Talk:0.999.../Arguments

MfD Result Notice

This page was the subject of an MfD discussion closed on 12 May 2007, with a keep result. Xoloz 18:36, 12 May 2007 (UTC)

1 does not equal .999 ... (moved from article talk page)

The fallacy, clearly, is that the notation ‘...’ upon which the proof relies, is intended to merely represent an undefined infinite sequence. It is a notation, not a factor. But it is used in this proof as an integral part of the value of the factor being compared to one, as though it was a defined mathematical quantity.

But the quantity ‘infinite’ is not specifically defined, (nor, I would argue, is it mathmatical). The result is nonsense. Would you like to have a cent for every decimal position in which the .9 has not yet reached the equivalent of one?

Look: If 1 = .999..., then 1.000... = 0.999... . Then subtracting 1 from both sides, 0.000...0 = -.000...1. Since .000...0 does not equal .000...1, the proposition is false. Professor Krepotkin (talk) —Preceding comment was added at 20:09, 13 June 2008 (UTC)

You might be interested in reading about finitism, it's a philosophy of maths that rejects anything infinite (I don't know if finitists generally reject infinite decimal expansions, though - someone else here will probably know). If you're going to reject the idea of having an infinite number of 9's, then 0.999... simply doesn't exist, so indeed it doesn't equal 1. However, if you do that, decimal expansions become pretty limited. If you limit yourself to terminating decimals, you can only describe numbers of the form ${\frac {a}{10^{n}}}$ for some integers a and n, which almost all numbers aren't (you're restricting yourself to a very small subset of the rational numbers). The expression "0.000...1" is meaningless even if you do accept an infinite number of digits - you can't have a 1 at the end, since an infinite sequence doesn't have an end (at least, not without getting into transfinite ordinals, which we don't want to do!). 1 - 0.999... is simply zero. There isn't a non-zero number infinitely close to zero, which is what you are trying to describe. We call such numbers "infinitesimals", and the real numbers don't contain any (see Archimedean property for the technical details). --Tango (talk) 20:31, 13 June 2008 (UTC)

Tango, you are overestimating the importance of almost all numbers. Most numbers are just as interesting as average novel written by one of the monkeys on typewriter. Almost all numbers belong to Poincaré's teratologic museum. Finite decimal expansions are limited in a good way. Tlepp (talk) 09:09, 14 June 2008 (UTC)

Sure, almost all numbers are never going to be used by anyone, but the numbers which can be written as finite decimal expansions are an extremely small set of numbers, far smaller than is required by the term "almost all". They are a tiny subset of the rational numbers. Even if you don't care about irrational numbers, can you really argue that you don't need all rational numbers? --Tango (talk) 17:42, 14 June 2008 (UTC)

I don't need infinite decimal expansions for rational numbers. When I think about two thirds, I prefer 2/3 over '0.666...'. When I want to discuss roots of $x^{2}-2$, I prefer $\pm {\sqrt {2}}$ over '1.414...'. For the circumference of a unit circle it's better to write $2\pi$ than '6.283...'. Three dots notation is rarely the best notation. How many different ways are there to assign 2.999... pigeons into 4.999... pigeonholes? Tlepp (talk) 18:15, 14 June 2008 (UTC)

I believe you said that Tango was "overestimating the importance of almost all numbers", not the decimal expansion of numbers. You appeared to have suggested that such numbers as $\pm {\sqrt {2}}$ and 2/3 are actually unimportant, however you notate them. In other words, you have started, unintentionally, perhaps, a semantical battle. Perhaps we should put that to rest before it gets to be a flaming war? ;) I will attempt to address what appears to me to be the actual issue at stake.

It is unavoidable that, in order to describe the real number set, an infinite process must occur. Some such infinite representations of the real numbers are Cauchy sequences and Dedekind cuts. Decimal expansions are another representation of the real numbers (see Construction of real numbers). If you don't accept the existence or usefulness of infinite decimal expansions, then you can hardly suggest that any of these other representations are much better, and by extension, you can't accept the existence of the real number set. In short, you shouldn't throw away infinite processes so quickly. --70.124.85.24 (talk) 19:48, 14 June 2008 (UTC)

In other words, you want to just ditch decimal expansions altogether? The good thing about decimal expansions is that, while they are probably never the best way to write any given real number, they can be used to write *any* real number. It's useful to have one way to write everything, since it allows direct comparisons. For example, which is bigger, ${\sqrt {10}}$ or $\pi$? It's difficult to tell without giving it some thought. Now, which is bigger, 3.162278... or 3.141593...? Now, it's easy. --Tango (talk) 21:44, 14 June 2008 (UTC)

Which is bigger, ${\sqrt {10}}$ or $\pi$? ${\sqrt {10}}\approx 3.16>3.14\approx \pi$. No need for three dots. I agree equalities look prettier than approximations, but the added beauty is an illusion with little practical value.

Important numbers are those that can be described with a finite number of characters, and it's a countable set. The real number set is uncountable. Almost all numbers (99.999...%) in our favorite number system are anonymous and can't be described.

99.999...% = 100% $\neq$ all

This is one of the few situations were '99.999...' seems better that shorter '100'. Somehow '99.999...%' notation seems to say a little more than it really does. Since 99.999...% equals 100% exactly, there's is no hint whatsoever that some meager set of numbers is not included in the 99.999...%. Tlepp (talk) 07:53, 15 June 2008 (UTC)

You need a better notation - $\approx$ doesn't tell you how close it is. If it's +/- 10%, say, then the inequality doesn't follow. You need something to specify that it's +/- 5 in the next decimal place, that isn't a standard meaning of $\approx$, to my knowledge. You also need a better notation than 99.999...%, it's better to just say almost all. It has a precise meaning that can't really be expressed in terms of percentages. 99.999...% of real numbers are not between 0 and 1, but not almost all. --Tango (talk) 14:37, 15 June 2008 (UTC)

Careless, careless percentages. Firstly, 100% = Almost all =/= all, infinite repeating nines never entering into it. Secondly, what kind of probablility measure are you using to talk about percentages of real numbers? Thirdly, let's forget about the utility of various sets of measure zero since the 'Professor' doesn't discuss anything more than a misunderstanding of simple decimal notation. Endomorphic (talk) 13:35, 16 June 2008 (UTC)

Not quite. We are not discussing probability measures, but just measures (the Lebesgue measure, say). 100% (or just a proportion of 1) would mean, if anything (though I haven't seen it written), ${\frac {\mu (A^{c})}{\mu (X)}}=0$. Thus $[0,1]^{c}$ contains 100% of the real numbers. Almost all means $\mu (A^{c})=0$. Thus $[0,1]^{c}$ does not contain almost all real numbers. -- Meni Rosenfeld (talk) 13:55, 16 June 2008 (UTC)

I take that back - specifically in real numbers, the most natural meaning of 100% would be $\lim _{x\to \infty }{\frac {\mu ([-x,x]\cap A)}{\mu ([-x,x])}}=1$. The rest remains unchanged. -- Meni Rosenfeld (talk) 15:16, 16 June 2008 (UTC)

You're right, it is a careless use of percentages - I explicitly said the percentages can't really be used to describe these things. But, if you really want to use them, then any reasonable way of doing so wouldn't work the way Tlepp wants it to. --Tango (talk) 15:04, 16 June 2008 (UTC)

The discussions almost all seem to be centered around mathematical theorems and proofs, yet whether 0.999... is 1 or not could be argued to be fundamental, and, as such, could not be proven mathematically, but dictated by logic/philosophy. If logic/philosophy dictates that 0.999... is not equal to one, then any theorem that 'proves' it does is wrong, rather than vice-versa. In terms of logic and philosophy, 0.999... can be very strongly argued to be distinct from 1. However, it is important to remember that mathematics does not really exist; it is a human construction to be a useful tool. As such, it may be useful to define 0.999... as being equal to one, even if it is distinct in terms of logic/philosophy. The sum of an infinite series is defined as its limit; it is useful for that to be so, and it is, for practical purposes, entirely accurate, even if it is philosophical incorrect. Hence the argument over whether it is philosophically correct or not is entirely mute. Captain Griffen (talk) 12:37, 20 July 2008 (UTC)

What alternative definition would you give for 0.999..., then? Oli Filth^(talk) 12:57, 20 July 2008 (UTC)

It's not about definitions of 0.999..., but whether infinitely close is the same as identical. Logically, it isn't, but practically, it is. Maths goes with the practical definition here, but that's arbitary, and doesn't mean that it applies outside of maths, nor does it mean that 0.999... == 1, only that they are taken as being the same. My point is that this debate cannot be argued from a mathematical stand point, as it is to do with something fundamental (ie: is infinitely close the same as identical). Captain Griffen (talk) 13:25, 20 July 2008 (UTC)

The question of whether 0.999... = 1 depends entirely on how one defines the symbol 0.999.... Without a definition, the question is meaningless (in the same way that asking whether k?37##!8 = 1 is meaningless). If one chooses to define the symbol 0.999... as the limit of the infinite series (i.e. the standard definition), then by definition, it will equal 1. If one chooses to apply an alternative definition, then a different result may arise (albeit a non-standard one). However, without such an alternative definition, this direction of debate is all but moot. Oli Filth^(talk) 16:08, 20 July 2008 (UTC)

All philosophical questions are meaningless. /sarcasm

Gian-Carlo Rota has criticized analytic philosophy with following words:

Whereas mathematics starts with a definition, philosophy ends with a definition. A Clear statement of what it is we are talking about is not only missing in philosophy, such a statement would be the instant end of all philosophy. - - The prejudice that a concept must be precisely defined in order to be meaningful, or that an argument must be precisely stated in order to make sense, is one of the most insidious of the twentieth century.

Captain Griffen's point should be interpreted in philosophical context. Tlepp (talk) 20:41, 20 July 2008 (UTC)

That might be relevant if we didn't already have a clear definition! If we go back to philosophy, then according to you, we work toward a definition, but we already have one! Why go backward? And more pertinently, where would we go back to? Oli Filth^(talk) 20:57, 20 July 2008 (UTC)

Why do math students bother with (verified) proofs? Why don't we just learn the theorems and formulas? Tlepp (talk) 21:20, 20 July 2008 (UTC)

No one I've ever spoken to has ever taken '0.999...' as anything except 0.9 recurring, which even the wiki article agrees with. 0.999... may be taken in maths generally to be the limit, but that does not mean it is defined as that. Hence it can be broken down to whether the infinitely small gap between 0.9 recurring and 1 should be considered, given it is infinitely small; this is to do with the philosophy or fundamentals of maths, and cannot be resolved by mathematical theorems, since mathematical theorems must be based on those same principles or fundamentals. If a theorem disagrees with the underlying philosophy, then the theorem is falsified - not vice versa. Mathematics is a tool, however, and so assumptions are made where they are useful (eg: 0.9 recurring being equal to 1, even if it is a philosophically contentious arguement). The reasoning behind 0.9 recurring being one in much of this debate is flawed for these reasons. Captain Griffen (talk) 21:13, 20 July 2008 (UTC)

According to the Wikipedia article on this, 0.999... is the same as 0.9 recurring, ie: 0.999999 to infinity. That disagrees with your assertion of what the standard definition is (in my work with mathematics, I've never been told that is the definition of '...', and it would be in direct contrast with the use in normal English). It is taken as being the limit, and it is for all practical purposes, but it is not [i]defined[/i] as being the limit, in my view (if it is, then that wiki article needs updating, as it is alledging 0.999... is synonomous with 0.9 recurring, which you are saying is not the case; 0.9 recurring is not the same as the limit of 0.9 recurring). Captain Griffen (talk) 20:24, 20 July 2008 (UTC)

The article is fine. 0.999... is the same as 0.9 recurring, and 0.9 recurring means by definition the infinite sum 0.9 + 0.09 + 0.009 + ..., which by definition means the limit of the sequence of finite partial sums 0.9, 0.99, 0.999 ... . I don't know what you mean by "0.9 recurring is not the same as the limit of 0.9 recurring" - the "limit of 0.9 recurring" is a meaningless statement AFAICS. Mdwh (talk) 00:01, 22 July 2008 (UTC)

You're right, it is a meaningless statement. I believe the article discusses the misconception that 0.999... refers to a process ("It gets closer and closer to one but never gets there" and similar). It is not, it is the limit of a process. --Tango (talk) 00:51, 22 July 2008 (UTC)

It's not really a matter of defining 0.999... to equal one, it's about defining the real numbers. If you take the standard definition of the real numbers (which turns out to be an extremely useful definition for all kinds of things, hence it being standard), then any reasonable definition of 0.999... as a real number will be equal to one. You either have to not have 0.999... as a real number (so it's basically just a meaningless sequence of symbols), or you have to change the definition of a real number (which would make maths useless for describing much of the real world). --Tango (talk) 17:14, 20 July 2008 (UTC)

The definition of real numbers (a useful tool) used in maths has no relevence to the actuality of it. What I'm saying is that arguing from maths about the basis of maths doesn't make any sense. Captain Griffen (talk) 20:24, 20 July 2008 (UTC)

I will give Captain Griffen and Tlepp this much: Yes, it's true that there are nontrivial philosophical issues involved in saying that the real numbers are the "correct" realization of the intuition of continuity, and it's true that the resolution of these issues is not simply a matter of mathematical proof in the ordinary sense, nor about stipulating a formal definition and seeing where it leads. It's also true that you don't have to have a precise definition before you can discuss something.

