Talk:0.999...

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Rational numbers are sufficient for this subject

(Excuse my English) Hello. In my opinion, any references to the real numbers should be deleted from this article, because

• 0.999... is a notation for the limit of a sequence of numbers
• every number in the sequence 0.9, 0.99, 0.999, ... is rational
• the sequence converges to 1, a rational number
• the convergence to 1 can be proved in a few lines, using only rational numbers
• therefore 0.999... = 1

That's it! That is all. No real numbers are used. Following Occam's razor we should not introduce entities here, which are not needed to understand the situation.

Don't get me wrong: Of course, real numbers exist, and the material on them contained in the article (Dedekind cuts, Cauchy sequences, nested intervals, etc.) is interesting. But it is not needed her. Things are presented much more complicated than they really are.

You don't need to understand the concepts of real numbers to understand 0.999... = 1. On the other hand, understanding 0.999... = 1 does not help you very much in understanding the real numbers. 0.999... is not the representation of a "typical" real number. It denotes a rational (even natural) number.

I hope, you find this not too polemic. Yours, Wasseralm 19:40, 1 February 2007 (UTC)

Real numbers are each defined as the limit of a sequence of rationals. 0.999... is defined as the limit of a sequence of rationals. Thus, although 0.999... is rational, the concept of real numbers is greatly helpful in understanding the topic at hand. Calbaer 20:16, 1 February 2007 (UTC)

I disagree. Using only rational numbers will at best obfuscate important details; more likely, it would incubate the "last 9" delusion. The reals are the natural setting for talk about convergence; the rational numbers aren't complete. Technically, your bullet points should be reordered: in a formal proof, confirmation that 0.999... is rational would *follow* from 0.999...=1, not the other way around.

Further, I do not consider it to be in Wikipedia's interests to present the bare minimum of useful information. The "skepticism in education" section is utterly irrelevant to proving 0.999...=1, but I consider it fundamental to this article. Wikipedia should be a fountain of understanding, not just a quarry of facts. That being said, I do agree that the current article is too long, especially for a layperson's first exposure to real analysis. I feel the flow might be improved if the article progressed directly from "Skepticism in Education" to "Popular Culture", with sections in between moved to a new "0.999... in other number systems" article or suchlike. The truly daunting list of references would be thinned out if the technical detail were to be moved elsewhere. If I met someone having difficulty with 0.999...=1, I wouldn't tell them about the p-adics. Endomorphic 21:14, 1 February 2007 (UTC)

I'll emphasize that every decimal expansion represents some real number, but not every decimal expansion represents some rational number. On the other hand, every rational number has a simple representation as a ratio of integers, which most real numbers don't have. This means that discussing decimal expansions only makes sense in the context of real numbers. Should we choose to restrict ourselves to rationals, there would be no need to discuss any decimal expansion, 0.999... included, in the first place. Conversely, if we do choose to discuss 0.999..., it would only make sense within the framework of real numbers.

Your English is fine, by the way. -- Meni Rosenfeld (talk) 13:02, 3 February 2007 (UTC)

More emphasis on exact

I think the word "exact" should occur more often in the article or some early comment should emphasize that "equal" means "exactly equal". People understand that 0.75 is exactly equal to 3/4 . The also know expressions that are approximately equal. If we mean exact, then we should often say exact. -- 64.9.233.132 03:46, 2 February 2007 (UTC)

"Equals" means "exactly equals". Same thing. Tparameter 05:50, 2 February 2007 (UTC)

As a high-school math teacher, I have to say that the exact meaning of "exact" is not easily understood by everyone. Therefore, I don't think there's much point to change "equal" to "exactly equal" - both are correct and will be understood by most, but not everyone.--Niels Ø (noe) 07:43, 2 February 2007 (UTC)

Why "exactly equal" - the expression

The beginning of the article says it's notable that 0.999 is exactly (in italics for some reason) equal to 1. I'm not aware of the mathematical meaning of "exactly equal to". However, if you want to say they're the same object, it should read "identically equal to", which can be found readily in mathematical literature. Tparameter 20:28, 4 February 2007 (UTC)

To support my position, note that the very next sentence points out that this is an "identity", which means each is "identically equal" to the other. Tparameter 20:38, 4 February 2007 (UTC)

"Equal" and "identically equal" are the same. "Exactly equal" is a dubious statement which some interpret to be the same as these. Since "identically equal" will not be understood by laymen, "equal" is the best choice IMO. -- Meni Rosenfeld (talk) 21:30, 4 February 2007 (UTC)

Equal is equal. You only need to modify it if it's something rather like but not quite the same as equal, like "approximately equal". Equal is equal, and for that matter exactly equal, to "exactly equal". It's somewhat unique in that sense. --jpgordon^∇∆∇∆ 21:38, 4 February 2007 (UTC)

"Equal" is definately better than "exactly equal". However, I disagree that "equal" and "identically equal" are the same. In this particular case, that happens to be true - so, if that's what you meant, then I concede that point. To the "laymen" point, I was not aware of that prerequisite. Tparameter 22:21, 4 February 2007 (UTC)

Not necessarily a prerequisite, but definitely something we should keep in mind. The mathematically educated, those who will understand the term "identically equal", have no need for most of this article anyway.

Now, can you explain why "equal" and "identically equal" are not the same? I assume that any notation means the object it represents, rather than the representation. For example, I take 0.999... not to mean the decimal expansion $\{(n,If(n>0,9,0))|n\in \mathbb {Z} \}$, but rather the real number it represents, 1. -- Meni Rosenfeld (talk) 12:53, 5 February 2007 (UTC)

Sure. Consider the difference between an equation and an identity. For example, 4x-3=0. The two sides are only equal under special circumstances. On the other hand, (sin x)^2 + (cos x)^2 = 1 is always true for any values of x. In the second case, the two sides of the identity are "identically equal". They are truly the same object. As for "exactly equals" - it's simply a redundancy, and it sounds ridiculous, sort of embarrassing, IMO. Tparameter 15:33, 5 February 2007 (UTC)

Suggesting for introduction

As far as I know, a system in which 0.999... does not equal 1 has never been formally constructed. As such, I suggest changing the last couple sentences of the second paragraph to

"At the same time, it is possible to construct a system of notation in which an object that can reasonably be called "0.999…" is strictly less than 1, though such a system has never been formally constructed, would sacrifice familiar features of the real number system, and would be of dubious utility."

This is better because the current information makes it seem like such a number system actually exists. Thoughts? Argyrios 03:43, 11 February 2007 (UTC)

Well, that wording doesn't really reflect the "Breaking subtraction" section. Not that we should be overemphasizing such a tiny mathematical idea, but it's not nonexistent either.

