-to-right 10
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mathematicians. Richman notes that taking Dedekind cuts in any
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is the infinite set of all rational numbers that are less than
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Motivated by this observation, they define Dedekind cuts, with
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left-to-right notation, as we've done so in that article.) —
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Foundations of Mathematics: The Real Number System and Algebra
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Or, absent an answer to that, what needs rewriting, and why?
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If an upper bound less than 1 exists, it can be written as 1−
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Proof that the sum of the reciprocals of the primes diverges
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Yeah, I deleted it! So, you put it back by accident, then?
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This archive page covers approximately the dates between
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Does this mean you want me to stop editing the article?
3271:(with a slightly different zero-element!) but that (A5)
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Hey, this is just the standard move template speaking.
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2994:'s changes to the introduction are harmful. I prefer
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you expect from an encyclopedia article on the topic?
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How about a conceptual "proof" by the Socratic method?
3655:. The lead will still need something, though... hmm.
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3062:. Surely no one will fault me for undoing this last?
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per Melchoir and the fact that it's easier to find. —
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in an earlier thread and I still support that title.
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using a comparison operator, because we haven't yet
3511:...I guess I'll take that as a no. Back to work...
2806:, you should take a look at your actions. Whatever
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763:rational numbers, when it actually is the lack of
3202:; presumably it happened while you were writing.
1214:is less than some other number of the form 1 − (⁄
299:Look carefully at the assumptions you have made.
211:Then the points they represent must be different.
2051:. I've never thought of it before, but now that
2557:, as per my statement in the previous section.
1815:0 < 3 < 6 < 1 < 4 < 7 < 2...
1160:For convenience, I'll stick the variation here.
2600:Original proof of Gödel's completeness theorem
1718:Standard reals can also be extended to become
1291:This will save digging through the history. --
279:But 0.999... is equal to 0.999..., is it not?
3950:"Mathematics Magazine:Guidelines for Authors"
3079:First; they look like subtitles, they should
2683:make the article more verbose, but how about
2524:per Melchoir: more encyclopedic and shorter.
1656:can do it if entered as an infinite series. —
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2983:headings by subsections. You seem to be be
2055:mentions it, the current title does have an
1400:, and so on. I could say much more, but why?
135:Talk:Proof that 0.999... equals 1/Archive10
2717:a proof that 0.999... = 1. As long as the
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3100:should probably be piped to specifically
1888:-adics now. This is going to kick ass...
703:… is the least upper bound of the set of
2972:is inappropriate and counterproductive?
2431:'s "vote". Could be 1, could be -1. —
1436:, where I will be eager to discuss them.
1080:⟩ there is one and only one real number
3954:The Mathematical Association of America
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3387:the article here. That said, I believe
2846:, could we have both: An article named
2822:to your own version of this article? --
1884:I've got three or four sources for the
771:Topoi, the categorial analysis of logic
673:, from MIT Press. In the first volume,
468:I'm not confident of my ability to put
344:Or an alternative way of looking at it
2020:Knowledge:Naming conventions (plurals)
1972:I'm interested in learning this: what
1577:After all that, I'm thinking GA -: -->
44:Do not edit the contents of this page.
2427:, per my comment in the thread under
2083:I also support moving the article to
1875:...and MacTutor comes through again.
207:What if the two numbers are unequal?
7:
2982:OK, first replace all < div : -->
659:that it precipitates any bloodshed.
139:Knowledge:How to archive a talk page
3289:OK, the "positive decimals" form a
2897:seems to be arguing that we should
2818:. Might it be that you have become
2713:. In its present state the article
1246:to be the rational number 1 / (1 −
789:is uniquely determined by the sets
513:(i.e., the Archimedean property).
3264:, suppose that the property (III)
24:
2810:'s motives (and I see no need to
2721:redirects here I see no proplem.
2595:Proofs of Fermat's little theorem
2043:I would support a move to either
1578:cook up some illustrations -: -->
3591:Lead section and popular culture
3445:03:19, 6 September 2006 Supadawg
2296:notation in which it's equal to
1946:Is This an Encyclopedia Article?
175:And when are two numbers equal?
29:
3261:With reference to the Appendix
721:has previously been defined as
420:), so it must be the case that
3006:) I reverted to is acceptable.
