Talk:0.999...

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view · edit Frequently asked questions Q: Are you positive that 0.999... equals 1 exactly, not approximately? A: In the set of real numbers, yes. This is covered in the article. If you still have doubts, you can discuss it at Talk:0.999.../Arguments. However, please note that original research should never be added to a Wikipedia article, and original arguments and research in the talk pages will not change the content of the article—only reputable secondary and tertiary sources can do so. Q: Can't "1 - 0.999..." be expressed as "0.000...1"? A: No. The string "0.000...1" is not a meaningful real decimal because, although a decimal representation of a real number has a potentially infinite number of decimal places, each of the decimal places is a finite distance from the decimal point; the meaning of digit d being k places past the decimal point is that the digit contributes d · 10^-k toward the value of the number represented. It may help to ask yourself how many places past the decimal point the "1" is. It cannot be an infinite number of real decimal places, because all real places must be finite. Also ask yourself what the value of ${\displaystyle {\frac {0.000\dots 1}{10}}}$ would be. Some people proposing this argument believe the answer to be 0.000...1, yet, basic algebra shows that, if a real number divided by 10 is itself, then that number must be 0. Q: The highest number in 0.999... is 0.999...9, with a last '9' after an infinite number of 9s, so isn't it smaller than 1? A: If you have a number like 0.999...9, it is not the last number in the sequence (0.9, 0.99, ...); you can always create 0.999...99, which is a higher number. The limit ${\displaystyle 0.999\ldots =\lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}}$ is not defined as the highest number in the sequence, but as the smallest number that is higher than any number in the sequence. In the reals, that smallest number is the number 1. Q: 0.9 < 1, 0.99 < 1, and so forth. Therefore it's obvious that 0.999... < 1. A: No. By this logic, 0.9 < 0.999...; 0.99 < 0.999... and so forth. Therefore 0.999... < 0.999..., which is absurd. Something that holds for various values need not hold for the limit of those values. For example, f (x)=x^3/x is positive (>0) for all values in its implied domain (x ≠ 0). However, the limit as x goes to 0 is 0, which is not positive. This is an important consideration in proving inequalities based on limits. Moreover, although you may have been taught that ${\displaystyle 0.x_{1}x_{2}x_{3}...}$ must be less than ${\displaystyle 1.y_{1}y_{2}y_{3}...}$ for any values, this is not an axiom of decimal representation, but rather a property for terminating decimals that can be derived from the definition of decimals and the axioms of the real numbers. Systems of numbers have axioms; representations of numbers do not. To emphasize: Decimal representation, being only a representation, has no associated axioms or other special significance over any other numerical representation. Q: 0.999... is written differently from 1, so it can't be equal. A: 1 can be written many ways: 1/1, 2/2, cos 0, ln e, i^4, 2 - 1, 1e0, 12, and so forth. Another way of writing it is 0.999...; contrary to the intuition of many people, decimal notation is not a bijection from decimal representations to real numbers. Q: Is it possible to create a new number system other than the reals in which 0.999... < 1, the difference being an infinitesimal amount? A: Yes, although such systems are not as oftenly used as the real numbers. They lack 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." There is no nonzero constant infinitesimal in the real numbers; quantities generally thought of informally as "infinitesimal" include ε, which is not a fixed constant; differentials, which are not numbers at all; differential forms, which are not real numbers and have anticommutativity; 0^+, which is not a number, but rather part of the expression ${\displaystyle \lim _{x\rightarrow 0^{+}}f(x)}$, the right limit of x (which can also be expressed without the "+" as ${\displaystyle \lim _{x\downarrow 0}f(x)}$); 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 application. Q: Are you sure 0.999... equals 1 in hyperreals? A: If notation '0.999...' means anything useful in hyperreals, it still means number 1. There are several ways to define hyperreal numbers, but if we use the construction given here, the problem is that almost same sequences give different hyperreal numbers, ${\displaystyle 0.(9)<0.9(9)<0.99(9)<0.(99)<0.9(99)<0.(999)<1\;}$, and even the '()' notation doesn't represent all hyperreals. The correct notation is (0.9; 0.99; 0,999; ...). Q: If it is possible to construct number systems in which 0.999... is less than 1, shouldn't we be talking about those instead of focusing so much on the real numbers? Aren't people justified in believing that 0.999... is less than one when other number systems can show this explicitly? A: At the expense of abandoning many familiar features of mathematics, it is possible to construct a system of notation in which the string of symbols "0.999..." is different than the number 1. This object would represent a different number than the topic of this article, and this notation is not commonly used in applied mathematics. Moreover, it does not change the fact that 0.999... = 1 in the real number system. The fact that 0.999... = 1 is not a "problem" with the real number system and is not something that other number systems "fix". Absent a WP:POV desire to cling to intuitive misconceptions about real numbers, there is little incentive to use a different system. Q: The initial proofs don't seem formal and the later proofs don't seem understandable. Are you sure you proved this? I'm an intelligent person, but this doesn't seem right. A: Yes. The initial proofs are necessarily somewhat informal so as to be understandable by novices. The later proofs are formal, but more difficult to understand. If you haven't completed a course on real analysis, it shouldn't be surprising that you find difficulty understanding some of the proofs, and, indeed, might have some skepticism that 0.999... = 1; this isn't a sign of inferior intelligence. Hopefully the informal arguments can give you a flavor of why 0.999... = 1. If you want to formally understand 0.999..., however, you'd be best to study real analysis. If you're getting a college degree in engineering, mathematics, statistics, computer science, or a natural science, it would probably help you in the future anyway. Q: But I still think I'm right! Shouldn't both sides of the debate be discussed in the article? A: The criteria for inclusion in Wikipedia is for information to be attributable to a reliable published source, not an editor's opinion. Regardless of how confident you may be, at least one published, reliable source is needed to warrant space in the article. Until such a document is provided, including such material would violate Wikipedia policy. Arguments posted on the Talk:0.999.../Arguments page are disqualified, as their inclusion would violate Wikipedia policy on original research.