I guess something like "dubious utility" could be tacked on, but to me that almost undersells the point. To me, it's more relevant that other possible notations have attracted no usage in either applications or education. Richman's decimal numbers "exist" in the sense that they've been published and they rank highly on a popular Internet search engine, but no one really cares about them. It's not so much that they've been systematically evaluated and found lacking; they just don't serve any purpose, so they attract no attention. Melchoir 05:54, 11 February 2007 (UTC)

I think that it would be more honest (or helpful) to start the article SOMETHING like "The mathematical symbol 0.999… is almost exclusively interpreted as the recurring decimal expansion of a real number. With this interpretation, it denotes the same real number as 1." That, IMHO, is pretty good. A little weaker (or better in the main article) is to continue: " The fact that (as decimal reals)0.999...=1 seems counterintuitive to many people when first encountered. Various responses are common. One persuasive tack is to demonstrate that the equality follows from previously accepted equations and arithmetic manipulations (and hence rejecting the equation entails breaking arithmetic.) Another is to show why it is an unavoidable consequence of the fact that real numbers "are" linear magnitudes as understood since the time of Euclid and Archimedes (essentially, two [real] numbers are either equal or else there is an integer n so that they differ by more than 1/n) "

The point is that by starting "0.999 is a (conventional) decimal real number " then OF COURSE it is equal to 1. The article is perhaps not POV but it has a feeling of attempting to bully the skeptic into silence. There is enough of that other places. The tone is an obstacle to the reader who sincerely wishes to understand this paradox (by which I mean `A seemingly false statement that nonetheless is true.`) They may come away with greater understanding, or at least less confident in their dissent. If not, we the writers should not feel threatened.

I would hesitate to put it in the article but 0.999..=1 is true BY DEFINITION. That is the way we define things. The fact is that it does not really work (for what we want) to define things any other way. (Buried in the article is the quote by Timothy Gowers which says that 0.99...=1 by convention but there are very good reasons for that convention.) Along with the Dedekind cuts and Cauchy sequences there is another 1792 construction due to Weierstrass which basically goes: A real number is an expansion a0+a1/10+a2/100+... where a1,a2,.. are 0..9; operations are what you expect and repeating 9's are dealt with by..." It is every bit as valid as the other two constructions. I am waiting to see the article:

Pierre Dugac

Eléments d'analyse de Karl Weierstrass

Journal Archive for History of Exact Sciences

Publisher Springer Berlin / Heidelberg

ISSN 0003-9519 (Print) 1432-0657 (Online)

Issue Volume 10, Numbers 1-2 / January, 1973

DOI 10.1007/BF00343406

Pages 41-174

before putting that in someplace.

To get back to why I want to start that way: There certainly are systems where 0.999.. is less than 1 but they are not the real numbers. The article refers to the combinatorial games situation (which is binary but could be tricked up to be decimal). This, and the Richman stuff (which really explores just how far we can go without making the convention that the equation is true) and the quote by Gowers only make sense if 0.999... is explored as a symbol which can be interpreted various ways.

For something completely different, can we agree that while p-adic numbers are fantastic, they really don't belong in this artice? Gentlemath 00:49, 12 February 2007 (UTC)

It's tempting to start the article with "The mathematical symbol 0.999…" but not very appropriate. The difference between a symbol and the number it represents is too subtle an idea for most readers, and it betrays the actual usage of those symbols. 0.999… is a number just as surely as 1+1 and zeta(3) are numbers, and this attitude is perfectly consistent with the sources that attempt to explain the problem.

I think it's also optimistic to say that if 0.999… is real then "of course" it's 1. The skeptics don't seem to think that's so obvious.

What you call a bullying tone I call the bold assertion of facts. This is an encyclopedia article, not a guide on a magical journey of discovery. There's no room for "it's true but maybe we shouldn't say so". I really think we do need a Wikibooks article or book on the decimal system. There, it would make sense to start out agnostic on the definition of an infinite decimal, and talk about the properties that numbers should have — to reinvent every wheel in sight. There's a book by A. Gardiner, Understanding Infinity: The Mathematics of Infinite Processes that spends dozens of pages on this project and phrases all its mathematics as consequences of decisions that "we" agree to make. It would be a useful guide.

I've defended the p-adics before, so suffice it to say that no, we can't agree that they don't belong here. Melchoir 01:59, 12 February 2007 (UTC)

The reals ARE subtle and the symbol 0.999... predates their definition. The people who come to this article are going to have to deal with the distinction between symbol and interpretation if they are going to gain any understanding rather than be silenced by "bold statements" of facts. The people who come to this article might not know the fine details of the reals. The article on construction of reals includes a brief but correct section which reads in its entirety:

Construction by decimal expansions

We can take the infinite decimal expansion to be the definition of a real number, considering expansions like 0.9999... and 1.0000... to be equivalent, and define the arithmetical operations formally.

Taking that as our construction (and why not? It dates back to Weierstrass and it is in no way inferior to Dedekind cuts. It is better in some ways, not as good in others.) it is definitely unquestionable that 0.999... and 1.000... are equal.

I did find your defense that the p-adics are appropriate because there are proofs of ...999=-1 in which mirror those that 0.999...=1. Do you really consider that appropriate for the readers? I'll admit that I've studied p-adic valuations but don't recall 10-adics. At any rate, we don't agree that it should be there either.

Finally, I am not personally insulted by the "magical journey" comment but I consider it in line with the problem I see in the article. The article has the tone "Only a fool would question 0.999..=1" and your comment says "Only a fool would question my article." Gentlemath 06:51, 12 February 2007 (UTC)

Yes, 0.999… predates a lot of things. It predates the reals, but more importantly, it predates the entire attitude that numbers are constructed, abstract objects. Eighteenth-century mathematicians didn't stare at a symbol and ask, how many interpretations can we bestow upon this thing?

I do find it appropriate to place the 10-adic subsection where it is; any reader who has survived "Infinitesimals" and "Breaking subtraction" can probably handle whatever else we throw at them. It definitely belongs somewhere in this article, as it's an interesting extension of 0.999…, and if you disagree with that then you can complain to DeSua and Fjelstad.

As for fools, I think you're reading way too deeply into several things. This article actually says "0.999… = 1, which means any one of half a dozen longer, easily proved statements, and most students question it for a range of educationally important issues. Representative attempts to break the equality have the following logical consequences. Blah blah Golden ratio Paul Erdős Cantor set." And I actually say "Only a fool would pursue any agenda in this article." The article is extremely simple; it repeats the contents of reliable sources. It doesn't pass judgement on which number systems we should be using, but it describes the system that we do use. It doesn't consider new ways to redefine symbols, but it uses symbols as they are presently used. It doesn't even emphasize these issues any more than they are already emphasized in the literature. Frankly I don't trust us to do fulfill more than these roles.

By all means, if you want to add a third treatment to "Real numbers" then go for it. I have a couple of sources that might even help in that direction, if you can do the history. But remodeling only this article is a little like building Camelot on a swamp… for the second time. If you want to help expose the surprising richness and subtlely of mathematics, may I suggest writing new Featured Articles on broader topics first? Construction of real numbers needs love too. Melchoir 08:19, 12 February 2007 (UTC)

The problem is in the reals set

From the words of a 15 year old idiot:

The steps defined here only works for the set of real numbers, and thereby you can ask yourself whether or not the axioms for the reals account for an infinitesimal or infinity or not. Limits, in my opinion, is not a good enough proof. Why? Because it is basically an approximation. I may not have recieved enough mathematical training to give a say in this, however, consider the following:

Set k = 1/infinity

k is an infinitesimal, 1 divided by infinity. Therefore, it cannot be less than or equal to 0 and cannot be larger than or equal to 1.

Set 0 < k < 1

Now we go to our culprit.

1 - k = 0.9999....

Add k to both sides.

1 = 0.9999.... + k

If we assume 0.9999.... is equivalent to one, as you have stated:

1 = 1 + k

Subtract one from both sides:

0 = k

Which counters our original truth statement of 0 < k < 1, therefore our assumption is false, which means 0.9999.... != 1.

