Talk:0.999.../Arguments/Archive 5

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Contents 1 A proposed solution to the conflict (for John Lattier and similar dissidents) 2 A message to John Lattier 3 Spoke with 2 professors 4 Deep thought 5 Rudeness by 41.etc 6 For the few who continue to reject the generally accepted definitions of mathemetics 7 Let's end this madness 8 Sources that 0.999... != 1 9 closest possible number to 1 10 Arguments irrelevant to question above 11 Attn: SCZenz and Calbaer 12 Infinitesimals 13 What should a number system satisfy? 14 Real number system does not have unique decimal representation 15 User:JohnLattier 16 How about this 17 An alternative way of defining the problem? 18 Just end it all 19 A FAQ for this page? 20 a new proof/disproof or just a plain huge error? 21 Smooth infintesimal Analysis 22 question 23 The Purpose of this Page 24 Definition of 'Limit', 'Sum of Infinite Series' 25 Conclusions, &c. 26 Yep, it's a question for John Lattier 27 Confusion about the definition of .9999...

A proposed solution to the conflict (for John Lattier and similar dissidents)

If (0.999...) is a real number, then it MUST be equal to 1, since there is no such thing as "the number prior to 1" by definition of the reals. Then, let's all agree that IF we define (0.999...) as a real, then it is equal to one. Then, if you choose to imagine your own number system, or if you choose to create a new number system, you can redefine (0.999...) there. Meanwhile, IN THE REALS, 0.999... = 1. Then we can put an end to this silly debate. Tparameter 17:26, 13 November 2006 (UTC)

Nonsense. Why must 0.999... = 1? There is no such number as the number prior to 1 in metric spaces. Contrary to popular (academic) belief, the reals are NOT a metric space. There is no such thing as the 'number before pi', is there? You don't know the limit of any pi expansion (you claim it is pi but cannot quantify it in by any means), yet you claim it is real because it is bounded above by a number you call pi. How are pi and 0.999... similar? They both cannot be expressed as rationals and both numbers are bounded, therefore both numbers are real. It is necessary that for two real numbers x and y, d(x,y) must be 0 if they are to be equal but it is NOT sufficient because the reals are not a metric space. As for definitions: If you define an infinite sum as being equal to its limit, then you better be consistent. Show me how pi is equal to some limit pi (you cannot represent it or quantify it in any way) and then I will accept that 0.999... = 1. Repeating fractions in certain radix systems do not respresent those fractions from which they are derived. Example: 1/3 is not equal to 0.333... in base 10 but it is exactly equal to 0.1 in base 3, exactly equal to 0.2 in base 6, etc. 0.999... (base 10) is exactly eqaul to 0.222... (base 3). However, both these numbers are strictly less than 1. If you fear calculus (particularly integration) falls apart, then you evidently have no idea what the Fundamental Theorem Of Calculus is about. Finding an area under a curve (evaluating an integral with or without infinite limits) means computing an average sum - you are not doing anything with 'infinity' as you so mistakenly think. Sending men to the moon does not require that 0.999... = 1. Whether you like it or not, arithmetic with irrational values in any radix system is always an approximation. 0.999... is irrational. The definition that states what an irrational is, applies to 0.999... otherwise it can't be real, can it? And yes, you can find infinitely manu numbers between 0.999... and 1. Neither does this violate any principle or the Archimedian property. There is sufficient proof of this in the archives. If you delete this post, I will post again and I will be extremely rude. 41.243.26.131 10:11, 30 November 2006 (UTC)

Just to reply a sec, you are incorrect in your introduction when you say above that "contrary to popular (academic) belief, the reals are NOT a metric space." The reals are a metric space, with the metric distance function d(x,y)=|x-y|. To see that d(x,y)=|x-y| is the distance function for the reals, and that the reals are therefore a metric space, consider the definition of a metric space:

A metric space is a 2-tuple (X,d) where X is a set and d is a metric on X, that is, a function

d : X × X → R such that

d(x, y) ≥ 0 (non-negativity)

d(x, y) = 0 if and only if x = y (identity)

d(x, y) = d(y, x) (symmetry)

d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).

Note that |x-y| meets all four criteria for a metric function on the reals. Specifically, for all real numbers x and y, |x-y|≥ 0, |x-y|=0 if and only if x=y, |x-y| = |y-x| and |x-z| ≤ |x-y| + |y-z|.

As to the rest of the paragraph, I'll point out that you are overlooking the Least Upper Bound property of the reals, from which you can prove that there are no non-zero infinitesimal numbers (the proof is in the article Archimedean property). If you create a new number system and remove that axiom, then you can potentially have an infinite number of infinistesimally small non-zero numbers, but that is not the standard definition for the reals. Dugwiki 16:06, 30 November 2006 (UTC)

You might want to reread my post Dugwiki - I stated that d(x,y) = 0 is necessary for reals but insufficient. A metric space does not accurately model or describe the real numbers. This debate is evidence of the same. Metric space theory attempts to describe reals in terms of the distance function and fails because real numbers do not have 'holes'. However, in distances, we can always find a location between any two locations. As for an example of a number between 0.999.... and 1, I suggested you read the archives as there are plenty of examples. I will state just one: Let x=99, then X/100+X/10000+X/1000000+... is between 0.999... and 1. You compare the partial sums, not the limits. 41.243.47.226 09:40, 1 December 2006 (UTC)

Again, the reals, as normally defined, ARE a metric space. What I posted above is the exact definition of what a metric space is. Specifically, a metric space is defined as a set X with an associated metric function d(x,y) such that d(x,y) obeys four specific properties (non-negativity, identity, symmetry, triangle inequality). In the case of the reals, the function d(x,y)=|x-y| obeys all four properties, and therefore the combination of the reals form a metric space under the metric function |x-y|.

Now, that being said, I think I understand you're complaint, which is a deeper issue. What you, and I think John Latier, take issue with is that you do not feel the existing axiom system commonly used to represent the real numbers actually reflects how you feel the real numbers SHOULD be reflected. For example, you presumably feel that the Least Upper Bound axiom for the reals, which states that every bounded subset of the reals has a least upper bound, is a bad axiom. The sort of number system you are proposing would have no guaranteed least upper bound for a bounded subset, but instead might have an infinite number of possible upper bounds that differ by infinitesimal amounts. Under THAT sort of system, it might even be possible that |x-y| fails to meet one of the properties for a metric function. Also under such a system, it might not be true the an infinite sum equals the limit of its partial sums.

The main issue, therefore, is that while you can probably construct such a system with no least upper bound property and it is apparently the sort of system you are promoting, it is not the system normally used to represent the real numbers. You might believe that your proposed system should be used, but it's not. In other words, the problem isn't that "you are right and everybody else is wrong", it's that you aren't accepting the same axiom system everybody else is using. The question of exactly why we're using the Least Upper Bound axiom and such is a fairly deep theoretical issue, though, and probably over my head. I'll simply leave it that the people who accept the Least Upper Bound property as a reasonable axiom logically conclude from that axiom that a) there are no non-zero infinitesimal numbers, and therefore b) an infinite sum is the limit of the corresponding series of sums in the reals. Dugwiki 17:07, 1 December 2006 (UTC)

LOL! If there is a number between .999... and 1, then it must be different than both of them. Then it must have at least one digit different than .999..., an infinite string of 9s. Whichever of these digits differs from 9 makes that number less than .999... This is simple arithmetic buddy. 72.193.74.36 15:55, 1 December 2006 (UTC)

So, can you please provide one number between 1 and .999... that is NOT a decimal followed by an infinite string of 9s? 72.193.74.36 16:28, 1 December 2006 (UTC)

Okay, you said, "yes, you can find infinitely manu (sic) numbers between 0.999... and 1". So, name just one. 131.216.2.83 17:55, 30 November 2006 (UTC)

If infinitesimal doesn't exist in real analysis, neither would 0.999... So either they are unequal because they are separate numbers or they are unequal because one is nonexistent. --JohnLattier 22:04, 13 November 2006 (UTC)

Agree Vantucci 15:05, 17 November 2006 (UTC)

Right. Reread my post. You can now say that "IF (0.999...) is a real, then..." and then add "but" you don't consider it a real. Do you see what I mean? This is key. Tparameter 23:26, 13 November 2006 (UTC)

First will you define 0.999... in the reals? --JohnLattier 00:00, 14 November 2006 (UTC)

This is unneeded for what I said. If you can simply recognize that **IF** (0.999...) is a real number, **THEN** it must be equal to 1, since there are an infinite number of reals between any two particular reals (non-equal). See what I mean? As for the definition, a long-bloated inelegant one is on the front page of this article. If you admit this minor concession, I promise to provide an elegant definition of 0.999... as a real number right here. Tparameter 01:08, 14 November 2006 (UTC)

If I have to agree to both "0.999... is among the reals" and "there are an infinite number of reals between any two particular non-equal reals," I can't. Either 0.999... is not among the reals, or there can be numbers not separated by infinite points. I can't agree with your ifs so I can't move to your conclusion. But if I had, what would be your elegant definition? --JohnLattier 03:35, 14 November 2006 (UTC)

There is an infinite number of reals between 0.999... and 1. Read the archives for examples John. The wiki puppets will tell you these numbers are 1 because their limit is 1. It all has to do with the problematic definition today's mathematicians have made that an infinite sum is equal to its limit. Isaac Newton had no idea what a limit is as it is known in today's terms. In fact, I am older than most people on this forum and can tell you that 25 years ago, it was understood that there is a clear difference between infinite sum and limit of infinite sum. NOt once in my entire education (and I have a BS) was it ever mentioined that 0.999... = 1. Make up your own mind. Again, contrary to popular belief, radix systems were designed to represent numbers uniquely. Just study the history of Roman Numerals and see what confusion they had... 41.243.26.131 10:21, 30 November 2006 (UTC)

I'm not sure what "puppets" you're referring to; it is fairly clear from writing styles, timestamps, aliases, other contributions, etc., that those posting under multiple aliases are you and JL (albeit both due to refusal to consistently use a username, not due to sockpuppetry). Anyway, either you are remembering your education poorly or you were educated poorly — I doubt you predate Augustin Louis Cauchy — because an infinite sum is the limit of finite sums and this has been well known for hundreds of years. And if we're going to be playing "I have a," I have a Ph.D. under one of the leaders of an applied mathematics subfield. But that wouldn't matter if I started to claim that the mathematics of the past few hundred years was wrong because I didn't like 0.999...=1. If you can prove it wrong or if think you have a useful, consistent alternative — and, as many have noted, there's not much consistency among "doubters" — write it up and get it published, but this is not the place for publishing new theories. Calbaer 18:09, 30 November 2006 (UTC)

It is you and others like you who are the problem. You stated that "because an infinite sum is the limit of finite sums and this has been well known for hundreds of years." - this is outright rubbish and it is evident your education was of an inferior quality but this is hardly amazing. There is not a single source or publication before 1960 that states clearly an infinite sum is the limite of its partial sums. You are misinformed and I pity those who are being taught by you! 41.243.47.226 09:46, 1 December 2006 (UTC)

Advanced Calculus by David Widder, page 285, from 1947 129.2.164.195 14:20, 1 December 2006 (UTC)

Introduction to the Theory of Fourier's Series and Integrals by H.S. Carslaw, page 49, from 1930.

Advanced Calculus by Sokolnikoff, page 6-7, from 1939.

The only two math books I seem to have from before 1960 - and they both have what you say "there is not a single source" for. It seems that YOU are in fact misinformed. 72.193.74.36 16:02, 1 December 2006 (UTC)

You said, "your education was of an inferior quality". I think the guy went to Stanford. If that's not good enough for you - then we're all in trouble. LOL! 72.193.74.36 16:11, 1 December 2006 (UTC)

Technically, I should have put "for well over a hundred years" to be safe; although the level of rigor with which we now examine this was found in the early 19th century, some might not consider the time period that has passed "hundreds of years." By the way, 72.193.74.36, you did indeed surmise my alma mater, while 41.243.47.226, I am currently not a professor or tutor so no one is being taught by me. As is often the case, the level of accuracy and logic of a contributor is reflected in historical and personal statements, not just in his or her mathematical analysis! Calbaer 20:22, 1 December 2006 (UTC)

Again, to 41.243.26.131, if you claim "There is an infinite number of reals between 0.999... and 1", than name one, so we can all be enlightened and graduate from puppethood. Just name one. 131.216.2.83 18:44, 30 November 2006 (UTC)

It seems to me that (0.999...) lies ON THE REAL NUMBER LINE somewhere between 0.5 and 1.5. Therefore, I conclude that it is a real number. Do you agree? Tparameter 04:09, 14 November 2006 (UTC)

If that's the only criteria, sure, but then so would infinitesimal, infinity, and any other infinity-defined number. --JohnLattier 04:29, 14 November 2006 (UTC)

It's obvious that you don't understand this simple fact, but I'll say it again. In calculus, we use limits to evaluate things related to infinity and other indeterminate cases related to 0 and infinity. In calculus, 1/inf is NOT an indeterminate form. It is easily evaluated using limits. Some examples of indeterminate cases:

1. 0/0

2. -0/-0

3. +inf/(+inf)

4. -inf/(-inf)

5. -inf*-0

6. +inf*+0

7. 1^(+inf)

8. 1^(-inf)

However, 1/inf is not one of them. lim 1/inf = 0. So, you can either reject calculus, or accept limits. Make your choice. Either way, (0.999...) IS DEFINED BY A LIMIT, and nothing can change that. You can make your own definition - but, that is original research, and doesn't apply here. Tparameter 14:12, 14 November 2006 (UTC)

How about L'Hôpital's rule? You can evaluate limits with indeterminate cases with that as well. Also, 0.999... is a real number. It isn't necessary forced to 1, because we're talking about real numbers, not integers. Real numbers encompass many different kinds of numbers. In real numbers, "the number prior to 1" could be 0.999...8. Because that is a real number, and it's less than 1, you could say 0.999... could possibly be prior to one. Of course, this sum:

converges to 1, so 0.999... really is 1. ~ EdBoy^{[p] [m]} 22:34, 27 November 2006 (UTC)

even though lim x->inf 1/x = 0, 1/x itself is undefined at 0. Anything with a vertical asymptote is undefined at the asymptote. --JohnLattier 02:45, 17 November 2006 (UTC)

I'm back. A workable way to say it might be "The concept represented by .999... is not supported in the set of real numbers. In this context it is considered to be equal to one." The square root of -1 is not a real number either, but it is nevertheless an important concept. BTW, infinity does not lie on the number line. It is next to it.  ;) Algr 09:06, 17 November 2006 (UTC)

Algr, that is a conclusion I can agree upon. --JohnLattier 16:37, 18 November 2006 (UTC)

Wouldn't infinity... Be the number line? I like to think of infinity as all numbers. *shrugs* ~ EdBoy^{[p] [m]} 22:34, 27 November 2006 (UTC)

John: What exactly do you mean by "Anything with a vertical asymptote is undefined at the asymptote." ? Consider the function defined by f(0)=0 and if x is not 0, f(x)=1/x. According to you does this have a "vertical asymptote" at x=0 ?

I would say yes that's what he's saying, since there is an asymptote there. Which brings me to a question for you guys. You say x≠0 for f(x)=1/x but you would say x=0 for f(x)=(1-(1/x)), since x=0 is the only way you'll get a value of 1. I'm not trying to be difficult, I'm really trying to see it, but I just see too many flaws and this causes my disbelief in .999...=1. Vantucci 18:14, 17 November 2006 (UTC)

I don't understand. is undefined for x=0, for the same reason is undefined for x=0. This does not mean we cannot easily prove things like . Am I missing your point? --jpgordon^∇∆∇∆ 02:59, 18 November 2006 (UTC)

Pardon me, but . Last I checked, this limit doesn't exist. Do you possibly mean ? --King Bee 19:18, 28 November 2006 (UTC)

A message to John Lattier

You are heavily outnumbered. I have a BS (Mathematics major) and do not agree that 0.999... = 1 for the very reason that I know Real Analysis is 'bunk' as one user commented. BTW: Real Analysis was one of the subjects I had to pass to obtain my degree. Fortunately, they did not ask me to prove that 0.999... = 1 because I would have had to respond with 'impossible' as an answer. I obtained first class passes for all my math courses. There is no doubt the limit of the series 0.9+0.99+0.999+ ... is 1 in my mind. However, that any mathematics breaks down (especially calculus) is absolutely rubbish. As an example, most Phds in Mathematics cannot state the fundamental theorem of calculus correctly, much less understand it or teach it. I know most of the individuals (or should I say pirranhas?) on this board and I can assure you and any one else that you are wasting your time. There are serious problems with real analysis: not only with definitions, but also with concepts. I have found the individuals in this discussion to be either highly ignorant or obnoxious and rude. I have tried all approaches and it is simply not possible to change the mind of someone who will not believe. At the conclusion of this discussion, these individuals will believe what they want to believe and what is socially acceptable in their academic circles. Any professor who dares to think otherwise would not be foolish enough to state an opinion because this would lead to he or she being ostracized from the community and in all likelihood end up without a job. The professors you spoke to do not know any real proofs. Evidence of this is in the fact they repeated the same illogical nonsense you find on this board. Sooner, rather than later, your status will be changed to troll or crank and you will be summarily dismissed. 41.244.196.64 06:13, 13 November 2006 (UTC)

Wow. Scary stuff there buddy. Hey listen, in your words "There is no doubt the limit of the series 0.9+0.99+0.999+ ... is 1". Well, now just relax for a minute and really try to think about this. Let me say it clearly. That is THE DEFINITION of 0.999..., period. It is the limit of that series, BY DEFINITION. It is a mathematical construct. It's not some intuitive thing that mathematicians stubbornly try to prove. It is a mathematical construct. Once you accept that, then you will understand. So, quit trying to deny THE ACTUAL DEFINTION. 72.193.74.36 15:16, 13 November 2006 (UTC)

The key word is limit. Let's look at some definitions.

1 Limit: a number whose numerical difference from a mathematical function is arbitrarily small for all values of the independent variables that are sufficiently close to but not equal to given prescribed numbers or that are sufficiently large positively or negatively

2 Limit: a number that for an infinite sequence of numbers is such that ultimately each of the remaining terms of the sequence differs from this number by less than any given positive amount

3 Limit: a number such that the value of a given function remains arbitrarily close to this number when the independent variable is sufficiently close to a specified point or is sufficiently large. The limit of 1/x is zero as x approaches infinity; the limit of (x − 1)2 is zero as x approaches 1.

4 Limit: a number such that the absolute value of the difference between terms of a given sequence and the number approaches zero as the index of the terms increases to infinity.

These all say pretty much the same thing. Arbitrarily close, difference less than any quantifiable amount, difference approaches zero. Hopefully this won't rattle people's brains too much, but 0.999... and 1 and 1.000...1 can all serve as the limit for the summation, based on the definitions. Does this mean each number is equal? No, it means they are arbitrarily close, with a difference less than any quantifiable amount, a difference approaching zero. I doubt this next sentence will rattle people's brains, rather just make them think: The sigma function that represents 0.999..., never ever ever reaches 1 for any value of independent variable. 1 acts as its limit but never acts as the output. So User 72.- please realize that 0.999... is not defined by its limit. Its limit is defined by it. --JohnLattier 22:27, 13 November 2006 (UTC)

I'm so glad someone else actually gets this. Vantucci 19:24, 14 November 2006 (UTC)

I'm so sad you and others misunderstand this. Tparameter 19:40, 14 November 2006 (UTC)

So hang on. Let me just ask you a question here. If 0.999... ≠ 1, does 0.333... = 1/3? Because that's a limit, too. And if not, can we define 1/3 in decmal? Edit: And if not, what does 0.333... equal, and wouldn't it be more useful to redefine the decimal expansion such that the limit is what counts and not the... whatever you say counts? That way we can define things like 1/3 precisely and we don't have any decimal expansions that equate to numbers we can't do maths with. Andrew 15:28, 14 November 2006 (UTC)

Techincally no .333...≠1/3, depending on who you ask. There was a comment made before about it not equalling because 1÷3 yields .3r1, or .33r1 or .333r1. No matter how far you take it there'll always be a remainder of 1 left over. Saying 1/3=.333... is throwing out the remainder thus not giving a true exact number, but rather an approximation. Vantucci 19:22, 14 November 2006 (UTC)

LOL! "Technically", huh? Actually, this is a technical matter. "Technically", 1/3 = 0.333... In fact, it is only by non-technical means that you disagree with that fact. Second, I agree that it "depends on who you ask". For example, if you ask any Mathematician, they will tell you that you are wrong. Tparameter 19:38, 14 November 2006 (UTC)

I have talked with people who have encountered mathematicians who do not agre that .999...=1. So be careful when you say any. Vantucci 21:12, 15 November 2006 (UTC)

Is this the new urban legend? Rather than believing a FOAF, how about a source for a mathemetician not agreeing that 0.999...=1? -- Schapel 21:22, 15 November 2006 (UTC)

Some of have trustworthy and dependable friends. Vantucci 15:01, 17 November 2006 (UTC)

Then go find those friends, and ask them to name the mathematicians. It's nothing to do with not thinking your friends are trustworthy, it's that Wikipedia does not accept as a legitimate citation, "I know a guy who knows a guy," for entirely obvious reasons. Maelin (Talk | Contribs) 09:13, 18 November 2006 (UTC)

Your division is incorrect: 1÷3 yields .3r.1, or .33r.01 or .333r.001. But assuming that indeed .333...≠1/3 because of some remainder: What's that remainder? Is 1/3-.333... = (1-.999...)/3? John would probably disagree. --Huon 19:44, 14 November 2006 (UTC)

As I understand it, 1/3 is exactly equal to 0.333..., but both hold a remainder with a value equivalent to (1/3)(1/∞). Thus 0.333...+0.333...+0.333... holds the remainder (1/∞) which is exactly the difference needed to push from 0.999... to 1. --JohnLattier 02:51, 17 November 2006 (UTC)

Huh? I thought 1-0.9999... was supposed to be the smallest positive number? And now we can divide it by three? If you no longer consider the "smallest positive number" a necessity, I again urge you to have a look at the hyperreals. They seem to satisfy most of what you expect.

On the other hand, maybe I misunderstand you completely. How is 1/3 supposed to "hold a remainder with a value equivalent to (1/3)(1/∞)"?

By the way, I find it vaguely reassuring that no two people disagreeing with 0.999...=1 agree with each other. --Huon 11:00, 17 November 2006 (UTC)

Any time you manipulate the infinitesimal or infinite in such a way like (1/3)(1/∞), or 2∞, it remains meaningless unless you can demanipulate it, as in 3*(1/3)(1/∞), or 2∞/2. That's exactly what 0.333...*3 does. -JL--68.158.207.15 06:14, 18 November 2006 (UTC)

"demanipulate"? What the heck sort of word is that? Is it a word that allows us to use divide by zero as long as we multiply by zero to cancel it out? --jpgordon^∇∆∇∆ 06:37, 18 November 2006 (UTC)

manipulate and "demanipulate": LOL! --Kprateek88(Talk | Contribs) 15:00, 18 November 2006 (UTC)

By Lattier's manip/demanip theorem, I can show that operations with 0 "remain meaningless".