Yet another anon

Moved to Arguments subpage

Is the Archimedean property an axiom of the real numbers?

The current version of the article contains the following phrase:

...the Archimedean property, a defining axiom of the real number system

I am unaware of any source that states the Archimedean property as an axiom of the real numbers. Conventionally, the axiom expressing Completeness of the real numbers is either Dedekind completeness or the Least-upper-bound property, although there are other equivalent statements. While the Archimedean property follows from any of these axioms, I don't think that it is equivalent and therefore could be substituted as an axiom.

If not, then we need to correct the article. If so, then we'd need a source and we're still faced with the issue that the article is at variance with the "usual" literature. Or at least with the ones I'm familiar with.

I'm correcting the language pending further discussion here. Mr. Swordfish (talk) 23:43, 21 June 2025 (UTC)[reply]

It is sometimes described as an axiom (the "axiom of Archimedes"), and I think axiom works better in this context than property. E.g., it is one of Hilbert's axioms. Tito Omburo (talk) 11:55, 22 June 2025 (UTC)[reply]

Many things are called "axioms" that are not a "defining axiom of the real numbers". This appears to be an example since although the Archimedean property can be proved from the Dedekind completeness or the Least-upper-bound property axioms, those do not follow from the Archimedean property. i.e. The Archimedean property is not strong enough to serve as an axiom of the real numbers. It's not, so let's not say that it is. Mr. Swordfish (talk) 12:59, 22 June 2025 (UTC)[reply]

I mean, by itself completeness is also "not strong enough to serve as an axiom of the real number system", viewed in isolation. Completeness axiom + ordered axiom. Or completion of rationals + Archimedean axiom. Both give you the real numbers. (The latter was Hilbert's viewpoint. And if we're worried about independence of the axioms defining the rals, something rarely discussed, here is a list of axioms, proven independent, which includes a property equivalent to the Archimedean property.) But my point is that the Archimedean property is an axiomatic property, in the colloquial sense: something assumed to be true of the structure. If some other set of axioms didnt imply the Archimedean property, we wouldn't accept them as defining a "real number". It was foundational, and preceded all other attempts to systematize the real number system. Tito Omburo (talk) 16:16, 22 June 2025 (UTC)[reply]

The usual axiomatic treatment of the reals is to list the field axioms, the order axioms that make it an ordered field, and some version of completeness. The three together characterize the reals up to isomorphism.

