Talk:0.999.../Archive 1

2901:

define 0 and 9 and what a fraction is. Then you have to define operations on these numbers. In fact, it is the stupidity of academics that has blinded them: by trying to carefully construct the real numbers, they have introduced circular definitions everywhere. What came first, numbers or their definition? Can you use numbers to define numbers? I can understand trying to use set theory, e.g. the cardinality of a set could be interpreted as a number. However, it becomes slightly more complicated to define parts of a number such as fractions. 1 is well defined, i.e. 1 x 1 unit. 10 is well defined, 1 x 10 + 1 x 1 unit. 1/2 is well defined, 5/10. All arithmetic operations work only on finite numbers. The dimensions of numbers such as 0.333..., 0.999..., pi, e, etc are infinite. Whatever 0.999... is, it cannot be comprehended. We can only state that it's infinite sum has a limit that it never reaches. Why do you want to define numbers as the limit of a number that immediately preceeds it? The difference between the two is infinitesimal. You would first have to accept infinitesimals. Real analysis rejects infinitesimals but then uses the same to prove a lot of things. Problem is: it is fraught with inaccuracies and contradictions. Yes, mathematics today is no more rigorous than it was in the time of Archimedes.

2694:-) They can't tell the difference between the above two. -) They will insist that (B) is equal to the number (Michael Hardy is an example) - a fallacy since *none* of the partial sums *ever* reaches the limit (supremum). Hardy will insist these are by *definition* the same thing. A common tactic by those who lack understanding is to use the phrase *by definition*. -) They feel comfortable with real analysis which is fraught with non-sense and inaccuracies. They do not fully understand many concepts but will do their utmost to convince others of their standpoint because they are arrogant and believe their viewpoints are probably more accurate. -) They will perpetrate non-truths (same as *lies* but then I risk being called a troll) to suit themselves and save face, not to mention the 'rigour' of mathematics. There is no rigour except what humans perceive as rigour. -) They feel comfortable with contradictions and confusion. The first universities were established by churches and the idea was to get rid of superstition and nonsense. Today the universities are controlled by a different kind of clergy - the ignorant PHds. 624:

FRENCH un numéro, LATIN numerus, (OLD LATIN) NUMERVS, FINNISH numero. You may plot your decimal "numbers" in Cartesian co-ordinates (X,Y), X-axis expresses "whole" numbers, Y-axis expresses desimals like (7,91), but no point presents (0,999...). "999..." impression does not tell where in the Y-axis "999..." ends. Did you think "0.9990999099909990999..." when you wrote (0.999...)? Do you think that "10000/9999" is number? Do you think that "0,333..." is number too? Do you think that "...666" is number too? Do you think that "...66,6*15=...333,0" is number too? Do you think that "1000/999/666" is number too? Do you think that "1000/666/999" is "number" too? Do you think that "1500" is number too? Do you think that "1000/999/999" is "number" too? Do you think that number "one" 1 is not divided (undivided) and not multiplied (unmultiplied) (Greek Atom). Do you know what means in France le numéro atomique (le nombre atomique, le nombre ordinal)? Did you know that multiplication is a making long division? Do not tell to a tennisplayer that you feel "love" for him or her (he or she may think that love is 0, nothing)... Santa Claus 22 Jun 2005

1822:

calculate it to as many places as you like. Everyone knows this. Rather tell me if you are able to calculate pi completely. Numbers are defined in real analysis as Cauchy sequences that are in fact sets in many respects (sets can be formed out of the partial sums) and thus capable of having least upper bounds. In fact, the completeness principle states this clearly. It is so easy to call someone a crackpot and/or a troll when you have shown how ignorant you are and can't respond in any sensible way. You became sarcastic and then expected the poster to remain polite? Oh forgive me my math god for I have sinned grieviously against your sarcasm and disdain. Provide *proof* and sensible rebuttals, not rhetoric and sarcasm. Finally, just because you say something, does not make it true or divine law. You are wrong and you have not answered any rebuttal satisfactorily. Again, prove to me that the sum of 9/10 + 9/100 + 9/1000 is 1. Be careful! Don't write nonsense. Think about every word you use. You are not even half as smart as you think and you do not know everything as well as you think.

2620:

exists and we can usually find it, pi has a least upper bound that we cannot find. In reality, pi (like e and sqrt(2)) are numbers that cannot be represented finitely in any number system we know - these numbers can only be approximated. They are mysterious and exotic numbers because they do not reveal their full extent. Well, I am personifying them but it's just for effect. There are not infinitely many numbers between two real numbers. This is in fact a complete contradiction and once again exposes the weaknesses and errors of real analysis. On the one hand its supporters will claim there are no infinitesimals and on the other hand they will make a statement to the effect that there are infinitely many numbers between any two real numbers. The sad truth is that mathematics has been hijacked by a lot of idiots with PHds in mathematics. Almost 99% of dissertations I have read are not even worth the paper they are written on. Yet these fools were granted PHds. The result? Millions of idiotlets believing that 0.999... = 1 !

2163:

have to occur at some point in the sum. It is clearly evident that a carry is *impossible* therefore the sum can never be 1. To write what you have written only shows that you and many of your classmates were not listening when these things were taught to you in high school. The completeness principle is *not* violated in this instance because there is no number between 0.999... and 1. 1 is a rational number but 0.999... is not a rational number, thus the completeness principle still stands and your arguments are false! 0.333 is an approximation for 1/3. 3.14 is ane approximation for pi. 2.718 is an approximation for e. 0.999 is an approximation for 1. BUT 0.333 is not a 1/3, 3.14 is not pi, 2.718 is not e and yes, 0.999 is not 1 ! 0.333... , 3.14..., 2.718..., 0.999... are all irrational numbers that are close approximations of other rational numbers.

2775:

forthcoming. In fact, Hardy will not prove this because it is evidently not true. Way I see it, 1 has a finite representation in base 10. There is no need to approximate it. 1/3, 1/7 and many other finite numbers have no finite representation in base 10. Whether you like it or not, any calculations involving infinitely represented numbers in base 10 have always been approximations. The completeness principle does not apply to infinitely represented numbers. Every proof in favour of 0.999... = 1 is false. 0.999... is an indeterminate number that is very close to 1 but is never equal to 1. Although the difference between the two is infinitesimal, there is a difference. 0.333..., 0.999... and any infinitely represented number in any base does not make sense. It can only be used as an approximation. --Unsigned by user at 192.67.48.22

1758:

expansion, then by default pi must have a least upper bound (sup). Uh oh! Pi does not have a supremum. Everything falls apart here: is the completeness principle still true? Yes. It's just the teachers who claim every number can be expressed as a decimal expansion that are liars. Be very careful when you are taught - do not only hear what you want, but what is actually spoken. Beware of teachers like Mr. Hardy. No wonder so many MIT grads don't know the difference between 0.999... and 1!

2077:

equal to 0.999... just as pi is not equal to 3.14..., just as 1/3 is not equal to 0.333.... 0.999... is an indeterminate quantity. 0.333... is an indeterminate quantity. It is *impossible* to represent 1/3, 2/3, 4/9, etc. in the decimal system except to approximate the same. There is no *known method* in any mathematics to compute the sum of an *infinite* number of terms. How is it that you are *interpreting* the *limit of a sum* to be the *actual* sum? What does

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base ten. Thus you cannot state that the quotient is 0.999... and then conclude that no number exists between 0.999... and 1, therefore these must be the same number. Please prove to me that 1.999... divided by 2 is 0.999... and then tell me the completeness axiom is the reason. Otherwise please do not post anything more - you will simply show how narrow-minded you and most mathematicians really are!

2652:

nines. They're hampering their own communication with the rest of the world of mathematics, since they've actually started to use a non-standard notation. I think mathematicians should respect the difference between things that are true by definition, and things with a deeper meaning (yeah, I know, if such a meaning exists is in itself a matter of discussion).