Let x = 2. 0 * x = 0. Then we "demanipulate", x = 0/0. Hence, 0 is not "demanipulable", and thus it remains meaningless. QED. 72.193.74.36 16:24, 18 November 2006 (UTC)

That's great news, further proof that the infinitesimal doesn't equal 0. --JohnLattier 16:32, 18 November 2006 (UTC)

Um, how are we disagreeing? The comment he just made was the first comment we did not agree on. I do not believe 1/3 has a remainder. 1/3*3 does equal 1, but I do believe he is correct with .333... having a remainder. 1/3 cannot be properly displayed as a decimal. Vantucci 15:01, 17 November 2006 (UTC)

disagree: to fail to agree. Huon's point is that no two people who think that 0.999... should be 1 can seem to agree on consistent properties for a number system to have, just as they can only cite misinterpreted sources from nonmathematicians and, worse, "a friend of a friend." No consistency, no reliable sources, the claim of being able to divide the indivisible — it seems one just can't avoid 0.999...=1 and have a sensible number system. Calbaer 18:01, 17 November 2006 (UTC)

I have not tried to come up with a number system, so that isn't a disagreement with JL. I don't believe anyone is trying to prove another number system, just that .999... doesn't belong in the reals. No consistency? when have I not been consistent? No reliable sources? That's a relative term. I would take the mathematical capabilities of some autistic kids over some proclaimed mathematicians, because the mathematically capable autistic kids simply do the numbers and understand them instead of trying to be known for something whether it makes sense or not. As JL mentions using infitesimal numbers doesn't work in the real system, thus why you can't use them in the real system. But yet, all the believers in the Real system, have no problem using inf to define the limit of numbers. Just makes no sense. Vantucci 18:41, 17 November 2006 (UTC)

Well, you're mixing apples and prunes here. "Infinity" is not a real number; and non-zero infinitesimals are also not real numbers. Infinity is a useful things to use to describe the behavior and properties of real numbers, as is the concept of infinitesimals. One does not have to believe the real system, anyway; it's just useful to understand it if you want to be talking the same language as, well, anyone who uses numbers for pretty much anything. --jpgordon^∇∆∇∆ 03:12, 18 November 2006 (UTC)

So if non-zero infitesimals are not real numbers, then .999... shouldn't be a real, right? Vantucci 22:44, 18 November 2006 (UTC)

Wrong. They're not needed to construct .999... at all. Here's a generating function approach: It can quickly be shown that for any n, g(n) is a real number, as it is the sum of two real numbers, for any value no matter how big -- in other words, to infinity. --jpgordon^∇∆∇∆ 23:11, 18 November 2006 (UTC)

The reason it doesn't make sense to you is because you haven't formally studied calculus. If you understood limits, this whole thing would make perfect sense. If I went and sat in some graduate nuclear physics class, what they were talking about might not "make sense" to me - but, I would be forced to be humble and recognize that I have no basis to argue. So, if my intuition told me something was wrong, then I would need to seek education in these matters to confirm or deny my suspicion. It wouldn't be enough to tell the professor he was wrong because of some intuitive feeling that I imagined or got from some imaginary "autistic kid". Beyond that, you should try to recognize that in math when something is defined or proved to be true, it's over. There is no more debate. Everything is built up from axioms and fundamental definitions, and there is a direct logical path. That is why this topic is not debatable in the context it is defined. 24.234.150.195 22:02, 17 November 2006 (UTC)

Last I looked at my transcript I had calculus and much further in college, since it's required for engineers. And as far as I remember, they never taught us that a limit is EQUAL to a summation. Have I ever tried to say that any of you are not educated in Calculus? Didn't think so, so I'd apreciate it if you don't claim that I am not, because you're wrong. Vantucci 22:44, 18 November 2006 (UTC)

Last I looked, , i.e., a limit is EQUAL to a summation. Sue your Calculus professors. 67.77.153.56 22:54, 18 November 2006 (UTC)

I was not referring to you specifically but the "doubters" as a whole. And Wikipedia:Reliable sources is less relative than you may think. FOAF is, IMHO, the least reliable source possible; at least with a confirmed crank, you know they're probably wrong. In order to deny 0.999...=1, you have to work within a number system of some sort, but trying to make 0.999... a different number than 1 results in undesirable properties that make a number system impractical, unintuitive, and/or inconsistent. By the way, the extended real number line is a valid system to use, but, if you don't like it, infinities can usually be avoided; Huon sketched a formal approach that avoids them, for example. Calbaer 19:59, 17 November 2006 (UTC)

I didn't do the division wrong, I was just displaying it wrong, thanks for that. I was displaying it as in I was still doing the math under the division sign. Anyways... I would agree that 1/3-.333... = (1-.999...)/3. Vantucci 21:12, 15 November 2006 (UTC)

Right. so if 0.333...≠1/3, please tell me the "correct" decimal expansion for one third. And if there isn't one, then please tell me why we can't define the decimal expansion such that 0.333... is one third. Andrew 13:37, 15 November 2006 (UTC)

Understood --JohnLattier 13:34, 13 November 2006 (UTC)

Spoke with 2 professors

Like I said I would, I engaged in email correspondance with two professors regarding 0.999... and 1. We were unable to come to any sort of agreement. I could see through the proofs they suggested. I'm happy they responded, and in case they read this: thanks. But I was disappointed not to come to any kind of conclusion. The only new thought I gained from the experience is that perhaps both 0.999... and 1 can be the limit for the same summation because as the number of summations increases, both 0.999... and 1 are approached and neither are reached. That still does not mean they are the same exact number. In the end, I of course am aware the professors I corresponded with are of very high intelligence and far superior to myself in mathematics, but I remain of the mindset that a majority of even the most intelligent individuals can still be wrong (aka Big Bang = beginning of time and matter; dimensions are bendable objects; string theory; many philosophical topics). It's an interesting debate, really shows a lack of understanding of infinity/infinitesimal/limits in our current culture, and a need for more work. This is one of the rare debates where the uninformed commonfolk, using dictionary definitions and human intuition, while not necessarily having a sharper understanding of the situation and being easily swayed, have more accurate immediate impressions as to the identity of the infinitesimal because of less exposure to mathematical trickery. --JohnLattier 18:52, 8 November 2006 (UTC)

Of course you could not come to an agreement with math professors. They told you that you were wrong, and you obviously don't agree! Do you really think two independent professors both gave you bad proofs? Could it possibly be that you're simply incorrect? Keep working on it! -- Schapel 19:00, 8 November 2006 (UTC)

John: You say both 0.999... and 1 can be the limit for the same summation because as the number of summations increases, both 0.999... and 1 are approached and neither are reached. That still does not mean they are the same exact number. Not true. Any real sequence cannot have two distinct limits (this is quite easy to prove). You have indicated several times that you want to work in some number system different from real numbers, and have implicitly stated some desired (by you) properties of that system. Please give a formal construction of the number system you want to work with. Others have also raised this point, but you have never replied to it. --Kprateek88(Talk | Contribs) 11:25, 9 November 2006 (UTC)

A formal construction? I would make no changes to the common number system, as real analysis has done. Every point on the numberline is represented by a zero-dimensional number. Between any two finite points, there are an infinite amount of numbers. From the reference point of a finite number, one can represent the neighboring number as a ...999... By definition there are no numbers located in between a finite reference point and its neighboring infinite decimal. I've said it all a thousand times, I don't know how many times you want me to repeat myself, or what exactly you expect me to say. I would make no changes to the universally understood number system I have just described. I would just shy away from the presumptuous real analysis postulates. --JohnLattier 13:08, 9 November 2006 (UTC)

You are going on in circles again. What do you mean by "common number system"? For most people it would be the real numbers. For you evidently it isn't. You have not defined your number system. number system I have just described No, that is a very vague description of a supposed "number system". If your system (whatever it is) is "universally understood" as you claim, could you please cite some sources where it is described in detail? Does base 10 have any special role to play in your number system? If we were born with six fingers on each hand would your system be different (ideally it shouldn't)? --Kprateek88(Talk | Contribs) 13:59, 9 November 2006 (UTC)

They used versions of the x=0.999... so 10x=9.999... etc. proofs. I do not agree that 0.999*10 can give a meaninful answer. I would see it as 10x<9.999... How much less? not a quantifiable amount rather an infinitesimal amount. --JohnLattier 19:05, 8 November 2006 (UTC)

• (a) There are no non-zero infinitesimal in the real numbers. (b) Both 0.999... and 10 are real numbers; the real numbers are a field; hence multiplying the two numbers gives a meaningful answer. Keep working on it, though. --jpgordon^∇∆∇∆ 19:59, 8 November 2006 (UTC)

Ask your self: who has more right? the two math professors, or you? _→AzaToth 20:01, 8 November 2006 (UTC)

A good friend of mine, have suggested that you should contect Gene Ray about your ideas. _→AzaToth 20:07, 8 November 2006 (UTC)

Well, I'll give him credit for at least admitting that he could find no one with any mathematical credentials who would agree with him. Tell me, did both professors seem to have a familiarity with the concept, or did they arrive at their proofs independently?

Anyway, let's get this straight: You, JL, believe that the "correct" number system would:

• Not have the same axioms as the real numbers
• Not have a closure property (which exists for real numbers under multiplication), in that there exist two numbers, 1.999... and 0.5, which do not have another number as their multiplicative product. (In your words: "It's illogical to add, subtract, multiply, divide, square, etc. a number with infinite decimal digits.")
• Have (unlike real numbers, in which all three values are equal)

Yes? I mean, there's no sense trying to convince you you're "wrong," but at least let others — those who recognize and don't recognize 0.999...=1 — know what you believe. This will help others to recognize whether or not they agree with you. Calbaer 20:22, 8 November 2006 (UTC)

• Have " this tells me you are way off. --JohnLattier 00:22, 9 November 2006 (UTC)

Just when I think you're seriously engaged, you go back to refusing to answer questions about your version of what mathematics should be. Sorry I wasted my time. Calbaer 00:45, 9 November 2006 (UTC)

John, having varified that your ideas are *without any doubt* Original Research, does this mean Wikipedia won't be hearing any more from you? Untill you're published, that is? —The preceding unsigned comment was added by 210.54.148.98 (talk • contribs) 20:29, 8 Nov 2006 (UTC)

John, what good is a number system if we can't even guarantee that we can multiply numbers? What good is a number system in which perfectly well-defined calculations are "illogical"? What good is a number system where before we do some calculations, we have to come and ask you, personally, JohnLattier, whether each step we perform is "logical"? You certainly haven't given any formal rules nor explanations about why certain calculations are "illogical", as you claim.

You want to reject huge, deeply-developed branches of mathematics because they don't sit with your intuitions about 0.999.... If mathematicians had always rejected ideas because they didn't square with intuition, we'd never have discovered negative numbers, we'd never have discovered irrational numbers, we'd never have discovered imaginary numbers. If scientists in general did it, we'd never have realised that the Earth was spherical, we'd never have formalised gravity, we'd never have discovered relativity. We'd certainly never have come up with quantum mechanics. Your inability to accept ideas despite all the evidence because they are mildly counterintuitive, and your refusal to formalise your objections to them, have no place in mathematics nor any other science. Maelin (Talk | Contribs) 01:16, 9 November 2006 (UTC)

• Or, perhaps more tersely: Common sense is that sense which tells you that the earth is flat. --jpgordon^∇∆∇∆ 01:27, 9 November 2006 (UTC)

Deep thought

I am copying here this question posted by 137.240.136.81 on the main page, since I read it to be a good faith query by the user, and not simple vandalism:

"Deep Thought - How do we get .333...? 1/3 yes? Let's go back to our simple division days. 1/3=.3 with a remainder of 1, or .33 with a remainder of 1, or .333 with a remainder of 1, etc. If we only designate it as .333... what happens to the remainder of 1? It is lost, just gotten rid of, thrown out the door. This means that .333... can't equal 1/3. You must have the remainder of 1 in there and we have no way to represent it. Please do not delete this section unless you can account for where this remainder goes."

My short answer to the query is as follows: It is true that 0.333...3, i.e. the decimal expansion with a finite numbers of 3's is not equal to 1/3, precisely for the reasons stated above. However 0.333..., with an infinite number of 3's, is exactly equal to 1/3. The reason for the latter relies on the correct interpretation of what 0.333... means in mathematics (rather real analysis) and is explained in the Calculus and Analysis section of the main page, or any basis analysis text. Anyone care to expand? Abecedare 18:20, 6 November 2006 (UTC)

You're both making good points. Abecedare is right that 1/3 does equal the infinite decimal 0.333..., whereas the original poster is making an interesting point about remainders. While the phantom remainder is not altering the 0.333...=1/3 equality, it plays a role in why 0.333...*3=1 rather than 0.999... --JohnLattier 18:04, 8 November 2006 (UTC)

An infinite decimal expansion is more formally defined as the infinite sum of its decimal fraction components. Specifically:

Under that definition, . So assume that . Then

• 
• 
• 
• 
• 
• 
• = 3 + y
• Thus 10y = 3 + y
• 9y = 3
• y = 3 / 9 = 1 / 3

Therefore . (I appologize for the weird format at the end. I'm not very good with the alignment here. Maybe someone can clean it up?) This proof avoids the informal use of "remainders" to calculate the real number value that 0.333... represents. Dugwiki 21:54, 8 November 2006 (UTC)

I realized you might instead be wondering about more formally deriving the 0.333... expansion from the fraction 1/3. I'll use a system similar to long division, but using more formal notation:

• 
• 
• 

And so on. At each step N of this process, you have an equation of the form:

• 

So, as N approaches infinity, you get

• 
• 
• 
• 
• 

Thus, informally, the "remainder" term in each step of the long division equations approaches zero as the number of steps approaches infinity. Thus that part of the expression vanishes to zero for the corresponding infinite series. Dugwiki 22:35, 8 November 2006 (UTC)

What is with everyone's tendency to equate "approaching zero" and "equaling (or vanishing) to zero?" Sure, if you are going to redefine the infinitesimal as zero, your 0.999...=1 will suddenly and magically appear true. --JohnLattier 13:10, 9 November 2006 (UTC)

We don't "redefine" it, we observe that in the real numbers it doesn't exist unless it's zero, due to the Archimedean property. You are repeating your mistakes after we have told you why you are wrong. Either listen, or go away. This article concerns the reals. There are no nonzero ininitesimals in the reals. They don't exist. We haven't redefined them, they just don't exist in the set, just like there are no fractions in the integers. Accept it. Maelin (Talk | Contribs) 13:18, 9 November 2006 (UTC)

Well, let's see, you follow a number system that sets the infinitesimal equal to zero. I'd say that's quite a redefinition. The Archimedean property just states that the infinite is not finite therefore it does not exist. I would venture to say whomever devised the Archimedean property had a real tough time understanding infinity and infinitesimals. Perhaps that disability is a not uncommon aspect of the human brain? I would not say common, since most people I have talked to understand 0.999... isn't 1. It sounds like real analysis is a type of brainwash. Why people so aggressively defend something that is only supported by its own redefinitions is astonishing. Listen, if you are going to make the infinitesimal equaling to zero as one of your unproveable definition foundations, why even have this article? It shouldn't suprise you so much that your 0.999... equals 1 if your infinitesimal equals zero. I'll welcome you all back to the real real number system when you're ready. --JohnLattier 14:12, 9 November 2006 (UTC)

FYI, the article Archimedean property provides a proof in the first section that there are no non-zero infinitesimals in the real number system. For reference, the proof is as follows:

The non-existence of nonzero infinitesimal real numbers follows from the least upper bound property of the real numbers, as follows: The set Z of infinitesimals is bounded above (by 1, or by any other positive non-infinitesimal, for that matter) and nonempty (because 0 is infinitesimal); therefore, it has a least upper bound c. Suppose that c is positive. Is c itself an infinitesimal? If so, then 2c is also an infinitesimal (since n(2c) = (2n)c < 1), but that contradicts the fact that c is an upper bound of Z (since 2c > c when c is positive). Thus c is not infinitesimal, so neither is c/2 (by the same argument as for 2c, done the other way), but that contradicts the fact that among all upper bounds of Z, c is the least (since c/2 < c when c is positive). Therefore, c is not positive, so c = 0 is the only infinitesimal. (The Archimedean property of real numbers holds also in constructive analysis, even though the least upper bound property may fail in that context.)

Thus the real number system has no non-zero positive infinitesimals. It is possible to construct other systems, such as the hyperreals and surreal number systems, which do not possess the least upper bound property and can have strictly positive infinitesimals. But when talking about the real number system, there aren't any. Dugwiki 16:07, 9 November 2006 (UTC)

But I disagree that you can even perform 2c. Or c/2. I challenge that. It has no meaning. Infinitesimal*2 should=error. (0.999...+1)/2 for example should = error as well. If it does not, why not, and what system is claiming these equations are valid? --JohnLattier 22:39, 9 November 2006 (UTC)

• Certainly not the real numbers, and that's the realm we're discussing here. There are no numbers in the reals that cannot be added, subtracted, multiplied, and divided (except by zero); see Field. There are no infinitesimals in the real numbers. --jpgordon^∇∆∇∆ 23:08, 9 November 2006 (UTC)

Rudeness by 41.etc

41.etc. is clearly contributing nothing to the discussion at this point, and is essentially just trying to waste our time. Therefore posts that are primarily insults should be reverted on sight rather than responded to. I've blocked the current IP for 3 hours for this behavior, but it appears the user hops IP's and will be back. -- SCZenz 17:16, 9 November 2006 (UTC)

Well, it's his abuse and reverting that are the main problem. Others are contributing little to nothing, but they're not abusive so we aren't blocking them. Calbaer 17:35, 9 November 2006 (UTC)

You are correct. It is the abuse that makes his contributions unacceptable, and which justify removing/blocking on sight if they continue as they have. -- SCZenz 17:54, 9 November 2006 (UTC)

For the few who continue to reject the generally accepted definitions of mathemetics

• The infinitesmal is a construct invented to help in the development and understanding calculus. However, for calculus to work, a nonzero infinitesmal can't exist, and, since calculus is the basis of most advanced applied mathematics, it doesn't exist in practice.
• You can't say real analysis is bunk without bothering to learn it. Read a book or take a class. Make sure you finish it, too.
• You can't propose a new number system on Wikipedia. If you reject all referenced number systems and lack a reference to the number system you're discussing, you are proposing a new number system. Read Wikipedia:NOR and find a reference defining the number system you're referring to, or accept that your intuition and the mathematical systems in use contradict. Move on to quantum mechanics, special relativity, the Monty Hall problem, and other topics which contradict intuition but are still true. Calbaer 17:35, 9 November 2006 (UTC)

The infinitesimal does NOT exist. You cannot define junk and then claim it to be true. 41.243.26.131 10:23, 30 November 2006 (UTC)

Let's end this madness

41.243.17.117 (talk • contribs) is obviously 198.54.202.254 (talk • contribs), who has been trolling the article under different IPs for a year now. If he is not already blocked, he should be. Should he return with a different IP, it should be blocked as well.

JohnLattier continues to test our assumption of good faith. While I cannot say that he is clearly a troll/crank, he certainly seems either incapable or unwilling to participate in an intelligent discussion here - especially seeing that he refuses to respond to (or even read) some very sharp replies to his complaints. I suggest we simply stop responding to him. If he still has any question after a (fairly long) period of cooling down and study, we will be happy to answer.

It is against the best interests of everyone to respond to the troll(s) that roam this talk page. -- Meni Rosenfeld (talk) 21:09, 9 November 2006 (UTC)

I'll go further: this page is a failed experiment and should be closed down altogether. Future non-editorial discussion on the main talk page can be moved to the refdesk. Melchoir 21:33, 9 November 2006 (UTC)

I don't agree. This page has very well served its (unstated) purpose, which is to ghettoize the cranks and keep them out of sight, or rather out of the sight of everyone except those foolish enough to venture here. Personally I really don't want them at the reference desk. A few of us can volunteer to come argue with them here occasionally, on a rotating basis, to keep them happy and out of everyone else's hair. --Trovatore 21:37, 9 November 2006 (UTC)

You allude to one of my worries: this page is occupying too much of the community's effort. I also worry that it isn't quite out of sight enough. Melchoir 21:54, 9 November 2006 (UTC)

I think the small number of people actually contributing this page can determine for themselves how much of their effort is too much. It's recreational typing, nothing more. --jpgordon^∇∆∇∆ 22:01, 9 November 2006 (UTC)

Excuse me, you said "seems either incapable or unwilling". Should you edit that to read "IS incapable and/or unwilling"? Tparameter 18:47, 11 November 2006 (UTC)

I make attempts to respond to everything, but being severely outnumbered, I just cannot keep up. And I'm sure most of you think I've had too much input. If there is a particular comment or question you had, place it here and I'll try to respond. Wikipedia discussion boards are not ideally organized, so I don't really know where questions have been asked of me. I have very little time, and am frankly spending more than I should visiting this discussion board. This board is just as you describe it, a place to imprison the unconvinced into obscurity. But it would be a shame to mute their thoughts altogether. Even I, for example, am still trying to put the pieces together, and am learning much from visiting. We should have a common goal of trying to, at least, decipher what the root of this debate is. That's the main thing I've been trying to do, and I'd similarly like to hear what you all think is the root of the debate, not just with myself but with the population as a whole. --JohnLattier 22:34, 9 November 2006 (UTC)

The root of the debate is that you only have pieces, yet you declare all other nonanons — most of whom have bothered to formally learn the topic — wrong. If you only have pieces, how could you be so convinced that you're right? An infinite sum equals its limit (if the limit exists), nonzero infinitesimals do not exist in standard mathematics, and you won't accept these results/properties, so what is there to debate? There is no debate if you won't accept the standard axioms and conventions of mathematics, nor the results that arise from these axioms. Good faith or none, there is just nowhere to go from here. The article page should continue to say that this is a result of mathematics and you can continue to think that mathematics is based on a faulty premise. The end. Calbaer 23:11, 9 November 2006 (UTC)

I think the main thing in your case, John, is that you want to introduce strictly positive infinitesimal numbers, but this disagrees with the standard accepted axioms of the real number system, which include that the reals are complete. This completeness property forces the reals to have the property that all bounded subsets of the reals have a least upper bound that is a real number, and this in turn forces the reals not to have any infinitesimal numbers greater than zero. (The article Archimedean property provides a proof for reference.) The only way to introduce non-zero infinitesimals into the system is to remove the completeness axiom and use a different number system such as the hyperreals, which is related to the real number system but is not identical.

It's much like discussing non-Euclidean geometry versus Euclidean geometry; both sorts of geometries are internally consistent and are closely related, but they aren't identical and USUALLY (not always) when people talk about geometry they are by default talking about Euclidean geometry. Similarly, while both the reals and the hyperreals are consistent and somewhat similar, it's almost always the case that when mathematicians talk about "reals" they mean the real number system and not something like the hyperreals.

Anyway, at this point, if you're still not convinced, I'm not sure I can do much else. But I will admit that this debate gave me a chance to brush up on my Wiki math formatting and fiddle around with some fun proofs. :) Dugwiki 23:23, 9 November 2006 (UTC)

I think it's a good deal worse than that he wants to introduce infinitesimals (what's this "positive infinitesimal" jazz? Zero is not an infinitesimal in standard usage.). The problem is that he has a very strange infinitesimal that doesn't seem to be good for anything. You can't divide it by 2, for example. You can subtract "the infinitesimal" from 1/2, giving 0.4999..., but you apparently can't subtract it from 1/3. Numbers whose decimal expansions end in 000... have these ghostly duplicates infinitesimally below them, but other numbers don't, and the ghostly duplicates don't enter into general arithmetic (can't divide them by 2). There's no sensible explanation, for what you can and can't do with them, that doesn't make specific reference to base ten. --Trovatore 00:14, 10 November 2006 (UTC)

Just to reply a sec, the reason I said "strictly positive infinitesimals" is to keep the argument clear to anybody who might consider zero an infinitesimal real number. Depending on what system you're using you can consider zero a trivial case of an infinitesimal number, since it meets the criteria that for all numbers n, n x |0| <= |c| for any given constant c in the set of numbers. Therefore whether or not you want to include zero in your list of infinitesimal numbers will depend more on convenience and aesthetics, since it does meet the primary basic property that all multiples of it are less than or equal to all other numbers in the set. Dugwiki 16:01, 10 November 2006 (UTC)

That's a good point. What is the base 10 equivalent of a base 3's 0.1222...₍₃₎ versus 0.2₍₃₎? Since we can't divide "...", the logical conclusion would be that we've found a real number that would no longer exist if we counted in base 3! Anyway, I think we all can agree there are more problems than just a refusal to accept the Archimedean property. It's that those who don't like 0.999...=1 are rejecting real arithmetic without presenting an internally consistent alternative. Calbaer 01:37, 10 November 2006 (UTC)

Melchoir: This page serves both as a lightning rod for trolls, and as a place where people with genuine questions can have answers. Seeing that this is a subject that gets more questions than most others, and answers to them are often rejected, it is a good idea to separate it from the reference desk (there's no room there for such long debates).

Trovatore: The definition I know is that an infinitesimal is a number which cannot be multiplied by an integer to obtain a number larger than any given integer. According to it, 0 is an infinitesimal.

John: Okay, I'll try this one more time. While I will not go into as many details as my previous replies, I will present here a few things you need to realize. If you do not accept something of these, please state what and why.

