Talk:Real number

Useful for measurement?
The line "In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more." should be edited or removed, as this is demonstrably false. No measuring instrument can access the infinite precision of real numbers (if such a thing even exists); they usually use decimal numbers which are like rational numbers, i.e. have a finite representation, can be represented as a ratio of integers. — Preceding unsigned comment added by 199.46.11.200 (talk) 20:37, 24 August 2022 (UTC)


 * Done.—Anita5192 (talk) 21:43, 24 August 2022 (UTC)

severe inaccessibility to a nonspecialist audience; entirely missing most of the fundamental concepts
This article does not do an adequate job of explaining what the real numbers are, because it presumes too much background and skips basic material.

The lead section of this article is too long and rambling, and spends too much time on advanced material explained using inaccessible jargon. The advanced material is fine to include in the article, but should be pared back if not eliminated from the lead section, which should should aim for legibility for a wider audience.

The first section in the article, about history, is (a) somewhat incomplete, and (b) should be deferred until at least after basic discussion of what real numbers are and why they matter. The history section as it is currently does not sufficiently define or motivate the ideas mentioned that someone who does not already know what real numbers are will get much out of it.

The important feature of the real numbers (per se) is right there in our current definition, a real number is a value of a continuous quantity that can represent a distance along a line. Namely, real numbers are continuous. But not only is "continuous" not wiki-linked, it is nowhere properly defined in this article, and not discussed in an accessible way. The non-continuity of the rational numbers is not described or discussed, nor is it explained why this non-continuity matters, what alternatives we might have to continuity, nor what the real numbers do to address this. (Aside: I think the basic definition here is a bit sloppy, since distance is usually an unsigned idea.)

I would have hoped that continuity (mathematics) would explain the basic idea, but it redirects to List of continuity-related mathematical topics which does not provide any basic conceptual description of what continuity means but just links to more advanced articles like continuum (set theory) and linear continuum which circularly describe a continuum as being "like the real numbers", continuous variable which just describes having an uncountable set of values (not quite technically complete/precise, or helpful as a basic idea), continuum (topology) which is absurdly terse and technical, the kind of definition you’d find in a journal paper for an audience of mathematicians, etc.

The notion of a line is repeatedly invoked, but again not really defined; at the article line there is no discussion of continuity. (Aside: real line redirects to number line, which also does not really discuss continuity, except via inaccessible jargon.) The idea that the real number line is infinite in extent is mentioned a few times, but this is not defined, nor are there any links to other articles where the concept of infinite extent might be found. (The article number line mentions but does not describe or discuss this either.)

The idea that real numbers are ordered is mentioned, but not described in an accessible way. Someone who follows the link on linear order which redirects to total order finds an incomprehensible wall of jargon and symbols.

The concept of irrational is invoked right off the bat in the lead (which describes real numbers as consisting of both rational and irrational numbers) and then in the history section, but is not properly defined or discussed, and if someone clicks through irrational number they get a circular definition that irrational numbers are real numbers that aren't rational, and are going to have to work hard to figure out what irrational really means. The concept of rational numbers is likewise invoked (as presumed background knowledge) but not really described or discussed.

The uncountability of real numbers is mentioned, but there is no explanation given for why real numbers are uncountable; this is just taken as an obvious fact, and readers curious to understand what "uncountable" means or why it matters are going to have to do a lot more searching to find out.

The basic property of real numbers (that they share with rational numbers, etc.) is that they can be added, subtracted, multiplied, and divided, but there is no explanation in this article of what that means or how it works in practice. The article field (mathematics) is too technical to be relied on for covering this. There should be, early in the article, some kind of explanation about different geometrical interpretations of these operations (e.g. subtraction as a measure of distance; addition as a kind of translation; multiplication/division as scaling). Some of this material is covered at number line. Some could maybe be pulled from addition, subtraction, multiplication, division (mathematics).

