Talk:0.999.../Arguments

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Comment This page is for mathematical arguments concerning 0.999... Previous discussions have been archived from the main talk page, which is now reserved for editorial discussions. Before posting, you may want to read the FAQ on Talk:0.999... and the following FAQ.
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Comment		 Frequently Asked Questions (FAQ)
Q: You guys talk a lot about real analysis, limits, and calculus; shouldn't this just be about arithmetic?
A: Unfortunately, in order to formally prove many qualities of numbers, one often has to resort to higher mathematics: real analysis in the case of real numbers, number theory in the case of integers, and so forth. The article and arguments page both aim to be understandable to all, but, since many skeptics ask for formal proofs, higher mathematics will inevitably come into play.
Q: Person X made a pretty convincing argument that 0.999... \ne 1! Those who say otherwise are giving arguments I can't understand.
A: Before believing an argument, check sources, responses, and record. The main article is well-sourced, whereas arguments against generally cite (if anything) non-mathematical sources such as online message boards and dictionaries. Also, although most people are trying to write something everyone can understand, some arguments, in relying on higher mathematics, will not be easy to follow for all. Still, try to follow those you can. Finally, those who firmly believe that mainstream mathematics is mistaken will generally reveal their lack of rigor and/or contempt for experts and others who disagree with them. Before replying, read their other contributions to make sure you aren't siding with someone you yourself would not trust.
Q: I have a mathematical question.
A: Please check the FAQ on the talk page.
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[edit] Publish the source!

Please show me where it is clearly published that an infinite sum is defined as the limit of an infinite sum and I will be the first of the unbelievers to support your article. None of your sources state this explicitly. They are all open to reader interpretation. Once again, you insist no original research - so practice what you preach. You are a laughing stock. Honestly, I warn everyone I know about reading anything on Wikipedia. You are sketchy and unreliable. Go ahead, do the right thing and publish a source - print every word as it appears in your source so that there is no doubt this article is not original research. 41.243.47.226 14:10, 1 December 2006 (UTC)

If you read 0.999...#Infinite series and sequences, your attention will be drawn to note 5, which reads "5. ^ For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31". Full bibliographic details are in 0.999...#References. I recommend trying Stewart first. Melchoir 16:38, 1 December 2006 (UTC)
You could also look at, for instance, this. CroydThoth 21:08, 1 December 2006 (UTC)
Just copying from the archive, there's also:
Advanced Calculus by David Widder, page 285, from 1947
Introduction to the Theory of Fourier's Series and Integrals by H.S. Carslaw, page 49, from 1930.
Advanced Calculus by Sokolnikoff, page 6-7, from 1939.
Calbaer 21:57, 1 December 2006 (UTC)
Sure, here's one from my first year Calculus textbook, Calculus: Concepts and Contexts by James Stewart, second edition (published 2001 by Thomson Learning Inc., ISBN 0-534-37718-1), page 574:
Definition: Given a series \textstyle \sum_{n=1}^\infty a_n = a_1 + a_2 + a_3 + \cdots, let sn denote its nth partial sum:
s_n = \sum_{i=1}^n a_i = a_1 + a_2 + \cdots + a_n
If the sequence {sn} is convergent and \textstyle \lim_{n \to \infty} s_n = s exists as a real number, then the series \textstyle \sum a_n is called convergent and we write
a_1 + a_2 + \cdots + a_n + \cdots = s or \sum_{n=1}^\infty a_n = s
The number s is called the sum of the series. If the sequence {sn} is divergent, then the series is called divergent.
Hope this helped! Maelin (Talk | Contribs) 08:40, 3 December 2006 (UTC)
There you go with those pesky facts again. Facts will never sway our favorite skeptics. 72.193.74.36 16:14, 3 December 2006 (UTC)