However: These issues have been resolved; there is a broad consensus to take the reals as, by default, the structure that realizes the intuition of a continuous line. This is for good reason, though the reasons are not as easily explained as the proofs in the article. The resolution is not the of sort of once-and-for-all definitive type that you get from a mathematical proof; in principle the question could be revisited if some genuinely new way of looking at things came to light. There is, however, no evidence of that here. And while you don't need a precise definition before you start, the continued failure to find a relevant way of making the doubters' ideas precise (or even minimally clear) is pretty strong evidence that they are not likely to be fruitful. --Trovatore (talk) 23:43, 20 July 2008 (UTC)

(Aside) When I wrote the above, I was under the impression that Tlepp was arguing on Captain Griffen's side. A more careful reading shows this is not true. --Trovatore (talk) 00:12, 21 July 2008 (UTC)

You almost seem to be agreeing that the way this is dealt with is not down to mathematical proof, but convention based on what is useful (which is my point), based on a contentious logical point (mainly that infinitely small = 0). This is useful, as it allows stuff like digit manipulation without maths breaking apart. (For example, the digit manipulation 'proof' assumes a 1:1 infinite-digit mapping after the decimal point, which isn't necessarily valid; without such an assumption the result could be used to argue subtraction of infinite length don't have any real meaning.) I agree in mathematics we should take 0.999... to be equal to 1, but purely because it is useful and as part of other assumptions made in maths, rather than any underlying logical reason. The same digit manipulation 'proof' can be used as good evidence of techniques which become valid thanks to that assumption. Captain Griffen (talk) 10:04, 21 July 2008 (UTC)

Wrong. Whether there are infinitely small non-zero numbers (i.e., infinitesimals) or not is simply irrelevant. In at least the common theories of non-standard reals, we have

• There exist numbers x such that 0 < x < 1/n for any natural number n.
• 0.999... = 1.

You're off on a red herring. The fact that there are no infinitely small non-zero distances in the standard reals is irrelevant to this proof, since there are theories in which such infinitesimals exist and 0.999... = 1.

To be sure, the section on infinitesimals isn't so clearly written. A few non-standard systems are mentioned, but for the most part, the article does not say whether 0.999... = 1 in these systems. Perhaps someone who knows this stuff better than me can fix that section? Phiwum (talk) 12:08, 21 July 2008 (UTC)

I'm not arguing that 0.999... isn't = 1, but that the arguments used to 'prove' it are erroneous when based purely on mathematical theorems. The article is fine to start with, particularly clear with "As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is the real number which is the limit of the convergent sequence (0.9, 0.99, 0.999, 0.9999, …)." - it is clear it is [i]assigned[/i] the value one as the limit by convention. Other parts, particularly in the 'proofs' are more confused on this (many of which are fatally flawed as proofs, particularly the 1/3 and digit manipulation based 'proofs', but others as well, given that they rely on assumptions that may or may not be considered valid). Speaking of the 1/3 based one, why is it even there, given that it assumes 1/3 == 0.333..., which is an assumption denied by anyone who does not accept 0.999... == 1. Captain Griffen (talk) 15:15, 21 July 2008 (UTC)

That's not a problem with 0.999..., it's a problem with all infinite decimals. An infinite decimal is defined as the limit of the sequence, it's the only reasonable definition. You could define it some other way, but you wouldn't have a useful notation (it wouldn't represent a real number). It's all just notation, so you're right that there is no underlying logic to it. The notation is defined to mean what it's most useful for it to mean, in this case, that results in 0.999... being an alternative notation for one. --Tango (talk) 16:00, 21 July 2008 (UTC)

I'd agree with all of that, except rather than the only reasonable [i]definition[/i], I'd say the only reasonable way to handle it (the difference in that being 3! is defined as 3 times 2 times 1, but has a value of 6; six can replace 3!, but 3! is not defined as 6). The arguments have mostly tended to revolve around mathematical theorems rather than arguing from maths-as-a-tool, including the wiki article, which, I'd say, is less effective in informing, and less accurate in explaining and showing the reasons behind it. Captain Griffen (talk) 23:27, 21 July 2008 (UTC)

Captain Griffen, you may be surprised just how many mathematical novices accept that 0.333... = 1/3 and doubt that 0.999... = 1, at least until they are shown that the former implies the latter. Of course, one cannot consistently believe that 0.333... = 1/3 and 0.999... = 1 (along with other basic facts about addition), but the fact is that, for whatever reason, the former is often intuitively regarded as less controversial than the latter. Phiwum (talk) 17:40, 21 July 2008 (UTC)

I agree with Phiwum (a few posts above) that the article is sadly weak in its coverage of decimal expansions in non-Archimdean fields. Back in the day, I did some ad-hoc browsing in a couple research libraries and Internet searches, but I couldn't find anything more solid than what you see. I'm not an expert in that area, so it's entirely possible that I just didn't know where to look. You might want to bring this up on the main talk page, where it might get more positive attention. Melchoir (talk) 22:27, 21 July 2008 (UTC)

I don't know of any other number system that uses decimal expansions, at least not routinely. You can use a similar system to describe p-adic numbers, but I don't think 0.999... would have any meaning there, I believe you can only have infinite expansions to the left, not the right, with p-adic numbers (the opposite of with real numbers). --Tango (talk) 00:51, 22 July 2008 (UTC)

The need for infinitesimals in the real world

THis discussion has an irrational fear of infinitesimals. But it is easy to come up with real world situations where infinitesimals are necessary. Consider the simple example of a tire (circle) on a road (line) In the real world, two solid objects cannot occupy the same space. So if the surface of the road is point 0, then how would you describe the surface of the tire at it's bottom point? The tire's vertical volume would be x>0, but if there is no "lowest value" to that equation, then the tire has no bottom edge - clearly an absurd failure to describe the situation. Thus values like .000...1 and .999... must exist in the real world, if not in real numbers. Algr (talk) 21:30, 13 June 2008 (UTC)

Considering solid objects is not really helpful here. In the real world, at the sort of microscales relevant to this article, the atoms of both the tire and the road are not at all "solid", and are best described by wave functions, and not only is interpenetration possible but the whole idea of "bottom edge" is also meaningless. Indeed, as the scale reduces further below the Planck distance, even ideas like time and space become difficult to reason about, and the actual nature of physical reality at those scales -- including whether or not spacetime is actually a continuum -- is as yet unknown. And all of this may happen no more than 30 or 40 places to the right of the decimal place: only a tiny way along the digits of 0.999...

Fortunately, none of this matters when we are discussing the reals, which we can reason about regardless of whether or not they actually describe physical reality at those scales. -- The Anome (talk) 22:00, 13 June 2008 (UTC)

If the real numbers have no connection to the real world, then why are we talking about them? Why not debate Star Trek transporters or which is the first Pokimon? In your effort to define yourself into correctness you've turned the whole Real Set into a useless fantasy that can't even deal with something as simple as a tire on a road. Algr (talk) 23:28, 13 June 2008 (UTC)

(Pure) Mathematics does not attempt to describe the real world. Scientists use mathematics to describe the real world, and that's why mathematicians get funding, but it's not why they come up with the maths. They do it for the sake of knowledge and beauty - it's all abstract. We don't need a real world application in order to consider it worthwhile to talk about maths. That said, the real numbers are very useful for describing the distance between a tire and a road - that distance is a non-zero finite real number. The electrons in the tire and the electrons in the road repel each other, so they are always separated by at least a small amount. If you want to ignore that aspect of the real world and consider an idealised tire and road, then you always need to ignore the bit about two objects not being able to occupy the same space, and the distance is just zero. --Tango (talk) 23:53, 13 June 2008 (UTC)

When the tire's lowest point is 0, the intersection of circle and line is a single point, which has volume 0, so there's no volume shared by tire and road. If the tire's lowest point were some infinitesimal, which infinitesimal would it be? What's at the point in between 0 and the tire's lowest point? Would there be an infinitesimal air-filled space between the tire and the road? Would they actually touch? Huon (talk) 00:12, 14 June 2008 (UTC)

The tire in your example doesn't "touch" the road. There is an electromagnetic field separating them. The distance is measurable. Tparameter (talk) 00:22, 14 June 2008 (UTC)

If one is more interested in philosophy than mathematics, http://plato.stanford.edu/entries/boundary/ is a better starting place than this article. Melchoir (talk) 00:46, 14 June 2008 (UTC)

Absolutely. Physical objects are not describable as point sets, made of continuous smooth "stuff" with clear boundaries, and it's not clear how to define these boundaries even if you could do so, so all arguments relating infinitesimals to physical objects are unfortunately irrelevant. Arguments based on common-sense understandings of reality sadly evaporate in the face of the weirdness of actual physical reality. Different physical regimes apply at different scales and energies, and since there is no practical reason to develop intuitive understandings of the universe as it behaves at large or small energies or scales (they're either irrelevant to ordinary life, inaccessible, or you're dead from heat/cold/vacuum etc.) evolutionary selection processes didn't bother to equip us with the necessary intuitions. -- The Anome (talk) 13:29, 14 June 2008 (UTC)

Off-topic but somehow related: intuitive vs formal continuity http://www.maa.org/devlin/devlin_11_06.html Tlepp (talk) 16:46, 14 June 2008 (UTC)

Nice article - thanks for the link. Now I don't feel quite so bad about not getting epsilon-delta proofs of continuity the first time they were explained to me! --Tango (talk) 17:50, 14 June 2008 (UTC)

Odd. I don't recall having any trouble with this back in the day. I guess either I have suppressed the memories of the difficult struggle, or they were better explained to me. I do agree, though, that the idea that "a continuous function is one that can be drawn with a single stroke of a quill" is a blatant lie - the Weierstrass function cannot be so drawn. -- Meni Rosenfeld (talk) 20:11, 14 June 2008 (UTC)

Better explained may well be it - I didn't get it when the lecturer went through it, but once my tutor did an example explaining it slightly differently, it all clicked into place (on second thoughts... I wonder if I was entirely paying attention the first time...). It gets worse when you get into general topology, though - "the pre-image of an open set is open". What on Earth has that got to do with anything? It works, and does everything you want of it, but it doesn't bear any relation to drawing lines on a pieces of paper without lifting the pen... For a start, it's talking about the inverse of the function, rather than the function itself, which is just weird... I think you just have to accept it as an arbitrary definition and get on with it... --Tango (talk) 21:31, 14 June 2008 (UTC)

Somewhat amusing to see point-set topology applied to the PA/MD border. But in any case the article seems to have made one fairly elementary topological blunder in claiming that the second realist theory requires one of the touching bodies to be topologically open and one of them closed -- why shouldn't the boundary be partitioned between them in some much more complicated way? --Trovatore (talk) 03:42, 14 June 2008 (UTC)

In Prof. Varzi's defense, he does say "These theories ... need not be exhaustive and can be further articulated...". It may simply be that partitioning the boundary is an unpopular option. And I can see a possible reason why: to partition a line in a fair and uniform manner you'd need to introduce non-measurable sets. For me, this is not much better than saying that a given boundary point might be assigned to one body or to the other, but we have no way of deciding which! And if you're going to allow that kind of indeterminacy then you might as well do away with the partition and let the whole line lie in limbo, at which point we've arrived full circle back at option 2.1.2 and therefore throw our hands up in the air and go get drunk. Melchoir (talk) 05:07, 14 June 2008 (UTC)

Actually I just thought of another argument against partitions: they're no use if the boundary in question is a single point. Which state does the center of the Four Corners Monument belong to?