258:"Consider the following proof that I propose be substituted for the one in this article: assume first that 0.999... and 1 are distinct. Between any two distinct real numbers lies another real number (infinitely many, in fact). Since there is no real number between 0.999... and 1, we have reached a contradiction and our original assumption must be false." 2193:

What's the point of having the -9 and 9? I find it demeaning, superfluous, and do to my overly analytical nature, confusing. The summation converges to 1 on it's own, why bother with adding the 9? Unless of course you're trying to make a parallel to the dumbed down version, which is neither necessary

1821:

So you say pi is the least upper bound? Interesting. But pi cannot be computed completely since it is infinitely represented. How then can it be the l.u.b ? You can say 1 is the l.u.b of 0.999... but you do not know exactly what pi is, so this is evidently non-sense. Please don't respond that you can

1753:

We can see that (B) has a maximum (clues: upper bound, completeness princple and *limit*) value if the second term is zero. Now does this ever happen? NO!!! However, as n gets infinitely large, the second term gets closer and closer to zero. In other words you can make the second term as close as you

1530:

I feel what means "limit" (i.e. finitus) when I drop myself from the sky and then hit the ground or when I hit my face on the wall. It's a kind of frustrating to type such a dialect (mutation) of Latin (e.g. "English") which contains so much/many synonyms and yet they are expressed visually different

1486:

No. For you to say it exists, means you already know its value. You don't. Pi can only be known as an approximation in a some radix system. According to the completeness principle, it exists only if it is a number whose full extent is known. We don't know the full extent of pi just like we don't know

2844:

Most students will first encounter this topic below the college level. For them, a proof using limits and Cauchy sequences is likely to be unhelpful. Others will revisit this topic at a more sophisticated level. For them, the arguments given were sorely lacking in substance. The completely rewritten

2774:

Look at the section titled Explanation in this page. He seems to have made a sufficiently solid case in my opinion. There is a clear difference between 'infinite sum' and 'limit of an infinite sum'. He challenged Michael Hardy to prove that 9/10 + 9/100 + 9/1000 + ... = 1. So far, no proof has been

1526:

If you read the posts carefully, you would have noticed I was talking about this right from the start. My response was clear: you cannot find the *exact* value of pi. I used extent instead because it explains it better. You are as open-minded as the rest - your mind has already been made up and even

1447:

because we do not know the full extent of pi. What are the *final* parts of pi? Where does it end? What exactly is the value of pi? We don't know!! By definition a supremum is the least upper bound and if it exists, then we can find it by the completeness principle. Consequently, not all numbers can

2162:

Are all these numbers the same as 1? You are telling me that if I sum the terms of the sequence 0.999... I will end up with 1 and I am telling you that I do not believe you. There is no way I will ever reach 1 even if I continue to sum the terms forever. In order for this to happen, a *carry* would

1740:

Your above formula is incorrect. The infinite sum is not equal to 10/9. I hate to break this to you but you cannot compute the above infinite sum or any other infinite sum. That's what Mr. Hardy would have taught you (wrongly of course). So where does the expression a/(1-r) come from? Well, the sum

1515:

And now you're adding the completeness principle to the list of concepts you're redefining or misunderstanding. The correct statement of the completeness property of the real number system is this: if a non-empty set of real numbers has an upper bound, it has a least upper bound. Nothing to do with

614:

Yeah. Thought you might be heading in that direction, which is why I chose my wording carefully in my last point 2. Thing is, 0.999... is the same value. They're different representations, but they express the same value. Besides, you're trying to make an argument by analogy between natural numbers

1873:

Same thing, different words. It is perfectly correct the way I stated it. No mathematician who *knows* what he is talking about will agree with you. And no, pi is *indeed* not the l.u.b of the set of all finite truncations of its decimal expansion. A l.u.b must be completely defined. pi is not. As

325:. Besides, the "..." is just shorthand. Furthermore, we have a proof that 0.999... = 1 in this article. It relies on a series converging. If there's a problem with any particular step in that proof, please reveal it. As it stands, we have a solid proof that runs counter to what you're claiming. -- 150:

But what is 1.999... divided by 2 equal to ? Please do not tell me the result is 0.999... because you cannot compute this quotient. It involves a quantity that is *infinitely* represented. All our arithmetic works only on *finite* numbers or approximations used for numbers we know are finite in

2900:

The very same proof that is used against it, proves it. Just read the above post. There is a difference between 'infinite sum' and 'limit of an infinite sum'. sup(sum of(0.999...)) = 1 but 0.999... is not equal to 1. If you want to go to such lengths to define the number 1, then you first have to

2651:

Hey guys, isn't the controversy _really_ just one of definition? If you define 0.999... as being equal to the limit of 0.999... as you let the number of nines "go towards" infinity, then 0.999... = 1. But people should be free to think of 0.999... as simply 0.999 followed by an infinite number of

2619:

I will answer this question for you. In fact, pi must have a least upper bound (even though we cannot calculate it) for it is *finite*. circumference = diameter * pi. The circumference of any circle as we know is finite. Unlike converging Cauchy sequences in which we can state a least upper bound

1993:

Wrong, 0.333... is an indeterminate number just like 0.999... In your example you are dealing with fractions that are well defined. 0.333..., 0.666..., 0.999... are not well defined numbers. (y/3)x3 proves nothing because all it means is a value of 'y'. y can be anything you want in this example.

817:

I don't know. I mean, it might convince some people, but to others it might seem like handwaving. In the proof currently given, it reduces the problem to the summation of a series (clearly leading into limits). The other proof is somewhat indirect and less likely to convince. We could put both of

670:

Either way, I'm through trying to convince here. We have an elegant proof that reduces the problem to a convergent series. We have 3 alternate proofs that illustrate the point further. If someone finds a flaw in the initial proof, I'll certainly take a second look at it, but for now I think we're

133:

Consider the following proof that I propose be substituted for the one in this article: assume first that 0.999... and 1 are distinct. Between any two distinct real numbers lies another real number (infinitely many, in fact). Since there is no real number between 0.999... and 1, we have reached a

2141:

To say that "the is no *known method* in any mathematics to compute the sum of an *infinite* number of terms" is either to say that you failed to learn such methods in secondary school when your classmates learned them, or to say that you reject that on some philosophical ground, about which you

2076:

So what you are saying Mr. Hardy is that we can only perform approximate calculations with any numbers represented *infinitely* in the decimal system - this is true. So which is a better representation of 1 - 0.999... or 1 ? I find it completely ridiculous to state that 1 = 0.999... It is *not*

623:

e.g. 999 is in ENGLISH figure, tally, count, SWEDEN tal, siffra) GERMAN Zahl, FRENCH chiffre), FINNISH OFFICIALLY luku. One of 0,1,2,3,4,5,6,7,8,9 is in ENGLISH OFFICIALLY digit MIDDLE ENGLISH nombre, GERMAN digiTAL, Ziffer, (SWEDEN OFFICIALLY siffra, UNOFFICIALLY numret, nummer, (SELDOM) numro,

2779:

But, as was previously pointed out in this discussion, numbers only look like approximations in certain bases. I think we'd both agree that pi is irrational in base 10, but pi is represented as 10 in base pi and is therefore an integer in that base. Semantics aside, all of these symbols we use

1757:

So, 10/9 is the value that your sum will never reach. What does this mean? It means that it has an upper bound or a limit. This is another reason why not all numbers can be expressed as a decimal expansion. Take pi for example. If this were true that all numbers can be expressed as a decimal

1019:

I very much agree. Maybe something as short as the top 3 misconceptions. No doubt our anonymous friend here is providing us with detail. The links I found these arguments taking place on should also provide a starting point. I'll look through later on and try to put a short summary together.