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884:if it satisfies the sentences
860:. In general an ordered pair ⟨
1:
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2673:Proof of Bertrand's postulate
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2590:Proof of Bertrand's postulate
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2363:article itself recently). —
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2197:The result of the debate was
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1714:Meh, I'll remove it to here:
1238:) for every positive integer
591:20:39, 1 September 2006 (UTC)
412:. But the rationals have the
3161:{\displaystyle {\sqrt {1}}1}
3134:{\displaystyle {\sqrt {1}}1}
2848:Proof that 0.999... equals 1
2769:Add any additional comments
2605:Proof that 0.999... equals 1
2221:Proof that 0.999... equals 1
1528:What are you talking about?
1394:construction of real numbers
712:." (For decimal expansions,
133:Please add new archivals to
3030:First: You could have just
3002:, although the version (of
2838:Following the example from
1902:18:09, 31 August 2006 (UTC)
1893:08:35, 31 August 2006 (UTC)
1880:21:02, 30 August 2006 (UTC)
1871:20:39, 30 August 2006 (UTC)
1862:19:50, 30 August 2006 (UTC)
1846:17:28, 30 August 2006 (UTC)
1837:17:25, 30 August 2006 (UTC)
1820:17:15, 30 August 2006 (UTC)
1808:, then on the multiples of
1804:, then on the multiples of
1791:16:52, 30 August 2006 (UTC)
1774:15:44, 31 August 2006 (UTC)
1765:15:12, 31 August 2006 (UTC)
1756:15:03, 31 August 2006 (UTC)
1738:06:36, 31 August 2006 (UTC)
1710:16:38, 30 August 2006 (UTC)
1680:05:54, 30 August 2006 (UTC)
1671:02:43, 30 August 2006 (UTC)
1648:21:28, 29 August 2006 (UTC)
1635:21:14, 29 August 2006 (UTC)
1614:20:57, 29 August 2006 (UTC)
1600:23:12, 25 August 2006 (UTC)
1590:18:39, 25 August 2006 (UTC)
1542:22:03, 24 August 2006 (UTC)
1533:20:39, 24 August 2006 (UTC)
1496:20:28, 24 August 2006 (UTC)
1463:15:04, 24 August 2006 (UTC)
1451:14:50, 24 August 2006 (UTC)
1408:12:41, 24 August 2006 (UTC)
1360:02:41, 24 August 2006 (UTC)
1331:02:23, 24 August 2006 (UTC)
1310:00:52, 24 August 2006 (UTC)
1305:before reverting anything.
1296:00:47, 24 August 2006 (UTC)
1182:approach, each real number
1154:00:36, 24 August 2006 (UTC)
671:Fundamentals of Mathematics
664:22:06, 23 August 2006 (UTC)
643:01:28, 10 August 2006 (UTC)
621:01:17, 10 August 2006 (UTC)
408:for every positive integer
376:for some positive rational
295:I must have made a mistake.
3989:
2842:'s list of articles named
2634:both content and title. --
2615:Proof that e is irrational
2610:Proof that 22/7 exceeds pi
1692:Dual numbers: not ordered?
1068:range over the members of
604:But what about 1=0.999...?
3106:combinatorial game theory
3086:Third; neither does one (
1965:As for what this article
1931:Expanding "Real analysis"
569:20:55, 29 June 2006 (UTC)
545:19:46, 29 June 2006 (UTC)
532:Actually, the very term "
518:19:41, 29 June 2006 (UTC)
486:19:18, 29 June 2006 (UTC)
353:16:33, 29 June 2006 (UTC)
329:16:06, 29 June 2006 (UTC)
313:19:23, 29 June 2006 (UTC)
3797:Perhaps this is better?
3595:Hey, hang on, Supadawg.
3000:08:48, September 5, 2006
2954:Please do not modify it.
2780:section for discussion.
2749:I support this change.
2189:Please do not modify it.
3198:By "popups" I refer to
2937:Knowledge:Summary style
2559:Confusing Manifestation
2094:Confusing Manifestation
1272:, which is targeted at
586:equivalent. *applaud*
436:({1}), and 0.9999… = 1.
166:How would you do this?
3881:
3840:defined it equals 1.)