(Nothing personal; it's obviously an interesting question...) Melchoir (talk) 05:24, 14 June 2008 (UTC)

Well, one of these things is not like the other. I've lived in New Mexico, I've lived in Colorado, I've lived in Arizona, but I've never lived in Utah. So obviously it belongs to Utah. --Trovatore (talk) 08:13, 14 June 2008 (UTC)

Ah yes, Brentano's classic 1903 Utahn Non-Residential Symmetry Breaking Argument. Melchoir (talk) 19:08, 14 June 2008 (UTC)

But what seperates the interior from it's own boundry? Endomorphic (talk) 14:26, 16 June 2008 (UTC)

Nothing. You can get arbitrarily close to the boundary while remaining in the interior. There is, however, a difference between "arbitrarily close" and "infinitesimally close". Any point in the interior is a finite distance from the boundary. --Tango (talk) 15:01, 16 June 2008 (UTC)

That's the mathematics of it, yes. But the link discusses the philosophy, a boundary as an ontological neccessity coming between distinct objects. Boundaries are things that seperate things; but if a boundary is a thing then there must be some thing seperating the boundary from each thing it seperates. And so on. /end{Devil's Advocate} Endomorphic (talk) 15:46, 16 June 2008 (UTC)

THis discussion has an irrational fear of modulo arithmetic. But it is easy to come up with real world situations where modulo arithmetic is necessary. Consider the simple example of a tire (circle) on a road (line) In the real world, when the wheel has rotated one whole revolution, it's in exactly the same orientation as when it had not rotated at all. Thus equations like 1=0 must hold in the real world, if not in the real numbers. Endomorphic (talk) 14:26, 16 June 2008 (UTC)

What does modulo arithmetic have to do with anything? Different situations in the real world need to be modelled using different mathematical objects. Distances are generally modelled using normal real numbers, angles are modelled using real numbers modulo 2*pi. That different things need different models doesn't make the models wrong. --Tango (talk) 15:01, 16 June 2008 (UTC)

Endomorphic's post is a parody of Algr's, designed to deliver roughly this message. -- Meni Rosenfeld (talk) 15:10, 16 June 2008 (UTC)

We've had this discussion before. 1=0 is correct if your variable is exclusively measuring the angle of the tire; Like Tango says; different things need different models. Hyperreals are inevitable as soon as you have inclusive and exclusive ranges. Algr (talk) 01:40, 14 July 2008 (UTC)

What would you model using hyperreals? Please give a specific example, including explicitly what hyperreal you would use. --Tango (talk) 01:57, 14 July 2008 (UTC)

Gee, I dunno, how about a tire on a road? Has anyone brought that example up? Algr (talk) 03:50, 14 July 2008 (UTC)

I asked for an explicit hyperreal that you would use for that scenario. --Tango (talk) 16:29, 14 July 2008 (UTC)

Hyperreals won't really solve ontological or epistemological problems of border. Hyperreal numbers are points on the hyperreal line and point is completely characterized by it's location. No two points can occupy same location. Two distinct hyperreal numbers can be infinitely close to each other, yet there's always a positive distance between them. If the border of a road and a tire isn't allowed to intersect, we can make their distance smaller than any positive rational number, but we can't make it zero. Thus the tire is not on the road, it is flying above the road. If an infinitesimally small bug is sitting on the road, you can drive(=fly) over it, and it won't get crushed. Tlepp (talk) 07:32, 14 July 2008 (UTC)

Tlepp is right, of course. To be a little more explicit: At the top of this section, you claimed the reals were not sufficient to describe a tire on a road. Let me quote the relevant section:

In the real world, two solid objects cannot occupy the same space. So if the surface of the road is point 0, then how would you describe the surface of the tire at it's bottom point? The tire's vertical volume would be x>0, but if there is no "lowest value" to that equation, then the tire has no bottom edge - clearly an absurd failure to describe the situation.

So let's try to describe the situation with hyperreal numbers. Again the surface of the road is point 0, which happens to be a well-defined hyprerreal number. But as with the real numbers, the hyperreals don't offer a "lowest value" to the (in)equation x>0, and in the hyperreals, the tire still has no bottom edge if you want it to be the smallest solution to x>0. What has been gained by using the hyperreals? Nothing. Huon (talk) 12:35, 14 July 2008 (UTC)

Well, to be fair, his point wasn't that the tire be 'as close as possible' to the road, but just 'infinitely close'. But it's still a ridiculous example, since, in the real world, objects are never infinitely close (at least not in the everyday world. I believe it's been suggested that inside a black hole, all particles are literally occupying the same point in space, which constitutes being infinitely close together). --69.91.95.139 (talk) 14:02, 14 July 2008 (UTC)

This may be a misunderstanding. Tlepp talks about things being infinitely close, Algr explicitly speaks of the "lowest value" to x>0, and claims that hyperreals would somehow help. They don't. Huon (talk) 15:26, 14 July 2008 (UTC)

Sorry, you're right. And I was referring to Algr's example when I said 'his point'. Incorrectly so, I'll admit. --69.91.95.139 (talk) 15:42, 14 July 2008 (UTC)

2

"Any point in the interior is a finite distance from the boundary." - Does this mean that a point on the boundary is not within either area? In other words, any point on a state border is not within any state, even though it is within the interior of the USA? Algr (talk) 09:45, 18 August 2008 (UTC)

In mathematics, a point on the boundary is not in the interior of either the set or its complement. It may (or may not) be an element of the set, but the interior of a set is characterized by each element having an open neighborhood entirely within that set. I doubt that point-set topology is the best model of real-world geography and politics, though - where sets of measure zero (such as boundaries) belong to is rather irrelevant in those contexts. Huon (talk) 11:49, 18 August 2008 (UTC)

If you're feeling up to it, you're welcome to take a crack at my argument in regards to "actual infinite sum" vs "limit of a sum to infinity", found at Talk:0.999...#Alternative_proof, and reposted here. I really look forward to finding out what you have to say about this.

In regards to what the underlying definitions of real numbers are supposed to represent conceptually and intuitively:

Ok, let's start over. Your first sentence was:

Both the ${\sqrt {2}}$ and the .333... proofs fail because they both make unjustified assumptions about what happens when a process is repeated infinitely.

So, can you explain how exactly a justified assumption about infinite processes can be made? If so, then please do tell. Otherwise, you're just going to have to face the facts: in mathematics, infinity is a key concept, and sometimes, assumptions about infinity are necessary.

In answer to your entire paragraph, however, the definition of real numbers is not the result of "repeating a process infinitely" (and therefore your argument about repeating a process infinitely is irrelevant); common definitions follow one of two approaches to defining the reals: A) as a limit (Cauchy Sequences) or B) as a gap (Dedekind Cuts). If you considered each term in a Cauchy Sequence and repeated the process to infinity, of course we have no idea what you would ultimately get. However, that is not what Cauchy Sequences represent. It is not what we get after the infinite process (which is more a matter of philosophical speculation than anything else) that they represent, but what we get when we take the limit to infinity - a well defined mathematical concept, out of which ultimately pops your disputed Archimedean property. To see how this applies to your argument above, notice that 0.333... is actually a Cauchy Sequence.

The Dedekind cut is similar, though perhaps a little easier to wrap one's mind around; none of the numbers in the cut actually represents the real number; it is the gap left by those numbers that represent a real.

I'm sure you will have a well-prepared and thought through response to my answer. You don't have to worry about addressing my first question, as I have rendered it irrelevant. There; I've reduced your workload a bit, haven't I? Of course, ∞-1 is still ∞, so... --69.91.95.139 (talk) 20:19, 14 July 2008 (UTC)

I'm waiting (but don't worry; I'm not holding my breath). --69.91.95.139 (talk) 14:18, 24 August 2008 (UTC)

Sorry for the wait, calculus wasn't built in a day. (And I do have a life outside of .999...) Algr (talk) 20:30, 25 August 2008 (UTC)

No need to apologize; take your time. --69.91.95.139 (talk) 00:28, 26 August 2008 (UTC)

Inaccuracy of Decimals

Personally, I think that this entire article describes the inability of decimals to represent certain fractions. You can't "start" at ".999...". You have to reach it doing other maths, and the way that happens is when you are converting fractions to decimals. All these proofs, arguments, counter-arguments, etc., are moot when you consider that if you had just stuck with the fraction notation you would not have this situation. Basically what I am saying is: .999... = 1 not by these proofs, but by the fact that you only arrived at ".999..." because you did 1/3 + 1/3 + 1/3 in decimal form (i.e. .333... + .333... + .333...). I know I can't be the only person to think of this, especially when you consider all the brilliant minds involved. --MadDawg2552 (talk) 14:24, 26 July 2008 (UTC)

I see what you're saying. Fractions are usually a far better way of representing rational numbers precisely than decimals, however decimals can also be used for irrational numbers and for approximations of numbers to a desired degree of accuracy, which makes them very useful. So, that's why we define decimals the way we do. Once you do that, you then want to try and make sense of every possible decimal expansion, including 0.999... . That doesn't really come from 1/3+1/3+1/3, it comes from 9/10+9/100+9/1000+... . It doesn't generally come up naturally, so in most cases it doesn't really matter what it equals, but if you want to be able to talk about all decimal expansions at once you need to be able to interpret all of them, even the ones that never really come up. Does any of that make sense? --Tango (talk) 15:09, 26 July 2008 (UTC)

While I was at the employment agency just now, twiddling my thumbs, I came to the realization that 0.999... is the limit as i approaches infinity of the following equation:

$9\times {\sum _{n=1}^{i}{10^{-n}}}\equiv 1-10^{-i}$

i	value	equivalence
1	0.9	1 − 0.1
2	0.99	1 − 0.01
3	0.999	1 − 0.001
∞	0.999999...	1 − 10-∞ == 1?

It easily rounds to but is not exactly unity. Interestingly, this appears to also indicate that $0.111...=\sum _{n=1}^{\infty }{10^{-n}}\equiv {1 \over 9}-{10^{-\infty } \over 9}$, is infinitesimally less than 1/9. Has no one else bothered to state the problem in this way? D. F. Schmidt (talk) 21:57, 28 July 2008 (UTC)

Lots of people have looked at the problem that way. Your mistake is in assuming that 10^-∞ is greater than zero - it isn't, it's exactly zero. See Archimedean property. --Tango (talk) 22:13, 28 July 2008 (UTC)

More precisely, the limit of 10^-x as x goes to ∞ is exactly 0. 10^-∞ is undefined in the real numbers, because ∞ is not a real number.

Of course, that doesn't blow the issue out of the water; it only redirects the problem: should decimal expansions really refer to the limit of their increasingly accurate finite expansions? The answer to that is, naturally, a definitive yes. The reason for that has a long history, which can be summarized by saying that the real numbers have turned out to be the most useful number sets throughout the ages, in both applied and abstract mathematics. And, within the real numbers, decimal expansions are most meaningful as a limit. --69.91.95.139 (talk) 23:00, 28 July 2008 (UTC)

Also, there really isn't an alternative definition. How else could you interpret them that would be even the slightest bit meaningful? --Tango (talk) 00:34, 29 July 2008 (UTC)

Yes, that's pretty much the gist of it. --69.91.95.139 (talk) 00:59, 29 July 2008 (UTC)

Well call me proud, prejudiced, or what-have-you, but none of the proofs given in the article are as succinct and definitive as the one I posted yesterday. The Cauchy sequence in the article right now has an introduction that's difficult to follow (for me, at least) -- and in fact such an introduction wouldn't have to be given if this other proof was given in the article, along with the simple table I constructed. The digit manipulation proof is a little awkward too, in my opinion, and I wouldn't be surprised if anyone that exercises logic would reject it. But anyways, my limited understanding of math would indicate that:

1. Practitioners of mathematics usually tend to prefer infinite precision, such that they prefer symbols (π, e, etc.) – and I don't think that's only for ease of use. (Scientists and engineers – applying science – might prefer symbols for the sake of pragma, but not mathematicians. Likewise, object-oriented programmers understand that even though two or more object referents may have the same value, it doesn't immediately follow that the two referents are the same object.)
2. All scientists (though not necessarily engineers) including mathematicians (as one might suppose) prefer to use precise language to the point of saying (as I said above) that $9\sum _{n=1}^{i}{10^{-n}}+10^{-i}\equiv 1$. If this wasn't so, we wouldn't have "divide-by-zero" problems, right? -- We might just as easily say that it is generically and absolutely true that $0^{-1}\equiv \infty$. As with all other sciences, assumptions must be declared -- especially when a "proof" is in question.

Ultimately, (in of course my own opinion) the proof that "0.999... = 1" could use my quite simple approach, and the only assumption to be declared is that the limit as i approaches infinity, 10^-i = 0. No need for any other clunky explanations like the one found under "fractions" or "digit manipulation" (which looks to me like garbage anyways, to be perfectly honest stating -- as a matter of course -- my own opinion). D. F. Schmidt (talk) 23:27, 29 July 2008 (UTC)

For a mathematician, you're absolutely right, that's the based way to look at it, however for the layman, understanding that the limit is zero rather than some positive infinitesimal is difficult - that's basically the Archimedean property. Once you accept the Archimedean property, the whole problem basically disappears, there's nothing really left to prove, but it's a difficult property to prove because it requires technical details about how the real numbers are defined. The digit manipulations proofs, while not wholly convincing (they are valid, but it takes some effort to prove it), are possible to understand with only a knowledge of basic arithmetic. --Tango (talk) 01:21, 30 July 2008 (UTC)

I would like to add one more remark on this topic: The precision of the equation I rendered above is absolute, to infinity. When i is 1, we can say that -- to infinite precision (which the mathematician prefers) -- the number of significant 9's is 1. When i is infinity, we can say that -- to infinite precision -- the number of significant 9's is infinite. Even when the number of 9's is infinite, there is one small (infinitesimal) value by which it must be increased in order to equate it with 1. I contend that to say otherwise is to say that -- to infinite precision -- ∞^-1 = 0. Remember that (from what I can tell) the Archimedean property is based on heuristics (which see).

However, when the precision is dropped from infinity, it is immediately rounded to 1. But mathematicians don't like to round numbers where they can avoid it. Am I right? D. F. Schmidt (talk) 01:31, 1 August 2008 (UTC)

No, you're wrong. The Archimedean Property is not based on heuristics, it's an absolute theorem. It's implied by the definition of the real numbers. ∞^-1 really is zero (if it's defined at all), see Extended real numbers#Arithmetic operations. --Tango (talk) 01:55, 1 August 2008 (UTC)

Am I wrong? What part of Archimedean property#Archimedean property of the real numbers is not heuristic? Quoting from that section, "Is c itself an infinitesimal? If so..." And if not? D. F. Schmidt (talk) 03:37, 1 August 2008 (UTC)

The first paragraph of that section is a mathematical proof of the fact that the real numbers are Archimedean.

