Talk:0.999.../Arguments

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Frequently asked questions (FAQ)
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.


Q: I have a mathematical question.
A: Please check the FAQ on the talk page.


Q: Do any reliable sources side with the students' stubborn feeling that 0.999... should be less than 1?
A: Yes. See the section on infinitesimals in the main article.


Nominated at MfD[edit]

I have begun a discussion of this talk page at MfD, because it is being used as a forum in violation of Wikipedia policy. This sort of discussion is fun, and there are lots of appropriate places on the Internet to have it. Wikipedia is not one of them. Lagrange613 01:20, 16 October 2014 (UTC)

MFD closed as Keep. — xaosflux Talk 12
58, 24 October 2014 (UTC)

Ellipses denote approximations which ignore infinitesimally small remainders[edit]

.333... is an approximation, valid only to the infinity decimal positions, of the precise value 3/9

Because our calculations will never reach infinity decimal positions we should say that 3/9 ≈ .333..., not 3/9 = .333...

For the sake of postulation, let's suppose that "..." denoted a specific number of decimal positions.

1/9 = .111... with a remainder of .000... followed by the precise fraction 1/9

3/9 = .333... with a remainder of .000... followed by the precise fraction 3/9

9/9 = 1 with no remainder

3 * (.333... with a remainder of .000... followed by 3/9) = 1

3 * (.333... with no remainder) = .999.. with no remainder

3/9 = .333... is a mathematical approximation, it is not an absolute value and cannot be multiplied by 3 to get 1. The approximation times 3 equals the approximation .999..., or approximately 1. It should be written 3/9 ≈ .333...

The use of the particular piece of mathematical short hand that allows 3/9 = .333... is flawed at its core. — Preceding unsigned comment added by 69.51.129.99 (talkcontribs) 18:10, 4 April 2016‎

This is not true within the real numbers, for a simple reason: For every positive real number x there exists a natural number n so that nx≥1 (the Archimedean property). If there were some non-zero difference x between 1/3 and 0.333..., what natural number n would do that? None, since the existence of such a natural number n would imply that there are only finitely many zeros in the decimal representation of x. This leads to a contradiction.
On a more basic level, there seems to be a misunderstanding regarding the meaning of "..." and infinite decimal representations. There are no "infinity decimal positions" that are treated differently in principle from the first, second, third and so on decimal positions. The "..." in 0.333... means that in each of the infinitely many decimal positions, each of which is itself finite (since there are infinitely many finite natural numbers), is a "3". Huon (talk) 19:20, 4 April 2016 (UTC)
I'm not entirely sure I follow the IP's post, but it seems to me that what it may have intended is that the expression 0.333... means that you write down some large-but-finite number of 3's. He/she may reject the idea that having infinitely many 3's after the decimal point is even possible. That is a possible position to take (see ultrafinitism). I'm not sure what, if anything, the article should say about the ultrafinitist view (presumably ultrafinitists could still allow an expression like 0.3̅that's supposed to be a 3 with an overbar; didn't come out as well as I hoped on my screen, and reason about it formally). --Trovatore (talk) 20:04, 4 April 2016 (UTC)
(And this was so beautifully quiet for so long...)
I confess that I really don't understand what the IP is trying to get at. If Trovatore has read him/her correctly, s/he thinks that the expression 0.333... must have only a finite number of threes, because if you start writing them out, one at a time, you'll never reach a point where there are an infinite number of threes. But there is no need to have an infinitieth 3 (which is good because there isn't one). You just need it to be such that if you consider any specific 3, there is always one after it. Like I said above: imagine you are immortal. You never actually reach an infinite age (since your age is a real number), nor do you need to. You just need to be 100% certain on every day that you will live to see the next one. Double sharp (talk) 17:06, 16 April 2016 (UTC)
Well, it depends on how you're understanding the argument. To me, the most straightforward approach is to take the eight characters "0.999..." to be a shorthand for a string containing infinitely many 9s. Then you argue that the interpretation of that string as a real number gives you precisely the real number 1.
The OP seems to be saying that no such infinitely long string exists, so 0.999... is shorthand for something that doesn't exist, and therefore is not a meaningful expression at all.
You on the other hand seem to be saying that you don't have to accept a completed infinite string of 9s, because the limit can be shown to be 1 without them. That is true, but now the story gets more complicated — 0.999... is no longer shorthand for an infinite string, but instead has to be taken to be eight literal characters which now have to be given an interpretation. It would be something like a specification for a computer program that, given a number n, returns the first n 9s. Then you have to give an account of the interpretations of such computer programs, which makes you address details that seem to be a bit of a distraction. (Also, it would be limited to the computable reals, and most reals are not computable.)
Note that a completed infinite string of 9s is not at all the same thing as an infinitieth nine. --Trovatore (talk) 20:19, 17 April 2016 (UTC)
I've always understood infinities in mathematics as processes rather than objects, which matches your concept of a computer program specification. The 0.999... is an infinite process because, no matter how many 9s you have added, you can always can add one more; this is not true of many other mathematical process which are finite and restrict how many times you can execute them (such as subtracting 1 from a natural number to get another natural). Thus, "infinite 9s" means "in any finite string, you can always have at least one more"; and that infinite process certainly exist.
Now, to make that string equal to the number 1, you have to define the limit of the process as the "smallest number above any of the partial, finite strings of 9s". Only when you make such definition you have the equivalence of the process with a number, any number, which in the reals happens to be the number 1. Diego (talk) 11:12, 18 April 2016 (UTC)
@Trovatore: I don't personally have a problem accepting a completed infinity myself, but given that the OP seems to be rejecting the existence of a completed infinity of 9s, I felt that it would perhaps be better not to insist on it in my argument. Double sharp (talk) 12:12, 18 April 2016 (UTC)

