Talk:0.999.../Archive 15

Image in lede
This image is a bit pointless (it doesn't provide any further clarification above that provided by the article text) and it looks pretty amateurish to boot. It should be removed again. Chris Cunningham (not at work) - talk 09:41, 1 July 2009 (UTC)


 * I disagree, as do many other editors. See the archives (not sure of the date) where this was discussed at great length. Any further consideration about removing or replacing the image should be discussed (again) here on the talk page first. — Loadmaster (talk) 18:53, 1 July 2009 (UTC)
 * I support keeping the image, and don't find it at all amateurish. Algr (talk) 21:09, 1 July 2009 (UTC)

I think the image is good to have, but ought to be revised to use a different font, perhaps a conventional sans-serif or serif font. I think this would reduce the "amateurish" impression. --216.171.189.244 (talk) 20:05, 6 September 2009 (UTC)
 * Anyone is welcome to try another font. I didn't experiment with many. POV-Ray is free software available for all platforms, and commons:File:999 Perspective.png contains the source code. Just change the two ttf fields and adjust "space" and "depth" to get the kerning right for the new font. Please upload any new version to a new filename, so that the versions can be compared side-by-side. Melchoir (talk) 20:32, 6 September 2009 (UTC)

A recent edit by 69.14.160.108
The user 69.14.160.108 added the following sentence to the digit manipulation proof:

"One supposed problem, however, does exist with the above proof. When multiplying .999... by 10, one of the nines is moved to the left of the decimal. The one "altered" set of nines is then equated with the "unaltered" set of nines. Some students struggle with this fact and thus do not accept the equality of .999... and the integer 1."

I'm not sure that the sentence belongs in the article. First, I would argue that the supposed problem does NOT exist, since the values after the decimal point in 0.9... and 9.9... are the same regardless of what process produced the numbers. Second, you could point to any proof in the article and say "this proof uses property X; some people doubt X"; I don't see the value in appending such messages in a mathematical article. Gustave the Steel (talk) 21:14, 16 July 2009 (UTC)


 * I agree. --Zarel (talk) 02:36, 17 July 2009 (UTC)

Circular reasonings
Yes 0./9/ = 1, no doubt about that, why is it one, because it's defined as one. That's all there is. The point is that 'a 0 with "infinite" 9's behind it' is hardly mathematical rigour. So you have to define infinite digits in some way, and they are defined quite simply in the epsilon-delta way. Say there is a repeating pattern behind it, the repeating digit is defined as the constant that can be arbitrarily approached by just repeating that pattern enough times. And that's just to make sense of a way to write down numbers, having nothing to do with what they really 'are'. We write in Arabic numerals, it cannot properly represent 1/3, so 0./3/ is just defined as being equal to 1/3, the constant it can approach as close as you like if you just add enough threes. And it all depends on this definition. All the arguments are begging the question.

First argument, it already starts with that 1/9 = 0./1/, it proves supposedly that 0./9/ = 1 (just another form of that 1/9 = 0./1/) by assuming that equality as the first given? It tries to prove that behaviour of digits by first assuming it? Circular reasoning.

Second argument: The digit manipulation, we say that x = 0.999... right? Who says that 0.999... is an object, a mathematically sensible thing? Shouldn't it be defined what it means first before we can just say that x equals it? And who says you can multiply it with ten to move all digits to the right. Yeah, it's proven that you can if you have a finite number of digits, not if they're infinite unless of course you use the property of the density of real numbers to define it as 1. Thereby begging the question.

The infinite sequences sums it up. It says it's simply defined as an infinite series how I explained above. A mathematical object means nothing until it's defined in other objects that are understood until you come at the very bottom of proof theory and then you just have faith people understand what you mean with rule of inference. I can also define in my own books that 0.ape, making no sense from here is equal to 0.5 and then baffle people with my circular proofs that it's so because they assume my definition. The difference here is that this definition is commonly accepted as it makes intuitive sense and it solves a handy inconvenience to write down objects like 1/3 in a positional numeral system.

The final point is that I find it a stupid way to test students aptness at maths, as they are often asked this before having been introduced to the formal definition of the limit, either in words or epsilon-delta terms. The point is they have not seen a definition of 0./9/ yet and if they have the proper question to ask them comes down to: 'If I give you a random real number x, can you get the difference between 1 and 0.999... smaller than x if you can just add as many 9's at the end if you like for any random real number x no matter how small?', and then of course they would about all answer 'yes'. Most in fact give the correct answer by just saying 'I don't understand.' and not getting it, because '0./9/' means nothing until you define it as $$\sum_{n = 1}^{\infty} 9 \cdot 10^{-n}$$

I didn't give any 'source' as neither do the arguments on this page, I belief the arguments are concise above and all the arguments on the page are circular reasonings. That 0./9/ = 1 is a dogma, an axiom, a definition, nothing more. Rajakhr (talk) 15:25, 30 September 2009 (UTC)


 * We all know the digit manipulation argument, etc., aren't rigorous proofs. They aren't meant to be. They are meant to convince people that don't have the necessary training to understand an epsilon-delta proof. --Tango (talk) 16:13, 30 September 2009 (UTC)


 * There's a difference between a colloquial argument and a circular reasoning. In fact, a circular reasoning can be quite rigorous and logically very valid, they always are extremely formal as an axiom is a theorem. The point is more that the article puts it as some universal truth that flows out of established mathematics and older results while it's a definition, nothing more. No one is 'wrong' to deny that 0./9/ = 1, they've just defined repeating decimals differently if they do. I personally chose to interpret that symbols aren't objects, they indicate them to human readers. '1' is by agreement the symbol for multiplicative identities, successors of the first natural or however you want it. And 0./9/ is by agreement equal to it. There is no mathematical object 'zero point nine and then "infinite nines". ' Rajakhr (talk) 18:55, 30 September 2009 (UTC)


 * This reminds me, I still haven't written the commentary bits we were talking about earlier... well, all things in the fullness of time. Maybe after I finish with that other article. Melchoir (talk) 16:34, 30 September 2009 (UTC)


 * I don't see the circular reasoning. 0.999... is not defined to equal 1; rather there's a general definition for all infinite decimals which happens to make 0.999... equal to 1. That needs to be proved. Especially I don't see the circular reasoning in the two proofs pointed out by Rajakhr. The first proof indeed assumes that 0.111...=1/9, but that result can be reached by long division without assuming 0.999...=1. The "digit manipulation" indeed assumes that infinite decimals behave similarly to finite decimals under multiplication by 10, but again, where's the circle in our reasoning? Huon (talk) 19:43, 30 September 2009 (UTC)


 * No mathematician in her right mind would dream of defining 0.999... to be one - that would be plain stupid. Next she'd had to define 1.999..., and at some point 4321.407999... etc. Instead, some definitions are made about series, the meaning of non-terminating decimals, etc. - and from these it follows that 0.999... = 1. To say that "it's just a definition" makes an important point, but also misses the point.Noe (talk) 06:52, 1 October 2009 (UTC)
 * I rephrase then, it's defined as the infinite series, that that infinite series is one is pretty trivially one. I quote myself: 'because '0./9/' means nothing until you define it as $$\sum_{n = 1}^{\infty} 9 \cdot 10^{-n}$$', it's a definition and students are asked what 0./9/ means without having been introduced to the infinite series and the limit and thus the question makes no sense to them. However they are led to believe, and this article leads people to believe, that '0./9/' is 'a zero with infinite nines behind the decimal', which is not mathematical rigour, you need to define what the symbol represents. That is, the infinite series and the article goes out proving this without even defining forehand what it represents. Rajakhr (talk) 11:15, 5 October 2009 (UTC)
 * Actually, the symbol 0.999... is defined, in the introduction section, as "the limit of the convergent sequence (0.9, 0.99, 0.999, 0.9999, …)." That, of course, does not help the layperson, and I'm not sure why you're suggesting that the article should be more mathematically rigorous than it already is, since it is directed, first and foremost, at laypeople. They have both right and wrong intuitions about decimal numbers, and this article has to try and knock down the wrong ones. --COVIZAPIBETEFOKY (talk) 21:02, 7 October 2009 (UTC)
 * I didn't mean to mark this edit as a minor edit; my mouse slipped. I need to get it fixed. --COVIZAPIBETEFOKY (talk) 21:04, 7 October 2009 (UTC)

My Insane Rantings
So 1/3 does equal .3..., so you could say the continuous threes are basically and infinite about of 3s. However infinity is a concept. Since .3...... involves a concept you cannot mulitply is by 3 and justify it equaling 1. Techiqually you cant multiply by a concept, so when you mulitply .3... by 3, youre actually multiplying 1/3, a non-concept number, by 3. However since a calculator can't understand concepts it gives you .9..., which is another concept.

SO taking all this in account, you can't actually multiply .3... by 3 because it is a concept. When you do that you are actually multiplying 1/3 by 3, which is not a concept, and gives you 1. —Preceding unsigned comment added by 75.189.196.97 (talk) 20:18, 21 October 2009 (UTC)


 * Ho ho! This person forgot to make his own section, instead inserting it into an old discussion where it didn't belong! Oh well, just gives me the right to title his post myself. Hope you like it, anon.
 * Anyways, here's your answer, anon (unfortunately, I could not find a font big enough):
 * NUMBERS
 * ARE
 * CONCEPTS!
 * HTH. --COVIZAPIBETEFOKY (talk) 21:37, 21 October 2009 (UTC)
 * CONCEPTS!
 * HTH. --COVIZAPIBETEFOKY (talk) 21:37, 21 October 2009 (UTC)
 * HTH. --COVIZAPIBETEFOKY (talk) 21:37, 21 October 2009 (UTC)
 * HTH. --COVIZAPIBETEFOKY (talk) 21:37, 21 October 2009 (UTC)

first paragraph should
mention that, like all fractions, 0.999... is a sum (9/10 + 9/100 + 9/1000), and, in fact, a sum of infinite addends, these adding up to "1". 0.999... is a sum just as 1.000... is a sum. (1 + 0/10 + 0/100 + 0/1000). Once the reader is reminded that decimals -- whether terminating or not -- are sums, it will no longer be surprising in the least that two of them (1 and 0.999...) can be the same, any more than it is surprising that 1, 1.0, 1.00, 1.000, and 1.0000 are the same sum.

By contrast with the above, correct initial reminder, the current phrasing is misleading. 1.0 and 1.00 don't "represent" the same number: they are sums whose value is the same number. The first "represents" 1 + 0/10 and the second "represents" 1 + 0/10 + 0/100. Likewise, 1 and 0.999... don't "represent" the same real number (as is currently stated). Rather, they represent the sum 1/10^0 (single addend) on the one hand, and 9/10^(-1) + 9/10^(-2) etc (infinite addends) on the other. As soon as the reader is reminded of this, 99% of the rest of the article is unnecessary. Decimals are, very simply, shortcuts for sums of powers of ten.


 * "Sum of infinite addends" usually isn't defined at all. It's a series (or, for full precision, the sum of a series), which is a special case of limit of a sequence. There is a significant difference between how finite and infinite decimals are constructed (although, of course, the finite decimals can be seen as a special case of the infinite ones, corresponding to the sequences which become constant - but then even the finite decimals cease to be seen as "sums"), and I don't think we should try to hide that difference. Huon (talk) 21:57, 2 November 2009 (UTC)

If 0.9999... equals 1, I cannot understand why a monkey typing for an infinite amount of time cannot re-write everything written by Shakespeare (see the 'infinite monkey' article). There seems to be a contradiction here.86.153.54.29 (talk) 21:12, 25 November 2009 (UTC)
 * A monkey typing for an infinite amount of time can re-write all of Shakespeare... (they aren't guaranteed to do so, just almost sure to, but that difference is a very subtle one). --Tango (talk) 21:25, 25 November 2009 (UTC)
 * Given enough time and a key typing process that is truly random, they will. (This assumes that the typing process is random enough to guarantee that all possible subsequences of keystrokes are equally likely.) — Loadmaster (talk) 23:50, 2 December 2009 (UTC)

How?
How can one number be the same as another? I don't doubt that 0.999999999 etc is close to 1 but it is not one. If that was the case you can say subtracting 1 from 0.999999999 would give you zero, but that's not true. Also is 1.9999999999999999 the same as 2? etc etc?

It is sufficient to say that it is close enough to, to say it equals 1, but technically it is not 1. 203.134.124.36 (talk) 03:07, 23 October 2009 (UTC)


 * I'll assume that you meant to add ellipses to the end of each of your trailing nines, to indicate infinitely many nines, rather than some large finite number of nines.
 * Can you, perchance, tell me exactly what number you believe 0.999... - 1 is, if not 0? Also, what is 1 - 0.999...?
 * In answer to each of your questions:
 * How can one number be the same as another?
 * It is important to make the distinction between a number and its representation. 0.999... and 1 are different representations for the same number.
 * Also is 1.9999999999999999 the same as 2? etc etc?
 * Yes, 1.999... = 2, as well as 99.999... = 100, 0.0999... = 0.1, 3.4999... = 3.5, 4.3879999999... = 4.388, etc.
 * It is sufficient to say that it is close enough to, to say it equals 1, but technically it is not 1.
 * No, it is not sufficient to say that 0.999... is 'close enough to 1' to say it equals 1. When we write that equality, we mean exactly. To take an analogy that was mentioned in the arguments page a little further, the difference between 1 and 0.999... can perhaps be thought of as being the difference between a whole pie, and a pie in which I've made a single cut from the center. I haven't actually removed anything, so there's the same amount of pie.
 * Hope that helps. --COVIZAPIBETEFOKY (talk) 04:44, 23 October 2009 (UTC)

It's important to remember that we're dealing with the decimal number system for representing numbers. we're so used to it that we often skip certain steps in our head without realizing what we're doing. For example, .1 is not really a number, it's just a decimal representation of a number. It says in the tenths value we have 1 quanity. Or to put it simply: 1/10. Or, if we have 1 pizza, it has been cut into 10 equal slices and 9 of those slices are missing, leaving us 1 slice.

When you look at a number like .999... and think think it's less than 1, your brain is skipping a step. It sees that .9 and doesn't really care about the rest because it knows no matter what - this number is less than 1. But that's because you're so used to working with the decinal number system that you jumped over actually looking at what .999... really represents. What it really means is every place value from the tenths, to the hundredths, on down, has the quantity 9 in it. Then you have to look at what that means.

Let's go back to our pizza. What does the number .938 represent? 9 tenths. 3 hundredths. 8 thousandths. On our pizza that means we cut it into 10 slices, save 9 of those 10. Then with the remaining slice we cut that into 10 equal slices as well, we save 3 of those slices, then we take one of the remaining 7 slices from that cut and cut that into 10 slices as well, and save 8 of those. We are left with the following: 9 slices 1/10th sized. 3 slices 1/100th sized, and 8 slices 1/1000th sized. As well as 6 slices 1/100th size and 2 slices 1/1000th sized we didnt use and will throw away. also notice if you add up .9 + .03 + .008 and + .06 + .002 (the leftover slices we didnt use) youll get 1.

This may seem trivial but there's something important to note here, what's the maximum amount of slices of pizza you can ever have in a given place value? The answer is 9. You may be tempted to say 10, but the 10th slice is used for the next place value. Remember how our 1/100th sized slices in the previous example were actually part of the 10th slice of 1/10th sized slices? (note if you actually do have 10 slices then you really still just have 1 slice of the previous place value (ie 10/10 = 1)

So .999... means every place value is maxed out. We have 9 slices, the maximum, for every single division on down, forever. So what does this mean?

Well let's take a look at our pizza and start at the top. We see we have 9 slices of 1/10th size. 9 slices of the 1/100th size. 9 slices of the 1/1000th size...etc You'll quickly realize that you're not actually missing any pizza here. The "left over" pizza (remember how we had .06 and .002 left over when we cut our pizza into .938?) after your divisions is actually being used to divide further. The "missing" 1/10 slice from our first 9 slices of 1/10th size that would give us 10/10 slices or 1 whole pizza, is actually there, it's just being used to hold all 9 slices of the hundreds. And the "missing" 1/100 slice from our 9 slices of the 1/100th size is being used to hold all 9 slices of the 1/1000s and so on. So in actuality, all "missing" slices are in use for dividing the pizza down to every place. Every place has its maximum 9 slices, and every places "missing" 10th slize that would "complete" the pizza is still there, it's just being used to divide down to the next lowest place value. There is no missing slices and we have a full pizza. .999... = 1.

If you want to prove this yourself. Start from the top down. Take an imaginary pizza and cut it into 10 slices for your first place value. Then cut one of those slice into 10 slices for your 1/100ths. Then cut one of those slices into 10 for your 1/1000ths. Then cut one of those slices into 10 for your 1/10,000ths. Now take a break and look at the pattern you have.

9 slices 1/10th size 9 slices 1/100th size 9 slices 1/1000th size 10 slices 1/10,000th size

Looking familiar? .9 + .09 + .009 +...But wait a minute you say, I thought we couldn't have 10 slices? That's right. So those 10 slices of 1/10,000th become 1 slice of 1/100th which add our total of 1/100th slices to 10 which becomes 1 slice of 1/10th and that means we now have 10 slices of 1/10th or 10/10 or 1.

But say we didn't take that break, we just kept cutting and took one of those 10 1/10,000th sized slices and cut it into 10 for 1/100,000th and cut one of those and so on forever....

I'm sure you now see that if you keep going forever what you really have here is just every possible division of the pizza by 1/10 (which is infinite) and what this leaves you with is 1 whole pizza cut in a way that every possible decimal slice is available to you, and no slices are missing.

You have 1 pizza divided into every possible division of 10. This just happens to give you 9 slices of each division, which just happens to be the same thing as .999. repeating decimal.

You could also have 1 pizza divided into every possible division of 2, which would give you only 1 slice of each division. This gives you 1/2 + 1/4 + 1/8 + 1/16...(forever) = 1, but there's no provocative way to write that so most people don't have a problem with it. Whereas writing out 9/10 + 9/100 + 9/1000...(forever) = 1 as .999... = 1 is provocative and grinds some gears, but now you know what it really means and perhaps it isn't so scary :) PS If this was too long winded for this discussion page i apologize, dont mind if it's deleted. 76.103.47.66 (talk) 00:28, 25 October 2009 (UTC)

One number may be the same as another number if they represent the same thing. "4/4" and "1" look differnent on the page, but they are simply different ways of representing the same thing, they are both equal to one. Similarly, 0.9999999... is just another way of representing the number 1.--RLent (talk) 21:16, 31 December 2009 (UTC)

Really, guys
I've learnt this in highschool. And I mean learnt by being proven logically to me, using Aristotelian logic, common sense and mathematic rigour, not taught by professoral authority. Maybe here in Europe we have a higher level of highschool teaching than in other places (no offence) but we're giving Wikipedia such a bad name by arguing about such a common knowledge and common sense thing.