At one point in the proof, a certain number c is constructed. The goal is to prove that c = 0. The rest of the proof then proceeds by contradiction. It is assumed (by contradiction) that c is positive. First, in the case that c is infinitesimal, a contradiction is reached. Second, in the case that c is not infinitesimal, another contradiction is reached. This last case is the "if not" you refer to. Melchoir (talk) 04:28, 1 August 2008 (UTC)

I don't understand how the fact that 2c > c contradicts the fact that c is a least upper bound for Z. That seems to me like a counterpart statement is that 2 > 1 means my head will implode in 5 seconds. The rest of the "proof" is not at all edifying to me except to indicate to me that one's words are somehow heavier than another's. I can't comprehend how the sum of all infinitesimals could be any less than infinity. After all, what is differentiation but the study of infinitesimals?! What am I missing here? D. F. Schmidt (talk) 05:51, 1 August 2008 (UTC)

If c is a positive infinitesimal, then 2c>c is a positive infinitesimal, too. Then c is no upper bound for the set Z of positive infinitesimals, because there's an element of Z larger than c. Huon (talk) 10:35, 1 August 2008 (UTC)

Ok, I guess I can understand the explanation now, but for the sake of other people who might have my same problem comprehending this, I guess it'd be like saying since 2c is a positive infinitesimal then it too belongs in the set Z, which increases the size of Z and with it, the lowest upper bound. Still: what's the rationale that the least upper bound for all infinitesimals is any less than infinity? After all, thinking this way would lead me to conclude that you believe that the least upper bound for all real numbers is less than infinity. D. F. Schmidt (talk) 18:10, 1 August 2008 (UTC)

By definition, all infinitesimals are less than all finite numbers. That's what it means to be infinitesimal (the technical definition is phrased a little differently, but that's what it means). Therefore 1 (or 2, or 1/2 or any other finite real number) is an upper bound for the set of infinitesimals and 1 is less than infinity, so the least upper bound must be less than infinity. --Tango (talk) 20:07, 1 August 2008 (UTC)

By that definition, I guess my argument holds no more water. But do you think it might be possible, say, to define a number by its reciprocal? Such as, 2 = 2 because 1/2 = 1/2? And 3 = 3 because 1/3 = 1/3? If so, 0.333... = 0.333... because 1/0.333... = 1/0.333....

As trivial as it sounds, this form of definition might be worth something. (Even if it's not worth something, bear with me; what preceded this may be completely irrelevant to the rest of my comment, which may have merit.) If you've never thought about it, as you start from addition and subtraction and progress to algebra and then calculus, the steps taken to arrive at the higher level of math that you've attained look trivial and differential rather than immense as those steps may have looked before you took them. Well, at least that's how I remember it. Then again, the farthest I ever went in a meaningful way with math is calculus II. (I did take III, but I didn't retain any of that, and I took D.E. but I comprehended so little in the first place.)

Well consider that 0 is a legitimate value, and 0 = 0. There are many numbers that resemble 0. By way of relation, the ratio of Avogadro's number (6.022·10^23, which is far greater than 0) to a googol is rather close to 0: $6.22*10^{23}*10^{-100}=6.022*10^{-77}$. Certainly, (hypothetically speaking) if you were to take a googol number of objects minus Avogadro's number, the change would probably go unnoticed when observing a physical change except by observing the difference itself -- such as if the Avogadro's number of objects were found in another place within the arena of observation. Bear with me. Even if 100 of these clusters were taken from the googleplex, it would still go unnoticed unless you were specifically looking for it. $10^{100}-100*6.022*10^{23}=10^{100}-6.022*10^{25}=10^{100}$ (observably).

Given the vast differences between these very large numbers – not to mention the even more vast difference between a googol and the very small constants, such as Coulomb's constant – remember that just as the difference (using any value x ≥ 1) between 10^x and x is so large, would it be at all fair to say that 10^∞ is equal to ∞? Even with a value of x as low as 20, the ratio of 10²⁰ / 20 is great. In calculus I, I was taught (if memory serves) that if problems come up such as this – $10^{\infty }/\infty =?$ — you take the derivative of each, such as $10^{x}/x$ and determine whether the value then is infinity or 0.

But as I reckon it, it is a ratio between infinities. And speaking of them as such, they do have reciprocals. Those reciprocals are asserted to have infinitesimal values (that is, values equivalent to 0). What, then, is the ratio of 0:0? Well this is my assertion: That whether or not 10^-∞ is infinitesimal, it is not exactly 0, nor is it equivalent to 0 except by rounding or for purposes of practical usage. It is a near-zero value which is impractical to calculate or use, which when necessary rounds to (but is not equal to) 0. After all, if it (10^-∞) was zero, it would not have a reciprocal of 10^∞, right? Well, somehow I feel like I haven't conveyed my point very well, but one of you might get the idea. At least, I hope so. 68.60.8.26 (talk) 02:28, 2 August 2008 (UTC) -- D. F. Schmidt (talk) 02:31, 2 August 2008 (UTC)

All the stuff about various constants made no sense at all, I'm afraid. As for $10^{\infty }$, I guess it depends on what you mean by the expression. The most obvious way to interpret it is as $\lim _{n\rightarrow \infty }10^{n}$, in which case it is just infinity, however your sense that it should be bigger than infinity isn't entirely misplaced. Take a look at aleph number - the integers have a cardinality of $\aleph _{0}$, the real numbers have a cardinality of $2^{\aleph _{0}}$, which is indeed a bigger infinity (for information about which one it is, see continuum hypothesis). Changing that 2 to a 10 won't make any difference, as far as I can see. As for $10^{-\infty }$, if you're working in the real numbers, it really is exactly 0, not just approximately but precisely. It's clearly not finite, and there are no non-zero infinitesimals in the real numbers, so it must be 0. --Tango (talk) 03:48, 2 August 2008 (UTC)

As with most Wikipedia articles covering math (any math higher than someone might already have studied), those for the aleph number and continuum hypothesis are beyond my comprehension, and even my interest, for that matter. If I had studied it, or if I might later study it, I might enjoy it. So I guess this is as far as I go in this discussion.

I do understand sometimes the necessity of using higher math in order to evaluate singularities, such as x/0. The truth is, as is mentioned, subtracting x − 0 any number of times will not reduce x to a remainder of 0 or a number having the absolute value less than 0. But just as I mentioned that 10^x/x is >>> 1 (when x > 1), it would seem to me that that any similar evaluation would be similarly obvious. Likewise, 10^-x*x (which is the reciprocal) should indicate that it is indistinguishable from but non-zero. I suppose my point is that higher math is like higher theology (which is a topic close to me): the higher math builds upon the lower math, and sometimes it is necessary to use higher math in order to explain lower-math singularities. But when the lower math (probably at least one order above where the singularity occurs) does already explain a lower singularity, why do you need to appeal to even higher orders of math? If this comment doesn't ring any bells of truth, then I don't suppose I belong here discussing this matter. Thanks for your time and labor in your attempt to show me why 0.999... = 1. D. F. Schmidt (talk) 04:30, 3 August 2008 (UTC)

From elementary school up through high school and even some of college, mathematics is simplified to a low level for us, but requires higher level math to fully justify. It's perfectly normal. As for the aleph numbers: they aren't that complicated; let me give it a shot.

The aleph numbers are numbers which refer to the size of infinite sets ("cardinality" is just a fancy term for size). The natural numbers, integers and rational numbers are all the same size (a bit of a paradox, but fully justifiable; see Countable set if you're interested), represented by the aleph number $\aleph _{0}$. The Axiom of choice ensures the existence of a next cardinal number (the cardinal numbers could have been infinitely dense, like the rational and real numbers, but they're not if we accept the axiom of choice as true), which we denote $\aleph _{1}$. The Continuum hypothesis can be expressed in several different ways, as shown in the article. However, the key idea is that the cardinality of the real number set is the same as that of the power set of the natural numbers, that is, $2^{\aleph _{0}}$, a provable fact without the continuum hypothesis. The continuum hypothesis takes the next step by suggesting that the real number set represents the next infinity, that is, $2^{\aleph _{0}}=\aleph _{1}$.

HTH. --69.91.95.139 (talk) 12:02, 3 August 2008 (UTC)

See also article on well ordering. Why is the ordering well? I guess for same reason the real numbers are real and right-wing is right. Is cardinality and cardinal number a better term?

If we don't assume Continuum hypothesis, what ordinal number $\aleph _{1}$ is explicitly? (Compare: what (hyper)real number 0.999... is explicitly? Assume 0.999... hypothesis = Archimedean property) Tlepp (talk) 21:59, 3 August 2008 (UTC)

Of course a "well" ordering is not necessarily "better" than any other ordering; it's a technical usage. This is one of the reasons I prefer the modern spelling wellordering with no hyphen and no space—I think that makes it clearer that it's a technical term rather than a value judgment. --Trovatore (talk) 22:04, 3 August 2008 (UTC)

Were those questions intended for me? --69.91.95.139 (talk) 00:05, 4 August 2008 (UTC)

$\lim _{x\rightarrow \infty }{\frac {10^{x}}{x}}=\infty$ and $\lim _{x\rightarrow \infty }{\frac {x}{10^{x}}}=0$. That's something you would cover in a first course in real analysis (1st year of Uni, possibly even before that in a non-rigorous fashion). It's important to remember that ${\frac {\infty }{\infty }}$ isn't 1, it's indeterminate and can turn out to equal anything depending on where it came from. While 10^x and x both go to infinity as x increases, 10^x does so much faster, so the ratio of the two ends up being infinity. The reciprocal case works exactly the same way. --Tango (talk) 20:37, 3 August 2008 (UTC)

Yeah. Well, don't be surprised that your fairly short essay does not enlighten me, 69.91.95.139. It's probably not your fault. Tango, I completely understand that infinity/infinity is indeterminate without context, and as such, is a singularity (without such context, as I assert). And I understand that the reciprocal winds up, well, the reciprocal. And division by zero is a singularity. But 10^∞ and its reciprocal are neither simply "infinity" or "zero". Ratios or subtractions may cause them to evaluate (or be roughly equivalent) to zero or to infinity, but still are not identical with either. It's like you guys want to do mid-calculation rounding, which is a grave no-go in the math and science world. This is my assertion, and I think that's all I have to add. D. F. Schmidt (talk) 04:47, 4 August 2008 (UTC)

Well, it's not really defined at all, but if you want to define it, the only meaningful way I can see is to define it as the limit of 10^x as x goes to infinity, in which case it really is infinity. There is no approximation going on. (It's worth pointing out, we're implicitly working with the extended real numbers now, rather than the real numbers, but that's not really important except for precision.) --Tango (talk) 21:56, 4 August 2008 (UTC)

What seems to be missing from your consideration here is that you can't really evaluate ∞/∞; it is beyond evaluation. 10^∞/∞ on the other hand is--it evaluates to something that you would assert is equal to ∞. The reciprocal evaluates to something that you would assert is equal to 0. So now consider ${10^{\infty }/\infty } \over \infty$. If you had previously evaluated the numerator, the result of the overall equation would then be ∞/∞ which is again beyond evaluation. However, if you evaluated it properly like any algebra student knows to do, it would be obviously evaluated to ∞. I agree that when facing infinite or infinitesimal values, they are essentially infinity or 0; however, in order to continue to use these values practically (or at least more so), you need to not "round" (or whatever word you choose) them mid-calculation.

