Talk:0.999...

Nonzero or non-zero

I would be writing the term as "nonzero." Any opinions, or arguments against that? Twipley (talk) 01:14, 30 March 2011 (UTC)

My dictionaries show "nonzero" as the standard spelling, so go for it. 28bytes (talk) 03:29, 30 March 2011 (UTC)

While that may be the case with your dictionary, non-zero is also correct. It appears both ways all throughout WP, oftentimes in the same article, as this Google search and this Google search demonstrate. So it's not really clear at all which form is better. — Loadmaster (talk) 17:46, 27 April 2011 (UTC)

New thread

1/9 does not equal .111.... It is an approximation. —Preceding unsigned comment added by 165.155.196.69 (talk) 15:26, 14 April 2011 (UTC)

More precisely, it is an adequality. Tkuvho (talk) 15:32, 14 April 2011 (UTC)

The article could devote more attention to the student intuition that "0.999..." is adequal, rather than equal, to 1. Before the number system is specified (e.g. as being the real numbers), such intuitions are not erroneous but rather nonstandard, see R. Ely's article. Tkuvho (talk) 18:34, 16 April 2011 (UTC)

That's precisely what the article already says. Giving more prominence to the Ely article would be undue weight. Regarding the re-addition of 165.155.196.69's comment, please have a look at WP:TALK: "Article talk pages should not be used by editors as platforms for their personal views on a subject", and "[i]rrelevant discussions are subject to removal." This clearly includes your n-th attempt to proclaim that students somehow intuitively think of hyperreals instead of reals. I'd say the arguments page is a nice case study to the contrary: Hardly anybody who disagrees with the equality instead thinks of the hyperreals - the hyperreals just violate their intuition in other places than the reals. Huon (talk) 18:44, 16 April 2011 (UTC)

OK, I agree with what you wrote. At any rate, it was a legitimate issue to be raised, and I see no reason to delete it summarily as some kind of an expletive. Tkuvho (talk) 19:48, 16 April 2011 (UTC)

The mention of adequality was just deleted from the lede, which is a pity. The comment deleted nicely summarized the infinitesimal section. I have mentioned numerous times that it is not hyperreals that the students intuit (that would be remarkable indeed!), but rather a number system containing infinitesimals, as envisioned by Fermat and Leibniz. The latter certainly were not thinking in terms of hyperreals. Tkuvho (talk) 12:42, 17 April 2011 (UTC)

Well, in the decades in which I taught mathematics, I never once came across a student who appeared to "intuit" either hypereals or a number system containing infinitesimals. What I saw year after year was students who perceived decimals as strings of figures, rather than as an abstract concept which has strings of numbers as a convenient concrete representation. If your concept of what a decimal number actually is is a string of figures, then clearly 0.99999... is not the same as 1. And if you think of the order relation on decimal numbers being defined in terms of that string in some such tems as "Compare the numbers digit by digit from the left until you find the first difference: then the number which has the larger digit in that place is the larger number" then clearly 0.99999... < 1. Since these students have learnt the properties of decimals by rote from a very early age, they have internalised some process along those lines. From the point of view of a person with a high level of mathematical understanding it is clear that logically the difference in that case would be infinitesimal, but in my experience that is not how it is perceived by the vast majority of people who cannot accept that 0.999... = 1. They simply are not thinking in such terms. Yes, you can push them in the diection of thinkinbg in such terms by, for example, asking them what 1 - 0.999... is, but I have never seen any evidence that they think in such terms spontaneously. However, in my opinion this is not the main point. The main point is that the article essentially is about what 0.999... represents in the real number system, about the popular miconception that it does not represent the number 1 in that system, and about what reasons there are for accepting that in fact does. To insert stuff about hyperreals or infinitesimals into such an article confuses and muddies the issue for the average reader, and detracts from the clarity of the essential point that the article seeks to convey. Mathematicians are not the principal readership. JamesBWatson (talk) 18:11, 17 April 2011 (UTC)