1. Keeping in mind Trovatore's advice against "escaping to formalism", I still find it important to clarify that over the course of mathematical study, it has been observed that the only way to do mathematics without introducing errors and contradictions, is to start with some axioms and definitions, and rigorously deduce theorems that follow from them. We then obtain a certain mathematical structure or theory, which can be used to model real-life situations, or to construct other mathematical theories. The question of whether a given structure is suitable to model some physical phenomenon or not, is debatable; The question of whether a given theorem holds in some structure is not detable once the theorem has been proven. A rigorous proof is a technical thing which consists of a sequence of statements, each of them justified by technically applying an inference rule to some statements which we have previously proven or taken as axioms. This is so technical that a computer can check if a proof is valid or not (the hard part is actually finding a proof). The interesting part is choosing a set of axioms which lead to beautiful and useful structures, and interpreting the real-life meaning of the validity of the theorems.
2. An example of such a structure is the real numbers (the specific axioms which lead to it are in that article). Note that:
   1. It is evidently extremely useful in modelling the physical universe (both nature and human life) and in constructing other mathematical theories (which themselves are both beautiful and applicalble to real life). It is no exaggeration that probably, no branch of science would have gotten very far without them.
   2. It is regarded as the rigorous and consistent stucture which most closely resembles what people intuitively think of when they talk about "numbers".
   3. A convenient notation for representing real numbers is decimal expansions, which are based on the number ten (picked solely because we have 10 fingers on our hands). Similar notations can be used where ten is replaced by some other integer larger than 1, and some significantly different notations can be used as well. The important thing is that this notation can be defined precisely and its properties can be studied and proven rigorously.
   4. One such property is that there can be two decimal expansions which represent the same real number. For example, both 0.999... and 1.000... (these are just shorthand for the precise notation) represent the real number 1. Again, this rigorously follows from the axioms of real numbers and the definition of the decimal expansion.
   5. Therefore, the claim that "in the structure of the real numbers, 0.999 ≠ 1" is simply wrong, just as the claim "in the structure of the real numbers, 4/2 ≠ 2" is wrong. You cannot expect to be taken seriously if you make such a claim.
   6. The reason you believe this claim is that you think that decimal expansions have a deep meaning on their own, and let your intuition guide you to concluding that two different expansions must represent two different numbers. This is wrong. Decimal expansions are just a tool for analyzing real numbers.
   7. On the other hand, the claim "the structure of real numbers is not as great as people say it is" is legitimate, but in light of the above, it is rather dubious. If you make this claim, you should accompany it with some very good reasons (since there are many extremely good reasons for the contrary).
3. You keep saying that the real numbers "redefine" something. They do not redefine anything. Strictly speaking, there isn't such a thing as "redefine"; Even if we accept a loose interpretation of the term, the real numbers are pretty much the "original" definition. Can you provide a precise definition for the "original" term that the real numbers "redefine"? I guess that you cannot, because you do not have any precise notion of "numbers" but only a lot of confidence in your intuition.
4. You seem to think very little of experts. It is of course true that experts are neither ominiscient nor infallible; But they know much more about their field than a layman, even an intelligent one (that's why they're called "experts"). Denouncing something that every expert in the world agrees on requires more than just handwaving and intuition. This of course applies to our current debate (a layman usually just doesn't know what mathematical rigor means) and to anything else. To take your example of the big bang, there is no contemporary theory that states that the big bang was a local event. There is the big bang theory, which states that it was the beginning of space and time (at the very least, for the current cycle of the universe). As a physical theory, it can have varying degrees of accuracy, and can be completely wrong - however, there does not currently seem to exist any other explanation for the universe. A layman may have been taught at school that it was just some local event, because this is easier to visualize and agrees with the intuition. An expert, on the other hand, will know exactly what the theory says and all the evidence which supports it.
5. You have earlier presented a list of questions about infinitesimals. I have already replied to it there. Please read it, and respond here (to keep the discussion in one place) if you have any other questions.

That's enough for now. I'll be happy to respond if you have further questions. -- Meni Rosenfeld (talk) 10:52, 10 November 2006 (UTC)

Thank you for addressing my concerns. I agree with #1, but I don't have the higher mathematical training to produce a rigorous proof. My reasoning goes as follows. I start with a foundation that all numbers are represented by one and only one decimal expansion. Excluding any zeroes you may add. As it is a foundation, I don't see any way to prove it right or wrong. Next, I hold the definition of infinitesimal as being the unit of distance between any 2 zero dimensional points on a one dimensional number line. To visualize this, we must first start at a point, then move 1 infinitesimal length from it. This is only doable by first starting with a finitely defined number. I don't have a well-defined system of manipulating the infinitesimal (add, multiply, etc.) yet. I strongly suggest one is made, sometime. I think part of our limitation in performing some of these equations like 0.999...^2 is that we do not have a notation defined to handle such answers. And if we did have a notation, we would unfortunately face further infinities, to the point where it would be unwriteable. Anyway, I don't know where to look for well known axioms / foundations about numbers or infinities.

2-1: I would like to know some examples about how setting infinitesimal equal to zero (and therefore any two numbers infinitely space to be equal) provides us with beneficial physics or mathematic output. I have taken the calculus courses, and I understand well the concept of a dy/dx. I agree that the limit used for this is 0, but that does not make the infinitesimal equal to zero.

2-2: I still have not decided if 0.999... should be considered a number (though I have called it a number up to now) OR a concept. While it would hold a theoretical place on the number line, it is not a pinpointable number. It is not something I think exists outside of our own constructs.

2-4: Which axioms does it follow from? I have seen many, and so far I disagree with every last one I have seen. I have not yet found any conclusions about this that logically flow from the axioms.

2-7: Like I've said, I do not think the real numbers accurately represent what society has commonly considered the infinitesimal (or 0.999... which is 1-infinitesimal). Most importantly, it cannot be denied that there is such a concept as a smallest number greater than zero. Even if the number itself does not exist or cannot be proven to exist, the concept still does. The concept exists because I just said it, I have the concept in my mind, as do many people, as do you I presume. That concept is commonly called the infinitesimal (just as 0.999... is commonly called the largest number less than 1). Now, if real analysis were to set the infinitesimal equal to zero, that does nothing to erase the concept from my mind. The concept, a smallest number greater than zero, *needs a name.* I call it the infinitesimal, because I am not convinced that the proofs against it are mathematically sound as they use manipulation of the infinitesimal in ways that produce meaningless results.

3: I have checked many sites looking for the definition of infinitesimal. I have come up with a variety of responses. Ranging from strictly greater than zero to strictly equal to zero. I'd venture to say the definition is not unanimous, which is not a good thing. If an infinitesimal is defined as zero, then just call it zero, and let's find a name for the smallest number greater than 1. I nominate "infinitesimal."

4: No, I respect experts. I only acknowledge that there is occasionally, not often, a tendency for a scientific theory to be prematurely raised to universal fact. As I mentioned the Big Bang Theory, I hold the position that it was only an occurrence and that scientists overwhelmingly believe it was the beginning of time and space because they feel the need to believe in some kind of beginning. I see a lot of great mathematics and physics built from the basis of it being the beginning of time, but I see absolutely no reason to use that as a basis. There are a handful of enlightened cosmologists, such as William C. Mitchell. Was he a PhD? I don't think so. If I remember right he got a BS and went on to NASA. It is often those with less credentials that pave the correct road, as those with the highest credentials are stuck playing the game.

Thanks Meni --JohnLattier 02:53, 11 November 2006 (UTC)

Pardon me, but you were absolutely wrong when you said "it cannot be denied that there is such a concept as a smallest number greater than zero" (infinitesimal). In the calculus of Newton and Liebniz, and in infinitesimal calculus of various forms since then, squaring an infinitesimal is allowed, disproving your strange claim. I think Calbear's earlier comment has merit, (possibly implicitly) suggesting that taking the time to formally study math is a prerequisite to deny sophisiticated mathematical principles - and even then, one must use standard terms with standard definitions to do so. Tparameter 18:59, 11 November 2006 (UTC)

I don't know of anyone in the world who uses your definition of the infinitesimal. Dense number systems, even those constructed with infinitesimals, generally have no "smallest number greater than one." If they did, I believe you'd have to throw out closure, which is an important property of all number systems I've seen; without closure, other important properties would stop holding. In addition, you are pretty much basing your number system of off there being no two decimal representations for the same thing. In other words, the first axiom of your number system that this article must not be true! That's an extremely arbitrary first axiom. Every worse than its being an arbitrary first principle in general, in particular it is one premised on our having ten fingers and using a number system based on that. Number systems need to be universal, not based on quirks in human anatomy. Surely you must recognize this.

More formally: Assert a number system with closure in addition, subtraction, and multiplication; the distributive property; and a smallest number greater than 0, x, such that x is between the additive identity 0 and the multiplicative identity 1. We know (by closure) that there must be some y such that x+y = 1. (You call this y 0.999....) Thus x*(x+y) = x, so x*x+x*y = x. Clearly, x*x, x*y, and x are all positive. But that means that x*x < x, so x is not the smallest number greater than 0. So our assumptions must have been wrong. Therefore there is no "smallest number greater than 0" in any number system you might like to describe. Again, if you throw out closure, a system starts to break down, because you can't do too much with numbers if it no longer makes sense to add, subtract, or multiply them.

You say you've found several definitions of "infinitesimal." Did any define them as you did? If they didn't, it's because your definition is inconsistent. Sort of like, "the smallest human being over 10 feet tall"; you can say it, but that doesn't mean it must exist. Calbaer 04:38, 11 November 2006 (UTC)

Certainly the majority of online definition sites list the universally understood definition of infinitesimal,

"In mathematics, an infinitesimal, or infinitely small number, is a number that is greater in absolute value than zero yet smaller than any positive real number. A number x ≠ 0 is an infinitesimal iff every sum |x| + ... + |x| of finitely many terms is less than 1, no matter how large the finite number of terms. In that case, 1/x is larger than any positive real number."

"The mathematical inverse of infinity, an abstract description of nearly nothing."

"A very small quantity, approaching zero"

"An immeasurably or incalculably minute amount or quantity."

"Mathematics. A function or variable continuously approaching zero as a limit."

" in mathematics, a quantity less than any finite quantity yet not zero."

"A hypothetical number that is larger than zero but smaller than any positive real number. Although the existence of such numbers makes no sense in the real number system, many worthwhile results can be obtained by overlooking this obstacle."

Sites like dictionary.com, mathwords.com, britannica.com, definitionsfor.com. I think the only site giving me trouble is wikipedia, but it is probably edited by the same people who edited this article? --JohnLattier 06:26, 11 November 2006 (UTC)

None of these agree with your definition, the "smallest number greater than zero." Note that this is different than the "a number smaller than any positive real number" (not a real number itself) or a "function approaching zero" (not a mathematical constant), Those exist. None of the definitions have the word smallest in them, largely due to the contradiction I mentioned. (Someone stronger than I am on abstract algebra and/or nonstandard analysis can confirm.) More to the point, these are dictionary definitions, thus the fuzziness. The term can be used for multiple applications, which is why we have to peg down what you mean by it. For example, the hyperreal infinitesimal (a distinct number not present in the reals) is different than the infinitesimal in real analysis (usually a real quantity that can be considered arbitrarily small, like a function). You've invented a new definition with a similar flavor, but, unfortunately, it's something that can't exist, at least not in real numbers or any similarly useful mathematical construct. Calbaer 07:22, 11 November 2006 (UTC)

All above definitions are consistant and equal the universal understanding of the infinitesimal. It is clearly not zero. And it is clearly the number that is as close to zero as infinitely possible without equaling it. No ambiguity or fuzziness about it. You know, I'd be perfectly content if real analysis claimed the infinitesimal and 0.999... didn't exist at all as "real" numbers. But to claim they do exist AND equal to the number nearest to them is illogical. JL--68.211.195.141 07:33, 11 November 2006 (UTC)

0.999... isn't "next to" 1. It is 1, just as 2/2 and (3-2) and 1.000... are 1. The positive constant infinitesimal isn't "equal to" 0. In real numbers, it doesn't exist; in other systems, it isn't the smallest number. It can't be, as I've shown. And a real number is not the same as a real function, which is not the same as a real random variable. Anyway, I tried to show you that your definition isn't self-consistent to your notions of what a number system is. If you want to explore nonstandard analysis further, you should read up about it. That way you can reject the real analysis and see what the self-consistent alternatives are. Calbaer 07:52, 11 November 2006 (UTC)

You say "0.999... isn't next to 1. It is 1" then "positive constant infinitesimal isn't "equal to" 0" This is a contradiction. The only way 0.999... equals 1 is if your infinitesimal equals 0. JL--68.211.195.141 08:06, 11 November 2006 (UTC)

1. Like I said, your preliminary assumption that decimal expansions have a meaning on their own is flawed. First, whatever the case, you must provide a rigorous definition if you want your structure to have any meaning. Without such a definition, we cannot even discuss what is good or bad about it. It is also possible that there is no way to define a structure with the properties that you expect of it. Even if you did manage to work out a precise definition, it would lead to absurd results. In such a structure, you will not be generally able to add, subtract, multiply and divide (by nonzeroes). That means that any theorem of mathematics or formula in applications will not work. What good is that?
2. About definitions: A definition is seen as precise if it can easily and unambiguously restated in the language of formal mathematical logic. For example, the definition "x is an infinitesimal iff for every integer m, mx < 1" can be written as . The other definitions you quote have no mathematical meaning - they are just intuitive explanations for laymen.
3. About infinitesimals: We have not "set the infinitesimal equal to 0". We have defined an infinitesimal as above; and we have shown that from the other axioms of real numbers, there are no real nonzero infinitesimals.
4. It is not the lack of nonzero infinitesimals that is useful everywhere; It is the existence of a well-defined structure. Every single place where you have seen any formula, assumes that we have the real numbers at our disposal. Take away the real numbers, or replace them with an inconsistent or ill-defined structure, and the formulas will be worthless, along with the theories they represent. While you could probably use the hyperreals or something for most purposes, they are too complicated to be of real use, and they, too, don't allow such strange creatures as "the smallest positive numbers". There is at least one structure where this thing exists - the integers. So, unless you wish to use only them, you should drop your hopes for its existence.
5. 0.999... is well defined as a part of the general definition of decimal expansions (e.g. the supremum of the set of truncated expansions). Just like you have any other terminating or recurring expansion, and many non-recurring ones, you have this as well.
6. Providing a really rigorous proof of the equality will be very long, and require the use of formal language which I'm not sure you'll understand. That's where trusting experts comes in habdy. If you feel you're up to the challenge, you really should pick any good textbook which develops these concepts and follow through (perhaps Melchoir will have one to recommend). Another approach is taking a look at the #A new proof for the skeptics section above. I am willing to provide whatever details are necessary for whichever step you disagree with.
7. I have no problem with giving a name to the "smallest positive number". Since "infinitesimal" is already taken, I suggest we call it "the SPN". So, in the integers, the SPN is 1; in the set , the SPN is 2; in the real numbers, there is no SPN. Not in the hyperreals, either. And not in any other densely ordered stucture.
8. No, the definition of an infinitisemal is what I said above (or something similar). Again, it is not "defined" as zero, but rather there is a theorem which says that in the real numbers, the only infinitesimal is 0. Other structures may have several infinitesimals.
9. You don't know the reason why cosmologists say that the big bang is (in some sense) the beginning of space and time, so you assume it is because of some need to believe in something. The real reason is that the formulae obtained from observations don't make sense any other way. It may be that there was "something" before the big bang, possibly a previous cycle of the universe. That's a question we probably will never be able to answer. But the statement that the big bang is the beginning is a quantitative one, meaning, is true as far as the formulae are concerned. There's no way to describe it as just some explosion in a single point in an infinite universe which just waited for it to explode. Of course, a layman will not be able to appreciate any of this without going through the math.
10. Actually, I am comfortable too with not defining infinitesimals or 0.999... in the context of real numbers. The only infinitesimal is 0, and 0.999... is just 1, so we could do without those definitions. Sure, this would limit the generality of decimal expansions, and force us to put some restrictions on theorems regarding them, but it seems workable. That doesn't change the fact, though, that if we do choose to define the real number 0.999..., it must be 1.

-- Meni Rosenfeld (talk) 08:49, 11 November 2006 (UTC)

What Meni Rosenfeld wrote contains everything I was going to say, so I'll just say to read carefully what Meni says and think about it. Once again: There is no dense, meaningful number system with a "smallest positive number." It may seem like a handy concept, but it falls apart if you try to formally analyze it, no matter what your system of analysis is. Calbaer 09:02, 11 November 2006 (UTC)

The picture on heaviside step function will help you visualize the infinitesimal. 0.000...1 returns a value of 1, 0 returns a value of 0.5, and -0.000...1 returns a value of 0. --68.158.133.18 05:37, 12 November 2006 (UTC)

Your typical response: ignore all explanation and logic, and stick to your "intuition math." Oh well. I hope someone is gaining some understanding from all this, especially Meni's lengthy, well-written and well-thought out response; otherwise, it's just pearls before swine. Calbaer 05:59, 12 November 2006 (UTC)

What is this talk about 0 being an infinitesimal? Infinitesimal denotes having an amount. Zero has no amount. So what's the verdict about 0.999... is it a real number or not? Is infinity a real number? If infinity is not a real number, how can you use it above the sigma to defined 0.999... then say 0.999... is a real number?--JohnLattier 22:37, 13 November 2006 (UTC)

Infinity is not a real number, but the sum of an infinite series is defined. It is defined as the limit of the partial sums as the number of terms goes to infinity. That definition is what gives 0.999... any meaning at all, and by that definition, it is equal to 1. If you reject that definition, you will need to provide one to replace it.--Trystan 22:44, 13 November 2006 (UTC)

If the reals do not include infinity, they cannot include 0.999... --JohnLattier 03:36, 14 November 2006 (UTC)

They can, and do. The definition of 0.999... does not require infinity. The use of infinity in a limit is merely a shorthand, it is not explicitly defined using infinity. Read the formal definiton of a limit of a sequence. Maelin (Talk | Contribs) 04:09, 14 November 2006 (UTC)

Sources that 0.999... != 1

• source
• source
• source

Got anything to fill in the blanks? Either on or offline. -- kenb215 talk 00:05, 10 November 2006 (UTC)

Here's something to start

Infinitesimal (Noun)

  1. An immeasurably or incalculably minute amount or quantity.

  2. Mathematics A function or variable continuously approaching zero as a limit.

from Yahoo! Dictionary Since the infinitesimal (noun) is defined as a quantity approaching zero, it is not zero, So 1-infinitesimal isn't 1-0. Straight from dictionary. --68.211.195.141 03:58, 10 November 2006 (UTC)

Since you seem to like the dictionary, you'll probably want to look up the definitions of "function," "variable," "number," and "constant." Shockingly, they're not all the same! Calbaer 04:02, 10 November 2006 (UTC)

Just in case the above wasn't clear, I mean to say that the dictionary itself does not support 0.999...=1. 0.999... is neither a function nor a variable, unless it is taken to be 0.999... for all inputs / outcome events. According to the definition, 1/x, e^-x, and 0 are all infinitesimals. That being said, a dictionary is not a mathematical source. And, as pointed out below, it contains terms for items that are not necessarily extant. I'm hesitant to point this out at all since it is so unbelievably obvious, but I feel I must dissuade those who would look at this and think that those who say, "[constant nonzero] infinitesimals don't exist [in the real numbers]" are wrong because of a lack of understanding of the [implied] context. Finally, even if all this weren't true, it is a huge leap to assert that 0.999... != 1. It would be like saying that infinity is defined in the dictionary, so infinity plus one isn't equal to infinity, which is wrong. Calbaer 18:11, 10 November 2006 (UTC)

A dictionary defines how language is used. It does not define the existence or non-existence of the objects described. Dictionary.com, for example, has the following:

ex·tra·ter·res·tri·al (kstr-t-rstr-l) 
n. 
An extraterrestrial being or life form.

This does not mean that such a being or life form actually exists.

Srpnor 07:56, 10 November 2006 (UTC)

Not an analogous comparison. Analogous would be redefining extraterrestrial to mean earth dweller. --68.211.195.141 12:07, 10 November 2006 (UTC)

The definition of infinitesimals does not change in the context of real numbers. It only happens to be provable that there are no strictly positive infinitesimals in the real numbers. But even in the context of the real numbers, "infinitesimal=0" is not itself a definition.

Whether you want to include 0 among the infinitesimals is another question, and if you prefer not to, I personally wouldn't object, but the only difference would be that then the reals contain no infinitesimals. --Huon 13:29, 10 November 2006 (UTC)

It seems to be a contradiction to say 0 is an infinitesimal. 0 is the lack of a quantity, the only number without a positive or negative because it is the origin point set on a number line. Infinitesimal describes a quantity, can be positive or made negative depending on the approach direction, and can be thought of as either a distance between two consecutive points on the numberline or the actual consecutive point from origin 0 (same thing). -JL--68.158.133.18 16:33, 11 November 2006 (UTC)

The only number without a positive or negative huh? Tell me, is (1 - i) positive or negative? Tparameter 18:41, 11 November 2006 (UTC)

I keep seeing this recurring theme of people stating equations and calling them numbers. For instance, people say "2/2 is 1, 2-1 is 1, etc." to say that 1 is represented by different numbers. These are equations, people. Not numbers. 1-i is an equation. Give the decimal expansion for 1-i, (as every number has only 1 decimal expansion, along with the rule that any nonwritten digit is zero). You won't be able to give me a decimal expansion for i or 1-i because there is none. It does not have a place on the numberline. If it has a place on the numberline then you can deem it positive or negative. --68.158.133.18 19:20, 11 November 2006 (UTC)

Disregarding for a moment Tparameter's digression into the complex numbers:

• what you call an "equation" above is what mathematicians would call an algebraic expression: and every notation in the decimal numbers with more than one digit is an algebraic expression in terms of sums of products of powers of ten: and it is not particularly surprising that a number can be represented by more than one expression, as 2/4 = 2/1 = 1+1 = 2 shows.
• your assertion above that "every number has only 1 decimal expansion" is simply incorrect, with 0.999... = 1 being the classic counterexample. Indeed, every finite decimal expansion has a matching infinite expansion with trailing 9's. -- The Anome 19:40, 11 November 2006 (UTC)

I'll assume 68.158.133.18 is JohnLattier. You continue to treat decimal expansions as being somehow special. They are not. You have not presented us any logical foundation for a system where decimal expansions are the fundumental notion. You continue to use some vague folk notion of infinitesimal instead of referring to some accepted mathematical definition. If my explanations above were clear, you should know that this is wrong. If not, you should specify what part wasn't clear or you disagree with.

You are correct of course that the complex number 1-i does not have a decimal expansion, for the simple reason that it is not a real number, and decimal expansions are for real numbers (again: they are just a convenient tool to represent them!). Nor does it have a place on the "number line", which is just shorthand for the real number line, a part of the complex number plane.

"2/2" is a string of symbols, which represents an algebraic expression, which is equal to some number. "1" is also a string of symbols, which represents an algebraic expression, which is equal to some number - and it happens that this is the same number as before. "0.999..." is another string of symbols, which represents an algebraic expression, which is equal to the same number again. Same goes for 1-i (only that this time, the number is different). But that's all just syntactics. It doesn't really change anything as far as our discussion is concerned.

And for the last time, no, an infinitesimal cannot be thought of as a distance between two consecutive points on the numberline, because there are no two consecutive points on the number line. Please stop repeating this nonsense, unless you have the ability to justify it.

I'll be straightforward. It is this kind of behavior - continuing to ramble on instead of addressing the problems that have been raised in your arguments - that makes people suspicious of you and reluctant to respond to you. As far as I'm concerned, your choices are either:

• Accept what we are trying to explain to you,
• Ask substantial questions that will help you understand the matter,
• Decide that you do not currently have the mathematical skill to participate in this discussion, and return when you do (of course, by that time there will be no need for this discussion), or
• Decide that the people here are incapable of having this discussion with you, and move on to someplace else.

Any other choice (such as, continuing to fill this page with nonsensical claims) will be seen as trolling and reacted upon as such. I apologize if you actually are acting in good faith, but I feel these things need to be said. -- Meni Rosenfeld (talk) 21:04, 11 November 2006 (UTC)

JL, let me give you a hint for future reference: EQUATIONS USE THIS SYMBOL: = . It's called an "equals sign". Next, (1 − i) IS IN FACT a number. It is a complex number. The "decimal expansion" of it is (0.999... − i * 0.999...) (LOL!). In case you didn't know, the real numbers are a subset of the complex numbers, kind of like how the integers are a subset of the rationals, and so forth. So, I ask again, is (1 − i) positive or negative? Tparameter 23:14, 11 November 2006 (UTC)

JL, next, you said "It does not have a place on the numberline. If it has a place on the numberline then you can deem it positive or negative." This raises two natural questions:

1. Please point to the exact place on the numberline (using any scale) where your infinitesimal lies. Please?

2. 0 has a place on the number line, do you "deem" it positive or negative?

Tparameter 23:18, 11 November 2006 (UTC)

"EQUATIONS USE THIS SYMBOL: =" ok true. I should have said manipulation. It doesn't change the fact that saying 2/2=1 is no logical basis that a decimal expansion can overlap with another. To display the value of infinitesimal on the number line, simply consider (0,1], it is represented by the "(" or another way to represent it is drawing a bubble at zero and not filling it in and extending a line in the positive direction, which means positive numbers greater than zero and infinitesimal is the first of these numbers.

Meni, perhaps you can help me find a foundation upon which we both agree, and then move from there. ? I think we'd agree that infinitesimal is also known as the manipulation "1/infinity" and that the limit of 1/x as x approaches infinity is 0. Do we agree that 0.999... is represented by 1-1/infinity and that the limit of 1-1/x as x approaches infinity is 1? Where we disagree is that I see 1/inf as uniquely different than limit to infinity of 1/x, this flows directly from the definitions that infinitesimal is strictly not zero. gotta run -JL logging in is so tedious --68.158.133.18 00:09, 12 November 2006 (UTC)

Simple question: Since "infinitesimal is the first of these numbers" next to 0 on (0,1]:

Let I = your infinitesimal.

Where on the number line is I^2?