And if I'm right in saying the above, I suspect that perhaps no value which involves infinity is truly 0 or simple infinity. D. F. Schmidt (talk) 01:07, 5 August 2008 (UTC)

Yes, I see what you're saying. Evaluation of limits and basic arithmetic operations do not commute when infinity is involved (they do otherwise - I learnt it as the Calculus of Limits Theorem (COLT), although I suspect it has an alternative name, since a google search gives 5 results, 3 of which are on my uni's website and one is me on a Wikipedia talk page about 0.999...!). That's evidence in favour of simply not counting infinite limits as existing and just saying 10^∞ is undefined and leaving it at that. If you want to define it, though, you do need to be careful about when you take limits (which is what you're referring to as rounding - it's not a bad analogy, I suppose). --Tango (talk) 01:30, 5 August 2008 (UTC)

PS: It's not quite as bad as rounding early, though - with rounding, you'll get an answer at the end and it will be wrong, with this you just don't get an answer. --Tango (talk) 01:31, 5 August 2008 (UTC)

Lack of Knowledge

Moved from main talk page. --Tango (talk) 22:32, 20 September 2008 (UTC)

Is it just me, or is this discussion page just full of people talking non-mathematical non-sense? People saying that when one squares the square root of two one doesn't get two. There is too much non-technically, informal, non-rigorous non-sense on this page. It seems (from reading some of the user pages) that many people that have contributed are little more that, if at all, undergraduate students. People; please don't write anything unless you're sure... It's confusing, and a little annoying. For me 0.999... does not equal one. This is informal shorthand. The true statement is that

$\lim _{k\to \infty }\sum _{n=1}^{k}{\frac {9}{10^{k}}}=1.$

That does not say that the sum is ever equal to one for any k. It simply says that the sum tends to one as k becomes very large. To say that something is "equal to" and that something "tends to" are very different statements. For example, people have proved that almost all of the zeros of the Riemann Zeta function fall on the critical line. As the number of zeros tends to infinity the ratio of zeros on the line to those off it tends to zero. That does not say that all of the zeros fall on the critical line. Limits and equalities are very different things. So there! ;o) Declan Davis (talk) 22:08, 20 September 2008 (UTC)

So what you define "0.999..." to mean? --Tango (talk) 22:32, 20 September 2008 (UTC)

Sorry, I don't understand the question. Do you mean to ask "So what do you define "0.999..." to be?" Well, to me, it is a zero followed by a series of nines. You could write as many nines as you like but you will never end up with one. If, however, you want compute the limit, then the limit will tend to one. I suggest that you read "The Prime Number Theorem" by G. J. O. Jameson, London Mathematical Society, 2003, for some interesting discussion on the difference between limits and equality. It is a very subtle difference, I agree. Declan Davis (talk) 23:43, 20 September 2008 (UTC)

So you don't think 0.999... is well defined? In that case, it's a useless concept, so it's much better to stick with the standard definition which is that something ending with "..." refers to the limit. The limit is equal to 1, therefore 0.999... is equal to one. --Tango (talk) 00:00, 21 September 2008 (UTC)

Exactly! The limit is well defined and is equal to one. I just don't like people writing a zero followed by lots of ones. I'd mark that wrong in every exam. It's a little bit like root two: it doesn't have a decimal expansion, and writing ever more accurate decimal approximations to root two and then saying "oh look, it's root two" is all wrong. If the convention is to assume that

$0.999...:=\lim _{k\to \infty }\sum _{n=1}^{k}{\frac {9}{10^{k}}}$

then I guess the statement is correct, although I don't for one second condone the abuse of notation. I would still strongly recommend that you read the above book.

Declan Davis (talk) 00:08, 21 September 2008 (UTC)

That's the convention I've always known, and I think it's pretty universal. "..." means the obvious pattern continues to infinity - more rigorously, you take the limit as the length of the expansions tends to infinity. If you object to the notation entirely, then the question of what it equals doesn't even exist. However, the notation is standard, so if you're marking it wrong in exams I would expect students to appeal (assuming they get to see the scripts - my Uni won't show us them...). --Tango (talk) 00:19, 21 September 2008 (UTC)

No-one needs to appeal: I drum it into them not to write such things. Also, on second reading I'm getting a little worried about your authority on matters mathematical. From reading your posts you seem to have problems with εδ-proofs, not to mention the defintion of a continuous function. What did you say? "It's just plain wierd"? May I ask: what is your mathematical background, and what makes you feel that your are such an authority on all things mathematical? Declan Davis (talk) 00:54, 21 September 2008 (UTC)

Seeing as you like εδ-proofs so much, I thought I'd rewrite the limit it terms of an εδ-statement. Let

$a_{k}:=\sum _{n=1}^{k}9/10^{n}.$

Then the actual statement should not be that 0.999... = 1, but should be that for each ε > 0 there exists a natural number $K_{\varepsilon }$ such that for all $k>K_{\varepsilon }$ we have $|1-a_{k}|<\varepsilon .$ Declan Davis (talk) 01:30, 21 September 2008 (UTC)

Because 0.999... is usually defined as the limit of the sequence (a_k), that's the same statement as 0.999... = 1, except that you avoid writing 0.999..., which misses the article's point. As an aside, given that Tango has quite a history of contributions on these talk pages, and given that he was awarded reference desk barnstars for his help on math topics, I don't think one needs to worry about his credentials. Huon (talk) 01:39, 21 September 2008 (UTC)

Thanks for the advise Huon, but I was asking Tango and not you. Besides I was questioning his mathematical authority, and not his skill as a Wikipedia editor. He is obviously a very skilled editor. Thanks again for your thoughts. Declan Davis (talk) 01:54, 21 September 2008 (UTC)

While I agree that it is important to emphasize that the statement "0.999... = 1" is actually

$\lim _{k\to \infty }\sum _{n=1}^{k}{\frac {9}{10^{k}}}=1$

I do not agree that you should force your students to rewrite every single decimal expansion as its corresponding limit. That's ridiculously and unnecessarily dull repetitious work. And yes, infinite decimal representations are defined as the limit of their successively larger representations, by convention, in the vast majority of mathematical contexts (as are infinite series). --69.91.95.139 (talk) 01:47, 21 September 2008 (UTC)

I did not say that I force my students to rewrite every single decimal expansion as its corresponding limit. If I did then please leave a link. Please don't put words in my mouth (or should that be words in my hands seeing as I'm typing, I don't know). I was simply saying that if a student were to write, for example π = 3.142, then this would be incorrect. I shall repeat, once again, my objection: limits and equalities are different. Declan Davis (talk) 01:54, 21 September 2008 (UTC)

There is a big difference between π = 3.142 and π = 3.141..., the former is, of course, incorrect. The latter is entirely correct (although not very precise since being irrational there is no pattern to continue on). --Tango (talk) 11:02, 21 September 2008 (UTC)

I know there's a difference, but thanks for pointing it out just in case I didn't. I didn't ever say that there wasn't. People really need to stop trying to put words into my mouth. Instead, people need to read what I have written and stop infering things which may or may not be true. Once again: thank you for your thoughts, I found them most enlightening.  Declan Davis   (talk)  23:49, 21 September 2008 (UTC)

You have an extremely unpleasant argument style, Declan. Stop accusing people of "putting words in your mouth" every time they try to emphasise a point that you appear to be disregarding. Stop questioning other editors' "mathematical authority" (as if appeal to authority weren't even more invalid in mathematics than it is elsewhere). Stop mocking editors for typoes. Please read WP:CIVIL. This is not the place for you to smugly belittle others, it is a place to discuss mathematics, in particular the equality 0.999... = 1, in a civil manner. Maelin (Talk | Contribs) 00:10, 22 September 2008 (UTC)

The evidence suggests you don't know the difference. Why else would you compare 0.999... to 3.142? --Tango (talk) 13:42, 22 September 2008 (UTC)

Declan Davis has been perfectly civil. You guys are misrepresenting (or misunderstanding) what seems to be perfectly valid points of his. Algr (talk) 17:17, 22 September 2008 (UTC)

Of course he has! Every bit as much as you have! --69.91.95.139 (talk) 21:48, 22 September 2008 (UTC)

Thanks Algr, at least someone's reading what's actually been written. For the other editiors, please allow me a few words. As far "accusing people" of putting words into my mouth, let me present two cases in hand (both can be found above). The IP user 69.91.95.139 said "I do not agree that you should force your students to rewrite every single decimal expansion as its corresponding limit." I had never for one moment said that I force my students to rewrite every single decimal expansion as its corresponding limit. So that is putting words into my mouth. Tango said that "There is a big difference between π = 3.142 and π = 3.141...", which implies that I had made the point that that there wasn't a difference, which I hadn't. So that's coming close to putting words in my mouth. I have questioned people's mathematical authority because they have used phrases such as "it's pretty universal" when I don't think that they are in a position to comment on what is, or is not, universal. I find IP user 69.91.95.139's reaction quite funny: his way of saying that I haven't been civil is to be sarcastic and be uncivil towards Algr. I find that most amusing. Finally, Tango: it is clear that π ≠ 3.142. I made the point in an earlier post to tell IP user 69.91.95.139 that I didn't "force my students to rewrite every single decimal expansion as its corresponding limit" but that if they tried to use some truncated decimal expansion in place of a proper limit then their answer would be marked as incorrect.

 Declan Davis   (talk)  00:51, 23 September 2008 (UTC)

If people are misinterpreting you, it's because you are explaining yourself extremely poorly. If you're not getting students to rewrite recurring decimals, what did you mean by that comment? And, since no-one suggested a truncated decimal was correct, what was the relevance of your comment about 3.142? --Tango (talk) 00:58, 23 September 2008 (UTC)

I think that I have explained myself perfectly well: limits and equalities are different things. I have made this point several times. My point about π was that formally it does not have a decimal expansion. So to attempt to write a decimal expansion of π is incorrect, even though π is a well defined number. For me, I don't like the practise of writing 1/3 = 0.3(3). One third is 1/3 and it's best left as an object with the property that three times it is one. Real analysis can be a bit wishy-washy. Proving the completeness of the reals involves showing that a certain number's less than this, and more than that, without actually getting a grip on the number itself. This is much the same. Trying to put decimal expansions to limits, irrational numbers, or transcendental numbers is a bit sloppy. These numbers exist, and have formal definitions (using limits) but you can't put a string of numbers to them... or at least that's what I think.

 Declan Davis   (talk)  01:09, 23 September 2008 (UTC)

I disagree completely with your assertion that π does not have a decimal expansion. Every real number has a decimal expansion (indeed, many have two, as this article demonstrates). It may not have a finite decimal expansion, but that is another matter. I can even give you a closed form for each digit of π in any base of the form 2ⁿ, if you like. A third is indeed 0.333..., and 1/3, and also 3^-1 and all of these have the property that three times them is one. Maelin (Talk | Contribs) 01:57, 23 September 2008 (UTC)

Maelin: Well, that's your opinion, and you're allowed to have and express that. I disagree with you. We shall have to agree to disagree. I have been teaching in university for four years now, and one of my modules is "Fourier series and iteration" which includes a large amount of real analysis. I have been thinking about these things for almost ten years now. I suggest that you show this discussion thread to one of you professors at university, and see what they say. The whole idea behind a transcendental number is that it doesn't have a well defined decimal expansion. Please go and ask a teacher and let me know what they say. Until then: good luck in your studies, and have a nice week.  Declan Davis   (talk)  02:04, 24 September 2008 (UTC)

Declan, if you have been telling students that transcendental numbers do not have a well-defined decimal expansion, then you've been doing them a disservice. This is just not correct. The decimal expansion of pi, say, is perfectly well-defined, even in a constructivist sense (there is an algorithm for computing the digits of pi). You are welcome to your own private definitions, but it's a damn shame when you try to appeal to your own authority when others object. Phiwum (talk) 11:03, 24 September 2008 (UTC)

Declan, I realize you have a PhD in mathematics, so I don't have any idea why you think that these notions of yours are relevant to classical mathematics. It's a plain fact that every real number has a decimal representation and it's a plain fact that infinite decimal expansions are commonplace in classical analysis. You may have your own personal definitions and philosophy of mathematics, but they don't hold any particular weight for others. Phiwum (talk) 03:16, 23 September 2008 (UTC)

Once again, this is a case of people not reading what has gone before. I realise that it's tempting to skim-read a post and get straight down to writing your point. I have never said that "these notions of [mine] are relevant to classical mathematics". I've already made my point above. The article is too informal, it makes something out of nothing. The use of informal shorthand becomes so ingrained that people forget that it is shorthand. Then they start to believe that there is some kind of mathematical voodoo. For me, this whole article is vacuuous. If, instead of using the shorthand, we wrote out the limit then there'd be no problem. students don't understand the shorthand. A large chunk of the article is about why people don't understand or why they disagree. If we wrote out the truth first time then there'd be much less trouble. It's like an idiom in a foreign language. It make perfect sense to the native speaker, but to a student it seems crazy and needs to be explained.  Declan Davis   (talk)  01:56, 24 September 2008 (UTC)

Declan... You forgot to ask Algr about his (her?) mathematical background. Was that really because of Maelin's advice, or did you just decide give up on that argument technique when it started working against you? ;) --69.91.95.139 (talk) 11:10, 23 September 2008 (UTC)

IP user 69.91.95.139: Not at all friend. I didn't find any need to question Algr's mathematical authority since s/he didn't make any statements like "it is universal". They didn't profess to know the whole picture. When people without the requisite experience and/or knowledge make claims (without acceptable sources) then I will naturally question their knowledge and authority. I am a scientist. If I did not question the validity of the information presented to me then I would not be doing my job.  Declan Davis   (talk)  01:56, 24 September 2008 (UTC)

It's true that the "..." notation is not a rigorously defined mathematical symbol, as far as I know (and indeed, Ellipsis#In_mathematical_notation states this). However, the article clearly states the recurring decimal, and also lists the other notations such as using a dot or bar over the 9 (which I have always been taught as denoting a recurring decimal).

As it happens, I raised the possible ambiguity over using ellipses, way back in Talk:0.999.../Archive_6#Use_of_ellipses. But do you agree that $0.{\bar {9}}$ equals 1? Or if we were speaking it, that "0.9-recurring equals 1"?

