Misplaced Pages

Revision as of 21:36, 17 December 2005

Because of their length, the previous discussions on this page have been archived. If further archiving is needed, see Misplaced Pages:How to archive a talk page.

Previous discussions:

• Archive 1 (2005-05-06 to 2005-11-16)
• Archive 2 (2005-11-16 to 2005-12-07)
• Archive 3 (2005-12-07 to 2005-12-09)

Merge proposal

I have proposed that this page be merged with positional notation, as it is only really interesting because it expresses the peculiar property of positional notation that there are multiple distinct sequences of digits representing identical real numbers. I also would consider making it briefer. Deco 02:15, 10 December 2005 (UTC)

• Disagree. If anything, this article should be longer. The topic may have limited interest in real mathematics, but it's a fascinating educational issue. You're welcome to expand Positional notation, but this article couldn't be merged there without destroying it. Melchoir 02:59, 10 December 2005 (UTC)

I have to question the appropriateness of this page. I also find the question fascinating as an educational issue, and I think a coherent evaluation of the topic is a very useful thing for a lot of students. However, the article seems a bit too "instructive," and I wonder if it really belongs in an encyclopedia. --Monguin61 03:08, 10 December 2005 (UTC)

Monguin61, maybe the article gets "instructive" at times, but surely that can be fixed, leaving an informative article, without deleting it? Melchoir 03:26, 10 December 2005 (UTC)

That might be possible, but its the topic itself that bothers me more than the style. There aren't many other articles that go into such depth on a specific problem or example, are there? Removing the content entirely is unnecessary, but I do think that in light of the nature of the article, merging it with positional notation or a similar article might be in order. --Monguin61 03:31, 10 December 2005 (UTC)

Yes, I doubt that there is a comparable article of comparable length. Prompted by the merge suggestion, I've been looking for a better possible target for the merge. Positional notation is a poor choice, since it doesn't address infinite tails, and it doesn't specialize to the decimal system. I've found Recurring decimal, which is a much more natural home -- and similar content is already there! The arguments at Recurring decimal are a mess, so if you want to merge this article there, I won't complain. However, someday I'll take a crack at writing an encyclopedia article on 0.999... myself. Melchoir at 85.195.123.22 04:18, 10 December 2005 (UTC)

Scratch that, Monty Hall problem is longer, and has a section on "aids to understanding." Considering that it was a featured article, I guess this one is alright. --Monguin61 10:16, 10 December 2005 (UTC)

Oppose. So you find this topic boring and trivial. I find Madonna boring and trivial. Each still demands a dedicated Misplaced Pages article because they draw notable attention. This topic is raised repeatedly: on the sci.math newsgroup, in schools around the world, and all over the net. It is, in fact, a Frequently Asked Question for sci.math; but nowhere, that I have found, is the topic treated nearly as well as here.

And why is that? Because people with a course in real analysis under their belts dismiss it with a handwave, and elementary school teachers who know the psychological obstacles their students face do not have the expertise in foundations to handle it securely.

Or look at the endless discussions on this very talk page. (First, notice that half the posts ignore the most basic rules for all talk pages and lack signatures, indenting, and civility. Second, notice that, contrary to the stated purpose of talk pages, the discussions ignore the article itself.) Most telling, neither side in the debate argues well; the advanced proofs in the article are noticeably more careful than those seen on the talk page.

It takes a certain amount of mathematical sophistication to appreciate the difference, to get beyond the handwave "it's a limit, deal with it" argument repeated over and over to no avail. Also, as with many sensitive topics (abortion, for example), we must be careful in more than mere facts; we must choose our words carefully, because psychology and emotional impact matter. Which is also why it would be insensitive and misguided to shorten the article. Mathematically, a proof page is needed for no theorem, certainly not this one, because Misplaced Pages includes no original research. Nevertheless, we find proof articles valuable. On such pages we do not limit ourselves to a single brief proof; consider Proofs of Fermat's little theorem, which has five proofs.