I know that Wikipedians must be in consensus when something is "disputed" and that this usually sounds like "this is true because all the authorities say it's true" but this isn't so. Actually no one with an inch of reason and sense who has actually given some thought to this problem can say that 0.(9) does not equal 1. All those who have other (academic) preocuopation and first see this due to reason well explained by cognitive psychology feel that this isn't right. Some react emotionally and write down their opinions on Wikipedia.

Again, I know that we should be in consensus in case of "disputed" points of view, but really is this actually a "dispute" or a "point of view" or is it common sense and evident due to the flawless logical demonstration?

Would we for example take seriously anyone arguing on this pages that Washington DC is located in China or that London is the capital of Russia? Do we need to proove here that grass is a plant and not an animal? Do we need consensus that the city of Paris is not located on Mars but actually in France? We don't. If anyone would assert those kind of things on these talk pages they would be at best blatantly ignored. SO why not use the same policy here? —Preceding unsigned comment added by 82.77.239.121 (talk) 12:55, 30 December 2009 (UTC)


 * There is no dispute among mathematicians, European or otherwise, and I see no substantive contribution to this discussion in your comment above (notably, no reference to a reliable source). --macrakis (talk) 14:49, 30 December 2009 (UTC)
 * A bit of a dispute is evident in archive 14. Tkuvho (talk) 08:52, 1 January 2010 (UTC)


 * Well, indeed, but I don't think there is any dispute here - it's long been established that we can present this as factual in the article, and any dispute gets farmed off the talk page to Talk:0.999.../Arguments (I guess there's the debate about whether we should simply delete those comments instead, but either way, it doesn't clutter up this page). Regarding things that seem obviously true, such as Paris being in France, see When_to_cite. Whilst common knowledge doesn't need a source (the article gives Paris as an example), it's not clear to me that this is common knowledge. But still, we have references anyway. Mdwh (talk) 14:59, 1 January 2010 (UTC)

I'm just asserting, Kyrie Makraki, that the fact that 0.(9)=1 is subject-specific common-knowledge per When_to_cite. We appear laughable if we allow it to be disputed, even on archived talk pages. I never said it's disputed by mathematicians, I said that it's disputed by some on these talk pages and that diminishes Wikipedia's credibility. Personally I was abhorred that some good-faith knowledgeable mathematics contributors on these pages were taking the dispute seriously and were trying to prove the thing, not only as an exercise of mathematical rigour, but actually as support for a presumably unusual statement, in an almost journalistic way. just check out the talk pages if you don't believe me. —Preceding unsigned comment added by 82.77.239.121 (talk) 03:25, 8 January 2010 (UTC)


 * Are you referring to formulas of the sort $$.\underset{H}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{H}}$$ with infinite hypernatural H? Tkuvho (talk) 09:08, 8 January 2010 (UTC)


 * To me it sounds as if 82.77.239.121 is advocating wholesale removal of any discussion about the merits of 0.999...=1 on the talk page. I disagree. The widespread confusion about the equation is is part of what makes 0.999... notable in the first place, and it's no surprise that some of those who doubt the equaity make their way to the talk page. Of course I prefer them going to the arguments page and we should move such discussions there, but more importantly I prefer them discussing the merits of the equality anywhere but in the article proper. If we tried to stifle the debate on talk pages, it would probably lead to increased good-faith wrong edits on the article. And who knows, maybe some of those who doubt the equality can even be convinced by those discussions (though others probably are here just to troll). Huon (talk) 19:47, 8 January 2010 (UTC)

Repeated wikilink
Minor edit: removed repeated wikilink for Mathematical rigour. EngineerFromVega (talk) 09:28, 25 January 2010 (UTC)

Typo on the page
Unless I'm mistaken, there's a typo in the text of the Algebraic/Digit manipulation page--it says "Let the decimal number in question, 0.999…, be called x. Then 10x − x = 9x. This is the same as 9x = 9." By my reckoning, 10x - x = 9x is true, but the sentence should say 10x - x = 9, which is actually meaningful.

I'd go ahead and edit it, but my track record in successfully parsing and editing brain teasers isn't so good lately, and I'd rater not muck up the article on the off chance I'm wrong. Noliver (talk) 01:31, 22 January 2010 (UTC)


 * If you read the sequence of equations, you'll notice that the left column is all presented in terms of x, whereas the right column is expressed in constant values; substituting 9 for 9x, while mathematically correct, would lead to these concluding steps:



\begin{align} 10 x - x   &= 9.999\ldots - 0.999\ldots \\ 9          &= 9 \\ 1           &= 1 \end{align} $$


 * I consider this conclusion to be immensely inferior to the one presently displayed. Gustave the Steel (talk) 03:48, 22 January 2010 (UTC)


 * I think I failed to express myself clearly. I don't think there's any problem with the equations as displayed on the page--the issue is that the sentence before the equations doesn't seem to match the equations.  Specifically, I think "10x − x = 9x" should be replaced by "10x − x = 9" in the sentences just before the equations.  My reasoning is that if two equations are stated to be "the same", then they should give the same result, but 10x − x = 9x reduces to 1=1, while the equation that is "the same", 9x = 9, reduces to x=1.


 * If the sentence was trying to make a different point, or if I'm still not getting it, I'll stop arguing the point, but when I read your response, you seem to be saying the same thing in a different way. Noliver (talk) 01:38, 24 January 2010 (UTC)


 * The sentence "10x - x = 9x" is meant to describe the change in the left column, from step 3 to step 4. If you replace 9x with 9, the result is the following statement:
 * "Let the decimal number in question, 0.999…, be called x. Then 10x − x = 9."
 * This is only true if you already assume 0.999… = 1 (or learn it from other proofs). In an attempt to prove that 0.999… = 1, we are not allowed to assume that 0.999… = 1 for any intermediate step; otherwise, we have employed circular logic.  Gustave the Steel (talk) 05:53, 25 January 2010 (UTC)
 * If you perform the "column subtraction", then you get 10x - x = 9, not 10x - x = 9x, as Noliver already pointed out. Meanwhile, "10x - x = 9x" is a tautology, and does not need any column subtraction.  Tkuvho (talk) 08:59, 25 January 2010 (UTC)
 * If x is not equal to 1, then 10x - x does not equal 9. Therefore, asserting that 10x - x = 9 is the same as asserting x = 1 before the proof is done.  You can't assume that what you are trying to prove is true in the middle of your proof.
 * The sentence, at present, describes the change in the left column from the third step to the fourth step. It does use a tautology, which is perfectly reasonable to employ in a proof.  Perhaps you need to refresh your knowledge of tautologies. Gustave the Steel (talk) 17:05, 25 January 2010 (UTC)
 * There seems to be some miscommunication going on here, but let's not get into that. I think the current explanation in the article is awkward whether the sentence in question says 9 or 9x. Can we agree to consider rewriting the subsection in a simpler fashion? Melchoir (talk) 22:14, 25 January 2010 (UTC)
 * I'll second that. My suggestion would be to remove the entire text that clumsily tries to express what we then repeat more clearly as a sequence of equations: Leave out everything between "requires algebra" and the equations themselves. Huon (talk) 23:05, 25 January 2010 (UTC)
 * The "third step" Gustave referred to contains "9" in the right-hand-side, not "9x". Besides, if you start with a tautology, what is Gustave's explanation for the deduction "Then 10x − x = 9x. This is the same as 9x = 9" ?  Tkuvho (talk) 09:23, 26 January 2010 (UTC)
 * You're an odd one, that's for sure. Anyway, I agree with Melchior and Huon.  Let's just take the stuff out altogether. Gustave the Steel (talk) 16:36, 26 January 2010 (UTC)

<-- Actually, I think the proof given on the page is correct. 10x - x is equal to 9x. What else is wrong with the proof? If you do the subtraction on the other side, it gives 9, resulting in 9x=9, or x=1, therefore proving that 0.999...=1. — MC10 ( T • C • GB •L)  16:37, 20 February 2010 (UTC)
 * There is nothing wrong the proof displayed on the page, MC10. Apparently there was some text above it causing debate and/or confusion and has since been deleted. I have read these posts above and cannot grasp why someone thought 10x - x = 9 would be better than 10x - x = 9x. The whole argument makes little sense. Perhaps if we saw the original text we would understand what went on here, but without it I am just as in-the-dark about what the fuss was about.Racerx11 (talk) 21:45, 7 March 2010 (UTC)
 * I found the original text in the history. Jan '10. After the phrase "The final step uses algebra" we had: "Let the decimal number in question, 0.999…, be called x. Then 10x − x = 9x. This is the same as 9x = 9. Dividing both sides by 9 completes the proof: x = 1.[1] Written as a sequence of equations," then the proof. Still don't see the issue with this, but someone thought it should read "10x - x = 9" and so it was decided to just delete the whole sentence.Racerx11 (talk) 00:01, 8 March 2010 (UTC)

Function manipulation
I've (again) removed the section on function manipulation added by 150.156.217.147. It is basically the same as the digit manipulation proof; I don't think the function adds anything to comprehension. Huon (talk) 18:46, 2 February 2010 (UTC)

Finitism
I wonder whether some of the common objections to this can be seen as an instinctive form of finitism: that is, the objecting students simply do not believe that limits of infinite series actually exist, except as a number to which the sum of the partial series is getting ever closer without ever really reaching it. Of course, this also means that they don't believe in the standard construction of the real numbers. -- The Anome (talk) 14:29, 24 February 2010 (UTC)


 * "Getting closer" is wrong; "never reaching" is wrong. limx&rarr;&infin; sin(x)/x = 0, but sin(x)/x does not keep getting closer to 0 as x grows; rather it alternately gets closer and farther away; and it certainly does reach 0 every time x is an integer multiple of &pi;. Michael Hardy (talk) 04:05, 25 February 2010 (UTC)


 * ...and maybe I should add that 1/x gets closer to &minus;1 as x grows, and never reaches it, but nonetheless its limit is not &minus;1. Michael Hardy (talk) 04:43, 25 February 2010 (UTC)


 * Just to clarify things, I know that both "getting closer" and "never reaching" in the sentence above are wrong, in the senses you describe above: I was explaining how things might look to the equally (in this context) mistaken non-infinite-sum-believers being referred to in the same sentence, when they are thinking about the particular case of 0.999... Perhaps I should have put them in quotes? -- The Anome (talk)


 * The response by Michael Hardy is an example of the misplaced mathematical put downs that go on here. limx&rarr;&infin; sin(x)/x is not the limit of partial sums of an infinite series (of rational numbers).  In the context of infinite decimal expansions "Getting closer" is NOT wrong; "never reaching" is NOT wrong. It is true that for a sum of positive terms  which is bounded above (such as 0.9+0.09+0.009+... ) that there are numbers (any upper bound) to which  the sum of the partial series approach without ever reaching  (unless the series is all 0 from some point on). The limit IS " a number to which the sum of the partial series is getting ever closer without ever really reaching it." True it is not the only one, just the least one.  So perhaps the distinction which Anome refers is this: A thoughtful student of a certain age might say: I learned to measure including 5750 mm= 5.750 M  and I learned division like 3/4=0.750. I see that doing division for 1/3 gives 0.333... and I was ok with saying 1/3=0.333...  but now that I see that tripling gives 0.999...=1 I am troubled. To such a student one does not say: "Well, it is! After calculus 3 and differential equations you can take real analysis and the textbook will perhaps sketch cuts and might even have a detailed appendix about cuts and THEN you will know why." You say "Yes it seems weird but one has the choice of saying that there is no decimal equal to 1/3 or saying that decimal is 0.333... One can either say that 0.333... is outside our system or that 1-0.333..=0.666...=2*(0.333...) etc." Look at the discussion elsewhere of why in algebra 0^0 = 1 (we want 3x^2+5x+7=3x^2+5x^1+7x^0 to work even when x=0). It doesn't say that integer exponentiation involves the RATIONAL INTEGERS and LIKE EVERYTHING ELSE is defined inductively using the Peano Axioms. --Gentlemath (talk) 19:12, 28 February 2010 (UTC)


 * Yes. Initial skepticism about 0.999... = 1 is an entirely reasonable stance for a bright student to take, and should lead naturally into a discussion of how the reals are constructed in terms of limits of sequences: at which point, it should no longer be an issue for the student. There are then two problems to be considered:
 * the student who is just bright enough to raise the initial doubts ("what do you mean by infinite series?", "how can you possibly sum them if they don't ever stop?", "how do you define equal?"), but not capable of accepting the explanation when offered.
 * the teacher who does not understand that the student's doubts are reasonable, stemming from genuine mathematical curiosity, and thus need proper explanations to satisfy them, and the student might actually be smart enough to understand those explanations
 * -- The Anome (talk) 17:30, 1 March 2010 (UTC)
 * The reals are constructed that way (in part) because the "number line" is Archimedean. It is not an arbitrary choice, it really doesn't work any other way. One perfectly fine way to construct the reals (Due to Weirstrass) is as infinite decimals with 0.999...=1 built in. There are difficult series which take careful definitions. A test of the definitions is that they do the right thing for the well understood case of convergent geometric series. Maybe I am wrong but I think Wikipedia is highly unusual in its emphasis on axiomatic construction of reals as the justification of these convergent geometric series. It is absurd to take that approach with high school students. I'll stop with a random page I just found indicating how this is treated in a foundations of mathematics class for math majors at Villanova http://www41.homepage.villanova.edu/robert.styer/Bouncingball/HowInstructorsCanUseGeomSeriesProofs.htm
 * especially http://www41.homepage.villanova.edu/robert.styer/Bouncingball/geometric_series_3.htm --Gentlemath (talk) 07:34, 8 March 2010 (UTC)

Division by zero article
Here is an article that handles the issue of real numbers exactly as I proposed a while ago: Division by zero  Rather then demanding that any perspective other then the real set be buried at the end, it acknowledges at the outset that the meaning of the subject depends upon the mathematical setting. The situation with 0.999... is the same, so why can't we do this? Algr (talk) 00:24, 1 March 2010 (UTC)


 * It isn't the same at all. Division by zero is meaningless in the real numbers, so of course an article on it needs to concentrate on other number systems. 0.999... is perfectly meaningful in the real numbers. Also, the number systems in which division by zero works are very commonly used. The number systems in which 0.999... is anything other than 1 are very obscure and are usually just proofs of concept rather than anything that anyone uses for anything. --Tango (talk) 00:28, 1 March 2010 (UTC)


 * So you agree that an article should not primarily view its subject in a context in which the subject is rendered meaningless. I think that that is progress.  So what about a context in which it's subject is rendered trivial?  0.999... isn't meaningless in the Real set, but it is trivial. (See my example about the article 20/20 above.)  Algr (talk) 21:52, 8 March 2010 (UTC)

An obvious explanation for the confusion
This is not about arguments about the mathematical statement 1 = 0.999999..., but about understanding why this fact should be so disturbing to many people. I miss on the page mention of what would seem to me an obvious source of this confusion: many people think of real numbers as being defined as infinite sequences of decimals, rather than as abstract mathematical objects that can be represented by infinite sequences of decimals. Indeed, although it can be argued that such sequences are no less abstract mathematical objects than real numbers, they seem more concretely approachable by going on and on and on and... giving more decimals. There are many circumstances that would produce such a suggestion, from pocket calculators, which pretend to compute real numbers but in fact manipulate digit strings, to statements like π = 3.1415926535..., which appear to equate a real number to an infinite sequence of digits. Of course if real numbers are defined as infinite sequences of decimals, then the idea that two obviously different sequences are equal real numbers becomes absurd. [original message continues below]


 * Absurd, unless, of course, more than one decimal digit sequence is allowed to map to the same real number, i.e., if multiple "definitions" of a real number are allowed to represent the same number. Obviously, this requires an understanding of mapping, as well as deeper understanding of number. — Loadmaster (talk) 05:09, 20 February 2010 (UTC)


 * No, that is misunderstanding the point. If real numbers were defined as digit sequences, then one could not make the distinction. Mapping digit sequences to real numbers would mean mapping digit sequences to digit sequences. The fact that one can map "0.999999..." → "1.000000..." and "1.000000..." → "1.000000..." (or why not "1.000000..." → "0.999999...") makes these sequences no more equal than the fact that one can map "0.999999..." → "3.141592..." implies an equality. If you define some notion as something, then you are stuck with the properties of "something", whether you like them or not. So defining real numbers as digit sequences is a really bad idea (and no decent arithmetic could be defined in that case). Of course they could be defined as equivalence classes of infinite digit sequences (with at most two members in each class), but I think that point of view is probably already too abstract for those suffering from the mentioned confusion. Marc van Leeuwen (talk) 11:05, 20 February 2010 (UTC)


 * Okay. But to be clear, you must make the point that each distinct digit sequence defines a single distinct and unique real number. Otherwise, there is nothing inherent in the definition to preclude multiple equivalences. On the other hand, if you extend the definition to define "equality" as requiring identical digit sequences, then you would indeed have 1.000... ≠ 0.999... within such a system. — Loadmaster (talk) 19:51, 20 February 2010 (UTC)

I would bet (and the discussions on this subject provide ample confirmation) that people who believe that real numbers are by definition infinite sequences of decimals also believe that arithmetic on real numbers is defined in terms of manipulating such sequences as if they were finite. They do not realize that such manipulation is not even possible in an ideal (Platonic) sense (because carries are propagated to the left, one must start at the right end of the sequence, but being infinite means there is no right end). The fact that 3×0.333333... = 0.999999... is proposed (without explanation) as an argument illustrates this: apparently that computation is seen as more obviously valid than 0.999999... = 1 or even than 3×0.333333... = 1.000000..., indicating a belief that this particular multiplication on digit strings can be preformed in some definite way (it cannot, since one needs the hypothesis that there will be no carry in any digit position; this hypothesis is self-confirming, but so is the hypothesis that there will be a carry in every digit position). So the real problem is that people are insufficiently critical of their own beliefs. (As an aside, if I were asked to convince somebody who believes that 0.999999...≠1, I would ask to compute the average of the two numbers.) Marc van Leeuwen (talk) 13:56, 19 February 2010 (UTC)
 * The expression ".999..." has infinity inherent in it's definition and is therefore a form of infinity. (Or at least an object outside of the real set, which does not include infinity.) So asking for the average of .999... and 1 is like asking for the average of 0 and infinity. Algr (talk) 02:58, 20 February 2010 (UTC)


 * No offense, but that's complete nonsense - at least the part where you say '".999..." has infinity inherent in it's definition and is therefore a form of infinity'. Certainly, it is an abbreviation of a representation that, by necessity, contains an infinite number of symbols, but by the same argument so is "1.000..." and no-one ever seems to take issue with the possibility that 1.000...=1, and no-one ever claims that 1.000... "is a form of infinity". Confusing Manifestation (Say hi!) 09:27, 20 February 2010 (UTC)
 * Indeed. Computing the average of .999... and 1 is very simple in view of the fact that .999... = 1, and it's by no means comparable to operating with infinity. — Anonymous Dissident  Talk 09:34, 20 February 2010 (UTC)