Set k = 1/infinity
Set 0 < k < 1
1 - k = 0.9999....
1 = 0.9999.... + k
1 = 1 + k
0 = k

In short, real numbers do not span infinity. There were not made to account for infinitesimals.

Frankly, I've gotten quite tired of these occasional ramblings on why 0.999... is equal to 1, what happens when you try to equate 0.9999...8? Perhaps you people may want to use a better set of numbers, ever heard of the hyperreal set? It accounts for the infinitesimal and infinity, something the reals lack. To sum it up, this is my little contibution to the mess of nonsense going around. It's about time we move on.

-too lazy to log in Kia Kroas direct flames to kiafaldorius [AT] gmail.com (and no I haven't bothered to read all the discussions that are here)

What? You do know that: 1 = 0.999..., then the equation: 1 - k = 0.9999..., is only valid if k = 0. which you figured out. What's the problem? Mr Mo 22:45, 15 March 2007 (UTC)

Yep, we've heard about the set of hyperreal numbers. And while it is certainly fascinating, it is pretty much useless for any other mathematical investigation or real-world application. Also, decimal expansions do not adequately describe every hyperreal number, so it is silly to use decimal expansions in the context of hyperreal numbers. 0.999... is a decimal expansion, so it only makes sense in the context of real numbers. Also, there is no element called "infinity" in the hyperreals. There's ω, but where did you get the idea that 1 - 1/ω = 0.999...?

While certainly true that the set of real numbers contains neither infinitesimals nor infinite quantities, how exactly is that a problem? Finally, strictly speaking, limits can be defined rigorously and there is nothing "approximate" about them. -- Meni Rosenfeld (talk) 11:12, 12 February 2007 (UTC)

From the "Related questions" section of this article:

Division by zero occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as complex analysis, where the extended complex plane, i.e. the Riemann sphere, has point "infinity". Here, it makes sense to define ¹/₀ to be infinity; and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.

If infinity does indeed equal 1/0, then in order to divide by infinity, you would multiply by 0/1. Thus, k = 0, your second statement is incorrect, and your proof falls apart. 69.110.37.56 04:09, 28 March 2007 (UTC)

...which illustrates beautifully why infinity and infinitesimals are not part of the standard theory of ordinary numbers: There are so many different ways of including such quantities, appropriate for different contexts, and leading to no end of contradictions if mixed freely. For instance, the Riemann sphere includes one quantity "infinity". In the complex plane, that means that if you go infinitely far in any direction, you wind up at the same place. In other contexts, you may wish two infinities, called "plus infinity" and "minus infinity". (With the Riemann sphere, "minus infinity" would be no different from "plus infinity".) Or, in the complex plane, you might want a different "infinity" for each direction you can go, sort of like "infinity with an argument" (in the complex-number-sense of argument). And how about infinity-squared? In some contexts, it could be convenient to consider that to be of a different "order" of intinity; in others not. Then there are the transfinite cardinals; that's a different kind of infinity - in fact a hierarchy of infinities. - As for the infinitesimals, for a start you could consider the reciprocal of each of those different infinities, and then, you could add infinitesimals like dx and dy from calculus, and what not.

So what's my point? This: We need a consistent theory of the numbers we work with commonly. If you are a carpenter, say, you can't limit yourself to integers; you need reals as they are the smallest closed set of numbers that simultaneously can represent quantities like the side and diagonal of a square object, and the radius and circumference of a round object. We have that theory - we call it real numbers - and we have a convenient representation for the numbers included in that theory - decimal numerals. There are a couple of imperfections that we cannot avoid, though: (1) The set is not closed with respect to division (a/b is not always a real, even when a and b are, because you cannot divide by zero). (2) In the aforementioned "convenient representation", some numbers do not have a unique representation; hence this article.

Sometimes we do need to include infinity and infinitesimals - sometimes in one sense; sometimes in another sense. If we included one sense in our standard theory, it would be much more confusing to deal woth situations where they needed to be included in a different sense. Therefore, we leave them out, till we actuially need them. And carpenters don't need them.--Niels Ø (noe) 07:14, 28 March 2007 (UTC)

All problems except one has been pointed out: 0.999...8 doesn't exist. "0.999..." means the digit 9 is repeated infinity times (correct me if I'm wrong), and you cannot add an 8 to the end. Why? Because there is no end! That is why "0.000...1" doesn't exist. From the words of a 13 year old idiot. Chrishyman 02:03, 20 April 2007 (UTC)

Physical Reality

This sentence should be added as most people do not realize the full meaning of "in mathematics" and automatically assume that it always reflects physical reality (though it does most of the time), so it is POV by ommision not to clarify this to the general public which Wikipedia is supposed to be accesible to. There is a lot of unnecessary arguing on the arguments talk page on use of number systems and it is necessary this be mentioned for a thorough article. It has been reverted two times but one author incorporated it in:

It is important to note, however, that mathematics does not always correspond to physical reality.

--JEF 22:48, 12 February 2007 (UTC)

If we add it here, I suggest we put it in the introduction of every mathematics article on Wikipedia. We can't have a user reading the entry on ring theory and have them think that it reflects realty (like wedding rings), can we? –King Bee ^{(T • C)} 22:53, 12 February 2007 (UTC)

It is more appropriate for the controversial ones.--JEF 23:00, 12 February 2007 (UTC)

If by "controversial" you mean that people without math training don't understand, then you pretty much mean every math article. It doesn't make sense to point out this very obvious statement in this context. Put it in the mathematics article if you wish. Tparameter 23:04, 12 February 2007 (UTC)

As Tparameter suggests, there's a difference between "controversial" meaning that it's disputed amongst mathematicians, and "controversial" in that some non-mathematicians don't understand it. I also dislike the way it's written - it implies to me that this result is worthless, and that people who dispute it are correct after all "in reality". If we have the link, it would be better to place it under "Skepticism in education", as an explanation of why people have trouble with it (i.e., they have trouble with it because they think that the result should reflect some notion of reality). Mdwh 23:21, 12 February 2007 (UTC)

I will mention it in "skepticism in education" then, but as that section points out, many college students that, by the description in the article, at least seem to be knowledgable in mathematics disagree with this as well as knowledgable people (though rare) on the arguments page. It would seem then that it is not just controversial among the general public.--JEF 23:43, 12 February 2007 (UTC)

The problem I have with any mention of "physical reality" in this context is the inherent suggestion that 0.999... has some meaning outside mathematics. Unless you can source that, I doubt it. When was the last time you stumbled upon 0.999... outside mathematics, be it physics or everyday life? Can you give any source where 0.999... appears outside mathematics and is not equal to 1? --Huon 00:04, 13 February 2007 (UTC)

Suprisingly .999... can be seen in particle accelerators as the limitation of the speed of light lets the electron come as close to the speed of light as possible without actually reaching it (which would be .999... of the speed of light .--JEF 01:27, 13 February 2007 (UTC)

You mean that scientists use "0.999..." to represent the fact that the speed of an electron must be less than the speed of light? Do you have any sources for this terminology? Mdwh 02:29, 13 February 2007 (UTC)

The word "scientists" isn't in JEF's latest contribution to this page; I think he/she means that's how he/she sees it (in reaction to my comments to the contrary). If I misinterpret this, I too welcome a source on this terminology. Calbaer 03:07, 13 February 2007 (UTC)