Both the merge proposal and the shortening proposal are ill-considered, and should be rejected. --KSmrq 06:12, 10 December 2005 (UTC)

I appreciate your sentiment above, but as you mention, the topic is a Frequently Asked Question, and it seems that the article is there to attempt to resolve that question for anyone who might be looking for an answer. Is that really what Misplaced Pages is for? On the other hand, you do have a good point that the topic is treated better here than elsewhere, and to me that is reason enough for the content to remain. --Monguin61 06:36, 10 December 2005 (UTC)

• Oppose. This article is far from complete. Among other things, it fails to point out that the z-transform of a p-adic expansion of a real number has an image that is a Cantor space. What does this mean? Consider the 10-adic (aka "decimal") expansion of a real number, with digits ${\displaystyle d_{n}}$. The z-transform is then

${\displaystyle \sum _{n}d_{n}z^{n}\,}$.

Then, for z=1/10, one gets exactly the decimal expansion of a real number, but for other z-values, even z-values infinitessimally close to 1/10, one gets Cantor sets of various shapes. This is a non-trivial observation, and seems to underly the properties of fractals in particular, and chaos in general. The resulting topologies are non-trivial in a variety of ways. Note, in particular, the z-transform of 0.9999... is not at all equal to the z-transform of 1.00000... except when z=1/10 (precisely and exactly). (An example of a high-falutin' version of this is the Alexander horned sphere.) linas 07:45, 10 December 2005 (UTC)

Oh, and why should one mention this? I think this will make a lot of the people who don't believe that 0.999..=1.000... a lot happier, and give them something to think about. Although it is "technically" true that 0.999..=1.000.., the z-transform makes it clear that decimal expansions are actually very pathological. Its a mathematical slight-of-hand that holds true only for the very special value of z=1/10, and makes something look smooth when its really not. So I'm hoping that this extra tit-bit can make everyone go home happy. linas 07:55, 10 December 2005 (UTC)

Are you really? And I'm hoping for world peace. ;-) --KSmrq 08:49, 10 December 2005 (UTC)

In particular, one can construct maps from the Cantor set onto the unit interval, in such a way that 0.9999... and 1.000 correspond to two distinct points of the cantor set. (note the word "onto": the cantor set really does have the cardinality of the real number line, plus, you could say, "a little bit more", an extra countable infinity of points).linas 08:06, 10 December 2005 (UTC)

That's a pretty interesting topic in itself. As far as satisfying those who don't believe that 0.999...=1, though, I think it would work more just to confuse. Correct me if I'm wrong, but this article only covers z=1/10, right? Using any other value for z is simply asking a different question, and only tangentially related. --Monguin61 08:08, 10 December 2005 (UTC)

Removing tongue from cheek, you do make a valid point, linas, about fun diversions. The article mentions that the construction of reals from rationals depends on ordering, so there is a hook there for the valuation that leads to p-adic numbers. It could also be instructive to delve into non-standard analysis, or the impact of replacing the usual Set topos with something more interesting. For example, Dedekind cuts and Cauchy sequences can give different "reals" in a non-Set topos. But I don't think I'm clever enough to write such an article without losing either the audience or my composure. And given the fun the forces of chaos have without encouragement, I'm not inclined to hand them more toys. --KSmrq 09:16, 10 December 2005 (UTC)

To respond to my various protestors: I don't find this topic boring at all, I have a Bachelors degree in mathematics. I proposed the merge because I thought it was a reasonable compromise between those who wanted the page removed altogether and its supporters — I'd rather see the content preserved than lost. At the very least I think positional notation should include a brief summary of this article and a link to it. Deco 20:05, 10 December 2005 (UTC)

Another response: You say "Positional notation is a poor choice, since it doesn't address infinite tails, and it doesn't specialize to the decimal system." It certainly should discuss infinite tails, and the 0.999... = 1 issue is not at all specific to the decimal system. For example, in octal, 0.777... = 1, and in binary, 0.111... = 1. Moreover, it extends to any real number ending in an infinite sequence of (largest digit), such as 0.263999... = 0.264. The example discussed by this page is just a very specific case. Deco 19:08, 13 December 2005 (UTC)

I agree about expanding Positional notation to discuss the issue of infinite tails, and in all bases, with a link here. I think I meant to say simply that Positional notation would be a poor choice for a merge, especially since there are better homes. But it's been awhile, and after reading Monguin61's, linas', and KSmrq's replies, I would no longer support a merge to any article, so my original comment is kind of moot. Melchoir 00:09, 14 December 2005 (UTC)

In light of the views stated here, I'm removing the "proposed merge" tag. If Deco or anyone else thinks I'm acting too hastily, feel free to revert. --KSmrq 00:31, 14 December 2005 (UTC)

