Talk:Real number

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D. what grade level are you aiming this article at?[edit]

CorvetteZ51 (talk) 09:36, 10 June 2015 (UTC)

Decimal digits[edit]

A real number is one that can be expressed in the form 'DDD.ddd'. DDD is zero or more decimal digits ddd is zero or more decimal digits Of course, DDD must be finite in length. This restriction does not apply to ddd.

Why must DDD be finite in length? If a sequence of real numbers goes to infinity, then there must be an (countably) infinite number of digits in ...DDD. What am I missing?

I don't really understand the question. The sequence 101,102,103,104,... goes to infinity, but none of the numbers have infinite digits.
In effect, a sequence of numbers may go to infinity, but a single number can't. (Consider the problem of comparing two such "infinite integers". How could you decide which was bigger without calculating all the (infinite) digits ..DDDD for both numbers.
On ...DDD.ddd...: ...DDD.ddd, for a finite .ddd defines the 10-adic numbers. p-adic numbers form a field only when p is prime. The real numbers are an ordered field. How do you order the repeating numbers a=...101010. and b=...010101. so that field order is preserved, giver B=10A+1 and A=10B? — Preceding unsigned comment added by (talk) 18:24, 16 June 2011 (UTC)
Hello, I would like to make a few suggestions for this article. First, I would recommend removing the "or" in the introduction after the semicolon (where it's talking about irrational numbers being things like the square root of two and pi) because it suggest a nonexistent duality (e.g. I can write pi as 3.14159265358979323846264338...). I would also recommend editing the example because it's too close to e (it looks like someone was trying to write e but couldn't remember the digits). There are also too many dots after that decimal expansion too. I would change it myself, but for some reason Wikipedia wont let me edit this page at the moment (it's just this page; I don't know why). —Preceding unsigned comment added by (talk) 20:40, 20 October 2008 (UTC)


The term "real number" is a retronym coined in response to "imaginary number".

I doubt this, and it is unsourced. The OED quotes Descartes: " sont pas tousiours reeles; mais quelquefois seulement imaginaires; " which suggests that the terms are twins. (This refers to real and imaginary roots, of course; but the earliest uses will be in that context.) Septentrionalis PMAnderson 22:11, 13 December 2006 (UTC)

The retronym question is something like a hoax, in my opinion. The term "real number", in various languages, appears well before 1872: Hamilton used it in English, Liouville in French, Leibniz and Gauss in Latin (realis quantitas). It comes straight from Descartes, the quotation from OED is correct: he divided the solutions of equations in "réelles" and "imaginaires", and the former into "vraies" (positive) and "fausses" (negative or zero). He also used "quantité réelle" and "quantité imaginaire". Later on, but already in the 18th century, mathematician started to say "number" instead of "quantity". But the contraposition between "real" and "imaginary" is there from the start. Bballmath (talk) 10:32, 3 July 2009 (UTC)

Symbols used for the set of real number[edit]

Hello, I just did a modification to the page, adding the original character that represents the real numbers: ℝ this is a Unicode character called the set of real numbers.

Should all the reference to R be changed to ℝ or this additional note in the article is enough ?

Erik Garres 08:34, 8 January 2007 (UTC)

It's not a good idea to use ℝ, as some browsers won't display it. Also, it's not the "original character" for the reals - people were using R for the reals long before anyone used blackboard bold, and many people still prefer R (except, perhaps, on blackboards). --Zundark 10:56, 8 January 2007 (UTC)

What is ? This is used for real axes on the argand diagram, so why not in say sets or other references to --

is sometimes used for denoting the real part of a complex number (although is more common for this purpose), which is why you've seen it used to label the real axis of the Argand diagram. It's sometimes used for other (unrelated) things as well. I'm not sure why you think it should be used in references to . --Zundark 14:30, 12 March 2007 (UTC)

Would someone please mentions the symbol "ℝ" Unicode number next to it ? --DynV (talk) 07:19, 25 October 2009 (UTC)

The article start: "A symbol of the set of real numbers (ℝ)", and the proceeds to use R for ℝ. Weird. Can someone point me to a list of "some browsers won't display" (ℝ), it has been over 5 years since this original sub-heading "symbols used for the set of real number" and (correct me if I am wrong) I suspect that wikipedia has moved on in the mean time and now officially supports Unicode 6.2 fully.