3791:
3738:
3653:where you're going now
3611:Knowledge:Lead Section
3597:Knowledge:Lead section
3162:
3135:
2049:Proofs that 0.999… = 1
3879:
3789:
3736:
3163:
3136:
2026:lead as you see fit!
1824:Interesting! Anyway,
1347:I have already asked
42:of past discussions.
3145:
3118:
2677:Bertrand's postulate
1812:; so in the 3-adics,
1398:Archimedean property
1269:Mathematics Magazine
1263:.) So 0.999... = 1.
882:Dedekind real number
443:Archimedean property
414:Archimedean property
126:Post replies to the
1072:. For such a pair ⟨
1052:denote the subsets
1004:)) "open upper cut"
969:)) "open lower cut"
404:must be less than ⁄
3972:Richman pp.398-399
3894:Looks good to me.
3882:
3792:
3739:
3158:
3141:to of the decimal
3131:
3037:Second: Thank you!
2711:Conditional Oppose
2628:reasonable version
2547:Williamborg (Bill)
2263:argumentative. —
2016:An earlier version
1579:Peer Review -: -->
1434:User talk:Melchoir
1250:), one would have
1044:where the symbols
1039:) "close together"
3527:
3488:
3419:
3153:
3126:
3096:Further details;
2850:(or should it be
2632:wanting to change
2494:
2453:
2253:0.999... equals 1
2117:Fredrik Johansson
1662:
1626:
1282:decimal fractions
649:Back on the horse
633:
614:1 equals 0.999...
416:(they contain no
103:
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48:current talk page
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2990:Second, I think
2956:
2812:assume bad faith
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1582:peace and love.
1581:Main Page -: -->
876:) of subsets of
781:, a real number
641:
640:
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392:can be at most ⁄
275:That is correct.
227:That is correct.
162:Yes, I think so.
81:
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3083:like subtitles.
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2579:Strongly oppose
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2335:P-adic#Notation
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1948:
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1817:Septentrionalis
1784:
1782:And the p-adics
1732:
1694:
1657:
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1564:
1562:Moving along...
1441:reliable source
1390:Cauchy sequence
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424:= 0. Therefore
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3920:Rudin pp.17-20
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3568:Infinitesimals
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2626:If you read a
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3939:Richman p.399
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2190:
2184:
2183:
2173:
2170:
2165:
2164:
2163:
2162:
2161:
2160:
2159:
2158:
2150:
2146:
2142:
2138:
2133:
2132:
2131:
2130:
2129:
2128:
2121:
2118:
2114:
2110:
2109:
2108:
2107:
2106:
2105:
2098:
2095:
2090:
2086:
2082:
2081:
2080:
2079:
2078:
2077:
2070:
2066:
2062:
2058:
2054:
2050:
2046:
2042:
2041:
2040:
2039:
2038:
2037:
2032:
2029:
2024:
2021:
2017:
2013:
2008:
2007:
2006:
2005:
2002:
1999:
1994:
1990:
1986:
1985:
1982:
1979:
1975:
1971:
1968:
1964:
1960:
1959:
1958:
1957:
1954:
1945:
1943:
1942:
1939:
1930:
1928:
1927:
1923:
1919:
1915:
1906:
1904:
1903:
1900:
1895:
1894:
1891:
1887:
1882:
1881:
1878:
1873:
1872:
1869:
1864:
1863:
1860:
1856:
1847:
1844:
1840:
1838:
1835:
1830:
1827:
1826:p-adic number
1823:
1822:
1821:
1818:
1814:
1811:
1807:
1803:
1799:
1798:ordered field
1795:
1794:
1793:
1792:
1789:
1781:
1775:
1772:
1768:
1766:
1763:
1759:
1757:
1754:
1749:
1748:
1746:
1742:
1741:
1740:
1739:
1736:
1731:
1729:
1725:
1721:
1715:
1712:
1711:
1708:
1699:
1698:
1697:
1691:
1681:
1678:
1674:
1673:
1672:
1667:
1663:
1655:
1651:
1650:
1649:
1646:
1642:
1638:
1637:
1636:
1631:
1627:
1618:
1617:
1616:
1615:
1612:
1604:
1602:
1601:
1598:
1592:
1591:
1588:
1583:
1575:
1573:
1568:
1561:
1543:
1540:
1536:
1534:
1531:
1527:
1526:
1525:
1524:
1523:
1522:
1521:
1520:
1519:
1518:
1517:
1516:
1515:
1514:
1513:
1512:
1497:
1494:
1490:
1489:
1488:
1487:
1486:
1485:
1484:
1483:
1482:
1481:
1480:
1479:
1478:
1477:
1464:
1461:
1457:
1454:
1452:
1449:
1445:
1442:
1438:
1435:
1431:
1430:
1429:
1428:
1427:
1426:
1425:
1424:
1423:
1422:
1421:
1420:
1409:
1406:
1402:
1399:
1395:
1391:
1387:
1383:
1379:
1378:
1377:
1376:
1375:
1374:
1373:
1372:
1371:
1370:
1361:
1358:
1354:
1350:
1346:
1345:
1344:
1343:
1342:
1341:
1340:
1339:
1332:
1329:
1325:
1321:
1320:
1319:
1318:
1317:
1316:
1311:
1308:
1303:
1302:
1301:
1300:
1297:
1294:
1290:
1287:
1286:
1285:
1283:
1279:
1275:
1274:undergraduate
1271:
1270:
1264:
1262:
1258:
1255:10 for every
1253:
1249:
1245:
1241:
1232:
1228:
1224:
1219:
1213:
1209:
1206:< 0.9, or
1205:
1201:
1197:
1191:
1189:
1185:
1181:
1177:
1172:
1170:
1162:
1159:
1158:
1155:
1152:
1147:
1144:
1141:
1136:
1132:
1127:
1123:
1118:
1112:
1108:
1104:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1071:
1067:
1063:
1059:
1055:
1051:
1047:
1043:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
1003:
999:
995:
991:
987:
983:
979:
975:
971:
968:
964:
960:
956:
952:
948:
944:
940:
936:
933:
929:
925:
921:
917:
913:
911:) "non-empty"
910:
906:
902:
898:
894:
890:
886:
885:
883:
879:
875:
871:
867:
863:
859:
855:
851:
847:
842:
838:
834:
830:
825:
821:
818:
815:
811:
807:
803:
798:
794:
791:
790:
788:
784:
780:
776:
775:
773:
772:
766:
765:infinitesimal
762:
758:
754:
750:
746:
742:
738:
734:
728:
724:
719:
715:
710:
706:
699:
693:
687:
680:
676:
672:
668:
667:
666:
665:
662:
657:
648:
644:
638:
634:
625:
624:
623:
622:
619:
615:
611:
603:
592:
589:
584:
583:
582:
581:
580:
579:
578:
577:
570:
567:
563:
558:
557:
556:
555:
554:
553:
546:
543:
539:
535:
531:
530:
529:
528:
527:
526:
519:
516:
512:
508:
504:
499:
498:
497:
496:
495:
494:
487:
484:
480:
476:
471:
467:
464:
460:
456:
453:
448:
444:
440:
435:
431:
427:
423:
419:
415:
411:
403:
399:
391:
388:less than 1,
379:
375:
371:
370:
368:
364:
363:
362:
361:
360:
359:
354:
351:
347:
343:
339:
338:
337:
336:
335:
330:
327:
323:
322:
314:
311:
308:
307:
306:
305:
302:
301:
300:
294:
293:
292:
290:
282:
281:
280:
274:
273:
272:
266:
265:
264:
258:
257:
256:
250:
249:
248:
242:
241:
240:
234:
233:
232:
226:
225:
224:
218:
217:
216:
210:
209:
208:
202:
201:
200:
194:
193:
192:
186:
185:
184:
178:
177:
176:
169:
168:
167:
161:
160:
159:
153:
152:
151:
145:
142:
140:
136:
131:
129:
124:
122:
118:
113:
112:
105:
99:
96:
93:
91:
88:
86:
83:
80:
76:
74:
71:
69:
66:
63:
61:
58:
57:
49:
45:
41:
40:
35:
28:
27:
19:
18:Talk:0.999...
3968:
3957:. Retrieved
3944:
3935:
3925:
3916:
3884:How's this?
3883:
3837:
3811:
3807:
3740:
3666:
3594:
3580:
3571:
3556:unprotection
3553:
3481:
3447:instead...?