If there's a source that develops real analysis from the Archimedian property as an axiom we might be technically correct calling it an an "axiom of the real numbers", but we'd be promoting a viewpoint that sufficiently unusual that it is possibly WP:FRINGE. Mr. Swordfish (talk) 16:39, 22 June 2025 (UTC)[reply]

I would encourage you to retract your viewpoint that this is WP:FRINGE. That is simply not tenable. It was one of the first axioms in history! (And the "usual axiomatic treatment" is a bit of a lie, isn’t it. The "usual axioms" that appear in many real analysis texts are not independent! Nor are they usually referred yo as "axioms".) Anyway, would you settle for "a defining property" rather than "axiom"? Tito Omburo (talk) 16:45, 22 June 2025 (UTC)[reply]

I would suggest "a foundational property": Although the Archimedean axiom is not always used in formal definitions of the real numbers, it is an axiom in all axiomatic definitions of the geometric line (the continuum, in the teminology of the 19th century, the real line in modern language). It is clear that Dedekind had this axiom in mind when defining the real numbers for modeling the continuum. If you do not like "foundational property", one may use also "fundamental property" D.Lazard (talk) 17:04, 22 June 2025 (UTC)[reply]

Foundational property is good, but I still think defining property would be more suitable for the basic audience (with axiom even better). (Although I'm also quite happy with Swordfish's latest, with my small copy edit.) Tito Omburo (talk) 17:45, 22 June 2025 (UTC)[reply]

My take is that I don't think we need to add flowery language like "fundamental property" or "foundational property", but I'm not going to object to either of those. Mr. Swordfish (talk) 18:27, 22 June 2025 (UTC)[reply]

IMO, this is precisely why axiom is better. "Foundational property" feels "flowery", but isn't. Axiom is correct and immediately understandable. Tito Omburo (talk) 18:48, 22 June 2025 (UTC)[reply]

The problem with "axiom" is that it assumes a particular axiomatic setup (or at least one of a particular proper subset of possible axiomatic setups) for describing the same thing (the real numbers). The issues being discussed are not specific to the axioms used. Even "foundational property" suggests that you're coming at the question from a foundationalist perspective. I would just go with "property". --Trovatore (talk) 19:14, 22 June 2025 (UTC)[reply]

"Axiom" is an ordinary English word. It means, specifically, something whose truth is taken to be self-evident and is not questioned. The word "axiom" does not mean "axiom of set theory". It certainly is an axiom of the real numbers. It does not assume any particular setup. Indeed, as an axiom it is more primitive than even completeness (see Hilbert). If a model of the "real numbers" didnt satisfy Archimedes axiom, it wouldn't be considered the real numbers! Tito Omburo (talk) 21:38, 22 June 2025 (UTC)[reply]

It's true that there is a sense of "axiom" that is an ordinary English word, but using that sense in a mathematics article invites confusion, and anyway it's not totally clear that the Archimedean property is an axiom in that sense.

Linguistic aside: Generally after "model of" you name a theory, not a structure. The real numbers are not a theory but a structure. So "model of the real numbers" doesn't exactly make sense in the usual usage, though there are available meanings that I would let you get away with. --Trovatore (talk) 22:21, 22 June 2025 (UTC)[reply]

Except that the real numbers existed before anyone formalized them. When they were formalized, if they did not satisfy the Archimedean axiom, that formalization would have been rejected. Thus Archimedes is an axiom. (See the article axiom.) Tito Omburo (talk) 09:35, 23 June 2025 (UTC)[reply]

I would say it was a discovered truth about them rather than an axiom. --Trovatore (talk) 17:29, 23 June 2025 (UTC)[reply]

It is explicitly on of Hilbert's axioms. Tito Omburo (talk) 17:48, 23 June 2025 (UTC)[reply]

Sure. There are probably lots of axiomatizations of the theory of the reals that include it as an axiom. That's a fact about those axiom systems, not a fact about the reals. The property itself is a fact about the reals, but calling it an axiom is not. --Trovatore (talk) 17:50, 23 June 2025 (UTC)[reply]

This is an argument that the reals have no axioms then, just properties. Do I understand correctly? Tito Omburo (talk) 17:55, 23 June 2025 (UTC)[reply]

Correct. There is no such thing as an "axiom of the reals" separate from a particular axiomatization. --Trovatore (talk) 19:12, 23 June 2025 (UTC)[reply]

Whether the real numbers "exist" is a controversial philosophical question. –jacobolus (t) 17:57, 23 June 2025 (UTC)[reply]

When I started this discussion I was not aware of any axiomatic presentation of the real numbers that included Archimedes property. This paper provides an axiomatic presentation of the real numbers that does include Archimedes property as an axiom. So, I've learned something here.

But this presentation is a rather unusual approach that leaves out much of the field axioms, replacing them with axioms based on ordering that then imply the usual field axioms. While it's not WP:FRINGE it's not how the axioms defining the reals are conventionally presented.