Tparameter 02:20, 12 November 2006 (UTC)

I^2 is meaningless. So is I/2. It's like saying "infinity plus one." --68.158.133.18 03:41, 12 November 2006 (UTC)

Maybe according to you, but Newton and Liebniz recognized this value. Tparameter 03:59, 12 November 2006 (UTC)

1/infinity is zero, not a positive value. And infinity plus one is infinity, not an undefined value. Since infinity is not a real number, the addition of infinity bring in some kinks to the number system, which is why it's best to do rigorously, not informally. In real analysis, infinity is only approached, and "approaching infinity" is just shorthand for getting larger without bound.

It has been said by Meni (and many others) that your basis for a number system is flawed. It is based on the notion that the decimal representation is somehow "special." I know you don't see this, but it is true in what you believe should be the axioms of a number system, e.g., that 0.999... and 1 should be distinct numbers with no numbers between them. And you indicated that 2/2, by contrast, is an "equation [sic]." This indicates that you would see 1/2 (one half) as an expression and 0.5 (five tenths) as an actual number. But in elementary school, surely you learned 1/2 before you learned 0.5. Fractional notation is much more intuitive and natural than decimal. In fact, fractions are how you go from integers to rationals, just as limits are how you go from rationals to reals and the square root of -1 is how you go from reals to complex numbers. None of these need decimal representations. For example, using continued fractions allows numbers to be expressed, compared, calculated without the need for decimals, thus "solving" the 0.999...=1 "problem." (Instead you get that , which is a lot easier to believe.) But decimals are more convenient for human beings (just as binary is more convenient for fast operations on computers), so that's what we use. As Meni said, there's nothing fundamental about it.

You however, see things otherwise. That is why "a foundation upon which we [all] agree" will be impossible to come to as long as you hold onto the talismans — that Meni so thoroughly listed — of what a number system should be. And if you still hold onto them today, after all this discussion, you will tomorrow and the next day as well. Calbaer 04:59, 12 November 2006 (UTC)

There is a glaring difference between 1/2 and 0.5. One contains an operation and the other does not. I don't want to hear that decimal notation can be transformed into a summation operation, that matters not.

Calbaer, it's impressive you understand we use base 10 because of our fingers, but looking at other base systems doesn't change anything. 2.222... doesn't equal 3 in base 3. --68.158.133.18 05:28, 12 November 2006 (UTC)

Quite simply, you are wrong. "Five tenths" (0.5) contains no fewer operations at "one half" (1/2). Your concept of rational numbers deviates from both an elementary school understanding and a formal understanding. And until you understand and recognize the difference between a number (an abstract quantity) and its representation (the words and symbols we use to represent numbers), talking to you about mathematics will be as effective as talking to a wall about mathematics (for you, me, readers, contributors, and the wall). Calbaer 05:55, 12 November 2006 (UTC)

Wow. Nobody said "five tenths." 0.5 might equate to five tenths ( I told you not to do this ), but the decimal notation is 0.5 not five tenths.

Anyway, I was looking at http://www.neowin.net/forum/lofiversion/index.php/pricing/t28370-0.html and I notice people stating the rule that you cannot multiple an infinitely repeating decimal by a finite number, or something along those lines, and that if possible you may convert the repeating decimal to fraction notation to perform the calculation. I like this line of thinking. Now where could I find some kind of "official" document regarding this rule and other similar rules to appease those who only listen to "official" arguments? --68.158.133.18 06:06, 12 November 2006 (UTC)

You know what, don't refer to me as a wall. I could easily have said the same for certain people on here. I'm listening attentively to what you say. I just haven't seen anything that equates 0.999... to 1. I keep hearing about these holy axioms but still no one has produced any. --68.158.133.18 06:11, 12 November 2006 (UTC)

Reading other message boards, I found this, I love this:

"0.999... You could sneeze, and if you added the amplitude of the sound that reaches Alpha Centauri, it would be enough to push the value greater than 1." http://www.ambrosiasw.com/forums/index.php?s=82d900be17fd31e895f4a96e2ac4f163&showtopic=96544&st=0&p=1444584&#entry1444584 --68.158.133.18 06:26, 12 November 2006 (UTC)

The only way to show you are listening here is to address the points made, which you haven't. Look at Meni's explanation. Look at your response. You ignore everything said, instead repeating your "intuition." This is why people are fed-up with trying to explain this. Now, if someone were to step forward and say that they're learning something from all this, it might be worthwhile to continue. Otherwise, it's not.

And, for the record, I wasn't saying you were a wall. I said the discussion is as effective as talking to a wall — both as effective as you talking to a wall and as others talking to a wall. I made that quite clear, although I could see how you would misinterpret this. You don't recognize such subtle differences as the difference between a number and its representation. "0.5," in words, is "five tenths." You say "point five," but that's no different from saying "five tenths." That's how decimal representation works. The definition of a rational number is the quotient of two integers (where the denominator can't be 0). You can't talk about a rational non-integer without division being implicitly or explicitly involved. Again, you must understand that a number and its representation are not identical concepts. Saying so would be like asserting that by your reading the word "strawberry" here you could reach out into the monitor and have a fruity snack.

The page you cite is no more or less official than this Arguments page. You aren't able to find an "official" document because you are presenting an illogical idea, one you continue to propose, ignoring the very contradictions that are so explicitly given to you. You ignore our ideas, and it's about time we reciprocated. Calbaer 07:07, 12 November 2006 (UTC)

The guy is a legendary troll. That much is clear at this point. Now he's arguing that 0.5 != 1/2. Holy moly! I've never seen anything like it. Tparameter 08:18, 12 November 2006 (UTC)

Tpara your post is very antagonistic fluff, doesn't serve well. You obviously didn't read what I wrote. Who said 0.5 doesn't equal 1/2? They are equal in value but they aren't the same notation. One uses an operation, and the other is the digit value. So the argument "A number can have multiple decimal representations" simply cannot be proven by pointing your finger a division operation. Last I checked 1/2 is not a decimal representation. --JohnLattier 09:45, 12 November 2006 (UTC)

The problem here, John, is this: "My reasoning goes as follows. I start with a foundation that all numbers are represented by one and only one decimal expansion. Excluding any zeroes you may add." -- from there it follows very trivially that 0.999... ≠ 1, simply because they're different decimal expansions, but yours isn't one of the "foundations" that the number system known as "the real numbers" is built on. You may or may not be able to constuct an internally consistent number system based on it, but if you did, it wouldn't be the reals. In real number theory, two different decimal expansions (read: infinite series) can indeed be equal, and there's a dozen tortoise kebabs to prove it (search that page for the word "amphora" and read from there). Assuming otherwise while talking about reals is like a child asking "if 1+1 was 3, what would 1+2 be?". It makes as much sense in real number theory as "what's 0.999.../2?" does in your 'system'. Andrew 13:09, 12 November 2006 (UTC)

And furthermore, if you -did- manage to construct an internally consistent number system based on the foundation that every number has exactly one decimal expansion, you would then have to convince people that it is actually useful for anything, particularly, in ways that the reals are, in your view, not useful. Which I expect will be even harder than making such a consistent system with any power in the first place. Maelin (Talk | Contribs) 13:59, 12 November 2006 (UTC)

One of the points here is that the reals are clearly only "an internally consistent number system" if you don't get bothered by the fact that their axioms (as everyone keeps pointing out) yield the obvious nonsense that 0.999... = 1, when everybody knows that this should come out as less than 1. This is not a contradiction because the axioms don't yield both, but it does show that something has gone awry somehwere.Davkal 14:16, 12 November 2006 (UTC)

You'll get nowhere in science or mathematics if you confuse what is obvious with what is true. Andrew 14:43, 12 November 2006 (UTC)

Davkal: Not, it only shows that intuition is often incompatible with truth. You, too, are making the mistake of treating decimal expansions as something fundumental (probably they emphasize them too much in school or something).

John: You say, "I keep hearing about these holy axioms but still no one has produced any". Did you simply ignore the part where I said that the axioms can be found at the article real number? Do you expect me to copy-paste the axioms here? Why should I put that much effort (and waste space) if you can't put enough effort to follow a simple link?

You've made suggestions for things we should both agree on; unfortuantely, introducing infinity as an element of our system will only complicate matters. I prefer to stick to the real numbers, where there is no such thing as "infinity". And the foundation I will accept for them are their axioms.

You have complained about not hearing how the axioms prove that 0.999... = 1, but have done everything except for stating "I believe I am mathematically mature enough to understand a rigorous mathematical proof, and would like to see one, even though I know that it can be long and difficult, and will take a lot of time for all parties involved". Of course, what you really should do is pick up a textbook about the topic, which will be able to provide much more detail than we could within our time frame. But I leave the above as an option, so we cannot be accused of being unable to reply to your query. Of course, this falls under the second choice from the list presented in my previous post; you can feel free to choose any of the other three, and I'll be happy to hear which one you've chosen. -- Meni Rosenfeld (talk) 15:03, 12 November 2006 (UTC)

Meni, and which of your beloved axioms do not, in the end, boil down to intuitions (some much less intuitive than 0.999... is less than 1). The point I made, the point not yet addressed, is that the axioms (the intuitions) yield 0.999... = 1, but since it should be less than 1 (another intuition, granted, but one which can be backed up by very obvious logical arguments much more straightforwrd than any "proof" to the contrary)we can: a) choose to just accept it as a problem and not get bothered by the fact that this shows a problem exists somewhere; or, b) pretend it isn't a problem and throw about words like "truth" to hide our embarrassment. Davkal 07:08, 13 November 2006 (UTC)

Depends on what you mean by "problem". If counterintuitive results are included in what you regard as problems, then I agree. 0.999... = 1 is a problem, and I am not bothered by the existence of such problems.

And indeed, choosing one axiom system over another involves intuition (among other things), but we must ultimately find a structure which is coherent, consistent and preferrably useful. There are such structures other than the reals; but there are no such structures where 0.999... < 1 (a few coherent and consistent ones, such as Richman's decimals, fail the usefulness test). -- Meni Rosenfeld (talk) 20:15, 13 November 2006 (UTC)

These axioms, in particular: "The set R is a field, meaning that addition and multiplication are defined and have the usual properties" leads to the conclusion that 0.999... is not even a member of so-called real numbers. The common rules of addition and multiplication do not hold up with any decimal number ending in infinite 9's as each 9 awaits carry over and error is only alleviated when a non-9 number is present. If 0.999... is not a member of so-called real numbers, how can it be said to equal 1, a real number? JL--68.158.133.18 17:19, 12 November 2006 (UTC)

Yet, in a prior breath you claim 0.999... is on the real number line "right next to 0." Let me give you another hint: if it's on the real number line, it's a real number. LOL! Unbelievable. Tparameter 18:59, 12 November 2006 (UTC)

Addition of numbers is independent of the notation used to represent them. You are confusing "adding numbers" with "using a certain algorithm to find a representation of the sum of two numbers given by some representation of them". The long addition algorithm may not be convenient (or, with a naive definition, not even applicable) to add non-terminating decimals, but that doesn't mean that their sum doesn't exist.

Asking questions is okay (though this particular question does not seem to indicate a great deal of effort to understand on your part). Filling the page with garbage, like you've done below, is not okay, and I've warned you about it. Please tell me a reason why I should not believe that you are trolling. -- Meni Rosenfeld (talk) 20:16, 12 November 2006 (UTC)

closest possible number to 1

I'm not arguing the fact that 0.999... = 1 (I believe it), but if it does, what is/are the closest number(s) to 1 without being 1? Is there any way to express it/them? (sorry, forgot to sign this earlier...) Qhiiyr

There is no closest number to 1, as illustrated in the above arguments. Simply put, for any real number a (other than 1), (1+a)/2 is closer to 1 than a, (3+a)/4 even closer than that, and so forth. So a cannot be "the closest" to 1 or even closer than a finite number of numbers. This is thus like asking what the "largest integer" is. For any integer k, I can find a larger k+1. So even though we have concepts like "the closest" or "the largest," they need not exist for a particular set of numbers (the reals and integers, respectively). Calbaer 17:13, 12 November 2006 (UTC)

• It's fascinating observing the various sorts of confusions and misunderstandings that this topic brings up; I imagine that's why some of us are here. A lot of the confusion seems to be a matter of generalizing properties of the integers into the reals, and an inability to comprehend the concept of density. --jpgordon^∇∆∇∆ 17:45, 12 November 2006 (UTC)

Arguments irrelevant to question above

I'm sectioning this off since it probably won't help the anon poster, who, unlike John, accepts that 0.999... = 1 Calbaer 19:23, 12 November 2006 (UTC)

Note: This paragraph was originally intended for "closest possible number to 1" but several people have been involved in censoring my response to this individual because of their misunderstanding of the infinitesimal. I have made an attempt to revert this sectioning originally titled "feeding JohnLattier" which is one of many references to "feeding trolls" that Calbaer has used. I cannot continue to revert their malicious edit as I am severely outnumbered by aggressive mathematicians (lol @ that) on this message board. So bear in mind my intended audience with the following paragraph: --JohnLattier 19:42, 12 November 2006 (UTC)

Real analysis is based on the faulty premise that between any two numbers on the numberline, there will always be another number wedged in between. The problem with this is that the numberline is constructed as an infinite string of zero dimensional points (no density) spaced infinitesimally apart from one another in two polar directions. It is easy to visualize this fact by first starting at any finitely defind number such as 718.90215, then replacing the final digit with one digit lower followed by all 9's, like so: 718.90214999... This is defined to represent the neighboring number on the numberline. The ...999... represents that which is closest, and any manipulation to it, such as tryting to push it closer to its limit, is error mathematics. The closest numbers to 1 without being 1 are 0.999... and 1.000...1, the latter has notation which is debated simply because no standard notation is set for it, but 1.000...1 remains the inverse of 0.999... nonetheless, and it is also an equal distance from 1 in the opposite direction. The ...1 at the "end" is of course never reached (many people have a hard time understanding this part), it is simply to signifity that it is greater than 1.000... which is exactly 1. JohnLattier--68.158.133.18 17:58, 12 November 2006 (UTC)

Surely you must realize that any attempt to do any math without a well-defined number system is a meaningless jumble of symbols. If you throw out the axioms of the reals, it makes no sense to start describing high-level characteristics of a new system without first defining its basic foundation.--Trystan 18:03, 12 November 2006 (UTC)

(Edit conflict) ...the numberline is constructed as... By whom? By you? Also, getting rid of density is hard, because the rational numbers are also dense. That is to say, never mind the decimal expansions, even in the regular fractions you can write out as the ratio of integers there is no closest number to the number one. -- SCZenz 18:04, 12 November 2006 (UTC)

Stop imagining the real numbers are a discrete set. They are a continuum. There is a number wedged between any two real numbers in the same way that if you have two points in space you must always have a point between them, as space is not made up of a series of discrete points, but is a continuous field. Oh, and "numberlines" are something that only happen to infant school pupils. Andrew 18:32, 12 November 2006 (UTC) Oh! And don't call real analysis "faulty" when you are a layperson and the greatest mathematical minds in history have failed to find fault with it. It smacks of supreme and hateful arrogance. Andrew 18:38, 12 November 2006 (UTC)

Based on the definitions I have mentioned before, there can be no numbers wedged between an infinitesimal and 0. It is defined as the number closest to zero without equaling zero. (The same relationship between 0.999... and 1). It doesn't get any simpler than that. If one cannot grasp that, one will need to keep trying to grasp it before proceeding with any mathematics that utilizes infinitesimals, lest you desire to give birth to trickery such as this for the ooh and awe effect.

To: whoever posted this topic, be wary of the ones who post to defend this agenda article. When a pretty proof seems to give a nonintuitive result, look harder at the proof until you realize the step(s) that are candidate tricks. Be particularly wary of when people manipulate the infinitesimal with seemingly elementary mathematics. JL--68.158.133.18 19:07, 12 November 2006 (UTC)

There are indeed no numbers "wedged between an infinitesimal and 0" in the real numbers, because the infinitesimal does not exist in the real numbers.Andrew 19:32, 12 November 2006 (UTC)

If infinitesimal does not exist in real numbers, then neither does 0.999..., and this whole article would be claiming two numbers equal when one of them doesn't even exist in the number set. --JohnLattier 19:42, 12 November 2006 (UTC)

Just to be clear: you are now claiming that (0.999...) is not on the real number line, right? Tparameter 19:46, 12 November 2006 (UTC)

You're aware, then, that nobody is claiming two numbers are equal. The statement in the article is that two decimal expansions denote the same number. And if 0.999... and 1 denote the same number, then it is logically impossible for one to "exist" and one to "not exist" -- something can't cease to exist just because you write it differently. You are trying to argue based on the presumption that 0.999... is a number strictly less than one, but what you should be arguing is that decimal expansions do not work the way we say they do. And they do, because we (by which I mean "mathematicians") invented them and ergo they mean whatever we decree them to mean. And that's what you miss: mathematicians invented decimal expansion. It's an entirely artificial construction. And therefore mathematicians can decide what is and is not true, and they decided it worked by limits, not by protracted summation. so it does. You keep referring to this "reality" that you think real analyisis deviates from, but that reality does not exist. It's like reading Lord Of The Rings and saying "that's not how it happened at all! And hobbits are taller than that!". Tolkein has literal authority just as mathematicians have literal authority over decimal expansions. Andrew 16:30, 27 November 2006 (UTC)

A common theme in this is the concept of "tricks." For instance the use of "real" in real analysis, is the opposite of real in the common sense. In reality, 0.999... is on the numberline. But in your "real" analysis, it would not even exist. --JohnLattier 19:50, 12 November 2006 (UTC)

It's not a difficult question to answer directly. I'm not talking about "real analysis", which it is clear you have not studied formally. Instead, using simple logic, let's examine 0.999... Let me pose the questions this way:

1. Yes or No, is (0.999...) on the real number line?

2. If so, is (0.999...) a real number? (again, yes or no)

3. If so, then do you agree that it must satisfy the definition of "real number", and all that it entails?

Tparameter 19:59, 12 November 2006 (UTC)

John: you say you have not studdied mathematics. You disagree with the vast majority of the people on this page. And every source available. And you disagree with the two experts you yourself contacted. And Newton, Leibniz, and Euler, Gauss and Cartan. That's all cool; leypeople are fully allowed to disagree with experts, and with common opinion, and with all available sources, and they can even be right - just not on Wikipedia, which is all about verifiability. Also, when you're doing mathematics, your intuition don't count for jack.

Regarding infantesimals: you like analogies so we'll try that. You want to buy two apples for $2 each. 2 is an integer. 2x2=4 in an equation entirely about integers. There are no dollars in the integers. How will you decide to pay $4 for your apples? Answer: $'s are an extra structure, not part of the integers, but added to make integers more usefull. Splitting $ off from 2 is cool, because then you can do 2+2=4 with litres, donuts, e.t.c., and not have to worry about inventing yet another number system to calculate 2 onions + 2 onions = 4 onions.

Similarly, infintesimals are not part of the Reals but an extra structure added to increase the functionality of the Reals. Splitting the infintesimals off from the Reals is cool, because then you can do calculus on surfaces, volumes, the complex numbers, the projective plane, all manner of manifolds that aren't the Real numbers. There's differential geometry for rigor, but it's quite impenetrable for undergraduates. Not having infantesimals in the Reals also means you can do stuff without having to worry about infantesimals turning up, for instance doing linear algebra and proving that 0.999.. = 1 among other things.

One more time: complaining about not having infintesimals in the Reals is like whining about not getting chocolate sprinkles inside your tub of vanila ice cream. Ice cream is ice cream. No sprinkles. Doesn't mean you have to eat it without sprinkles; there's a bowl of sprinkles just over there, go on and help yourself; but it's a *tub of ice cream* and the sprinkles come seperately. Armed with only ice cream, we can prove that 0.999... = 1, and adding sprinkles doesn't change it. Endomorphic 21:18, 12 November 2006 (UTC)

John, you keep claiming that real analysis is "contrary to reality". Firstly, this is because you are confusing a rechnical term ("real number") with normal English usage ("real" as in "actually exists"). Real numbers don't actually exist. For that matter, rational numbers don't actually exist. Not even the natural numbers actually exist. They're just models that we've developed, which for some reason nobody can really explain, turn out to be inordinately useful and incredibly accurate when we measure and count things in the real world. 2 objects plus 2 objects is always four objects, and this works when counting apples, oranges, trees, people, countries, you name it. It's an unreasonably effective system. But the numbers themselves do not exist. You can't point your finger at a two. You can only point at two things, and say, "that bunch of things has two things in it".

You furthermore claim that 0.999... is equal to 1 in real analysis, which is entirely true (although real analysis doesn't redefine anything, it works from the axioms that form the reals and investigates the results of those axioms), but then you make a completely unjustified claim that 0.999... is not equal to 1, "in the real world". What are you talking about? Where is 0.999... in the real world? Where is 1 in the real world? Have you ever held 0.999... in one hand, and 1 in the other hand, and compared them? Of course not. Numbers do not physically exist in the real world. They are just a modelling system that turns out to be astonishingly accurate and useful for dealing with the real world. But they don't "really" exist, and they have no other properties besides those they inherit from their definitions and the mathematical structures in which they reside. Maelin (Talk | Contribs) 02:46, 13 November 2006 (UTC)

I never claimed 0.999... equals 1 in real analysis. It either doesn't equal 1 or doesn't exist. Further, a noun doesn't have to be tangible to exist. A noun can also represent the relationship between nouns. Distance, thoughts, heat, speed, these exist. They are intangible but would continue to exist whether or not humans were here to describe them. Infinitesimal and infinity exist because of each other, they are just as natural as any number we can dream up. Mathematics as we know it breaks down in the face of infinities. It's very inconvenient. That may be why real analysis was even formed. That's all fine and good. But 0.999... would not exist in such a number system and therefore could not be equal to zero. --JohnLattier 22:51, 13 November 2006 (UTC)

JohnLattier, perhaps Mathematics as *you* know it breaks down at "infinity". Mathematicians, however, have a whole bunch of tools for playing with various infinite things. For completeness, mathematicians don't even use infintesimals for serious calculus any more: rather, sections of tangent bundles. Oh, and Maelin: less is more. Endomorphic 23:15, 13 November 2006 (UTC)

Attn: SCZenz and Calbaer

I insist you refrain from editing my posts. According to the guidelines on talk page guidelines:

1. Don't edit others' comments: Refrain from editing others' comments without their permission (with the exception of prohibited material such as libel and personal details). It is not necessary to bring talk pages to publishing standards, so there is no need to correct typing errors, grammar, etc. It tends to irritate the users whose comments you are correcting. Never edit someone's words to change their meaning.

--JohnLattier 19:48, 12 November 2006 (UTC)

I did not edit your comments. I have never edited your words. I merely subdivided a section and added my own words. (Here again is the problem of your missing a subtle but crucial difference, as with that between called a wall and being told that discussions with us are as effective as your talking to a wall, or between a number and its representation.) You then edited my section title. That edit can stand, as all I wanted to do was make the point — which you seemingly agree with judging from your retitling — that the original poster is not helped by your comments, which are instead a continuation of your earlier arguments.

Or maybe you already know this and just want more attention. It is not for me to say. Calbaer 19:59, 12 November 2006 (UTC)

I removed some of your comments as irrelevant. Such removals are also permitted by the talk page guidelines. -- SCZenz 20:28, 12 November 2006 (UTC)

Calbaer, removing my writing from its appropraite and intended topic to a new topic named "feeding JohnLattier," is a violation of the rules of Wikipedia (as listed above). You have done this several times, in addition to blatantly saying I am a troll. Do not do this again.

So you first write, "talking to you about mathematics will be as effective as talking to a wall" then you claim it says, "discussions with us are as effective as your talking to a wall." Not as pretty of a manipulation as the 10*0.999... but still a manipulation. I formally invite you to return from personal attacks and censorship to join us in logical debate. --JohnLattier 21:18, 12 November 2006 (UTC)

John, I formally invite you to refrain from personal intuition and join us in logical debate. Endomorphic 21:32, 12 November 2006 (UTC)

Infinitesimals

John, you seem to agree with the rest of us that 9.999... = 1 - ε, where ε is a very tiny number: and we both agree that ε is so tiny it's infinitesimal, in the sense that it has a magnitude less than any positive real number. The only thing that we disagree on is one small but crucial detail: that, in the real numbers, there is no infinitesimal real number (in the sense above) other than zero, and so ε must equal zero. Substituting in ε = 0 gives us 0.999... = 1 - 0 = 1. Would you agree with this, if we can prove that ε = 0? If so, we've made some progress, and we can go on from there. -- The Anome 21:20, 12 November 2006 (UTC)

It should also be noted that he said that 0.999... is itself not a real number. This is important. Tparameter 22:17, 12 November 2006 (UTC)

Ah. Then he should read Dedekind cut. -- The Anome 15:56, 13 November 2006 (UTC)

Your rational approach is a breath of fresh air. We may make progress. I have no time right now, but I plan to respond tonight. --JohnLattier 22:09, 12 November 2006 (UTC)

(edit conflict) I dare predict that John won't agree with this, because he prefers having infinitesimals to having real numbers. Now infinitesimals might be nice (compare hyperreals), but if we want to have "the smallest positive number", we need a strange kind that is not readily accessible to arithmetic (for example, ε/2 is undefined). Such a number system might be conceivable, even though some of its numbers don't have a decimal representation (0.0000...1 with infinitely many zeros isn't one), but I fail to see any advantage in having these strange infinitesimals. What, except part of our intuition, would go wrong if we just forgot about them? Would our number system lose any practical uses? John, could you help me? --Huon 22:23, 12 November 2006 (UTC)

I know you meant 0.999... is equal to 1 - ε. We agree on that of course. I say ε (infinitesimal) is defined multiple times in the dictionaries as strictly nonzero. I also say real analysis ignores these definitions and defines a number system where the infinitesimal does not exist (discontinuity). If the axioms of real analysis fail to describe the system already in place, then any claims it makes should stay in the realm of real analysis. I am very aware the article slightly hints that this is true for real analysis, but it overwhelmingly comes off as a generalized claim for all of mathematics. If you are able to show that ε is equal to zero, either your proof is wrong or the dictionaries are wrong. --JohnLattier 13:48, 13 November 2006 (UTC)

Dictionaries are not mathematics texts, and we are talking about real analysis here (see the intro sentence, which says "In mathematics, the real number denoted by the recurring decimal 0.999...", and the FAQ on Talk:0.999...)