It seems that one of the biggest hangups behind most people who don't accept the equality 0.999... = 1 is that they expect that you can start from the beginning and number the nines starting 1, 2, 3, getting every positive integer along the way (this is fine so far), and then (this is the crucial misunderstanding), getting to a last nine numbered ∞ (though I guess it ought to be called the ωth 9 instead). Such a thing does not exist in the real numbers. And even if it did (which seems to move straight into the hyperreals), we run into another problem: if this ωth 9 comes after all the 9's numbered with positive integers, what comes after the ωth 9? Shouldn't there be an (ω+1)th 9 as well? And an (ω+2)th? And so on. So even if you would allow hyperreal-style infinite nines, the hyperreal with nines going all the way (analogous to the one real with nines going all the way) is still one (because there is no first thing greater than zero, even in the hyperreals). (Assuming my understanding of hyperreals is correct, since I only started on that recently.) Double sharp (talk) 05:56, 17 April 2016 (UTC)

Why is there so much talking about it?[edit]

Consider the difference 1-0.(9). If you do this subtraction left to right, this difference is ten times smaller on every new step than it is on the previous step: 1, 0.1, 0.01, and so on. The difference is no more than any number that you achieve in this sequence, and therefore less than any previous number in this sequence. The difference is less than any positive number and more than any negative number. But this is exactly what zero is. Now, if the difference of two numbers is zero, then how on Earth these numbers can be anything else than equal? That's what I don't understand. - 91.122.7.245 (talk) 14:05, 27 April 2016 (UTC)