It is very unfair of you to call me a troll. I am your friend and hope that Wikipedia will someday be taken seriously. However, your rules state that no original research is allowed and information must have been published before it can be considered encyclopedic. How long must it have been published for? And by whom does it need to be blessed? The source referenced by Maelin (James Stewart, Calculus and Concepts) has not been around before 1960. What's so special about 1960? Well, a lot. Let me begin by stating that many 'mathematicians' born after this year have (shall we say) not been of a very good caliber in general. Most progress in Mathematics (including differential equations) has been made through the age old method of experimentation. Much experimentation and very little thought. But before I go off on a tangent, let me return to this article. Until 1980, I had never heard any professor of mathematics ever refer to a limit of an infinite sum (obviously a sum that converges) as its definition. There was always a clear distinction between partial sums (which approach the limit) and the actual sum of a series that was considered unattainable. Will I ever state in public that a sum is not defined as the limit of its partial sums? Of course not! Why? Not because it makes sense, but because the powers that be have decided to make it doctrine. Mathematics worked just fine for several hundred years without this ridiculous article and without any real analysis. Is real analysis bad? Well, of course not, but it is by no means beyond reproach and debate. Nothing is beyond debate - ever. You will retard the progress of many aspiring learners who cannot understand this rubbish because indeed that's exactly what it is. Your publication of this article supports the views of individuals such as Prof. Hardy, Kmsrq, etc who are all important contributors in your team. By publishing this article, you are solidifying the viewpoint of these and similar minded individuals. You constantly refer to academia who support your view but do you pay any attention to the hundreds of opposing views? It seems that anything becomes encyclopedic only if it is in line with your views. Repeated comments and posts have shown this article to be so full of holes. Yet you continue to maintain the stance that it is encyclopedic. What an irony! Encyclopedia is a Greek word literally meaning - 'encircling children'. The only real audience you have for such an article is the most foul and obnoxious individuals like Ksmrq and Prof. Hardy. Just because you cannot understand why 0.999... < 1 does not mean that it is untrue. Make sense? You cannot just decide upon a definition and then make it official doctrine. Why, even Newton did not dare publish knowledge just in case it was wrong and he might end up misleading many. Newton was not only one of the greatest mathematicians but no doubt the greatest scientist. Mathematicians today are so full of arrogance and obnoxiousness, they would not be able to recognize a 'truth' even if it were forced up their rear-ends! Back to the article: You discuss so many concepts that do not exist either in theory or practice. A good example is your so-called 'infinitesimal' - this is an oxymoron. Then you try to back your claims up by categorizing 0.999.... Only problem is you misplace this real number in your classification. It is in essence very similar to an irrational number.

The definition of rational number is a number of the form a/b (b not 0) and a,b integers. Has it crossed your minds that for the ancient Greeks a recurring decimal may not have been a rational number? Furthermore, when they designed arithmetic, there was no such concept as a limit. Indeed, how can any recurring decimal be a rational number unless it is actually possible to perform an infinite sum? So the Greeks not knowing anything about limits could not have had the same definition of rational as we have today. What I am saying is that just because a number might have a recurring pattern, this does not mean it is rational.

Once debate is dead, so is knowledge. This is why today so many graduates leave college even dumber than they were when they first started their education. Until there is no doubt about what an infinite sum is, this article should be deleted. You cannot just decide to define an infinite sum as a limit, it is already defined as an 'infinite sum'. 198.54.202.246 12:14, 5 December 2006 (UTC)