If you Google for "non-zero infinitesimal", you'll see that this phrase is in common use in mathematics, to distinguish between zero (which is a trivial case of an infinitesimal) and infinitesimals other than zero (which are perfectly reasonable to talk about in systems such as the hyperreals and surreal numbers, but don't exist in the reals.)

As far as I can see, all of your arguments are based on one idea: that the "really real" numbers are not the same as the real numbers. Since we can prove -- rigorously -- that 0.999... = 1 in the axiomatic system called the real numbers, you must be talking about some other number system, based on your personal mathematical intuitions. Can you tell us what that number system is, and what its properties are? -- The Anome 15:43, 13 November 2006 (UTC)

These exact points and questions have already been posed, in the above and archived text, many times to JohnLattier. You're not going to get a better answer from him by asking them again. Calbaer 17:13, 13 November 2006 (UTC)

More of those pesky facts. Don't you understand, in math, intuition is all the proof you need? For example, I believe pi is 3.14, therefore it is. <sarcasm> 72.193.74.36 15:42, 13 November 2006 (UTC)

Answer this: Is 0.999... a real number? If so, then this discussion is over. If not, then what is it? Tparameter 15:03, 13 November 2006 (UTC)

To answer that, we must answer if infinity is a real number, as it is used to define 0.999... What do you think? --JohnLattier 22:54, 13 November 2006 (UTC)

• I have my own question. Since John's told us we can't do basic arithmetic on infinitesimals, and since he's defining ε as being the infinitesimal, how can he refer to 1 - ε? --jpgordon^∇∆∇∆ 15:52, 13 November 2006 (UTC)

Some basic arithmetic holds up in the face of ∞. Some does not. Starting with ∞ itself, ∞+1 means nothing. ∞*2 also means nothing. ∞/∞ is 1. 0/∞ is 0. 1/∞ = defined as infinitesimal (which has no symbol as far as I know but deserves one). But then 2/∞ is meaningless as it would be 2*infinitesimal, though while meaningless it would be the "next point" after infinitesimal. It's such ugly mathematics. That's probably why it seems no one has actually set the record straight on manipulating infinities.--JohnLattier 23:04, 13 November 2006 (UTC)

John, you might want to take a look at the surreal numbers, a superset of the reals where expressions similar to those you cite above actually do have formally defined meanings. However, you should be prepared for strangeness, as many of the properties of the surreals are even more contrary to intuition than those of the reals. -- The Anome 02:57, 14 November 2006 (UTC)

"∞+1 means nothing. ∞*2 also means nothing. ∞/∞ is 1. 0/∞ is 0. 1/∞ = defined as infinitesimal"

That is almost the exact opposite of anything I have heard of outside of this page. Did you come up with this? If not, what book/web page did you read it in? I would like to read it myself. -- kenb215 talk 20:34, 19 November 2006 (UTC)

He makes up his own rules of math. If you're going to visit this page, get used to it. 131.216.2.83 21:02, 19 November 2006 (UTC)

He's already answered that, too. He refers to it as 0.999... and to epsilon as 0.000...1. The faults of this thinking (that you can subtract infinitesimals from certain numbers but not divide them by others) have already been pointed out. I really think that we're at the point that nothing new is being asked or answered here. Calbaer 17:13, 13 November 2006 (UTC)

Anome, I am still interested to hear your explanation of how ε is equal to 0. --JohnLattier 23:04, 13 November 2006 (UTC)

See Dugwiki's comment in the section below for details. -- The Anome 03:01, 14 November 2006 (UTC)

JL, I am still interested to hear your explanation of how 0.999... is not a real number. Tparameter 23:44, 13 November 2006 (UTC)

*Sigh*. Archimedean field. We've been there already. Hey John, in dy/dx, you want the dy and dx to be infintesimals, right? (If not, then what on earth are you infantesimals good for?) But apparently dy/dx is just

infintesimal/infintesimal = infantesimal * (1/infantesimal) = (1/infinity) * infinity = infinity/infinity = 1

So if dy/dx was always 1, I'm interested to hear why you spend so much time calculating it in physics papers. Endomorphic 23:51, 13 November 2006 (UTC)

dy/dx refers to the ratio of a dependent variable's rate of change over an independent variable's rate of change as the independent variable goes toward zero. No one said this is poor mathematics or that all manipulations of infinity are poor mathematics. A ratio of two infinities may be permissible, for instance while 3/∞ and 2/∞ are themselves meaningless, (3/∞)/(2/∞) would produce a useable result. That's what makes calculus so beautiful. --JohnLattier 03:46, 14 November 2006 (UTC)

No.

The reason calculus is beautiful is because it employs limits keep all values finite and managable, when lesser minds would try to imagine how a ratio of infinities might meaningfully cancel. Officially, 3/∞ is not undefined but meaningless (since ∞ is not a real number), similarly 2/∞ is meaningless, (3/∞)/(2/∞) is likewise meaningless and calculus was invented to ensure the various ∞ *never even apear* in the calculation.

You also say earlier that "∞/∞ is 1". Does this mean that the infintesimals you need for 0.999... \neq 1 are *not* the same things as dy and dx that every other mathematician uses when they have to say "infintesimal"? What good are these new "Johnfintesimals" then? Endomorphic 05:42, 14 November 2006 (UTC)

What should a number system satisfy?

I doubt JohnLattier is talking about real analysis. If I understood him correctly (correct me if I'm wrong!), he concedes that in real analysis 0.9999...=1, but that leading to such an equation should be considered a disadvantage of real analysis and that there is a better "system already in place" where infinitesimals exist and 0.9999...<1. But what is the "system already in place"? I believe John will have difficulties to give some sort of reference, because the system in place usually is real analysis. There's a reason why any approach using infinitesimals is called "non-standard". Nevertheless, John, you might try to go ahead and describe what your number system should satisfy. In particular:

1. Are the rational numbers a subset of your numbers? How about the standard real numbers?
2. I assume your system will include infinitesimals, but will the existence of numbers such as "the largest number less than 1" be required?
3. Will every number have a decimal representation? If you want to include numbers with descriptions such as 0.000...1 with infinitely many zeroes, you should also define (or should I say re-define?) "decimal representation".
4. What kind of arithmetics is allowed? That's a tricky question; feel free to leave any arithmetics you are not sure about undefined: We can't divide by zero, so why should we be able to, say, substract an infinitesimal?
5. For questions of continuity and convergence, a notion of "open set" would be nice.

The problem is that adding infinitesimals leads to other counter-intuitive results and/or makes arithmetics so limited as to make the number system pretty useless. If you answer as many of the above questions as you can (and provide any additional properties you deem necessary for your numbers), I'll try to conjure up a number system satisfying what you want and show some of the strangeness it leads to - or I'll try to show that your prerequisites contradict each other. Yours, Huon 15:47, 13 November 2006 (UTC)

I see you acting in good faith Huon, but must have better things to do than try to explain a new topology to someone who believes that \sum_{i=0}^{\infty} a_{i} \neq \lim_{N \rightarrow \infty} \sum_{i=0}^{N} a_{i} Endomorphic 22:51, 13 November 2006 (UTC)

Huon, as far as I know, I never said 0.999... exactly equals 1 in real analysis. I may have said it doesn't exist in the reals as they are defined. Rational numbers and real numbers are a subset of the number system that does include infinity and all infinity-defined numbers not already included such as "largest number less than 1," "smallest number greater than zero" etc. Decimal representations apparently can use the ... although I always used the bar over the top. I propose no changes to any math, except in the case of infinity-defined numbers. For example, ∞/2 would be meaningless. Infinitesimal/2 would also be meaningless. These types of expressions have been thrown around and they shouldn't be. --JohnLattier 23:13, 13 November 2006 (UTC)

There is no such thing as the "largest number less than 1" in the reals. Reals are a subset of Complex Numbers, and your statement isn't true there either. So, name this mythical number system!!! Actually, what you say is impossible. There is no way for the reals to be contained by an ordered number system. Tparameter 23:32, 13 November 2006 (UTC)

Such expressions get thrown around, JohnLattier, because mathematicians like to be able to do maths with their numbers. If $x$ and $y$ are real numbers, we should be multiply them without the universe imploding. You would otherwise exile the humble parabola $y=x^2$ - is x an infinitesimal? what if x is infinite? is the quadratic valid for some Reals and not others? Quagmire! Endomorphic 02:37, 14 November 2006 (UTC)

Real number system does not have unique decimal representation

Just wanted to reply quickly to a specific comment John made in a recent thread above. He said:

"Thank you for addressing my concerns. I agree with #1, but I don't have the higher mathematical training to produce a rigorous proof. My reasoning goes as follows. I start with a foundation that all numbers are represented by one and only one decimal expansion. Excluding any zeroes you may add. As it is a foundation, I don't see any way to prove it right or wrong."

In fact, although it's counter-intuitive, it follows from the fundamental axiom of the real numbers, the Least upper bound axiom, that in fact all rational numbers with a terminating decimal expansion (one that ends in all zeroes) also have another decimal expansion that ends in all nines.

The reason is as follows. The Least Upper Bound axiom says that all upper bounded subsets S of the reals have a real number which is a least upper bound. That least upper bound does not have to be in S, but it does have to be a real number. For example, the subset of all numbers strictly less than 1 has a least upper bound of 1, but 1 is not included in that subset. Using that axiom, you can directly prove the Archimedean property that there are no positive infinitesimal numbers c such that for all n>0 nc>1 (the proof is included in that article). From that result, it follows that if two real numbers differ by at most an infinitesimal number c, then since c must equal 0 the two numbers likewise must be equal.

So from the Least Upper Bound axiom it follows that since 0.999... and 1 differ by at most an infinitesmal amount, and there are no non-zero infinitesimals in the real numbers, it follows that in fact 0.999... and 1 must be equal. And, in fact, you can show that all real numbers with a finite representation also have an equal alternate representation where the last digit is reduced by one and all zeroes are replaced by 9's. For example, 0.25 = 0.24999... .

Thus, although it's not intuitive, decimal representation in the real number system under the Least Upper Bound axiom is not unique. The only way to have truly unique decimal representation for all numbers is to remove that axiom, in which case you are allowing subsets to exist that have no least upper bound. It is possible to construct such a number system, but as above that is not the standard axiom system used for the reals. Dugwiki 16:27, 13 November 2006 (UTC)

Excellent post. This is irrefutable. Tparameter 17:09, 13 November 2006 (UTC)

I should point out that the above mentioned proof included in Archimedean property also relies on the other axiom of the reals that states that the real number system is an Ordered field, which means that it is assumed that for all real numbers a and b, the product a times b exists and is a real number. Again, it is possible to construct a number system in which there exist two numbers in that system such that their product is not a defined number in that system (ie the system isn't closed under multiplication), but that's not the axiom used. Dugwiki 16:59, 13 November 2006 (UTC)

If you propose that 0.999... is not the upper bound of [0,1), then what is? An upper bound should be included in the set; if it is external to the set then it is ineligible to be a bound. If I say 0.999... is the greatest number less than 1, and someone else says they can wedge a number in between 0.999... and 1, then they are using faulty mathematics. (1+0.999...)/2 = error. The "..." cannot be manipulated in this manner. --JohnLattier 23:17, 13 November 2006 (UTC)

You say an upper bound should be included in the set, but this isn't even true in the rationals! What's the least upper bound of the set {x|0 <= x < 1, x is rational} ? -- SCZenz 23:27, 13 November 2006 (UTC) (Terminology corrected. -- SCZenz 23:37, 13 November 2006 (UTC))

The least upper bound of the set of rational numbers in [0,1) is 1. 1 is not a member of that particular set, but it is a member of the set of real numbers, which is what is assumed under the Least Upper Bound axiom for the reals. In other words, every bounded subset of the reals has a least upper bound that is a real number. That least upper bound doesn't have to be in the subset in question, but it does have to exist in the reals. Hope that clears up your question. Dugwiki 16:01, 14 November 2006 (UTC)

It was more of a question for JL, to help him understand the phallacy of his line of arguments. -- SCZenz 16:14, 14 November 2006 (UTC)

You can't define an open-interval on the 1 side, [0,1), and then claim it's closed!!! It's an open set, which means the (least) upper bound is not in the set, by definition. Please learn basic math. Tparameter 23:29, 13 November 2006 (UTC)

Guys, sort out your labels. Subsets of real numbers never have unique upper bounds. If $x$ is an upper bound for a set $S$ then every $y>x$ is also an upper bound of $S$. You guys mean least upper bounds, which (for the Reals) always exist and are always unique. JohnLattier further confuses bounds with maximal elements, which is where a least upper bound for a set $S$ happens to lie within $S$ itself. Endomorphic 23:34, 13 November 2006 (UTC)

You're right. I corrected my remark above. -- SCZenz 23:37, 13 November 2006 (UTC)

+1. Tparameter 23:46, 13 November 2006 (UTC)

User:JohnLattier

Anyone who has followed the discussion knows that there is no reason to believe that JohnLattier isn't a troll. He was given several chances to correct his ways, and has failed us every time. This time it's for real: Unless there is a really serious change of attitude on John's part, we should simply stop responding to him. We have already explained everything to him in every possible way; why would trying again lead to any different result? -- Meni Rosenfeld (talk) 20:33, 13 November 2006 (UTC)

Agreed with the almost all of this, although Hanlon's razor may be a refutation of the first sentence. However, whether it's stubbornness, stupidity, or maliciousness, the effect is the same. It seemed as though progress had been made by him accepting that the statement is true in the real numbers and even starting to discuss new number systems. This got us all thinking about properties of number systems and might have even been useful for explaining the reason behind the properties of a useful system. But the refusal to either present or follow logical arguments means that this conversation is ultimately not useful. (By the way, if JL starts to say I accused him of stupidity because I said that was a possibility, I'm going to ignore it. Honestly my feeling on this can best be stated as follows: "Never attribute to stupidity that which can be adequately explained by stubbornness.") Calbaer 21:34, 13 November 2006 (UTC)

I think paraphrasing Stephen Colbert can neatly sum up this counter-equality camp. "Anyone can can explain the math to you. I promise to feel the math at you."--Trystan 22:13, 13 November 2006 (UTC)

Further, whether or not JohnLattier is right is irrelevant, as are his "logical arguments". He provides no sources that specifically advance the claim 0.999... != 1, nor any sources indicating that infintesimals are required to resolve 0.999... ?= 1, nor any sources advocating the inclusion of non-zero infintesimals in the Reals. Both experts he contacted explicitly contradicted his position. He admits his opinions constitute original research. Even if he could show that his own private infantesimals were consistent and usefull, Wikipedia is not the place to do so. JohnLattier knowingly and repeatedly violates 3 out of 4 founding Wikipedia principles: WP:NOR, WP:VERIFY, WP:WWIN (in particular, Wikipedia is not for things made up in school one day). His opinions re: 0.999... have no place on Wikipedia. This isn't bias, it's official policy. Endomorphic 22:44, 13 November 2006 (UTC)

This board is for the discussion of whether 0.999... equals or does not equal 1. It is a place to ask questions, give answers, and try to find a truth if one exists. It is a place for those who are right and those who are wrong to gather and share ideas. It is not a place to label people as trolls if their thought process differs. It is not a place to censor anyone's thoughts. Neither is it a place to provide links to a "you're stupid" message. All of these violations can and will go unreported because I am only here to figure out the cause behind the bold statement proclaimed by the 0.999 article. Endo, this is an arguments board, not an article. Statements made here do not have to be sourced each and every time. I may try to find sources for my reasoning if I have the time, be patient. I have not violated any Wikipedia rules. Someone even tried to report me to the administrators and they quickly declined it. Let's just try to discuss this topic in peace?? And I can't answer each and every question. When I log in there are 15+ new messages. I simply don't have time for everything. And many posts are not even worth responding to. --JohnLattier 23:30, 13 November 2006 (UTC)

• Report you to the administrators? Hardly necessary; the administrators have been here all along. Me, for example. Other than being consistently and vehemently not-even-wrong, you've not done anything (on this page, anyway) worthy of admin attention; and the whole reason this page exists is so people who doubt the validity of 0.999... = 1 can be heard (as opposed to on the the usual Talk page, which is reserved for discussing the article, not the basis for the article.) Rather, someone requested mediation, but did so in such a way that the request was summarily dismissed; see Wikipedia:Requests for mediation/Talk:0.999.../Arguments. It's possible a legitimately formed and phrased request might be entertained, but even so, that's not an administrative issue, but rather, one of discussions among equals, editor to editor. --jpgordon^∇∆∇∆ 23:49, 13 November 2006 (UTC)

And we are simply saying that many to most of us who have been following this now think your posts are not worth responding to either. I've given up on discussing math with you, and I've given up at refuting your personal accusations. Those posts that actually have detailed analysis you generally deem those "not even worth responding to." That indicates that you are not serious about exchanging ideas, whether that be due to trolling or due to stubbornness, and whether that stubbornness is refusal to learn or refusal to admit what you do not understand. In any event, nothing useful is being generated, and that is usually a good indication to stop something. Better we use our time to explain this problem to new people, and better you use your time to learn more and come back after educating yourself sufficiently. That is, not in days or weeks, but months or years. Calbaer 00:00, 14 November 2006 (UTC)

No one is forcing your hand. --JohnLattier 00:01, 14 November 2006 (UTC)

Sorry, JohnLattier, my earlier comment was a little harsh. All I meant was that your views will never see the light of day via any Wikipedia article. How about this: the first step in learning is admitting you're wrong. A good step towards that is entertaining the posibility that you might be wrong. Rather than trying to build another explination for a conclusion you want to retain, why not write down what irks you about 0.999...=1, or which lines in which proofs you don't understand? That way, instead of "NO!" you might recieve some usefull information. I suggest you search this page for (other) occurances of "ice cream" and read & contemplate that post if you haven't already. It may address some of your concerns. Endomorphic 02:16, 14 November 2006 (UTC)

The problem here, JohnLattier, is that you come here to this page, saying not "I find this result conflicts with my grasp of mathematics, please help me improve my understanding," but rather, "All of you, and all professional mathematicians, are wrong about mathematics. You are using a foolish system with bad properties and you should stop." You then fail to respond to any post that might seriously challenge your claims, you fail to read the articles we repeatedly direct you to, you yourself acknowledge that you do not have the skills to properly utilise rigourous mathematical methods, and then you tell us we are wrong, and finally you continue to make up ideas based on your intuition with no formal mathematical basis. This debate is pointless. We can not help you if you refuse to listen and learn. Your stubbornness and refusal to entertain the notion that your intuition might be wrong means there is no point in this discussion continuing. If you are not going to try to improve your understanding of mathematics, please leave. Maelin (Talk | Contribs) 03:48, 14 November 2006 (UTC)

So, if 0.999... is the number immediately below 1, what's the number immediately below 0.999...? And how many numbers are there between 0 and 1, exactly?

How about this

Considering:

Is infinity a member of the reals? Because if infinity is not in the reals, then 0.999... is not in the reals (unless you know of some other way to define it without using infinity), and it cannot equal 1 if it doesn't exist in that number system. --JohnLattier 23:36, 13 November 2006 (UTC)

is shorthand for the limit of the sequence -- SCZenz 23:39, 13 November 2006 (UTC)

1. Is ∞ a member of the "reals?"
2. Can 0.999... be defined without using ∞?
3. Is 0.999... a member of the "reals?"

--JohnLattier 00:05, 14 November 2006 (UTC)

1. No.
2. Yes. You have given a possible definition without using ∞ above, because as I said the ∞ symbol does not mean the number infinity but is rather short hand for the limit of a sequence.
3. Yes.

Does that help? -- SCZenz 00:08, 14 November 2006 (UTC)

The limit of a sequence "as n goes to infinity?" --68.158.133.18 01:16, 14 November 2006 (UTC)

That is also shorthand language. We may write/say "the limit as n goes to infinity," but in fact limits are defined without treating infinity as a number in any way. Please read the article on the limit of a sequence if you want details; that's why I'm linking it. -- SCZenz 01:31, 14 November 2006 (UTC)

Of course you even can define "limit as n goes to infinity" in a way that treats "infinity" as a genuine object, not as shorthand. It's still not a member of the reals, of course. How is that relevant? Not everything that you make reference to, in discussing the reals, needs to be itself a real. --Trovatore 02:08, 14 November 2006 (UTC)

You will find, on the limit of a sequence article, a section entitled formal definition. This gives a formal definition of a limit as n approaches infinity, and you will see that the definition does not make any reference to infinity at all. If you don't like the phrase "as n approaches infinity" because it refers to infinity which is not a natural number, you can mentally substitute it with "as n gets arbitrarily large" and it means the same thing. The "n approaches infinity" thing is just a bit of shorthand, which we use because it is intuitively agreeable with what we actually mean formally. Maelin (Talk | Contribs) 03:21, 14 November 2006 (UTC)

Yes, I know how to do all that. But you don't have to. Take the natural numbers, plus an extra element labeled "infinity" greater than all the naturals, and give them the order topology (boils down to, a set is open if it does not contain infinity, or if it contains all naturals greater than a given one). Then works just the way you want it to, by the usual topological definition, and it's arguably more natural. --Trovatore 03:36, 14 November 2006 (UTC)

Perhaps, but I suspect that John's continued misconception that 0.999... cannot exist in the reals unless nonzero infinitesimals also exist in the reals is better combatted by removing all explicit use of infinity as an actual number, rather than officially condoning a few select ones. By the way, my post was directed at John, not at you. I wasn't implying you needed to learn how limits of sequences work, since you obviously understand them already. Maelin (Talk | Contribs) 04:20, 14 November 2006 (UTC)

Combat? Listen, you can't escape the fact that infinity is being used to defined 0.999... whether it be called "all reals greater than 1" or "as n gets arbitrarily large." I would like to discuss the interesting point Trovatore brought up that not everything you make reference to in discussing the reals needs itself be a real. Are you suggesting that even though 0.999... requires the use of a non-real to describe its existence, it would still be a real number? I'd need some time to think about that. --JohnLattier 04:57, 14 November 2006 (UTC)

you can't escape the fact that infinity is being used to defined 0.999... whether it be called "all reals greater than 1" or "as n gets arbitrarily large." That's pure rubbish. Very large numbers are in no way the same as infinity. Why don't you actually read the definition of the limit of a sequence before you comment further? -- SCZenz 05:18, 14 November 2006 (UTC)

Two things. 1.The article you refer to uses infinity just as I said it does. 2.What is this very large number you speak of where the summation reaches 1? --JohnLattier 05:54, 14 November 2006 (UTC)

No, it does not say that. If uses the symbol for infinity, but it does not in fact use it formally as a number (or anything else). Please read the formal definition. If you don't understand it, ask questions; without you knowing what the limit of a sequence is, there is no possible way you can understand any of the math involved in this topic. -- SCZenz 06:01, 14 November 2006 (UTC)

Are you implying the word "every" imposes a boundary? --68.158.133.18 06:42, 14 November 2006 (UTC)

Fine. Since you will not read the article yourself, here is a definiton of 0.999... with no mention of infinity anywhere. I have taken the formal definition from the article you refuse to read, and have replaced generals with specifics to tailor it to this situation (e.g. using |x - y| instead of d(x,y))

• We define a sequence a_n by
• Then we can see that a_n is the sequence {0.9, 0.99, 0.999, 0.9999, 0.99999, etc...}.
• Now, choose any positive real number ε (that is the Greek letter epsilon). This is our ε > 0, and it can be as arbitrarily small as we like, as long as it is a nonzero, positive real number.
• Now we look for a real number x that satisfies the following property:
  • For any ε, no matter how small, there exists a sufficiently large finite natural number N, so that for every natural number m that is greater than N, it is true that |x - a_m| < ε.
• That is, no matter how tight we pull our ε value, no matter how close to our number x we want to look, there is some term in our sequence (a_n), which we have called the Nth term (i.e. a_N), so that every term after the Nth term is within an ε-distance of x. There can only be one number x, because any other number could be excluded just by making epsilon sufficiently small.