Speaking directly about the article, I'm not sure that adequality is worth mentioning in the lede. We do link the term lower in the article, in the section on infinitesimals. Because infinitesimals are already not directly the subject of this article, I personally prefer to keep the lede sentences about them very tight. — Carl (CBM · talk) 18:12, 17 April 2011 (UTC)

Yes, that is very much in keeping with my own thoughts. It is mentioned, but it is not germane to the central point of the article, and should be resricted to the place where it is most relevant to the context. Certainly not in the lead. JamesBWatson (talk) 18:20, 17 April 2011 (UTC)

OK, whatever consensus emerges here is fine. Tkuvho (talk) 07:32, 18 April 2011 (UTC)

To respond briefly to JamesBWatson's detailed remarks above: Huon pointed out correctly that R. Ely's paper should not be given undue weight. On the other hand, it cannot be ignored altogether, either. JamesBWatson argues that student misconceptions about .999... result from their thinking of a number as being a string of digits. This is one possible interpretation. However, his claim that there is no evidence for any other interpretation is not correct, as Ely documents in his field study. The claim that some students think of 1-0.999... as an infinitesimal is documented in the education literature and is no longer in the realm of pure speculation. Tkuvho (talk) 10:17, 20 April 2011 (UTC)

I did not say that "there is no evidence for any other interpretation". I said that I had not seen such evidence. JamesBWatson (talk) 22:42, 20 April 2011 (UTC)

If you want students to intuitively grasp .999... = 1 there's an easy way imo, tell them a story about a King that wishes his cooks to make him a pie but to pre-slice it in diminishing divisions 1/10th the size of the last because the King wishes the option to eat any sized slice he wants without having to cut the pie again. The 1 single pie would come out and be presented to the king on a plate in an infinite amount of smaller slices starting with 9 slices of 1/10th size each, 9 slices 1/100th sized each, 9 slices 1/1000th sized each and so on. If you looked at the pie with your eyes squinted you wouldn't see the slices but the whole of what they form - 1 single pie. And there is no infinitesimal bit of pie missing because the cooks did nothing but take a knife to the existing pie and were careful to not let any crumbs fall out :P 76.103.47.66 (talk) 23:23, 25 April 2011 (UTC)

I like it. Cliff (talk) 00:00, 26 April 2011 (UTC)

Each slice the cook makes does not even come close to the "infinitesimal bit" at the end. At what stage do you claim that your "infinitesimal bit" disappears, if not when you apply the limit to sweep all infinitesimals under the rug? Tkuvho (talk) 04:59, 28 April 2011 (UTC)

I find this is a particularly smart metaphor. There are not infinitesimal pieces of cake that disappear, there are infinitesimally closer and closer cuts - each cut represented by a digit in the secuence. When you apply the limit, the final cut is performed at exactly 1.0 distance from the origin. Wait, this is now the Zeno's paradox!! Diego Moya (talk) 08:51, 28 April 2011 (UTC)

Please keep in mind that it was the previous editor, rather than myself, who pointed out that "there is no infinitesimal bit of pie missing". If you assume from the start that there are no infinitesimals, of course there is going to be none left at the end of the process. What editors keep trying to do all over again is prove "from first principles" the non-existence of infinitesimals. This is circular reasoning. Tkuvho (talk) 08:56, 28 April 2011 (UTC)

I understand what each of you said. But this is this what I find clever about the metaphor. There 'are' infinitesimals in it, but they are not used to describe the pie (the real number 1) but the cut actions (each of the "0.9...9" fractions). With this intuition there can't be an infinitesimal missing part of the pie which makes the series distinct from a whole pie; all the "payload" -the pieces of pie are always part of the cake. Diego Moya (talk) 10:16, 28 April 2011 (UTC)

It is unlikely we will get a culinary degree this way, but if you do assume that there are infinitesimals in the pie, it is difficult to claim that "there is nothing left". As you say, the pieces of the pie are always part of the cake, but at each cut there always remains a last uncut slice. If one assumes that there are infinitesimals there, then an infinitesimal "at the end of the pie" is always in the remaining slice. Therefore it is never eliminated. Bon appetit. Tkuvho (talk) 10:23, 28 April 2011 (UTC)