That's not the idea at all. An encyclopedia is a place to collect other people's research. There's no magic date something has to have been published and there's no one person or body who needs to "bless" it. It just needs to have been published somewhere, or even just reported upon someplace in the media. Things are reported as "fact" only when the research leaves no reasonable doubt. Even if 0.999... did not equal one (which is plainly does) it would still be wrong (by Wikipedia's policies at least) for Wikipedia to say so because that is not encyclopedic. That would be original research. Wikipedia is a website that can be edited by anyone, with or without any understanding of the topic, and therefore the only way it can ever hope to be respected is by consistently citing reputable sources. That way it can become a huge repository of knowledge. But it was never intended to be a source of knowledge, and that's an important disticntion.
And I keep being told that there are many mathematicians who dispute that 0.999...=1 just as I keep been told there are many scientists who reject evolution, and I have never met even one person from either category. I suspect this is because their numbers are hugely exaggerated.
Lastly, it is entirely reasonable for mathematicians to define terms like "sum" however they please. They're just words. They don't alter the underlying mathematics - you can invent another type of sum if you please, and coin your own term for it. But you can't dispute the ones we have because we've defined them quite precisely already.
Your little paragraph about rational numbers was interesting, though. I think it shed light on some of the problems here. If you'll indulge me, can I just spin the camera on this one and show you it from another angle? I think it would be a useful exercise for all of us...
Let's forget decimals for a moment. Pretend they haven't been invented yet. Let's just look at the maths. A rational number is defined as a/b (with caveats). Let's say a=1 and b=3. So, one third is a rational number. π, on the other hand, is not. Five is a rational number. e is not. The definition works perfectly, and every number we know of falls neatly into the rational or irrational category. There's nothing in there so far the Greeks couldn't have done, and I suspect nothing they didn't do. They were busy.
Now let's introduce decimal expansions. Remember, we've started from the maths and then brought in decimals. They're not numbers; they're not even things. They're just a convenient way of writing the numbers we just defined. Well, five is easy. That's "5". π is a bit trickier. We can Do the first bit, but we need infinite digits to get it right. But because it's irrational we couldn't write it in the old way either, so that's no great loss. The same goes for e — and it turns out that this is true of all irrational numbers. We have a handy second definition here, perhaps.
No. Hang on. One third is infinitely long in decimal too. But a closer look reveals that all of its infinite digits are threes. So let's say that a zero and a dot and an infinite string of threes (which we can just write as "3 reccurring") represents a third. Now we have a handy number system where we can write all rational numbers precisely. It's more writing than the old a/b method, but it's more obvious how large a number is just by looking at it. Let's keep this system and a/b, and from now on we'll use whichever system is more convenient at the time.
Later on, some mathematicians who had the added adantage of a few centuries of extra research formalise this system by introducing concepts such as the limit of an infinite sum. All I did there was start with maths and then simply use decimals as a way of writing that maths. Your "rational numbers" speech implied that decimals are the starting point and that if the Greeks couldn't write a number in decimal then they couldn't use it. We don't know what the exact value of π is, decimal or not, but that doesn't matter: we have a definition for it and we can still use it in algebra. We can even use numbers like i, that patently aren't real. Imaginary numbers can be used in mathematics. Why can't the Greeks use a number, just because they didn't know how to write it down?
What's so unlikely about that? If you look at the decimal system now, and think, "gosh, that's a complex system that involves a lot of things the ancient Greeks wouldn't understand. There's no way that could have come about before this century. There must be some mistake," then you may as well say the same thing about evolution. Only looking at the final step in the process evolution looks unlikely, but looking at each step on its own suddenly it's obvious. Looking at mathematics as it's now defined, yes, it certainly couldn't have been concocted by one Greek guy of an evening. But that's because that's not what happened. It was built up over centuries, and refined, and by now I think we've got a pretty good understanding of how the thing works, I'll thank you very much.
Andrew 14:14, 5 December 2006 (UTC)
Above there are references from the 1930s and 1940s which 198.54 chooses to ignore, still maintaining that no one thought infinite sums were defined as their limits before 1960. And he/she is simultaneously claims that the article should be deleted but "Once debate is dead, so is knowledge." Moreover, in this post he/she is declaring himself/herself our friend, yet the archives show numerous instances of personal attacks. And he/she wonders why he/she is called a troll.... Calbaer 17:35, 5 December 2006 (UTC)
So what's wrong with the sources listed above? 131.216.2.83 18:06, 5 December 2006 (UTC)
*Sigh* That's a long reply, Andrew. Surely you have better ways to spend your time than feed trolls? Apparently I don't either, for I made the effort to copy from the main atricle: "...proof (actually, that 10 equals 9.999...) appears as early as 1770 in Leonhard Euler's Elements of Algebra." Endomorphic 20:25, 5 December 2006 (UTC)
If everyone who posts here is a troll or feeding a troll, close the sodding page. It's arguing about a known and repeatedly proven fact, anyway -- clearly it's pointless at best. There's no Talk:France/Arguments. Considering how much stuff is deleted from Wikipedia every day for being "non-notable" this page is indefencible. Sometimes I think Wikipedia just deletes things not because it improves the service but because it considers itself "above" them somehow, but that's quite besides the point. Now, do you want a discussion, or do you just want a page for trolls to post on, safe in the knowledge that nobody will ever read it?
At least, this page should be renamed to something other than "arguments", because the fact is there is no argument. There are just some ignorant people.
On both sides. Andrew 00:27, 6 December 2006 (UTC)
Renaming the page has been suggested before :)
Also note that no less than 9 different sources (one quoted in full) have been provided in response to 189.54's initial "show me where it is clearly published" challenge. Still they disagree. Here's an excersise for the reader: Form a list of all claims made in the second 189.54 post. Then classify each claim as either mathematical, historical, cultural, or ad hominem. We can discuss any *mathematical* claims made. Endomorphic 01:48, 6 December 2006 (UTC)
We said that 198.54.202.246 is a troll (which is a fact), not that everyone who posts here is a troll. Both trolls and genuine questioners are bound to post their question somewhere, and it's better in this page than in Talk:0.999.... Genuines and those that are in doubt should be replied to; obvious trolls should not.
An argument is, by definition, an occasion where people are arguing. For better or worse, this is exactly what is happening here, so I don't really see how the name of the page is inappropriate. It doesn't "create" fruitless discussions, it just makes sure they are made here rather than someplace else. -- Meni Rosenfeld (talk) 07:25, 6 December 2006 (UTC)
Let me add a couple more sources:
"A Course in Pure Mathematics", G.H. Hardy, 1908. Ever heard of him? Also, please note that 1908<1960, though, I still don't understand why that matters.
"What is Mathematics", Courant and Robbins, 1941. 1941<1960.
72.193.74.36 16:33, 6 December 2006 (UTC)