3295:Arthur Rubin
3272:
3269:
3265:
3262:
3258:
3243:Arthur Rubin
3175:Arthur Rubin
3169:
3080:
3058:
3057:
3031:
3015:Arthur Rubin
3009:Third, both
2984:
2967:
2953:
2950:
2898:
2858:
2853:
2851:
2847:
2843:
2800:
2777:
2773:
2768:
2726:
2722:
2714:
2710:
2684:
2582:
2578:
2566:
2554:
2542:
2521:
2466:
2444:
2433:Arthur Rubin
2424:
2407:
2386:
2365:Arthur Rubin
2339:Arthur Rubin
2302:Arthur Rubin
2289:
2276:
2265:Arthur Rubin
2260:
2246:
2245:
2240:
2219:
2198:
2196:
2188:
2185:
2111:I suggested
2012:one year ago
1973:
1966:
1949:
1934:
1910:
1896:
1885:
1883:
1874:
1865:
1854:
1852:
1809:
1805:
1801:
1785:
1744:
1733:
1727:
1723:
1720:dual numbers
1717:
1713:
1703:
1695:
1608:
1593:
1584:
1576:
1571:
1569:
1565:
1386:Dedekind cut
1352:
1348:
1278:dense subset
1267:
1265:
1256:
1251:
1247:
1243:
1239:
1230:
1229:< 1, but
1226:
1222:
1220:
1211:
1207:
1203:
1199:
1195:
1192:
1187:
1183:
1180:Dedekind cut
1176:construction
1173:
1168:
1166:
1139:
1134:
1130:
1125:
1121:
1110:
1106:
1102:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1065:
1061:
1057:
1053:
1049:
1045:
1036:
1032:
1028:
1024:
1020:
1016:
1012:
1008:
1001:
997:
993:
989:
985:
981:
977:
973:
966:
962:
958:
954:
950:
946:
942:
938:
934:) "disjoint"
931:
927:
923:
919:
915:
908:
904:
900:
896:
892:
888:
881:
880:is called a
877:
873:
869:
865:
861:
857:
853:
849:
840:
836:
832:
828:
823:
819:
813:
809:
805:
801:
796:
792:
786:
782:
778:
769:
764:
760:
752:
744:
740:
736:
732:
726:
722:
717:
713:
708:
704:
697:
691:
685:
678:
674:
670:
655:
652:
607:
566:Silly rabbit
561:
542:Silly rabbit
537:
515:Silly rabbit
510:
506:
478:
474:
469:
462:
458:
451:
450:there is no
433:
429:
425:
421:
409:
401:
389:
384:, which is ⁄
380:. To bound ⁄
377:
373:
350:Raoul Harris
345:
341:
333:
310:Silly rabbit
298:
288:
286:
278:
270:
262:
254:
246:
238:
230:
222:
214:
206:
198:
190:
182:
174:
165:
157:
149:
132:
125:
114:
110:
109:
78:
43:
37:
3774:Any ideas?
3098:Game theory
2970:this revert
1605:Calc Thingy
1261:Archimedean
1242:. Defining
1202:< 0, or
1169:Order proof
848:called the
618:82.135.2.13
562:consequence
400:shows that
283:Yes, it is.
36:This is an
3959:2006-08-23
3651:Oh, I see
3011:User:KSmrq
2765:Discussion
2423:Change to
1907:Archive 08
1225:such that
1198:such that
610:1=0.999...
475:do include
446:technique.
267:Yes, I do.
121:2006-09-09
117:2006-06-29
98:Archive 15
90:Archive 11
85:Archive 10
2964:Reverting
2816:vandalism
1580:FA -: -->
1324:talk page
1178:. In the
854:lower cut
473:but that
441:Here the
398:induction
326:He Who Is
79:Archive 9
73:Archive 8
68:Archive 7
60:Archive 5
3904:contribs
3896:Supadawg
3886:Melchoir
3859:contribs
3851:Supadawg
3842:Melchoir
3824:contribs
3816:Supadawg
3799:Melchoir
3776:Melchoir
3762:contribs
3754:Supadawg
3744:Melchoir
3715:contribs
3707:Supadawg
3698:Melchoir
3688:contribs
3680:Supadawg
3669:Melchoir
3657:Melchoir
3638:Melchoir
3623:contribs
3615:Supadawg
3601:Melchoir
3583:Melchoir
3560:Melchoir
3542:Melchoir
3513:Melchoir
3504:Melchoir
3458:Melchoir
3449:Melchoir
3397:Melchoir
3389:Melchoir
3361:Melchoir
3351:Melchoir
3342:Melchoir
3340:effect.