Ok, I checked and v = 0.99999999995*c for electrons in particle accelerators [1]. I concede that .999... is still a theoretical number but small differences in velocity have huge effects under special relativity and there is a difference at least in special relativity between 1*c and .999... *c (this is based on my understanding of relativity).--JEF 03:37, 13 February 2007 (UTC)

And speaking of education, you'll need a source in order to make the nontrivial claim that a distinction between mathematics and reality is an educationally relevant barrier to understanding 0.999…. By the way, I wouldn't assume that students taking real analysis are knowledgable in mathematics. I don't have numbers, but I imagine that an introductory upper-division analysis course is where lots of students discover that they should be majoring in something else. Melchoir 00:15, 13 February 2007 (UTC)

• Please wait to add your paragraph until some kind of consensus has been reached. It seems to be the consensus that it doesn't belong, not that it does, at least right now. –King Bee ^{(T • C)} 01:43, 13 February 2007 (UTC)

Seems like it should be on some philosophy of math article, or something very general. This little article doesn't seem like the place for such a comment. Besides, last time I checked, 1 is very useful in "reality". 63.224.186.83 14:12, 13 February 2007 (UTC)

"there is a difference at least in special relativity between 1*c and .999... *c" - The rules of a mathematical system still apply when that system is used to model the physical world. You're still using the reals, under which 0.999... = 1.--Trystan 16:37, 13 February 2007 (UTC)

Huh? Please explain why you believe that 0.999... is any different in "special relativity" than it is here. I suspect that your claim is spurious, and based on bad logic. I suppose you think that "as something approaches infinity...", as in the light speed claim above; but, remember, 0.999... doesn't approach anything. It is 1. 63.224.186.83 22:28, 13 February 2007 (UTC)

It would be different if you define .999... as the closest a number can come to one without actually touching it. This would apply to reality if we describe the closest a particle can get to the speed of light without actually reaching it as .999...*c .--JEF 23:03, 13 February 2007 (UTC)

This is the same idea you expressed on Talk:0.999.../Arguments, so I suggest we end this line of discussion on this page and direct all further discussion to Talk:0.999.../Arguments. Calbaer 23:55, 13 February 2007 (UTC)

Who says there exists "the closest" speed less than the speed of light? That's quite a stretch, and, as I suspected, a spurious claim indeed. I'm wondering how a theoretical thing in your imagination exists in "reality" (as you put it). Do you have some evidence of the existence of this particle that you speak of - and, who has shown that the speed of this particle is "the closest" speed next to the speed of light? Nice try, pal. 63.224.186.83 00:00, 14 February 2007 (UTC)

Last word here: Are you asserting that scientific theories are supposed to describe something outside of reality? Theory is supposed to describe reality and well founded ones such as special relativity have a lot of research behind them that are based on physical reality and not postulates (though postulates are supposed to describe physical reality and usually do). There seems to be a conflict here between the assertion of a limit being eventually reachable or it being impossible to reach as in special relativity.--JEF 01:31, 14 February 2007 (UTC)

Moved to Talk:0.999.../Arguments#Decimal Representation Calbaer 02:52, 14 February 2007 (UTC)

Yet another proof variant

I was thinking about this article the other night and I came up with my own approach. It's superficially similar to this approach, if perhaps closer in fact to the "Cauchy sequences" proof. I don't see this specific approach discussed anywhere on the existing page. Yes, it lacks mathematical rigor, but it might help to sway those on the fence. (I contend that this is not "original research", because what are the odds that I am the first dumb ape to come up with it?)

If 1 and 0.999… are different numbers, then the definition of real numbers tells us there must be an infinite set of numbers between them. If they are the same number, there can be no numbers between them. Therefore, demonstrating at least one number between them is a valid counter-proof.

The most reliable way to compute a number between two numbers is to take their average. For two different numbers a and b, (a+b)/2 computes their average. If a and b are the same number, their average will also be the same number: (a+a)/2 = 2a/2 = a.

Attempting to produce the average of 1 and 0.999…:

(1 + 0.999…)/2  =  1.999…/2
 
    0.999...
  ________
2 ) 1.999...
   -0
   --
     19
    -18
    ---
      19
     -18
     ---
       19
      -18
      ---
        1

The result is 0.999…, which is one of the inputs, therefore the two inputs are the same number.

Any good? --Larry Hastings 15:28, 14 February 2007 (UTC)

Nice argument. However, since it's original research and therefor unappropriate to the article, you should probably post it at Talk:0.999.../Arguments instead, since this talk page is only for discussing the article itself, not new proofs (or counter-proofs) --Maelwys 15:34, 14 February 2007 (UTC)

It is a nice proof and certainly appears many places. I think it should go in in the following form (which also appears several places)

 0.999... + 0.999... =1.999...
        1 + 0.999... =1.999...

I'll do it sometime unless I hear objections. Here is why I think it is worth the extra space:

1) No multiplication or division (and the addition is not problematic).

2) This is the context of "breaking subtraction" mentioned elsewhere.

3) More technical reasons that I'll mention if desired Gentlemath 18:12, 14 February 2007 (UTC)

Judging from the myriad of objections to 0.999...=1 I've heard, these arguments will convince no one new. Most objectors seem to regard 0.999... as "the closest number to 1" and thus you'll lose them at the infinite number of numbers between them. Also, if persons won't believe that 0.999... = 1, they generally won't believe that twice 0.999... is 1.999.... Finally, as stated, new proofs are considered "original research." I could see bending that if the proof were rigorous, similar to a reliable source, and more convincing, but, although I admire the motivation, I don't believe this has been shown to be any of these. Calbaer 18:39, 14 February 2007 (UTC)

If you want a reference then pretty much exactly what Larry Hastings suggests is in the paper by David Tall (D.O. Tall and R.L.E. Schwarzenberger (1978). "Conflicts in the Learning of Real Numbers and Limits". Mathematics Teaching 82: 44-49.). I thought that the form I mentioned was in Richman. It is, but not as prominently as I thought. He does point out that (using grade school addition) 0.999...+x=1+x for any non-terminating decimal (that is the crux of the "breaking subtraction" or more precisely "breaking cancellation"). That assertion might not be obvious but the addition 0.999...+0.999...=1.999... is. Well, I wouldn't think of putting it in unless it is in a reliable and respected source (and that I reference it). Gentlemath 19:18, 14 February 2007 (UTC)

Many of the "doubters" think that 0.999...+0.999... is either 0.999...8 or undefined. As ridiculous as that might sound to you or me, that's what they think. So the question is: Of what benefit is the proof you present? Namely, whom would it help to supplement or replace the current content with it? It seems to me that it's similar to the first proof, only with a multiplication by 2 rather than 3. I can't picture anyone who didn't believe one believing the other. Again, my primary objectives aren't my concerns about OR and rigor, but about whether or not this would help. Does anyone else think it would? If so, where best to put it in the article? Calbaer 21:36, 14 February 2007 (UTC)

By the way, as someone new to Wikipedia, you might not be familiar with the article history. A look at the archive shows that the discussion is actually less heated than it has been in the past, so the constant barrage of new discussion is not due to article degradation but rather the nature of the topic itself. Calbaer 21:43, 14 February 2007 (UTC)

I never thought there was degradation. I'm not convinced that another proof should be there although I'm also not convinced that it shouldn't, What I had in mind was very short and uses only addition (and perhaps subtraction.) Roughly something like:

${\begin{aligned}0.999\ldots +0.999\ldots &=1.999\ldots \\0.999\ldots +0.999\ldots &=1+0.999\ldots \\0.999\ldots &=1\end{aligned}}$

For the last step cancel the + 0.999... from both sides.