My take is that referencing such an unusual and unconventional approach in the lead of a general-interest article is not ideal. Whether the real numbers actually exist, or which set of axioms to use to define them is outside the scope of the lead section of this article.

With that said, I'm now of the opinion that referencing the Archimedes property in the lead is an unnecessary complication, and would propose the following third paragraph for the lead:

An elementary proof is given below that involves only elementary arithmetic and the Archimedean property: that there is no positive real number less than the reciprocal of every natural number. There are many other ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The intuitive arguments are generally based on properties of finite decimals that are extended without proof to infinite decimals. Other proofs are generally based on basic properties of real numbers and methods of calculus, such as series and limits. A question studied in mathematics education is why some people reject this equality.

— Preceding unsigned comment added by Mr swordfish (talk • contribs) 18:51, 23 June 2025 (UTC)[reply]

I would skip the wikilink to mathematical rigor, which seems more distracting than helpful if we are linking mathematical proof immediately afterward. –jacobolus (t) 19:06, 23 June 2025 (UTC)[reply]

I have no objections to that. Mr. Swordfish (talk) 19:15, 23 June 2025 (UTC)[reply]

Any objections to the proposal above with the mod suggested by –jacobolus ? If none, I'll go ahead and make the change. Mr. Swordfish (talk) 15:19, 25 June 2025 (UTC)[reply]

Also "intuitive arguments" should link to the title mathematical intuition (which currently redirects to logical intuition). A link to intuition in general is not that useful. (Or we could skip wikilinking anything from there.) –jacobolus (t) 16:43, 25 June 2025 (UTC)[reply]

I oppose strongly to the removal of the sentence that is struck out in the above quotation. The Archimedean property is a fundamental and very elementary; no reason for hiding it. And this is true that the given proof is elementary. As quoted above, the paragraph suggest that readers must learn calculus before having a rigorous proof. This is a form of unacceptable pedantry: "if you do not know much mathematics, you cannot have a rigorous proof".

This being said, I don't oppose to replace the statement of the Archimedean property by its original formulation: "for every real number, there are integers that are greater". This would need to add in the proof a short paragraph showing the equivalence of the two formulations (to take the multiplicative inverses). D.Lazard (talk) 17:30, 25 June 2025 (UTC)[reply]

Agree with all of Lazard's points. Tito Omburo (talk) 17:48, 25 June 2025 (UTC)[reply]

If we're going to mention the Archimedes property in the lead, then the simpler version as stated by D.Lazard is preferable to what is there now.

The recent edit [1] is an improvement, and perhaps better than what I proposed above. Combining that with D.Lazard's and jacobolus' suggestions gives:

There are many ways of showing this equality, from intuitive arguments to rigorous proofs. The intuitive arguments are generally based on properties of finite decimals that are extended without proof to infinite decimals. An elementary but rigorous proof is given below that involves only elementary arithmetic and the Archimedean property: for every real number, there are natural numbers that are greater. Other proofs are generally based on basic properties of real numbers and methods of calculus, such as series and limits. A question studied in mathematics education is why some people reject this equality.

Opinions? Mr. Swordfish (talk) 00:17, 26 June 2025 (UTC)[reply]

Agree, I thanked the editor who made the change, and did not yet have a chance to opine here. Tito Omburo (talk) 00:19, 26 June 2025 (UTC)[reply]

Is that in fact the usual statement of the Archimedean property? I always thought of it as "for any real positive ε and any natural number N, if you add ε to itself enough times, you get something bigger than N". Certainly this is equivalent given the field properties for the reals, but that requires getting into algebra for something that seems more order-theoretic. --Trovatore (talk) 00:34, 26 June 2025 (UTC)[reply]

I don't know what the usual statement is, but it's the version that is given in the book we cite, and I think it's the most convenient for the purposes of an intro paragraph, where we want a statement whose meaning is clear and whose truth seems as obvious as possible. Stepwise Continuous Dysfunction (talk) 00:43, 26 June 2025 (UTC)[reply]

I changed the link at "intuitive" to point to mathematical intuition and swapped out the explanation of the Archimedean property, since D.Lazard's suggestion is simpler and makes for a more digestible introduction. I think the longer discussion of the Archimedean property below may need some modifications. For one thing, the current text says "there is no positive number that is less than ${\textstyle {\frac {1}{10^{n}}}}$ for all ⁠${\displaystyle n}$⁠" and calls this "a version of" the Archimedean property. Bringing in the specific number ${\textstyle {\frac {1}{10^{n}}}}$ here makes this seem more like a consequence of the property than a "version" of it. I am not aware of a source that defines the Archimedean property using a specific base like that. Stepwise Continuous Dysfunction (talk) 00:32, 26 June 2025 (UTC)[reply]

I have done a little rewriting to try and address this. Stepwise Continuous Dysfunction (talk) 00:54, 26 June 2025 (UTC)[reply]

Intuitive explanation

In the section "Intuitive explanation", we say:

Meanwhile, every number larger than 1 will be larger than any decimal of the form 0.999...9 for any finite number of nines.