And therefore movement is impossible? :-) Of course you always have infinitesimals - before you reach the infinite, while you have "arbitrarily small but bigger than zero" uncut slices. But when you 'jump' to the infinite you arrive at exactly 0 distance from the origin to make the 'last' cut, so there can't be any uncut pie left. This jump is more difficult to understand when the limit is explained as 'what you have when you add an infinite amount of increasingly small fractions'. Is there a referenced demonstration that uses sectors of a circle? If so it could be used in the article to illustrate this insight. Diego Moya (talk) 11:31, 28 April 2011 (UTC)

On the contrary, if you do have a last cut, then you have a precisely measurable infinitesimal leftover, see 0.999...#infinitesimals. Tkuvho (talk) 12:28, 28 April 2011 (UTC)

I thought we were talking about real numbers (physical cake and all), where there are no non-zero infinitesimals? I don't think the pie metaphor is a good fit for alternate numeral systems. Diego Moya (talk) 12:56, 28 April 2011 (UTC)

Tkuvho, the cake analogy is not offered as a proof, but as a way of helping ones students overcome their misunderstanding when it comes to the real numbers, and the 0.999...=1 statment. Diego Moya, yes. We are working from the assumption that our number set is the reals, this is why Tkuvho's arguments make no sense here. Underlying the statment 0.999... = 1 is an assumption that our set is the real numbers, (it works in the rationals too). Cliff (talk) 14:52, 28 April 2011 (UTC)

To clarify my original pie story, I am not starting with the assumption that there is no infinitesimal missing pie. I started with the assumption that there is 1 pie and you can cut it up in to any number of slices you wish. If you cut it into 10 slices you have 10/10 = 1. If you cut one of those slices into 10 you have 9/10 + 10/100 = 1. Cut one of those and get 9/10 + 9/100 + 10/1000 = 1. Next is 9/10 + 9/100 + 9/1000 + 10/10000...If instead of this linear geometric approach you simply think of every possible division existing at once without having to cut down to it you can see that any number of slices in any the particular division would be 9. So another way to think of 1 whole pie is at 9 slices each in infinite descending place values in base 10. Or, .999....76.103.47.66 (talk) 00:37, 2 May 2011 (UTC)

What Tkuhvo is saying, anon 76, is that according to some number systems, your story leaves an infinitely small slice out of the sum. Such number systems are usually doctoral level study and not widely known. Look for information on "non-standard analysis" to see what he's saying. Cliff (talk) 03:22, 2 May 2011 (UTC)

Thanks, a fair summary except for the word "doctoral". Recent education studies report successful undergraduate calculus teaching using infinitesimals. Tkuvho (talk) 04:14, 2 May 2011 (UTC)

Yeah most of that goes over my head. Like a lot of people I found the concept of .999... = 1 to be counter-intuitive and was never completely convinced by any of the proofs (even the ones i understood thoroughly) until I approached the problem in a way that I could visualize the equality rather than simply acknowledge it, and that's how I came up with the pie idea. I thought it might help others as it did me in solidifying the equality intuitively without having to resort to complex theories. As for the non-standard analysis stuff, I cannot comment on that and leave it for smarter people than me :) 76.103.47.66 (talk) 06:07, 2 May 2011 (UTC)

Another way of seeing it geometrically is by flipping the problem to the other side of 1, namely by considering 2-0.999... What kind of number could it be? Here the decimal digit 1 at rank n gets pushed off further and further to infinity. Since there are no infinite ranks in the ral number system, in the limit we get 1.000... on the nose. Tkuvho (talk) 12:03, 2 May 2011 (UTC)

Oh good grief. 1/9 is equal to 0.111..., in the sense of being simply different textual representations of the same real number (just like 0.5 + 0.5 is equal to 1). It is not an adequality, as defined by that article, except in the trivial sense of being an identity William M. Connolley (talk) 16:09, 3 May 2011 (UTC)

fuzzy rendering... (not the same old question)

In the section on non-standard systems, the N renders on my browser as fuzzy. not clear or sharp. Anybody know why this might be? Cliff (talk) 16:55, 22 April 2011 (UTC)