Sure, any number that has zero, many nines after the dot, and then only zeros is less than one. But this is not the number that is denoted by the notation, because it has all zeros and not all nines at the end. If we can't reach the limit, then we have a totally different string of digits, and the question is different. So, I just don't understand what causes the confusion this time. Lack of imagination? - 91.122.7.245 (talk) 16:17, 27 April 2016 (UTC)
In my experience a common misconception is that people think of 0.999... as some kind of process that never quite "reaches" 1. Others argue that the difference should be some non-zero infinitesimal and are more willing to abandon the real numbers than accept that 0.999...=1. Huon (talk) 00:26, 28 April 2016 (UTC)
So, the confusion seems to be about the denotation… What does the denotation mean, what is that object whose properties are to be investigated… The “process” is a rather unclear object, because it's not static and is not all in vision, but the root of the confusion is clear, it seems… I. e., like with the diagonal argument, the cause is the wrong question again: a question that is asked about an incomplete state of the things which cannot be static and therefore cannot provide an answer. (Like just one real number to enumerate instead of the complete set that needs to be enumerated.) The reason why I was wondering was that I didn't believe that some people had better logical abilities than others. If someone insists to be wrong, it's probably not a failure of the “grasp of elementary notions“ when developing an answer, like Trovatore suggested somewhere, but a failure to ask oneself the right question… - 91.122.0.103 (talk) 14:49, 28 April 2016 (UTC)
That confusion is inherent to everything labelled as "infinite" in math. Does an infinite set exist if it can't be computed? Is there "a member at the infinite"? Does it have an infinite amount of members, or is it just that you can compute any finite member? These questions are intuitive and reasonable to ask, but in the end the only effective way to handle them is to use a formal approach and ask "what axioms define the properties of the the infinite object?" Depending on the axioms chosen, the answers to those questions may vary. Diego (talk) 15:25, 28 April 2016 (UTC)
While different questions, if they are correctly made and concern static objects, can indeed yield different static answers, I don't believe that these questions must necessarily be formulated in a formal language to yield meaningful answers. Perhaps a formal language is just a convenient tool of communication for mathematicians. But this is, of course, a very different question. And I am not prepared to go in the depth of details, because I am not a mathematician… - 91.122.0.103 (talk) 15:36, 28 April 2016 (UTC)
You can formulate them with the language of philosophy as well, which resembles natural language. But in the end, to resolve ambiguities you need to reach a level of detail not different from formalism. Using natural language merely gets you some shortcuts at the steps where precision is not required, but it can be tricky to assess where those shortcuts can be made safely. Diego (talk) 15:57, 28 April 2016 (UTC)
P.S. I like your description of the differences getting smaller and smaller until they "disappear" below any positive number you may think; it reverses the common misunderstanding of "always having a small but non-zero amount". In fact, that is how the limit of the sequence is defined formally. Diego (talk) 16:01, 28 April 2016 (UTC)
While the level of detail may need to be the same with the two approaches to exposition, the method of exposition of that detail is different. I don't think that resolution of ambiguities necessarily makes formal the language to use. Basically, the question is: does my thinking happen naturally and without myself being aware how it really happens, or I need to put something in paper in correspondence to my thought as it should happen? In the first case, the use of formal language is not a pre-requisite to get something right: I just use natural language to only point at the thought process rather than mirror it in part or in whole, like it happens in real life, too. Rather, the pre-requisite is to ask the right question, as one's mind does its own independent work to arrive at questions and find answers. However, the use of natural language may probably be tedious for large volumes of mathematics… - 91.122.0.103 (talk) 18:15, 28 April 2016 (UTC)

Interpretation within the ultrafinitistic framework[edit]

See here:

" Why the "fact" that 0.99999999...(ad infinitum)=1 is NOT EVEN WRONG

The statement of the title, is, in fact, meaningless, because it tacitly assumes that we can add-up "infinitely" many numbers, and good old Zenon already told us that this is absurd.

The true statement is that the sequence, a(n), defined by the recurrence

a(n)=a(n-1)+9/10^n a(0)=0 ,

has the finitistic property that there exists an algorithm that inputs a (symbolic!) positive rational number ε and outputs a (symbolic!) positive integer N=N(ε) such that

|a(n)-1|<ε for (symbolic!) n>N .

Note that nowhere did I use the quantifier "for every", that is completely meaningless if it is applied to an "infinite" set. There are no infinite sets! Everything can be reduced to manipulations with a (finite!) set of symbols."

Count Iblis (talk) 22:32, 13 July 2016 (UTC)

And he is doing very poor mathematics as he rejects the axiom at hand to refute the statement within those axioms, hence his statement is not even wrong. It is just stupid. Just claiming "infinite sets don't exist" is wrong, because they do in mathematics wether he likes it or not, they are defined to be as anything else. TheZelos (talk) 14:45, 14 February 2017 (UTC)

Blindly accepting decimal as the representative numeric system for all numbers and situations[edit]

Just a quick thought. Having .999~ represent 1 bases itself on the assumption that .333~ is the correct representation of 1/3. (We can not and do not have an accurate representation of something between a 3 and a 4 when writing decimal numbers)

This kind of fractions are what I would consider a defect of the decimal number. So we just have an article about one of the artifacts of a deficient numeric system(deficient at least for the task of dividing and multiplying by 3).