From this we conclude that 0.999... cannot equal a number larger than 1. However, one could replace "number larger than 1" with "1", and thus seemingly conclude that 0.999... cannot equal 1. Can we strengthen the wording to get rid of this problem - without making it too complex to warrant the heading? Nø (talk) 09:35, 14 August 2025 (UTC)[reply]

We don't need to consider numbers larger than 1. The argument shows that 1 is the least upper bound of the sequence, since it is an upper bound and no number smaller is. Since this is an intuitive argument, I think we could just gloss this point without trying to be too precise. No one would argue that 0.999... could be greater than one. Tito Omburo (talk) 10:01, 14 August 2025 (UTC)[reply]

Ambiguity

0.999... is ambiguous. An infinite number of sequences could start with nines, yet only one of them has all nines. Furthermore, even if it were guaranteed to continue with nines, it is ambiguous whether it refers to the limit of the summation of a geometric series as the number of terms approaches infinity or some other definition. Lxvgu5petXUJZmqXsVUn2FV8aZyqwKnO (talk) 22:14, 18 August 2025 (UTC)[reply]

It is no more or less ambiguous than 0.33333... or 0.50000... which are clearly equal to 1/3 and 1/2.

It is just as unambiguously clear that 0.9999... is equal to one. Mr. Swordfish (talk) 22:57, 18 August 2025 (UTC)[reply]

I think you're missing the OP's point. In the lead sentence, it does say that this is notation for a repeating decimal, but it could maybe stand to be a bit more explicit that this is shorthand for a repeating sequence of 9s, rather than simply a truncation of some other decimal that starts with three 9s (repeating or not). 35.139.154.158 (talk) 23:08, 18 August 2025 (UTC)[reply]

The lead sentence says that it is a repeating decimal, which is what the "..." notation means. Perhaps we should explicitly state that it is an infinite sequence of nines, but it seems unnecessary to me. Mr. Swordfish (talk) 23:25, 18 August 2025 (UTC)[reply]

The caption to the figure in the lede does say so. Tito Omburo (talk) 12:10, 19 August 2025 (UTC)[reply]

I have pushed into a footnote the altenative notations and the claification of the meaning of the ellipsis. IMO, adding these details to the text is distracting; nevetheless, they may be useful to some readers. D.Lazard (talk) 12:56, 19 August 2025 (UTC)[reply]

What do folks think of this opening sentence:

In mathematics, 0.999... is a repeating decimal (i.e. there is an infinite string on 9s after the decimal point) that is an alternative way of writing the number 1.

Instead of making the reader click on a wikilink or a footnote to see the meaning of "repeating decimal", it is explained right in the text. As MOS:LINKSTYLE says:

Use a link when appropriate, but as far as possible do not force a reader to use that link to understand the sentence. The text needs to make sense to readers who cannot follow links.

If we assume that some readers don't already understand what "repeating decimal" means, then the parenthetical sentence above would conform to the MOS. Mr. Swordfish (talk) 14:10, 19 August 2025 (UTC)[reply]

Someone who doesn't know what a repeating decimal is is unlikely to know that a "string" means a list of symbols. You should use a common word like "list", "sequence", or "succession" instead of a computer science jargon word. I'd also recommend offsetting such an explanation by commas rather than parentheses, or using a separate sentence, and avoiding "i.e.". So something instead like:

In mathematics, 0.999... is a repeating decimal that is an alternative way of writing the number 1. The three dots represent an infinite list of "9" digits.

would be better than this proposal. I'll leave it to others to decide whether such an explanation is necessary. –jacobolus (t) 14:43, 19 August 2025 (UTC)[reply]

From experience with cranks on sci.math, I think "unending" is better than "infinite". Otherwise there is always the "after the infinite sequence of xs there must be a y" line; with "unending" you simply point out that this means there isn't an end. Imaginatorium (talk) 16:16, 19 August 2025 (UTC)[reply]

I think it is fine as is. Tito Omburo (talk) 16:43, 19 August 2025 (UTC)[reply]

I prefer the version proposed by jacobolus (t) over my proposal. "Unending" is also probably better than "infinite".