Shows up as a blackboard bold N on my browser. 28bytes (talk) 17:03, 22 April 2011 (UTC)

mine too, but it's not clear. It's fuzzy. Cliff (talk) 17:06, 22 April 2011 (UTC)

same person, different browser. It renders fuzzy in chrome and in firefox. 134.29.231.11 (talk) 17:11, 22 April 2011 (UTC)

Hmm. It shows up as hollow but clear on my machine on both IE and FF at standard resolution, but if I zoom in or out even slightly, it does fuzz up a bit. 28bytes (talk) 17:16, 22 April 2011 (UTC)

Ah, that's probably it. I think I had zoomed in a bit because my eyes were bothering me...time to visit the opthamologist again I think. Cliff (talk) 04:55, 23 April 2011 (UTC)

Nope, that's not it. The N in text renders nicely at default zoom. The N in the equation still seems out of focus. Where does it come from, does anybody know? Is it a font on my machine or is it a "picture" that is downloaded with the webpage? Cliff (talk) 21:03, 25 April 2011 (UTC)

The N in the text (... where $[\mathbb {N} ]$ is...) is significantly larger than those in the equation (e.g. ${\frac {1}{10^{[\mathbb {N} ]}}}$); maybe that's why it renders better at default zoom? Both should be images; the TeX code used to write them is, I believe, turned into PNG images. The large one can be seen here. Huon (talk) 21:16, 25 April 2011 (UTC)

Is the small one just a shrunken version of the large one or its own image? Cliff (talk) 00:02, 26 April 2011 (UTC)

There's more to this. It's not just the $\mathbb {N}$. All the text in the math equations are fuzzy. I guess they are PNG images, I don't know, but at 150% zoom, all the math symbols and digits in between the standard text appear with the same fuzziness as other images, like for example the Wikipedia logo at the upper left of all pages and also the Wikimedia and Mediawiki banners at the bottom of most all pages. The $\mathbb {N}$ seems to appear even more out of focus because of the double-barred diagonal. More of an optical illusion going on there. Zoom to 150 and you will see many examples of other text and symbols with the same fuzziness.

For an even more curious example, zoom on the following, the lead from this article:

In mathematics, the repeating decimal 0.999... which may also be written as 0.9, $\scriptstyle \mathbf {0} .\mathbf {\dot {9}}$ or 0.(9), denotes a real number that can be shown to be the number one.

The $\scriptstyle \mathbf {0} .\mathbf {\dot {9}}$ appears fuzzy when zoomed to 150%, but all the other numbers and letters in this sentence remain clear and sharp on my browser. Kinda weird eh? Racerx11 (talk) 04:05, 26 April 2011 (UTC)

I can explain that one, at least... The 0.9 with the dot over it is the only one of the three rendered as a PNG; the others are text. Previously the 0.9 with a dot over it was also rendered as text using either CSS or a Unicode combining dot character (I forget which), but the dot couldn't be positioned correctly (depending on the browser, it would be shifted to the left or right) so that was changed to use the PNG method. 28bytes (talk) 04:18, 26 April 2011 (UTC)

I think you are exactly right. The PNG characters all appear fuzzy when in zoom. The $\mathbb {N}$ is particularly troublesome because, like I said, it has that double-barred element, in addition to it apparently also being rendered as a PNG image (at least it looks the same as the PNG images). This double diagonal makes it appear more out of focus even at low and normal zoom. Racerx11 (talk) 04:33, 26 April 2011 (UTC)

And to follow up on the original point; when certain PNG characters are shrunk down such as in (e.g. ${\frac {1}{10^{[\mathbb {N} ]}}}$), the small $\mathbb {N}$ in particular looks even more out of focus. Racerx11 (talk) 04:54, 26 April 2011 (UTC)

Is there a way of "fixing" this? can the images be replaced with higher resolution ones that don't have this problem on zoom? Cliff (talk) 14:54, 28 April 2011 (UTC)

Probably not without modifying of the MediaWiki software. The images are automatically created from the TeX code - I don't think there's a way to manually change that creation process to increase image resolution. Huon (talk) 11:55, 2 May 2011 (UTC)

There is mathJax, which instead of PNG generates SVG, and arguably looks better if scaled. It's not built into MediaWiki, but can be added manually with a couple of steps described at User:Nageh/mathJax.--JohnBlackburne^words_deeds 12:05, 2 May 2011 (UTC)

1 = .999 . . . is technically a mistatement

A better statement is that the limit of 1 - 1/n as n approaches infinity is equal to 1.