3308:Melchoir
3291:semiring
3280:Melchoir
3213:Melchoir
3204:Melchoir
3064:Melchoir
3043:Melchoir
2974:Melchoir
2941:Melchoir
2859:0.999...
2844:Proof...
2808:Melchoir
2791:Melchoir
2778:separate
2751:Peter O.
2740:Melchoir
2719:0.999...
2698:contribs
2690:Supadawg
2652:Melchoir
2534:contribs
2526:Supadawg
2510:Melchoir
2476:Jarlaxle
2412:Melchoir
2391:Melchoir
2352:Melchoir
2257:0.999...
2229:Melchoir
2225:0.999...
2149:contribs
2141:Supadawg
2137:0.999...
2053:Chadbald
2028:Melchoir
1998:Chadbald
1978:Melchoir
1953:Chadbald
1938:Melchoir
1922:contribs
1914:Supadawg
1899:Melchoir
1890:Melchoir
1877:Melchoir
1868:Melchoir
1859:Melchoir
1843:Melchoir
1834:Melchoir
1788:Melchoir
1771:Melchoir
1762:Melchoir
1753:Melchoir
1735:Melchoir
1707:Melchoir
1677:Melchoir
1645:Melchoir
1611:Melchoir
1597:Melchoir
1587:Melchoir
1539:Melchoir
1530:Melchoir
1460:Melchoir
1448:Melchoir
1357:Melchoir
1307:Melchoir
761:infinite
661:Melchoir
593:Piepants
588:Piepants
507:equality
459:smallest
106:Untitled
3833:darker?
3435:Trystan
3274:fails."
2903:Niels Ø
2863:Niels Ø
2801:Comment
2774:Comment
2727:Support
2685:editing
2567:Support
2555:Support
2543:Support
2522:Support
2479:Artemis
2467:Support
2445:Support
2425:Support
2408:Support
2387:Support
2290:Comment
2277:Support
2089:decimal
1962:anyway.
1140:defined
756:theirs.
454:number.
452:largest
289:unequal
137:. (See
39:archive
3808:proves
3399:). --
3393:Arthur
3299:(talk)
3247:(talk)
3239:WP:3RR
3179:(talk)
3102:Conway
3059:popups
3019:(talk)
2731:Maelin
2437:(talk)
2369:(talk)
2361:P-adic
2343:(talk)
2306:(talk)
2269:(talk)
2255:, but
2247:Oppose
2237:Survey
2113:0.999…
2085:0.999…
2045:0.999…
1234:1 − (⁄
612:" or "
534:decide
3930:cut".
3241:. —
3032:asked
3004:08:27
2895:KSmrq
2852:Proof
2840:KSmrq
2804:KSmrq
2636:KSmrq
2393:. --
2337:. —
2300:. —
2294:-adic
2169:KSmrq
2057:ORish
1974:would
1654:TI-89
1493:KSmrq
1456:There
1405:KSmrq
1328:KSmrq
1293:KSmrq
1254:: -->
1233:: -->
1151:KSmrq
1088:with
1019:: -->
965:: -->
872:)×𝒫(
868:⟩∈𝒫(
850:upper
839:: -->
812:<
483:KSmrq
171:less.
16:<
3900:talk
3855:talk
3820:talk
3812:more
3758:talk
3711:talk
3684:talk
3619:talk
3531:talk
3524:Mets
3492:talk
3485:Mets
3423:talk
3416:Mets
3405:talk
3395:and
3200:this
3170:does
2921:talk
2899:only
2880:talk
2824:Huon
2755:Talk
2694:talk
2675:and
2661:talk
2530:talk
2498:talk
2491:Mets
2457:talk
2450:Mets
2416:Huon
2414:. --
2410:per
2399:talk
2389:per
2333:See
2325:talk
2261:more
2207:talk
2199:move
2145:talk
2065:talk
1918:talk
1666:talk
1659:Mets
1630:talk
1623:Mets
1129:and
1101:and
1064:and
1056:and
1048:and
1000:<
852:and
753:most
748:10.)