That positive real numbers have real square roots is considered an "advanced property" in this article, and what this means is not described. But the history section mentions this several times (though without really going into much detail), implying it is important to understanding the motivation for real numbers. This article should at least point out that a "square root" gets its name from being the side length of a square of a given area, and that the Pythagorean theorem gives a way of finding triangle side lengths in terms of square roots, so that square roots come up all the time as lengths in analytic geometry.

That real numbers contain the limits of converging sequences of rational numbers is mentioned several times and is clearly very important, but sequence is not linked and the notion of a sequence is not defined. The term used is Cauchy sequence, but that page does not describe what a sequence is, and is too technical for much of the presumed audience of this article. There are links from here to completeness of the real numbers but except for one sentence that article does not describe concretely what completeness means, except in very technical terms.

In my view there are a few primary uses of real numbers all over science and mathematics which should be discussed prominently:


 * As an abstract model for arbitrary kinds of measured quantities in which basic arithmetic operations are well defined so that people don’t need to worry about too many caveats when doing basic symbolic manipulations. This is discussed in the "Applications and connections to other areas > In physics" section, but the immediate invocation of "physical constants" and "physical variables" seems less clear than just saying "measured quantities", and there's not too deep a discussion. This seems to me much more important than the other parts of this applications section. In particular, scalar quantities in the sense of scalar (physics).
 * As the coordinates of some coordinate system, e.g. for analytic geometry. There is a brief mention of this with links to real coordinate space, dimension, Cartesian product (frankly unnecessary technical jargon in this context), Cartesian coordinate system. This article should summarize more of this material directly, perhaps including a definition real coordinate space, and should clarify (which the real coordinate space doesn't do a very good job of) that it can be used for purposes other than Euclidean geometry, e.g. for other coordinate systems, for generalized coordinates, etc.
 * As the domain or codomain of functions. (Also cf. e.g. scalar field, parametric curve). This is important historically in the development of the function concept, and underlies many other kinds of mathematical objects used as domains/codomains that are defined in terms of real numbers. This is not really mentioned here at all. This article does state that real numbers are used in calculus and mathematical analysis, but does not explain how or why, except to say that real numbers were historically used without being entirely well defined.
 * In particular as the codomain of the distance function any metric space.
 * As a field of scalars, e.g. for use in vector spaces (cf. scalar (mathematics), scalar multiplication, scaling (geometry)). There should be some discussion of why real numbers are used for this and how it works, since there are surely links to real number from many other articles in this context.
 * As the "real part" of complex numbers or various kinds of hypercomplex numbers. This article should define complex numbers and discuss the relationship between real and complex numbers in more detail.

Much of the rest of this article seems more or less okay. It is not very accessible to non-experts, but doesn’t necessarily have to be. –jacobolus (t) 20:40, 20 October 2022 (UTC)
 * On a first reading, I generally agree with all of that. Paul August &#9742; 21:29, 20 October 2022 (UTC)
 * In mathematics, "continuous" is an adjective most commonly applied to a function. It is not standard to say that a number is continuous, or that a set of numbers is continuous.  What definition of continuous are you thinking of when you say that "real numbers are continuous"? Ebony Jackson (talk) 03:56, 21 October 2022 (UTC)
 * Here we are talking about a linear continuum. But what do you think this article means when it defines real number as "the value of a continuous quantity"? (This is formalized as the least-upper-bound property, as this article states, but just stating that is too formal/technical and not in itself all that useful to non-specialists; it is important to explain why this is the definition used, what intuitive notion it is getting at. There is discussion of this under the section "completeness" and the article completeness of the real numbers, but they are pitched at too high a level for the widest audience that might be expected to read this article.) –jacobolus (t) 06:04, 21 October 2022 (UTC)
 * I think the closest you can get to capturing the intuitive concept we're trying to get at by a precise topological notion is actually connectedness (our more specific article is at connected space). That too is, as you say, pitched too high for the general introduction.  But we might be able to get across the informal idea that you can't pull it apart into two separate pieces without breaking something, unlike, for example, the rationals, which you can pull apart by dividing them at an irrational. --Trovatore (talk) 06:39, 21 October 2022 (UTC)
 * Yes, that’s right. What the article currently says about this is The reals form a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals. Which to an interested layperson is a more or less incomprehensible wall of jargon. I'm not suggesting we should turn this into a whole mini-course on point-set or metric topology. But I think we should spend a section or three at the start of the article trying to explain what the issue is in the most accessible way we can. –jacobolus (t) 07:24, 21 October 2022 (UTC)