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No. You are sorely mistaken. Pi is not 10 in base pi because pi is not completely defined. The only bases that are permissible are those that are defined. In fact, pi cannot be fully denoted in any base. It is an irrational number. 0.999... in base 10 is also an irrational number. pi is an

2088:

never exceed* 1! It is *not* saying the sum is 1. Although the completeness axiom is true, a result of it that is used to show a contradiction here is untrue because arithmetic can only be performed on *finite* numbers. I believe this article is non-sense and should be deleted. It is not

1041:, you are done. WP should consist of true facts; but doesn't in general need to convince people of themm if they insist on being wrong. There is no alternate POV to be represented. Other pages that are just magnets for discussion threads have in the end been dealt with in a similar way. 2845:

article separates these two audiences, and offers proofs at both levels. Given the history of edits to this page, it would not surprise me if some other editor soon trashes it; but I hope this more honest acknowledgement of the importance of foundations makes that less likely. Enjoy. --

2123:

number that it will never exceed. But since some terms in some series are negative, it is better to define it thus: you can make it as close as you want to 1 by using a large enough number of terms (how large is large enough depends on how close you want to make it), and 1 is the

1582:

Today I reverted some new content added to this article by anonimous user 130.230.47.130. My comment was "reverted ignorant contribution". I must say that I was wrong, actually that is meaningful text. But looks kind of long. Would anybody take a look at that contribution

951:), you can put it on a different page, but I'd just put them on the main page, I think. People don't have to scroll down if they don't want to. Maybe we could create a suppage of this talk page and put all the proofs there while discussing which should be included. -- 1809:; pi is a number. (And in the paragraph above, when the anonymous poster uses the word "liars", and in the subsequent rhetoric, you begin to see that that poster is a crackpot or a troll (and I suspect the latter because of the claims to be a retired supermodel).) 1800:

the least upper bound of the set of finite truncations of its decimal expansion, or of the set of circumferences or regular polygons inscribed in a circle of diameter 1, or of any of various other sets. The things that have least upper bounds are non-empty bounded

1719: 333:

You should tell me how should understand in unambiguous manner these three dots. As far as I am concerned that "0.999..." ends to third (from left to right)dot. (in Europe decimals are separated from whole numbers by "," ...expressed 0,999 not 0.999) -Santa

1535:

observe. I was frontier (border etc.) guard (observer) during the military service and I sure can tell you that there truly is very really fine (infinitesimal) line between the border zone and the non-border zone :) And yet they keep claiming that you need

1210:

I learned in my topology classes that an infinite number of finite things is a completely different beast from an infinite thing. There are an infinite number of numbers like 0.9, 0.99, 0.999, 0.9999, etc. in the domain [0, 1[. All of these numbers have a

2835:

What good is a number system in which representation is not unique? It leads only to chaos and confusion. Just look at the confusion caused when someone tries to have 0.999... = 1. This statement is false whichever way you look at it. -- 192.67.48.22

646:

BradBeattie, I've read through this "Santa Claus" person's arguments, and they show that he either doesn't know what people are talking about, or pretends not to. I don't think he's being serious here, he is very probably a troll. He is best ignored.

787:

The problem with that argument is that it doesn't bring convergence to the table. The summation of the infinite series is key to understanding this equation. The 10x-x proof could be construed as handwaving by those that don't already understand it.

1754:

want to zero BUTTTT you cannot make it zero. Why? Coz if you do, you are assuming that the infinite sum is equal to a/(1-r) and that is clearly *wrong*. You cannot discard the second term from (B) and innocently claim the infinite sum is a/(1-r)!

1862:

of Cauchy sequences. And yes, pi is indeed the least upper bound of the set of all finite truncations of its decimal expansion, and it is also the least upper bound of the set of all perimeters of polygons inscribed in a circle of diamter 1.

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combinations of symbols (not to mention about the enunciation, pronunciation, vocalization i.e. making a sound based on symbols (phonemes, letter, symbols of alphabet). Infinite (endless, boundlessness) "is" something you can't

609:

Okay, true then. So could tell me any real number (X) differentially less than 1 (and greater than 0) so that there is no other real number (B) "between" X and 1. Define X, when 0<X<B<1 that there is no B? -Santa

250:"I do not want to sound rude by saying this, but please do not try to refute this (universally accepted among mathematicians) proof unless you have some background in mathematics that goes beyond your mere intuition." 2780:

comprise the language of mathematics. They represent the idea of a number. The symbols 1/2, 0.5, 0.1 in base 2, 0.4999... represent the same value. It's important not to confuse the denotation with the connotation. --

2487: 435: 1850:

There is no danger that any mathematician reading this will conclude that I am the one who is ignorant. No, I did not "become sarcastic and expect the poster to remain polite"; rather, I responded to the poster's

998:

I think the page needs a section on common misconceptions and flaws in the reasoning of these misconceptions. If the abracadabra section on the talk page says anything, it's that some people don't really get this.

2763:

Could you please provide proof that shows the two are not equal? I'd think that if all experts on the matter agree that they do, that the burden is on you to provide solid evidence against their equality.

2758:

That's correct. I typed this up too quickly. Aside from this error, the rest seems okay. Not that it matters much as far as most are concerned: they don't see a difference between the two. 68.238.110.56

2703:

Unfortunately, it is not one of definition. Finally, no one can just make up a definition - it has to be logical and must be justifiable. 0.999... has always meant the sum of 9/10 + 9/100 + 9/1000 + ...

602:

True, but you're picking 2 seperate values here. You initially asked for one quantity that is greater than 5, less than 6 and within the natural number set. No such number satisfies those properties. --

969:, what do you think? If we agree that that subpage is unnecessary, then BradBeattie, you will need to ask it to be speedy deleted. That is as simple as going to that page, writing in the following: 2156:

Thanks, but I don't need anyone to defend me. Your argument "can make it as close as I want" makes no sense whatsoever. I can make anything I like as close as I want to 1. Here are some examples:

319:"a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range)." 910:

Can we just move the alternate proofs to the talk page? As it is, the main page could easily be cluttered with dozens of proofs. While they all might be valid, we really only need one proof. --

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I object to the 'Property of real numbers' proof, though. It is correct that there are infinitely many real numbers between any given two distinct ones. But why is there no number between

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While I suppose that is true, this page is not understandable for people who haven't studied mathematics on a high level (me included). For them, the 10x-x proof is more understandable. --

1620: 1516:"extent". Given the rest of the discussion, including your recent rant against "PHds" (I am not one.), I will not further participate in this discussion, having deemed it fruitless. 1356: 734: 2592:

There is no real/rational number between 0.999... and 1 because 0.999... is not a rational number. If pi had a least upper bound say x, would there be a number between x and pi ?

1111: 768:

I'm not convinced this is the most intuitive proof available ... wouldn't the x = 0.999..., 10x-x = 9.999... - 0.999... = 9x = 9, x = 1 proof be more appropriate? Just a thought

1006:

But the new section should probably be not too long. (The set of all common misconceptions of people is by several orders of magnitude larger than the set of all knowledge. :)

1152: 2419: 1271:

This is an infinite number of finite-length numbers. 0.9 (one nine) 0.99 (two nines) 0.999 (three nines) 0.9999 (four nines) 0.99999 (five nines) etc. (or ... if you prefer)

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I agree that we can include the 10x-x proof in a section called say "Alternative proof" or "Another proof", after the series proof. It certainly adds value to the article.

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in that given set of reals. But then elementary calculus tells you that multiples and sums of converging sequences are again converging and the obvious rules, i.e.

584:

You don't set k; it is declared in the summation. If you chose k=3 in your example above, how does the sigma work? As it reads currently, it doesn't make sense. --

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Of course this is biased since I am speaking from personal experience; this is how I convinced myself at the age where I asked that question for the first time.

341: 1476:

Okay, so if sup(partial sums(3.14...)) is not pi, what is it? According to the completeness principle, it exists, and hence, according to you, we can "find it".

700:

I think you are right. I submitted it first for deletion because the title looked a bit misleading. This is not a series of nines, the series is if you wish of

1874:

for sarcasm and abuse, I think you may have a bad memory in this regard too. Why don't you reread the posts and see who started to be disdainful and nasty?