Keep in mind that in the majority?many? of wikipedia's <math> LaTeX sections </math> a ℝ is being displayed. This is producing a weird inconsistency in ℝ notation between wikipedia's text and LaTeX content.

NevilleDNZ (talk) 08:09, 29 August 2013 (UTC)

Zundark's more important point is the second one — blackboard bold is an expedient to put bold on blackboards. Some workers do use it as "the symbol for the reals" even in books and papers, but I do not think this is the majority usage. My preference is just plain R. There's no reason that can't be used in LaTeX as well. --Trovatore (talk) 16:07, 29 August 2013 (UTC)

To be frank, I have not done a literature search on R for ℝ. But as I said above "This is producing a weird inconsistency in ℝ notation between wikipedia's text and LaTeX content."

Re: "people were using R for the reals long before anyone used blackboard bold" ...

Interesting to encounter this line of argument: To paraphrase "people were using stones for the counting long before anyone used computers"...

Re: "some browsers won't display it"

ℂ, ℍ, ℕ, ℙ, ℚ, ℝ & ℤ are widely supported. c.f. Blackboard bold
FYI: Here is the complete list of bold characters: 𝔸 𝔹 ℂ 𝔻 𝔼 𝔽 𝔾 ℍ 𝕀 𝕁 𝕂 𝕃 ℕ 𝕆 ℙ ℚ ℝ 𝕊 𝕋 𝕌 𝕍 𝕎 𝕏 𝕐 ℤ

Maybe this is a favorite vs centre kind of issue? The "inconsistency in ℝ notation" is at the discretion of the individual editor...?

NevilleDNZ (talk) 22:58, 29 August 2013 (UTC)

reals, defined to non-imaginary ?[edit]

or am I missing something. Wiki readers who are non-mathmeticians need to know how reals are different. CorvetteZ51 13:50, 22 May 2007 (UTC)