An alternative I like less is:

${\begin{aligned}0.999\ldots &=0.999\ldots +0\\&=0.999\ldots +(0.999\ldots -0.999\ldots )\\&=(0.999\ldots +0.999\ldots )-0.999\ldots \\&=1.999\ldots -0.999\ldots \\&=1\end{aligned}}$

Gentlemath 07:15, 15 February 2007 (UTC)

My question still remains: Whom do you think this presentation would help, e.g., who would be able to follow/believe this proof but wouldn't follow/believe those in the current article? Again, it's my contention that anyone who might have trouble with the current proofs wouldn't follow why/how/that 0.999... + 0.999... = 1.999.... Anyway, I could be wrong; it might be good to have some other opinions.... Calbaer 16:59, 15 February 2007 (UTC)

I agree with Calbaer. Probably we'll get reasoning along the lines "0.999...+0.999... doesn't equal 1.999..., there's an infinitesimal tiny bit missing! The sum's digit at infinity should be an 8, not a 9!" Adding yet another digit manipulation proof won't convince those unconvinced by those we already have. --Huon 22:13, 15 February 2007 (UTC)

Yeah, you're both probably right. There might be a more persuasive way to phrase it, but it's not going to win people over any more than the existing arguments. I'm just happy to have come up with it on my own. --Larry Hastings 20:38, 19 February 2007 (UTC)

Minor Math Formatting

At the start I put the very first 0.999... and also the 0.(9) in math blocks

$\left.0.999...,0.(9)\right.$. For me this makes it look better (I was trying to figure out the MathML if that is what it is. ) For most of the choices of math preferences the page otherwise shows up for me with three different looking things in different sizes and looked bad. I realize that might not show the same for everyone. What do people suggest? I had the problem too that at some resolutions the 0.(9) broke to a new line after the decimal point. I fixed that by moving it to the start of its list.

Does everyone know that in preferences one can set ones math preferences? I didn't although of course many users don't have accounts anyway.Gentlemath 18:27, 14 February 2007 (UTC)

My preferences > Math

By the way, I reverted your edits because at WP:MOS, it says that the first occurrence of the article name (in this case, 0.999...) should be bolded. Not between <math> tags. For one thing, it looks bad on those who force it to render as HTML, and doesn't work for those who disable images. x42bn6 Talk 23:37, 14 February 2007 (UTC)

ok then I'll just make the 0.(9) look like the other two. Gentlemath 01:32, 15 February 2007 (UTC)

Inclusion of the fractional proof

I'm a little concerned about the fractional proof given. While it properly acknowledges that ${\frac {1}{3}}=0.{\dot {3}}$ (an infinite string of threes), it seems to employ the logic used for rational numbers (assuming it has a determinate length). Should it really be included? --59.154.24.148 04:30, 20 February 2007 (UTC)

Are you assuming all rationals have a finite number of decimals only? That is not correct; any decimal that ends in a periodic repetition of digits is rational. Examples:

• x=34.213400000... is rational (10000x=342134, hence x=342134/10000)
• x=0.333... is rational (10x=3.33..., 9x=10x-x=3, hence x=9/3=1/3)
• x=0.142857142857... is rational (1000000x=142857.142857... , 999999x=1000000x-x=142857, hence x=142857/999999=(3*3*3*11*13*37)/(3*3*3*7*11*13*37)=1/7)
• x=4.321720720720720... is rational (1000x=4321.720720720720..., 1000000x=4321720.720720720..., 999000x=1000000x-1000x=4317399, hence x=4317399/999000)
• x=0.999... is rational (10x=9.99..., 9x=10x-x=9, hence x=9/9=1)

In general, muliply x by a power of 10 to bring aperiodic part in front of decimal point (y), then multiply by power of 10 to bring the first period in front of the decimal point (z), then subtract z-y to obtain an integer.--Niels Ø (noe) 08:06, 20 February 2007 (UTC)

I think I might be able to clarify a little here, because, yes, the question is badly worded. I think the question being asked is, isn't the fractional proof using the logic of a finite decimal to address the similarity between $0.{\dot {9}}$ (an infinite decimal representation) and 1? (There's no need to worry about the algebraic proof; it's logic is fairly straightforward and understandable.) --JB Adder | Talk 12:39, 22 February 2007 (UTC)

If 0.999... does equal to 1 then i can prove that 0.99=1 by the same method

.99 = x 10x = 9.99 (decimal point left once) 9x = 10x - x = 9.99 - .99 = 9 9x = 9 x=1,(X=0.99)

Uh, no. if x=.99, then 10x equals 9.9, not 9.99. That effect does not appear with .999... because there's no last 9 - after shifting the decimal point by one, there are still infinitely many nines left. If you have further doubts, you should probably take them to the arguments page - this talk page is for discussing the article, not the math. --Huon 19:56, 22 February 2007 (UTC)

More Emphasis of the Cantor set and P-ary expansion

I would like to ask for more emphasis on the Cantor set and P-ary expansion in this article. In particular, please make it painfully obvious that ANY real number can have TWO expansions and emphasize the difference between a string and a number and HOW numbers are represented. Page 40 of Royden's Real Analysis, problem 22 is a good, no GREAT, problem and IMHO (well maybe not so humble), throws this concept into the student's face. If they still can't be convinced...I...am at a loss...but for me it was awesome :) —The preceding unsigned comment was added by 67.9.140.19 (talk • contribs) 04:36, 23 February 2007 (UTC)

Okay, now I'm curious: what's in page 40 of Royden? (By the way, only decimal fractions have two decimal expansions; other numbers, such as the irrationals, have just one.) Melchoir 04:57, 23 February 2007 (UTC)

Change to lead?

Hi, I'm the kind of editor who rarely checks sources; instead I write what I'm convinced is true. Since I'm usually right(!), most of my edits survive, if neccesary backed up by references by others. But changing the lead in this particular article is a serious matter, so, havng no sources, I'll suggest my change here instead of being bold. Here's a quote from the lead as it stands:

The students' reasoning for denying or affirming the equality is typically based on one of a few common erroneous intuitions about the real numbers; for example, a belief that each unique decimal expansion must correspond to a unique number,

I think that, for a lot of students, the confusion is more accurately described by saying that they identify numbers and their decimal forms. Actually, common terminology promotes this misunderstanding: "101101 is a binary number", say. No it's not, it's a number written in binary notation, or a binary numeral. (Unless of course it is in fact a different number written in a different notation; it really needs context.) So I suggest adding to the quote above:

often caused by the student identifying numbers as such with their decimal expansions,

Alternatively, one could replace everything n the quote following for example, by:

an identification of numbers as such with their decimal expansions,

I know that as a student, I myself was confused in exactly this way for many years.--Niels Ø (noe) 09:22, 23 February 2007 (UTC)

I like it. Black Carrot 08:27, 7 March 2007 (UTC)

Problem in the representation?

Maybe part of the disagreement is caused by an ambiguity in the figure 0.999...

Doesn't this all depend on a purely conventional approach as to what notion we consider 0.999... to represent? There aren't any mathematical rules behind this. It's not about a relationship of actual numbers. It's about the relationship you see between the figurative representation you are given (0.999...) and the abstract mathematical entity that you hold the figure to represent.