108: 1263:

That's not what I am talking about. I might have been unclear in my message. Sorry about that. Here is an attempt at a more helpful way of putting it:

573:{\displaystyle F(k)=-9+9\times \sum _{k=0}^{\infty }\left({\frac {1}{10}}\right)^{k}=9\times \sum _{k=1}^{\infty }\left({\frac {1}{10}}\right)^{k}} 1282:

Although there are an infinite number of the things (put in rows) in the first example, they're all in [0, 1[. In contrast, the second example is

2119:

Sorry, you're wrong. It does not mean that the sum will never exceed 1. Since all terms are positive, it could be taken to mean that 1 is the

2426: 2917: 2871: 2636: 2010: 1838: 894:

Ah, right. You mentioned this before but I forgot. I'll try to keep it in mind. Thanks! Just trying to clean this page up at the moment. --

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You need to take a calculus course, to learn what limit is. Then, a good exercise would be to prove that if one considers the sequence

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I feel that this article should be marked as "under debate". There is a difference between the terms "converges to" and "equals". --

1963:

yes, this is correct, but this is just another form of the 1/9 = 0.1..., 2/9 = 0.2... 9/9 = 0.999, 9/9 = 1, which is already there :)

1774: 237: 202: 167: 134:

contradiction and our original assumption must be false. Therefore, 0.999... and 1 are not distinct (i.e. they are equal in value).

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number of nines, despite the fact that there's an infinite number of the numbers themselves. In contrast, in 0.999..., there's an

636:. If you find a flaw in that I'll continue this discussion. At this point, I'm tired of arguing something I've already proven. -- 1727:

Yep, that's true. Check the article page for how that's multiplied by 9 and then 9 is subtracted from that. It all works out. --

2392:

After taking everything into account which necessarily has to be known or accepted to be able to 'ask' the question whether

805: 1737:

Timeout! It's time for a lesson on high school limits for Mr. Hardy and those who agree with BradBeattie the masculist.

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and those who actually know how to prove it can easily see how to modify that proof for one to the original question.

301:

So 0.999... is a number. As per the proof in the article, it describes the quantity 1. Furthermore, 0.999... is not a

1714:{\displaystyle \sum _{k=0}^{\infty }\left({\frac {1}{10}}\right)^{k}={\frac {1}{1-{\frac {1}{10}}}}={\frac {10}{9}}} 2354: 38: 1747:

If we let n go to infinity, all we can do is speculate about what happens to (A). So we break up (A) as follows:

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is simply one that cannot be written as a ratio of two integers. Bases have nothing to do with the definition.

2496: 2310: 2913: 2867: 2731:

You are wrong about (A). You can find the limit of an infinite sum. It's an infinite sum you cannot calculate.

221:

Sarcasm is the lowest form of wit. 1.999... is not equal to 2 for the same reasons 0.999... is not equal to 1.

2632: 2006: 1834: 1421:

If you don't understand what limit is, no offence, but it is pointless to have this discussion to start with.

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Please. There is no such thing as "irrational in base 10" as opposed to irrational in some other base. An

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That's only an issue with one proof though, the article itself is sound. What is the difference, by the by?

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actually say? For 0.999... it says that no matter how many terms we sum in this sequence, the sum will

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The 9 and -9 are put in so that you have the 9 x sum etc. bit, which is neccesary for the proof.

2143: 2068: 1864: 1810: 1574:"This is an infinite number of finite-length numbers" What a jargon. Do I need to say more(?) ;D 1192: 948: 212: 177: 2679:

No. It is *never* true that 0.999... = 1. The confusion is really about the difference between:

2543: 2287: 2264: 2796: 2584: 2216: 1859: 1793: 1611: 1064: 1057: 802: 116: 418:{\displaystyle 0.999\ldots =-9+9\times \sum _{k=0}^{\infty }\left({\frac {1}{10}}\right)^{k}} 2781: 2765: 2142:

should be explicit instead of pretending that the mathematics is what you're writing about.

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and a convincing explaination. It has to be you to ask that, because you created that page.

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to help keep the main page brief, but also allow us to post as many proofs as we'd like. --

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infinitesimal points (amorphous dots) to draw a line (segment) between them. -Santa Claus

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I think this is fruitless. If you define the recurring decimal as a limit, and believe in

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proof. It all depends on what machinery you have available. When writing things such as

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if you did understand, you would refuse to accept truth. Suit yourself. 68.238.110.56

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Seems that some user didn't like the alternate proof page idea. Thoughts on this? --

2821:, you can have bases of complex (or at least irrational) numbers. Take for example 2050: 1517: 1477: 798: 286: 176:

Using the proofs in this article, your question can be reduced to 2/2. This is 1.

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remains. We could argue indefinately about semantics of math, but it won't change

2855:

Nonsense. There is no proof at undergraduate or college level that 0.999... = 1.

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It is not correct that real numbers are defined as Cauchy sequences, but rather,

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of a series in which there is a common ratio r, a first term 'a' and n terms is:

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If you wish to start a new discussion or revive an old one, please do so on the

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Please read the first proof provided on this article. It unambiguously defines

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I saw and the confusion that arose. Figured it was something worth noting. --

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limit(infinite sum(0.999...)) IS NOT equal 1 (In fact it never reaches 1)

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The key step to understanding this proof is to recognize that the following

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Natural number 5 is less than 6 and 6 is greater than 5, true? - Santa Claus

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Above seems to me a function of k (0.999... expression does not include k)

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be expressed as the supremum of the partial sums of decimal expansions.

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This archive page covers approximately the dates between DATE and DATE.

2822: 2688:"The limit of the partial sums of a sequence" (B) 2682:"An infinite sum" - impossible to calculate (A) 2482:{\displaystyle 1=3\cdot {\frac {1}{3}}=3\cdot 0.333\dots =0.999\dots ,} 2049:

How else would you define the result of an infinite decimal expansion?

104:, copying or summarizing the section you are replying to if necessary. 751:

True, the title was a little slap-dash. Thanks for the improvement. --

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Point 2: There is no natural number greater than 5 and less than 6. --

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This is an infinite-length number. 0.999... (infinitely many nines)

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What means "infinite-length"? -Santa Claus At 1:06 Northpole Time

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DO NOT EDIT OR POST REPLIES TO THIS PAGE. THIS PAGE IS AN ARCHIVE.

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to the real number that the sequence of its finite initial parts

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and real numbers, but the two have very different properties. --

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approximation in any well defined base you like. 192.67.48.22

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That is nonsense: There is no mention of ∞/∞ in this article.

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Santa Claus, as much as I'd love to continue this discussion,

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Doesn't sound like 0.999... is a function. Sounds more like a

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The number represented by any infinite decimal expansion is

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a/(1-r) - ar^n/(1-r) (B)

1744:(a - ar^n)/(1-r) (A) 1586:

and see if anything can be included in the article? Thanks.

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Using three dots makes it (0.999...) a function -Santa Claus

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is an abstract entity used originally to describe quantity."

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is totally nonsense. -Santa Claus 21:02 At Northpole Time.

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Subpages aren't allowed in the main article namespace (see

2067:. Therefore the objection expressed above is not cogent. 1909:

Unlike all the items listed in that article, this one has

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Wrong. It cannot be reduced to 2/2 - this is the point.