It's not about what the majority of editors think, it's about verifiable sources and not personal opinions. And you may have been asking Tango, however, this is a public page addressing issues about a Wikipedia article, so it is fair for any editor with an interest to respond. Mdwh (talk) 02:47, 21 September 2008 (UTC)

You guys always resort to "verifiable sources" when you run out of arguments, but this page is not the article, it is for people who practice scientific skepticism of what they read. Such people will reject .999... = (1/3)x3 as circular logic, and you have defined limits and the Archimedean property in ways that make those arguments equally circular. Algr (talk) 07:45, 23 September 2008 (UTC)

I only brought up verifiable sources in response to him asking about people's authority. I'm happy to try to explain the mathematical proof for those who do not understand it. Mdwh (talk) 01:37, 24 September 2008 (UTC)

Not a Proof?

The 0.999... = 3(1/3) argument is not a proof. It is a heuristic argument designed to show how a potentially counterintuitive result can be seen as a simple consequence of a hopefully less counterintuitive result. Someone who rejects that 1/3 = 0.333... recurring is perfectly entitled to reject that it demonstrates 0.999... = 1, but this does not mean they are entitled to reject proper proofs. Limits are defined via epsilon delta proofs and are fully formal, fully rigourous and fully legitimate definitions. The Archimedean property of the real numbers is a direct result of its construction. You are simply practising intellectual dishonesty by refusing to try to understand the arguments raised here, and you should not pollute Declan's entirely unrelated discussion with your little sniping comments. Maelin (Talk | Contribs) 08:38, 23 September 2008 (UTC)

O'RLY? The article has 0.999... = 3(1/3) as the first example in the proofs section and presents it as a perfectly valid reason why people ought to believe that .999...=1. You don't see anything "intellectual dishonest" about that? I see that as a perfectly valid reason to distrust the premise and suspect that all the other proofs are equally flawed. If I submitted a research paper for peer review and had this kind of error at the top, would you expect people to take my other findings at face value? Algr (talk) 18:23, 23 September 2008 (UTC)

A research paper is aimed at an audience of experts. Wikipedia is aimed at an audience of laymen. The article isn't intended to be a rigorous mathematical paper, it's intended to be an encyclopaedia article. If we started with a detailed rigorous proof referencing the constructions of the real numbers and the Archimedean property and limits and convergence and everything else we would just scare away our readers. The first "proof" is intended to convince readers of the equality and it is a valid argument it just misses out a load of steps (it relies on the fact that those kind of arithmetic manipulations work on convergent series, a fact that needs proving, as does the fact that the series converges - neither of those are within the scope of that first argument, though). The article then moves on to more rigorous approaches for those readers that want them. --Tango (talk) 18:42, 23 September 2008 (UTC)

Laymen can still understand logic. .999...=1 and .333...=1/3 both invite the exact same question: What does it mean for a decimal to extend to infinity? To answer one by assuming the other is not "leaving out steps", it is circular logic and insulting to the reader who wants to understand the real question. Algr (talk) 19:13, 23 September 2008 (UTC)

Most laymen cannot, however, understand the answer to your stated question, because the answer invokes a bit of high-level mathematics. It is most certainly not insulting to say that most laymen would not come even close to understanding the underlying reason for 0.999 and 1 being the same number, as the reason is a tad bit complicated.

And no, the 'reason' I refer to is not the Archimedean property; I refer to the property of completeness. The former is a consequence of the later. --69.91.95.139 (talk) 22:29, 23 September 2008 (UTC)

Intuitively, it's very easy to see what it means for a decimal to extend to infinity - it means it keeps going on forever and never stops. Making that rigorous is a little more difficult (it requires some 1st year analysis), which is why we skip it. --Tango (talk) 22:38, 23 September 2008 (UTC)

Yes, the process is easy to understand, it is the result which is the problem, I've already explained why. Completeness is just more circular logic because the Real space is only complete if you assume that .999...=1. If non zero infinitesimals exist, then the real set is not complete. Algr (talk) 00:27, 24 September 2008 (UTC)

No. The real number set is complete because of its construction. 0.999... = 1 is a result of its completeness. Nobody sat down and said, "Okay, let's make sure 0.999... = 1 is true just to offend the sensibilities of people without formal maths education. Hum, that means we have to make the real set complete. Okay, complete it is." We don't construct the real numbers in the way we did simply to make 0.999... = 1, we construct them that way to make an interesting and useful number system. Completeness is part of that construction. And the equality arises from that. Maelin (Talk | Contribs) 00:38, 24 September 2008 (UTC)

If the real number set is defined as complete, then you can't say that any describable value is not a part of it. "The highest value of x<1" is just as valid a description of a number as "The square root of two". Both values have unambiguous positions on the number line, (actually √2 has two positions.)... Algr (talk)

Sure I can. I can describe a value, "a number x that squares to -1", and yet, that number is not a part of the real number set. Your confusion here arises because you have absolutely no idea what complete means, and are instead pretending that you do for the sake of continuing the argument. Maelin (Talk | Contribs) 05:42, 26 September 2008 (UTC)

Congratulations! You have at last recognized that the Real set is not the appropriate venue answer to every mathematical question. Had you failed to make this leap you would have had to insist that there is no √-1, just as there is no highest value for x<1. Algr (talk)

But you have never described an extension of the real numbers that would contain a "highest value for x<1". And please don't bring up the hyperreals: they do not hold that property, and wishing they did will not make it true. --69.91.95.139 (talk) 12:18, 28 September 2008 (UTC)

No-one has claimed that the real numbers are the appropriate venue for every mathematical question, but they are the appropriate venue for this one since decimals are a notation for real numbers. --Tango (talk) 14:18, 28 September 2008 (UTC)

... But that is beside the more serious point. I can't believe you are defending teaching laymen a clearly wrong proof (.333...x3) just because it makes them think what you want them to! That is no better then if I made a fake moon rock and and started using it to "prove" that the moon landings were real. Once you descend to that kind of tactic, you have no right to expect anyone to trust what you say. Algr (talk) 05:18, 26 September 2008 (UTC)

The proof isn't wrong, it just has a few gaps in it. It's not intended to be rigorous, it's just intended to convince laymen of the result. The rest of the article goes on to more rigorous proofs. --Tango (talk) 15:29, 26 September 2008 (UTC)

Gaps? Yes, you left out the part where you assume that .999... = 1. Algr (talk)

Where, exactly, is that assumption hidden? Please specify PRECISELY where we have assumed that 0.999... = 1. You need to point out exactly which particular inference we make, whcih requires 0.999... = 1 to be true, and which will fail to be true if the equality is untrue. Don't say "it's an underlying assumption of the whole proof," this will merely prove you have no idea what you are talking about, as usual, and that you are just trying to be inflammatory. Don't pick apart this post. Don't accuse me of bullying. Just answer the question. Where did we make that assumption? Maelin (Talk | Contribs) 06:50, 27 September 2008 (UTC)

It is by extension of assuming that .333=1/3. Also, both http://www.dpmms.cam.ac.uk/~wtg10/roottwo.html and Construction of the real numbers show that the equivalence is simply written into the definition of what a real number is, so any proof involving real numbers is circular logic. Here is the exact quote:

__

So when you say "the real number x" what you really mean is "the decimal expansion x"?

Well, it's not always what I think of when I talk about real numbers, but if you insist on a precise definition, then I can fall back on this one.

Does that mean that 0.999999.... and 1 are different numbers?

Oh yes, I forgot about that. Different decimal expansions correspond to different real numbers except in cases like 2.439999999.... equalling 2.44. So I suppose my definition is that real numbers are finite or infinite decimals except that a finite decimal can also be written as, and is considered equal to, the "previous" finite decimal with an infinite string of nines on the end. Happy now?

__

Firstly 0.333 does not equal 1/3, 0.333... does. You need to be more precise in your notation if you want to be taken seriously. Secondly, long division proves that 1/3 is 0.333..., you don't need 0.999...=1, you just need that "10 divided by 3 is 3 remainder 1", do you dispute that? I don't see what that quote has to do with anything, that's not a rigorous definition of the real numbers. The real numbers are defined either in terms of Cauchy sequences or Dedekind cuts, neither or which rely on 0.999...=1. --Tango (talk) 16:23, 27 September 2008 (UTC)

Precise? You and Maelin make all sorts of spelling errors, and I never comment on them. I may not always succeed, but I do my best to understand what you are really trying to say. Assuming that a remainder will just disappear if you push it far enough away doesn't sound very rigorous to me. Finally, the first line of Dedekind cut includes " and A contains no greatest element." Where does that come from? Algr (talk) 02:50, 28 September 2008 (UTC)

The "no greatest element" line is to avoid an ambiguity. If you omitted it, you could construct two "Dedekind cuts" for every rational number, say 1/3: One with A containing the rational number, the other with B containing it. I assume that you're about to say that this artificial construction is what forces 0.999...=1, but that's not the point: If you tried to construct a number system out of partitions of the rationals without this restriction, you'd double more than just those numbers with a terminating decimal expansion. There's not even a decimal representation for most of the numbers you'd get (say, for the doppelganger of 1/3). If you tried a straightforward generalization of addition, you'd also realize that some of those numbers don't have an additive inverse. Huon (talk) 11:16, 28 September 2008 (UTC)

Infinite decimal expansion?

Every real number has a decimal expansion (indeed, many have two, as this article demonstrates). It may not have a finite decimal expansion, but that is another matter.

Well this brings up an interesting point. If you start with the decimal expansion for π, changing any digit in that expansion will result in a value that is not π. Correct? So what happens if you change an infinitely distant digit? If π has no last digit, then do further digits become irrelevant at some point? And if the idea of an infinitely distant digit somehow has no meaning, then how could any decimal expansion be described as infinite? Algr (talk) 07:35, 23 September 2008 (UTC)

For exactly the same reason that the set of natural numbers is infinite, but no individual natural number is infinite. --Trovatore (talk) 07:43, 23 September 2008 (UTC)

So we are back to "infinity only works when we want it to." If infinite digits exist, then you have non-zero infinitesimals. If they don't, then .333... approaches 1/3, but can never equal it. Algr (talk) 07:51, 23 September 2008 (UTC)

0.333... can't "approach" anything, it's a fixed number. The main difference here is between "infinitely many" (as in "infinitely many digits") and "infinitely large". Obviously there can be sets containing infinitely many elements without any of those elements being infinitely large. The set of real numbers between 0 and 1 is one example, the set of natural numbers is another. The set of digits of 0.333... is bijective to the set of natural numbers, as there's a first digit, a second digit and so on. Since there are no infinitely large natural numbers, there's no "infinite" digit. Huon (talk) 10:35, 23 September 2008 (UTC)

But that's the point Huon: "0.333..." is not a fixed number, it's short hand! It's actually a limit. Let $a_{k}:=\sum _{k=1}^{n}{\frac {9}{10^{n}}}$, then when you write 0.333... you actually (should) mean $\lim _{k\to \infty }a_{k}$. In much the same way that when we write $\int _{0}^{\infty }f(x)\ dx$ we actually mean $\lim _{a\to \infty }\int _{0}^{a}f(x)\ dx$. Any good calculus text book will make the second point quite early on, but to save on paper and ink we write the upper limit as $\infty$ with the understanding that we really mean the limit.

You're right when you say that (in the limit) there is no infinite digit. In $\lim _{k\to \infty }a_{k}$ we do indeed get a bijection between the digits in the decimal expansion and the (non-zero) natural natural numbers, this is given by $n\mapsto {\frac {9}{10^{n}}}$.

 Declan Davis   (talk)  11:37, 23 September 2008 (UTC)

Quite right, Declan, $0.333...=\lim _{k\to \infty }\sum _{i=1}^{k}3^{-i}$ by definition. But you're quite wrong when you suggest that 0.333... is therefore not a number. It is trivial to prove (as you know) that

$\lim _{k\to \infty }\sum _{i=1}^{k}3^{-i}=1/3$

and hence 0.333... is a number. It's the same number as 1/3.

You seem to think that limits are different than numbers, but when a limit $\lim _{k\to \infty }a_{k}$ converges to, say, $b$, then we write $\lim _{k\to \infty }a_{k}=b$ for a reason. Namely, in such situations, $\lim _{k\to \infty }a_{k}$ means $b$ in exactly the same sense that $2+2$ means $4$. (Admittedly, such limits aren't usually terms in first-order logic, because function symbols are taken to represent total functions, but this is a technical fact of no essential consequence.) Phiwum (talk) 11:55, 23 September 2008 (UTC)

Sorry, I didn't explain myself properly. What I meant to say is that the shorthand "0.333..." itself is not a number. It's just that: shorthand. The limit that it represents is of course a real, well defined number. I still probably haven't explained myself totally, but I hope you can see what I'm trying to get at. And don't worry, I am a mathematician and I fully understand limits. That's why I have written that $\lim _{k\to \infty }\sum _{n=1}^{k}{\frac {9}{10^{n}}}=1$. For me, the whole point is that this quirk of notation has lead to people thinking there's some kind of mathematical voodoo going on. The use of such shorthand leads to interesting expressions such as "0.999... = 1".  Declan Davis   (talk)  12:16, 23 September 2008 (UTC)

The situation with 0.333... is no different than the situation with $\lim _{k\to \infty }\sum _{n=1}^{k}{\frac {3}{10^{n}}}$ or, indeed, with 1/3 or 2 + 2. In each case, we have a term ("0.333..." or "2 + 2", say). These are syntactic objects which represent certain numbers. In each case, the representation depends on relevant definitions. There is, perhaps, a sense in which the ellipsis is a bit informal (I don't know of any formal definition of the ellipsis), but in practice the meaning is clear.