Then 0.999... is x. Now, this is a definition of 0.999... that uses the formal definition of a limit of a sequence. Note that it does not use infinity, just like we said. We do not need infinity to exist in the natural numbers or the reals in order to discuss 0.999..., we just use it as shorthand because it makes life easier. Maelin (Talk | Contribs) 08:18, 14 November 2006 (UTC)

Do you not see your use of "any" and "every" is equivalent to using infinity? I thought maybe you had some kind of deeper understanding of that linked article, but you repeat the same thing it says yet still don't even notice the implied infinity? It's not even really implied since they outright say it in the article. 0.999... will not be able to escape infinity, the "..." itself is a measure of that. How about you make life easy on the rest of us and just tell us what value of N represents your Nth term without using words like "every" "any" or anything else that leaves it open-ended? --JohnLattier 12:45, 14 November 2006 (UTC)

I'll give an explicit formula for the N and the Nth term used above: For a real number x, let [x] denote the largest integer less than x. That's a function on the reals that will take the integral part of non-integers and whose restriction on integers is the map . The N given above depends on ε: Let N be [1/ε]+1. Then, since ε was a positive real number, N is an integer greater than or equal to 1. The Nth term is given by . Sure, that depends on ε, but that was not forbidden (and is, in fact, required).

I'm not sure I understand your remark about "any" and "every" being equivalent to using infinity. Surely you'll agree that I can make claims about every member of the set {1, 2, 3} without the "every" amounting to a use of infinity. Do you mean that infinity is involved because there are infinitely many reals, or because some of the numbers we speak about could be infinitely large (or small)? The latter is false, there are neither infinitely large nor infinitely small numbers involved (we use real numbers). And while there are infinitely many of them, a statement like "For every real number we have property P." still does not involve infinity; it's equivalent to the statement "The set of real numbers without property P is empty.", and here surely is no infinity any more. --Huon 14:25, 14 November 2006 (UTC)

It depends if there is a limitation established. If you mean {1,2,3}, this is bounded on both sides and exclusive of all but three terms. There is a finite number of terms. But to say every real number, that would be open-ended, unbounded. As for real numbers, while it may not use infinities to describe any individual number (lets hope it doesn't), an open-ended set of real numbers, such as "all real numbers," or "every real number greater than 1," or "every point between 1 and 2," is going to be an infinite amount of members of that set. To say "arbitrarily large" N or "arbitrarily small" ε most certainly invokes infinities. Unless, as I have asked 8 times already, someone can name for me the N or ε not defined by infinities (open-endedness) that they are using to define 0.999... and infinitesimal. --JohnLattier 02:19, 15 November 2006 (UTC)

---

Responding to John's question to me above. John, while the others are of course correct that you don't need a number called infinity to take the relevant limits -- suppose they were wrong? Suppose you really and truly did need a number infinity, outside the real numbers, to explain this limit, which is a real number? What exactly would be so strange about that?

You seem to be claiming that a limit as n goes to infinity can't be a real number, if infinity is not a real number. That strikes me as analogous to claiming that you can't say a pride of lions is two or more lions -- because the number two is not a lion. --Trovatore 07:17, 14 November 2006 (UTC)

Your analogy would be true if I were trying to proclaim a pride is a lion. A pride is an amount of lions, as is two. --JohnLattier 12:45, 14 November 2006 (UTC)

I think John's question actually indirectly relates to what I posted above about the reals having no non-zero infinitesimals because of the Least Upper Bound axiom. In fact, I believe that if that axiom is removed, then it is possible to construct a system where . The reason is that technically if you construct a new number system without that axiom, you might be able to redefine the symbol using, for example, the infinite union of corresponding sets. In that case, you could end up with an infinite sum which differs from the limit of the appropriate finite sums by an infinitesimal number under that new number system which is not strictly zero. That is, if , but for some strictly positive infinitesimal c, or even maybe that no unique limit exists but rather the finite sums approach an infinite number of values that differ by a set of possible infinitesimal values.

That's all speculation, though, and it doesn't apply to the real numbers as normally defined. Rather, it would be a system that uses some sort of set construction, for instance, and creates a labelling between those sets and either real numbers or infinitesimal numbers. In the reals, though, (see Series (mathematics) for other information on infinite sums in the reals).Dugwiki 16:23, 14 November 2006 (UTC)

Would that make sense of the fact that as the summations go to infinity, both 0.999... and 1 are approached but never "reached?" Both act as a limit? --68.158.207.15 01:12, 15 November 2006 (UTC)

An alternative way of defining the problem?

This 'problem' (okay okay you qualified mathemeticians out there I know it's not a problem for you, just my intuition ;p) has fascinated me since I read about it on a forum I frequent. I have flash-learned math just to be able to fully grasp what the issues are and in doing so very much enrichened my understanding of numbers.

During the thread I came up with (with the help of a pure math graduate) a different way of describing the problem I would have to solve were there to be a proof that 0.999.. does not equal 1. Just wondered what you people 'in the know' on here thought of it and whether you thought it was solvable/provable. (I suspect I already know the answer, but here we go anyway...)

Now I'm about a week new to anything over basic GCSE math, but I've been reading like its gonig out of fashion on the subject and as far as I can see, if you can prove the above it follows that the hypothesis that must be false.

I get the feeling I must be wrong somewhere, but I don't know where... any pointers? Archgimp 08:12, 14 November 2006 (UTC)

Well, there's two things there. If you want it to be a valid disproof, you need to prove firstly the premise (that is, prove that the set you described is nonempty), and then you also need to prove the logical implication (i.e. prove "if this set is nonempty, then 0.999... is not 1"). It seems like the logical implication is fairly obviously true, although I can't see how to prove it at first glance. But nonetheless, your set is empty. I can't think of a good proof that doesn't rely on the fact that 0.999... = 1 (which is hardly going to be convincing for you), but someone else might be able to whip up a proof of the emptiness of your set. Maelin (Talk | Contribs) 08:30, 14 November 2006 (UTC)

I don't mind being proven wrong at all, after all the only thing at the moment which tells me that is intuition, and intuition can get your legs broken. (I wonder how many people laying in traction in hospital say to themselves "I know I checked both ways before crossing, I always check both ways"). The interesting part for me is that the method described above could lead to a new understanding of the issue, and I came up with it all by my lonesome (although undoubtably, someone else has also come up with it before me, I haven't seen it written anywhere else). I think of it this way, even if the mountain doesn't have a summit, it's satisfying to climb higher up than you managed last attempt. Archgimp 08:55, 14 November 2006 (UTC)

It could be useful, yes, if understanding that set was easier than understanding the problem... and, of course, if what you want to prove were true. -- SCZenz 09:10, 14 November 2006 (UTC)

So let's see... if x is in the set you define above, then 1/x < 1 and 1/x > 999...9/100...0, where the fraction has as many 9's and zero's, for any number of repeated 9's and 0's. We can rewrite the fraction, more concretely, as for any n. so we have (by flipping the inequalities) that for all n. Define y = x-1, so that we have for all n. Thus we have an x in your set only if there's a positive number y that's less than for all n. There is no such number, so the set is empty. -- SCZenz 09:18, 14 November 2006 (UTC)

Continuing my musings... If what I said above is true (i.e. if your set is non-empty), then from the field axioms we have that a number y^-1 exists such that. y*y^-1 = 1. Multiplying the inequality by y^-1*10ⁿ, we get that y ^{− 1} > 10ⁿ for all n. This is effectively equivalent to saying there exists a largest number, or at least a number bigger than every integer; it should be even more clear that this is false (it's a direct consequence of how the natural numbers are defined, in fact). -- SCZenz 02:13, 15 November 2006 (UTC)

If you are trying to squeeze a point x in between 1 and 1/0.999... it won't happen. It's defined as being infinitely close, and infinities are not subject to further magnitudinal change. --JohnLattier 02:31, 15 November 2006 (UTC)

Thanks everyone for taking the time to explain to me why the set must be empty. I am getting up to speed on this as fast as I can without formal education in the subject. I think I may have to give up on my intuition and bow at the feet of logical proof on this one. Now onto the easier problem of finding a way of quantifying y where for every N there are y composite numbers =< N heh ;) Archgimp 08:55, 15 November 2006 (UTC)

You're welcome. Thanks for asking, and listening; it ended up being a useful illustration of how trying to stuff numbers between 0.999... and 1 requires infinity, which I think most of us can agree shouldn't be a number. (And indeed isn't, in the usual real analysis.) -- SCZenz 17:08, 15 November 2006 (UTC)

Just end it all

Look, talking with JohnL about this is unnecessary -- and he may as well just go away. The first sentence of the article is explicit: In mathematics, the real number denoted by the recurring decimal 0.999… (also written ) is exactly equal to 1. Every argument John makes requires not using the real numbers; that being the case, there really isn't anything to argue about, since he doesn't accept real numbers (they're "trickery"). Now, if he wants to assert that the equality is untrue in the reals, that would be a different matter entirely; and nothing in the article extends the equality past the reals. --jpgordon^∇∆∇∆ 23:23, 14 November 2006 (UTC)

He asserted both that it was true and false in the reals. He went to professors but refuses to read short articles on the subject, let alone books. He has enough time to "contribute" a dozen times a day, but claims he doesn't have time to respond to those questions that seriously analyze the issue or to those that find holes in his arguments. He says falsehoods with a enough confidence to make people unfamiliar with the subject believe. Like I said before, it doesn't matter whether he's a crank, a troll, or something else. The effect is the same and both old and new contributors should concentrate on those who have both the desire and the capacity to understand the concepts of this article, rather than those who don't. Calbaer 00:15, 15 November 2006 (UTC)

Agreed. His post above, after the one of mine giving a formal definition of limits, has made it apparent that he has no ability to follow a formal mathematical argument nor the desire to learn how. He comes bearing serious misunderstandings of mathematics but refuses to release his intuition-borne misconceptions. He repeatedly tells people with more mathematical knowledge than himself that they do not understand infinity but that he does, and finally, as mentioned, he ignores posts that find the holes in his arguments or could make serious progress in improving his understanding. I will not respond to him again until he demonstrates a willingness to learn and an appreciation that his intuitive ideas about maths are more fallible than rigourous definitions and proofs. Maelin (Talk | Contribs) 00:32, 15 November 2006 (UTC)

Jp: 0.999... does not conform to the requirements to be a member of the real number system as it is defined.

Calbaer: I never assert that this is true for reals. The closest I came was when I discovered this entire debate is restricted to real analysis, and I ask, oh this is true only for real numbers? Then I come to find out it's still not true because 0.999... is not considered a real number.

Maelin: I successfully showed that 0.999... is reliant upon infinity (a non-real) to be defined. Your article backed up my statement, then you backed it up as well, unwillingly (I dare say unknowingly?). In essence I have shown that these two numbers, 0.999... and 1, cannot be equal because the same axioms that lead to your proofs lead to 0.999... not being a member of the reals whatsoever. You can continue to back out of this debate if you wish, I won't stop you. --68.158.207.15 00:48, 15 November 2006 (UTC)

Actually, you claimed it. In maths, showing something is when you make an assertion with some kind of convincing logical argument to support it. You didn't show anything because you have not even once formulated a structured mathematical argument yet for any of your many assertions on this page. I'm not going to try to explain to you why a "for any real number" type statement does not require infinity to be a member of the reals, go and ask a professor about it. Just wanted to make clear the difference between an unsubstantiated claim and actually showing something. By the way, that "feel free to give in and leave if you can't stand up to my arguments" card you pulled is a pretty clear pointer to trolling. Feel free to try it again, I won't respond next time. Maelin (Talk | Contribs) 03:26, 15 November 2006 (UTC)

Why ask him to waste someone else's time? He already tried two professors. And he won't follow or read, so what's the point? If he were serious, he'd realize that, in the hours he's spent Wiki-ing, he could've read a book to better arm himself mathematically, or, better yet, learn how the current Information Age is a product of the "trickery" of rigorous mathematics. And, yes, he likes the double-bind. If you don't respond, clearly (he says) he has won the argument. If you do, don't blame him for wasting your time, because he's not forcing you to do so, and — poor guy — he can't be expected to reply because there's just so much to respond to! Calbaer 03:56, 15 November 2006 (UTC)

When I say trickery, I am referring to manipulations like 10*0.999...=9.999... That is trickery. That is blatantly incorrect, the *10 cannot be applied to the "..." There is no answer to 10*0.999... but giving it such an answer results in the truncation of an infinitesimal quantity. Then, adding up three 0.333... is trickery. The remainder goes unaccounted for, and again an infinitesimal is truncated. Then, setting a sigma to go to infinity-1 is trickery. Infinity-1 is not valid mathematics. It does not equal infinity. Then, using a non-real to define 0.999... results in the birth of a number which automatically conflicts with the definition of the reals (that any two non equal points are separated by an infinite number of points) from which it was borne. It cannot exist among that number set. It's an internal contradiction. Then, saying the summation that provides 0.999... is exactly equal to its limit, and then saying its limit is 1 is trickery. The limit is the number(s) which are being approached. In this case, all three of 0.999... and 1 and 1+1/inf are being approached, although given its approach direction one can rule out 1+1/inf. That does not mean the summation itself ever, upon any value of independent variable, actually equals 1. Then, to say that 0.999... is truly a member of [1,2] is trickery. The reasoning? Because 0.999... equals 1? That's completely wrong. 0.999... can never exist in the set defined by [1,2]. 0.999... is, in a sense, a living number, it is forever approaching 1, that's what the ... stand for. It's part of the definition of the number itself, as being a product of a summation that never ends. And it approaches 1 from a very specific direction. From <1. It does not approach 1 from >1. Only <1. Therefore, since it is a number defined by a neverending summation, it is always approaching but never exactly equal. Then, saying that there are no non-zero infinitesimals in the reals is nice, but zero is not an infinitesimal either. Based on the definition of infinitesimal, it is not zero. If a number system pulls and tugs on the infinitesimal to redefine it and call it equal to zero, that number system is flawed. But you know what? The real number system DOES NOT do this. The real number system defines the infinitesimal as NOT a MEMBER of its set. It is those who interpret these axioms that falsely categorize the infinitesimal as exactly equal to zero. I have not even seen any valid explanation why they do this. And I await that explanation with an open mind, but fear that the explanation will manipulate infinities or perform some other kind of "trickery" as I so aptly put it. Trickery? Not the number system itself. The trickery is performed by those who misunderstand infinities and use faulty mathematics to produce pretty results. --68.158.207.15 01:07, 15 November 2006 (UTC)

From above, JL formally defines what it is to be a mathematician: "those who misunderstand infinities and use faulty mathematics to produce pretty results." You go on believing that JL; and the rest of us will stick with these "tricks" that have allowed for the invention of everything from the standard model to television and even the internet itself, which of course creates a place for people like you to expose the "trickery" we call mathematics. Tparameter 01:59, 15 November 2006 (UTC)

If you are a mathematician you would surely know if A is a member of B, that doesn't mean B is a member of A. Those that use trickery might be mathematicians, but does that mean a mathematician is defined by the use of trickery? Of course not. Tell me, what good has ever come out of setting the infinitesimal exactly equal to zero? I'm not talking about the limit; please do not get these confused. All of mathematics, all the known universe, continues to exist without two separate numbers eclipsing, so what benefit would it be to promote them from convergant to equal? --JohnLattier 02:11, 15 November 2006 (UTC)

Simple. Provide a counter-example. Name one, JUST ONE, mathematician that disagrees with our "trickery" of limits. Tparameter 02:49, 15 November 2006 (UTC)

I wonder what John L. Bell would have to say about this. He seems to have an understanding of the infinitesimal. "If two points on a continuous line R are indistinguishable this will not in general imply their identity here." “In practical approaches to the differential calculus an infinitesimal quantity or number is one so small that its square and all higher powers can be neglected, i.e. set to zero.” “From the principle of infinitesimal affineness it also follows that all functions on R are continuous, that is, send neighboring points to neighboring points. Here two points x,y on R are said to be neighbors if . . . x and y differ by an infinitesimal.” I'm curious about this smooth infinitesimal analysis. It seems to be going in somewhat the right direction. Although it proclaims the infinitesimal to be both equal and unequal to zero simultaneously, which, in my understanding, defies the definition of equality. Still interesting. --JohnLattier 03:54, 15 November 2006 (UTC)

Tell me, what good has ever come out of setting the infinitesimal exactly equal to zero?

A well-defined, internally consistent, and extraordinarily powerful number system called the reals.--Trystan 02:33, 15 November 2006 (UTC)

All of mathematics continues to exist *because* two numbers "eclipse". John, you alone are the source of confusion. The infintesimal is not "set to zero" in real analysis; "set to zero" were *your* words, already refuted by Meni, and Trystan is lax to accept that wording. Analysis employs n-tuples, metrics, balls, measures, open covers, functions, families of functions, all manner of things which are themselves not members of the Reals. Infintesimals are not included in the Reals. The singular broken item is JohnLattier's irrelevant intuition. Also, make arguments with equations, not wordy hand-waving. For instance,

.

Until you can put maths into an equation, you're running around with a pitchfork, flaming torch, and mob chorusing "trickery! trickery!". Not the path to enlightenment. Endomorphic 02:40, 15 November 2006 (UTC)

Again, truncation. Manipulation of infinities. It should've read:

Your equations are easy to see through. If you claim infinity-1=infinity, you're sadly mistaken.

"Infintesimals are not included in the Reals. The singular broken item is JohnLattier's irrelevant intuition." Neither are infinities, that's why 0.999... isn't a member of the reals either. And why 0.999... cannot equal 1. It's not a matter of intuition. The only role intuition plays in this entire debate is that proponants of 0.999...=1 rely on shock value of opposing peoples' intuitions. Irrelevant? Hardly. I have demonstrated a plethora of excellent points, while showing the faultiness of the many mathematical manipulations of infinities. In fact, I have yet to see anything convincing from anyone that 0.999... could equal 1. I'm still waiting. --JohnLattier 02:54, 15 November 2006 (UTC)

And you can define for us what a "sum to infinity minus 1" means? Mdwh 03:05, 15 November 2006 (UTC)

What's meant is if you are going to multiply this sum by 10, and you are going to compare the sum before and after this multiplication by 10, you have one sum with relatively less terms. The magnitude of ∞ has been tampered with, so the sum after being multiplied by 10 is either in error or is subject to a yet-undefined set of rules regarding the manipulation of ∞. Can one say ∞-1=∞? Or ∞_A-1=∞_B? ∞-1>∞? What can be said is that once this ∞ is manipulated, it has a profound affect on the remaining sequence, that is, it truncates an infinitesimal value leading to 0.999...=1. Is this truncation sound? Can it not be said that if a truncation of infinitesimal is used to set 0.999...=1, then they are in fact an infinitesimal distance (0<x<N where N equals all real numbers) apart? Any number system that allows 0.999... should also allow 1/∞. --JohnLattier 04:13, 15 November 2006 (UTC)

I asked for a definition of "sum to infinity minus 1", and you haven't given it. A "sum to infinity" is defined as the limit of the partial sums, so what's your new definition? Mdwh 00:46, 16 November 2006 (UTC)

• You'll wait forever, then, since you disbelieve in the fundamentals of mathematics and have instead an intuitive misunderstanding thereof. Since you don't understand the reals, and since you don't understand how non-reals can be used to describe the behavior of the reals, you will never be convinced; you're talking a different language from that of mathematicians and others who use the results of mathematicians' work (which includes pretty much every single scientist that lives or has ever lived). Be comfortable with your unique mathematical formulations, if you want; but it's really time for you to leave this discussion now. --jpgordon^∇∆∇∆ 03:07, 15 November 2006 (UTC)

Exactly; and if he doesn't leave, we shouldn't respond. To anyone with sufficient mathematical and/or logical literacy, this section alone illustrates how useless it is to respond to JL. Someone with insufficient mathematical knowledge may have to look in the archives to see this. In any event, there are very interesting concepts here to discuss with someone with an open mind and a willingness to learn. In spite of protestations to the contrary, JL has shown no evidence of possessing either and plenty of evidence of enjoying being the center of attention. And this page was not created to stoke the egos of cranks. Calbaer 03:22, 15 November 2006 (UTC)

+1. Excellent Post. In addition to what you said, I am amazed at the confidence he seems to have. (I say "seems" only because I still think he's a high-level troll.) He abuses notation and creates contradictory "rules" of math on the fly, making sweeping generalizations without a sliver of understanding about the things he speaks. It's amazing. When a first rate mathematician presents something as a classical theorem, for instance, I assume it's true and verify the proof later. I don't charge in armed strictly with "intuition" and tell him he's wrong. There is a certain humility required to really learn these things. Analysis is often "counter-intuitive", whatever that means. The reason is because we usually just see the synthesis, while the analysis was the actual path of discovery. The synthesis, however, is usually the tough part, and it doesn't always make intuitive sense. Tparameter 03:35, 15 November 2006 (UTC)

Continue to dodge the topic at hand in pursuit of personal attacks; I don't mind. Anyway, back to topic again. Infinity-1 is (or should be defined as) undefined. I still have not been shown any set of official rules regarding the manipulation of magnitudes of infinity. I have legitimate concerns and questions, so if there are any takers who have not scampered off, please help a fellow math enthusiast out with some infinity rules. Thanks. Oh, and if they have not been formally defined, shall we make suggestions? What is infinity? How is it defined? What number sets is it a part of? Which operations can be performed on it? Discuss the topic of changing the magnitude. Can two infinities of differing magnitude be compared? Can they be reinserted into equations? (2∞ > ∞)? That looks like a face. 2∞=∞? Let's discuss? Anyone? --JohnLattier 04:04, 15 November 2006 (UTC)

See the article entitled "infinity". This topic is about a real number. Better yet, take Calc I. From the aforementioned article:

Operations involving infinity and real numbers.

If then

If then

Certainly they are defined. I suggest you read the textbook A Transition to Advanced Mathematics by Douglas Smith, Maurice Eggen, and Richard St. Andre if you are interested in the answers to your questions. The book will patiently tell you all the answers you are looking for. I used that book to begin my studies in upper division mathematics in college, and it served as an excellent foundation to understanding many basic concepts in mathematics, including proofs, set theory, and cardinality (including infinities). If you can't find that textbook, I'm sure you can find many others that will also suit your purposes. -- Schapel 04:29, 15 November 2006 (UTC)

Thanks Schapel, I'll try to see if my school has that book. I looked at the Wiki site for those infinity rules, and I don't see it sourced. What is the reference for that? I disagree with most of those rules. I would say, for instance (where N is any number):

or a different magnitude of infinity

Because if and N − N = 0 then and So it's only true if x=0.

or a different magnitude of infinity

Because if and N * 0 = 0 then So it's only true if x = 0.

or a different magnitude of infinity

Because? For the same kind of reasoning, it's only true if x=1.

Don't get me wrong, I know infinity alone has no defined magnitude. But when comparing infinities, it's very important to take this into account. I see 2∞ > ∞. Because if it weren't, then for instance f(x)=2x/x would = 1 for the limit x->∞. --JohnLattier 05:10, 15 November 2006 (UTC)

If you read that book, or any other math book that rigorously defines infinity, you'll see why your statements about infinity are incorrect. Go and learn the material before you start debating it. -- Schapel 13:39, 15 November 2006 (UTC)

JL said, "I disagree with most of those rules" (infinity). Then go there and argue about it. This isn't the right place, that's why I gave you the link. Tparameter 05:41, 15 November 2006 (UTC)

Okay. It seems like 0.999... hinges on manipulation of infinity-defined values. --68.158.207.15 07:17, 15 November 2006 (UTC)

John's problems with manipulations of infinity are solved differently: While we do set ∞+x = ∞ for every real number, what we have to leave undefined is ∞-∞. Similarly, for positive real x, we may set ∞*x = ∞, but have to leave ∞/∞ undefined. Finally, 0*∞ also is undefined, but 1/∞ can be set equal to 0. One reason for these conventions is the arithmetic of limits: We do have a formal notion of sequence "going to infinity" (and no, even that definition does not involve infinity itself), and if we consider all real numbers as limits of sequences and then add an element "∞" as the "limit" of the sequences "going to infinity", arithmetics of sequences and those rules for the limits are consistent. All this can be made more precise than I did here, but that requires more topology than I believe useful. --Huon 11:24, 15 November 2006 (UTC)

Wow. Read the article. Yes, 0.999... "hinges" on limits. It is interesting that you debate these things obviously without having any background in calculus. 72.193.74.36 15:35, 15 November 2006 (UTC)

To reply to one of John's statements, he said that . The symbol isn't normally defined, since infinity isn't a real number. Rather, the infinity symbol is actually shorthand for "including all the members of an infinite set". So "infinity minus 1" isn't normally defined.