Adding the z-transform

KSmrq is worried, but I say go for it, linas. Rather than encourage the forces of chaos, I think such examples might help convince our doubters that, yes, mathematicians have seriously considered the possibility of setting 0.999... not equal to one, and the consequences thereof. It shouldn't cause trouble as long as you write it after the discussion of the decimal series, and you don't reuse notation. Melchoir 18:04, 10 December 2005 (UTC)

Delete: ... 0.999... is equal to 0.999... - this is its sum. It is not equal to 1. In fact, simple induction shows it is less than 1 in the decimal system which is how most people interpret 0.999... People do not think of 0.999... as the limit of its partial sums any less than they think of 4 as the limit of its partial sums (4 + 0/10+0/100+...) This article ... should be deleted or rewritten to show that in a positional system, 0.999... is less than 1. ... 0.999... = 1 is an ambiguous and chancy statement. Either qualify it or discard it completely. Do you want people to be attracted to Mathematics ...? Limit are fairly easy to understand. If you want to talk about limits, then you should make it clear you are talking about limits. In every other sense, 0.999... is treated as a number just like 0.333... just like 4, just like any other number in the decimal system. Do the right thing - DELETE! Let KSmrq write another article... — Preceding unsigned comment added by 68.238.101.29 (talk • contribs) 2005 December 13 (UTC)

Anon, if you honestly want to discuss this article, you're going to have to be a lot more polite about it. Melchoir 18:20, 13 December 2005 (UTC)

No, he has a good point. There should be a section at the top giving a precise definition of the terms and notations used. With that, no more argument would be possible. --Monguin61 23:28, 13 December 2005 (UTC)

He may have a good point, and we should be more precise in the article. But have you seen anon's above post before I got to it? I won't encourage that sort of behavior. Melchoir 00:03, 14 December 2005 (UTC)

In what way should we be more precise? Each of the advanced proofs defines reals, equality of reals, and the interpretation of a recurring decimal in terms of reals. However, I would vigorously disagree with putting such material at the top. Quoting from Misplaced Pages:Manual of Style (mathematics):

• A general approach is to start simple, then move toward more abstract and general statements as the article proceeds.
• It is a good idea to also have an informal introduction to the topic, without rigor, …

The article follows the guidelines appropriately, down to the concluding "Generalizations" and "See also" sections. --KSmrq 02:33, 14 December 2005 (UTC)

Anon's post wasn't appropriate, obviously. The thing about the general guidelines is that they're general, and they don't take into account the nature of this particular article/debate. This debate is going to continue on the internet as long as there are people that do not understand the entire question being asked. I think, for the sake of driving away the impolite anons that have some insatiable desire to argue here, an exact definition of terms and notation should be introduced in the article, very clearly. I think if this article exists at all, it should do everything possible to quell the discussion. Unlike some of the other subjects of debate on this site, the specifics of the decimal system do not depend on your politics, or your opinions. There is one, only one, right answer. It is unfortunate that so much must be written to convince some people of this, but it is true. --Monguin61 19:15, 14 December 2005 (UTC)

...you have stated that you "love to mercilessly edit and correct these"... 158.35.225.231 13:36, 14 December 2005 (UTC)

Isn't it ironic that after you maliciously misquote me, I delete everything else you say? Gosh, the world's a funny place. Melchoir 18:42, 14 December 2005 (UTC)

EEK

I'd just like to take a timeout and say...ouch. This talk page makes my head hurt. Since when is mathematics so...political. I love math because there are right and wrong answers, and I seriously can't believe this has progressed over 2 archives. Oh, and psst.. .999...99 = 1. :D Arkon 05:14, 15 December 2005 (UTC)

If you think this is crazy you should see Misplaced Pages: Lamest edit wars ever. :-) Deco 05:28, 15 December 2005 (UTC)

Thanks for the reminder; we were due for a new archive!