The distinction is clear if you consider the limit of (1 - 1/n)^n as n approaches infinity. The value of this limit is 1/e or approximately .38787. . .

On the other hand the limit of (1)^n as n approaches infinity is 1. In this case 1 does not give the same limit as .999 . . ., which shows that the equivalence depends on the context.

Prokrop (talk) 01:18, 17 May 2011 (UTC)

Actually the statement is correct as long as you interpret 0.999... to be the real number whose decimal expansion consists entirely of 9's, for all negative powers of 10. (And 0 otherwise of course.) But if you read the history of this page, you'll find many discussions on the subject. Thenub314 (talk) 01:29, 17 May 2011 (UTC)

Especially, .999... is the limit of 1 - 1/n (or more precisely, the limit of 1-1/(10^n)) as n approaches infinity; it's not the sequence itself. The limit of $(\lim _{n\to \infty }1-10^{-n})^{m}$ as m approaches infinity is just as much 1 as the limit of (1)^m as m approaches infinity. Huon (talk) 02:48, 17 May 2011 (UTC)

Hi Huan, can you summarize my position for me here? Perhaps we can write a separate FAQ to address this type of comment, which as you know I know you know, is not completely misguided. Tkuvho (talk) 05:18, 17 May 2011 (UTC)

Sorry, but I don't think I know your position on this issue well enough to summarize it for you; if I had to speculate I'd assume that you'd want to say that modulo the choice of an ultrafilter, interpreting 0.999... as a sequence will lead to a hyperreal. But that ultrafilter step is essential and non-trivial, and I don't think it's what those who think about 0.999... as a sequence have in mind. A hyperreal isn't more of a sequence of reals than a real number is a Cauchy sequence of rationals. Anyway, an addition to the FAQ regarding the "sequence vs. number" question may be useful. Huon (talk) 12:16, 17 May 2011 (UTC)

Thanks :) However my starting point wouldn't be to jump to ultrafilters, but rather to relate to the fact that many students think of an infinite string of 9s as a terminating one, i.e. there is a "last" one. That particular intuition can be fruitfully implemented in an infinitesimal-enriched continuum. The particular construction you choose (such as ultrafilters, etc) is not so important, as such systems can be described axiomatically, as well. Tkuvho (talk) 12:58, 17 May 2011 (UTC)

A "last 9" doesn't seem to be the issue here. If I understood Prokrop correctly, he didn't argue that 0.999... is different from 1 because there is a "last nine at infinity" or something like that, but rather says that 0.999... is a sequence of reals which simply never reaches its limit (which he agrees would be 1, not something a little short of 1). Those are two different misinterpretations of 0.999... Huon (talk) 13:17, 17 May 2011 (UTC)

will 0.999... always equate the same as 1 in all instances?

1 raised to the power of any number = 1 as in 1ⁿ as n→∞ = 1. Does that hold true for 0.999...ⁿ as n→∞ = 1. I have tried to work the math behind this but have as yet been unable to determine how to fit an infite series into an infite series. Any help here would be great. —Preceding unsigned comment added by 68.170.209.249 (talk) 08:21, 17 May 2011 (UTC)

Short answer: Yes. If you have an expression like $\lim _{n\to \infty }(\lim _{m\to \infty }1-10^{-m})^{n},$ you should evaluate the limits from the inside out. And since $\lim _{m\to \infty }1-10^{-m}=1,$ you end up with $\lim _{n\to \infty }(\lim _{m\to \infty }1-10^{-m})^{n}=\lim _{n\to \infty }(1)^{n}=1.$