In any case, I must admit this isn't what I thought you were saying. Seemed to me that you were saying 0.333... is "really" a limit and hence not a number and certainly not 1/3 (although it "approached" 1/3). That opinion is, of course, contrary to standard mathematical exposition. But now you seem to be saying merely that 0.333... is an informal (though nonetheless well-defined) shorthand for a particular limit and this fact may escape mathematical novices. Perhaps so, though I don't see it as a persuasive argument that we should drop the shorthand and write the limit out instead. Phiwum (talk) 13:40, 23 September 2008 (UTC)

I fail to see your point. We can agree that "0.333..." is a notation that denotes a limit and thus a number, but that's more semantics than mathematics. Still neither 0.333... nor any limits tend to anything, and for the sake of extreme precision, I'd also disagree with Phiwum's notation that "$\lim _{k\to \infty }a_{k}$ converges to, say, b": The sequence (a_k) may converge to b, the limit, if it exists, is a number and can't converge. Huon (talk) 12:34, 23 September 2008 (UTC)

Well, that's your opinion and you're entitled to it. We shall have to agree to disagree. Have a nice day Huon.  Declan Davis   (talk)  12:41, 23 September 2008 (UTC)

Declan, I'm still trying to understand exactly what your contention is here. Which, if any, of these statements do you find objectionable?

• "0.999..." is a shorthand for an infinitely long string, consisting of a zero, a decimal point, and an infinite string of nines
• That infinitely long string is a legitimate representation of a single real number
• The real number represented is $\textstyle 0.999...:=\lim _{k\to \infty }\sum _{n=1}^{k}{\frac {9}{10^{n}}}$
• This interpretation is, in some sense, the "correct" interpretation that we should promote on Wikipedia
• Since the infinitely long string represents this real number, then so too does the shorthand 0.999... for that string
• The real number represented, that is, the limit described above, is equal to the real number 1

I must confess I'm not sure exactly what it is that you dislike about the article. Please try to summarise your opinion in a self-contained form, and we will hopefully be able to avoid any further issues of people misinterpreting you as above. Maelin (Talk | Contribs) 13:58, 23 September 2008 (UTC)

Maelin, my objection was posted a long time ago way up top. It's that I don't like the lack of formality and the use of shorthand; well shorthand's okay provided people remember that it is just that. Out of your list I would object to

• That infinitely long strings are legitimate representations of real numbers.
• This interpretation is, in some sense, the "correct" interpretation that we should promote on Wikipedia.

I agree with the other points. For me it's a bit like power series. In Taylor's work he actally says that a smooth function can be written as a polynomial plus some smooth remainder. It's the same with numbers like π, people talk about the k-th digit of π but when they're saying is that we can write π as a finite desimal plus some other transcendental remainder. With irrational numbers, we can write root two as a finite desimal plus an irrational remainder. With numbers like 1/3 we can write it as a finite decimal plus a rational remainder. That's what we formally mean when we try to write down decimal expansions of π, root 2, 1/3, etc. In the case of 0.999... we write it as a finite decimal plus some remainder expressed by a limit. People are mixing up equalities and limits. It's a subtle difference but people were moving between one and the other as if they were the same. I am a pure mathematician and I prefer formal rigour to handwaving. Now I'm sure that some people will immedietly attack that comment, by saying "It's not handwaving, it's a fact you idiot", or something along those line. But it's not a fact, it's shorthand. The discussion has totally got out of hand, and is nothing to do with the article anymore. People started to put words into my mouth, so I replied to that. Then some people didn't think that people were putting words into my mouth, so I replied with evidence, and this has continued ad nauseum. Most of my replies that been to clear up things people have said about what I have said. I have tried to bow out of this discussion a few times, since it has degenerated into nonsense. I shall attempt to do the same thing again... although I am sure that someone will have something to say that prompts a response from me.  Declan Davis   (talk)  14:16, 23 September 2008 (UTC)

"1" is a shorthand for "the successor of the natural number postulated to exist by the fifth Peano axiom" (based on the version of the axioms in that article, I know not everyone formulates it exactly the same way). Are you suggesting we shouldn't use the notation "1"? Mathematics is full of shorthands, they are what make it feasible to actually use the concepts for anything. As long as the shorthand is well defined, as "0.999..." is, there is no reason not to use it. --Tango (talk) 14:42, 23 September 2008 (UTC)

What I understand Davis to be saying is that you should only use shorthand if you are certain that the reader will understand what said shorthand really means. It is unfair of you, Tango, to take this to the ludicrous extreme of suggesting that a wikipedia reader might not know what "1" means. There are reasonable and unreasonable suggestions for confusion. If you are roman you might think that IV=4. Others might say that IV=Intravenous therapy. Many writers here seem to have no regard at all for their readers. But an encyclopedia exists to inform people; It's pointless to be "right" in a way that does not inform.Algr (talk) 18:55, 23 September 2008 (UTC)

Tango: I've just read the article on the Peano axioms, and the choice of 0 or 1 is simply a choice of the base case for the inductive argument. The article also says that the axioms have been used "almost unchanged", i.e. they have been changed, and that mathematical logic was in "its infancy" when they were constructed. I reject this as a useful source. Algr: Please, call me Declan.  Declan Davis   (talk)  01:32, 24 September 2008 (UTC)

I have no idea what but the limit could be meant by that shorthand 0.999... And the article's first sentence and the "Introduction" section should make it crystal clear. Huon (talk) 19:23, 23 September 2008 (UTC)

Well this is what I meant by my earlier comment [1] about ellipsis - is it simply that he objects to the use of ellipses as "shorthand"?

Would he agree that $0.{\bar {9}}=1$?

Would he agree that $\sum _{n=1}^{\infty }{\frac {9}{10^{n}}}=1$?

I see that he agrees with $\lim _{k\to \infty }\sum _{n=1}^{k}{\frac {9}{10^{n}}}=1$, so it seems to be he's not disputing the mathematical facts here, he's just saying that people shouldn't use certain notation. Is this correct? (Although I'm not sure why one shouldn't use notation that is well defined - I can see a plausible argument regarding the use of ellipsis, but not with these other notations. If you are worried that the reader might not understand, then why do you assume that they do understand the notation that you are happy to use, such as summation signs, infinity, and limits?) Mdwh (talk) 02:17, 24 September 2008 (UTC)

Algr, as an aside, you seem to suggest in the above discussion that limits are somehow ill-defined. Could you elaborate? For you convenience, I will provide you with definitions. I will even use words instead of mathematical symbols, just for you.

$\lim _{X\to p}f(X)=L\Leftrightarrow \forall \epsilon \in \mathbb {Q} [\epsilon >0\Rightarrow \exists \delta \in \mathbb {Q} (\delta >0\land \forall X[0<|X-p|<\delta \Rightarrow |f(X)-L|<\epsilon ])]$.

$\lim _{n\to \infty }X_{n}=L\Leftrightarrow \forall \epsilon \in \mathbb {Q} [\epsilon >0\Rightarrow \exists N\in \mathbb {N} (N>0\land \forall n\in \mathbb {N} [n>N\Rightarrow |X_{n}-L|<\epsilon ])]$

--69.91.95.139 (talk) 11:14, 25 September 2008 (UTC)

You shouldn't use real numbers in the definition until everybody agrees what a real number is.

if and only if for every rational number ε > 0 there exists a rational number δ > 0

Although this leaves uniqueness unanswered. Tlepp (talk) 11:54, 23 September 2008 (UTC)

Good point. Changed accordingly. To that last sentence, I assume you mean the question of the uniqueness of a limit, as in, there is only one L which meets the definition? I would say that is implied by the equality, although, before a particular limit can be used, of course, it would have to be proved that that limit has exactly 1 solution L to the above definition. --69.91.95.139 (talk) 12:01, 23 September 2008 (UTC)

∏håπks ƒδr µs¡n∂ wδr∂s ¡πs†æd δƒ må†hεm冡çål s¥mßδls. Algr (talk)

I hope that's not sarcasm. I can always replace the definition that uses words with that using mathematical symbols if you want.

How long did you spend on that comment, anyway? --69.91.95.139 (talk) 10:08, 24 September 2008 (UTC)

There's probably a webpage somewhere that does it for you. --Tango (talk) 12:39, 24 September 2008 (UTC)

No web page. It's my personal spin on L33T speak. And it is sarcasm. How about words for ε, δ, and the intent for L and p?. As far as I can tell, it just looks like you are going on about completeness some more. Algr (talk)

I have addressed your needs above. However, in answer to your questions:

ε represents how close we want f(X) or X_n to get to the limit L.

δ represents how close we have to get X to p to get f(X) within ε of L.

L represents the value of the Limit.

p is the x-value approached to get the limit of f(X).

You didn't ask, but:

N represents how large n has to be in order for X_n to be within ε of L.

The definition works a bit like a really bad game: I give you an ε, and you give me an N or a δ to get within ε of the limit.

In a way, yes, I am going on about completeness. However, it really has more to do with you having a problem with the limit definition that you don't even know.

Happy bashing. :) --69.91.95.139 (talk) 11:09, 25 September 2008 (UTC)

Algr, it's time to stop this

"as far as I can tell". Algr, let's face facts. At this stage, your convictions here are stronger than your knowledge of maths is capable of rectifying. Someone to whom the 0.999... = 1 equality has little importance may be convinced by one of the "simple proofs" on the article. You, however, evidently believe the equality false with such conviction that it will be impossible, given your currently inadequate understanding of real analysis, for us to present a strong enough argument to convince you otherwise. You don't know enough maths for us to change your mind. So we have two reasonable options here to avoid this pointless conversation going around in circles forever.

• The first option is for you to leave, carry on believing whatever you want regardless of whether it's correct, and sit content in your ignorance-borne beliefs. This option will be easy but unsatisfying, and you will still be wrong.
• The second is for you to learn the maths. This will be more difficult for you as you have demonstrated a strong resistance toward learning any relevant maths in previous exchanges. Every time someone finally puts an unavoidable, unambiguous challenge to you to explain something or confess you don't know, every time someone finally lays out clearly exactly where your error in reasoning lies, you simply leave that thread and carry on elsewhere. So you will have to make a genuine attempt to learn a bit of maths BEFORE we can get to the question of 0.999... = 1. You will have to admit that you know less than other people regarding this issue. Don't worry, it won't be too hard. Nothing involved is beyond a first year university student level. Once you have learned the maths, if you still disagree, we will listen. But only after you have shown the commitment to actually learn what you're talking about.

So you have to make a choice now. I, and I'm sure many other editors here, are getting tired of you skimreading each post, trying to find a particular sentence or two, some little snippet that you can creatively interpret into something you can pull apart, while disregarding everything else. Ask yourself why you are still here, arguing this after all these months. Is it because you are after a bit of inflammatory internet argument with all its laughs, or do you really want to understand the mathematics involved and figure out the truth here? Maelin (Talk | Contribs) 14:22, 25 September 2008 (UTC)

Awww... You had to take the fun out of it!

Sorry, you're right. Of course you are. I absolutely agree that he should get some ecumicashun before he writes another word on this talk page.

But you still took the fun out of it. :( --69.91.95.139 (talk) 22:56, 25 September 2008 (UTC)

Declan Davis had plenty of "ecumicashun", and what good did it do him? You bullied him off of the forums by sheer volume, and pulled me back in because I had to tell you to be civil. So you are stuck with me. Forever. You guys think that the mathematical community is on your side, but your understanding is a parody of real mathematics, and you have all but admitted that this whole article is a con. BTW: When I abandon discussions, it is usually because they are going round in circles, or becoming distracted by the point, or I have a real life to attend to. Algr (talk) 04:58, 26 September 2008 (UTC)

We didn't bully Declan off, he left due to a disagreement of opinions. If you had any idea what he had been arguing, which you don't, you would understand that his issue is totally separate and distinct from yours. The mathematical community is on our side. Go ask any proper mathematician to look at our 0.999... article and then ask them whether 0.999... = 1 is true in the real numbers. I notice that you seem to conclude that discussions are going around in circles just when someone has finally refuted your point in a manner that you can't easily ignore. You are a troll, Algr. You aren't interested in getting to the truth, you are just interested in endless argument. You don't want to learn, you want to argue. Maelin (Talk | Contribs) 05:48, 26 September 2008 (UTC)

I want to learn, but not from people who directly advocate misinforming laymen and intentionally misinterpret what others have said. If you understood Declan Davis then why did you keep doing that? He was perfectly clear to me. The only reason you outnumber us is because every time someone who knows what they are talking about shows up, you guys overwhelm them and drive them away. You guys still have no idea what my objection is, nor do you care. It takes two to tango, and more then two to bully. Algr (talk)

We've watched you long enough to know that your objection is whatever lets you argue with us. For a while it was "we need the hyperreals so that we don't have 0.999... = 1." Before that it was "the real numbers are useless". You swing wildly between whatever claim you can use to float your arguments for a little while.