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Your "proof" lies on two notions. First there is that

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This is also wrong. It would only be correct to say:

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The second one on the Dr. Math page now I look at it

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As I see it, there is no fault or handwaving in the

1219:number of nines, and therefore it's not in [0, 1[. 2575: 2555: 2524: 2481: 2413: 2381: 2343: 2299: 2276: 2253: 1713: 1409: 1350: 1179: 1146: 1105: 728: 572: 417: 1923:I disagree as well. I'm reverting the change. -- 1376: 2491:since most readers probably believe the lemma 2382:{\displaystyle 2\cdot 0.333\dots =0.666\dots } 1206:I originally wrote this, but it got deleted. 1063:I think this page should just be merged into 926:Proof that 0.999... equals 1/Alternate proofs 8: 1410:{\displaystyle \lim _{n\to \infty }x_{n}=1.} 1180:{\displaystyle {\frac {\infty }{\infty }}=1} 2525:{\displaystyle {\frac {1}{3}}=0.333\dots ,} 2421:, a useful proof from my point of view is 2344:{\displaystyle 0.3,\,0.33,\,0.333,\,\dots } 1437:but it is incorrect to write Lt x<n: --> 254:-Why/What are you so afraid of? Santa Claus 109:Talk:Proof that 0.999... equals 1/Archive02 1245:Doesn't etc. (et cetera) stand for "..."? 129:Keep in mind your knowledge of mathematics 2568: 2545: 2500: 2498: 2442: 2428: 2397: 2356: 2312: 2289: 2266: 2237: 1701: 1685: 1673: 1664: 1650: 1639: 1628: 1622: 1395: 1379: 1373: 1330: 1324: 1161: 1159: 1118: 1086: 718: 709: 707: 564: 550: 539: 528: 509: 495: 484: 473: 437: 409: 395: 384: 373: 343: 984:Seems to have been taken care of. :) -- 285:5 and 6? Do you even understand what is 265:0,999... is not a number at all, it's a 2336: 2328: 2320: 1858:of the way of characterizing them is a 2159:0.9 0.99 0.999 0.9999 0.99999 1947:1/3=0.3333... 0.3333... x3 =0.9999... 44:Do not edit the contents of this page. 2189:Do Not Feed The Troll (and a comment) 7: 2691:Common misconceptions amongst PHds: 1544:Your statement is FALSE, because... 1351:{\displaystyle x_{n}=0.999\dots 999} 113:Knowledge:How to archive a talk page 2883:I'd like to see you prove that ... 729:{\displaystyle {\frac {9}{10^{n}}}} 683:I created this page in response to 2697:sup(partial sums(0.999...)) = 1 1640: 1444:sup(partial sums(3.14...)) = pi 1386: 1166: 1163: 1141: 1100: 1088: 634:the proof displayed in the article 540: 485: 385: 24: 2215:Not convinced. Removed the -9. -- 2080:Lt (a - ar^n)/(1-r) n-: --> 1788:It is not correct to say that pi 1106:{\displaystyle \infty +1=\infty } 1434:sup(partial sums(0.999...)) = 1 1202:Infinite number of finite things 852:And by the way, one should write 29: 1913:proofs. Not appropriate there. 246:The disagreement of Santa Claus 1981:15:03, 29 September 2005 (UTC) 1968:19:13, 27 September 2005 (UTC) 1958:19:01, 27 September 2005 (UTC) 1383: 1147:{\displaystyle XXX...=\infty } 994:New section: Arguments against 448: 442: 1: 2414:{\displaystyle 0.999\dots =1} 2198:03:50, 16 October 2005 (UTC) 2128:number of which that is true. 216:03:59, 10 November 2005 (UTC) 123:13:37, 16 November 2005 (UTC) 2888:11:47, 29 October 2005 (UTC) 2850:05:11, 27 October 2005 (UTC) 2830:07:36, 27 October 2005 (UTC) 2804:21:08, 25 October 2005 (UTC) 2785:15:55, 25 October 2005 (UTC) 2769:13:09, 25 October 2005 (UTC) 2752:18:17, 24 October 2005 (UTC) 2725:11:01, 24 October 2005 (UTC) 2673:08:35, 24 October 2005 (UTC) 2641:21:30, 21 October 2005 (UTC) 2613:16:50, 20 October 2005 (UTC) 2588:15:29, 20 October 2005 (UTC) 2220:15:29, 20 October 2005 (UTC) 2207:20:30, 17 October 2005 (UTC) 2147:23:30, 17 October 2005 (UTC) 2072:02:56, 11 October 2005 (UTC) 2054:02:43, 11 October 2005 (UTC) 2043:23:18, 10 October 2005 (UTC) 2033:23:08, 10 October 2005 (UTC) 1895:21:49, 22 October 2005 (UTC) 1868:18:24, 22 October 2005 (UTC) 1843:01:00, 22 October 2005 (UTC) 1814:23:05, 21 October 2005 (UTC) 1779:18:42, 21 October 2005 (UTC) 1521:03:48, 25 October 2005 (UTC) 1508:13:16, 23 October 2005 (UTC) 1487:the full extent of 0.999... 1481:01:53, 23 October 2005 (UTC) 1469:14:56, 22 October 2005 (UTC) 1077:infinity divided by infinity 181:21:20, 7 November 2005 (UTC) 107:Please add new archivals to 2556:{\displaystyle 0.999\dots } 2300:{\displaystyle 0.333\dots } 2277:{\displaystyle 0.333\dots } 1362:where the number of 9's is 965:No, subpages are not good. 211:Well, you've convinced me! 2941: 1565:IS FALSE also.-Santa Claus 1441:It is incorrect to write: 1928:13:58, 24 July 2005 (UTC) 1918:16:24, 23 July 2005 (UTC) 1902:Disagree with merging to 1590:18:47, 13 Jun 2005 (UTC) 866:== Alternative Proofs == 860:== Alternative proofs == 675:13:24, 22 Jun 2005 (UTC) 640:13:52, 16 Jun 2005 (UTC) 619:21:15, 15 Jun 2005 (UTC) 588:13:47, 16 Jun 2005 (UTC) 329:20:19, 15 Jun 2005 (UTC) 309:17:02, 15 Jun 2005 (UTC) 147:1 + 0.999... = 1.999... 2920:) 14:33, 2005 October 29 2874:) 11:05, 2005 October 27 2182:) 13:54, 18 October 2005 2108:) 17:41, 17 October 2005 2013:) 00:53, 2005 October 31 1731:13:18, 16 Jun 2005 (UTC) 1247:(anonymous comment from 915:19:49, 19 May 2005 (UTC) 664:12:54, 22 Jun 2005 (UTC) 170:) 17:53, 17 October 2005 1950:then 0.999... equals 1 1934:Another proof... Maybe? 