Isn't precisely which number 0.999... should be the decimal expression of a little ambiguous? I'm not sure that we've developed a definite convention on how to correspond this kind of notation with some actual number. It's clear that there isn't any mathematical reason why the figure '1' should represent the first positive integer, it's merely convention that we do it that way, so that we all do do it the same way. In the case of 0.999..., we are required to decide for ourself precisely what actual number this 'figure' represents. There doesn't appear to be any set convention for this. If you can cite some definite convention here, please correct me but otherwise: We have to somehow grasp this number - the actual abstract mathematical entity that we hold our term (0.999...) to refer to - from the information in front of us.

I think part of the cause of dispute is that 0.999... could also be used to represent the number in the following example:

1. I'm gonna start at 0, and aiming towards 1, add 90% of what I need to get there each time, and repeat that an infinite number of times. So 0, 0.9, 0.99, 0.999, and so on. Each time I get a little closer to 1, but each step can only take me most of the distance I need to go. The distance diminishes from 1 at the first instance, to 0.1 in the second, and 0.01 in the third, etc. An infinite number of nines would represent an infinite number of such steps, which implies that, with proportion, the remaining distance has been fractioned an infinite number of times. So the distance remaining is infinitely small, and may, for all intents and purposes, be considered equivalent to zero. If the distance is zero, we've reached our point, and we're at 1.

The thing is, even if this remainder is infinitely small, the fact that there is this positive remainder means that if you square rooted it an infinite number of times, the final product would be 1.

But if you square root zero, you get zero. So doing so an infinite number of times would still always result in zero.

so 1 - 0.999... > 0, regardless of how small it is, even if it is infinitely small.

Let's call 1 - 0.999... 'n'

n > 0 1 - 0.999... = n so the difference between 1 and 0.999... is greater than 0, so there is a difference.

0.999... might not have to represent 1 - n, but it can. So there's definitely ambiguity here. What you need to do, I think, is explain explicitly what number '0.999...' should be considered as representing - how we summon the idea of the number that doesn't leave this infinitely small - but positive nonetheless - difference between it and 1.

00:08, 12 March 2007 (UTC)jonbeer

The section "Infinite series and sequences" already explicitly sets out what "0.999…" means, and the section "Analogues in other number systems" begins by acknowledging the viewpoint that this meaning is a human convention. Is there something inadequate there?

By the way, if you'd like to discuss your example, it would be best to take it to Talk:0.999.../Arguments. I'll just briefly say that the example does not define a number. Melchoir 00:32, 12 March 2007 (UTC)

To start out with, we really do have a formal definition of what 0.999... is. We can approach that definition several ways, but the simplest way is that 0.999... is the limit of the sequence {0.9, 0.99, 0.999, ...}. Limits of sequences are well defined using epsilon-delta proofs that never involve infinity and so there is no problem here.

The problem with saying things like, "Let's think about starting at 0 and moving to 0.9, then moving to 0.99, then moving to 0.999, etc" is that you then need to invent vague and ill-defined notions like "where I am once I've done this an infinite number of times". This is the cause of the problem that many people have with the equality. They think of constructing 0.999... by moving along the sequence {0.9, 0.99, 0.999, ...}, and seeing what they have reached when they are at some hypothetical "end", presumably after an infinite number of terms. This is a problematic view that can cause much confusion. It's far better to understand the flaws in this vague definition and then move to the limit-based definition of decimals. Maelin (Talk | Contribs) 04:41, 12 March 2007 (UTC)

Arbitrary conventions are not always bad things. Language itself is but a collection of conventions. In this case the mathematics is well established, and there is no ambiguity as to which decimal "0.999..." represents. Endomorphic 20:26, 12 March 2007 (UTC)

so 1 - 0.999... > 0, regardless of how small it is, even if it is infinitely small

I think you missed something out, 0.999... equals 1, so 1-0.999...=0. It is like saying 2-1 < 0, it is wrong, and you can not go any further.

so 1 - 0.999... > 0, regardless of how small it is, even if it is infinitely small.

So what number can you put between 1 and 0.999...? None? Surprise, it is the same number :) It's like saying what's between 2 and 4/2.

This page should redirect to 1(number)

This page should redirect to 1(number). If they are indeed equal and all. Maybe merge the two?

I think that is a bad idea. This is not an article about the number 1; it's really about the fact that all terminating decimals except 0 have an alternative decimal representation ending in 999... This fact is usually exemplified by the pair of representations "0.999..." and "1", but we could also consider e.g. -23.03999... and -23.04. Arguably, the article should have a name better reflecting its contents. However, this has all been discussed before, and the present stat of affairs is the result of near-concensus.--Niels Ø (noe) 08:13, 15 March 2007 (UTC)

Another, albeit not a formal proof

I have always enjoyed this method which provides little room for arguement:
1/9 = 0.111...
2/9 = 0.222...
3/9 = 0.333...
4/9 = 0.444...
5/9 = 0.555...
6/9 = 0.666...
7/9 = 0.777...
8/9 = 0.888...
and 9/9 = 0.999... from the pattern. Though the fraction's value is 1 by simple divison.
Since 9/9 can be only one value under group theory rules then 1 = 0.999...-- User talk:207.210.20.56 21:50, 21 March 2007

Your argument is not as full-proof as you think because how do you know that 1/9 = .111... , that 2/9 =.222... , etc. ? I would argue that 1/9=.111...+a remainder of 1 at infinity and thus is impossible to put into an exact decimal form (divide by nine and get a remainder of 1, divide by nine and get a remainder of 1...). This would apply to all repeating decimals.--JEF 02:22, 22 March 2007 (UTC)

What are we after here, mathematical proof or arguments to convince sceptical students? I think 207.210.20.56's idea is in the latter category, and I've never met a student denying 1/3=0.333..., so I think it's a fine argument. It's a less formal version of the other argument using 1/9. (Both can be changed into arguments using 1/3 instead of 1/9.)--Niels Ø (noe) 07:56, 22 March 2007 (UTC)

Well, you have met one now. I had been arguing this for weeks on the arguments page, but because of how the "real" number system is defined, I can't argue that this is how the reals work, though my argument seems to have some validity. My debate is now archived.--JEF 13:49, 22 March 2007 (UTC)

"I've never met a student denying 1/3=0.333..." I did, when I was about 10. In fact that was what my whole objection was based around. I argued that 1/3 couldn't equal 0.333... because if you then multiply it by 3 you get 0.999... as opposed to one. I see it differently now, but only because I've seen it demonstrated why 0.999...=1 (the argument which swayed me was the "algebraic proof" as stated in this article, though I first saw it somewhere else), which then shows that 0.333...=1/3. Raoul 13:37, 3 April 2007 (UTC)

It's always good to keep in mind a simple rule that the decimal values of fractions are the remainders of the numerator divided by the denominator (e.g. 1/3 = 1 divided by 3 = 0.333.. or 3/3 = 3 divided by three = 1). --88.193.241.224 16:40, 6 May 2007 (UTC)

And thank you JEF for finally identifying your misunderstanding (entry 02:22, 22 March 2007). Your's is not with the real numbers but with your concept of infinity. JEF, infinity does not exist; it is merely a concept. An interesting verification goes like this: if you think you've found infinity you are mistaken, because by adding just one to it, you now have larger than infinity, which is a contradiction; so infinity does not actually exist. In arguing that 1/9 = 0.111... there is no remainder (as you quite rightly asserted at infinity) because there is no end. User talk:207.210.20.56 22:35, 28 March 2007

Kind of. If infinity doesn't exist and is merely a concept then 4 doesn't exist and is merely a concept. Secondly, there are formulations of set theory where you can add one to infinity, and you do get infinity again. An example: there are infinitely many odd numbers. Now consider the set of all odd numbers and two, ie, {2, 1, 3, 5, 7, 9...} - there are infinitely many members, again. This doesn't mean that infinity doesn't exist, it just means you can't do the usual arithmatic on it. Which is why infinity and "1/infinity" (whatever that could be) are kept out of the reals. Lastly, 207.210 is right in one context: there is no "infinity-ith" decimal place containing a remainder for a real number. Ever.