On ternary system .333~ or 1/3 would be .1, on a 30 digit system it would be 0.a and we wouldn't be having dumb articles like this. — Preceding unsigned comment added by RenatoFontes (talkcontribs) 21:25, 20 July 2016 (UTC)

I have moved this here from the talk page. KSFTC 21:38, 20 July 2016 (UTC)

Try to figure out the base-30 representation of 1/29. Now multiply by 29. --Trovatore (talk) 22:22, 20 July 2016 (UTC)
True: decimals are not numbers. The fact that decimals represent numbers is actually a theorem. But it is not written down in stone that every number corresponds to one and only one decimal. That would actually be wrong: the subject of the article is an example of how and why this is wrong. Sławomir Biały (talk) 22:34, 20 July 2016 (UTC)

.333... x 3 = 1, NOT .999... What does this imply?[edit]

1 ÷ 3 = .333...

therefore

.333... × 3 =1 (NOT .999...)

So wouldn't that imply 0.999... and 1 are different things? Not saying that this proves 0.999... < 1, but just that t's just something else.

Are there calculations that give a result of 0.999...? That is, one where we are compelled to give the answer explicitly as "0.999..."? If not, it seems as though "0.999..." only exists fictionally for the sake of us arguing about it. --96.35.2.199 (talk) 18:32, 12 September 2016 (UTC)

You could say that as well of any infinite number, like "pi" or "e"; there's no way you can write them in full. That doesn't make them any more or less "existing" nor "fictional". Infinite numbers are typically described by the operations used when manipulating them through arithmetic and calculus, not by enumerating them to the end. In this case, 0.333... x 3 is clearly = 0.999... as can be seen from the basic digit-by-digit calculation, and it's also clear that 0.333 x 3 = 1 as well (since it's the inverse of 1 ÷ 3 ). Diego (talk) 21:51, 12 September 2016 (UTC)
Thanks but I think you misunderstood part of my question. I'm not talking about writing the entire expression out in full with an endless string of 9s. What I meant was, when would one need to give the representation "0.999..." as a result? In other words, why in any practical application would one feel the need to write out a zero, a dot, 3 nines and 3 dots when the result can just be given as "1"? --96.35.2.199 (talk) 23:40, 12 September 2016 (UTC)
By definition 0.999... is the sum . This is an infinite sum that has a meaning independently of whether it is equal to one or not. It happens to be a mathematical theorem that this sum is equal to one. But is meaningful apart from its 1ness. Sławomir Biały (talk) 00:21, 13 September 2016 (UTC)

Article is overtly biased toward the veracity of 0.999... = 1[edit]

This article is propagandist and does not adopt an unbiased view of the subject "0.999...". 1. It offensively belittles "students" as the group mostly holding to the "wrong" view that "0.999... < 1", with complete disregard to the possibility that the intuitive result may be right.

2. It overtly treats proofs supporting the "right" result more favorably than proofs supporting the "skeptical" result. The very word "skeptical" is used in the overall tone of the page as a pejorative. The correct headings would be "Arguments supporting "0.999...=1" and "Arguments supporting "0.999...<1" with equal treatment of both.

3. I have attempted to add a reference to a blog post containing a robust (and I might add, formidable) proof that very clearly demonstrates (possibly rigourously) in elementary school math that "0.999... < 1". This reference has been excluded on the basis that (in the excluder's opinion) the poster "does not understand limits". That may well be the case, however, the proof makes no reference to limits and has not dependency upon them. My reading of the blog post is that the discussion regarding Limits" is merely an opinion piece to promote debate, not offered as any part of the proof. Consequently, the reason for exclusion is both spurious and irrelevant, and simply reinforces my feeling that this page is far from objective. Alex Alexander Bunyip (talk) 15:07, 24 July 2016 (UTC)