1426:22:29, 9 May 2005 (UTC) 1304:07:51, 9 May 2005 (UTC) 1237:15:30, 8 May 2005 (UTC) 1196:21:22, 8 May 2005 (UTC) 1154:But Your second notion 1072:16:17, 8 May 2005 (UTC) 1046:18:01, 8 May 2005 (UTC) 1025:22:30, 6 May 2005 (UTC) 1011:22:24, 6 May 2005 (UTC) 989:18:34, 8 May 2005 (UTC) 980:16:49, 8 May 2005 (UTC) 960:16:13, 8 May 2005 (UTC) 942:15:56, 8 May 2005 (UTC) 933:15:39, 8 May 2005 (UTC) 899:14:43, 8 May 2005 (UTC) 878:14:41, 8 May 2005 (UTC) 840:14:41, 8 May 2005 (UTC) 823:14:13, 8 May 2005 (UTC) 813:12:00, 8 May 2005 (UTC) 793:23:35, 6 May 2005 (UTC) 782:23:24, 6 May 2005 (UTC) 773:23:23, 6 May 2005 (UTC) 756:19:03, 6 May 2005 (UTC) 743:19:01, 6 May 2005 (UTC) 695:18:58, 6 May 2005 (UTC) 143:01:27, 7 May 2005 (UTC) 2577: 2557: 2526: 2483: 2415: 2383: 2345: 2301: 2278: 2255: 1805:numbers. pi is not a 1715: 1644: 1411: 1352: 1181: 1148: 1107: 730: 679:Creation of this entry 574: 544: 489: 419: 389: 2578: 2558: 2527: 2484: 2416: 2384: 2346: 2302: 2279: 2256: 2254:{\displaystyle 10x-x} 2194:or has been noted. -- 1976:Yeah, you're right.-- 1716: 1624: 1412: 1353: 1182: 1149: 1108: 731: 671:done here. Cheers. -- 575: 524: 469: 420: 369: 42:of past discussions. 2567: 2544: 2497: 2427: 2396: 2355: 2311: 2288: 2265: 2236: 1621: 1372: 1323: 1158: 1117: 1085: 706: 436: 342: 100:Post replies to the 1860:equivalence classes 1249:User:213.216.199.18 818:them, I suppose. -- 2573: 2553: 2522: 2479: 2411: 2379: 2341: 2337: 2329: 2321: 2297: 2274: 2251: 1711: 1407: 1390: 1348: 1177: 1144: 1113:. You can you say 1103: 949:Knowledge:Subpages 764:Alternative proofs 726: 570: 415: 240:) 2005 November 10 2922: 2908:comment added by 2876: 2862:comment added by 2797:irrational number 2755: 2738:comment added by 2728: 2711:comment added by 2676: 2659:comment added by 2644: 2627:comment added by 2616: 2599:comment added by 2576:{\displaystyle 1} 2508: 2450: 2184: 2170:comment added by 2110: 2096:comment added by 2015: 2001:comment added by 1898: 1881:comment added by 1846: 1829:comment added by 1794:least upper bound 1782: 1765:comment added by 1709: 1696: 1693: 1658: 1511: 1494:comment added by 1472: 1455:comment added by 1375: 1169: 1065:recurring decimal 1058:recurring decimal 811: 724: 558: 503: 403: 242: 228:comment added by 207: 205:) 2005 November 9 193:comment added by 172: 158:comment added by 85: 84: 54: 53: 48:current talk page 2932: 2921: 2902: 2885:Charles Matthews 2875: 2856: 2754: 2732: 2727: 2705: 2675: 2653: 2643: 2621: 2615: 2593: 2582: 2580: 2579: 2574: 2562: 2560: 2559: 2554: 2531: 2529: 2528: 2523: 2509: 2501: 2488: 2486: 2485: 2480: 2451: 2443: 2420: 2418: 2417: 2412: 2388: 2386: 2385: 2380: 2350: 2348: 2347: 2342: 2307:as the limit of 2306: 2304: 2303: 2298: 2283: 2281: 2280: 2275: 2260: 2258: 2257: 2252: 2183: 2164: 2109: 2090: 2014: 1995: 1897: 1875: 1845: 1823: 1781: 1759: 1720: 1718: 1717: 1712: 1710: 1702: 1697: 1695: 1694: 1686: 1674: 1669: 1668: 1663: 1659: 1651: 1643: 1638: 1510: 1488: 1471: 1449: 1416: 1414: 1413: 1408: 1400: 1399: 1389: 1357: 1355: 1354: 1349: 1335: 1334: 1298: 1295: 1292: 1231: 1228: 1225: 1186: 1184: 1183: 1178: 1170: 1162: 1153: 1151: 1150: 1145: 1112: 1110: 1109: 1104: 1069:Charles Matthews 1043:Charles Matthews 1039:geometric series 801: 735: 733: 732: 727: 725: 723: 722: 710: 659: 656: 653: 630:my initial proof 579: 577: 576: 571: 569: 568: 563: 559: 551: 543: 538: 514: 513: 508: 504: 496: 488: 483: 424: 422: 421: 416: 414: 413: 408: 404: 396: 388: 383: 241: 222: 206: 187: 171: 152: 63: 56: 55: 33: 32: 26: 2940: 2939: 2935: 2934: 2933: 2931: 2930: 2929: 2903: 2857: 2842: 2733: 2706: 2701: 2698: 2695: 2689: 2683: 2654: 2622: 2594: 2565: 2564: 2542: 2541: 2495: 2494: 2425: 2424: 2394: 2393: 2353: 2352: 2309: 2308: 2286: 2285: 2263: 2262: 2234: 2233: 2230: 2191: 2165: 2160: 2091: 2082: 2030:192.233.129.254 2026: 1996: 1936: 1915:Oleg Alexandrov 1907: 1876: 1824: 1760: 1751: 1745: 1678: 1646: 1645: 1619: 1618: 1601: 1596: 1588:Oleg Alexandrov 1580: 1562:999...=1000... 1489: 1450: 1445: 1435: 1423:Oleg Alexandrov 1391: 1370: 1369: 1326: 1321: 1320: 1296: 1293: 1290: 1275: 1272: 1229: 1226: 1223: 1204: 1156: 1155: 1115: 1114: 1083: 1082: 1079: 1061: 1008:Oleg Alexandrov 996: 977:Oleg Alexandrov 922: 875:Oleg Alexandrov 867: 861: 837:Oleg Alexandrov 766: 740:Oleg Alexandrov 714: 704: 703: 681: 657: 654: 651: 546: 545: 491: 490: 434: 433: 391: 390: 340: 339: 283:natural numbers 248: 223: 188: 153: 148: 131: 90: 59: 30: 22: 21: 20: 12: 11: 5: 2938: 2936: 2928: 2927: 2926: 2925: 2924: 2923: 2910:68.238.102.180 2893: 2892: 2891: 2890: 2878: 2877: 2864:68.238.102.180 2841: 2838: 2833: 2832: 2811: 2810: 2809: 2808: 2807: 2806: 2788: 2787: 2772: 2771: 2699: 2696: 2693: 2687: 2681: 2649: 2647: 2629:68.238.104.130 2572: 2552: 2549: 2521: 2518: 2515: 2512: 2507: 2504: 2478: 2475: 2472: 2469: 2466: 2463: 2460: 2457: 2454: 2449: 2446: 2441: 2438: 2435: 2432: 2410: 2407: 2404: 2401: 2378: 2375: 2372: 2369: 2366: 2363: 2360: 2340: 2335: 2332: 2327: 2324: 2319: 2316: 2296: 2293: 2273: 2270: 2250: 2247: 2244: 2241: 2229: 2226: 2225: 2224: 2223: 2222: 2210: 2209: 2190: 2187: 2186: 2158: 2154: 2153: 2152: 2151: 2150: 2149: 2134: 2133: 2132: 2131: 2130: 2129: 2112: 2111: 2089:encyclopedic. 