 * As far as I can tell, everything Marc van Leeuwen is saying here is true. Unfortunately, the fact thay he (and I) miss this approach in the article is not enough to include it - it requires a source. In general, I don't bother much with sources as long as material is true and fairly uncontroversial, but this article is a bit of a battle ground, and a good source is required. Anyone?--Nø (talk) 14:27, 20 February 2010 (UTC)


 * The first bullet in the Skepticism section notes, "Students are often 'mentally committed to the notion that a number can be represented in one and only one way by a decimal.' Seeing two manifestly different decimals representing the same number appears to be a paradox...", and it does have citations. Are you envisioning a different approach? Melchoir (talk) 23:33, 20 February 2010 (UTC)


 * Well, I agree that that statement comes close to what I wanted to say. But that is such fluffy language. What does it mean to be mentally committed to a notion? Does than mean believe the stuff, or having made a commitment (why? to whom?) to believe it? And "that a number can be represented in one and only one way" already seems to me a more mature point of view than I think many people actually have: they just believe real numbers are infinite decimals, because nobody ever told them an alternative way to view them (unlikely they've studied Dedekind cuts or equivalence classes of Cauchy sequences or such). [more comment a bit below] Marc van Leeuwen (talk) 11:00, 21 February 2010 (UTC)


 * The average of 0 and infinity is obvious; it's $$\frac{(\infty+0)}{2} = \infty.$$ --Zarel (talk) 21:06, 20 February 2010 (UTC)


 * Your explanation is nice, but I don't really see what your point is. Wikipedia talk pages are for discussing how to improve the article (see WP:TPG). If you're proposing to add this to the article, you need a reliable external source. --Zarel (talk) 21:12, 20 February 2010 (UTC)


 * Agreed, I know of no cognitive psychologist paper that studies what people who insist that 0.999999...≠1 actually think that real numbers are. Maybe it ought to be done. As for adding to the article, I have no intention. However, the article might be a improved by removing a lot of stuff. For one thing, those arguments ("proofs") that are based on the false assumption that one can compute directly with infinite decimals (like the one I mentioned above, or equally well 9.999... −0.999... = 9; why is the result not 8.999... because of a "borrow"?), which only reinforce the misconception that underlies the confusion. Marc van Leeuwen (talk) 11:00, 21 February 2010 (UTC)


 * It's true that no algorithm can compute the nth digit of a decimal expansion of the sum of two arbitrary numbers, provided only a finite number f(n) of digits of their decimal expansions. This observation is well-known, and I assume it's what you're referring to. Fortunately, it's moot in this case because we know every digit of the inputs.
 * On the other hand, I can see why you might be concerned, as some of the article's language suggests following digit-by-digit algorithms which would return non-digits in the general case. I'll clean it up. Melchoir (talk) 23:16, 21 February 2010 (UTC)


 * I say something stronger: no algorithm can guarantee the computation of any decimal digit of a sum, even if the values of all decimal digits of the arguments are available. Your statement requires information to be truncated at a position f(n) independent of the actual arguments, but even if the arguments are known, the algorithm must decide at some point to stop pondering how much input it will actually inspect and start producing some output, so it it ever produces the nth digit it must do so based on some finite part of the arguments (but which can depend on other things than n). Here's a sample case of adding two numbers: the first number is 0.555... repeating, the second starts 0.444... but has its n-th digit after the point equal to 4 if the first n digits of π differ from the next group of n digits of π (positions n+1 to 2n), while if these two groups of digits do happen to be equal, it is 3 in case n is even and 5 in case n is odd (the specification of the second digit sequence is complete, since π, being irrational, has a unique decimal representation, and each of its digits is explicitly computable). No algorithm can guarantee the computation of any digit after the decimal point. Marc van Leeuwen (talk) 13:46, 23 February 2010 (UTC)
 * Good point, I neglected to make my first statement strong enough. But it's still irrelevant if we aren't claiming the existence of such an algorithm. Melchoir (talk) 19:40, 23 February 2010 (UTC)


 * "Certainly, it is an abbreviation of a representation that, by necessity, contains an infinite number of symbols, but by the same argument so is "1.000..." and no-one ever seems to take issue with the possibility that 1.000...=1"
 * The distinction here is that if you truncate 1.000... to an arbitrary finite number of digits, it always equals 1. But .999... never equals one for any finite number of nines.  Thus the equality  depends upon infinity being treated as an accomplished goal with uniquely defined properties, even though in the next breath people will say "infinity is not a number".  To me this is a blatant contradiction.  Algr (talk) 10:00, 23 February 2010 (UTC)
 * The "accomplished goal", in this case, is a limit, which of course has uniquely defined properties and indeed is a number. Real analysis (and 0.999...) can make do without ever invoking an object "infinity", and from an abstract point of view that's preferable. When we speak of "infinity", we usually denote some intuitive idea which has a precise definition behind it. Examples: There are "infinitely many" objects in a set if there's an injective map from the natural numbers into that set, or a sequence "tends to infinity" if for every positive real number C the elements of that sequence eventually become larger than C. Indeed "infinity" is not a number, but whenever we invoke it, it's code for something well-defined. Most examples of "infinity" in real analysis should be explained in a first-year calculus course. Huon (talk) 12:32, 23 February 2010 (UTC)
 * Of course infinity is a number! Whoever told you it isn't? There are many infinite cardinal numbers, ordinal numbers, hyperreal numbers, surreal numbers, and so on.
 * Oh, wait, you meant a real number. "Number" ≠ "real number", hence the need for the qualifying "real" in front. --COVIZAPIBETEFOKY (talk) 13:22, 23 February 2010 (UTC)
 * The fact that there are infinite cardinal numbers, infinite hyperreal numbers, etc., doesn't mean that "infinity" is a number. Michael Hardy (talk) 13:34, 23 February 2010 (UTC)
 * Returning to the section heading "explanation for the confusion", somehow it has not come through clearly enough that the issue is the absence of infinitesimals as far as the common number system is concerned. Those of us who are familiar with the details of the construction of the real numbers, have no difficulty with 0.999...=1; those who are not so familiar, find the insistence on unital evaluation, confusing.  The issue here is not so much WHICH infinitesimal-enriched system one uses, but rather the presence of any nonzero infinitesimals.  Education research shows that interpreting an unlimited decimal string of 9s as falling infinitesimally short of 1 can be a useful educational tool in understanding the ideas of calculus.  Tkuvho (talk) 13:55, 23 February 2010 (UTC)
 * I assume your quibble is with the fact that "infinity is a number" seems to suggest that there is only 1 infinity, when all those number systems have many infinite numbers. Perhaps a better way of wording it would have been "there are infinite numbers", rather than "infinity is a number". --COVIZAPIBETEFOKY (talk) 18:05, 23 February 2010 (UTC)

In no way did I intend to mention the presence/absence of infinitesimals as a source of confusion. Quite to the contrary, I think it is a red herring; people incapable of accepting 0.999...=1 are certainly not ripe for understanding infinitesimals. Saying that "an unlimited decimal string of 9s is falling infinitesimally short of 1" is fooling yourself for comfort, and I don't think it is educationally useful for anything. If you must, why take this example and not that an unlimited decimal string of 3s is falling infinitesimally short of 1/3 (which is exactly similar)? The source of confusion I meant is failure to conceive the concept of real numbers as anything distinct from that of infinite sequences of decimals. Marc van Leeuwen (talk) 16:08, 23 February 2010 (UTC)
 * To respond to your two points: student intuitions about infinitesimals, specifically in the context of .9.., are not in the domain of speculation but rather in the realm of published educational research, see e.g. Robert Ely's recent work. As far as .3.. is concerned, it can similarly fall short of 1/3.  Tkuvho (talk) 16:12, 23 February 2010 (UTC)
 * I can't imagine anyone accepting .333...=1/3 after questioning .999...=1. The only reason anyone believes .333...=1/3 is because it is taught in grade school that way.  (And even then I found the equality to be vaguely imperfect somehow.)  Limits appear to me to be nothing but a way to talk around the problem.  Here are four examples:
 * x= The square root of negative one.
 * x= The number with an absolute value of negative one.
 * x= The number of nines necessary to make .999...=1
 * x= The lowest value in the range y>0
 * What makes some of these ideas "accepted" and others not? It seems like little more then a popularity contest.  Algr (talk) 21:03, 23 February 2010 (UTC)

"Algr", are you suggesting that the square root of &minus;1 is an example of a limit? Are you suggesting that someone (besides yourself maybe?) believes there is a "lowest value in the range y>0"? Are you suggesting that someone (besides yourself) thinks there is a "number of nines necessary to make .999...=1"? Are you suggesting there are people (besides yourself and maybe some confused undergraduates who'd rather not have to take math) who think there's a number whose absolute value is &minus;1? All of those propositions are false. √&minus;1 is not defined as a limit. No mathematician thinks there is a "lowest value in the range y>0". No mathematician thinks there is a "number of nines necessary to make .999...=1". No mathematician thinks there is a "number with an absolute value of negative one". Michael Hardy (talk) 21:25, 23 February 2010 (UTC)
 * Good job of evading the point through insults and proof via popularity. Those are four examples of talking around problems.  I'd expected that you would at least recognize that two of them WERE accepted mathematics, but in your rage you denied all four.  Calm down and at least try to understand the question before answering it.  Algr (talk)
 * Algr, you are confused. I phrased my points carefully.  I am a mathematician and I am familiar with the situation.  You are the beneficiary of my kindness.  &radic;&minus;1 is a commonplace, with practical applications in many fields.  It is not defined as a limit.  No mathematician says there is a lowest value in the range y>0; they say instead that there is a largest lower bound, which is 0, and which is not in that "range".  No mathematician says there is a "number of 9s necessary to make 0.9999... equal to 1; they say that 0.9999... is equal to 1 and that what that means is that the limit of a certain sequence of terminating decimals is equal to 1.  And Algr, if you're just here to pick fights with people who are kind to you, then go away. Michael Hardy (talk) 04:00, 25 February 2010 (UTC)
 * What makes some of these concepts accepted is that they are derived from sound mathematical logic. The square root of −1 does not exist as a real number, but it does as an imaginary or complex number, based on sound mathematical definitions. The number with an absolute value of −1 is a contradiction of the terms used, so therefore does not exist. The number of 9 digits in 0.999... is greater than any finite number, so therefore must be an infinite (countable ordinal or cardinal) number. The lowest real value y>0 involves the well-defined concept of limits, and does not exist. What makes the answers exist is the application of well-established mathematics and logic. "Popularity" is irrelevant. — Loadmaster (talk) 21:59, 23 February 2010 (UTC)
 * Why are complex numbers accepted? There are many reasons, but most notably, the fundamental theorem of algebra (a result which is accessible to high school students, though the proof is not) and analyticity of holomorphic functions (not accessible to high school students) are extremely useful, but do not hold for real numbers.
 * Is there a number with an absolute value of -1? Of course not! We define absolute value as the distance from the number to 0; distances are positive.
 * Is there a number representing the number of nines it takes to get .999... = 1? Not really; it's a notion which can be described, but the language of numbers doesn't seem sufficient. The best answer I can think of is $$\aleph_0$$, but that doesn't eliminate, eg, 0.909090909..., since the number of 9's there is, in fact, $$\aleph_0$$.
 * Is there a smallest number greater than 0? Well, yes. It's called 1. In the integers. In the reals, of course, there is no such number. --COVIZAPIBETEFOKY (talk) 22:11, 23 February 2010 (UTC)
 * (edit conflict, this responds to Loadmaster.) I brought up popularity because Michael Hardy kept emphasizing "No one else but you" in his post for some unknowable reason. √-1 and |x|=-1 seem to me to be identical contradictions of the terms used.  Complex numbers are built around the premise that √-1 has an answer.  You could do exactly the same thing with |x|=-1 or the lowest real value y>0.  The only difference is that √-1 happens to lead to some practical uses involving rotating objects in a two dimensional plain.  In my board game I used .999... as a tool to distinguish between inclusive and exclusive ranges, so that too has a practical use. And unlike √-1 or |x|=-1 .999... as an infinitesimal can be approximated on a number line.  Algr (talk) 22:18, 23 February 2010 (UTC)
 * The idea of a "lowest" positive number y, and similarly the "last" ".999..." before reaching 1, is an intriguing one, but so far mathematicians have not been able to develop an operative number system of this sort, which would correspond to centuries-old intuitions about "indivisible infinitesimals". The one who came closest is Lawvere, who developed a system containing nilsquare infinitesimals.  A nilsquare infinitesimal is still going to have infinitesimal smaller than itself, such as its own half, but at least there are no nonzero infinitesimals "of higher order" such as epsilon2.  Tkuvho (talk) 09:57, 24 February 2010 (UTC)
 * And perhaps more to the point at hand, infinitesimals don't work within the real number system (which is what the article is about). These other concepts likewise either don't work for the reals or are simply contradictions in terms. — Loadmaster (talk) 03:48, 25 February 2010 (UTC)

2

 * The article isn't about real numbers, it is about the symbols "0.999..." and how they might be interpreted. Therefore the article needs to explain up front why the real set is the appropriate framework and not integers, surreals, or color theory.  Any variation of "We've always done it that way." or "The experts say so." is a betrayal of scientific thought.  In the whole numbers, fractions are as meaningless as infinitesimals.  But that doesn't mean it is appropriate to have the article 1/2 spend any time on whole number analysis.  The symbols "0.999..."  invariably imply infinitesimals, even if they don't correctly define one.  The use of the real number set without justification is therefore going to do nothing but frustrate any reader who actually needs this article to understand 0.999...  The reason I keep coming back here is because I feel that this article insults my intelligence.  I proposed a way to fix this in the intro a while back.  Algr (talk) 22:42, 25 February 2010 (UTC)
 * Sorry, but I don't follow you. The article clearly states in the very first sentence that 0.999... represents a real number. Are you suggesting that decimal fraction 0.999... represents something other than a real number in normal common usage? — Loadmaster (talk) 05:31, 26 February 2010 (UTC)
 * No, he's talking about the difference between "represents a real number" and "is a real number". This is much clearer if one discusses such representations of a real numbers as "three" or "the square root of two" - these are representations of specific real numbers, and they are clearly not unique representations of those numbers. "0.99999...", or similar strings, or a conceptual endless string of similar form are representations of a specific real number as well, and clearly not unique representations. The proper approach is something like "0.99999... is a representation of the number 1 that is possible in the decimal number system. We know it is a representation of 1 because [blah blah blah]". — Gavia immer (talk) 05:51, 26 February 2010 (UTC)

I think it can legitimately be asked what this article is about: is it about the symbol ".999..." and how students view it, or is it about the real numbers and the subtleties of decimal notation, such as its non-uniqueness. Now what makes this page significant, and also the reason it has the featured article status, is that students are experiencing difficulties with the unital evaluation .999...=1. These students are typically being exposed to unital evaluation before they learn about either the rigorous notion of limit, or the construction of the real number system. In fact, the real number system is typically not constructed until a real analysis course. This means that students go through calculus pretty much accepting unital evaluation on faith. Limiting this page to the real numbers may defeat the purpose of having such a page in the first place. Tkuvho (talk) 09:04, 26 February 2010 (UTC)