I'm afraid the formulation added was not very accurate. All real numbers are also complex numbers (in other words, the set of complex numbers contains the real numbers), that is how the term "complex number" is defined. The number zero is both a real number and an imaginary number, though the latter may be a matter of definition and not accepted by everybody.
Nevertheless, something along the lines you propose should perhaps be added, like "The imaginary unit i, which satisfies the equation i2 = −1, is not a real number." I'm not sure what the best place would be. -- Jitse Niesen (talk) 16:14, 27 May 2007 (UTC)
reals definition --> no imaginary components.the BS about what a real is, does not change what thereal is not, the real is not imaginary, please allow non-mathemeticitians to know what a real is, the real has no imaginary parts. CorvetteZ51 16:45, 27 May 2007 (UTC)
There are lots of things a real number is not. The important idea to get across is what a real number is. You seem to be wanting to emphasize the use of "real" as distinctive from complex numbers that aren't real -- but the notion of a complex number is more complicated than, and comes logically after, the notion of a real number. First we have to explain what a real number is, not what it isn't. --Trovatore 18:48, 27 May 2007 (UTC)
my understanding is... any number with non-zero 'i' is non-real. Others are real. Not sure about 0i or 0 + 0i. Please enlighten me, if what I have wrote is not, correct and entirely complete. CorvetteZ51 12:05, 28 May 2007 (UTC)
Well, that's correct, if by "number" you mean "complex number". But then you have to define "complex number" first; how are you going to do that?
here is how Mathworld defines 'complex number', , here is how Mathworld defines 'reals', , who dissagrees with that?CorvetteZ51 10:19, 29 May 2007 (UTC):
So first of all you should be aware, in general, that Mathworld, while sometimes a useful resource, is, let's say, "quirky". In the case at hand, though, there's nothing I disagree with in the definitional part of these two articles -- it's just that they never get to an actual definition.
To summarize: The Mathworld articles say that a complex number is x+iy where x and y are real numbers; that's fine. "Real number" is linked and says that a real number is either a rational number or an irrational number; this is true but not a good definition as it appears to make the notion of an irrational number more fundamental than that of a real number, which is backwards. But it could work if "irrational number were suitably defined. Click on the link to "irrational" and you discover that an "irrational number" is a "number that cannot be expressed as p/q for any integers p and q." So what then is a "number"? No link to it, and the chain of definitions has terminated with an undefined term, "number", that as I pointed out above, needs further explanation. --Trovatore 17:39, 29 May 2007 (UTC)
Trovatore, I think there needs to be pointed out that there are multiple definitions for 'real numbers'. For a high school student, the definition would be,,, 'Real' numbers are the union of rational and irrationals, and are essentially the 'number-line' numbers. The term is needed as a contradistinction of imaginary. Somebody else can write the mathmatician's version.CorvetteZ51 11:34, 30 May 2007 (UTC)
The term is in contradistincton to "imaginary"; that's true, and worthy of mention in the lede, I think. However that doesn't define it. The "union of the rationals and the irrationals" would define it if you could define the irrationals first, but how are you going to do that? The "number-line" version is a good intuitive motivation, one that I think deserves more emphasis than it currently has (look back in this page or its archives for the discussion of the "Kantian" perspective, which I actually agree with in this case, as little love as I have for Kant's dour philosophy in general). --Trovatore 17:37, 30 May 2007 (UTC)
You seem to have the idea that the word "number", by itself, is non-problematic, but that isn't so. There are all sorts of things that are or have been occasionally called "numbers" that are not real or complex numbers. A few examples: Transfinite cardinal numbers, transfinite ordinal numbers, hypercomplex numbers, extended real numbers (these include +∞ and −∞), and "numbers" like XBQIRT that identify your reservation at the airport. And even if there weren't all these other meanings of the word "number", there's still the problem that, before it's meaningful to say that a real number is a "number with zero imaginary part", you first have to say what a number is in the first place. --Trovatore 21:09, 28 May 2007 (UTC)
A real number has the property of being a complex number with an imaginary part equal to zero. The formal definition of a real number is a set of numbers that have a bijection with points on a line extending towards infinity in both directions, and is closed under multiplication and addition. From the reals, one can define imaginaries as the square roots of nonpositive reals, and complexes as the sums of reals and imaginaries. (talk) 17:27, 21 February 2012 (UTC) Michael Ejercito
Actually I just looked back in the history: Corvette's actual point seems to be about the etymology of "real". I think it is probably accurate to say that "real" is a retronym in contradistinction to "imaginary", and I do think it would be reasonable to say something about that in the lede, if we're sure it's true (anyone have a source?). --Trovatore 18:57, 27 May 2007 (UTC)
I don't think real numbers should be defined as non-imaginary. But I do think that there should be something in this article that gives examples of non-real numbers. It is quite similar to saying an object is a tree if it has leaves and grows. While this is true of a tree, bushes also fit this definition yet are not trees. While it is difficult to make a fully correct definition that does not lead to misnomers, we can still close that gap. I'm mainly pointing this out because the definition says that "real numbers include both rational numbers...and irrational numbers, such as pi and the square root of two" but does not point out that the square root of a negative number (square root of negative two) is not a real number while it can still be irrational. I do see that the next paragraph does note that it is difficult to define a real number, but this seems like something that can help clarify rather than including a statement that needs citation after the definition ("The term "real number" is a retronym coined in response to "imaginary number".").
I will note that I do not have a degree in mathematics, but I think it's still a valid point. I just feel that something so dynamic needs a slightly better definition that may encompass what it isn't instead of only what it is." Xe7al (talk) 04:25, 15 June 2009 (UTC)


The article currently claims:

The term "real number" is a retronym coined in response to "imaginary number".

However, I checked the Oxford English Dictionary, and it's not clear that this is quite true.

In particular, the OED has an entry for the mathematical usage of "real" as an adjective to describe "quantities (Opposed to IMAGINARY, or IMPOSSIBLE)" (emphasis added). In particular, the earliest usages they cite actually seem to be using the term "real" to describe only positive roots, and in another case it seems to be restricted to only rational roots:

1727-41 CHAMBERS Cycl. s.v. Root, If the value of x be positive, i.e. if x be a positive quantity,..the root [of an equation] is called a real or true root. 1841 Penny Cycl. XX. 150/2 Here a and b are meant to be real algebraical quantities, that is, reducible to positive or negative whole numbers or fractions.