2079: 2057: 2056: 2046: 2045: 2025: 2022: 2021: 2020: 2019: 2018: 2017: 2016: 2003:68.238.109.228 1986: 1985: 1984: 1983: 1971: 1970: 1935: 1932: 1931: 1930: 1906: 1900: 1871: 1870: 1852: 1831:68.238.104.130 1819: 1818: 1817: 1816: 1807:set of numbers 1749: 1743: 1735: 1734: 1733: 1732: 1722: 1721: 1708: 1705: 1700: 1692: 1689: 1684: 1681: 1677: 1672: 1667: 1662: 1657: 1654: 1649: 1642: 1637: 1634: 1631: 1627: 1600: 1597: 1595: 1592: 1579: 1576: 1573: 1571: 1570: 1569: 1568: 1567: 1566: 1560: 1557: 1554: 1551: 1548: 1524: 1523: 1484: 1483: 1443: 1433: 1429: 1428: 1419: 1418: 1417: 1406: 1403: 1398: 1394: 1388: 1385: 1382: 1378: 1360: 1359: 1358: 1347: 1344: 1341: 1338: 1333: 1329: 1311: 1310: 1309: 1308: 1307: 1306: 1273: 1270: 1269: 1268: 1267: 1266: 1265: 1264: 1256: 1255: 1254: 1253: 1240: 1239: 1203: 1200: 1199: 1198: 1176: 1173: 1168: 1165: 1143: 1140: 1137: 1134: 1131: 1128: 1125: 1122: 1102: 1099: 1096: 1093: 1090: 1078: 1075: 1060: 1054: 1053: 1052: 1051: 1050: 1049: 1048: 1030: 1029: 1028: 1027: 1014: 1013: 995: 992: 963: 962: 921: 918: 908: 907: 906: 905: 904: 903: 902: 901: 885: 884: 883: 882: 881: 880: 865: 859: 858: 857: 856: 855: 854: 853: 845: 844: 843: 842: 830: 829: 828: 827: 826: 825: 765: 762: 761: 760: 759: 758: 746: 745: 736: 721: 717: 713: 701: 680: 677: 668: 667: 666: 665: 626: 625: 612: 611: 600: 599: 582: 581: 567: 562: 557: 554: 549: 542: 537: 534: 531: 527: 523: 520: 517: 512: 507: 502: 499: 494: 487: 482: 479: 476: 472: 468: 465: 462: 459: 456: 453: 450: 447: 444: 441: 412: 407: 402: 399: 394: 387: 382: 379: 376: 372: 368: 365: 362: 359: 356: 353: 350: 347: 336: 335: 315: 314: 291: 290: 289:? -Santa Claus 256: 255: 247: 244: 219: 218: 184: 183: 146: 130: 127: 126: 115:.) Thank you. 102:main talk page 89: 86: 83: 82: 77: 74: 69: 64: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 2937: 2919: 2915: 2911: 2907: 2899: 2898: 2897: 2896: 2895: 2894: 2889: 2886: 2882: 2881: 2880: 2879: 2873: 2869: 2865: 2861: 2854: 2853: 2852: 2851: 2848: 2839: 2837: 2831: 2828: 2824: 2820: 2817: 2816: 2815: 2805: 2802: 2801:Michael Hardy 2798: 2794: 2793: 2792: 2791: 2790: 2789: 2786: 2783: 2778: 2777: 2776: 2770: 2767: 2762: 2761: 2760: 2756: 2753: 2749: 2745: 2741: 2737: 2729: 2726: 2722: 2718: 2714: 2713:68.238.110.56 2710: 2692: 2686: 2680: 2677: 2674: 2670: 2666: 2662: 2661:194.192.22.33 2658: 2648: 2645: 2642: 2638: 2634: 2630: 2626: 2617: 2614: 2610: 2606: 2602: 2598: 2590: 2589: 2586: 2570: 2550: 2547: 2538: 2535: 2532: 2519: 2516: 2513: 2510: 2505: 2502: 2492: 2489: 2476: 2473: 2470: 2467: 2464: 2461: 2458: 2455: 2452: 2447: 2444: 2439: 2436: 2433: 2430: 2422: 2408: 2405: 2402: 2399: 2390: 2376: 2373: 2370: 2367: 2364: 2361: 2358: 2338: 2333: 2330: 2325: 2322: 2317: 2314: 2294: 2291: 2271: 2268: 2248: 2245: 2242: 2239: 2227: 2221: 2218: 2214: 2213: 2212: 2211: 2208: 2205: 2201: 2200: 2199: 2197: 2188: 2185: 2181: 2177: 2173: 2169: 2157: 2148: 2145: 2144:Michael Hardy 2140: 2139: 2138: 2137: 2136: 2135: 2127: 2122: 2118: 2117: 2116: 2115: 2114: 2113: 2107: 2103: 2099: 2095: 2087: 2086: 2085: 2078: 2074: 2073: 2070: 2069:Michael Hardy 2066: 2062: 2055: 2052: 2048: 2047: 2044: 2041: 2037: 2036: 2035: 2034: 2031: 2023: 2012: 2008: 2004: 2000: 1992: 1991: 1990: 1989: 1988: 1987: 1982: 1979: 1975: 1974: 1973: 1972: 1969: 1966: 1962: 1961: 1960: 1959: 1956: 1951: 1948: 1945: 1942: 1939: 1933: 1929: 1926: 1922: 1921: 1920: 1919: 1916: 1912: 1905: 1904:invalid proof 1901: 1899: 1896: 1892: 1888: 1884: 1883:68.238.110.56 1880: 1869: 1866: 1865:Michael Hardy 1861: 1857: 1853: 1849: 1848: 1847: 1844: 1840: 1836: 1832: 1828: 1815: 1812: 1811:Michael Hardy 1808: 1804: 1799: 1796:; rather, pi 1795: 1791: 1787: 1786: 1785: 1784: 1783: 1780: 1776: 1772: 1768: 1764: 1755: 1748: 1742: 1738: 1730: 1726: 1725: 1724: 1723: 1706: 1703: 1698: 1690: 1687: 1682: 1679: 1675: 1670: 1665: 1660: 1655: 1652: 1647: 1635: 1632: 1629: 1625: 1617: 1616: 1615: 1613: 1609: 1606: 1598: 1593: 1591: 1589: 1585: 1577: 1575: 1564: 1563: 1561: 1558: 1555: 1552: 1549: 1546: 1545: 1543: 1542: 1541: 1539: 1534: 1528: 1522: 1519: 1514: 1513: 1512: 1509: 1505: 1501: 1497: 1496:68.238.110.56 1493: 1482: 1479: 1475: 1474: 1473: 1470: 1466: 1462: 1458: 1457:68.238.110.56 1454: 1442: 1439: 1432: 1427: 1424: 1420: 1404: 1401: 1396: 1392: 1380: 1368: 1367: 1365: 1361: 1345: 1342: 1339: 1336: 1331: 1327: 1319: 1318: 1316: 1315: 1314: 1305: 1302: 1288: 1285: 1281: 1280: 1279: 1278: 1277: 1276: 1262: 1261: 1260: 1259: 1258: 1257: 1252: 1250: 1244: 1243: 1242: 1241: 1238: 1235: 1221: 1218: 1214: 1209: 1208: 1207: 1201: 1197: 1194: 1193:Michael Hardy 1190: 1189: 1188: 1174: 1171: 1138: 1135: 1132: 1129: 1126: 1123: 1120: 1097: 1094: 1091: 1076: 1074: 1073: 1070: 1066: 1059: 1055: 1047: 1044: 1040: 1036: 1035: 1034: 1033: 1032: 1031: 1026: 1023: 1018: 1017: 1016: 1015: 1012: 1009: 1005: 1004: 1003: 1002: 993: 991: 990: 987: 982: 981: 978: 973: 970: 968: 961: 958: 954: 950: 946: 945: 944: 943: 940: 935: 934: 931: 927: 919: 917: 916: 913: 900: 897: 893: 892: 891: 890: 889: 888: 887: 886: 879: 876: 873: 872: 871: 870: 869: 868: 864: 851: 850: 849: 848: 847: 846: 841: 838: 834: 833: 832: 831: 824: 821: 816: 815: 814: 810: 807: 804: 800: 796: 795: 794: 791: 786: 785: 784: 783: 780: 775: 774: 771: 763: 757: 754: 750: 749: 748: 747: 744: 741: 737: 719: 715: 711: 702: 699: 698: 697: 696: 693: 689: 686: 678: 676: 674: 663: 649: 645: 644: 643: 642: 641: 639: 635: 631: 622: 621: 620: 618: 608: 607: 606: 605: 597: 596: 595: 594: 589: 587: 565: 560: 555: 552: 547: 535: 532: 529: 525: 521: 518: 515: 510: 505: 500: 497: 492: 480: 477: 474: 470: 466: 463: 460: 457: 454: 451: 445: 439: 431: 430: 429: 428: 410: 405: 400: 397: 392: 380: 377: 374: 370: 366: 363: 360: 357: 354: 351: 348: 345: 332: 331: 330: 328: 324: 320: 312: 311: 310: 308: 304: 300: 298: 288: 284: 280: 276: 272: 268: 264: 261: 260: 259: 253: 252: 251: 245: 243: 239: 235: 231: 227: 217: 214: 210: 209: 208: 204: 200: 196: 192: 182: 179: 175: 174: 173: 169: 165: 161: 157: 145: 144: 141: 137: 128: 125: 124: 121: 118: 114: 110: 105: 103: 98: 95: 94: 87: 81: 78: 75: 73: 70: 68: 65: 62: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 18:Talk:0.999... 