 * The article already mentions other number systems. But despite Algr's doubts the real numbers are the default number system in which 0.999... is considered - so much so that whenever someone interprets 0.999... as anything but a real number, he explicitly has to say so. Since this is at least partly a convention, it does follow from "everybody says so". Should we be more explicit in addressing why everybody says so, except because everybody else also says so? While desirable, I believe such an explanation might go beyond the scope of this article and should rest with the decimal representation article. The short answer would be: We don't start with 0.999... and wonder about the appropriate number system; we start with the real numbers, want to represent them and run into this peculiarity. Introducing infinitesimals would simultaneously render the decimals worthless for representing the real numbers and leave us with a number system which cannot be represented by decimals, either - and the subset of representable numbers lacks any nice properties whatsoever. Huon (talk) 11:36, 26 February 2010 (UTC)
 * If you start with Idaho, you can show that Moscow is in the United States, but that is not a helpful thing to do to people who type "Moscow" into a search bar and want info. The article must start with 0.999... because that is it's name.  If you think that the Real set is important, then you must introduce it and explain it's relevance before relying on it for anything.  If you assume that the reader already understands the real set, and yet somehow doesn't understand 0.999... then you have written an article that can never honestly inform anyone. Algr (talk) 03:54, 28 February 2010 (UTC)
 * Just as "Moscow" by default gives information on the capital of Russia, so 0.999... should explain the properties as (representation of) a real number. That is the default meaning, and as I said the reasons why go beyond the scope of this article; we link real number in the very first sentence for background information. The Moscow article also doesn't contain an explanation why the capital of Russia is more important than any of the other Moscows which might exist, and the reader is assumed to understand its relevance himself. We don't write a math textbook, but an encyclopedia article. Someone who looks up 0.999... is extremely likely to have happened upon it in the context of real numbers; if his context is another, he probably knows so. Huon (talk) 09:58, 28 February 2010 (UTC)
 * On the contrary, before even the introduction it says "This article is about the capital of Russia. For other uses, see Moscow (disambiguation)." They correctly deal with the different levels of understanding that readers might bring to the article. 0.999... by contrast is written for someone with an impossible combination of knowledges and unknowns. To communicate, you must understand who you are talking to.  Algr (talk) 23:18, 28 February 2010 (UTC)
 * I don't quite understand what you're saying here. There are no other articles on 0.999..., so we don't have a disambiguation hatnote. If we had separate articles on, say, 0.999... (Katz), 0.999... (decimal number) and 0.999... (Hackenstring), we'd also have a disambiguation, but there's not enough materiel to write any of those articles - each would be based on a single source, provide a definition, and nothing else. And someone looking for 0.999... will either have advanced mathematical knowledge, or he won't look for any of those alternative meanings. People may not be aware, but unless they're mathematicians, most likely all numbers they know are reals (or possibly complex if they've heard of i). If we were to suggest that other number systems are more important than the reals in the context of 0.999..., or just equally important, we'd mislead our readers. Mentioning other number systems more prominently than we already do would give them undue weight. Huon (talk) 01:00, 1 March 2010 (UTC)
 * In the real numbers 0.999... and 20/20 are both trivial restatements of one. But in other contexts they have dramatically different meanings.  Would you insist that the article 20/20 be devoted to mathematics instead of television or optometry?  Of course not!  The fact that 20/20=1 is so trivial that a mathematical interpretation doesn't even warrant an appearance on 20/20 (disambiguation).  Why then should 0.999... be locked into a context that renders it as trivial as 20/20=1?  That serves no purpose except to suppress knowledge on infinitesimals and hackenstrings. This goes back to what I said about 1/2 in the Natural set.  This article is more about denying meaning then truly explaining anything.  Algr (talk) 21:41, 8 March 2010 (UTC)
 * I'd say that an article with a given title should provide information on the subject most often associated with that title. People looking for 0.999... won't be looking for the Hackenstring, nor for Richman's decimal number, nor for Katz' interpretation. They will also not be looking for the interpretation necessary to make your game work without open intervals. You may not like it, but the real number is the default meaning. (Just for fun, I compared Google searches. 0.999... and "real number": 36,400 hits; 0.999... and "surreal number": 1,740 hits; 0.999... and Hackenstrings: 899 hits.) Huon (talk) 22:55, 8 March 2010 (UTC)
 * I would like to add to points by Tkuvho and Algr, by noting that students will often go on to be taught to conceive of integral calculus in terms of the summation of infinitesimals. If Archimedean numbers are clearly the default in mathematical reason, then why do we ask students to reason in terms of infinitesimals, and generally without mention of their impropriety? If this article should have value for the mathematically naive, then it seems hypocritical to dismiss infinitesimals a priori, when we teach their heuristic value up through sophomore level calculus. JSchilz (talk) 04:40, 25 March 2010 (UTC)
 * I don't know who the "we" are that teach students calculus in terms of infinitesimals. I have taught calculus for many years, and I have certainly never even considered teaching it in terms of infinitesimals. In fact I am somewhat taken aback to discover that there is someone who regards doing so as standard practice. JamesBWatson (talk) 14:40, 25 March 2010 (UTC)
 * The "slices" concept draws from the infinitesimal interpretation. If you talk about adding up a bunch of infinitely tiny slices, that's infinitesimal. Most texts I've seen do set up the foundation for the limit definition--as the limit of ever finer Riemann sums--but then have students imagine adding planar slices when they're setting up the integral without discussion of that method's inapplicability in the reals. JSchilz (talk) 19:54, 25 March 2010 (UTC)
 * The "slices" concept is simply a motivational picture for defining the integral as a particular limit. I don't see anything about that picture that requires infinitesimals.  I certainly don't recall my text ever suggesting that integration involves adding rectangles of infinitesimal width or something similar.  I suspect this is your own personal interpretation or intuition (but perhaps you can cite a textbook to show I'm mistaken). Phiwum (talk) 20:03, 25 March 2010 (UTC)
 * Calculus Made Easy is a text that talks explicitly about infinitesimals. That's a poor example, though. I pulled down a copy of Stewart, and in volumes he's at times clearly depicting planar slices. And of course there's the history of infinitesimal calculus preceeding Weierstrass. I don't know if we disagree, and if we do then I'm not sure on what point. Let me be clear, I'm not suggesting that your text ever mentioned infinitesimals. But if you imagine integrating an area as gluing together an infinite number of linear slices, that's an infinitesimal interpretation. It may be purely motivational, we wouldn't disagree. Since this has apparently been dealt with to the satisfaction of many, I don't want to take this too far off track. So I welcome your response but will concede all points to the personal bias you mention. JSchilz (talk) 20:42, 25 March 2010 (UTC)
 * Actually this issue is already treated in the current version of the article (see the section "further reading" and the discussion of Robert Ely's paper). Tkuvho (talk) 13:04, 25 March 2010 (UTC)
 * Very well. I did see that it had some treatment. I don't think it's quite sufficient, but I won't press the issue alone. JSchilz (talk) 19:54, 25 March 2010 (UTC)
 * You're right, a Further Reading section is always going to be insufficient. :-) I think we would all agree that relevant, reliable sources should be integrated into the article proper; it's just a matter of doing the work. (That said, we also want to ensure that experimental or minority programs aren't given undue weight, so even after that's done it's probable that some editors will still feel like the coverage is insufficient. We'll burn that bridge when we get to it.) Melchoir (talk) 20:45, 25 March 2010 (UTC)
 * I'm not suggesting that we should give greater coverage to infinitesimals, though I can see why I gave that impression. In my singular observation, .999...=1 is generally pulled out for its shock value, to demonstrate some mastery over intuition, and perhaps ostentatiously to teach something "deep" about numbers. And yet the accompanying proofs almost always entirely lack rigor. For these students, the central question is not "What does .999... represent?" but, "How do I go about interpreting .999...?" I argue it should not be an assumption that .999... represents a real number, as it is taken in the first sentence of this article. Rather, we should give it as a hypothesis under which the statement is true and justify the conventional use of that hypothesis. Specifically, I contend that:
 * "In mathematics, the repeating decimal 0.999… which may also be written as $$0.\bar{9}$$, $$0.\dot{9}$$ or $$0.(9)\,\!$$, denotes a real number that can be shown to be the number one."
 * Should be written as:
 * "In the real numbers, the repeating decimal 0.999… which may also be written as $$0.\bar{9}$$, $$0.\dot{9}$$ or $$0.(9)\,\!$$, can be shown to equal the number one."
 * Then I would also suggest a paragraph in the introduction about why we generally interpret 0.999...as a real number. Yet similar suggestions seem to have raised contention among critics, arguing from apparently platonic notions that 0.999... should a priori have an Archimedean interpretation. In my mind, this denies the object of contention rather than illuminating it. JSchilz (talk) 00:08, 26 March 2010 (UTC)
 * I'm not quite sure whether you already include me among the critics since I did object to similar proposals before; I again object now. Not for "platonic reasons", but because the real number interpretation is so vastly more common than any other (and all others combined) that in my opinion not stating bluntly that 0.999... denotes a real number gives undue weight to fringe interpretations. That's as if I'd object to calling 7 (number) the natural number between 6 and 8 because it may also denote an element in one of several finite fields. Huon (talk) 01:34, 26 March 2010 (UTC)
 * I think that's a valid perspective. My concern, and maybe I'm imagining the wrong audience, is that students come to this article with the notion that 0.999...=1 completely betrays their intuition. And the answer they find is basically that 'the man' has decided that 0.999... is a real number, and that 'the man' says that real numbers have such and such properties. There are a couple of proofs that they can comprehend, but seem just as dubious as the statement in question. And there are a couple of proofs that they absolutely can't comprehend. I'll drop the notion of changing the first sentence, if no one else feels the same way and the issue has been hashed out. However, I feel that if we're documenting for the audience I imagine we are, then part of that should include justification for the use of the reals. Otherwise the student can't be faulted for asking why 'the man's' numbers are any better than he might come up with. I use 'the man' tongue in cheek, btw, but it might capture the student's frustration. It sounds like something I can cook up and show off. I just want to proceed cautiously because this is such a controversial topic. 71.137.5.216 (talk) 06:38, 26 March 2010 (UTC)
 * A few thoughts:
 * Questions that have answers should be answered. In an encyclopedia, those answers should be given immediately.
 * Noting that a question has an answer doesn't prevent one from going back and explaining why. I expect anyone who truly thinks that knowledge impedes learning to instinctively avoid Wikipedia anyway.
 * Anyone is welcome is make controversial edits to controversial articles; you don't have to build a consensus first. If an article is controversial, that simply means your edits are likely to be reverted or repainted over. (Also, anonymous strangers might say unkind things about your work, but that's always going to happen on the Internet.) Your personal decision to seek consensus ahead or time, or not, should be a function of your ego's ability to absorb such blows.
 * Even reverted edits are useful in the sense that they give the community a concrete proposal to weigh on its merits, rather than the open-ended philosophizing that usually happens around here. See BOLD, revert, discuss cycle. If everyone is cautious, nothing gets done!
 * Do keep No original research in mind...
 * Melchoir (talk) 07:39, 26 March 2010 (UTC)

3
I'll throw in that I still think throwing in the real numbers and suggesting that "the equation can be postponed until one proves the standard theorems of real analysis." is misplaced. It might (attempt to) shut up questioners by saying "until you take a course in foundations of analysis, you can't REALLY grasp this " It makes the equation seem more mysterious and difficult than it is. The real numbers as an object of thought are 120 years old or so. They did not clarify the issue of why 0.999...=1 (which, of course it really does). From before the time of Euclid it was understood that, for "linear quantities", IF a < b THEN there is an integer q large enough that q(b-a)>1. Reworked slightly, there is a integer q large enough that a 0, there exists an integer N such that 1-10^(-n) ≥ 1 − 1/q for all n > N   You are saying "a rational separates them." So you are saying the true thing "that is how we do it" but saying it by saying "we have this intricate construction (actually we have 3 or so) from the late 1800's which, inter alia, has built into it this way that we did it since Euclid." The real question is WHY do we do it this way. And this article does that reasonably well but with the defensive attitude of "these are baby explanations for the unsophisticated" --Gentlemath (talk) 06:29, 31 March 2010 (UTC)
 * You guess that the phrase "can be postponed" is an attempt to shut up questioners. I wrote it, so I can tell you that you're wrong.
 * Every history page contains a link to Revision history search. Using it, we discover that the phrase was introduced with this version of the article, when the order of proofs was quite different. The role of the sentence in that context was to transition from a section working with intricate constructions to a section that applies an abstract principle (which may be proved from a variety of foundations, so the reader doesn't need to care about what exactly happened in the late 1800's). Melchoir (talk) 07:20, 31 March 2010 (UTC)

I was intemperate in that remark. But I do think it is misplaced and unfortunate. I maintain that the abstract principal ( if by that you mean that "0<x implies 0<1/n<x for some integer n" is true in the place the decimals live) is not accepted because it is proved from the foundations, rather it is more or less explicitly included in the foundations because it is an accepted property of the objects which one is trying to capture. One can prove it from Euclid. I see that you did fine work on 1/4+1/16+... without a need to invoke constructions of the reals.--Gentlemath (talk) 05:33, 1 April 2010 (UTC)
 * No argument there, at least not from me! Melchoir (talk) 06:29, 1 April 2010 (UTC)

It seems obvious that 0.9 recurring is infinitely close to 1. Does that make it equal to 1? If you accept the ideas of limits, which is the basis of differentiation, then it is equal to 1. Another proof that 0.9 recurring equals 1? - Any number is either rational or irrational. Therefore, 0.9 recurring must be rational or irrational. As it is a recurring decimal, it theoretically should be able to be expressed as a whole fraction. But what would it be? Does that imply that it would be irrational? But it is a recurring decimal...   If 0.9 is not equal to 1, how can that be if it is neither rational or irrational? It seems that when you talk to people about 0.9 recurring equal to 1, they will fly into the same "rage" when calculus was introduced. My 2 Cents&#39; Worth (talk) 15:21, 19 May 2010 (UTC)


 * I agree with My 1.999.... Cents&#39; Worth. Phiwum (talk) 19:43, 19 May 2010 (UTC)

But is that me, or a different person? That is the question... :) My 2 Cents&#39; Worth (talk) 06:24, 20 May 2010 (UTC)
 * I similarly agree with My 2 Cents&#39; Worth. Note that your argument is somewhat redundant, as once it has been admitted that "0.9 recurring is infinitely close to 1", it immediately follows that they are equal, as the real numbers have been specifically designed in such a way as to exclude infinitesimals (the only infinitesimal is zero, hence .999...=1).  Thus the rest of your argument is unnecessary.  The limit is the basis for differention, as you mentioned, in the Weierstrassian approach to the foundations of infinitesimal calculus.  In a Leibnizian approach as clarified by Robinson, the limit of a sequence  would be the standard part of the value of un for an infinite value of the index n, and the derivative would be the standard part of the infinitesimal ratio.  Tkuvho (talk) 08:16, 20 May 2010 (UTC)
 * Exactly. If you differentiate x^2, you say that it is 2x, not infinitely close to 2x. My 2 Cents&#39; Worth (talk) 08:30, 20 May 2010 (UTC)
 * Does anyone disagree with that? Certainly not Leibniz.  Tkuvho (talk) 08:42, 20 May 2010 (UTC)
 * Or Newton. Calculus has been on a pretty solid footing for well over 200 years; I doubt that there are many who would still disagree with Leibnez or Newton.My 2 Cents&#39; Worth (talk) 11:55, 20 May 2010 (UTC)
 * Well, technically speaking, while we do agree with Leibniz that the derivative of x^2 should be 2x, we disagree with his definition of the derivative.  He defined the derivative as an infinitesimal ratio, but as you pointed out, the infinitesimal ratio will only be infinitely close to 2x, rather than 2x on the nose.  This problem most famously bothered George Berkeley, and of course numerous mathematicians ever since.  What Robinson pointed out is that the definition of the derivative should be modified by applying the standard part function to the infinitesimal ratio.  Tkuvho (talk) 12:20, 20 May 2010 (UTC)

Well, here's a thought. If you take the function x^2, and differentiate it by the formula f(x+h)-f(x) / h, you obviously get 2x+h. But when you say that x=0, you get f '(0)=h. Now, using limits, it would have shown that the h 'disappears', but in the case of f '(0)=h, isn't it obvious that h=0? Because if you plot x^2, the gradient at (0,0) obviously must be equal to 0, as the curve is a tangent to the x axis. So that implies that h must be equal to 0, which in turn must imply that the infintesimal must be equal to 0. Does this make sense?My 2 Cents&#39; Worth (talk) 14:21, 20 May 2010 (UTC)
 * Your objection does make sense. It is a good question that will shed light on the definition of the derivative.  The answer to the question is the following.  The infinitesimal ratio will in general depend on the infinitesimal h chosen.  In the case of x^2, it will be slightly bigger than 0 if h is positive, and slightly less than zero if h is negative.  However, under suitable circumstances, the standard part of the ratio will be independent of the infinitesimal chosen.  What are these circumstances?  Precisely when the function is differentiable.  It is a theorem of infinitesimal calculus that the function is differentiable if and only if the infinitesimal ratio exists for all h, and all ratios are infinitely close to each other. Tkuvho (talk) 14:28, 20 May 2010 (UTC)

Here's a practical thought about .999...=1. If you can't travel at the speed of light, but you can go infinitely close to that distance, does that imply that you are moving at the speed of light?My 2 Cents&#39; Worth (talk) 09:04, 21 May 2010 (UTC)
 * Well, not really. You would only be moving at .9.. times the speed of light :) Tkuvho (talk) 09:19, 21 May 2010 (UTC)

Yes, but if we agree with the article saying that .9 recurring is equal to 1 .... My 2 Cents&#39; Worth (talk) 15:07, 21 May 2010 (UTC)
 * You're bringing gross physicality into this. --jpgordon:==( o ) 15:12, 21 May 2010 (UTC)
 * You can't travel at the speed of light, and you can't get infinitely close to the speed of light. What you can do (in theory, of course) is get arbitrarily close to the speed of light (theoretically). Meaning, you can get to 0.9 times the speed of light, 0.99 times the speed of light, 0.999 times the speed of light, etc. --COVIZAPIBETEFOKY (talk) 16:04, 21 May 2010 (UTC)
 * "Infinitely close" is meaningless. --jpgordon:==( o ) 01:08, 22 May 2010 (UTC)
 * No, it isn't. It just happens to be the same thing as traveling at the speed of light. --COVIZAPIBETEFOKY (talk) 01:53, 22 May 2010 (UTC)
 * But that contradicts the statement that nothing can move at the speed of light (apart from light, and some would argue, gravity. And probably some other stuff that I'm not aware of).  Also, if you can get arbitrarily close to the speed of light, can you choose to go infinitely close to it?My 2 Cents&#39; Worth (talk) 15:26, 22 May 2010 (UTC)
 * To answer the second question: no. Think of an analogy.  I can choose an arbitrarily large natural number.  It does not follow that I can choose an infinitely large natural number.  Phiwum (talk) 15:44, 22 May 2010 (UTC)
 * But the thing that I don't understand is that can you choose infinity as the arbitrarily large number?My 2 Cents&#39; Worth (talk) 17:26, 22 May 2010 (UTC)
 * Phiwum didn't say arbitrarily large number; he said arbitrarily large natural number. Infinity is not a natural number. Similarly, when I say one can get arbitrarily close to the speed of light, I mean this in the implicit context of real numbers, so you can't pick "infinitely close" for arbitrarily close. --COVIZAPIBETEFOKY (talk) 18:05, 22 May 2010 (UTC)
 * Here My 2 Cents' Worth may have been asking about getting infinitely close in the context of a discussion of a number system containing infinitesimals, where x infinitely close to y simply means that x-y is infinitesimal. Similarly, the reciprocal of an infinitesimal will be infinite, and one can arrange for an infinite tail of 9s to fall short of 1 by an infinitesimal amount.  Tkuvho (talk) 15:15, 23 May 2010 (UTC)

Possible introduction edit
I have issues with this part
 * These various identities have been applied to better understand patterns in the decimal expansions of fractions and the structure of a simple fractal, the Cantor set. They also occur in a classic investigation of the infinitude of the entire set of real numbers.