Note that while $\lim _{n\to \infty }(\lim _{m\to \infty }1-10^{-m})^{n}=\lim _{n\to \infty }\lim _{m\to \infty }((1-10^{-m})^{n}),$ in general $\lim _{n\to \infty }\lim _{m\to \infty }a_{n,m}$ and $\lim _{m\to \infty }\lim _{n\to \infty }a_{n,m}$ will differ, and neither need be equal to $\lim _{n\to \infty }a_{n,n}.$ Huon (talk) 12:37, 17 May 2011 (UTC)

Saturday Morning Breakfast Cereal

See a Saturday Morning Breakfast Cereal cartoon: [1]. Fridek (talk) 13:20, 18 May 2011 (UTC)

0.999_ does not equal 1

http://uncyclopedia.wikia.com/wiki/0.999...

In mathematical theory .9 is a specific number. It is not half or a fraction of a number. It is .9 and it remains .9 in the equation. Unless an equation calls for rounding of the numbers, then .9 remains solely .9 in the equation. Therefore 0.999_ is a infinitely repeating number.

In mathematical practice there are no exact numbers. As reality is incapable of producing an exactly equal number in relation to something. So in mathematical practice 0.999_ cannot equal 1 regardless.

In mathematical theory where rounding occurs then 0.999_ equals 1.

If rounding is not established or .9 is specifically established as being an exact number, then no amount of crying and bitching changes it to 1.

You've been Uncyclopediaed, yo. 58.7.214.181 (talk) 03:50, 19 May 2011 (UTC) Harlequin

The "rounding" you referred to can be mathematically implemented either by Fermat's adequality or the standard part function, you may find those pages helpful. Tkuvho (talk) 03:59, 19 May 2011 (UTC)

Perhaps you should go back and read where I specifically, in detail, referred to equations where it wasn't being "rounded". But if all you can come up with is "no, no...if they are not infinitely repeating, then it totally equals 1", it's just further proving my point. 58.7.214.181 (talk) 04:03, 19 May 2011 (UTC) Harlequin

And yet you're the one trying to use Uncyclopedia like it's a reliable reference or something. Do you realise that most of that article was written by people who wrote this article, who needed to blow off some steam after arguing with "equality deniers" on this very talk page? Confusing Manifestation(Say hi!) 04:11, 19 May 2011 (UTC)

I'm using Uncyclopedia as a reference? Haha, well done at further proving my point and again failing to read. Strange, seems I explained exactly how 0.999_ does not equal 1 without anything relating to the Uncyclopedia article on the subject. I merely pointed out another site that makes fun of the bitching and crying those like yourself are doing.

I gave the equation separate from any other site. Which you should have known had you, you know, actually read anything. Hahaha.

0.999_ repeating in mathematical theory can only be 1 if it is not infinitely repeating. You're claim here is that somehow an infinitely repeated number will magically have an end and that it will magically round up once it has reached that end, even when the equation it is in specifically states it doesn't. Like ive stated, no amount of bitching and crying changes that fact that an infinitely repeating number does not have an end.

It's pretty hilarious really. I even explain that where it isn't infinitely repeating and is in an equation that "rounds" the numbers that it would equal 1 (again, had you actually read anything) and yet you seem to think that an equation where it can't even reach an end to "round" would somehow still do so even when it's specifically mentioned that none can occur?

Try again. This time without trying to take credit for other sites material. Especially when the best "argument" (or lack of) you can come up with is that "some guys" created the other article (despite it's refutation of "their" supposed claim that 0.999_ ALWAYS equals 1) and because "they did" it is therefore magically wrong. 58.7.214.181 (talk) 04:39, 19 May 2011 (UTC) Harlequin

Yo Har, is there a last "9" in your 0.999_ after infinitely many 9s? Uncyclopedia says there is. Do you agree? Tkuvho (talk) 04:15, 19 May 2011 (UTC)

I notice you don't know what infinite means. There is no "end", so there can't be a "last 9". Try again. Especially since I didn't state anything about Uncyclopedia being correct and specifically provided a detailed explanation of why the claim "hurr 0.999_ always equals 1 no matter what" is a false statement. 58.7.214.181 (talk) 04:39, 19 May 2011 (UTC) Harlequin