I won't comment on the sociology or politics. Mathematics does have right and wrong answers; it also has choices. Over the millennia some of the choices troubled people. Should we write numbers using powers of 60, as in Babylon? Should units, tens, and hundreds have their own symbols, as in Rome? Are negative numbers really numbers; can they be correct solutions to equations? What are we to make of incommensurable numbers like √2, which caused consternation in Greek religion? And yes, the solution of some real equations seems to pass through "imaginary numbers" to get real answers, but are those "imaginaries" really numbers? Following the quaternions, can we make up a number system following crazy rules like "multiplication doesn't commute"? Does it make sense for a proof to be able to conclude anything from a falsity, to use the law of the excluded middle; or should proofs deny existence without construction? If a statement is true, will we always be able to prove it?

Are these old, settled questions? Our clocks and angles still use 60. Accountants use double-entry bookkeeping and parentheses so they never touch negative numbers. We still call √(−1) an imaginary number. Is it really that surprising that repeating decimals trouble some people?

I don't think that's what generated the volume of discussion on this page, but it has done so elsewhere.

Mathematics is a human endeavor. So is Misplaced Pages. So is life. Ain't it grand? :-) --KSmrq 07:20, 15 December 2005 (UTC)

I say the whole topic is simultaneously grand and lame. Melchoir 07:24, 15 December 2005 (UTC)

...Who the hell do you think you are Melchoir? .... AS for me, I do have better things to do. --anon

I am neither an admin nor a sysop; I practice soft power and I get away with it because I am right. Please, don't let me hold you up on your business elsewhere. Melchoir 20:06, 15 December 2005 (UTC)

The anon is being rude, but I personally disagree with this practice of editing others' talk page comments solely to remove mildly offensive rantings, even if they are off-topic. The beauty of free speech is that when everybody speaks their mind, ridiculous things like trolling and personal attacks just get ignored. Don't worry about it. Deco 20:40, 15 December 2005 (UTC)

Well, I started removing the off-topic stuff because I thought it would discourage the addition of more. It seems I have failed in that aim. Melchoir 21:52, 15 December 2005 (UTC)

You seem to have failed at a lot of things and now you are trying to forcibly silence your opponents? This is what a dictator does - so you are not wrong by calling yourself one.

Are you done? Melchoir 23:35, 15 December 2005 (UTC)

What's it to you? Don't read the posts if they are too harsh for your wimpish character.

Now are you done? Melchoir 00:16, 16 December 2005 (UTC)

Again, what's it to you?

I am interested in discussing the article. Melchoir 00:18, 16 December 2005 (UTC)

Well, now I am done.

New content

I've removed the following recently added "Counter Argument" from the article:

"Consider this: 0 is nothing, 1 - 0.9_ is not nothing, but 0.0_1 . 0 is not equal to something infinitely larger than 0, it's smaller. Therefore 0<0.0_1, 1>0.9_ . 1 is larger than the indication of a number approaching 1, no matter how close to 1 it is, because of this, it is never equal, always greater. The problem with the argument of whether .9_ = 1 is the definition/workings of the decimal, non-rational form of expressing rational numbers. The only situation where you could say .9_ = 1 is if you assume that by definition, that numbers follows a repetitive series infinitely, but infinity has a limit, which is where the number is equal to it's limit. This is a flawed interpretation of the notation; Really what the notation would mean is that the number continues as stated by the repeater indicator, but continues INDEFINITELY, NEVER reaching it's limit, which is why this notation is inaccurate to a rational notation.

If you say you have 1 of something, you have 1. you do not have an infinitely repeating quantity of numbers which approach 1. If you wanted to say you have an infinitely close quantity to 1, then you would use .9 +repeater symbol (underscore in this case (0.9_) and NOT 1."

I'm not sure what that passage is trying to say, but it contradicts the rest of the article. Whoever boldly added it didn't do anything wrong, but if you want to keep it in the article, you'll have to defend it here first. Melchoir 20:21, 16 December 2005 (UTC)

I concur. An article should never contradict itself. It also doesn't really make much sense to me. The phrase "infinitely close to" is meaningless, at least in this setting, many sequences never reach their limit, and the rest of it seems to contradict the usual definition of the intended meaning of infinite expansions in positional notation. Deco 20:34, 16 December 2005 (UTC)

Actually, they did do something wrong. Misplaced Pages has a strict policy of No Original Research. The proposed material violates that policy, just as surely as a statement that the Earth is flat or denial of the Nazi Holocaust. Attempts to insert these kinds of things in any article are routinely reverted, and repeated attempts will result in a block on the perpetrator. --KSmrq 23:37, 16 December 2005 (UTC)