Before reading this thread, I rewrote the lead (IMO, the two last paragraph do not belong to it, and must still be removed). I hope that this solves 's concerns about the lead.) D.Lazard (talk) 10:19, 21 October 2022 (UTC)


 * That is definitely an improvement. I added mentions of real analysis and analytic geometry; hopefully those are clear enough.
 * Does anyone have ideas for how to organize 2–3 new sections at the top of this article elaborating on the motivation and some of the uses of real numbers for a general audience before we get to technical definitions? (I think the 'history' section can probably then be moved much further down the page.) –jacobolus (t) 22:02, 21 October 2022 (UTC)
 * I'm not super comfortable with a description that would treat the rational numbers as "continuous". I think what we want to get across here, intuitively, is that the real line "doesn't have any holes in it".  Completeness will imply that, but it's really closer conceptually to connectedness.  --Trovatore (talk) 23:40, 21 October 2022 (UTC)
 * Do you want to take a shot at rewriting that part of the lead to establish that point? I think the first section after the lead should be something like "Motivation", which could (among other topics) discuss the use of numbers to represent geometric distances, lengths of curves, areas, etc., among which some are clearly not rational and thus "missing" from a number line which only includes rational numbers. –jacobolus (t) 00:06, 22 October 2022 (UTC)
 * I agree with your concern about "continuity" of rational numbers. However, the lack of holes in the real line is not sufficient, for most readers, to explain the difference between real and rational numbers. This is because of this that the first line does not talk of "continuity" of the reals, but talk only of continuity of measured quantities. This is because I do not know any non-technical way to explain the "non-continuity" of the reals, that I have added an explanatory footnote. This leaves the first paragraph non-misleading, although mathematically correct.
 * I do not think that the main motivation of introducing real numbers is geometry. IMO, this is calculus. This is the reason for which I have added a "motivation" paragraph in the lead about that. About a section "Motivation": As WP:NOTTEXTBOOK, such a section must be written for most readers, even those who are not interested in geometry. As limits, continuity and derivatives require to know of real numbers, I do not imagine how to write such a section before describing the main properties of real numbers. So, your "Motivation" section would be better as an "Application" section, placed after the "properties" section. D.Lazard (talk) 13:08, 22 October 2022 (UTC)
 * Yes, the purpose is to make a number system where the solution to calculus problems are “numbers” (instead of just, say, sequences of approximations good to any desired precision). (I think we can address the issue of connectedness/completeness of the real numbers better in a dedicated "motivation" section than in a sentence or two in the lead section.) But calculus was developed to solve problems arising from analytic geometry and mechanics. Anyway, I agree that a "motivation" section should talk about calculus! Something like π doesn’t really make sense except in the context of calculus. But (a) I think it should go before a “properties” section, and (b) I think the properties section should be broken up into a few sections. I can try to come up with some draft idea by pulling images from other wikipedia articles, but am also glad to hear what other folks think the scope here can be. My goal would be that someone who does not know what the “real numbers” are could read a couple sections into the article and end up with a reasonable not-too-technical summary. –jacobolus (t) 18:11, 22 October 2022 (UTC)
 * As for WP:NOTTEXTBOOK... what it says is, The purpose of Wikipedia is to summarize accepted knowledge, not to teach subject matter. Articles should not read like textbooks, with leading questions and systematic problem solutions as examples. I am not suggesting adding leading questions or problem sets, nor is the goal here to “teach” the subject (for that, someone can consult an analysis textbook). The goal would be to introduce/explain the idea first for a wider audience curious to know what a real number is, before diving into a wall of more advanced jargon useful to the “mathematically mature”. –jacobolus (t) 18:25, 22 October 2022 (UTC)