However, in the reals, . Therefore one interpretation of what John means might be the following - . You can prove, though, that using the definition of limits. Therefore, in the reals, under that definition of the symbol. Dugwiki 16:23, 15 November 2006 (UTC)

Also, John Lattier asked "I wonder what John L. Bell would have to say about this. ... I'm curious about this smooth infinitesimal analysis. It seems to be going in somewhat the right direction. Although it proclaims the infinitesimal to be both equal and unequal to zero simultaneously, which, in my understanding, defies the definition of equality. Still interesting." The adobe acrobat file referenced in the article smooth infinitesimal analysis has a good presentation of what John Bell is describing.

Note, though, that if I'm reading it correctly this is essentially constructing what is basically a new mathematical system and isn't standard mathematical analysis. For example, under the axioms of smooth infinitesimal analysis, discontinuous functions are apparently prohibited. If x and y differ by an infinitesimal amount, the f(x) and f(y) must likewise differ by an infinitesimal amount. For example, there are no discontinuous functions such as f(x) = {1 if x=0; 0 otherwise}. Likewise, under the axioms of smooth infinitesimal analysis, the law of the excluded middle does not apply, meaning that it is not always the case that a proposition is either true or false.

So it's important to keep in mind that what Bell has constructed is essentially a very different system from standard real analysis. I'm sure there are places where Bell's model provides useful information or simplifies analysis, such as when dealing with specifically continuous functions. But at the same time it appears to lose the ability to analyze things like point-wise discontinous functions. As to the question at hand of whether or not 0.999... = 1, my guess is that in smooth infinitesimal analysis they probably differ by a non-zero infinitesimal. Dugwiki 17:08, 15 November 2006 (UTC)

A FAQ for this page?

The main page has a FAQ, but I think that this page could use one too. I propose the following start:

Q: 0.9 < 1, 0.99 < 1, and so forth. Therefore it's obvious that 0.999...<1.

A: No. Something that holds for various values need not hold for the limit of those values. For example, f(x) = x³ / x is positive (>0) for all values in its implied domain (). However, the limit as x goes to 0 is 0, which is not positive. This is actually an important consideration in proving inequalities based on limits.

Q: You guys talk a lot about real analysis, limits, and calculus; shouldn't this just be about arithmetic?

A: Unfortunately, in order to formally prove many qualities of numbers, one often has to resort to higher mathematics: real analysis in the case of real numbers, number theory in the case of integers, and so forth. The article and arguments page both aim to be understandable to all, but, since many skeptics ask for formal proofs, higher mathematics will inevitably come into play.

Q: Person X made a pretty convincing argument that ! Those who say otherwise are giving arguments I can't understand.

A: Before believing an argument, check sources, responses, and record. The main article is well-sourced, whereas arguments against generally cite (if anything) non-mathematical sources such as online message boards and dictionaries. Also, although most people are trying to write something everyone can understand, some arguments, in relying on higher mathematics, will not be easy to follow for all. Still, try to follow those you can. Finally, those who firmly believe that mainstream mathematics is mistaken will generally reveal their lack of rigor and/or contempt for experts and others who disagree with them. Before replying, read their other contributions to make sure you aren't siding with someone you yourself would not trust.

What do you think? Calbaer 20:58, 15 November 2006 (UTC)

Maybe add the following Q/A:

Q: Is it possible to create a new number system other than the reals in which 0.999... < 1 by an infinitesimal amount?

A: Yes, it is possible to create a system other than the standard real number system such as the Hyperreal numbers in which that system's equivalent of "0.999..." differs from that system's equivalent of "1" by an infinitesimal amount. Such a system would have to include non-zero infinitesimal elements. The standard real number system most commonly used by mathematicians, though, has no infinitesimally small numbers (other than zero) due to the Least upper bound axiom, and so for almost all mathematical applications 0.999... = 1. Dugwiki 21:54, 15 November 2006 (UTC)

The problem with that phrasing is that people will think that (0.999...)*(0.999...) = 0.999... < 1, which is true for no number system I know of (or could even imagine). Calbaer 22:20, 15 November 2006 (UTC)

Be very, very careful with an answer like this. If you use another number system, you have to be sure that 0.999... represents something analogous in definition to what it represents in the reals if you want this problem to be solved in an interesting way. Also, people are regularly offering the hyperreals as the solution to all our problems, but nobody has proven that 0.999... ≠ 1 in the hyperreals, either. The mere existence of infinitesimals does not cut it. Maelin (Talk | Contribs) 01:02, 16 November 2006 (UTC)

Good point, Maelin. You'd first have to prove that 0.999... ≠ 1 in your alternate system. Infinitesimals are obviously a necessity, but they might not be sufficient by themselves. A rewording of the above Q/A is probably a good idea.

As far as Calbaer's comment about (0.999...)*(0.999...), if you have a system in which 0.999... = 1-c for some infinitesimal c, then it would follow that (0.999...)² = 1 − 2c + c² (or something like it), which would in fact not equal either 0.999... or 1 but would differ from them by an infinitesimal amount slightly different than c. Obviously in the reals, though, c=0. Dugwiki 16:24, 16 November 2006 (UTC)

That's my point; those who believe the infinitesimal cannot be doubled, halved, or otherwise manipulated would have trouble. Although I know little about the hyperreals, the smooth infinitesimal analysis, etc., I do know that they have no "smallest infinitesimal." In fact, those who cite calculus yet believe in the smallest infinitesimal would have quite a lot of trouble differentiating x/2. My example, in retrospect, wasn't the most illustrative, though, and Maelin makes a similar point better than I did. Calbaer 19:09, 16 November 2006 (UTC)

Looking at it off-handedly, it looks like the Surreal number system might be a good candidate to try. Under system, you have the non-zero infinitesimal ε = {0|..., 0.001, 0.01, 0.1, 1} and the number 1. I'm pretty sure it follows that 0.999... = 1 - ε , and since ε ≠ 0 it would follow that 0.999... ≠ 1 in the Surreals. It would be interesting, though, if someone else could double check that, since I'm still getting my head around how surreal numbers work. Dugwiki 16:53, 16 November 2006 (UTC)

Don't bother with SIA. I've made some referenced comments on it below; it's consistent, but it breaks too much remain usefull or intuitive. For instance, you can't prove that any of the infintesimals aren't zero. As for the surreals, they contain the real numbers as a subfield. Which means either 0.999... = 1, or you have to give a *different* definition of how a decimal works for the surreals. I can't see how to make the other terminating decimals match, eg, 0.999 (surreals) correspond to 0.999 (reals) while making 0.999... (surreals) distinct from 0.999... = 1 (reals) without breaking something like the metric, or cauchy sequences converging. Endomorphic 22:16, 16 November 2006 (UTC)

The reals are presumably a subfield of the surreals, but I'm not so sure that in cases where a real number has multiple decimal representations that both those representations map to the same surreal number. So while the symbol "0.999..." maps to 1 in the reals, it might be that 1 maps to 1 in the surreals but the symbol "0.999..." maps to 1 - ε. Also, note that the least upper bound axiom doesn't hold in the surreals, so it's not clear to me that cauchy sequences of surreal numbers converge to a unique limit. Dugwiki 22:38, 16 November 2006 (UTC)

How about:

Q: Is it possible to create a new number system other than the reals in which 0.999... < 1 by an infinitesimal amount?

A: Yes, although such systems are neither as used nor as useful as the real numbers, lacking properties such as the ability to take limits (which defines the real numbers), to divide (which defines the rational numbers, and thus applies to real numbers), or to add and subtract (which defines the integers, and thus applies to real numbers). Furthermore, we must define what we mean by "an infinitesimal amount." These is no nonzero constant infinitesimal in the real numbers; quantities generally thought of informally as "infinitesimal" include ε, which is not a fixed constant; dx, which is not a number at all, dx, which is not a real number and has anticommutativity, that is, dx dy = - dy dx; 0⁺, which is not a number, but rather part of , the right limit of x (which can also be expressed without the "+" as ); and values in number systems such as dual numbers and hyperreals. In these systems, 0.999...=1 still holds due to real numbers being a subfield. As detailed in the main article, there are systems for which 0.999... and 1 are distinct, systems that have both alternative means of notation and alternative properties, and systems for which subtraction no longer holds. These, however, are rarely used and possess little to no practical applications.

Calbaer 22:18, 18 November 2006 (UTC)

a new proof/disproof or just a plain huge error?

I was just fooling with different numbers and equations trying to use values as you guys see them. Forgive me, I don't know how to write the nice equations in wiki: You all agree that x/inf=0, since that's it's limit and you do not accept .000...1, yes? Please look at this and let me know what you think. I may have screwed something up, I don't know, but it looked right to me.

x=.999...

thus

x/inf=.999.../inf

so

x-(x/inf)=.999...-(.999.../inf)

x-0=.999...-0

x=.999...

So with using the limits as being equal, this shows to me that it proves otherwise. Please any help with understanding this is appreciated. Vantucci 18:28, 17 November 2006 (UTC)

While your equations seem to be equivalent to each other, I don't see what you try to prove: You start with x=.999... and end with x=.999... Nothing gained in either direction.

A word of warning though: Infinity is tricky. It is (obviously) not a real number. There are several methods to "add" a notion of infinity to the reals, but each is problematic. The one I gave below has the advantage of being compatible with limits and with Hilbert's hotel. One could also, as JohnLattier is proposing, add different types of infinity. Two systems of that kind are the surreal numbers and the hyperreals, but then other important properties (such as uniqueness of limits) are lost. The infinity that seems to be used in limits and series is just notation: There are precise definitions which don't mention infinity at all. So if we are to use a notion of infinity (which should be avoided, since .999... doesn't require it), we should be precise in stating what notion of infinity we use. --Huon 20:14, 17 November 2006 (UTC)

Smooth infintesimal Analysis

John L. Bell was sourced by John Lattier as someone who said agreeable things about infintesimals. Things gleaned directly from Invitation to Smooth Infinitesimal Analysis (PDF file linked at the bottom of SIA):

1. SIA is introduced as an extension to the Reals. Which means that if decimal representations are still valid in SIA, and convergent series still converge, then 0.999... = 1 holds for SIA, because it holds in the Reals.
2. Function continuity falls directly out of SIA axioms: equation 4, f(\epsilon) = f(0) + \epsilon D, "is taken as axiomatic". So in SIA, every function is linear at every point at which it is defined. Sure. Discontinuities (even points of non-differentiability) either prevent function eligibility, or are surrounded by enough little infintesimals to force linearity, regardless of the implications for convergence of limits. Doens't sound like an advantage.
3. In the third paragraph: "in standard mathematical analysis... 0 is the sole infantesimal". The claim 0.999... = 1 is about standard analysis. SIA is very much *non-standard*; Bell has to break a bunch of sensible properties to get SIA.
4. Straddling pages 5 & 6, "in SIA no microquantity (apart from zero itself) is provably not equal to 0" - so what microquantities does SIA have? Introduce an axiom to the effect of "There exists a non-zero infintesimal", otherwise you're just playing dress-up with the Reals.
5. Top of page 6: "microquantities do not have multiplicitave inverses, and R is a field" - what? Are these microquantities Reals in SIA or not? Recall from page 2: the infintesimals \Delta = \{x:x \in Reals and x^2=0 \} supposedly reside in the Reals. As do the other infintesimals in the sets I and J defined on page 8. They don't all have multiplicitave inverses. But the Reals are still a field. How?
6. Top of page 7: "In some models of SIA, R satisfies the Archimedian property" - no kidding! That's when 0 is the only infintesimal and you're doing standard real analysis, but without the tools that SIA breaks.

To be honest, SIA looks like a poor attempt to do calculus in the spirit of o(h) without explicitly using o(h), breaking the law of the excluded middle, the total ordering of the Reals, and limit convergence along the way. The derivative f'(x) can be defined in standard real analysis (if/when it exists) as the unique D for which f(x+h) = f(x) + Dh + o(h^2) holds for suitably small, non-zero h. SIA just invents some new objects to sweep o(h^2) under the rug. Says so in the very first paragraph. There may be some justification for SIA, but John L. Bell doesn't provide it. The "L" in John L. Bell doesn't stand for Lattier, does it? Endomorphic 22:28, 15 November 2006 (UTC)

Haha. That would be ironic. --JohnLattier 04:01, 16 November 2006 (UTC)

Yeah, I wondered if the paper was a hoax initially. He's got some *really* loose notation in places. Some of his "results" are strictly less usefull than his axioms. I dug around and found a bunch of other stuff by JLB, he's apparently a distinguished member of some Canadian university, I forget which one. JLB appears a philosopher rather than a mathmatician, which explains his awful use of pictures in proofs and his willingness to throw away the law of the excluded middle. I just liked the coincidence - JL vs JLB :) Endomorphic 22:02, 16 November 2006 (UTC)

question

How would one go about proving/disproving that for N=all real numbers?--JohnLattier 06:29, 16 November 2006 (UTC)

2nd Try: Examining the RHS:

Examining the LHS:

So, the sequences would look like:

{8.91,8.991,8.9991,...} < {8.9,8.99,8.999,...}

Which is not true. —The preceding unsigned comment was added by 72.193.74.36 (talk • contribs) 07:18, 16 November 2006 (UTC).

recheck your LHS--JohnLattier 07:21, 16 November 2006 (UTC)

Wait... What's the mistake? Step-by-step

Right? Then, simply adjusting the index to match the RHS.

Then, factoring out the constant 8.1.

Okay? I'm not doing anything fancy here. EDIT: Oh, yeah, the upper index is wrong. Mine only works as N->inf.

would be {LHS, RHS} {8.1, 8.9} {8.91, 8.99} {8.991, 8.999} The reason I posted it is because the 8.999...1 describes the 10A-A when A=0.999... and it was said to be equal to 9. Even though if you use sigma notation, at a given number N, 10A-A < 8.999... < 9 --JohnLattier 02:39, 17 November 2006 (UTC)

What exactly do you mean by "8.999...1"? Tparameter 04:55, 17 November 2006 (UTC)

The summation produces 8.1, 8.91, 8.991, 8.9991, 8.99991, etc. such that 9's are infinitely accumulated before a one. at all values N, the summation that gives 8.999...9 is greater than the summation that gives 8.999...1. "At infinity" the one is never reached, but the number is still less than 8.999...9.. because it always is. This is what I see when I read about 10*0.999...-0.999... equalling 9. I see it as less than 9 for all possible representations. (I used 8.999... rather than 9, but it wouldn't matter which you use. If it's less than 8.999... it's less than 9.) --68.158.207.15 06:30, 18 November 2006 (UTC)

By your reasoning then,

since, "they never get there" and, the LHS is "always less". Unfortunately, you are wrong.72.193.74.36 08:03, 18 November 2006 (UTC)

EDIT: This is what I was trying to say, If , then then which is clearly wrong, as it is =1/2. so must be true.--JohnLattier 16:29, 18 November 2006 (UTC)

EDIT comment: (1) Please don't do that "EDIT" thing; it totally screws up the flow of conversation, because now it looks like people are responding to something that they're not. (2) Nope. You can't use . It's undefined, so is undefined. (Simply put, if you're willing to use undefined operations, you can prove anything you want. Like 0 = 1.) --jpgordon^∇∆∇∆ 16:29, 19 November 2006 (UTC)

Excuse me. is not "undefined". I think you meant "indeterminate". 72.193.74.36 16:37, 19 November 2006 (UTC)

No, I meant undefined. It's indeterminate as well. --jpgordon^∇∆∇∆ 17:31, 19 November 2006 (UTC)

Sorry to pick, but isn't it defined as indeterminate? Yet to be determined, so to speak? Tparameter 17:47, 19 November 2006 (UTC)

Are you kidding? Listen man, you don't even know freshman calc. Please quit. 72.193.74.36 16:49, 18 November 2006 (UTC)

Let me give you a little hint about these limits. Try pluggin in 0. As x -> 0 the LHS -> 0 and the RHS -> 0. Sit in any high school AP calc class, and that teacher will gladly prove that for you. {sigh} Tparameter 17:01, 18 November 2006 (UTC)

John Lattier, you are saying that 0/0 = 1. Do you take this as fact? The truth is that 0/0 is indeterminate. 131.216.2.83 20:24, 18 November 2006 (UTC)

Are you saying "0+" doesn't mean "greater than zero?" -JL--68.158.207.15 01:16, 19 November 2006 (UTC)

Actually, what does mean? Is the intent something like "x approaches some smallest possible positive number"? --jpgordon^∇∆∇∆ 01:24, 19 November 2006 (UTC)

x->0+ means "goes to zero from the positive side". Replace with 0 if you wish - it makes no difference in this case. The point is that you don't know the rules of limits, so this whole thing is a waste of time. Study and learn limits, or don't talk about limits. 72.193.74.36 01:26, 19 November 2006 (UTC)

I understand limits. I just wasn't familiar with the "x->0+" term. I thought it meant as x gets greater from zero, as in "infinity." Which is what I'd like to talk about anyway. So disregard the fact that I misunderstood it, and look at my comment using infinity in place of 0+. -JohnLattier --68.158.207.15 04:48, 19 November 2006 (UTC)

Please read Galileo's paradox; it's a similar concept involving infinities. Calbaer 07:40, 18 November 2006 (UTC)

at all values N, the summation that gives 8.999...9 is greater than the summation that gives 8.999...1. "At infinity" the one is never reached, but the number is still less than 8.999...9.. because it always is. Something that is true for all natural numbers N does not necessarily hold at infinity, which is not a natural number.--Trystan 17:38, 18 November 2006 (UTC)

JohnLattier, do you have any comments about my proof of below? Eric119 18:08, 17 November 2006 (UTC)

It looks good, I reached the 72/10^n also on my own while just looking at it, but I think I lost you after the 1/9 comment. -- how did you get that first equal sign? Can you please explain? (Knock some sense into me)--68.158.207.15 06:30, 18 November 2006 (UTC)

I assume you mean all natural numbers, not all real numbers. Let and . Let u_n = s_n − t_n Note that and for n > 1. From this it follows that . Thus, u_n > u_{n − 1}, and u_n is strictly increasing. Suppose that for some n, s_n > t_n. Taking ε = u_n, we now see it cannot be the case that . If we have s_n = t_n, then s_{n + 1} > t_{n + 1} and the same conclusion follows. Now, we have . From this we easily compute that , which we can write as . Hence, s_n < t_n for all natural n. Eric119 07:27, 16 November 2006 (UTC)

The Purpose of this Page

It clearly says at the top that "this page is for mathematical arguments..." So, how do we characterize the actual dissident arguments below? The "autistic kid" doesn't believe it argument, or the "Lattier Demanipulation Theorem", or the similar arguments denying very basic rules of calculus or laws of real numbers - they're not "mathematical" by nature. They are purely intuitive arguments presented by laypeople who have not even mastered arithmetic, much less Calculus. Is this the right forum, or should this introductory statement be changed to allow for delusional intuitive arguments? 131.216.2.83 20:20, 18 November 2006 (UTC)

And this comment is a mathematical argument? Vantucci 22:05, 18 November 2006 (UTC)

Perhaps we should name this the personal attack forum? People get very sensitive when their magic tricks are exposed as fraud. For newcomers such as 131.- I suggest you read back into the archives for mathematical arguments. I don't feel like repeating myself, especially when greeted with disrespect. --JohnLattier 03:18, 20 November 2006 (UTC)

It's not fraudulent magic tricks, it's mathematics as universally accepted by mathematicians. People actually find it annoying when other people accuse formally defined and universally accepted mathematics of being "trickery". Particularly when those other people consistently fail to justify their claims, and when they ignore the thrust of long, carefully constructed arguments made specifically to help them understand, choosing instead to latch onto some small and irrelevent side-point. Consider Meni Rosenfeld's incredible but futile attempts here. Maelin (Talk | Contribs) 03:25, 20 November 2006 (UTC)

Exactly! I have yet to see one sound mathematical argument by JL. Wiki isn't the place to publish the ideas of dreamers and skeptics. It is an encylopedia meant to convey well-sourced facts. Until JL has some sources for his CLAIMS, there is no reason to even read his drivel. 131.216.2.83 03:33, 20 November 2006 (UTC)

This has been discussed already. Being an arguments rather than edit discussion page protects JohnLattier who repeatedly violates 3 of 4 Wikipedia founding principles - WP:OR, WP:VERIFY and WP:WWIN. Personally I blame the naming of this page. "Arguments" makes it sound like we would welcome people who stubbornly insist on refuting established, accepted, well documented, and rigerously proven facts - sorry - theorems (which are even more true than facts). "Justification" or "Interactive Proof" are perhaps preferable. We're to help people who want to know *why* 0.999... = 1. We shouldn't be discussing whether it is or not. Endomorphic 03:18, 23 November 2006 (UTC)

Definition of 'Limit', 'Sum of Infinite Series'

Assumption: any purported proof that 0.999...=1 requires calculus, since simple addition of the members of an infinite series is impossible.

Definition of limit, informally:

The limit of a function f(x) as x approaches a is L

iff given a number greater than zero between f(x) and L, there is a number greater than zero between x and a.

Definition of sum of an infinite series:

The sum of an infinite series is the limit of the sequence of partial sums.

Can we restate this definition such that 0.999...=1? Interpret '0.999...' as the sum of the infinite series denoted by '0.999...' and we have a contradiction, for '0.999...' can't both denote the sum of an infinite series and the infinite series itself. So we must take '0.999...' to denote the infinite series only. In that case, '0.999...=1' has to be rewritten as

The limit of the sequence of partial sums denoted by '0.999...' is equal to 1.—The preceding unsigned comment was added by Shawn Fitzgibbons (talk • contribs).

You're basically saying, "The number represented by '0.999...' is equal to the number represented by '1'." That's the same as saying "0.999...=1." So this doesn't seem to be as much an argument as either an objection to needing to resort to limits to understand some nonterminating representations or a confusion between a number and its representation. In the former case, you need limits in order to represent some numbers; using decimal is otherwise quite clumsy. In the latter case, I think, once pointed out, it is fairly clear, for example, that "1" or "1/2" or "0.5" or "0.999..." are all representations of numbers, but that saying "0.999...=1" is a lot more convenient than the verbose alternative(s). Calbaer 20:13, 19 November 2006 (UTC)

'You're basically saying' has no place in this kind of discourse. What I said can't be said any clearer while adhering to the specific language of the calculus. What is slightly amusing, yet annoying, is that we have this wonderful mathematical language and people attempt to obfuscate it by saying things like: 'respresentation of a number', 'the sum of an infinite series is an infinite series'. The sum of an infinite series is quite a different animal than the sum of a finite series. The former requires taking the limit of a function, the other does not:

The limit of an infinite series is sometimes called its "sum at infinity," but of course this is not a sum in the usual arithmetical sense when the number of terms is finite. You can't obtain the "sum" of an infinite series by adding, because the number of terms to be added is infinite. When we speak of the "sum" of an infinite series, this is just a short way of naming its limit. (fr.Gardner. Calculus Made Easy. p19.)

What confuses the majority of people is that they do not know we are talking about a limit. They look at the equation and go "huh?", and that they should, because they haven't taken calculus yet. Should we laught at them and fail to explain what we are talking about like some snooty math dork who thinks he's the shite because he can confuse people with math? —The preceding unsigned comment was added by Shawn Fitzgibbons (talk • contribs).

This is what you're saying, right:

1. This definition is in the article: :

It's already in the article. Though, the sequence definition should IMO be at the top. 131.216.2.83 22:06, 19 November 2006 (UTC)

What I'm saying is that the following is the only equation needed on the entire page: :—The preceding unsigned comment was added by Shawn Fitzgibbons (talk • contribs).

I agree completely. That is, after all, the definition of 0.999... It could be put like this:

(by definition)

(by rules of limits)

Hence, 0.999... = 1.