Whoever boldly added it has more common sense than the lot of you combined. He does not need to defend it - it is self-explanatory. KSmrq needs to defend the article. Every one of his so-called 'proofs' has been refuted by me. Again, I ask: "what does infinity working behind the scenes mean"? Melchoir says the phrase "infinitely close to" is meaningless - what non-sense. It's as meaningless as when you talk about "as close as you like" or "as small as you want". This article is about as useful as me stating that there are not 360 degrees in a circle. Of course there are not 360 degrees in a circle - we have defined it this way. However, there is circular measure that can't be disputed (radians). 1/3 is not equal to 0.333... so Ksmrq's first proof fails. The limit of 0.333... is 1/3 but its sum is 0.333... Under Advanced proofs there is really nothiing but a whole lot of handwaving. As for the order proof or Archimedean proof - this too is false for you can find infinitely many numbers between 0.999... and 1 as I have demonstrated clearly. The limit proof is nonsense because the actual sum (0.999...) is not equal to its limit. KSmrq gets horribly confused under generalizations by misunderstanding that this happens in every radix system simply because many numbers cannot be represented finitely in all radix systems. This is a limitation of using radix systems. This article says nothing: it starts off with a vague claim without any definitions. It is very badly written and assumes that the reader thinks along the same lines as Ksmrq - wrong! Unfortunately this will probably also be edited because it's true and Misplaced Pages only publishes inaccurate information. Until just a few years ago, anyone who would have said there were more than 9 planets in the solar system would have been condemned as a fool. Today he would have been correct. Once again, this article is Original research. Show me one reliable source - an encyclopedia or a textbook wrwitten by more than one authority that proves what this article claims. You have not been able to do this because this article is false. — Preceding unsigned comment added by 71.248.136.72 (talk • contribs) 2005 December 17 (UTC)

First of all, I made the statement that "infinitely close to" is meaningless - or at least, very informal. The inserted text isn't exactly wrong, but it's so informal and vague that it makes no verifiable or meaningful assertion. However, describing our assumptions — in particular, describing the definition for the meaning of an infinite decimal in positional notation — is a great idea and I'll do this. Deco 02:19, 17 December 2005 (UTC)

I inserted a bunch of new text. The idea behind everything I wrote is that "it's natural to define the value of an infinite decimal using infinite sums". I think this is the point of contention here. I also tried to make it clear that the point of the proofs is not so much to prove this fact, but to justify the choice of definition; with a different definition, these proofs would fail, and we'd have to throw all sorts of familiar proof techniques out the window. I hope this helps to find some consensus. Deco 03:13, 17 December 2005 (UTC)

I've moved the new text to the end, and restored the previous intro. Here's why.

My first major objection is that it is not suitable for kids. In its first paragraph it uses exponential notation, 10, including a negative and a variable. In the second paragraph, two sentences later, it talks about countable and uncountable sets. The third paragraph introduces infinite summation, with sigma notation, and limits. Such material at the top ignores the guidance found in Misplaced Pages:Manual of Style (mathematics):

• Probably the hardest part of writing a mathematical article … is the difficulty of addressing the level of mathematical knowledge on the part of the reader.
• A general approach is to start simple, then move toward more abstract and general statements as the article proceeds.
• It is a good idea to also have an informal introduction to the topic, without rigor, …

This article will be seen by students in elementary school. The notation and level of mathematics used in the original intro (now restored) try to respect that; this attempt doesn't even come close.

My second objection is that it clashes with the careful definitions and proofs already given in the body. It is fine to give yet another set of definitions and proofs in the advanced section, if that will help some readers. But in the intro, this kind of text gives the sense that we must use infinite series and limits, which is neither true nor helpful. It's especially troubling that it does not even lay the groundwork for a naive reader to understand what these mean. Most readers do not have a background in analysis, and Melchoir dug up some informal surveys of college mathematics students, discussed in an archive, that shows widespread confusion.

My third objection is that, despite the length, the new material does not leave me with a sense of greater confidence and clarity. The language is not simple, the concepts are not simple, and both pile on. One minor, but perhaps telling, example of overly complicated language is the following sentence:

• This non-uniqueness of nonterminating representations is often counterintuitive for people more familiar with terminating representations.