 * For a rough outline, I am thinking about something like the following, with everything before 'formal definitions' trying to stay somewhat accessible to a broad audience (say, someone with a high school or at most undergraduate engineering/science background) :
 * Lead section
 * Motivation (not too long, just explaining why the rationals are insufficient and the rough concept of the reals)
 * Structure
 * Fundamental operations (+, –, ×, /)
 * Order (<)
 * Completeness
 * Cardinality
 * Intervals
 * Subsets (N, Q, etc.)
 * Basic applications
 * Arithmetic
 * Analytic geometry
 * Calculus
 * Complex numbers
 * Linear algebra
 * Formal definitions
 * Advanced properties
 * Use in analysis
 * Metric spaces
 * Measure theory
 * Extensions
 * Affinely extended real numbers
 * Projectively extended real numbers
 * Projectively extended complex numbers
 * Generalizations
 * Other applications
 * History
 * What do folks think? jacobolus (t) 16:05, 24 October 2022 (UTC)
 * IMO, the "Motivation" section must be based on the Intermediate value theorem that implies directly that the intersection points of two (continuous) curves have real coordinates, that a continuous function that changes of sign has a zero, and, in particular, that a univariate polynomial of odd degree has a root (an important part of the fundamental theorem of algebra). These properties are fairly intuitive, and, as they are wrong when restricted to rational numbers, show clearly that one need more numbers than the rational ones. D.Lazard (talk) 17:17, 24 October 2022 (UTC)
 * This sounds like a fine plan. My understanding is that historically, the development of analytic geometry was delayed (until Descartes or later) by the use of rational numbers in arithmetic/algebra. Geometry dominated mathematics of the time and could "exactly" intersect circles, forming segments satisfying quadratic constraints (which might be incommensurable, but that was considered fine), while algebra/arithmetic seemed like a substantially separate topic. Irrational numbers were treated with some suspicion, and it was hard to convince mathematicians to accept square roots as numbers per se in solutions to geometric problems. The development of real numbers, mathematical notation, analytic geometry, calculus, the function concept, and mechanics were all intertwined. This idea that continuous curves (either plane curves or smooth functions) must have well-defined intersections is essential to the real number concept.–jacobolus (t) 22:24, 24 October 2022 (UTC)
 * IMO, the "Motivation" section must be based on the Intermediate value theorem that implies directly that the intersection points of two (continuous) curves have real coordinates, that a continuous function that changes of sign has a zero, and, in particular, that a univariate polynomial of odd degree has a root (an important part of the fundamental theorem of algebra). These properties are fairly intuitive, and, as they are wrong when restricted to rational numbers, show clearly that one need more numbers than the rational ones. D.Lazard (talk) 17:17, 24 October 2022 (UTC)
 * This sounds like a fine plan. My understanding is that historically, the development of analytic geometry was delayed (until Descartes or later) by the use of rational numbers in arithmetic/algebra. Geometry dominated mathematics of the time and could "exactly" intersect circles, forming segments satisfying quadratic constraints (which might be incommensurable, but that was considered fine), while algebra/arithmetic seemed like a substantially separate topic. Irrational numbers were treated with some suspicion, and it was hard to convince mathematicians to accept square roots as numbers per se in solutions to geometric problems. The development of real numbers, mathematical notation, analytic geometry, calculus, the function concept, and mechanics were all intertwined. This idea that continuous curves (either plane curves or smooth functions) must have well-defined intersections is essential to the real number concept.–jacobolus (t) 22:24, 24 October 2022 (UTC)

