Talk:0.999...

Rationality proof and proof by long division

As I have already indicated, I support relabeling the proofs offered in the Algebra section to something other than proofs (See discussion above). But I would now like to focus attention on two purported proofs which are particularly weak. Those are the long division "proof" and the "proof" assuming rationality. Long division is an algorithm, not a method of proof. In fact in this case the algorithm never terminates and therefore never has an final output. In the article this proof doesn't even have a final statement just an ellipsis indicating the algorithm goes on forever. Furthermore, the proof assuming rationality is also very weak. 0.999... is known to be rational only because it can be proven to be equal to 1 which is rational. Proving that 0.999... = 1 assuming that 0.999.... is rational is similar to proving 0.999... = 1 assuming 0.999... = 1. I strongly suggest the removal of these proofs.NereusAJ (talk) 21:26, 2 January 2012 (UTC)

I've been trying to tell them that for years. Algr (talk) 22:22, 2 January 2012 (UTC)

That's impossible. The two proofs I am referring to were only added last month.NereusAJ (talk) 22:51, 2 January 2012 (UTC)

Algr, I think you misunderstood me. I am not denying the equality. I think this article contains excellent rigorous proofs of the equality. I am just complaining about two specific proofs that were added very recently. The long division proof and the proof assuming rationality. NereusAJ (talk) 05:53, 3 January 2012 (UTC)

Seconded. The rationality proof was discussed already and it is seriously flawed, proving only that 0.999... cannot be rational, not that it's equal to 1. I am therefore removing it, based on the views expressed as and no-one has suggested a way to fix or improve it. The long division proof makes a bit more sense, is properly sourced, but it's using a very non-standard way of doing long division which really needs explaining, as it is in the source. The other proofs don't depend on such mental stretches, and so are I think much clearer, and the first of which uses similar arithmetic renders it unnecessary.--JohnBlackburne^words_deeds 08:23, 3 January 2012 (UTC)

See my comment above in Talk:0.999...#Proof assuming rationality. Tkuvho (talk) 11:14, 3 January 2012 (UTC)

I just noticed that the non-standard division proof was still in the article, even though we seem to agree it has too many issues, so per the objections above I've removed it.--JohnBlackburne^words_deeds 20:16, 22 January 2012 (UTC)

Cannot be written as stated.

From the article: "In mathematics, the repeating decimal 0.999... (which may also be written as ... "0.9" followed by any number of 9s." No it can't. That would represent a completely different number that is a terminating decimal as opposed to recurring decimal and would definately not equal 1. Zibart (talk) 19:57, 22 January 2012 (UTC)

It says up to the end of the bracket "or as "0.9" followed by any number of 9s in the repeating decimal", which means it could be

0.9... or 0.99... or 0.999... or 0.9999..., and so on.

It's simply stating that there's nothing special about 0.999... and its three nines. This is covered more fully at repeating decimal, which ends the sentence but is not linked as it's linked immediately before.

Does that make sense? I don't know if we want to add a detailed explanation, i.e. the above or even more from repeating decimal, as it would make the lead too long, and any reader who's not familiar with repeating decimals should start there before tackling this article. Or any suggestions on what could be added to clarify this?--JohnBlackburne^words_deeds 20:22, 22 January 2012 (UTC)

That was the intent of the sentence, but if Zibart has been confused about its meaning, maybe it could be replaced by the more literal: ...may also be written as "0.9" followed by any number of 9s and a final ellipsis. Diego (talk) 09:46, 23 January 2012 (UTC)

I was not confused I was simply worried that other people may write the number incorrectly. Zibart (talk) 14:33, 23 January 2012 (UTC)

The following edit was recently deleted: "(which may also be written as 0.9, $\scriptstyle \mathbf {0} .\mathbf {\dot {9}}$, 0.(9), indicating an infinite tail of 9s)". This captures the essence of the matter. Tkuvho (talk) 16:51, 23 January 2012 (UTC)

Yes, it removed Diego Moya's edit which I also agreed with (his edit was a good answer to my last question posed above). 'infinite tail' is much less clear: "tail" is not a mathematical term, and it is not necessarily infinite, just recurring i.e. repeating without end.--JohnBlackburne^words_deeds 17:11, 23 January 2012 (UTC)

I agree with JohnBlackburne and Diego Moya that we should mention "0.9999..." and similar notations as equally valid and that Diego Moya's edit does a good job. Huon (talk) 17:16, 23 January 2012 (UTC)