In short, the term "real" was originally a broader reflection of the prejudices of pre-modern mathematicians against certain classes of numbers as "impossible" or "unreal". As time went on, eventually negative and irrational solutions were considered "real", and only the imaginary solutions were excluded...which apparently became fixed terminology by the end of the 19th century (when most of these prejudices had been cast aside in favor of a more axiomatic rigor).

—Steven G. Johnson (talk) 00:45, 17 August 2008 (UTC)

Interesting. Of course in some sense the Chambers quote is not directly on point, since he's using a distinct notion of real from the one treated in the article; in another sense, the example actually supports what is being claimed about real being coined in contradistinction to imaginary; and in still another sense, you're right, it makes the current text problematic. I wonder if we could find a less dictionary-like source that gives more details about the history of the usage. --Trovatore (talk) 00:52, 17 August 2008 (UTC)
The point is that the term "real number" did not originate as an antonym to "imaginary number", it originated as a classification of polynomial roots, and gradually evolved from there to the modern definition. —Steven G. Johnson (talk) 06:41, 17 August 2008 (UTC)
Well, it isn't clear that the modern sense of real number actually evolved from the usage being cited, but if it did, it was apparently in contradistinction to an earlier sort notion of imaginariness. But in any case this is all speculative, and dictionaries are not good references. It would be nice to find something about this in a history of mathematics; that would be a good source. --Trovatore (talk) 08:38, 17 August 2008 (UTC) claims that "the terms IMAGINARY and REAL were introduced in French by Rene Descartes (1596-1650) in "La Geometrie" (1637)" and also gives an English citation dated 1685. In both cases real means not imaginary (or not complex). I've no idea how reliable that is, but the quotation looks pretty convincing to me. As Trovatore says, a maths history would be better. -- Jitse Niesen (talk) 10:11, 17 August 2008 (UTC)

complete implies Archimedean[edit]

The lead says that the reals can be defined axiomatically as the "complete Archimedean ordered field." The main body of the article, however, gives an axiomatization that simply defines them as a "complete ordered field." The second version is correct, and the word "Archimedean" in the lead should be deleted. The Archimedean property is not independent of completeness. If you make a conservative extension of the reals to a nonarchimedean system, you get the hyperreals, which lack completeness. The Archimedean property is important, but it shouldn't be given in the lead as if it were independent of completeness. You only have to specify whether it's Archimedean if it's not complete. The rationals are Archimedean and not complete; the hyperreals are non-Archimedean and not complete.-- (talk) 15:16, 22 March 2009 (UTC)

I opened this talk page to suggest this very issue. Just because the hyperreals are not complete does not mean that no other non-Archemedean ordered field is not complete. — Preceding unsigned comment added by (talk) 18:36, 16 June 2011 (UTC)
I changed "Archimedean ordered" to "totally ordered", which should resolve this issue. Lapasotka (talk) 09:18, 18 June 2011 (UTC)

Too much jargon[edit]

Whilst a science article can't always subscribe to the axiom if you can't explain it to a layman, then you don't know your subject, I'd go as far as to say that this article is obfuscatory in the extreme. I'm not a stupid person (my mother once told me), but I still fail to understand what the difference between a real and unreal (or is it imaginary) number is from the first paragraph - and that is something I'm afraid I always complain about! We have a duty to non-specialists to inform, as well as provide the specialists with hard reference material. Blitterbug (talk) 07:29, 9 July 2009 (UTC)

The article is not (and should not be) primarily about the difference between real and imaginary numbers. It is, and should be, primarily about the real numbers, which are a much subtler and more difficult concept than it would first appear.
Still, I kind of agree that the article could do a better job of explaining, at least, why it's a subtle and difficult concept. The geometric nature of the reals is underemphasized, and Zeno's paradoxes, which are one of the key first indications that something non-obvious is needed, are not mentioned at all. --Trovatore (talk) 16:08, 9 July 2009 (UTC)
Iam so lost  —Preceding unsigned comment added by (talk) 21:34, 20 October 2009 (UTC) 

rotational expression —Preceding unsigned comment added by (talk) 00:23, 15 July 2010 (UTC)

Meaning of "strictly more"[edit]

What is the meaning of "strictly more". Isn't it just the same thing as "more"? —Preceding unsigned comment added by (talk) 05:41, 29 September 2010 (UTC)

It means "more, and not the same". Yes, that is just the same thing as "more". The "strictly" is for emphasis, to make it crystal clear that we don't mean "at least as many as". --Trovatore (talk) 08:16, 29 September 2010 (UTC)

Countable or uncountable number of algorithms?[edit]

In the "Uses" section, we have the following statement:

"Because there are only countably many algorithms, but an uncountable number of reals, "most" real numbers fail to be computable."