 * I agree that there is a connection to the ternary Cantor set, the prototypical fractal.
 * The first claim probably refers to things like 8/37+29/37=0.216216...+0.783783..=0.999... but that is just one of the multiple representation numbers so the infinitely many does not come into it.
 * I assume that the final claim is about the diagonal proof that the reals are uncountable. There our numbers do not occur, we shun them to avoid confusion. Mentioning them here is not needed or helpful. --Gentlemath (talk) 07:59, 17 March 2010 (UTC)
 * Have you read the article? Melchoir (talk) 08:12, 17 March 2010 (UTC)
 * Which article are you referring to? The introduction does not mention any.  Later in the text, the relevant footnotes seem to be Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise. ^ Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method. ^ Rudin p.50, Pugh p.98.  Tkuvho (talk) 10:42, 17 March 2010 (UTC)
 * This article. Here, I'll provide a link to the section: 0.999.... To avoid misunderstanding, editors discussing changes to the relevant part of the lead section should be familiar with that section as well. Melchoir (talk) 18:29, 17 March 2010 (UTC)

Why three 9s?
Possibly a small point, but I couldn't help wondering why it's 0.999... and not simply 0.9... ? We don't do this for the other definitions (it's 0.(9), not 0.99(9), for example). Do we have a reference that for the ellipsis notation, three repeating digits should first be used? I guess part of the problem is that the ellipsis notation is rather ambiguous here - e.g., consider 0.454545... - is it 0.(45), or 0.45454(5)? - and I still say we should be using the more well-defined notations first, but if we're going to use ellipsis notation, the extra 9s seem redundant to me. Mdwh (talk) 16:51, 21 March 2010 (UTC)
 * There need to be enough 9s to establish the pattern. "0.9..." would not be enough, and there is something aesthetically pleasing about having three 9s for the three dots. I don't think anyone is likely to misread the current title. &mdash; Carl (CBM · talk) 16:58, 21 March 2010 (UTC)
 * If we need a reference using precisely three nines, we can easily find one; probably most of the references we already have do so. Finding one that explicitly discusses the number of nines to be used might be significantly harder, but repeating the pattern thrice seems to be standard to reduce ambiguity. Huon (talk) 18:17, 21 March 2010 (UTC)
 * I think if people meant 0.45454(5), they would write 0.45454555... instead. In addition to the aesthetic appeal, three establishes the pattern fairly clearly. Of course, it'll always be a little ambiguous when there isn't some established agreed-upon notation. --COVIZAPIBETEFOKY (talk) 23:45, 21 March 2010 (UTC)
 * Of course, if the number were 0.444555444(5), then the ellipsis notation would require more 5's to establish the pattern, so repeating 3 times isn't always perfect. --COVIZAPIBETEFOKY (talk) 23:47, 21 March 2010 (UTC)
 * With three nines you can stand on your head and discover that real numbers are the antichrist. ;) Algr (talk)
 * The questioner hits the nail on the head precisely. The confusion here is simply that the symbol 0.999... doesn't mean anything in mathematics, so it can be considered to be 1, or a number less than 1, or a smelly blue thought in a summer glass.  Once you start to work out precisely what you mean by 0.999..., all the confusion disappears.  0.(9) is precise mathematics, 0.999... is "making stuff up with no meaning".  Historically, mathematicians have been as confused about such issues as the general public still seems to be now (for example, arguing about the limit of 1 + 1 - 1 + 1 - 1 + ...), and the answer was and still is: Write down what it means, precisely and axiomatically, and all the confusion disappears.  The article is way too long for something actually pretty simple as this.  And to see why intuition on such issues is wrong, one just needs to refer to Zeno's paradoxes, 2500 years old.  Geez.  —Preceding unsigned comment added by 87.139.38.177 (talk) 03:19, 25 March 2010 (UTC)


 * 0.(9) is no more precise than 0.999... "0.999..." is a perfectly matematically valid representation. --RLent (talk) 17:32, 29 March 2010 (UTC)

Another point is that virtually anyone will realise what "0.999..." means, whereas 0.(9) is more obscure. It may be well known in some parts of the world for all I know, but it is certainly not in general use where I live. JamesBWatson (talk) 13:00, 30 March 2010 (UTC)
 * It's just that 0.999... makes the recursion look more aesthetically pleasing than just 0.9... — MC10 ( T • C • GB •L)  18:18, 26 July 2010 (UTC)

Removing the proof "Fractions and long division" under "Algebraic proofs"
I would vouch for removing the proof, because it's a tautology and does not really prove anything useful. If you start assuming that 1/9 equals 0.111... you are already assuming the problem solved that most people (I would guess) perceive to be the main problem. If you do not agree that 0.999... equals 1, then why would you agree that 1/9 equals 0.111...? I would guess most people think of 0.999... to be "approaching" 1 as limit, and thus 0.111... would also approach 1/9, but not be equivalent to it. So I don't see how that proof provides any useful insights -- the proof under "Digits manipulation" is, in my opinion, clearer and less debatable, and thus, should be preferred. Indeed I think that the proof under "fractions and long division" should only be provided if at the same time supplied with a proof that 1/9 equals 0.111.., for instance through a supremum proof, otherwise it will probably just confuse students. 88.89.239.218 (talk) 09:56, 23 July 2010 (UTC)
 * As the section name implies, showing that 1/9=0.111... is supposed to be done by long division. Some people who never heard of suprema may still accept long division (and as a consequence 0.999...=1). We already say that there are proofs with varying degrees of mathematical rigour; this is one where all the intricacies are "hidden" in the long division. Huon (talk) 12:34, 23 July 2010 (UTC)
 * Many people would not understand proofs with infinite geometric series or limits; this proof, although technically not a "valid" proof, would help convince them that 0.999... actually equals to 1. — MC10 ( T • C • GB •L)  18:20, 26 July 2010 (UTC)
 * This is also answered by the FAQ, second-to-last question. — MC10 ( T • C • GB •L)  18:28, 26 July 2010 (UTC)

Should we use "…" or "..."?
The MOS recommends using ... for the ellipsis, so should we use three dots rather than the "…" HTML character? — MC10 ( T • C • GB •L)  18:25, 26 July 2010 (UTC)
 * Specifically, it's MOS:ELLIPSIS. — MC10 ( T • C • GB •L)  21:35, 28 July 2010 (UTC)
 * I have changed all of the "…" to "...", per MOS:ELLIPSIS. If you wish to change it back, feel free to revert, but discuss here. — MC10 ( T • C • GB •L)  21:39, 28 July 2010 (UTC)

MOS rules like this don't always apply to mathematical notation the way they do to other things. WP:MOSMATH also exists. I haven't looked at whether it says anything about this. Michael Hardy (talk) 17:47, 16 August 2010 (UTC)

(Regarding 0.999.... supposedly being equal to one) I'm not a mathematician, but...
It's my understanding that .9 is 9/10 (nine tenths) of one. And .99 is (9/10 of one) plus 9/10 of the remaining 1/10. And .999 is (9/10 of 1) + (9/10 of 1/10) + 9/10 of 1/100 (which was the remaining one tenth of one tenth) and so on.

Now let's say the distance between point A and point B is one unit (meter, mile, cubit, whatever). And something goes nine tents of the distance from A to B, and then nine tenths of the remaining distance, then nine tenths of the remaining distance, then nine tenths of the remaining distance, and so on ad infinitum. Then wouldn't that thing with each step get closer and closer to point B, but never touch it, no matter how many times it went nine tenths of the way? (Of course in the real world you would get down to the subatomic level and humans would soon reach the limit of how small of a distance they could observe. But in the abstract world of numbers, one could infinitely "zoom in" farther and farther and quantify this minute distance as it got ten times smaller with each step.)

Now, like I said, I'm not a mathematician, and this is just based on my understanding of logic and numbers and how they work. But I'm pretty sure that my logic here is sound. I'm also pretty sure that I'm not the first one to come up with this idea.

Has anyone offered a refutation to this logical critique? Would be interesting to see, and if such discussion has been published, then I think it should be included in the article. -Helvetica (talk) 16:28, 15 August 2010 (UTC)
 * It has been, see section on infinitesimals. Tkuvho (talk) 16:36, 15 August 2010 (UTC)
 * You may also want to sift through the archives for this talk page as well as the arguments archive for this page.Racerx11 (talk) 00:24, 16 August 2010 (UTC)
 * You may also be interested in Zeno's paradoxes, where similar reasoning is employed to "prove" that motion is an illusion. Huon (talk) 02:11, 16 August 2010 (UTC)

You're basically restating the problem, which is that no amount of divisions by 10 by themselves can add up to 1. It would be the same to say that if I was given immortality and existed in a non expiring universe I could get to work writing .99999999... and so on and never have any hope of reaching 1 ever, just as your example never has hope of reaching point B. But that's because you're trying to reduce an infinitely divisible quantity into a dynamic linear expansion of a finite stream of digits.

You need to switch your thinking, it's not that 9/10th + 9/100th and so on will eventually add up to 1. It's that 1 itself contains every possible division of itself in base 10. If you take a pizza and cut it into 10 slices you now have the 1 pizza quantified into tenths, if you take one of those 10 slices and divide that into ten, you have the pizza divided down to hundredths. If you divide one of those slices it's down to thousandths. Look at your pizza so far, 9 slices in the tenths, 9 in the hundredths, and 10 in the thousandths. But you can quickly see you don't actually have to cut the pizza, you can see in each slice contains the divisions innately, so the tenth slice of the thousandths could be imagined as 10 slices of the ten-thousandths and the 10th of that as the hundred-thousandths so on and now your pizza looks like this: 9/10, 9/100, 9/1000, 9/10000, 9/100000 etc. Now you can see a single pizza can be thought of as having infinite divisions of base 10 and if you look at each division it is always 9 slices because the 10th slice is itself cut into 10 smaller slices 1/10th the size so you're not missing 1/10th your pizza, it's all there, and it all equals 1 whole pizza. 76.103.47.66 (talk) 07:19, 13 September 2010 (UTC)

It's a falacy
Talk:0.999.../Arguments


 * Perhaps, but it's certainly not a fallacy. — Loadmaster (talk) 17:27, 17 October 2010 (UTC)

lead edit
I have WP:boldly added a sentence to the lead, based on a recent discussion with an IP, concerning student unease about the limit definition. Should we mention also that they tend readily to agree that the sequence .9, .99, .999, ... tends to 1? Tkuvho (talk) 12:34, 20 October 2010 (UTC)
 * While I tend to agree with the sentiment expressed, it sounds like original research, though some rather vague references were given. If there are reliable sources about student opinion, we can add them, but if not, we shouldn't add our personal speculation or anecdotes. Huon (talk) 13:45, 20 October 2010 (UTC)

Flaw in the "digit manipulation" proof?
I think I may have identified a flaw in the famous "digit manipulation" proof which is included in the article:



\begin{align} x          &= 0.999\ldots \\ 10 x      &= 9.999\ldots \\ 10 x - x   &= 9.999\ldots - 0.999\ldots \\ 9 x        &= 9 \\ x          &= 1 \end{align} $$

The second step (10 * 0.999... = 9.999...) takes advantage of the fact if there are an infinite number of nines then if you add or subtract one more (or 300 more or whatever) there are still an infinite number of nines. Essentially the math for that would look like this: ∞ -1 = ∞ And, though this is true, one could use the same principle to "prove" any number of patently absurd claims.

For example:

4 + ∞ = ∞ 5 + ∞ = ∞ 4 + ∞ = 5 + ∞ 4 = 5 (subtracting ∞ from both sides) 2 + 2 = 5 (or even an Orwellian conclusion ;-)

I could go on with more examples, but I think you get the idea. The point being that if we include this unique property of infinity in a proof then it can lead to conclusions which are clearly wrong. So therefore a proof cannot depend on this property the way the "digit manipulation" proof does and still be considered reliable. Helvetica (talk) 17:23, 16 August 2010 (UTC)
 * Not quite. The "infinity" here is a cardinality - there are (countably) infinitely many digits. "∞ -1 = ∞" in this context means that the sets of positive integers and the set of non-negative integers are of equal cardinality, even though the latter has all the elements of the former and one more. They are, since there's an obvious bijection. But subtracting infinite cardinalities is not well-defined - if you have a countably infinite set and remove infinitely many elements, there's no a priori way of knowing how many elements will be left. Huon (talk) 17:34, 16 August 2010 (UTC)

That is not a flaw in the argument. It is true that you can't subtract in the manner used above to show that 4 = 5, but that's not what is being done at all.

And no one said that 9 &times; 0.999... = 9.999... . Rather it was asserted that 10 &times; 0.999... = 9.999..., and that 9 + 0.999... = 9.999... . Michael Hardy (talk) 17:45, 16 August 2010 (UTC)


 * MH, that was an error in my re-writing of that part of the proof. I"ll go ahead and fix that.  I realize that the manipulation proof doesn't use the same exact process as the one I made, but it includes the same step of adding or subtracting from infinity.

I suppose I should have constructed a proof that started with more precisely the same process. Then it would be something like this:

∞ - 1 = ∞ ∞ = ∞ + 1 (add 1 to both sides) ∞ = 1 + ∞ (commutative property) 0 = 1 (subtract ∞ from both sides)

4 = 5 (add 4 to both sides; if you want to do a bit more digital manipulation to get to the same conclusion as my other example)

-Helvetica (talk) 18:00, 16 August 2010 (UTC)


 * Your problem is that $$\infty - \infty$$ isn't necessarily equal to zero. You can still subtract $$\infty$$ from both sides, but you can't cancel it with itself, since you can only do that with real numbers (and several other types of numbers, but not infinity).
 * Let's try this again, remembering that $$\infty - \infty = indet$$ (an indeterminate form).

\begin{align} \infty &= 1 + \infty \\ \infty - \infty &= 1 + \infty - \infty \\ indet &= 1 + indet \\ indet &= indet \end{align} $$
 * Nothing wrong here. --Zarel (talk&sdot;c) 18:27, 16 August 2010 (UTC)
 * A real decimal is defined as the limit of its truncated decimals. In this setting, it can be shown that multiplying by 10 corresponds to shifting the decimal point one position to the right.  Believe me, a lot of people have checked this and there is no contradiction (unless ZFC contains one).  Unless you get out of the framework of the reals and limits, you can't challenge the apparently paradoxical conclusion that .999...=1.  Instead of trying to prove everyone wrong, try learning something new in the infinitesimal section.  Tkuvho (talk) 18:50, 16 August 2010 (UTC)


 * It seems to me that: an infinite string of ditits, mulitply by 10, shift the decimal and you still have an infinit string of digits after the decimal; is not quite the same thing as: the concept of an infinite number - 1 = the concept of an infinite number.


 * 0.999... * 10 = 9.999... is shifting decimal point to the right. You are looking at the number of nines to right of the decimal and saying one nine has been subtracted and there are still infinite nines then assuming thats the same thing as a line in a proof ∞ - 1 = ∞.


 * In the former we are talking about the number of digits implied to complete a particular decimal representation of a "non-infinite" number and the number of digits required to indicate the result of a mathematical operation on that finite number. While the latter is a mathmatical operation on the conceptual quantity of infinity itself.


 * One more thing which is somewhat implied in the article as well as in a comment above. If one accepts that .999... * 10 = 9.999... ; then you can quickly show this itself proves 0.999...=1: If a number multiplied by 10 equals the same number + 9, then that number must be 1.
 * So if 10x = 9 + x, then x is 1
 * plug in 0.999... for x
 * 9.999...= 9.999...
 * Both sides equal therefore 0.999... is 1
 * Racerx11 (talk) 14:27, 21 August 2010 (UTC)
 * This reasoning also works for this if:

\begin{align} 10x &= 8 + x, then x is 1 \\ 10(0.888...) &= 8 + 0.888... \\ 8.888... &= 8.888... \\ \end{align} $$ 18:46, 20 October 2010 (UTC) —Preceding unsigned comment added by 210.4.96.72 (talk)
 * Well no, of course. "10x = 8 + x, then x is 1" is false. Would be 10=9. I was just illustrating a simple psuedo-proof based on a prior acceptance of 10(0.999...)=9.999... and the fact: if 10x = 9 + x then x must be 1. Clearly its no good if you start with the false statement: if 10x = 8 + x, then x is 1.
 * Your example works as psuedo proof for .888...= 8/9 if you start with: if 10x = 8 + x, then x must be 8/9
 * plug in 0.888... for x
 * 10(0.888...)= 8 + 0.888...
 * 8.888... = 8.888...
 * both sides equal therefore 0.888...= 8/9
 * Just to state my original point another way: With 10x = 9 + x, there can be but one unique solution for x, or only one number for x in which both sides of the equation are equal. Since both the number 1 and 0.999... each work when plugged in for x, then they must be the same number. However, as pointed out this is assuming one accepts 10(0.999...) = 9.999...
 * Racerx11 (talk) 06:00, 30 October 2010 (UTC)

Here's my counter arguments for the first two proofs: (1/9) = (0.1111...) 9(1/9) = 9(0.1111...) 1 = 0.9999...

is similar to this argument (1/4) = (0.2500...) = (0.2...) 4(1/4) = 4(0.2...) 1 = 0.8...

Which is erroneous.

Second proof: Digit Manipulation (x) = (0.9999...) 10(x) = 10(0.9999...) (10x) = 9.999... (10x)-x = (9.999...) - (0.9999...) 9x = [9] [9 is ERRONEOUS, the answer is 8.9999... look at the argument below] x = 1

is similar to this argument: (x) = (0.9999) 10(x) = 10(0.9999) (10x) = 9.999 (10x)-x = (9.999) - (0.9999) 9x = 8.9991 x = 0.9999

See what I did there. —Preceding unsigned comment added by 210.4.96.72 (talk) 12:22, 16 October 2010 (UTC)
 * Yes, you made some elementary errors. 0.25 and 0.2 are not the same so that is clearly wrong. In the second you misunderstand what 0.9999... means. It is a recurring decimal so does not behave like 0.9999.You are not alone in this as it's a common way to misunderstand this:see the section skepticism in education for details.-- JohnBlackburne wordsdeeds 12:48, 16 October 2010 (UTC)

Hi, yes, these errors are what I was highlighting. You see, someone truncated the infinite number of 9s. It is like truncating 5 in the second example. These infinite 9s are important to the identity of 0.999...

The second similar argument shows that (9.999...) - (0.9999...) in fact is not 9 but instead 8.999...1 —Preceding unsigned comment added by 210.4.96.72 (talk) 02:10, 17 October 2010 (UTC)

Guys, big flaw in the digit manipulation proof:

Number of 9's in the decimal equals 4 (x) = (0.9999) 10(x) = 10(0.9999) (10x) = 9.999 (10x)-x = (9.999) - (0.9999) 9x = 8.9991 x = 0.9999

Number of 9's in the decimal equals 5 (x) = (0.99999) 10(x) = 10(0.99999) (10x) = 9.9999 (10x)-x = (9.9999) - (0.99999) 9x = 8.99991 x = 0.99999

Number of 9's in the decimal equals 6 (x) = (0.999999) 10(x) = 10(0.999999) (10x) = 9.99999 (10x)-x = (9.99999) - (0.999999) 9x = 8.999991 x = 0.999999

The pattern: 9's=4; 9x = 8.9991 9's=5; 9x = 8.99991 9's=6; 9x = 8.999991 9's=10; 9x = 8.9999999991 9's=infinity; 9x = 8.999...1 —Preceding unsigned comment added by 210.4.96.72 (talk) 14:40, 17 October 2010 (UTC)
 * But 8.999...1 is meaningless. --jpgordon:==( o ) 15:47, 17 October 2010 (UTC)


 * There is no flaw in the digit manipulation proof. Because 0.999... has no last digit, there is no place for that "1" in 8.999...1 to be. For every natural number n, the n-th digit (after the decimal separator) of 0.999... is 9. And for every natural number n, the n-th digit of 9.999... = 10 * 0.999... is 9, too - because the n-th digit of 10 * 0.999... is the n+1-th digit of 0.999... Thus, for every natural number n, the n-th digits of 0.999... and 9.999... are equal, making the n-th digit of the difference equal to 0. Thus, every digit of 10 * 0.999... - 0.999... after the decimal separator is zero; the difference is indeed 9. The trick is that infinitely many digits less one are still infinitely many.
 * Maybe it's clearer if you do it the other way around: We probably agree that 9+0.999... = 9.999... And what's 9.999.../10? The answer is 0.9999... = 0.999..., and it's rather obvious that "0.999...9" doesn't make any sense whatsoever because there's no place for "0.999...9" to have that last 9 where 0.999... didn't have a 9 already. Huon (talk) 15:56, 17 October 2010 (UTC)

8.999...1 is like a infinitely deep hole with a goblin at the bottom. If we are given eternity to reach the goblin, we'll never reach it, but we know the goblin is in the bottom. Similarly, in 8.999...1, we just know that the last digit is 1, but we will never reach it because of the infinite non-terminating 9s.