But the 0.999... notation already appeared before the parenthesis. Why do we need to do it again? On the other hand, the fact that there are infinitely many 9s should be mentioned. A recurring 9 produces infinitely many of them. Tkuvho (talk) 17:23, 23 January 2012 (UTC)

The sentence in parentheses is about the symbolic notation, which is neccesarily finite. The "any number of 9s" is to distinguish between "0.9...", "0.999..." or "0.999999..." which are all valid. Maybe its a good idea to add that all notations are meant to represent an infinite quantity of nines, but that's a separate idea (which applies to all notations, not only "0.999..."). If we agree to include it, it should be a separate sentence. What do you think of this version below? The new sentence is a way to write one level down by explaining how the technical term "repeating decimal" applies to this topic. Diego (talk) 18:40, 23 January 2012 (UTC)

In mathematics, the repeating decimal 0.999... denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigor, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience. The number may also be written as 0.9, $\scriptstyle \mathbf {0} .\mathbf {\dot {9}}$, 0.(9), or as "0.9" followed by any number of 9s and a final ellipsis; all these notations are ways to indicate that the dot is followed by an infinite number of 9s.

It's not necessarily infinite though. I mean recurring decimals are learned quite early, long before the formal notion of infinity is studied. So readers may have a clear concept of recurring decimals, that they repeat indefinitely/as many times as necessary. But that's not the same as an infinite number of 9's, and many readers won't understand it.--JohnBlackburne^words_deeds 19:02, 23 January 2012 (UTC)

I don't follow this; how would someone think that "indefinitely recurring decimal 9 digit" is different from an "infinite number of 9 digits"? And what does "as many times as necessary" mean when referring to an unending sequence of decimal digits? — Loadmaster (talk) 23:30, 23 January 2012 (UTC)

See e.g. the start of repeating decimal:

In arithmetic, repeating decimal is a way of representing a rational number. Thus, a decimal representation of a number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely.

It's possible to define repeating decimals, and so define 0.999..., without using 'infinity', which makes sense as repeating decimals are generally encountered by students long before infinity and infinite series. Again, in repeating decimal infinite series appears only towards the end, so the rest of the article does not depend on it.--JohnBlackburne^words_deeds 00:09, 24 January 2012 (UTC)

Excuse me, but nobody's talking of infinite series now. Recurring decimals is usually the first time students find infinite, in the form of an infinite string of characters (not a series). Repeating decimal just mentions infinite in the first paragraph and describes 0.99... as "an infinitely repeating sequence of nines". I agree with Tkuvho and LoadMaster that the ammount of repeating 9s in the number is best described as infinite even if any particular representation is not. I'm afraid the distinction you try to make between an infinite created with series and some kind of "not-infinite" created by an infinite repetition is too subtle and doesn't hold merit here. The problem you seem to have is the difficulty in understanding the idea of "an infinite number of 9s". That's one more reason to explain that infinite here means not having an ending condition that would produce a final '9' with nothing to the right; not to fully scrap the foundational idea of infinite. Diego (talk) 07:43, 24 January 2012 (UTC)

I think it is a good idea to mention the ellipsis, but the current sentence is a bit unwieldy. Why should the ellipsis be mentioned after all the alternative forms of denoting periodic decimals? It should be mentioned right after the notation 0.999... is first used. It seems to me that there is a consensus of scholars that there is a 9 for each of the natural numbers, which form an infinite set, which would make it an infinite tail of 9s. Students are reportedly confused in thinking that there are only finitely many 9s; shouldn't it be mentioned that there are infinitely many? Tkuvho (talk) 13:14, 24 January 2012 (UTC)

Any alternate wording is welcome anytime, of course. Diego (talk) 13:47, 24 January 2012 (UTC)

Except when repeating decimals are taught it is without referring to them as infinite. here e.g. is Wolfram (the second result after repeating decimal in this search) which describes repeating decimals fully and quite technically without using infinity. The concept of infinity is introduced somewhat later, after compulsory maths education ends, at least in the UK. So many if not most readers will understand 'repeating' but not 'infinite'. Per WP:MOSINTRO the lead should be as accessible as possible, so we should not use words in the definition which a large portion of the readership have not encountered, unless it is unavoidable.--JohnBlackburne^words_deeds 13:21, 24 January 2012 (UTC)