There is a nice discussion of some of the history of these concepts in Hartshorne (2000) "Teaching Geometry According to Euclid". –jacobolus (t) 11:24, 20 January 2023 (UTC)

Complete ordered field
The article suggests that the axiomatic characterization of R as "the complete ordered field" is a definition of R, and that the explicit constructions are alternative definitions. This is wrong, since without the constructions one would not know that such a field existed. Instead it is only the constructions that give definitions. That R is the unique complete ordered field up to isomorphism is a property of R, after it has been defined. Alternatively, it is a theorem that there is at most one complete ordered field up to isomorphism. Ebony Jackson (talk) 23:05, 23 October 2022 (UTC)
 * I agree that the alternative "definitions" are only constructions, but I think that starting form Cauchy completion would be clearer and closer to the common use of the reals (it is fundamental that $$\R^n$$ is complete): The Cauchy completion is the solution of a universal problem, and, as such, if it exists, it is unique up to an isomorphism. So, it is natural to define the reals as the completion of $$\Q.$$ Cauchy's and Dedekind's constructions are proofs that this completion exists (once $$\R$$ is defined, nobody uses these constructions). Then, it is a theorem that this completion is an ordered topological field. The fact that $$\R$$ is the unique Dedekind-complete ordered field is an interesting theorem, but, IMO, not a fundamental one, as rarely used in practice.


 * So I suggest to rewrite along this line the parts of the article that are devoted to the definitions of the reals. D.Lazard (talk) 11:36, 24 October 2022 (UTC)
 * I agree with your assessment of the fact about the unique Dedekind-complete ordered field.
 * Do I understand correctly that the notion of Cauchy completion requires the notion of a complete metric space, which in turn requires the notion of Cauchy sequence? In spelling out the details of Cauchy completion, one would then already be most of the way to the definition of R as a set of equivalence classes of Cauchy sequences.  Given this, I'd probably start with the classical definition in terms of those equivalence classes (easier for most readers to understand), and then mention the universal property later on (more elegant, and simplifies the work in constructing the field operations I suppose, but perhaps not enough is gained over the direct construction). Ebony Jackson (talk) 05:45, 28 October 2022 (UTC)
 * That R is the unique complete ordered field up to isomorphism is a property of R, after it has been defined. – What counts as a definition or an axiom vs. a theorem is to a large extent an arbitrary conventional choice. Mathematicians love to cut out as many properties of objects as they can when making definitions (for a variety of reasons including e.g. avoiding contradictions, making the definitions easier to remember, joy at proving even basic properties as theorems, rhetorical flourish, and personal vanity) so among alternative definitions the one with the fewest essential features is usually preferred. But we shouldn’t confuse how something is defined with what something is. If we choose a different (logically equivalent) definition of a structure or a different (logically equivalent) set of axioms for a mathematical discipline it doesn’t change the scope or nature of the subject, but only re-orders some of the subsequent statements, switches a few labels between 'axiom' vs. 'theorem', and forces some rewriting of some of the proofs. But those are all surface-level changes. –jacobolus (t) 17:09, 4 April 2023 (UTC)

Lead characterization
The lead reads "continuous means that values can have arbitrarily small variations". Though this language has been in the article for some time, it's not at all clear what it means. What does it mean for a value to "have variations"? I realize that this sentence is not intended to define the reals (which is more subtle) but as it stands, it doesn't even make sense. How about "continuous means that between any two real numbers there is another real number" -- which of course is necessary but not sufficient to define the reals. --Macrakis (talk) 20:34, 3 April 2023 (UTC)


 * Agreed, the current one is insufficient and misleading (and "between any two real numbers there is another real number" would be similarly misleading). It should include something about containing its limit points or the like. It’s tricky to come up with a clear/precise enough wording which is also accessible to non-technical readers. It would be good for someone to do some searching through past sources for a legible definition. –jacobolus (t) 16:57, 4 April 2023 (UTC)