9+0.999... is not equal to 9.999...; 9+0.999...=9+0.999... It's like 9+x, which is not equal to 9.x, but instead equal to 9+x. —Preceding unsigned comment added by 210.4.96.72 (talk) 02:38, 18 October 2010 (UTC)

Also if 0.999...9 doesn't make any sense, then why add it in the article's proofs? I've seen it countless of times used in the article. —Preceding unsigned comment added by 210.4.96.72 (talk) 03:24, 18 October 2010 (UTC)


 * Concerning 0.999...9, the article uses that notation to denote a number with finitely many nines; I found a single instance of use, and there the number of nines was explicitly given. Here, a notation like "8.999...1" is used to denote a number with infinitely many nines.
 * As I mentioned above, every digit (after the decimal separator) corresponds to a natural number. There is a first digit, a second digit, a third digit, and so on. For every natural number n, there's an n-th digit. And conversely, these are all digits there are. If a notation such as "8.999...1" were to make any sense, you'd have to be able to say to what natural number that "1" digit corresponds. Especially, there is no last digit of 0.999...; every nine is still followed by infinitely many more nines.
 * And concerning 9+0.999..., if it's not 9.999..., what is it? Every real number has a decimal representation, so what's the representation of 9+0.999...? You'll also note that for numbers of the form 0.x, where x is some string of digits, we indeed have 9+0.x = 9.x. Huon (talk) 07:43, 18 October 2010 (UTC)
 * Huan, your comments are patient and technically correct. However, user 210.4.96.72 is clearly groping for a notion of a terminating decimal with infinitely many digits.  Such a notion is available in the hyperreals, where a decimal can have an infinite hypernatural's worth of digits, where the last 9 occurs at an infinite rank H, say.  Let's give the students the credit they deserve, and let's stop assuming that their intuitions are erroneous.  Tkuvho (talk) 13:56, 18 October 2010 (UTC)


 * I've read what Katz had to say about terminating hyperreal decimals with infinitely many digits, but I still don't see why we should call any of them "0.999..." - there are infinitely many candidates, and none of them seem a more likely candidate to be called "0.999..." than any of the others. The natural candidate of a hyperreal to be called "0.999..." to me seems to be the one which has nines all the way - and that's again equal to 1. Besides, I doubt many of those who argue intuitively for a notion of a terminating decimal with infinitely many digits would actually be happier with the hyperreals than the reals - the hyperreals probably just violate their intuition in other places. Huon (talk) 15:40, 18 October 2010 (UTC)

Anyway, if 8.999...1 is not a number, at least consider that the answer to (9.999...-0.999...) is closer to 8.999... than it is to 9 (based on the pattern). And thank you for being patient with me. —Preceding unsigned comment added by 210.4.96.72 (talk) 14:51, 18 October 2010 (UTC)

Also, 0.999...9 was used in the limit formula where n->infinity
 * 9.999... - 0.999... is exactly the same distance from 8.999... as it is to 9, as 8.999... and 9 represent the same number, in exactly the same way 0.999... and 1 are the same number. --jpgordon:==( o ) 15:33, 18 October 2010 (UTC)


 * Well, since 8.999... equals 9, arguing whether 9.999...-0.999... is closer to one than to the other is a little pointless, but this begins to become circular reasoning. If you want to argue based on the pattern, you have to be careful in the very last step - the one leading from finitely many digits to infinitely many digits. This would require limits for full precision, but it's rather easy to see that some statements that are true for all numbers of the form 0.9, 0.99, 0.999, and so on no longer hold for 0.999... The most obvious such statement is that 0.9 < 0.999..., 0.99 < 0.999..., 0.999 < 0.999... and so on - but obviously we can't conclude that 0.999... < 0.999...
 * Concerning 0.999...9, that's the instance I noticed, too. There, it was explicitly said that 0.999...9 was a shorthand for a number with n nines, a finite number of non-zero digits. But if I understood you correctly, 8.999...1 was supposed to have infinitely many nines followed by a 1. As Tkuvho noted above, there exists a number system, the hyperreals, which actually allows for such numbers, but I find its relation to 0.999... rather forced. In the context of real numbers, the context most widely used, such a notation would be ill-defined. Huon (talk) 15:40, 18 October 2010 (UTC)

Thanks for the reply. This is exactly where the error in the proof which I like to highlight: when we assume that 8.999... already equals to 9. The proof is like a circular argument - it already carried the mathematical prejudice that 0.999... already equals 1. And we already know that the answer based on the pattern we observe is not exactly 8.999... but rather 8.999...1 (if it makes sense).

Moreover, in that same reasoning as the argument, how can we conclude that 0.999... = 0.999...? We already agree that we can't conclude that 0.999... < 0.999... but how did we conclude that the number of 9s are equal? I mean if we represent "n" as the infinite number of 9s and just subtract n from n, it would be infinity minus infinity, which is undefined. Wouldn't the answer better be left undefined?

For 0.999...1 analogy: take for instance an immortal horse in a treadmill: 0.999... is when the 9s are represented by the horse's every step - the horse running for eternity. In 0.999...1, a carrot (...1) is tied to the head of the horse, so that the animal would never reach it even if it runs for eternity. In this picture, we don't know if the number of steps of a horse with a carrot is greater than the number of steps of a horse with no carrot - but we know that the horse with a carrot is a different horse than one which has no carrot. Moreover, the step of a horse who only took only one whole gallop (1) is different from both horses.

Moreover, 0.999...9 was a limit were infinity substitutes "n" - not a finite number. Aside from the limit formula, 0.999...9 was also used in Nested intervals and least upper bounds section, infinitesimals and p-adic numbers.

Anyway, don't get me wrong. I see no flaw with the limit proof - since a limit of a sequence only says that it converges to, but does not equal, one. In a graph, the limit function would approach one but would not touch it. —Preceding unsigned comment added by 210.4.96.72 (talk) 16:51, 18 October 2010 (UTC)
 * Circular reasoning is indeed a threat, but the proof does not rely on it. We don't argue that 0.999...=1 because 8.999...=9. Let me present you with another pattern:
 * 10*0.99 - 0.9 = 9
 * 10*0.999 - 0.99 = 9
 * 10*0.9999 - 0.999 = 9
 * 10*0.99999999999 - 0.9999999999 = 9
 * Thus, 10*0.999... - 0.999... = 9.
 * Since the sequence (0.9, 0.99, 0.999, ...) tends to 0.999..., so does the sequence (0.99, 0.999, 0.9999, ...).
 * For the digits at infinity and your horse in a treadmill example, I'll just note that if 9.999...-0.999... actually were 8.999...1, wouldn't we have to denote the first two numbers as 9.999...0 and 0.999...9, respectively? That's actually what Tkuvho suggests; when we denote hyperreal numbers by decimals, there's not just a digit for every natural number, but also digits for certain infinite "hyperintegers". He'd indeed like to call a number "0.999..." which has infinitely many nines, and nines up to a certain infinite hyperinteger, and zeros from then on. The article discusses that number system in its section on infinitesimals; using that notation, he'd say that 0.999... should be defined to be some number of the type 0.999...;...999000... In this context, it's a good question whether 0.999... and 9.999... have the same number of digits; there seems no a priori reason (actually neither 0.999... nor 9.999... are well-defined in this context; both are no longer single numbers, but classes of similar numbers that differ by the (infinite) number of nines). That's why I'd consider his preferred definition rather useless. Besides, in that notation there's still a number where all digits are nines, and that number is still equal to 1. But in the context of real numbers, "infinitely many" nines just means that the cardinality of the set of digits which are nines is (countably) infinite. If we add or remove such a digit (say, by adding 9 or subtracting 0.9 or something like that), the resulting set of digits which are nines is still countably infinite; there's a bijection between the sets. In that sense, the numbers of nines are "the same". But in the context of real numbers, there's not actually an "infinity-th" digit; there are just digits for all the natural numbers.
 * We also don't have to subtract the numbers of digits for the digit manipulation proof. When we subtract 0.999... from 9.999..., we'll just do a regular subtraction, digit by digit. Since 9.999... has nines in all relevant digits (actually, in all digits after the decimal separator), we don't need to worry about carrying and can start on the left instead of the right: The first digit (after the decimal separator) of 9.999...-0.999... is 9-9=0. So is the second digit, so is the third digit, and so on. There's no natural number n for which we don't find that the n-th digits of both 9.999... and 0.999... are nines; thus, there's no n for which the n-th digit of 9.999...-0.999... isn't zero.
 * I'll also have to note that you seem to have some misconceptions about limits. A sequence cannot equal 1; that would be a type mismatch. A sequence simply is not a number. Its limit, on the other hand, is a single, fixed number and does not converge to anything. In our case, 0.999... is the limit of the sequence (0.9, 0.99, 0.999, ...), and while that sequence converges to 1, that's just saying that 0.999... equals 1. It's also not true that in a limit "as n tends to infinity" one actually substitutes infinity for n - while that intuition is often helpful, ultimately there's a precise definition of limit that doesn't involve anything resembling a number "infinity". The shortest definition I could give is that "the limit of a sequence an as n tends to infinity" is, by definition, a number a such that for every open set U containing a, there are at most finitely many natural numbers i such that ai is not an element of U. Thus, $$\lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n}$$ does not imply that $$0.\underbrace{ 99\ldots9 }_{\infty}$$ is a meaningful notation. And the "nested intervals" section says: ... [0.99...9, 1] for every finite string of 9s - it explicitly says that 0.99...9 refers to a number with just finitely many nines. That's indeed a standard notation. Maybe I should have been clearer when I said that 0.999...9 doesn't make any sense that I was speaking of the result of dividing 9+0.999... by 10 - that number would have infinitely many nines, but just as 0.999... doesn't have a "last nine", neither would (9+0.999...)/10. Huon (talk) 18:00, 18 October 2010 (UTC)

Alright, thanks for clearing that up. Not everyone is as patient as you guys in explaining these things and I commend you for that. I hope you guys don't mind if I ask another to clear up another doubt of mine: If both 1 and 0.999... are equal, it would also mean that they are interchangeable. If so, it should be possible to interchange them in the limit formula: $$\lim_{n\to\infty}\underbrace{1}_{n}$$ and they both should have the same graph. However, I don't even think that the limit function would make sense if I interchange 0.999... to 1.

Also, it looks like the proof is only a one way process -> I can't prove 1 to equal 0.999... by reversing the same process.210.4.96.72 (talk) 04:47, 19 October 2010 (UTC)



\begin{align} x          &= 1 \\ 9 x        &= 9 \\ 9 x + x   &= 9 + 1 \\ 10 x          &= 10 \\ x      &= 1 \\ \end{align} $$
 * Tell me would you agree that the sequence .9, .99, .999, ... gets closer and closer to 1 in such a way that it gets arbitrarily close to it? Tkuvho (talk) 05:19, 19 October 2010 (UTC)


 * I agree that the sequence's limit is 1. The sequence approaches 1 but does not reach and equal it (best described by the limit's graph). Hence the reason why one (1) limits the sequence - it's the brick wall. 1 is a number, while 0.9999... is an infinite sequence - isn't that a type mismatch to say that they are equal? Btw, may I know how to add graphs here? Also, is there a proof that 1=0.999... (reverse process)? Instead of the proof that 0.999...=1. This way, I think students would understand the proof better. And this would help improve the article. —Preceding unsigned comment added by 210.4.96.72 (talk) 06:13, 19 October 2010 (UTC)

Same digit manipulation proof, but different answer:

\begin{align} x          &= 0.999... \\ 10(x)        &= 10(0.999...) \\ 10(x)- x  &= 10(0.999...)-0.999... *Factor \\ x(10-1)     &= 0.999...(10-1) \\ x      &= 0.999... \\ \end{align} $$

Here, I've factored out 0.999... so that the "magic" in the proof would not happen. 210.4.96.72 (talk) 06:41, 19 October 2010 (UTC)
 * What you say is very interesting. Let's go back one step.  You say you agree that the limit of the sequence .9, .99, .999, ... is indeed 1.  You also made some interesting comments about the symbol "0.999..."  How do you think of that symbol exactly?  Do you think of it as (a) an infinite process, or (b) the infinitieth term in the sequence, or (c) some other way?  Tkuvho (talk) 07:41, 19 October 2010 (UTC)
 * Hi, I view it as a number similar to Pi and e. Our mathematical symbols can only approximate it but not point exactly to it. Unlike 1, wherein I can use the symbol 1 to point to it.210.4.96.72 (talk) 08:26, 19 October 2010 (UTC)

Ok, here is the difference between 0.9999... and 1 using this proof:

\begin{align} 0.99999...          &= x \\ 0.99999...+0.1        &= x+0.1 \\ 1.09999...  &= x+0.1 \\ 1.09999...+0.01  &= x+0.1+0.01 \\ 1.10999...-0.1  &= x+0.1+0.01-0.1 \\ 1.00999...+0.001  &= x+0.1+0.01-0.1+0.001 \\ 1.01099...-0.01  &= x+0.1+0.01-0.1+0.001-0.01\\ 1.00000...  &= x+ (sequence of 0.1+0.01-0.1+0.001-0.01+0.0001-0.001...)\\ 1 &= x + 0.0000...1\\ 1 &= 0.99999... + 0.0000...1\\ \end{align} $$ 210.4.96.72 (talk) 09:10, 19 October 2010 (UTC)
 * This is very interesting. I would like to pursue this a little further if you don't mind.  You wrote above that "Our mathematical symbols can only approximate it but not point exactly to it", but you also described it as an infinite sequence in an earlier post.  Would you say this is closest to (a) viewing it an an infinitieth term in the sequence .9, .99, .999, ..., or (b) an infinite process, or (c) something else?  I think everyone here believes that it is a number like pi, the question is what does that mean exactly?  Tkuvho (talk) 09:58, 19 October 2010 (UTC)


 * I think 0.999.. is the summation of the sequence (0.9+0.09+0.009+0.0009+0.00009...) 210.4.96.72 (talk) 10:07, 19 October 2010 (UTC)


 * I note that you used the term "summation" rather than "sum". Indeed, the sums we learn about ordinarily are finite sums, so what is immediately accessible is the sequence of partial sums of the expression you wrote down, which brings us back to the sequence .9, .99, .999, ...  In other words, restating the problem in terms of summation does not really clarify what is going on.  I won't pursue this any further but if you would like to respond to the a,b,c dichotomy above it may be interesting to compare your experiences with other students'.  Tkuvho (talk) 10:19, 19 October 2010 (UTC)
 * I'm sorry but the best description I can give to 0.999... is the summation of the sequence (0.9+0.09+0.009+0.0009+0.00009...). If the summation is indeed the infinitieth term in the sequence .9, .99, .999, ... then maybe it is (a). But I am not sure, and will never be sure, even if God gives me eternity - because I will never know what the infinitieth term in that sequence exactly is and what the summation of the sequence exactly is; I just know that one (1) is the limit (would never touch it in a graph).

210.4.96.72 (talk) 11:09, 19 October 2010 (UTC)


 * OK, that's fair. If you think that 0.999... is the summation, and the summation is perhaps the infinitieth term in the sequence .9, .99, .999, ... then your intuition is similar to those of many students who have grappled with this problem, who feel that the sequence does get arbitrarily close to 1 so that 1 is the limit, but feel as you do that 0.999... is something different, i.e., they are not convinced by the limit definition of 0.999...  As documented in recent education literature, these students intuitively feel that such an infinitieth term falls infinitesimally short of 1.  Also documented in the literature is the fact that such intuitions, far from being erroneous, are actually useful in the learning of the calculus.  The traditional approach to teaching calculus cannot distinguish between two numbers that are infinitely close, therefore the limit concept collapses all infinitesimal differences and declares 0.999... to be equal to 1 (such "collapse" is represented by the standard part function).  Have you started learning calculus yet?  If you are at a typical state university, you will find that the instructors are generally unreceptive to ideas about infinitesimals (you may be called "fringe" or worse).  Meanwhile, the latter are a useful tool in understanding calculus.  You may want to consult Keisler's book Elementary calculus, but I would not mention infinitesimals on any tests as this may lower your grade.  Tkuvho (talk) 11:20, 19 October 2010 (UTC)


 * I'm a graduate already. And sad to say but during my days in college, I've just accepted everything and tried my best to get the highest letter possible. And since we have so much subjects to tackle during our calculus classes, as a student, I didn't really have the time to ask the "whys" deeply. Either for these reasons, or I'm just forgetful. Also, I don't remember our class tackling infinitesimals...


 * Right now, I'm just concerned that the graphs of f(x)=1 compared to f(y)=0.999... is not the same.210.4.96.72 (talk) 12:22, 19 October 2010 (UTC)


 * Interesting. So graduated from college?  Were you a math major?  Tkuvho (talk) 12:58, 19 October 2010 (UTC)
 * Yup, graduate from college. And no, not a math major. I just love math. 210.4.96.72 (talk) 13:39, 19 October 2010 (UTC)
 * Interesting. If you get a chance to check out Keisler's book, I would be curious how it compares with your undergraduate experience. Tkuvho (talk) 13:51, 19 October 2010 (UTC)

Alright, thanks for the recommendation. I'll get back after I finish reading the book online. Again, thanks. 210.4.96.72 (talk) 13:57, 19 October 2010 (UTC)


 * OK, I will see you next year. The book is over 900 pages :) Tkuvho (talk) 14:00, 19 October 2010 (UTC)

Btw, while I'm reading the book, can anyone be brave enough to prove that:

\begin{align} summation of the sequence (0.1+0.01-0.1+0.001-0.01+0.0001-0.001...)=0\\ \end{align} $$ I think this will help improve the article. I can't find any other work which I can cite that has proven that it is equal to zero. —Preceding unsigned comment added by 210.4.96.72 (talk) 15:40, 20 October 2010 (UTC)
 * I'll tackle the desired proof on the arguments page since it's getting a little off-topic. Huon (talk) 16:15, 20 October 2010 (UTC)


 * The answer to this query is similar: I assume you would readily agree that the sequence of partial sums tends to zero (becomes arbitrarily small). This by definition means that the limit is zero.  The resistance to such a conclusion comes from a not unreasonable intuitive perception of the infinite sum as the infinitieth partial sum; then indeed it is a nonzero infinitesimal.   Tkuvho (talk) 17:21, 20 October 2010 (UTC)


 * My proof is now available at the arguments page, but indeed I show that the limit of the sequence of partial sums equals zero. If that's not what was meant by "summation of the sequence", that term would have to be defined. Huon (talk) 22:15, 20 October 2010 (UTC)

Can someone please verify the correctness of this proof:

√0.9999... = (3/√10)+(3/√10^2)+(3/√10^3)+(3/√10^4)+...+(3/√10^∞)\\ √0.9999... = 0 *3/∞ is zero

But

√1=1 √0.9999...=0 And 1!=0, thus 1!=0.9999...