We're looking at three negations in a row (non-uniqueness, nonterminating, counterintuitive); that's not reader-friendly. It's especially unfriendly for the elementary school students who are likely to be exactly the ones "more familiar with terminating representations". Compare this to the simple, everyday language of the restored intro.

My fourth objection is to this statement:

• Note that the point behind these proofs is not to actually show that some distinct nonterminating decimal sequences represent the same value, but rather to justify our definitions above.

That is not my view, and I find it confusing. I agree that it may be helpful and appropriate to explain why we define reals and the meaning of decimal notation as we do, but that is not what the proofs are about. The proofs take standard definitions and show how they lead to the asserted fact, 0.999… = 1.

I believe the new material needs to be simpler, shorter, and clearer, even in its new location. If the point is really to justify the standard definitions, then state that at the beginning of the new section, and tackle it directly.

And even if the justifications are wonderful, some folks will always object. --KSmrq 07:42, 17 December 2005 (UTC)

Thanks for the in-depth analysis, but it seems like you expect to be attacked on this! I for one am fine with it. Melchoir 07:58, 17 December 2005 (UTC)

Er, by "it" I mean your move. Melchoir 07:59, 17 December 2005 (UTC)

Trying to be courteous and constructive. When someone has gone to the trouble to write in good faith, however well they've succeeded, they would probably appreciate some explanation for a dissent, especially a revert. If there is no objection, unless the new material is substantially revised I'm inclined to proceed to actual removal.

Doing mathematics requires different knowledge and skills than writing about it in an encyclopedia. My analysis may turn out to be overkill, but (before it joins the archives) it may also be instructive. Suppose I say: (1) know your audience, (2) start simple, and (3) be clear and direct. Which is more helpful: the generic advice, or seeing it applied?

Besides, isn't it nice for a change to see a talk page used for its intended purpose? ;-) --KSmrq 08:27, 17 December 2005 (UTC)

It's quite the revolutionary concept, yes. Melchoir 09:28, 17 December 2005 (UTC)

Wheeee!

Of course the whole article is counter-intuitive. We can do almost anything we like in mathematics (as long as it is logical): we can have circles of 240 degrees. We can even define bases where transcendental numbers are no longer transcendental. Not only children are confused but many adults with Phds in mathematics. I have communicated with accomplished Phds in Mathematical Statistics, Applied mathematics and Pure Mathematics. Some of them agree that 0.999... is not equal to 1. Fine you say, and then use an Archimedean corollary to claim that a number can be found between the two if they are not equal. But of course you can find numbers between 0.999.. and 1 - infinitely many of them. The Archimedean property is one that supporters of the argument in favour of 0.999... = 1 like to use to show that 0.999... must be equal to 1. I would hardly call the limit argument a proof at all. It is simply irrelevant. However, it is misunderstood and incorrectly used. For example, any arithmetical operation with recurring decimals is not possible (except as an approximation) just as any exact operation with pi, e, etc is not possible either. To show that 1.999.../2 = 1 or 0.999..., you first have to demonstrate that the quotient is possible. There are problems with any radix system so that not all laws are satisfied and certain anomalies always exist. To make statements such as "infinity behind the scenes" is not only vague and ethereal but it implies a certain offensive arrogance that states the author understands infinity and we all know that the human mind can only speculate about infinity. Any profound results regarding infinity are arrived at through experimentation and speculation involving limits. The sum of 0.999... (since you are treating it as a sum) is 0.999..., not 1, not approximately 1 but certainly less than 1 is all we can say. This is how radix systems were constructed. — Preceding unsigned comment added by 68.238.103.100 (talk • contribs) 13:12, 17 December 2005 (UTC)

That we consider so many things in mathematics makes it all the more important that we stick to our definitions. Melchoir 18:37, 17 December 2005 (UTC)

Concerning the Archimedean property cited above: It is, iirc, not used to show that there are no numbers between 0.999... and 1, but to show that the difference between 0.999... and 1 cannot be greater than 0. A more detailed proof along these lines can be found in Archive 2. A short summary:

The difference is either 0 or an infinitesimal. 
By the Archimedean property, there are no infinitesimals in the set of real numbers. 
Thus, the difference must be 0. 

If one intends to claim that 0.999... is not 1, one would have to doubt one of these statements. Due to its shortness here, the first one is most open to criticism, but I refer to the more detailed proof cited above.--Huon 21:36, 17 December 2005 (UTC)