But maybe you've got a point. If you really think we haven't given you a fair go, prove that you're interested in one by making an effort to meet us halfway. You learn a bit of mathematics, we'll listen carefully and fairly to what you have to say. Or, if you prefer, we could just keep arguing pointlessly in this thread. Your call. Maelin (Talk | Contribs) 09:25, 26 September 2008 (UTC)

Just thought I'd drop by and see how all my old friends are getting on. I'm glad to see you're all getting on well. I shall not discuss any topic of the article since I have declared myself out of the debate, although I would like to say a couple of things. Maelin, you need to read WP:CIVIL, your last comments to Algr were just damn rude. I find you comment "You learn a bit of mathematics..." worst of all. As a 21 year old undergraduate you yourself have a lot of mathematics to learn. I had been tempted to mention this before, and thought it jolly bad form. There's a saying, how does it go? Oh yeah, don't throw stones if you live in a greenhouse. Also Algr was right, I left because I found it impossible to talk sense to anyone, it was like beating my head against a brick wall. Most people didn't seem to even read my posts fully, let alone try to sympathise with them, before getting on to their high horses and giving us all a speech. I have since been warned of the pitfalls to be encountered on these arguments pages.  Δεκλαν Δαφισ   (talk)  18:42, 3 October 2008 (UTC)

I should probably update that. I'm 22 years old now with a bachelors degree majoring in Pure Maths, completing an Honours year. But this is irrelevent because, as I said before, credentials are irrelevent in discussions of mathematics. Furthermore, Declan, you haven't been on this article for the two years that we've been going over the same old ground, over and over, with Algr. We've tried all manner of things. He ignores the rigourous proofs because he doesn't understand them, whilst claiming that the simpler, casual-layperson-targeted proofs are based on faulty reasoning. He latches onto ideas of which he has no understanding (the hyperreals and completeness, to name just two) and tries to use them to support his claims. Then he disappears once you've finally rebutted him in a way he can't weasel around, and turns up a while later jumping on somebody else's posts, claiming that he was busy. There are pages and pages in the archives of editors patiently trying to explain things to him while he employs standard troll tactics and ignores/misuses everything he doesn't understand. We have experienced his "discussion" style often enough to know that he is not interested in argument-as-a-means-of-establishing-truth, he is only here for argument-as-a-form-of-entertainment. Maelin (Talk | Contribs) 06:00, 4 October 2008 (UTC)

I see. Well you're right: I haven't been on this discussion page for two years. If what you say is the truth - which I'm not saying it isn't - then you are right to be annoyed. On a more touchy-feely note: we shouldn't allow ourselves to express these feeling of annoyance. If someone doesn't understand then we should try to be nice and help them understand. If they still disagree then it's often better to agree to disagree, move on, and focus on something more constructive.  Δεκλαν Δαφισ   (talk)  16:24, 5 October 2008 (UTC)

Unfortunately it is often not a case of people not understanding but rather of them not wanting to understand, which there is nothing much we can do about. Moving on is, indeed, often the best option. However, some of these people refuse to move on as well and will continue trying to argue their incorrect points of view in other sections. --Tango (talk) 16:28, 5 October 2008 (UTC)

"not wanting to understand"? I've explored every explanation you've come up with, and went through the trouble of researching all sorts of new ideas just to be sure I understood what you were talking about. You on the other hand don't even seem to know what my position is. I'm not saying 0.999... = 1 is false, I am saying it is declaratory. The article is written to tell people that =1 is observational, like the speed of light, when it is really more like the speed limit on a highway Do you understand how important that distinction is? Algr (talk) 08:01, 11 October 2008 (UTC)

0.999... = 1 is declaratory in the same way as 1+2 = 3 is declaratory or as it is declaratory that the sum of a triangle's angles is 180°. There are lots of times we've been through this. Still, this is an encyclopedia, and we report on "declaratory" truths just as on observational ones. Huon (talk) 08:36, 11 October 2008 (UTC)

The End (for me at least)

I am glad that so much conversation and debate has been generated about this article. I have learned a lot about myself, Wikipedia, and the human disposition over the last few days on this page. I think that we can all learn some lessons. Many of us (including myself) are too quick to judge, and too quick to dismiss. Many of us (including myself) have been rude and offensive. Many of us (including myself) have tried to put our opinion forward as fact. These are all natural things to do. I know that I shall try to learn from this discussion, and try to stop doing some of the bad things and try to do more of the good things. I guess one of the problems with mathematics is that it's too black or white, too right or wrong, and so our natural human competitiveness can come out in a bad way. I have seen this in every mathematical discussion page that I have read. All I can say is that some of us should agree to disagree. We should carry on working to improve Wikipedia. And finally, I hope that there are no hard feelings between any of us. After all: we're all on the same team.  Declan Davis   (talk)  02:39, 24 September 2008 (UTC)

I'm sorry to see you go, Declan. I think that a paragon shift is in the cards for mathematics soon. (Or at least the teaching of it.) It certainly won't be lead by me, but you may be a part of it. Algr (talk)

You right Declan. Lot of the people are fixed with their ideas and are not ready to change how they think. They like very much to say that they are correct and others are incorrect. I think you are wasting your effort trying to be nice. You have offered an 'olive branch' and no one (except Algr) have accepted it. Instead they ignore and they carry on to argue, almost like the wolfs that they eat the sheep. This is for why I do not write on talk pages anymore. Dharma6662000 (talk) 18:54, 24 September 2008 (UTC)

"This is for why I do not write on talk pages anymore.". Oops. 82.41.244.42 (talk) 20:38, 24 September 2008 (UTC)

Sorry. It is my bad English. What I wanted to say was that I do not write my ideas anymore. I had to write here because I have strong feelings about this and I need to say. Actually, you help to demonstrate my point. You rather laugh at what I write than understand what I mean. You understand the point I made, but you are happier to laugh and make yourself feel happy. This is exactly the point I was making, and you give example... thank you. Dharma6662000 (talk) 20:46, 24 September 2008 (UTC)

Declan's opinion simply does not reflect mathematical usage and terminology, despite his educational background. I can't imagine that we should pretend this is all a matter of opinion (rather than a matter of well-settled convention) for the sake of being nice. You may feel otherwise. Phiwum (talk) 21:27, 24 September 2008 (UTC)

Your profile it say that you study philosophy. For why you write mathematics page? Dharma6662000 (talk) 21:40, 24 September 2008 (UTC)

Because he/she is interest in mathematics, perhaps? The two subjects are actually quite closely related - they both depend heavily on logic. --Tango (talk) 23:30, 24 September 2008 (UTC)

As Tango says, it is possible to study both. You say something about my "profile", but I don't know what you mean. Perhaps you found a link to my home page, but if you did, you would learn that I'm not a mathematical novice (though I also do not have a PhD, as Declan has). In any case, I certainly don't ask that anyone rely on my authority. Phiwum (talk) 23:56, 24 September 2008 (UTC)

82.41.244.42 should take a look at WP:CIVIL, that comment was unnecessary. But since when has it become a custom to ask editors about their background? Editors should be judged by their contributions. Phiwum and Tango discuss mathematics, and to the best of my knowledge, their opinion is the mainstream one. If necessary, they could probably cite books to that effect. On the other hand, Declan's claims (for example, that irrational numbers shouldn't have decimal expansions) are anything but mainstream. For example, Cantor's diagonal argument for the uncountability of the real numbers exlicitly requires us to assign decimal expansions to (at least some) irrational numbers - in fact, it requires that any infinite sting of digits is indeed a representation of a real number. Appeals to authority won't help his case. Huon (talk) 23:29, 24 September 2008 (UTC)

It is the same. Declan have left the talk, you all say you are correct, so why you need to carry on talking? You are sure that you is write and he is wrong. Why you carry on to eat the sheep like a wolf? I do not see that you disagree with fact, but only with notations and conventions. This is a small thing. Why not move on? This is what he say earlier: "We should carry on working to improve Wikipedia." You say that you are right and he is wrong, so move on! Dharma6662000 (talk) 23:53, 24 September 2008 (UTC)

It's not really relevant to your point, but what a standard convention is is a fact. --Tango (talk) 23:57, 24 September 2008 (UTC)

No, Tango, this is not true. For some the natural numbers include zero, for others they do not include zero. A convention is a choice. You all agree on fact, but Declan does not like decimals and he likes limits. Others they say that decimals are OK. The fact is the same, the choice of view is different. I asked my professor about the limit question and he has just relied: "Yes, certainly, infinite decimals are limits. That is the only way to make sense of them. They say that, given ε > 0, there exists n(ε) such that the finite decimal with n places differs from a particular real number by less than ε. Of course this begs the question of the definition of a real number, but that is another story..." Dharma6662000 (talk) 00:09, 25 September 2008 (UTC)

No one disagrees that they are limits. I think you may have overinterpreted what your prof said. --Trovatore (talk) 00:13, 25 September 2008 (UTC)

Mike, you may missed something higher. Declan said (a long time ago) that infinite long decimals aren't defined and they are really limits. Most people said no. This is what I say. I know they are limits, you know, my prof know, Declan know. But people they like to argue, they like to be right, they like to be king. This is why they are how they are. Dharma6662000 (talk) 00:22, 25 September 2008 (UTC)

An infinitely long decimal, strictly speaking, denotes rather than is a limit, but most informed people asked whether it "is" a limit will elide that point and say yes. The limit that it denotes is a real number. There is no contradiction between something being a limit and being a real number. --Trovatore (talk) 00:28, 25 September 2008 (UTC)

I didn't say "convention", I said "standard convention". There is no standard convention on whether or not zero is a natural number, there is a standard convention on how to interpret a recurring decimal. Whether or not a standard convention exists is not a matter of opinion, it is a matter of fact. You can be of the opinion that the standard convention is a bad one, but you can't be of the opinion that it doesn't exist. --Tango (talk) 01:41, 25 September 2008 (UTC)

No I agree. The existance of a convention is a fact: it exist or no exist. But earlier you say "but what a standard convention is is a fact." You say that a standard convention is a fact. This is why I say that no is true. Existance and validness are different. Dharma6662000 (talk) 10:04, 25 September 2008 (UTC)

No, you're misinterpreting me. I think we agree completely. When I say "what a standard convention is is a fact" I mean the fact is "what the standard convention is", not "the standard convention". "It is a standard convention that recurring decimals denote a limit" is a fact. "Recurring decimals denote a limit" is not. --Tango (talk) 14:40, 25 September 2008 (UTC)

No, I no think that I misinterpreting. Please read what you has written in above. Maybe you talk too heavy and maybe you is changing mind, but that OK. I have dictionary and some time I understand what you say better that you do. What man write in moment is better shine on soul than what what man say after he thinks! I no like these English pages. For me it look like every English man he hate his brother. I never will understand the anglosaxon man. You was warriors in past, but time changes, why you no all be friends? Declan he try to make peace, but you all like to argue more. It very crazy. I hope you no insult by that what I say, but is what I say, is truth! Dharma6662000 (talk) 22:37, 25 September 2008 (UTC)

Where did your ability to speak English go? :-/ I'm pretty confident that I wrote what I meant, it's just a little confusing. Perhaps if I add some brackets it will be clearer: "(what a standard convention is) is a fact". That's distinct from "(that which is a standard convention) is a fact". --Tango (talk) 23:08, 25 September 2008 (UTC)

And where go your understanding of Civility? I may no be English, but I am not idiot. You write what you write. Maybe I should ask "Where your ability to write what you mean go?" Fact is fact friend. But actually no. You all seem to change fact when it please. Before you answer please read what you has wrote. I know you will not, that is why your next answer will be as silly as last. Dharma6662000 (talk) 02:59, 26 September 2008 (UTC)

Sorry, Dharma, I think it's pretty clear that you are the one misinterpreting Tango here. He wrote "what a standard convention is is a fact". The repeated 'is' makes his meaning clear: "the standard convention is x" is a fact. Your claims that you understand what he wrote better than he does are sheer arrogance on your part. Maelin (Talk | Contribs) 04:39, 26 September 2008 (UTC)

What a stardard convention is by wiktionary?

standard (adjective) 1. Falling within an accepted range of quality.

convention (noun) 1. A meeting or a gathering. (The convention was held in the arguments page.)

The arguments page convention has fallen outside an accepted range of quality. And that is a

fact (noun) 1. An honest observation.

Tlepp (talk) 06:43, 26 September 2008 (UTC)

I'm sorry if my comment about your English offended you, I'm just confused - you were writing in perfectly good English before and now suddenly your writing is full of mistakes. What changed? --Tango (talk) 15:33, 26 September 2008 (UTC)

Tango, a year ago I was trying to tell you what you just said right here, and you totally rejected it. Algr (talk) 09:38, 25 September 2008 (UTC)

Really? Could you provide a link? --Tango (talk) 14:40, 25 September 2008 (UTC)

It takes two to tango. For others may I suggest an article by Timothy Gowers.

A dialogue concerning the need for the real number system

or

The existence of the square root of two

Tlepp (talk) 10:23, 25 September 2008 (UTC)

These articles are quite good. Will respond later. Algr (talk)

In fact, you guys REALLY ought to read them yourselves... Algr (talk) 02:33, 26 September 2008 (UTC)

Done. Pretty interesting. --69.91.95.139 (talk) 11:00, 26 September 2008 (UTC)