2843: 2834: 2827:24.126.30.46 2818: 2812: 2773: 2757: 2740:192.67.48.22 2730: 2702: 2690: 2684: 2678: 2650: 2646: 2618: 2601:192.67.48.22 2591: 2539: 2536: 2533: 2493: 2490: 2423: 2391: 2231: 2196:24.126.30.46 2192: 2172:192.67.48.22 2161: 2155: 2125: 2120: 2098:192.67.48.22 2083: 2075: 2065:converges to 2064: 2060: 2058: 2027: 1953:correct? -- 1952: 1949: 1946: 1943: 1940: 1937: 1910: 1908: 1872: 1855: 1820: 1806: 1802: 1797: 1789: 1767:192.67.48.22 1756: 1752: 1746: 1739: 1736: 1602: 1581: 1572: 1537: 1532: 1529: 1525: 1485: 1446: 1440: 1436: 1430: 1363: 1312: 1283: 1246: 1216: 1212: 1205: 1080: 1062: 1056:Merger with 997: 983: 974: 971: 964: 936: 923: 909: 862: 776: 767: 682: 669: 627: 613: 601: 590: 583: 580:-Santa Claus 337: 318: 317:And I quote 316: 294: 292: 287:line segment 278: 274: 270: 266: 262: 257: 249: 230:71.248.131.9 220: 195:192.67.48.22 185: 160:192.67.48.22 149: 132: 106: 99: 96: 92: 91: 60: 43: 37: 2904:—Preceding 2858:—Preceding 2782:BradBeattie 2766:BradBeattie 2734:—Preceding 2707:—Preceding 2655:—Preceding 2623:—Preceding 2595:—Preceding 2166:—Preceding 2092:—Preceding 1997:—Preceding 1925:BradBeattie 1877:—Preceding 1825:—Preceding 1761:—Preceding 1729:BradBeattie 1599:Explanation 1578:My reversal 1490:—Preceding 1451:—Preceding 1438:= 0.999... 1286:in [0, 1[. 1022:BradBeattie 1001:BradBeattie 986:BradBeattie 967:BradBeattie 939:BradBeattie 930:BradBeattie 912:BradBeattie 896:BradBeattie 863:instead of 820:BradBeattie 790:BradBeattie 753:BradBeattie 692:BradBeattie 673:BradBeattie 638:BradBeattie 617:BradBeattie 604:BradBeattie 593:BradBeattie 586:BradBeattie 427:BradBeattie 327:BradBeattie 307:BradBeattie 224:—Preceding 189:—Preceding 154:—Preceding 36:This is an 2024:Suggestion 1941:(y/3)x3=y 1612:convergent 1610:series is 1594:about math 1559:9999=10000 924:I created 1938:You see; 1608:geometric 293:Point 1: 80:Archive 5 72:Archive 3 67:Archive 2 61:Archive 1 2918:contribs 2906:unsigned 2872:contribs 2860:unsigned 2819:Actually 2748:contribs 2736:unsigned 2721:contribs 2709:unsigned 2669:contribs 2657:unsigned 2637:contribs 2625:unsigned 2609:contribs 2597:unsigned 2389:apply. 2204:Mallocks 2180:contribs 2168:unsigned 2121:smallest 2106:contribs 2094:unsigned 2040:Mallocks 2011:contribs 1999:unsigned 1965:Mallocks 1891:contribs 1879:unsigned 1839:contribs 1827:unsigned 1775:contribs 1763:unsigned 1605:infinite 1556:999=1000 1533:actually 1504:contribs 1492:unsigned 1465:contribs 1453:unsigned 1366:, then, 1217:infinite 920:Subpages 779:Mallocks 770:Mallocks 738:Cheers, 303:function 281:between 271:Secondly 267:function 238:contribs 226:unsigned 203:contribs 191:unsigned 168:contribs 156:unsigned 140:Shutranm 88:Untitled 2840:rewrite 2823:phinary 2228:10x - x 2051:Eric119 1803:sets of 1518:Eric119 1478:Eric119 799:Pidgeot 688:threads 277:number 275:natural 263:Firstly 111:. (See 39:archive 1978:MrBird 1955:MrBird 1851:abuse. 1553:99=100 1213:finite 972:{{D}} 953:Mike C 323:number 297:number 213:silsor 178:silsor 136:Q.E.D. 120:(talk) 117:Rasmus 2847:KSmrq 2583:? -- 2548:0.999 2514:0.333 2471:0.999 2462:0.333 2400:0.999 2374:0.666 2365:0.333 2331:0.333 2292:0.333 2269:0.333 2061:equal 1911:valid 1340:0.999 610:Claus 346:0.999 334:Claus 273:what 16:< 2914:talk 2868:talk 2825:. -- 2744:talk 2717:talk 2685:and 2665:talk 2633:talk 2605:talk 2563:and 2323:0.33 2176:talk 2126:only 2102:talk 2081:Inf 2007:talk 1944:y=1 1887:talk 1835:talk 1771:talk 1550:9=10 1500:talk 1461:talk 1301:Talk 1234:Talk 957:talk 662:Talk 305:. -- 234:talk 199:talk 164:talk 2585:Tob 2315:0.3 2217:Tob 1856:one 1790:has 1547:0=1 1538:two 1377:lim 1346:999 1284:not 809:(e) 806:(c) 803:(t) 685:two 295:"A 2916:• 2870:• 2764:-- 2750:) 2746:• 2723:) 2719:• 2671:) 2667:• 2639:) 2635:• 2611:) 2607:• 2551:… 2517:… 2474:… 2465:⋯ 2459:⋅ 2440:⋅ 2403:⋯ 2377:… 2368:⋯ 2362:⋅ 2339:… 2295:… 2272:… 2246:− 2240:10 2178:• 2104:• 2009:• 1893:) 1889:• 1841:) 1837:• 1798:is 1792:a 1777:) 1773:• 1704:10 1691:10 1683:− 1656:10 1641:∞ 1626:∑ 1614:: 1506:) 1502:• 1467:) 1463:• 1405:1. 1387:∞ 1384:→ 1343:… 1299:| 1289:— 1232:| 1222:— 1167:∞ 1164:∞ 1142:∞ 1101:∞ 1089:∞ 1067:. 1020:-- 955:| 788:-- 716:10 660:| 650:— 556:10 541:∞ 526:∑ 522:× 501:10 486:∞ 471:∑ 467:× 455:− 425:-- 401:10 386:∞ 371:∑ 367:× 355:− 349:… 279:is 269:. 236:• 201:• 166:• 138:. 76:→ 2912:( 2866:( 2742:( 2715:( 2663:( 2631:( 2603:( 2571:1 2520:, 2511:= 2506:3 2503:1 2477:, 2468:= 2456:3 2453:= 2448:3 2445:1 2437:3 2434:= 2431:1 2409:1 2406:= 2371:= 2359:2 2334:, 2326:, 2318:, 2249:x 2243:x 2174:( 2100:( 2005:( 1885:( 1833:( 1769:( 1707:9 1699:= 1688:1 1680:1 1676:1 1671:= 1666:k 1661:) 1653:1 1648:( 1636:0 1633:= 1630:k 1498:( 1459:( 1402:= 1397:n 1393:x 1381:n 1364:n 1337:= 1332:n 1328:x 1297:P 1294:I 1291:J 1251:) 1230:P 1227:I 1224:J 1175:1 1172:= 1139:= 1136:. 1133:. 1130:. 1127:X 1124:X 1121:X 1098:= 1095:1 1092:+ 720:n 712:9 658:P 655:I 652:J 566:k 561:) 553:1 548:( 536:1 533:= 530:k 519:9 516:= 511:k 506:) 498:1 493:( 481:0 478:= 475:k 464:9 461:+ 458:9 452:= 449:) 446:k 443:( 440:F 411:k 406:) 398:1 393:( 381:0 378:= 375:k 364:9 361:+ 358:9 352:= 232:( 197:( 162:( 50:.

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