Also agree that it's fine as it is. Mr. Swordfish (talk) 16:54, 19 August 2025 (UTC)[reply]

A few years ago, the article started:

In mathematics, 0.999... (also written as 0.9, in repeating decimal notation) denotes the repeating decimal consisting of an unending sequence of 9s after the decimal point. [...]

–jacobolus (t) 18:20, 19 August 2025 (UTC)[reply]

I think that phrasing is problematic. For example, File:Ordinal ww.svg with nines at every site. Tito Omburo (talk) 18:25, 19 August 2025 (UTC)[reply]

I have implemented this suggestion with "unending" instead of "infinite". Stepwise Continuous Dysfunction (talk) 22:44, 19 August 2025 (UTC)[reply]

I'm not a fan of having a footnote in the middle of the first sentence. That feels pedantic and distracting. Stepwise Continuous Dysfunction (talk) 22:37, 19 August 2025 (UTC)[reply]

(imo) mathematics, just like every other field of science, can have facts/theories that are counter-intuitive. however as wikipedia is an aggregation, not the source of the knowledge, defending / asserting its correctness is probably not our *primary* job, in so far as to spend too much room addressing it *up front in the intro*.

i’d write somewhere “…see [[below|#misconceptions]]” and then put the bulk of clarification there, e.g. infinite means “unending”, etc.. - the problem is we already addressed this into detail later in article, and some readers didn’t seem to reach that part. 海盐沙冰 (talk) 10:28, 20 August 2025 (UTC)[reply]

In this case, most of the sources addressing the subject of the article are correcting the misconception or highlighting its role in education, so the focus is appropriate. Tito Omburo (talk) 12:40, 20 August 2025 (UTC)[reply]

Problems with 10-adic section

The first thing I note here is that the section heading is "p-adic numbers", and the body of the section also keeps using the term p-adic. But when you say p-adic, it's understood that p is prime. The 10-adics are a legitimate algebraic structure but they aren't usually included in the study of the p-adics.

More generally the section suffers from a tenuous connection to the subject of the article. It's interesting stuff but it doesn't seem to have much to do with 0.999.... --Trovatore (talk) 23:16, 21 August 2025 (UTC)[reply]

Fwiw, the 10-adics are canonically isomorphic as a topological ring to ${\displaystyle \mathbb {Z} _{2}\oplus \mathbb {Z} _{5}}$. Tito Omburo (talk) 10:42, 22 August 2025 (UTC)[reply]

Also, the image is not only cryptic, but it is misleading, as ignoring that 4-adic numbers are the same as 2-adic numbers and are obtained by grouping the binary digits by two. D.Lazard (talk) 15:20, 22 August 2025 (UTC)[reply]

Maybe something showing the compactness of the set, as it can be homeomorphically embedded in R^2 (or even R^1), but not much value in a picture attempting to show some particular value. Tito Omburo (talk) 20:11, 22 August 2025 (UTC)[reply]

I think the first source does discuss how 10-adics can arise in education, with an anecdote about how this arose in an elementary school class in which the teacher was stumped by the apparent fact, raised by a student, that ...999 = -1, in a discussion about 0.999... = 1. I think the last paragraph of the section is undue weight/fringe. Tito Omburo (talk) 20:11, 22 August 2025 (UTC)[reply]

Besides JSTOR 2309468 and doi:10.1080/07468342.1995.11973659 cited already, here are a couple other sources: doi:10.1007/s00591-022-00322-1; doi:10.1111/j.1949-8594.1992.tb15623.x; doi:10.1080/07468342.2008.11922313; CORE output ID 83041701 (doi:10.17877/DE290R-17740), p. 134. –jacobolus (t) 20:38, 22 August 2025 (UTC)[reply]

Yes, I think these support relevance to the article, but (based on the previews I've seen), also support the last paragraph as undue weight (although the reference – your JSTOR link – should be retained to support the overall link with the subject of the article). Tito Omburo (talk) 20:54, 22 August 2025 (UTC)[reply]

Agreed. It may also be worth mentioning the Method of complements or two's complement in this section. –jacobolus (t) 22:27, 22 August 2025 (UTC)[reply]