637:talk
630:Mets
503:Meno
470:that
432:) =
235:Yes.
219:Yes.
203:Yes.
195:Yes.
154:Yes.
119:and
3526:501
3487:501
3418:501
3297:|
3259:20.
3245:|
3177:|
3081:act
3017:|
2583:not
2571:pom
2493:501
2452:501
2435:|
2367:|
2341:|
2304:|
2267:|
2259:is
2203:JPD
2047:or
1661:501
1625:501
1218:).
856:of
827:= {
800:= {
779:Set
777:In
735:=0…
730:= ∑
632:501
538:may
479:use
141:.)
3952:.
3906:)
3902:•
3861:)
3857:•
3838:is
3826:)
3822:•
3764:)
3760:•
3717:)
3713:•
3690:)
3686:•
3625:)
3621:•
3558:.
3407:)
2985:un
2923:)
2882:)
2757:)
2723:IF
2715:is
2700:)
2696:•
2663:)
2536:)
2532:•
2401:)
2327:)
2298:-1
2223:→
2209:)
2201:.
2147:•
2067:)
1967:is
1924:)
1920:•
1396:,
1392:,
1388:,
1353:no
1349:my
1236:10
1216:10
1149:--
1105:=
1092:=
984:≡∃
949:≡∃
918:~(
843:},
835::
808::
463:is
406:10
394:10
386:10
382:10
123:.
94:→
64:←
3962:.
3898:(
3853:(
3818:(
3756:(
3709:(
3682:(
3617:(
3533:)
3529:(
3521:—
3494:)
3490:(
3425:)
3421:(
3403:(
3257:"
3156:1
3151:1
3129:1
3124:1
3108:.
2919:(
2878:(
2854:s
2753:(
2692:(
2659:(
2528:(
2500:)
2496:(
2472:⇒
2459:)
2455:(
2397:(
2323:(
2249:.
2205:(
2151:)
2143:(
2063:(
1916:(
1886:p
1855:p
1810:p
1806:p
1802:1
1745:k
1728:b
1726:+
1724:a
1668:)
1664:(
1632:)
1628:(
1620:—
1572:D
1257:n
1252:q
1248:r
1244:q
1240:n
1231:r
1227:r
1223:r
1212:r
1208:r
1204:r
1200:r
1196:r
1188:x
1184:x
1135:r
1131:L
1126:r
1122:U
1114:.
1111:r
1107:L
1103:L
1098:r
1094:U
1090:U
1086:R
1084:∈
1082:r
1078:L
1076:,
1074:U
1070:Q
1066:w
1062:v
1058:L
1054:U
1050:L
1046:U
1037:L
1035:ε
1033:w
1031:∨
1029:U
1027:ε
1025:v
1023:⊂
1021:w
1017:v
1015:(
1013:w
1011:∀
1009:v
1007:∀
1002:v
998:w
996:∧
994:U
992:ε
990:w
988:(
986:w
982:U
980:ε
978:v
976:(
974:v
972:∀
967:v
963:w
961:∧
959:L
957:ε
955:w
953:(
951:w
947:L
945:ε
943:v
941:(
939:v
937:∀
932:L
930:ε
928:v
926:∧
924:U
922:ε
920:v
916:v
914:∀
909:L
907:ε
905:w
903:∧
901:U
899:ε
897:v
895:(
893:w
891:∃
889:v
887:∃
878:Q
874:Q
870:Q
866:L
864:,
862:U
858:r
841:c
837:r
833:Q
831:∈
829:c
824:r
820:L
816:}
814:c
810:r
806:Q
804:∈
802:c
797:r
793:U
787:R
785:∈
783:r
745:i
741:a
737:n
733:i
727:n
723:r
718:n
714:r
709:n
705:r
701:3
698:a
695:2
692:a
689:1
686:a
684:.
682:0
679:a
656:U
639:)
635:(
511:R
434:U
430:S
428:(
426:U
422:x
410:n
402:x
390:x
378:x
374:x
50:.
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