First of all, I kinda feel the truthines of this statement, but have never seen it proved. So, proof needed. Second, if this or these proofs exist, there should definitely be an inline citation, as this would be material enough for an entire article on its own (IMO).

Cheers! Trolle3000 [talk] 06:51, 23 October 2010 (UTC)

See for example this book as a citation.
Basically, the statement is obvious if you understand what "countable" means. Any computer program, Turing machine, etcetera, can be expressed as a finite string of symbols (from a finite dictionary), and such strings are countable. — Steven G. Johnson (talk) 19:19, 23 October 2010 (UTC)

4X[2+3]-4+3 —Preceding unsigned comment added by (talk) 13:11, 4 December 2010 (UTC)

Nelson or not Nelson[edit]

After two reverts to restore a previous controversial section on Nelson's theory of the reals, tkuvho(talk) send me a private mail saying that I am wrong in my edits and that I have to read the IST page (There is no IST page, but I guess he meant internal set theory).

Here is my answer, which may be interesting to everybody:

"May be your are right in your assertion about Nelson. But

  1. The section was tagged twice as not sourced and non neutral. The tags precise that both are valid reasons for deletion. Protonk(talk) did it and you have reverted it without good reasons.
  2. My edit is aimed to resolve these issues in a constructive (usual, not mathematical meaning) way, which you never try to do. Again you revert it, starting seemingly an edit war.
  3. May be you are right and I am wrong about about Nelson theory. But you do not provide any verifiable source supporting your opinion. If you have such sources, you may edit the first paragraph of the new version by adding, for example "Edward Nelson (not Ed Nelson, which is a showbusinessman) has introduced another non ZF theory of the reals [cite source], which differs by this specific feature".
  4. In any case WP:NPOV implies that the space devoted to Nelson's theory should not be much larger than the space devoted to constructivism."

Comments are needed to avoid an edit war. D.Lazard (talk) 14:43, 22 December 2010 (UTC)

My comments:

  1. This sentence seems very dubious to me: "Abraham Robinson's theory of nonstandard nonstandard hyperreal numbers, although based on Zermelo-Fraenkel axiomatization, is somewhat related to constructivism." Any connection to constructivism there is extremely tenuous. The fact that constructivists are interested in real numbers is certainly true, but it should not need to be backed up by an example of Robinson. I will add a proper reference to Bishop and Bridges.
  2. I think that the paragraph on Robinson's hyperreals should be in the "generalizations and extensions" section. Would anyone object to that?
  3. I don't know that Nelson's theory is very widely studied. Certainly it seems much, much less well known than Robinson's theory. I think that, if it were going to be mentioned here, Nelson's theory should be limited to a single sentence and a link. Robinson's theory maybe deserves a paragraph.

— Carl (CBM · talk) 16:15, 22 December 2010 (UTC)

Also, the article on constructivism links Nelson with constructivism for his study of predicative arithmetic. That's correct, but IST is not a predicative system, nor one that people in constructive mathematics seem to study very much. — Carl (CBM · talk) 16:29, 22 December 2010 (UTC)