It's a theorem that the real number represented by the infinite decimal expansion 0.999... is identical with the real number 1. There are high quality sources that have proofs of this, beginning with the axioms of the real number system. As a mathematical theorem, a disproof would essentially imply that all of mathwmatics involving real numbers is inconsistent. One of the pillars of Wikipedia is WP:NPOV, which in particular implies that subjects like this are discussed according to the weight of different viewpoints in reliable sources. There are various sectioning of the article that discuss septicism, alternative number systems in which 0.999... is different from 1. Sławomir Biały (talk) 15:21, 24 July 2016 (UTC)
@Abunyip: We rely on reliable sources as our primary basis for weighing claims in articles. The claimed proof you cite is on a self-publishing website, and thus does not count as being a reliable source. This is to be contrasted with the many proofs given in reliable sources which demonstrate that 0.999... = 1, which provide the basis for the article's presentation of 0.999... = 1 as established mathematical fact. Please do not re-add the material without providing a reliable source that supports it. (Also: have you see this proof, which demonstrates 0.999... = 1 from first principles?) -- The Anome (talk) 15:41, 24 July 2016 (UTC)
I think we should include a sentence about the formal proof, citing the metamath source. Also, a standard challenge to anyone claiming to have discovered a "watertight" proof that 0.999... ≠ 1 should be "ok, well formalize your proof in metamath" (or Coq or HoLight, etc) Sławomir Biały (talk) 16:09, 24 July 2016 (UTC)
I'm not sure we can cite Metamath directly, as it's not a WP:RS of itself, which is why it's in the external links section rather than the body of the article itself. Are there papers on Metamath that we cite in its stead? -- The Anome (talk) 16:22, 24 July 2016 (UTC)
But yes, inviting people to formalize their argument would go a long way to helping clarify things. Not least for them themselves. -- The Anome (talk) 16:37, 24 July 2016 (UTC)
I would consider Metamath to be a reliable source. It does not seem like a theorem proven in Metamath is likely to be challenged. Indeed, Metamath is probably much more reliable than many textbooks, etc. Sławomir Biały (talk) 17:47, 24 July 2016 (UTC)
The article should be biased in favor of the viewpoint that 0.999.. = 1, because that is the viewpoint of essentially every mathematics reference. The idea of neutral point of view does not mean that we are neutral between all viewpoints; it means that we are neutral between viewpoints to the extent that they are represented in high-quality sources. The viewpoint that 0.999... is the same real number as 1 is so overwhelmingly dominant in the mathematics literature that, even if some other viewpoint might be possible, this article should reflect the viewpoint that the numbers are equal. A better question to ask might be: why do so many sources say that 0.999... is equal to 1? What do they mean by "equal"? That will help clarify what is going on in the literature. — Carl (CBM · talk) 16:11, 24 July 2016 (UTC)
There is no such thing as a "disproof" of this equality. Once you have a correct proof one way, there cannot be a proof contradicting that proof using the same assumptions. @Abunyip: I advise you to read more on what a proof is. The "source" you cited is not only unreliable, but the poster clearly does not know what they're doing, because he blatantly fails to use a correct definition of convergence of a real sequence. Either that, or he's taking a fringe, unaccepted way of looking at real analysis.--Jasper Deng (talk) 16:55, 6 October 2016 (UTC)

There is a correct proof that every 9 fails to reach 1. It is so by definition. What else do you want? If there is a counter proof, then the theory is useless. — Preceding unsigned comment added by 84.155.143.190 (talk) 17:30, 6 October 2016 (UTC)

@84.155.143.190: But that proof is wrong, because that's not the meaning of convergence. One of the fundamental properties of the reals is that between any two reals, there's another. There is, however, no real number between ".999999999..." and 1.--Jasper Deng (talk) 17:32, 6 October 2016 (UTC)

Of course the sequence 0.999... converges to 1, but being a sequence, it is not equal to 1. Having limit 1 and being equal to 1 are two different things. — Preceding unsigned comment added by 84.155.143.190 (talk) 17:38, 6 October 2016 (UTC)