Other than a few people here, not many people argue with (a=b & b=c) -> (a=c). 131.216.2.83 02:29, 20 November 2006 (UTC)

That's interesting. Can we do the same thing with pi? I mean, since pi=3.14..., what is the "definition" of pi?Shawn Fitzgibbons 18:03, 21 November 2006 (UTC)

Pi is "defined" as the number such that the circumference of a circle is pi times its diameter. Asking how numbers are defined is a bit tricky, since numbers are a little more fundamental to mathematics than functions and properties, which are what we usually define. If we are using decimal representations, when we say "x is defined as some sum" what we really mean is that "this decimal representation can only be interpreted in this way." The reason it's okay to say 0.999... is defined as that sum of 9/10^k thing is that we need an unambiguous, formal interpretation of the ... recurring notation. It's no good whipping up a notation unless we clearly and unambiguously state what the notation means, that's the "definition" of the ... notation. Maelin (Talk | Contribs) 01:36, 22 November 2006 (UTC)

I'm not sure you get what I'm aiming at here: the previous poster said that the following was a "definition"

Now, my question is why can't we do the same thing with pi, since it is given as 3.14... ?Shawn Fitzgibbons 03:36, 22 November 2006 (UTC)

This question was answered in the text it was posted above (which is still below this). Again, pi is not defined as 3.14... and pi is only described as "3.14..." meaning that it's in [3.14,3.15), hardly a definition or even an adequate description. By contrast, 0.999... does not mean [0.999,1) (indeed, it isn't even in that interval!), but instead or , which have the definition you quoted. Pi cannot have such a description, since it's irrational. Calbaer 03:47, 22 November 2006 (UTC)

But Cantor defined irrational numbers as limits of convergent sequences of rational numbers. Doesn't that mean that pi can be expressed like you've been expressing 0.999...? Also, if there is a problem with ambiguity here, why do you continue to use the '...' notation if there is a more precise way of exressing the number?Shawn Fitzgibbons 05:18, 22 November 2006 (UTC)

Certainly pi can be expressed via a decimal, but it would require an algorithm, not a simple description ("closed form," loosely) like 0.999... or (which equals pi). Regarding why "..." is used, although potentially ambiguous, meaning is usually clear from context, and, unlike dot and bar notation, can be easily expressed in ASCII text alone. Calbaer 05:47, 22 November 2006 (UTC)

Of course, one of the problems with "..." notation — as opposed to the unambiguous and — is that it is often used to mean "and so forth," not just an implied repeating pattern. Thus, one might write, using this broader definition, , while clearly . But it's clear from the context that that's not what we mean, and, of all the potential confusions encountered here, that notational one seems to be of little concern. Calbaer 02:05, 22 November 2006 (UTC)

Please sign your comments when posting on talk pages.--Trystan 01:14, 20 November 2006 (UTC)

When your words are unclear, those who are trying to understand them need to understand what you're "basically saying." Yes, there is an implied limit in 0.999..., but that doesn't make the representation any less valid than that of 1, 1.000..., 1/1, etc. And people who are confused about the fact that we are taking a limit should read the main article before coming here. It's made rather plain there. It's in the introduction for heaven's sake. If not plain enough, propose a fix on the talk page or fix it yourself with a sentence on how repeating decimals are formally defined as a limit of the sequence of truncated decimals. The whole idea of the article is to not confuse people with math (which is why such advanced ideas as limits aren't discussed in detail until well into the article). Calbaer 21:42, 19 November 2006 (UTC)

This seems to be a fairly common objection, that there is some "real" value of 0.999... and we are fudging by "estimating" it with the definition that uses limits.

The phrase "sum of an infinite series" is redundant, as a series is the sum of a sequence of terms. It makes no sense to say that "the infinite series only" is something different than the sum of the infinite sequence of terms.--Trystan 17:25, 19 November 2006 (UTC)

I find it curious that you use the term 'redundant' to apply to the definition of 'sum of an infinite series' used by calculus textbooks. We need to start rewriting these books immediately.—The preceding unsigned comment was added by Shawn Fitzgibbons (talk • contribs).

I agree that the definition of an infinite summation as being equal to its limit is wrong. It's lazy. The definition you propose is superior because an infinite summation ever converges to but never reaches its limit for any value of independent variable. To understand this concept, one must understand the meaning of convergence. It's critical to understanding limits. --JohnLattier 17:31, 19 November 2006 (UTC)

What he proposed, substituting the definition of series, was that 0.999... not equal "the sum of the sum of an infinite sequence of terms", and instead be only equal to "the sum of an infinite sequence of terms." Perhaps you could explain this alternative definition to me in slightly clearer terms.--Trystan 17:42, 19 November 2006 (UTC)

That's not what I said. Go back and reread it.—The preceding unsigned comment was added by Shawn Fitzgibbons (talk • contribs).

You said "sum of an infinite series". A series is the sum of a sequence of terms. A series is already a sum; what's the difference between a sum and the sum of a sum? I agree it's natural -- though redundant -- to say "the sum of a series", but trying to formally distinguish it from the definition of a series doesn't make sense to me.

Sign your posts please.--Trystan 18:07, 19 November 2006 (UTC)

You've made your point that a series is defined as a summation. That doesn't change what he's trying to say, however: an infinite summation is not identical to its limit. They converge. Converge means to become infinitely close without ever equalling. It's part of the definition of a limit. Look up the definition of limit and look up the definition of convergence. --68.158.207.15 18:28, 19 November 2006 (UTC)

An infinite summation is defined as being equal to the limit of the sequence of partial sums as the number of terms goes to infinity. That's the only value it has. What are you comparing that value to? Manually adding infinitely many terms?--Trystan 18:38, 19 November 2006 (UTC)

That's the point he's making. Anytime you see an infinite summation defined as equal to its limit, it's wrong. Whoever is spreading that concept is in error. 0.999... is represented by an infinite summation. This summation proceeds infinitely, but never reaches 1, not at a finite number, not at an infinite concept, never. No one is able to say when this summation reaches 1 because it does not. It's limit, the value it approaches, can be said to be 1. By definition the summation does not have to actually reach 1 to have its limit equal to 1.

       1 Limit: a number whose numerical difference from a mathematical function is arbitrarily small for all values of the independent variables that are sufficiently close to but not equal to given prescribed numbers or that are sufficiently large positively or negatively

       2 Limit: a number that for an infinite sequence of numbers is such that ultimately each of the remaining terms of the sequence differs from this number by less than any given positive amount

       3 Limit: a number such that the value of a given function remains arbitrarily close to this number when the independent variable is sufficiently close to a specified point or is sufficiently large. The limit of 1/x is zero as x approaches infinity; the limit of (x − 1)2 is zero as x approaches 1.

       4 Limit: a number such that the absolute value of the difference between terms of a given sequence and the number approaches zero as the index of the terms increases to infinity.

       5 "(mathematics) A value to which a sequence converges." http://en.wiktionary.org/wiki/limit

http://mathworld.wolfram.com/ConvergentSequence.html

http://en.wikipedia.org/wiki/Limit_of_a_sequence "The limit of a sequence is one of the oldest concepts in mathematical analysis. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit."

Converge:

1. (of a sequence) to have values eventually arbitrarily close to some number; to have a finite limit.
2. Mathematics. To approach a limit.

how convergence can be represented visually: http://www.okstate.edu/sas/v7/sashtml/books/ets/chap14/images/modasy.gif

I apologize for the messy look of my post. I'm not great with HTML or whatever this is. --68.158.207.15 18:44, 19 November 2006 (UTC)

What is your definition for the value of an infinite summation? And the standard definition for an infinite series is not the limit of the sequence itself, it's the limit of the sequence of partial sums of the sequence, a rather critical distinction.--Trystan 18:47, 19 November 2006 (UTC)

The "..." represents the infinite nature of the infinite summation. Otherwise a parial sum is the best anyone can do. The definition of a value for an infinite summation would be the product of a series of terms without a termination step. Since there is no termination step (convenient for limits but out of the scope of infinite summations), the product is defined using infinity in some way. In the case of 0.999... we have the "..." to represent endless 9's that are never rounded up to 1.0 --68.158.207.15 19:11, 19 November 2006 (UTC)

The product? Do you mean the sum? So the sum of an infinite sequence of terms is the sum of each term? Are you saying that the sum can not be calculated?--Trystan 19:15, 19 November 2006 (UTC)

I wanted to say sum, but people had a problem with that term. I guess I could say the "output." Product often refers to multiplication so I should avoid that. The output of an infinite summation. Yes it's the infinite sum of infinite terms. Can it be calculated? Not to a finite number. Any finite representation you may want to impose would only be an approximation for the infinite summation. --JohnLattier 20:22, 19 November 2006 (UTC)

No! There are multiple representations for the same number -- in fact, an infinite number of them. The simplest representation for the one in question is 1. Another representation is 3 - 2. Another representation is 3/3. Another representation is 0.5 + 0.5; and another is 0.999...; and another is . Each one of these represents the unique number 1, the multiplicative identity -- and the notation used to represent the value has no impact upon what the actual value is. --jpgordon^∇∆∇∆ 20:32, 19 November 2006 (UTC)

That makes way too much sense. 131.216.2.83 21:03, 19 November 2006 (UTC)

If the value can't be calculated, then how can you say whether or not the value assigned by definition is accurate or not? There is nothing to compare it to. Precisely because it can not be calculated, it is assigned a meaning, and what meaning would you assign an infinite series other than the value the partial sums approach as the number of terms goes to infinity? If that definition doesn't suit you, you are free to come up with another one that makes more sense. But it makes no sense to say that the defined value is inaccurate, when there is no other value to compare it to.--Trystan 01:11, 20 November 2006 (UTC)

Conclusions, &c.

1. 0.999... is not a member of the set of real numbers, because its definition relies upon infinity, a non-real number. Because it is not a member, it cannot be said to be equal to anything, in the reals.
2. Any attempt to define 0.999... without using infinity becomes simple rewording of infinity, using terms such as "all" or "every" in place of infinity.
3. There is no real number N for which
4. The limit of an infinite summation provides a finite approximation (asymptote) for the summation. (Not an equality!)
5. When a series fails to terminate, but is convergent, it approaches the convergence value as the number of summations tends toward infinity, but it is never equal to that convergence value.
6. 0.999... only represents a conceptual number that is dependent on finite numbers for its representation, such that it can be said to be the largest number less than one, or the upper, last or infinitieth number of the set [0,1).
7. 0.999... is not contained in the set [1,2]
8. Proofs in the article use simple mathematics to truncate an infinitesimal quantity, such as 10*0.999... which only equals 9.999... if a value 9/inf is truncated, after which subtracting 0.999... gives 8.999...1 rather than 9.
9. There is a general misunderstanding about how one can manipulate infinitely-defined values. People whimsically manipulate the "..." without regard to consequences, such as 0.999...*10 which does not equal 9.999... Manipulations such as these should remain undefined. Someone needs to come up with a set of rules regarding these types of manipulations.
10. The numberline is a 1-dimensional infinite string of infinitely many zero-dimensional points. Each point has one and only one decimal representation. The distance between any two points is infinitesimal. Between any two finite points are an infinite number of zero-dimensional points. In order to visualize consecutive points, one must first pick a finite point, such as 3.1415926535, then replace the final digit with either the adjacent less number followed by 9's, such as 3.1415926534999... or add the given representation of 1/inf, which becomes 3.1415926535000...1
11. If a number system includes 0.999... which represents the consecutively less point than 1, it must also include 1.000...1 which represents the consecutively greater point than 1. If the number system's definition of decimal expansion does not allow for this, it must be rewritten.
12. Important terms for so-called mathematicians to brush up on: convergence, asymptote, equals, limit, infinity, infinitesimal, approximation
13. If every number had a second decimal representation, what would the second decimal representation be for 0.333...?
14. Any attempts to refute the fact that every number has one decimal representation (excluding adding 0's which are assumed in their absence): any attempts by pointing to operations such as 1/2 does nothing as it is not a decimal representation
15. This debate will continue to be a debate so long as people keep thinking the finite limit (1) is exactly equal to the infinite summation (0.999...). When people snap out of it and realize their relationship, we can finally close the books on this one. --JohnLattier 04:06, 20 November 2006 (UTC)

Hi, John. As to your point number 1 -- you never have really explained why it bothers you that something that's not a real number might be used to define a real number. Others have pointed out that the use of infinity as an object isn't necessary here, but that's all beside the point, for even if it were necessary, it still would be no bar to 0.999... being a real number. A crane is used to build a skyscraper, and yet a crane is not a skyscraper; so too can things that are not real numbers be used to define real numbers. --Trovatore 04:15, 20 November 2006 (UTC)

It follows straight from that Archimedean property people seem to love so much. --JohnLattier 04:27, 20 November 2006 (UTC)

How exactly does it follow from the Archimedean property? --Trovatore 04:30, 20 November 2006 (UTC)

In other words, JohnLattier, two weeks worth of mathematics drifted, cloudlike, over your comprehension. Endomorphic 04:41, 20 November 2006 (UTC)

JL, simple questions:

1. You mention 9/inf. What is the difference between 9/inf and 1/inf?

2. You mention 8.999...1. What position is the 1? Is it the "infinitieth? Is it fixed?

3. Is pi a real number?

4. Are all infinite decimal expansions NOT real numbers?

131.216.2.83 04:50, 20 November 2006 (UTC)

• I like these conclusions. They are a straightforward, honest, and fascinating summary of JL's positions; there might even be a mathematical system somewhere in which his concepts hold. I have trouble imagining it, but I'm just a so-called mathematician (or I can kinda pretend to be one; at least I have a BA in the subject), and it's possible my vision is clouded by my education and experience. However, as I said above, the article is about the real number system, and JL's not interested in working within the real number system. So we really have nothing at all to discuss here. --jpgordon^∇∆∇∆ 05:15, 20 November 2006 (UTC)

JL: In your "number system" (which you refuse to define even after repeated requests) does the number 10 have any special status? It seems so from what you say. Why did you choose 10? Surely not because of human anatomy? You have not even defined your number system as a set. You have a long way to go. First define the set (for example, real numbers can be defined as the set of equivalence classes of Cauchy sequences of rational numbers, where two Cauchy sequences are equivalent if their difference converges to 0). Then define operations like addition and multiplication, with some useful structure (a field would be nice). Then you have to contemplate over whether one can make useful definitions regarding convergence, continuity of functions etc. --Kprateek88(Talk | Contribs) 06:11, 20 November 2006 (UTC)

JL: How do you define the number 2? Since by the logic of conclusion 1, 1+1 cannot give a real number since it involves addition, and addition is not a member of the set of real numbers. 129.2.164.195 07:28, 20 November 2006 (UTC)

I'm not even going to bother arguing these points with you, John, because you have shown you don't listen, but I am so tired of you saying things like point 3, "There is no real number N for which ". Summations are only meaningful over Natural number indices. The analogous process over real number variables is called integration and it has no relevance to this discussion. Next time when you are about to assert, incorrectly, again, that we can't define 0.999... because infinity is not a real number, say instead, a NATURAL number. Then you will be one small step towards finally knowing what you're talking about. Maelin (Talk | Contribs) 11:36, 20 November 2006 (UTC)

Hey John, I just wanted to reply quickly that, first, you seem like a smart guy, and mentioning biologist Richard Dawkins on your talk page personally gains you points in my assessment. Biology aside, though, I'm afraid you are incorrect in your conclusions above.

1. "0.999... is not a member of the set of real numbers, because its definition relies upon infinity, a non-real number. Because it is not a member, it cannot be said to be equal to anything, in the reals." The fact that "infinity" isn't a real number doesn't mean that the concepts of infinite sets isn't related to the nature and definition of real numbers. In particular, the concept of a "limit as N approaches infinity" is well defined, and that limit definition is used as the exact value (not approximation) of infinite convergent sums of real numbers, which is ultimately what a decimal expression represents (ie the sum of its decimal components).
2. "There is no real number N for which " - Correct. But your next step is incorrect...
3. The limit of an infinite summation provides a finite approximation (asymptote) for the summation. (Not an equality!) - This conclusion is incorrect. In the real number system, the limit of an infinite summation is always equal to the "actual" infinite sum, assuming that the limit exists. This follows due to the lack of non-zero infinitesimals in the reals, because the limit and the "actual" sum could only possibly differ by an infinitesimal amount.
4. "There is a general misunderstanding about how one can manipulate infinitely-defined values. People whimsically manipulate the "..." without regard to consequences, such as 0.999...*10 which does not equal 9.999... Manipulations such as these should remain undefined. Someone needs to come up with a set of rules regarding these types of manipulations." Personally, I've tried to use more formal summations in my proofs on this talk page, as opposed to simply "manipulating dots".
5. "The numberline is a 1-dimensional infinite string of infinitely many zero-dimensional points. Each point has one and only one decimal representation...." Again, incorrect. Surprisingly, all real numbers with a terminating decimal expression have another alternate but equal decimal expression that ends in all 9s. The formal proofs that 0.999... = 1 can be altered to show that, for instance, 0.24999... = 0.25 .
6. "If a number system includes 0.999... which represents the consecutively less point than 1, it must also include 1.000...1 which represents the consecutively greater point than 1...." Here you are confused on the concept of "consecutive". There is no such thing as "consecutivity" between real numbers, in the sense that one real number is "followed consecutively with no interruption" by another neighboring real number. Between any two real numbers x and y there exists another real number z (eg. z = (x+y)/2). The real numbers are ordered, that's true, but they're not consecutively ordered.
7. "If every number had a second decimal representation, what would the second decimal representation be for 0.333...?" Only numbers with a terminating decimal expansion, that is one that ends in all zeroes, has two alternate but equal expressions. Non-terminating decimals such as 0.333... and pi only have one decimal expansion.
8. "This debate will continue to be a debate so long as people keep thinking the finite limit (1) is exactly equal to the infinite summation (0.999...). When people snap out of it and realize their relationship, we can finally close the books on this one." Well, honestly, the same can be said the other way. Once you understand that there are no non-zero infinitesimals in the real number system (as outlined in previous sections on the talk page), you will hopefully realize that you were incorrect in some of your assumptions about how the real number system works and stop debating that 0.999... = 1. Like I said, it's nothing personal; you're just confused. Dugwiki 23:01, 20 November 2006 (UTC)

Yep, it's a question for John Lattier

You hold the position that 0.999... is not a real number because it is defined using infinity, correct? Now, I'll try not to dispute you directly, and I apologize if you have already answered this question, but here it is: what is your definition of a real number? --Ihope127 20:33, 25 November 2006 (UTC)

Good luck. 72.193.74.36 02:03, 26 November 2006 (UTC)

I'm guessing that was sarcasm. Please elaborate. --Ihope127 02:21, 26 November 2006 (UTC)

Yes, that was sarcasm. It was used because, based on previous replies by John Lattier, 72.193.74.36 felt that it was almost certain that he would likely not reply with a coherent, consistant answer. -- kenb215 talk 06:01, 26 November 2006 (UTC)

I'm genuinely wishing you luck in your quest for the much sought after, and oft-requested, straight answer from the now legendary JL. 72.193.74.36 17:53, 26 November 2006 (UTC)

Is .333... a real number? It too requires infinity to be defined - otherwise it fails to equal 1/3. It seems to me that the more complexity and caveats you need to stick into a definition, the more useless the concept being defined is. For Real number, I like "the real numbers may be thought of as points on an infinitely long number line.". this would include .999, Pi, and .000...1, but not infinity, which would have some very interesting results if it ever appeared on a number line. Algr 07:06, 26 November 2006 (UTC)

BTW, for those of you who insist that ".000...1" is meaningless, consider this: Start at the first digit of Pi, then proceed to the infinity-th digit. What comes next?

• If you can never reach the infinity-th digit, (Or just can't discuss it) then an infinite progression in unachievable and .333... does not equal 1/3.
• If the answer is "nothing" then you can produce an infinitesimal by subtracting all but the last digit of Pi.
• If the answer is "more digits" (I favor this) then you have established that digits beyond infinity are meaningful, thus validating .000...1 Algr 07:20, 26 November 2006 (UTC)

"... the infinity-th digit". This is where I stopped reading. If you believe there is an "infinity-th digit", then you should either avoid math, or engage in formal study. At this point, however, you are ill-prepared for this topic. 131.216.2.83 19:55, 26 November 2006 (UTC)

And yet you are happy to criticize John Lattier for not reading links? If you don't even listen to the questions, don't expect others to listen to your answers. Algr

Huh? There is no such thing as the "infinity-th" digit. That seems to be the point. 72.193.74.36 08:10, 29 November 2006 (UTC)

A summation "to infinity" doesn't mean we go to some point called "infinity" and then stop and look around; it means that the summation continues without end. For any natural number n, you could identify the nth term and determine its value. Infinity is not a natural number, and there is no "infinitieth" term, but that doesn't mean than an infinite sum isn't well-defined and meaningful.--Trystan 07:33, 26 November 2006 (UTC)

This is simply the first option that I described above. If there is no end, then .333... never equals 1/3.Algr

You've got that exactly backwards! It is because there is no end that .333... does equal 1/3. --jpgordon^∇∆∇∆ 07:30, 29 November 2006 (UTC)

Pretty much what I wanted to say before the edit conflict. That said -- there is no objection to considering a formal string of infinitely many zeroes, followed by a one. It just isn't the decimal representation of any real number. Basically you (Algr) are looking at the problem backwards. You don't start with decimal representations and get real numbers from them; you start with the real numbers, and then if necessary discuss their decimal representations. The decimal representations are far less important than the numbers themselves. --Trovatore 07:38, 26 November 2006 (UTC)

Yes, much of the argument does seem to be based on whether or not real numbers are defined by their decimal expansions. --Ihope127 13:33, 26 November 2006 (UTC)

Ihope127, I don't know how much of the discussion you have followed, but every single way of viewing the problem has been presented to John, without any positive result. Starting new sections with questions addressed at him won't help, and is against the spirit of WP:DNFT. -- Meni Rosenfeld (talk) 17:48, 26 November 2006 (UTC)

I didn't start with the representations, I started with the need to differentiate between inclusive and exclusive sets. The only alternative to .999... that anyone here could come up with was an equation that I'd bet less then 1 lay-person in 20 would be able to make heads or tails of. In terms of communication, .999... is the only game in town. Algr 06:30, 29 November 2006 (UTC)

Hm? You don't need 0.999... to be different from 1.000..., to "differentiate between inclusive and exclusive sets". In standard notation, [0,1] is the closed interval from 0 to 1 (that is, 0, 1, and all points in between). On the other hand [0,1), with the round bracket on the right, is the same interval, still containing 0, but not containing 1. Nothing complicated here; I think it's usually taught in junior high school, or early high school. Most laypersons should be able to follow it.

Maybe what you want is a greatest element in [0,1), or a number x such that [0,1) is the same as [0,x]. Sorry, there is simply no such thing. Closed intervals and half-open intervals are different; you can't make a half-open interval into a closed interval by discovering a different endpoint for it. --Trovatore 07:40, 29 November 2006 (UTC)

Confusion about the definition of .9999...

In my opinion, as it stands now, I define .9999... 'as the greatest number which is less than 1'. Which can be written as .

This works on the principal that , as, if then then 1 = 0 which is incorrect.

This is further reinforced by the fact for the graph , there are asymptotes along both the x and y axis at 0. However, yes, the limits towards infinity do give zero, using the math above, hence saying that is incorrect, as 1 does not equal 0. Hence, we say that: heads towards but never reaches 0, or 'the smallest number which is greater than 0'.

Therefore, we can say that heads towards but never equals one.

My apologies about the lack of quality with my math script. First time posting on a wiki and all.

202.150.117.33 11:54, 29 November 2006 (UTC)

Well, you're welcome to define it how you want, but there are a couple of things you will want to consider.

• Firstly, your definition is nonstandard. Mathematicians universally define 0.999... as , and your definition is not equivalent. Whilst you are free to define things however you want for the purposes of your own mathematical workings, replacing standard definitions with homebrewed ones will make it exceedingly difficult to discuss mathematics with anybody else.
• Secondly, your definition denotes something that does not exist in the real numbers. That's like defining x to be the integer that, when squared, equals 3 - such a thing does not exist. Sure, there might be numbers that have that property in -some- set, but if you are trying to work in the integers (just as we are trying to work in the reals), non-denoting definitions are not very useful.

If you try to avoid using this "x heads towards but never equals y" idea, you will be much happier. There is no way to express this in formal mathematical language because numbers do not "move". As soon as you try to express the property in a more formal way, such as "for no real value of x does 1/x equal zero" suddenly your problems will start to disappear. Maelin (Talk | Contribs) 13:24, 29 November 2006 (UTC)

• One of the ongoing problems here with the "doubters" is that they confuse process with number. You've said it well -- numbers do not "move". Methods of calculating them may seem to be sequential, but they aren't -- a summation doesn't actually consist of "this step and then that step and then that step", but rather, all the steps simultaneously. So, as you say, there's no "getting to" the value. --jpgordon^∇∆∇∆ 16:56, 29 November 2006 (UTC)

• Something else to consider with the way you're trying to define 0.999... is that your definition would be difficult or impossible to apply to other decimal numbers. For example, how would you define 0.222...? If you try and define it as "the greatest number less than 2/9?". That's a bit of an awkward attempt to work around what the decimals represent. You can, though, say that "0.222... is defined as and have that statement make sense.

• In addition, your definition wouldn't work for other irrational numbers. For instance, how would you define 0.1121222122222221... which is 1 on decimals which are a power of 2 and 0 elsewhere? The only definition that works would be to define it as where f(i)={1 if i is a power of 2, 2 otherwise} Dugwiki 17:15, 29 November 2006 (UTC)

• • Heh, you changed that before I could show ? --jpgordon^∇∆∇∆ 17:34, 29 November 2006 (UTC)

• • hehe, yeah, I thought of that after I posted, so I changed the 0's to 2's since I didn't want to confuse the issue. Dugwiki 17:40, 29 November 2006 (UTC)
    • Of course, that's ; hm, out of practice, does ? --jpgordon^∇∆∇∆ 19:16, 29 November 2006 (UTC)
      • Yes, , assuming that both sums exist. To see that this is true, note that and that for functions F and G if both limits exist. Dugwiki 21:08, 29 November 2006 (UTC)