PS this should really be on the arguments page too. Endomorphic 21:43, 29 March 2007 (UTC)

Does this work with other repeating decimals?

I understand that 0.999...=1 and have no doubts about that. Does this work for other repeats such as 0.888...=.9 0.899...=.9?65.197.192.130 01:43, 22 March 2007 (UTC)

No. The easiest way to see this is that you can squeze 0.889000... in between 0.888... and 0.9, so they have to be different numbers. A different round-about way to proving 0.888... != 0.9 is to look at 88.888...; would this be 88.9 or 90? It certainly can't be both, but if it's one then the other is equally valid, so it can't be either. So .888... can't be 0.9. Endomorphic 02:06, 22 March 2007 (UTC)

But note that 0.8999...=0.9 - in fact, any terminating decimal (like "0.9") has a partner ending in an infinite string of 9's. Well, nearly any - "0" has no such partner.--Niels Ø (noe) 08:04, 22 March 2007 (UTC)

Sorry about that. I meant to put .8999..., but when I realized I typed it wrong, I tried to stop the page, but it had already gone through. Thanks for answering though. 65.197.192.130 18:49, 22 March 2007 (UTC)

For future reference, keep in mind that MediaWiki (the software behind Wikipedia) treats articles and talk pages the same way, so it is possible to simply edit your question, like you would an article. You can also use <s> and </s> tags to strike out an incorrect piece, as I have taken the liberty to do in your original question. -- Meni Rosenfeld (talk) 20:41, 22 March 2007 (UTC)

Unecessarily long references section

I feel that the references section for this page needs pruning back quite a bit. There only need to be a few key texts mentioned, really. Teutanic 10:53, 25 March 2007 (UTC)

If you see a specific reference that isn't being used, feel free to remove it. Melchoir 19:17, 25 March 2007 (UTC)

Ha ha ha

Hilarious. You can prove a lot of daft things using infinity, there ought to be a category for stuff like this. -88.109.136.33 20:12, 2 April 2007 (UTC)

Categorizing articles based on whether some editors think they're silly? Probably not the best idea. Leebo ^T/C 20:24, 2 April 2007 (UTC)

"every non-zero, terminating decimal has a twin with trailing 9s." (pedantry)

This could be clarified in parentheses to specify that the same applies in different number systems, with the highest single-digit numeral. for example, trailing F's in hexadecimal, trailing 7's in octal, or trailing 1's in binary. (binary fractions do indeed work that way) This is completely pedantry though, so don't take this seriously. It's just something that should be mentioned, and leaving it on the talk page is probably good enough. 68.93.32.12 22:10, 3 April 2007 (UTC)

• And in every base, 0.(base-1)... = 1. So, 0.111...₂ = 1. Which also gives us the lovely 0.111...₂ = 0.999...₁₀. I like that. --jpgordon^∇∆∇∆ 23:00, 3 April 2007 (UTC)

this seems to cause a paradox

the real numbers between 0 and 1 can be identified with points on a line of length 1

the numbers with 1 digit decimal expansions are 10 points with a distance of 0.1, or 1-0.9 between them

with 2 digit decimal expansions, the smallest distance between different numbers is 0.01, or 1-0.99

so with all the real numbers, the smallest distance between different numbers is 1-0.999...=x

so the difference between 0 and the next biggest number is x, and the distance to the next biggest number is also x, and so on until we get to 1

according to this article 0.999...=1, so 1-0.999...=1-1=0

therefore x=0

however 0+0=0,0+0+0=0,0+0+0+0=0, and carrying on like this, this distance from any

number between 0 and 1 to 0 is 0

wth? 10:46, 4 April 2007 (UTC)

There is no smallest distance between different real numbers. For any number x there is always x/2, and if x is positive so is x/2, but it's smaller than x. --Huon 12:50, 4 April 2007 (UTC)

You can avoid the "smallest distance" error by adding the lengths of the closed intervals [x,x] for all x in [0,1]. Each interval has length 0, because they're all just single points. But the sum of lengths over all x should be the length of [0,1] which is 1. This rephrased question hits on some subtle measure theory. While it is true that 0+0+0+...+0=0 for any countable summation, it is not generally true when dealing with uncountable summations, and there are uncountably many points x between 0 and 1. Basically, there are just too many x's to fit into an expression like 0+0+0+0+..+0 and so you can't expect the algebra to work. It's a lot like Zeno's paradoxes. Each instant of time has zero duration, but somehow when you consider together all the times between 11:00 and 12:00 you have a total of an hour. Endomorphic 12:15, 8 April 2007 (UTC)

Interesting point, although "length" isn't the word I'd use. "Measure" or "size" would be better. Linguistic nitpicks aside, it's similar to the following problem: I want to pick a number at random in [0,1] using the continuous uniform distribution. The chance that I pick a particular number in [0,1] is 0, but the odds that I pick any number in [0,1] is 1. Again, that's measure theory at work, showing that an uncountable number of 0 measures, grouped together, result in a positive measure. It's unintuitive, but if you were to modify the rules of math to prohibit that property, a lot of important things would break. Since much of today's technology was built with the aid of such advanced math, we might live in a much more primitive place if such mathematics were not allowed. Calbaer 16:39, 8 April 2007 (UTC)

1 and 0.999... occupy the same point though. In this case x is not the smallest possible distance between real numbers (as Huon explained this does not exist), it is no distance at all. Raoul 09:23, 9 April 2007 (UTC)

Yet another proof of 1 = 0.999...

Ohanian saids " I have a wonderfully elegant proof that .999... = 1 but this margin is too short for me to write it down, however I shall not be a Fermat person and WILL WRITE IT IN THIS DAMNED MARGIN anyway! "

Let $U={\mbox{unity}}=1\,$

Let $N={\mbox{nines}}=0.999...\,$

Proof by contradiction.

Assume that $N\neq U$. This means that there are only two cases that can follow.

CASE 1 : $N>U$

$N>U\,$

$\downarrow$

$\sum _{k=1}^{\infty }{\frac {9}{10^{k}}}>1$

$\downarrow$

${\frac {9}{10}}+{\frac {9}{100}}+{\frac {9}{1000}}+...>1\,$

$\downarrow$

Contradiction!!!