 * It's not clear that the first paragraph needs to include a rigorous definition. It should give the general, non-technical, reader a sense of what the topic is. Although I suppose "every number defined by a decimal fraction (possibly infinite)" is correct, though certainly not what a mathematician would use as a definition. --Macrakis (talk) 17:02, 5 April 2023 (UTC)
 * "Number represented by an infinite decimal fraction" probably is the characterization that's both (1) extensionally accurate and (2) quickly understandable by the lay reader. That would be a pretty good argument for leading with that, except that it doesn't get at the motivation at all, and it leaves the impression that the real numbers are somehow deeply connected with base 10. I note in passing that so-called "terminating" decimal fractions are still infinitely long; it's just that all but finitely many of the places are occupied by 0.
 * The first paragraph of the current version is not bad; I think it strikes a reasonable balance. The last sentence of the first paragraph might be tweaked to say a little more explicitly something along the lines of "If you don't want to get into it too deeply, you can think of real numbers as being the values of infinite decimal fractions" (obviously cleaned up into a more encyclopedic tone). --Trovatore (talk) 17:37, 5 April 2023 (UTC)
 * "certainly not what a mathematician would use as a definition”: I disagree. There are many textbooks written by mathematicians that use this definition. Also, p-adic numbers are very often defined by infinite p-adic expansions, which are similar to decimal expansions. I am pretty sure that Dedekind invented Dedekind cuts for having a definition that is base independent.
 * If infinite decimal expansions are mentioned in the first paragraph, this is not to provide a definition; this is because, they are often taught very early to kids (unfortunately, in my opinion). So, mentioning them provides an informal explanation that refers to the background of many. D.Lazard (talk) 17:40, 5 April 2023 (UTC)
 * I agree with all of that except this part of the current version: "Here, continuous means that values can have arbitrarily small variations", which makes no sense. How can a value have a variation? I think what it intends to say is that there are numbers that are arbitrarily close to any given number, but as was pointed out above, that doesn't distinguish the reals from the rationals. --Macrakis (talk) 18:33, 5 April 2023 (UTC)
 * I think what it's getting at is that it can describe physical quantities that aren't granular. Distance, mass, time, things you measure with no preset limit to precision.  This is at an intuitive level; it's not really about whether those things are in fact quantized. --Trovatore (talk) 19:44, 5 April 2023 (UTC)
 * I tend to agree that the first paragraph does not need to include a rigorous definition. The article already has a Formal definitions sections with rigorous definitions. If anyone has a problem with the sloppy descriptions in the lead, perhaps we could precede them with something like "Loosely, . . ." or "Loosely speaking, . . ."—Anita5192 (talk) 18:41, 5 April 2023 (UTC)
 * Based on the above discussion, I've reworded the lead. Obviously open to improvement... --Macrakis (talk) 21:12, 5 April 2023 (UTC)
 * I reverted this change: the removal of the explanation of “continuous” introduces a confusion with the technical meaning of “continuous”. Also the change of the last sentence of the first paragraph amounts to replace a characterization of the reals with a property of decimal expansions, which is out of scope. D.Lazard (talk) 08:23, 6 April 2023 (UTC)
 * I agree that "continuous" is problematic. But the explanation, as I said above, is meaningless both informally and technically. How can a value have a "variation"? As for characterizing the reals as the numbers represented by decimal expansions, that seems to be by far the best proposed explanation of the reals for the non-technical reader. The fact that "Every real number can be almost uniquely represented by an infinite decimal expansion" tells us that the reals are a subset of the numbers representable by an infinite decimal expansion (after all, that statement is also true of the rationals or the algebraics); it doesn't say that the infinite decimal expansions characterize all the reals. --Macrakis (talk) 14:35, 6 April 2023 (UTC)
 * A possible solution for explaining "continuous" is to link to linear continuum, but unfortunately that's a rather technical article. --Macrakis (talk) 14:50, 6 April 2023 (UTC)
 * “How can a value have a "variation"?”: I understand that the speed of your car cannot vary or that its variation is not continuous. D.Lazard (talk) 15:31, 6 April 2023 (UTC)
 * A value is something like 4096 or π/7. It does not have "variations". You apparently want to introduce not just the concept of a real number, but that of a real variable, which is not the same thing. --Macrakis (talk) 15:41, 6 April 2023 (UTC)