[edit] Trolls

The person who has been trolling this article for the past year is now posting with IP addresses that begin with 41.243. Please do not respond to him. He knows that what he posts is nonsense, so trying to talk sense into him is completely futile. -- Meni Rosenfeld (talk) 20:51, 1 December 2006 (UTC)

Good point. I suppose I thought it was worthwhile to help show others that this was the case, but hopefully that should be self-evident without any effort on my part or that of others. But what is the most succinct way of revealing trolls without wasting time feeding them and/or resorting to personal attacks? Calbaer 21:01, 1 December 2006 (UTC)

Deleting their posts, of course. But only for really serious cases, like this particular troll. Anyone who thinks this is inappropriate, probably just hasn't seen enough of his "work". -- Meni Rosenfeld (talk) 21:32, 1 December 2006 (UTC)

  • I am curious how you are capable of reading this editor's mind. --jpgordon∇∆∇∆ 04:18, 2 December 2006 (UTC)

Occam's razor. -- Meni Rosenfeld (talk) 15:16, 2 December 2006 (UTC)

  • Well, there is that...but...I'm remembering when I was introduced to limits, epsilon proofs, and so on in my high school calculus class (back in the age when calculus really meant pebbles.) One of my significant "a-ha" moments was when limits "clicked" for me; it didn't make any sense, then suddenly it did...I'm also remembering a time a few years before that, when I was 11 or so, on a starry cold spring night in Denmark, discussing the stars with a friend, and suddenly "clicking" on the seeming infiniteness off the universe; my friend kinda freaked out, unable to grasp the concept of unendingness. I can easily see how someone could fail to have either of those epiphanies, but rather resist them, and thus be unable to understand simple things like "no, there's no infinity-th digit you can point to", and thus "no, there's no largest real number less than another real number." So for me, Occam suggests it's more likely that someone simply not comprehend a couple of key concepts than that they'd waste so much time and energy trying to drive a few very patient mathematicians to distraction. See Hanlon's razor. --jpgordon∇∆∇∆ 15:46, 2 December 2006 (UTC)
When it didn't make any sense, did you log on to Wikipedia and declare to the world that basic mathematics was broken? Endomorphic 20:28, 5 December 2006 (UTC)
I very well might have had it been available; I, of course, was the smartest boy in the world. --jpgordon∇∆∇∆ 15:34, 13 December 2006 (UTC)

This may explain some of the apparent trolls. But not this one. I'm not sure I'll be able to convey this efficiently with words, but he's not some first-year student who's having trouble understanding limits. He has proven to be familiar with a lot of advanced mathematical topics (some of which I myself am not familiar with), and has always carefully chosen words which will be as irritating as possible while maintaining the impression that he knows what he is talking about (which he does, incidentally, only that what he knows and what he posts don't match). This cannot be "explained by stupidity", so Hanlon's razor doesn't apply. But I beg you to take a look at the archives for posts to this page by 41.243, by his previous incarnation 198.54.202.54, and by his other IPs which you will have no trouble recognizing by his malicious style. I have no doubt that you will reach the same conclusion I did. -- Meni Rosenfeld (talk) 16:10, 2 December 2006 (UTC)

  • Malice in mathematics. I love it... Yeah, you're probably right. I never stop being amazed at what wankers will do for entertainment. --jpgordon∇∆∇∆ 16:21, 2 December 2006 (UTC)

[edit] Archive proposal

How about we mark this page as an archive, i.e. no longer used, and refer anyone with actual questions on this subject to Wikipedia:Reference desk/Mathematics? -- SCZenz 02:29, 6 December 2006 (UTC)

And then all the trolls will fill either Talk:0.999... or the reference desk with their garbage. I don't think that's a good option. -- Meni Rosenfeld (talk) 07:11, 6 December 2006 (UTC)
Per Wikipedia:Talk page guidelines, discussion not relevant to the article is subject to removal; arguing about the veracity of properly-cited facts is, in fact, irrelevant. The reference desk, I admit, could be a bit more problematic. -- SCZenz 07:16, 6 December 2006 (UTC)
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