In simpler terms, if 1 is equal to 0.999... then the square root of 1 should also equal the square root of 0.999... but as you can see, the square root of 0.999... is 0, and the square root of 1 is 1. And because we know that 0 is not 1, then we just shown that 0.9999... is not equal to 1. 210.4.96.72 (talk) 19:16, 20 October 2010 (UTC)
 * Do you truly believe that the square root of 0.999... is 0? --jpgordon:==( o ) 20:08, 20 October 2010 (UTC)

Well, since 3/∞ is zero... so yes, the square root of 0.999... is 0. 210.4.96.72 (talk) 20:30, 20 October 2010 (UTC) Wait! It is also possible that I've committed an error and the summation of the square root of 0.9999... is ∞/∞ or undefined. Nevertheless, it still holds true that 1 is not equal to undefined thus 0.999... is not equal to 1. —Preceding unsigned comment added by 210.4.96.72 (talk) 20:35, 20 October 2010 (UTC)
 * I think it's time to draw a line under this. You are just tying yourself up in knots doing maths that seems well beyond your ability, as you keep making fundamental and basic errors. The point of the talk page is not to check and correct your maths, which you are anyway producing too quickly for anyone to reasonably deal with. The proofs in the article are correct. If you don't understand them perhaps say which bits aren't clear and you can be directed to another article or a reference that explains it more fully.-- JohnBlackburne wordsdeeds 20:44, 20 October 2010 (UTC)


 * Please understand that I'm trying to improve the article. People seems confused between what a limit of a sequence is and what a number is. And since it is not clear cut in the article that it is also possible for 0.999... != 1 "since the notation 0.999...

stands, not for a single number, but for a class of numbers, all but one of which are less than 1." as stated on http://www.math.umt.edu/TMME/vol7no1/TMME_vol7no1_2010_article1_pp.3_30.pdf I hope you understand and I do not force anyone to correct my maths. I will correct them myself once it is brought to my attention. —Preceding unsigned comment added by 210.4.96.72 (talk) 21:06, 20 October 2010 (UTC)
 * That article says, before the number system has been explicitly specified, one can reasonably consider that the ellipsis “...” in the symbol .999... is in fact ambiguous. From this point of view, the notation .999 . . . stands, not for a single number, but for a class of numbers,3 all but one of which are less than 1. However, this article says, In mathematics, the repeating decimal 0.999... which may also be written as 0.9, 0.9̇ or 0.(9), denotes a real number that can be shown to be the number one.  So, since we explicitly specify the number system, the ambiguity does not exist. --jpgordon:==( o ) 21:19, 20 October 2010 (UTC)
 * Oh yes, Katz and Katz. They change the standard meaning of 0.999..., as far as I can tell for the sole purpose of making 0.999... less than 1. In my opinion, instead of trying to improve the students' understanding of mathematics, they try to make the math fit the students' preconceptions, and they lose too much to make it worthwhile. Huon (talk) 21:58, 20 October 2010 (UTC)

Edit sentence about truth of assertion
I think the sentence "The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks." should be changed. It makes it sound like there's still doubt about the validity of this equality. My 7th grade son even debated the truth of the equality based on how this sentence was phrased. That's why I added the "and is definitely true" at the end in my edit. Someone please consider rewriting this.

Bill Kinney, 10/14/10 —Preceding unsigned comment added by 140.88.126.30 (talk) 16:36, 14 October 2010 (UTC)


 * For an alternative view, see Gualtieri. Tkuvho (talk) 17:57, 14 October 2010 (UTC)


 * I don't think such an addition as proposed by Bill Kinney would be helpful. We already say that 0.999... "can be shown to be the number one" in the very first sentence. That should be unambiguous, shouldn't it? The "accepted by mathematicians and taught in textbooks" sentence, on the other hand, is indeed the introduction to a paragraph about people who doubt that equality. They're wrong, but they exist. Huon (talk) 22:50, 14 October 2010 (UTC)


 * Are they "wrong" because the numbers called real, really are what they are made out to be? Tkuvho (talk) 09:19, 15 October 2010 (UTC)


 * How about "The existing proofs has long made mathematicians accept the equality 0.999... = 1", or "The equality 0.999... = 1 has long been accepted by mathematicians because of the existing proofs"?Diego Moya (talk) 17:32, 30 October 2010 (UTC)

Versions of the paragraph
Please keep the discussion in the previous section, place here variants of the proposed paragraph. Anyone is welcome to edit these or add new versions. Diego Moya (talk) 17:29, 30 October 2010 (UTC)

Based on limits
"One particular proof uses the definition of the symbol '...'. This symbol defines a real number as the sum of the infinite repeating decimals that the symbol represents. The value of an infinite sum is defined by a limit of a sequence, and the limit value of this particular sum of trailing 9s (0.9+0.09+0.009+...) is the real number 1."

Based on fractions, one paragraph
"For example: 0.333...=$1/3$ and 3*$1/3$=1. Since 3*0.333...=0.999... this shows that 0.999...=1."

Based on fractions, with equations
One simple way of showing that "0.999..." and "1" are the same thing is to divide them both by the number 3. When "0.999..." is divided by 3, the answer is "0.333...", which is the same as $1⁄3$ (the fraction one third).


 * $${0.999\ldots \over 3} = 0.333\ldots = \frac 13$$

When "1" is divided by 3, the answer is $1⁄3$. Since the answers are the same, that means that "0.999..." and "1" are the same.

Based on fractions, text only
"1 is three times $1/3$. The fraction $1/3$ can be written as 0.333... and three times 0.333... is 0.999... which thus is the same number as 1."

Enumeration of the concepts used in proofs
"Proofs of this equality have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience. Algebraic proofs use fractions, long division, and digit manipulation to build transformations preserving equality from 0.999... to 1. Other proofs use concepts from real analysis showing the equality between 1 and the limit of infinite series or nested intervals which correspond to the repeating decimals in 0.999... A third approach is based on the structure of various definitions of the construction of the real numbers."

Simplest reason for truth
"Proofs of this equality have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience. Simpler proofs use fractions, long division, and digit manipulation to build transformations that preserve the equality from 0.999... to 1."

Coverage of the proof sections in the lead
I think the lead should mention that the symbols '0.999...' are a concise notation for a limit ( $$\lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n}$$ ) and that this limit has number 1 as its value; that's why the symbols represent the number. Either that, or say that the repeating decimals represent the remainder of a division. Current lead is using the argument by authority that 'some proofs exist', but is not explaining why the '0.999...' notation is useful to mathematicians. The lead should mention how a repeating decimal represents a number to allow for a basic understanding of the issue. Diego Moya (talk) 12:44, 15 October 2010 (UTC)
 * I don't think that's needed. The amount of such technical exposition in the lede should be strictly limited as it makes it too long and difficult to read. It's also a bit fundamental for this article: I think it's fair to expect the reader to understand what decimal, in particular a recurring decimal, is. If not they can follow the link in the first line. For the same reason I don't think your addition is needed: 1 is not normally written as 1.000... – for obvious reasons such numbers can be terminated after the last non-zero decimal, or as soon as is appropriate. So it's a poor example of recurring decimals, nor does it explain what the number 1 is.-- JohnBlackburne wordsdeeds 13:31, 15 October 2010 (UTC)
 * This doesn't solve the issue of the lead using an appeal to authority instead of an explanation to introduce the subject. Also understanding a recurring decimal doesn't explain how that concept relates to the 0.999... = 1 equation, so I insist in providing some comment about this relationship - preferably in a non-technical way.
 * As for the 1.000... it's the same example given at Decimal_representation to explain the non-uniqueness. It may be redundant and obvious to someone who already grasps the concept, but it's helpful to someone reading an encyclopedia article to learn about concepts he doesn't fully domain. Who are you writing the article for? ;-) Diego Moya (talk) 14:28, 15 October 2010 (UTC)
 * If we are writing the article for the students, then the introduction could be edited to include sourced material to the effect that student intuitions that a number described as "zero, dot, followed by an infinity of 9s" can be usefully realized by a number falling infinitesimally short of 1 in an infinitesimal-enriched number system, and that the reason traditional teachers insist on such an equality is that the traditional number system has for about 140 years excluded infinitesimals. Tkuvho (talk) 20:26, 16 October 2010 (UTC)
 * We already mention infinitesimal-enriched number systems in that very paragraph. Discussing why the reals are preferred seems rather off-topic. Huon (talk) 21:29, 16 October 2010 (UTC)
 * Yes, but we don't mention the idea that student intuitions can be usefully implemented. The viewpoint is that students are in error, and those of us in the know try to understand why they are making such an error.  This is certainly a legitimate point of view, and also the dominant one, but it is not the only one in the literature. Tkuvho (talk) 21:41, 16 October 2010 (UTC)
 * @Huon: The lead section devotes more space to explain why students are wrong than to why the proofs are right (in a system without infinitesimals). This is an imbalance of the relative weights of content in the article, which according to wp:lead should be reflected in the lead section. I'm not trying to include in the lead a discussion of why reals are preferred, but why 0.999... can be proven equal to 1. If you read the lead section, you'll find that this is not really done anywhere in those three paragraphs. Diego Moya (talk) 08:16, 17 October 2010 (UTC)
 * I see your point, but I'm not sure what to do about it. As the lead currently says, there are various proofs, and the only thing they have in common is that they work because 0.999... is a real number. We even once had a proof in the archives of either this page or the arguments page that didn't rely on a definition of 0.999..., but rather on a list of properties any number called "0.999..." should have (namely, it's real, it's not greater than 1, and it's greater than any number with just finitely many nines). What, specifically, do you propose? The limit definition is currently hidden behind the link to "repeating decimal", and I'm not sure laymen really are helped by that much calculus. Huon (talk) 16:06, 17 October 2010 (UTC)
 * We could take one proof (or more) and explain in a few words, in natural language, the key insights of that proof. It shouldn't take more than one paragraph; the lead guideline admits up to four paragraphs, so there's room. The point is not to use calculus - we want an explanation, not a proof; those are already below. I was thinking something along the lines of (but much better worded): "the symbol '...' defines a real number as the sum of the infinite decimals. The value of an infinite sum is defined by a limit function, and the limit of this particular sum of 9s is the number 1." I think this is the best proof to be used because it uses the definition of '...'. This explanation doesn't solve the doubts that this proof might produce, but that's already covered in the 'has long been accepted by mathematicians...' paragraph. The purpose of this explanation is to give at this point a light insight as to why mathematicians accept this as fact. Diego Moya (talk) 18:10, 17 October 2010 (UTC)
 * I think what you are saying is that the sequence .9, .99, .999, etc is getting closer and closer to 1, and therefore 1 is the limit by definition. But this merely highlights the fact that many students are not willing to accept the definition of .999... as a limit, so again you are merely changing the subject.  Tkuvho (talk) 19:01, 17 October 2010 (UTC)
 * No, I'm saying 1) that the value of the symbolic expression '0.999...' is the same as the value of the symbolic expression $$\sum_{n=1}^\infty \frac{1}{10^n}$$ by definition, and 2) the value of $$\sum_{n=1}^\infty \frac{1}{10^n}$$ is 1. This is one possible explanation of 0.999... being the real number 1. These 2 assertions are not currently in the lead, and they should be; the lead doesn't say that students are not willing to accept the definition of .999... as limits are not mentioned at all there. What subject am I changing by doing these assertions? i don't understand what you mean. Diego Moya (talk) 19:50, 17 October 2010 (UTC)
 * Thinking about it... am I confounding series with limits? My last post is what I meant the first time. Maybe I should have talked always of 'series' to make my point. Diego Moya (talk) 20:35, 17 October 2010 (UTC)
 * The only advantage I see in Diego Moya's proposed explanation is an opportunity to add links to "limit" or "infinite series" to the lead. Ultimately, it would amount to saying "0.999... is magic word, and in this case magic word equals 1." We could explicitly write down the series, link to the limit article, and so on, but unless we want to sacrifice precision (which in my opinion we shouldn't do), I don't think there is an explanation without drifting into calculus. Huon (talk) 21:43, 17 October 2010 (UTC)
 * There is of course nothing "imprecise" about Diego's proposal, but limits are a difficult notion, and a student who is comfortable with limits will not have trouble with real .999... being equal to 1. To respond to Diego's question, consider items (a) and (b):  (a) students will have no difficulty with the assertion that .9, .99, .999, etc. is getting closer and closer to 1.  According to the standard "limit" definition, this means that .999... equals 1.  But (b) students do have trouble with .999...=1.  This shows that they are not convinced by the "limit" definition.  Thus talking about limits is in a way changing the subject.  From the hyperreal viewpoint, of course, applying the limit amounts to applying the standard part function, so it is not surprising that the infinitesimal difference disappears. Tkuvho (talk) 02:04, 18 October 2010 (UTC)

@Huon: The current version says there is a magic procedure (a "proof") showing that "0.999... = 1" is true. How is this situation any better? At least with my version students would know what kind of magic they must learn first. This satisfies the goal of a well written article introduction, which is to provide a global overview of the topic by summarizing the most important points covered in the article in such a way that it can stand on its own as a concise version of the article (i.e. without the need to read the whole article to get a rough understanding), not to introduce exact definitions - which can be done better with lots of available space in the article's body.

@Tkuvho: Again, the point is not to convince students by reading the lead alone but to point them in the right direction to be convinced. Note that I didn't discard using examples of other kind of proofs apart from limits (algebraic, analytic) as long as some explanation about the main topic in this article is included in the lead section. I want to know why the lead gives priority to explaining "a classic proof of the uncountability of the entire set of real numbers" or that "certain real numbers can be represented by more than one digit string is not limited to the decimal system", over explaining the proof that "0.999... = 1". Diego Moya (talk) 17:03, 18 October 2010 (UTC)


 * The point is not to convince students of what exactly? Tkuvho (talk) 17:28, 18 October 2010 (UTC)
 * You said 'This shows that they are not convinced by the "limit" definition.' The lead should not convince students that "0.999... = 1", it should explain why mathematicians are convinced that this is a fact. Diego Moya (talk) 17:48, 18 October 2010 (UTC)


 * Some of them are not. Furthermore, this page is not addressed to mathematicians.  Tkuvho (talk) 17:51, 18 October 2010 (UTC)
 * I agree that adding a link to the limit article to the lead would probably be an improvement. I also agree that indicating "why mathematicians are convinced that this is a fact" sounds like a good idea - but doing so in terms non-mathematicians will understand will be very hard. And while technically some mathematicians indeed are not convinced, those represent a fringe position which certainly doesn't need to be represented in the lead. Besides, those who disagree change the definition of 0.999... for this very purpose, undermining the connection between real numbers and decimal representations and thus the usefulness of decimal representations. Huon (talk) 18:14, 18 October 2010 (UTC)
 * I'll try to come with a wording that doesn't hurt too much and add it to the article. Diego Moya (talk) 16:33, 30 October 2010 (UTC)
 * I've just reverted it as frankly I was not sure what it meant. I.e. I can't immediately see how the definition "the symbol '...' defines a real number" leads to a proof. If it's e.g. the one here then that's far from the easiest proof, and directing readers of the lede to it will just cause confusion. But on its own it's just confusing anyway as it's not clear what it's referring to.-- JohnBlackburne wordsdeeds 16:49, 30 October 2010 (UTC)
 * I'll copy it here. It seems we have conflicting edits, I'll take it away from the article and discuss it here. Diego Moya (talk) 16:51, 30 October 2010 (UTC)
 * Done. So, how are you going to help me to clarify this paragraph? ;-) Both the Lead guideline and Huon agree with me that this is a needed adition to the article. What exactly do you find not clear?
 * The way in which that first sentence leads to a proof is exactly what the other two sentences explain. And yes, that proof is the one I chose for the reasons I explained in the previous discussion - namely that, although this is not the esasiest proof, it's the one closest to the core difficulties people have with the concepts. The algebraic proofs seem like cheating in this respect as they don't really address the infinite sequence at all. Diego Moya (talk) 16:56, 30 October 2010 (UTC)

Refining the paragraph
"One particular proof uses the definition of the symbol '...'. This symbol defines a real number which is the sum of the infinite repeating decimals that the symbol represents. The value of an infinite sum is defined by a limit of a sequence, and the limit value of this particular sum of 9s is the real number 1."


 * As I noted above I'm not sure this helps clarify things, as it refers to one of the more difficult proofs, and then not clearly. The easiest proof is one that appeals to a readers understanding of repeating decimals, linked in the first sentence, by e.g. comparing 1 / 9 and 0.999... / 9 and seeing that they are the same. This proof is given in full immediately below the lede. It's perhaps not as obvious as the lede is long as is the TOC, but it's as early as it can be without including it in the intro, which I think would be a bad idea.
 * On linking to limit of a sequence there's already a link to repeating decimal which is probably more useful to a non-advanced reader. If they don't understand repeating decimals, i.e. how to represent fractions like 1/3 as decimals, they are unlikely to understand infinite summation of a sequence.-- JohnBlackburne wordsdeeds 17:05, 30 October 2010 (UTC)
 * Including it in the intro is required by the lead guideline. Why do you think it's a bad idea, and why do you think the good ideas in the guideline do not apply here?
 * On WP:LEAD the most important guideline is Provide an accessible overview, i.e. avoid technical language and specialist terminology. There are exceptions for mathematical and scientific articles but they depend on the level of the article and readership. For an article such as this aimed at non-mathematicians the lede should keep mathematical terminology to a minimum, and then only ease readers into it in the body of the article.-- JohnBlackburne wordsdeeds 17:12, 30 October 2010 (UTC)
 * The point of having the proof in the lead, not below the lead or linked from the lead, is to "stand on its own as a concise version of the article" i.e. without the readers having to read a whole collection of separate articles to get all the relevant information. What do you have to say in this respect? Diego Moya (talk) 17:15, 30 October 2010 (UTC)
 * Also Where uncommon terms are essential to describing the subject, they should be placed in context, briefly defined, and linked, which I did. I could write the same explanation without the 'limit' word, but it would reduce its usefulness. Or we cound use a different proof instead, which one do you propose and how do you represent it in one paragraph? We can use advice from WP:TECHNICAL and WP:MANYTHINGS. Diego Moya (talk) 17:19, 30 October 2010 (UTC)
 * To "stand on its own as a concise version of the article" does not mean everything should be included in the lede. The lede should summarise the article in an accessible way, so should not include too much technical language. And it should stand alone so sentences should make sense on their own, i.e. it should be clear what is referred to, which I would say is not true of the above. E.g. "this particular sum of 9s" suggests to me 9+9+9+9. I know that's not what's meant, but writing it out properly as $$\lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k}$$ would be clearly inappropriate in the lede.
 * And please try and use fewer edits to make your point - I've had to submit this reply three times because of edit conflicts as you change your comments before I reply.-- JohnBlackburne wordsdeeds 17:32, 30 October 2010 (UTC)
 * Sorry, I was also editing other sections in the talk page. It's true that 'everything' can't be in the lead, but relative emphasis makes it clear that its content must have emphasis relative to its importance. You know, 'do not tease the reader by hinting at startling facts without describing them.' Clearly the proofs are underrepresented in the lead as they make for half of the articles contents and the only mention to them is one sentence. What do you think of version number 3 below, taken from Simple wikipedia? Diego Moya (talk) 17:42, 30 October 2010 (UTC)

I assume you're referring to the principle of least astonishment. But here the fact is that 0.999... equals 1, which is described already. I would also refer you to MOS:MATH which on the article introduction says the lead section should should include 'historical motivation', 'Motivation or applications' and 'An informal introduction to the topic, without rigor, suitable for a general audience'. I would take that as avoiding proofs in the lede, unless the article is about a proof. The article at the simple wikipedia also does not have the proof in the lede.-- JohnBlackburne wordsdeeds 18:01, 30 October 2010 (UTC)
 * The structure in the simple wikipedia article is not the same as this one, and the lead reflects the article structure. IMHO an informal introduction to the topic must explain the primary concepts in the topic, not just make us know that they exist. But this is going in circles. Why do you think the article would be worse having in the lead an extremely simple explanation which is not a formal proof by itself? Citing MOS:MATH, "include proofs when they expose or illuminate the concept or idea". Diego Moya (talk) 18:13, 30 October 2010 (UTC)
 * That quote appears after the section on the lede, so include proofs in the body, not the lede. Immediately after that it has "Since many readers will want to skip proofs, it is a good idea to set them apart in some way, for instance by giving them a separate section", as is done here, i.e. again not in the lede where all readers will encounter them whether they want to or not. Including lengthy mathematical working in the lede will make it to technical, but without the working, as in the text you added, it is unclear and confusing.-- JohnBlackburne wordsdeeds 19:35, 30 October 2010 (UTC)

The MOS:MATH section is separated from the one about the suggested structure; I think you're reading too much in the guideline by assuming that this gives advice against short expository explanations of proofs in the introduction.