I changed this sentence to "Here, continuous means that pairs of values can have arbitrarily small differences." This is technically correct and should be understandable by readers.—Anita5192 (talk) 15:47, 6 April 2023 (UTC)
 * I think something like “variable quantity” would be significantly better than “pairs of values”. But arbitrarily small differences is also not the key point here; the important feature is that the reals don’t have “gaps” the way the rational numbers do. –jacobolus (t) 18:32, 6 April 2023 (UTC)
 * Macrakis's objection seems to be to applying the word "vary" to a "value". I do think it's better to say that a "quantity" can vary.  In some sense that means that the value of that quantity varies, and that's the sense in which the value can vary.
 * Some of the mathematical objections are a bit beside the point when applied to this language, which is meant to appeal to physical or quasi-physical intuition (geometric intuitions count as quasi-physical). --Trovatore (talk) 20:48, 6 April 2023 (UTC)
 * There are two issues here. One is the "vary" language; let's leave that to later.
 * The other is "Every real number can be almost uniquely represented by an infinite decimal expansion." This is certainly true, but not very interesting. As I said above, that statement is also true of every rational number, every algebraic number, heck even of the empty set. The interesting statement is "Every infinite decimal expansion represents a real number." This is (a) correct and (b) the basis for the diagonal proof of uncountability, so it captures a deep property of the reals as distinguished from other numbers. Yes, it needs to be made more precise for the proof (that they almost always represent distinct real numbers), but it is an excellent first cut at an intuitive definition.  --Macrakis (talk) 03:04, 18 April 2023 (UTC)
 * If there are no objections, I will use the wording in the previous comment. --Macrakis (talk) 16:03, 20 April 2023 (UTC)


 * I object. I think the line about decimal expansions should be removed, as it does not precisely characterize the real numbers. I don't think it gives readers a clear idea of what the real numbers are. Decimal expansions are already treated in a separate section.—Anita5192 (talk) 17:01, 20 April 2023 (UTC)

Decimal representation
I find this section extremely difficult to follow. The summation defining $$S_n=\sum_{i=-k}^n a_{-i} 10^{-i}$$ is not clear because it uses negative indexing, which I think is unnecessary. The statement that $$S_n< 10^k$$ for every $n$ appears to be incorrect, but I am not sure because it is so difficult to understand. Finally, what is the point of this section? What is it trying to say?—Anita5192 (talk) 16:25, 19 May 2023 (UTC)


 * This section is required, and must be near to the top, since, most people identify real numbers and their decimal representations. Also, the section use implicitly the decimal (or binary) representation for Cantor's diagonal argument. So the section is important.
 * However, I agree that it is too technical. In fact, technical accuracy, is much easier to obtain this way. So, this section must be rewritten with examples and less proofs in view of a better compromise better accuracy and readibility. D.Lazard (talk) 17:25, 19 May 2023 (UTC)
 * I rewritten the section, making the hypothesis that readers well know decimal numbers. I hope that the result is clearer. D.Lazard (talk) 17:20, 20 May 2023 (UTC)


 * This is better, but still difficult to follow. The description in the lead of the decimal representation article is clearer. We should be careful how we define an infinite set of numbers, each with a (possibly) infinite number of digits.
 * This section only defines nonnegative real numbers. I think it should address negative real numbers and zero.
 * As for the bijection: Since the decimal fraction part is a sum of only a finite number of digits, how do we distinguish between, for example, 1, 1.0, 1.00, 1.000, . . . ?
 * I think this article should simply link to the decimal representation article and we should make any necessary improvements there.—Anita5192 (talk) 23:13, 23 May 2023 (UTC)