But .999... has to be understood as the limit of the corresponding sequence. It has no meaning as a real number.--Jasper Deng (talk) 18:01, 6 October 2016 (UTC)
Why not tell the truth? 0.999... is a sequence. It has no numerical value, but it has a limit. Writing 0.999... = 1 is sloppy, confusing, and lacking the precision required in mathematics. Further, according to set theory there are all terms. You cannot denote them if you use the correct notation for the wrong notion, i.e., the limit. These things should at least be described in an unbiased article. — Preceding unsigned comment added by 84.155.143.190 (talk) 18:16, 6 October 2016 (UTC)
And finally every mathematician can verify that 0.9, 0.99, 0.999, ... is abbreviated by ...(((0,9)9)9)... and this is abbreviated by 0.999... The first one is not understood as its limit. Why should the last one be? — Preceding unsigned comment added by 84.155.143.190 (talk) 18:19, 6 October 2016 (UTC)
No, we write repeating decimals to represent rational numbers that happen to be the limits. That's the way positional notation works, and the way it is understood. There's nothing ambiguous about that. Other notations of the same sequence might not be understood as such, but that's the way this notation is interpreted, period.
This article is fine as-is. The very first sentence of the article says that's the way we read something like .9999999999 or .33333333333.--Jasper Deng (talk) 18:41, 6 October 2016 (UTC)
I agree strongly with the revert of this edit. The article contains quite a few high quality references supporting the contention that 0.999... = 1. Without countermanding sources of equivalently high quality contesting this identity, it would be inappropriate to call it "erroneous" in Wikipedia's voice. If there are reliable sources that contest the identity of 0.999... and 1, then we can reference those in the article, being careful to emphasize their WP:WEIGHT appropriately. For the record, I actually think that the current article does a good job of accommodating dissenting viewpoints. Even when such views might fall on the wrong side of WP:FRINGE, they provide an interesting and balanced article. But we need references of a sufficiently high quality to merit inclusion, and very high quality references indeed are required to put anything into the lead (such as a standard textbook on Real Analysis, for example). Sławomir Biały (talk) 20:03, 6 October 2016 (UTC)
I agree.--Kmhkmh (talk) 00:02, 7 October 2016 (UTC)
Is a text book published by one of the biggest science publishers "high quality" enough? — Preceding unsigned comment added by 84.155.136.151 (talk) 06:55, 7 October 2016 (UTC)
It probably would be. But it would actually have to discuss the subject of this article. As far as I can tell, the book you cited earlier did not. Certainly, neither of your main points that Euler "erroneously claimed it" and that it "looks true to someone with a sloppy mind" seems likely to appear in a reliable mathematical source, and do not in the source you cited earlier in this discussion page. Finally, any source would need to be weighed against the other high-quality sources to see if the views it contains are appropriate for the lead of the article (which is where the edit under discussion is). The current article has many high-quality mathematical sources containing proofs that 0.999... = 1. Only if the dissenting sources carry a comparable weight to those in the current article can a view be added to the lead, per WP:FRINGE. Sławomir Biały (talk) 10:44, 7 October 2016 (UTC)
0.999... is not a limit and not a sequence. The pedagogical section of the article does not seem to be prominent enough. Hawkeye7 (talk) 21:04, 7 October 2016 (UTC)
This comment is puzzling. I agree that the literal string of symbols "0.999..." is not a limit. It is a zero, followed by a period, followed by three nines and an ellipsis. But the real number represented by this notation is a limit, namely the value of the infinite series . Without clarification, I have no idea if this is what you mean, though. To pose a question: if it's not a limit, then what is it? Sławomir Biały (talk) 22:10, 7 October 2016 (UTC)
Moreover notation is also a question of convention and the literature i've seen treats as notation for (the limit of) that infinite sum.--Kmhkmh (talk) 02:56, 8 October 2016 (UTC)
It's a real number. It's called "one". As you say, it is a matter of convention. We can write it as ١ or 1 or . It is the value of the infinite sum . Hawkeye7 (talk) 03:52, 8 October 2016 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── It's more than a matter of convention. The number is not by definition equal to one. It is a mathematical theorem that it is equal to one. The definition of this number is as a limit: the sum of an infinite series is one type of limit. It can be proved that the value of this limit is identical to the real number 1. and so the two numbers are equal. But it's really misleading to say that is "not a limit". It is a limit. It is also one. Sławomir Biały (talk) 12:05, 8 October 2016 (UTC)

Yes, as the article says:
The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven.
So simple. - DVdm (talk) 20:46, 8 October 2016 (UTC)

Infinitesimals[edit]

Is an infinitesimal the same as an infinitely small number? Is there a dispute whether it counts as a "number"? I had come up with the idea 1-1/∞=.999... on my own and was suprised to see it here in a slightly different format in 0.999...#Infinitesimals.--User:Dwarf Kirlston - talk 02:31, 15 February 2017 (UTC)

The defenders here confuse the "Real" in real numbers into meaning that that is the "right" number set and the others are all "wrong" somehow. This is why they constantly introduce real-set assumptions into discussions with people who are plainly not talking about Reals in order to bog down the discussions and discourage them. Algr (talk) 20:14, 10 June 2017 (UTC)