The guideline clearly distinguish between two kinds of proofs: those that illuminate an idea, and those included only for correctness. The reason many readers would want to skip the proof is mainly because of these "for correctness only" proofs; a simple semi-formal or natural-language explanation of a proof that introduces new basic concepts won't trigger this desire to skip it. I understand this guideline as a recomendation to avoid the theorem-proof-example style of math treaties. See versions 4 and 5 below; none of them can be seen as "interrupting the flow of the article", as they are part of the exposition of concepts. Diego Moya (talk) 09:20, 31 October 2010 (UTC)

(Also: you write "the fact is that 0.999... equals 1, which is described already" - That's my biggest complaint, that this fact is not described but only mentioned. The lead doesn't give any insight as to why this fact is true). Diego Moya (talk) 09:24, 31 October 2010 (UTC)
 * It's given in two different ways in the first two sentences. Stated twice if you like but it's clearly given, not just mentioned. As for the rest I don't think we're able to agree on whether to include a proof in the introduction so I'll try and get some other eyes on this.-- JohnBlackburne wordsdeeds 12:19, 31 October 2010 (UTC)
 * Are you referring to the sentences "can be shown to be the number one" and "Proofs of this equality have been formulated with varying degrees of mathematical rigour"? Because that's not what I could consider a description of the fact. At this point getting other opinions seems a good idea, I'll post a RfC. Diego Moya (talk) 12:26, 31 October 2010 (UTC)
 * I've already raised it here: Wikipedia talk:WikiProject Mathematics.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 12:39, 31 October 2010 (UTC)
 * And I've posted it to Requests for comment/Maths, science, and technology to be discussed below. That's good, both pages seem likely to gather people interested in the subject. Diego Moya (talk) 12:43, 31 October 2010 (UTC)

NPOV concern
Hello on this page I observed some extreme liberal biases involving the "relativity" of the numbers 0.999... and 1. The obvious fact remains that 0.999.. is NOT equal to 1, it's either one or the other so how come we are using both?? How can you ignore the differences in between, it is like skipping 2 when counting 1 to 3. When math students are coming to this page they should be able to see all sides of the issue. I am not trying to start an argument but I'd like to point out that the opposing view need to have adequate representation as well. —Preceding unsigned comment added by 130.126.215.48 (talk) 07:29, 30 October 2010 (UTC)
 * I can't quite follow you. The obvious fact is that 0.999... is a different representation than 1, but as the article shows, 0.999... and 1 represent the same real number. It's like the difference between 1/1 and 2/2 - different representations for the same rational number. There is no "opposing view" - at least, not concerning the real numbers, which is the main context for 0.999... Other number systems are duly represented in the article. I also couldn't find a single mention of "relativity" in the article. Huon (talk) 14:08, 30 October 2010 (UTC)
 * Yes it is clear that 2/2 is equal to 1/1, any middle school student would know that... they are two representations for the same number.. But 0.999.. is NOT the same representation for 1 and equality is also nonexistant. I am surprised at how these "proofs" try to change the meaning of equal sign using exotic jargons which benefits noone but the most radical mathemeticians who use the article. Please be aware that there are other viewers as well. A friend who first showed me this article also has the same sentiments that the contents are misleading and unnecessarily complex. It seems to me that anyone who objects is swiftly relocated to the discussion archive, where the objection is jailed for eternaty. Maybe we should reconsider this tyrant like policy and get contributions into the article from everyone not just people who enjoy flaunting their knowledge innappropriately.130.126.215.48 (talk) 14:37, 30 October 2010 (UTC)
 * Hi, there are two ways in which you can have your view to be represented in the article. First you need to find someone else who shares your same view and is famous enough to publish that view in newspapers or scientific journals. If you find a well-written article that explains your opinion in a way that we can all understand, a link to that article can be a valuable adittion to this page. Second, you need to convince the other editors here that your reasoning is clear enough to include it; in Wikipedia anyone can delete your writing, so first we all have to agree what is the best content to include. That's the reason why you see those long discussions here at the talk page. Diego Moya (talk) 15:25, 30 October 2010 (UTC)
 * 130, if you find the article too complex, there is a simpler version at simple:0.999... that may help you. 28bytes (talk) 16:37, 30 October 2010 (UTC)

I'm afraid that contrary to your assertions the article is correct, that 1 and 0.999... are equal. not only is it correct but in contains a number of proofs, any one of which is enough to confirm the equality. I suggest you study the article further if you do not understand this, or consult any of the references, many of which can be viewed online. And no, the article is not open to contributions from editors that do not understand the subject and who would like to insert incorrect reasoning into it.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 14:45, 30 October 2010 (UTC)
 * Instead of lecturing to the IP we could try to understand where he is coming from. Thus, I would like to ask the following: (1) do you agree that the sequence .9, .99, .999, ... is getting closer and closer to 1 in such a way as to get arbitrarily close to it?  (2) do you view .999... as the infinitieth term in the sequence above, or rather as its limit?  (3) would you say that the difference 1-.999... is infinitesimal, or not necessarily? Tkuvho (talk) 19:49, 30 October 2010 (UTC)
 * The difference 1-.999... is zero. Zero, of course, can be represented many different ways. I'd urge the IP and anyone else interested to run the "0.999... divided by 3" exercise mentioned in simple wiki if the proofs involving limits seem unclear or are too complex. 28bytes (talk) 20:59, 30 October 2010 (UTC)


 * You should realize that what is typically contested here is not "the proofs involving limits" but rather the definition involving limits. Tkuvho (talk) 07:50, 31 October 2010 (UTC)

The original post looks like obvious conservapedia trolling to me. See. I think the thread should be closed per WP:DENY. Sławomir Biały (talk) 12:38, 31 October 2010 (UTC)

RfC: coverage of proofs in the lead
There's an argument going on as to whether the |current lead is enough to represent the concepts covered in the article body with respect to the reasons why "0.999... = 1" is true. Several paragraphs has been proposed for inclusion in the Talk:0.999... section. Please express what you think of one or several of these paragraphs, or other paragraphs expressing similar ideas, being included in the lead. Diego Moya (talk) 12:39, 31 October 2010 (UTC)


 * Comment. I don't think the lead should include sketches of proofs. For one thing, the lead is already a little bit too long, and probably needs careful trimming rather than expanding. But more importantly, we can't really include proofs for the excellent reasons already stated in the lead: "Proofs of this equality have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience."  There is no way to accommodate every possible target audience.  Also, none of the simple arithmetic proofs really illustrate anything useful (indeed, one could legitimately object to calling them proofs at all).  For the same reasons, the lead shouldn't say what 0.999... "really is", because while most readers are aware that there is a real number with this designation, considerably fewer have a good grasp of the notion of a limit.  If I had to choose from the given options, only the Enumeration of the concepts used in proofs seems to have value, but it is quite long, and the only thing it does is to give more context for the statement I have already quoted.  I think there is a reasonable presumption that, to grapple with the details of these proofs, people are expected to read the article.   Sławomir Biały  (talk) 12:59, 31 October 2010 (UTC)
 * The proposal is not to include an sketch of a proof. It's to include text that explains concepts used in proofs, since proofs are aproximately 50% of the article contents. Giving more context for the statement you quoted is exactly what this proposal pursues; adding details of a proof to the lead seems to be a misunderstanding by user JohnBlackburne, that was never my primary goal but just one possible way to achieve it. So do you think a shorter version of the 'Enumeration of the concepts used in proofs' paragraph would be valuable? Diego Moya (talk) 13:48, 31 October 2010 (UTC)
 * I would consider most of the options given in the above section to be sketches of the proof. As I said, I don't think any further context is necessary, but if I were pressed to decide between the options presented in the RfC, the Enumeration... one is the only that potentially adds value.  If it could be shortened, that would be better. Also, I see the RfC is already something of a moving target: the recently-added Simplest reason... seems to isolate the digit manipulation proof for no clear reason.   Sławomir Biały  (talk) 14:29, 31 October 2010 (UTC)
 * I agree, there is no need for either sketches of proofs or classifications of proofs in the lead. In any case, all of the proofs boil down to the claim that the sequence .9, .99, .999, ... tends to 1, which is something students readily agree with.  Tkuvho (talk) 14:43, 31 October 2010 (UTC)
 * @Sławomir Biały: How would you make the Enumeration... shorter without isolating the first type of proof like I did in Simplest reason...? Also, if you prefer I can add new versions here to distinguish them from those at the time of posting the RfC.
 * @Tkuvho: Can we then include something along the lines of "proofs boil down to the claim that the sequence .9, .99, .999, ... tends to 1" in the lead, or some other wording that casts light on the nature of the proofs? Diego Moya (talk) 16:09, 31 October 2010 (UTC)

But they don't all "boil down" to that: only some of the proofs can be characterised that way, the first and easiest ones don't consider the sequence at all. Many readers will not know about infinite sequences and limits, so including anything about such a proof in the introduction will confuse them and make the section far less accessible.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 16:20, 31 October 2010 (UTC)
 * So what's wrong with choosing an *example* explanation as representative of the class of "all valid proofs", and then describing something about it? (Not necessarily the proof itself, only giving a mention of the simple concepts involved in it). As I said in the discussion this is intended to give one particular bit of information, a reason on which to ground the new concepts learned when reading the rest of the article.
 * Also, if we were to only include concepts that all readers will previously know about, we should eliminate most of the second paragraph - specially the references to the Cantor set or non-integer-bases. I think we can reasonably expect that a student will know of limits before those other concepts. Diego Moya (talk) 16:37, 31 October 2010 (UTC)
 * So? Is a short version of Enumeration... valid for the lead? Diego Moya (talk) 19:01, 5 November 2010 (UTC)
 * I don't think so, or at least the consensus is that nothing further needs to be added: as noted the last sentence in the first paragraph already summarises it very well. Anything else is unnecessary.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 19:10, 5 November 2010 (UTC)
 * Well, that couldn't be 'consensus' as I don't agree to it, is it? Since the provided content has been found valuable, and the major reason for opposition is that it's not needed in the lead, I will use it as an introduction to the 'proofs' sections. The current article structure jumps from a lead without details right into the body of the first proof. This way the lead section will not suffer, and some context is provided of the concepts I find missing, which will ease the transition for the interested reader. This compromise would be enough to address my concern even if those concepts are not included in the lead. Diego Moya (talk) 19:31, 5 November 2010 (UTC)

No, consensus as in you are the only editor thinking that something should be added, and you have persuaded no-one else, so the consensus is against you making any changes. Consensus does not require unanimity, especially when the consensus is for the status quo, i.e. for the existing article which has evolved through a process of consensus to the state it is in.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 19:41, 5 November 2010 (UTC)
 * That's why I stopped editing the lead a week ago. Since consensus per the discussion above is also that the content adds value to the article, I hope the latest edit satisfies all parties involved. Diego Moya (talk) 19:55, 5 November 2010 (UTC)

Lead sentence
The following sentence is out of place in the lead, and should be removed: First off, we don't otherwise include any specific references to authors in the lead, so this seems to place undue weight on several researchers. Secondly, the references named don't really seem to support the summary given, or at least this is not a fair summary of those references. Rather this appears to be original research. Thirdly, the sentence also just seems to disrupt the logical flow of the paragraph. I had boldly removed this sentence as out of place from the lead, but I was then reverted. I would like to submit this here for further discussion. Sławomir Biały (talk) 14:48, 31 October 2010 (UTC)
 * Many students interpret 0.999... as an "infinitieth term" in the sequence 0.9, 0.99, 0.999, ... and are unconvinced by the limit definition thereof, see recent work by Ely, Dubinsky, Tall, and others.
 * I haven't checked the references, but I believe Tkuvho here said more or less explicitly that the edit was based on a discussion on this very talk page. I agree that the lead is not the place for the references; if they are relevant, they should probably go into the section on education. Huon (talk) 16:19, 31 October 2010 (UTC)
 * Mentioning those researchers in the lead gives them undue weight. But mentioning some of their ideas is different; the paragraph is specifically referring to "the reception of this equality among students", so citing one case seems relevant and a natural continuation of the previous abstract sentence. Having one example serves to illustrate what kind of reception we're referring to. If an example is found that is a faithful representation its references, I support having it there. Every concrete description helps comprehension more than a general definition. Diego Moya (talk) 10:57, 1 November 2010 (UTC)
 * If you have a concrete and brief proposal, we can see if it can be sourced in the literature. Tkuvho (talk) 05:28, 2 November 2010 (UTC)

Technical problem with notation
There's some technical problem with the second notation in the "may also be written as..." sentence. It's showing in my browser as 0.9 (zero-dot-nine), without a dot above it nor an apostrophe (as in 0.9' -zero-dot-nine-Apostrophe-, a notation I've seen in the past which is not covered in Repeating_decimal). This happens both in the article and the edit page. Is there a special character used for "9 with a dot on it" that my browser doesn't support? Diego Moya (talk) 18:05, 30 October 2010 (UTC)
 * It shows up poorly in my browser as well (sort of a nine with a tiny bump at the top right), which is why I used 0.\dot{9} for the simple:0.999... article. 28bytes (talk) 18:22, 30 October 2010 (UTC)
 * We should use the same code here, as the problem is likely to affect many users. Diego Moya (talk) 18:28, 30 October 2010 (UTC)
 * I agree, with the caveat that it will look out of place if the 0. 9  and 0.(9) are left as text. A quick survey of some other wikis (Russian, German, Spanish) show that the PNG is used for the overline version as well. I'd recommend using PNGs for all of them, like so:
 * "In mathematics, the repeating decimal 0.999... which may also be written as 0.\bar{9}, 0.\dot{9} or 0.(9)\,, denotes a real number that can be shown to be the number one."
 * ...but there may be accessibility issues prompting the use of the decorated-text version here. I'll wait for a couple of days to see if there are any objections before making this change. 28bytes (talk) 18:37, 30 October 2010 (UTC)
 * It's using Unicode, the same as ḣėṙė, nothing too obscure but you would need a browser that supports Unicode to see it. The overline uses CSS but I don't think that's possible, or is probably much more complex if it is, so the inline TeX is probably best at least for that one. I would keep the other two as text though, as they don't seem to be causing problems and inline TeX should be keep to a minimum.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 18:44, 30 October 2010 (UTC)
 * Interesting: the dots over the "ḣėṙė" show up fine for me, but the 9 with overdot looks broken in both IE and Firefox on my PC (running latest versions of both). Perhaps because the "ḣėṙė" dots are precomposed and there's no dotted-9 in Unicode? (Or is there?) 28bytes (talk) 18:56, 30 October 2010 (UTC)
 * It does the same for me with Firefox at Linux. I'd also prefer using the math tags for the three notations, they are the most likely to be represented fine in the majority of browsers, since unsuported browsers get the converted image version. Diego Moya (talk) 19:01, 30 October 2010 (UTC)

I've applied the fix by changing only the affected representation to a math tag. The dot now displays correctly, but the numbers are distinctly shown in a different rendering style. Does someone know at least how to show them in bold face? Diego Moya (talk) 20:03, 5 November 2010 (UTC)
 * I see you fixed he bold issue, I've just added '\scriptstyle' to make the size more consistent. In general scriptstyle is a bad idea as it can render all sorts of maths formulae too small or unreadable but it seems to work OK here.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 21:55, 5 November 2010 (UTC)
 * What browser and OS are you using? In my system the scriptstyle version is being rendered at half size with respect to the surrounding text, and the previous version showed at the correct height. It seems like our respective versions are using a different base size for the math characters? Could you describe how are both versions rendered for you? Diego Moya (talk) 22:26, 5 November 2010 (UTC)
 * Safari, OS X, default skin. The modifier \scriptstyle is one solution to inline maths being larger than the surrounding text, as by default it often is. The other is \textstyle. For comparison here they are with the default, unmodified, version: $$\mathbf{0}.\mathbf{\dot{9}}$$, $$\textstyle\mathbf{0}.\mathbf{\dot{9}}$$, $$\scriptstyle\mathbf{0}.\mathbf{\dot{9}}$$. To me the first two, the normal and textstyle ones are too big, while the scriptstyle one is the right size although in a different font. What do other editors see?-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 22:43, 5 November 2010 (UTC)
 * Funny, I've tried on Chromium and Konqueror and in both I see now the same as you. So it's a problem of the Webkit engine vs everything else. Diego Moya (talk) 22:56, 5 November 2010 (UTC)

I should add that this is the reason that inline TeX is discouraged: apart from putting extra load on the servers and increasing page load time because of the extra images it looks different for everyone. For many it has the wrong font and is rendered with jagged text, i.e. without smoothing, compared to the surrounding text. For some it will be too large or too small. This is all dependent on what intersection of OS, browser, browser settings, skin and WP preferences a reader has. It's why I previously used text for it, though obviously something that less Unicode capable browsers could not cope with.

On browsers, anything Webkit based is generally very standards compliant and so should be rendering content properly. What browser is it that displays the scriptstyle too small ?-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 23:08, 5 November 2010 (UTC)
 * Firefox is the one that displays small scriptstyle. Both Firefox and Webkit are as standards compliant as it gets, but it's to be expected that their different quirks produce slightly different results. I haven't tried it on IExplorer, I propose checking how it renders there and using the rendering version that shows better in two of these three platforms. Diego Moya (talk) 09:04, 6 November 2010 (UTC)
 * Interesting. I downloaded Firefox for Mac myself, version 3.6.12 which I presume is the latest, and the sizes are the same as in Safari for me, the only noticeable difference is in the height, i.e. the baseline, which is much better with Safari.
 * It would be good to get a clear idea of what the differences are between different browsers. It should not be happening: they should all be the same size. If not, and if it's as big a difference as we're seeing, then maybe it needs fixing in WP, for example in the CSS. What skin are you using ? In both my cases it's the default skin, Vector, but there are others including ones with quite different looks.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 16:15, 6 November 2010 (UTC)
 * I'm using Vector, too. Diego Moya (talk) 10:51, 7 November 2010 (UTC)