The wolfram page you linked to confronts terminating and repeating decimals, so repeating decimals are thus non-terminating (a term that is also used to describe non-repeating decimals for irrationals, that are explained at the same instruction level). I coincide with Loadmaster that I don't understand how you can make a distinction between "not terminating" and "infinite" (which is exactly the same, but in latin). Anyway, it didn't take much work to find several text books that describe "infinite repeating decimals" (see [1], [2], [3]) . This one in particular is targeted to Elementary School Mathematics and describes repeating decimals as infinite. Diego (talk) 13:59, 24 January 2012 (UTC)

I agree. User:JohnBlackburne's assumption that the article should be addressed to highschoolers is not necessarily endorsed by WP:MOSINTRO . Tkuvho (talk) 14:47, 24 January 2012 (UTC)

it should be as accessible as possible, which in this case means to highschoolers as the topic is not too advanced for them, and the first two proofs in particular are accessible to anyone that's encountered repeating decimals as they otherwise use only basic arithmetic. See e.g. MOS:MATH#Article introduction: "The lead should as far as possible be accessible to a general reader, so specialized terminology and symbols should be avoided as much as possible".--JohnBlackburne^words_deeds 16:41, 24 January 2012 (UTC)

The term 'infinite' is not specialized terminology, as in this case is used with the common meaning of "unbounded or unlimited" and not with any particular mathematical definition. If 'infinite' is not acceptable neither is 'repeating decimal' that does not have a meaning in common language (since "decimal" is always a mathematical term). Diego (talk) 17:48, 24 January 2012 (UTC)

See again my comments above and the Wolfram example: infinity may be common to me or you but there are many readers who will not know what it means because they have not learned it yet or finished their mathematics education before encountering it. They may know is as a common word but may not know what it means (it's used often just as a superlative, or for dramatic effect), and are unlikely to appreciate its mathematical meaning.--JohnBlackburne^words_deeds 01:21, 25 January 2012 (UTC)

You don't have evidence that using the "infinite" word will cause problems to readers, several editors agree that it's an accurate description, and we have reliable sources using that word in the context of repeating decimals as long division. The Wolfram page that you referenced uses "terminating"; we could explain infinite as "non-terminating" to differenciate it from the other common meanings you mentionted, which is in accordance to WP:Explain jargon (do not depend on wikilinking), Wikipedia:TECHNICAL#Don't oversimplify (not "tell lies to children") and WP:EXPLAINLEAD (provide an accessible overview). The only remaining question is the exact words by which this is to be described. I suggest asserting that having "infinite digits [created by a non-terminating long division] means that any '9' will have another '9' to the right, so there's no final digit". This would address one of the common confusions of students (the "final 9 at the infinite"). Diego (talk) 09:54, 25 January 2012 (UTC)

Teaching math

I've found this source discussing 0.999...=1 with respect to a "teacher's mathematics" category. It poses that students will not be convinced by one single proof or explanation but that showing a collection of varied explanations is what provides grounding for the concept. It has 30 citations, should we add it to the 0.999...#Skepticism_in_education section? Diego (talk) 10:29, 25 January 2012 (UTC)

"This last explanation does not help students who have not seen some of the other explanations, but it does show the consistency of mathematics and helps to give closure on the idea. This is teachers’ mathematics."

I was unable to access the reference, nor to reproduce the google scholar stats. Tkuvho (talk) 10:53, 25 January 2012 (UTC)

Sorry, it looks like I posted a link to a cache. Try the link now. The reference is for "Teachers' mathematics: A collection of content deserving to be a field" by Zalman Usiskin.Diego (talk) 11:26, 25 January 2012 (UTC)

Since it has 30 scholar cites, I see no reason why it shouldn't be mentioned. Tkuvho (talk) 11:34, 25 January 2012 (UTC)

Clear example why 0.999... != 1

moved to Talk:0.999.../Arguments#Clear_example_why_0.999..._.21.3D_1 Bulwersator (talk) 10:21, 2 February 2012 (UTC)

Why .999... does NOT equal 1

moved to Talk:0.999.../Arguments#Why .999... does NOT equal 1 Double sharp (talk) 03:20, 4 May 2012 (UTC)

By Definition

This article may benefit from the addition of a simple definition explanation. That is, there is a definition for this number, and that definition is a limit. From here, we can see that this limit converges. And thus, buy a=b and b=c, we have a=c.

$0.999\dots =\lim _{n\to \infty }(1-{\frac {1}{10^{n}}})$ (by definition).

$\lim _{n\to \infty }(1-{\frac {1}{10^{n}}})=1$ (convergent).

Hence, $0.999\dots =1$ (a=c, because a=b and b=c from above).

This is irrefutable, true, simple, and easy to understand. Tparameter (talk) 03:08, 31 May 2012 (UTC)