CASE 2 : $N<U$

$N<U\,$

$\downarrow$

$U+N<U+U$

$\downarrow$

$U+N<2U$

$\downarrow$

$(U+N)\cdot (U-N)<2U\cdot (U-N)$

$\downarrow$

$U^{2}-N^{2}<2U\cdot (1-\sum _{k=1}^{\infty }{\frac {9}{10^{k}}})$

$\downarrow$

$U^{2}-N^{2}<2U\cdot (\lim _{k\to \infty }{\frac {1}{10^{k}}})$

$\downarrow$

$U^{2}-N^{2}<2\cdot \lim _{k\to \infty }{\frac {1}{10^{k}}}$

$\downarrow$

$U^{2}<N^{2}+2\cdot \lim _{k\to \infty }{\frac {1}{10^{k}}}$

$\downarrow$

$U^{2}<N^{2}+2\cdot \lim _{k\to \infty }{\frac {1}{10^{k}}}<N+2\cdot \lim _{k\to \infty }{\frac {1}{10^{k}}}$

$\downarrow$

$U^{2}<N+2\cdot \lim _{k\to \infty }{\frac {1}{10^{k}}}$

$\downarrow$

$U<N+2\cdot \lim _{k\to \infty }{\frac {1}{10^{k}}}$

$\downarrow$

${\mbox{ }}$

${\frac {U}{2}}<{\frac {N}{2}}+{\frac {2}{2}}\cdot \lim _{k\to \infty }{\frac {1}{10^{k}}}$

$\downarrow$

$0.5<0.499...995+0.000...001$

$\downarrow$

Contradiction!!!

Now since both CASE 1 and CASE 2 results in contradiction, the only conclusion we can come up with is that 0.999... = 1

I rest my case. Ohanian 01:01, 5 May 2007 (UTC)

Case one begins with a statement that I don't think anyone ever supported, and then ends with a trivial rewording of the starting point. I don't see what the contradiction is, nor the point. Case two dies in step four when you multiply both sides by (U-N), which I believe is equal to .000...1. Many here maintain that .000...1 = 0; which kills any equation! (It's the secret behind the infamous 1=2 proof.) Even if .000...1 is not zero, it is still a form of infinity so you can't expect the normal rules of algebra to work when multiplying both sides of an equation by it. There was a proof above that made this same mistake, but I never got a chance to comment on it. (Been away a while, sorry.) Algr 02:41, 8 May 2007 (UTC)

You already know this, but... We are working here with the real numbers, which form an ordered field. Among other things, this means that for any numbers a and b, exactly one of the possibilities hold: Either $a>b$, $a=b$ or $a<b$. Ohanian establishes that U = N by showing that neither U < N (a necessary step, regardless of whether anyone "supports" it) nor U > N.

Under the assumption U > N, $U-N$ is positive, and as a consequence of the real numbers forming an ordered field, you can always multiply an inequality by a positive value while maintaining its truth value.

Note that I am not saying that the other details of his proof are flawless in my opinion. -- Meni Rosenfeld (talk) 07:31, 8 May 2007 (UTC)

You are NOT working with real numbers because you invoke infinity to make repeating decimals mean what you insist they do. Without infinity, .333... will never equal 1/3. Algr 17:13, 8 May 2007 (UTC)

Evidently, the term "real numbers" means something different to you. In mathematics, it is clear that 0.333.... is a (representation of a) real number. Moreover, your comment that multiplying both sides of an equation by zero "kills" any equation is simply false. And, more to the point, the second case of the proof was under the assumption that U - N is not equal to zero.

All that said, I don't see that the above proof involving U and N adds anything substantial to the discussion. Indeed, it seems to go off on a very strange tangent when he uses the "fact" that N/2 = 0.49999....95. We may all agree that 0.499999....95 is not the representation of any real number at all. Phiwum 17:33, 8 May 2007 (UTC)

If you do not like the fact that

(0.499999....95 is not the representation of any real number at all),

how about

$U<N+2\cdot \lim _{k\to \infty }{\frac {1}{10^{k}}}$

$\downarrow$

$U-N<2\cdot \lim _{k\to \infty }{\frac {1}{10^{k}}}$

$\downarrow$

$\lim _{k\to \infty }{\frac {1}{10^{k}}}<2\cdot \lim _{k\to \infty }{\frac {1}{10^{k}}}$

$\downarrow$

$0<2\cdot 0\,$

$\downarrow$

Contradiction!!!

The biggest problem with this alternative explanation is that someone

may not accept the fact that $\lim _{k\to \infty }{\frac {1}{10^{k}}}=0$

Ohanian 01:01, 9 May 2007 (UTC)

That's fine, but now the proof is rather more complicated than necessary. How is it any clearer than the simple observation that $0.999...=1-\lim _{k\to \infty }{\frac {1}{10^{k}}}=1-0=1$? Phiwum 14:07, 10 May 2007 (UTC)

The reason I used that long and complicated route is to avoid asking/forcing the skeptics to accept $\lim _{k\to \infty }{\frac {1}{10^{k}}}=0$. You have no idea how hard it is to get them to accept the concept of mathematical limits. They kept on insisting that the limit is NOT ZERO. Ohanian 22:56, 10 May 2007 (UTC)

Well, the long and complicated route you took didn't seem particularly persuasive to me. Sorry. Phiwum 23:40, 10 May 2007 (UTC)

Alternative algebraic proof

One common objection to the proof that begins

  c = 0.999...
10c = 9.999...

etc.

is that 10c 'really' = 9.999...0. You might short-circuit this reaction by the following proof:

        c = 0.999...
     c/10 = 0.0999...
   c-c/10 = 0.9
(1-1/10)c = 0.9
   9/10 c = 0.9
    0.9 c = 0.9
        c = 1

Whaddaya think? CroydThoth 20:10, 12 May 2007 (UTC)

Not bad, though I don't know that the proof should be made more complicated, especially since the "smallest infinitesimal" crowd would just say that dividing by ten (or equivalently, multiplying by 0.1) can't be done, or that "there's another '9' added to the end." The idea is to help people learn, not to "short-circuit" bad reactions. Perhaps, though, it's worth adding to the original proof as follows:

An equivalent proof is obtained by multiplying using any power of ten, e.g.,

        c = 0.999...
     0.1c = 0.0999...
   c-0.1c = 0.9
 (1-0.1)c = 0.9
    0.9 c = 0.9
        c = 1

<-- The above variant might help those who ask, "What about the last '9'?" -->

I'm curious what others might think of this change. Calbaer 05:26, 13 May 2007 (UTC)

Yes, of course you're right; the deniers would simply insist that 0.1 * 0.999... = 0.0999...9. So this alternative wouldn't 'short-circuit' anything anyway. CroydThoth 15:01, 13 May 2007 (UTC)

IEEE reals

Thus, "negative zero" in IEEE floating-point numbers is not a bona-fide negative zero.

The same argument can be used to support that positive zero in IEEE floating point numbers is not a bona-fide positive zero. And there is an argument for that, but then you have to talk about what IEEE floating point numbers are, and what they are not. I think this is best done in the article about IEEE floats itself; here it is only a distraction. Shinobu 04:27, 16 May 2007 (UTC)

I cut this as well:

In the case of IEEE floating-point numbers, negative zero represents a value that is too small to represent in the given precision but is, nonetheless, negative.

This is one way in which a negative zero might arise, but there are other ways (e.g. division of a positive value by negative infinity, or multiplication of a negative value by positive zero) and there are reasons you might choose to use it (e.g. an indication of which branch to take at a branch cut), so I think the claim is misleading. Gdr 20:55, 22 May 2007 (UTC)

If 0.99999=1....

Does 1.999=2? CJ 12:10, 12 June 2007 (UTC)