I agree with your comment and your modification of the page: Before reading them, I have reintroduced Nelson in Robinson's paragraph. I intended also to remove the dubious link to constructivism, but I forgot to do it. Ignoring everything about Nelson's theory I have tried to extract relevant information from tkuvho(talk) edits and related wp articles. I was wrong. D.Lazard (talk) 16:38, 22 December 2010 (UTC)
Thanks, D.Lazard. Carl: internal set theory is not as obscure as you seem to think. Thus, in France it is actually more in vogue than Robinson's approach. As far as the present page is concerned, Nelson's theory is more relevant than Robinson's, since the latter is a theory of an extension of the reals, whereas the former is a theory of the reals themselves--with respect to a different background set theory (though of course still with respect to classical logic, making it a far more "conservative" theory than Lawvere's, for example). Tkuvho (talk) 17:06, 22 December 2010 (UTC)
I don't think that Robinson's approach is particularly in vogue, but at least it's often described in model theory books as an application of ultraproducts. On the other hand, Nelson's set theory doesn't seem to be discussed either in set theory books or in model theory books, at least not the ones I have seen.
There is a difficulty when you say "extension of the reals": the reals are a semantic object in the metatheory, like the natural numbers. So what happens (apparently? ) is that IST never defines the actual real numbers, just as an ω-inconsistent set theory would never be able to define the actual natural numbers. I'm actually somewhat confused on that point, because the article on internal set theory is somewhat fuzzy, and claims that IST is a conservative extension of ZFC. ZFC certainly proves that there are no infinitesimal real numbers, so if the article is right then IST also proves there are no infinitesimals. — Carl (CBM · talk) 17:41, 22 December 2010 (UTC)
Actually I was hoping you would explain some of these things to me rather than vice versa. Having said this, I was describing Robinson, not Nelson, as an "extension of the reals", so I don't understand your first comment. Nelson's numbers are not an extension, they are the reals themselves. When you speak of the "actual reals" you are apparently referring to what Nelson would call "standard" reals, i.e. reals with a truthful value of his "S" predicate. These are indeed not a "set" in "internal set theory". This corresponds to the fact that in Robinson, the reals do not form an internal set. With the "sets" that are allowed in Nelson, the least upper bound axiom is satisfied. But this upper bound may fail to satisfy to "standard" predicate, so to a naive observer it looks "infinite". Here the ZFC axioms will indeed prove that there are no "infinitesimals", but you will nonetheless have positive reals less than any standard real. If I said anything that's amiss I would appreciate if someone could correct this, as I am much more familiar with Robinson than Nelson. Tkuvho (talk) 18:29, 22 December 2010 (UTC)
I do not know enough of logic to have an advice on Nelson's theory. I may only said the following: When I was in mathematics department until the eighties (I am now in computer science department), I have heard of Cohen's and Robinson's results, but never of Nelson's. I guess, that, even in France, most young mathematicians (except those specialized in logic) have never heard of any of the three. In any case this convince me that the title of the section is the good one. CBM, my opinion is that Robinson is better here: Non standard analysis prove exactly the same results as classical analysis, but simplifies some (or many?) proofs. The difference lies only in the basic framework. D.Lazard (talk) 18:48, 22 December 2010 (UTC)
I had a chance to look at this paper [1] by Nelson, enough to see what is going on (to some extent). What Nelson does is to add a new predicate, "standard", so that only some real numbers are called "standard". But this term "standard" does not mean "is a real number in the standard model of ZFC", it's just a new predicate that could have been called anything.
What Tkuvho said about infinitesimals seems right to me.
Now, Nelson's results imply that if there is a model of IST in which the "external" real numbers really are the real numbers of the standard model of ZFC, then the "standard" real numbers in this model are actually only a finite subset of the actual real numbers. In that case it cannot be said that the "standard" real numbers are meant to replace what we normally think of as real numbers. But it seems unlikely to me that IST has such a model, in which case the things that IST calls "external" real numbers will never be the set of actual real numbers (i.e. the ones in the standard model of ZFC.) In that case, which I think is quite likely but don't have the desire to check, IST does indeed entail an extension of the real numbers, because IST has no model in which the "real numbers" of the model are the actual real numbers.
In any case, ignoring all that, I think that linking to the article on IST from this article is completely appropriate, since it's a relevant link, but it needs to be kept lower in the article, with the more technical topics, and the discussion should be kept very brief. I view it almost like a "see also" link, but included in the article text instead of the see also section. — Carl (CBM · talk) 20:42, 22 December 2010 (UTC)
Existing text is still problematic, though. I have to object in particular to the claim that the classical treatment of the reals is "based on" ZF. I think it would be more accurate to say that Cantor's and Zermelo's work was motivated by considerations arising from the study of the reals. --Trovatore (talk) 20:48, 22 December 2010 (UTC)
Yes, I agree. How about "The real numbers are most often formalized using the Zermelo-Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics. In particular, the real numbers are also studied in reverse mathematics and in constructive mathematics." — Carl (CBM · talk) 21:10, 22 December 2010 (UTC)
An alternative wording would be to say that ZF is the background set theory of the traditional treatment of the reals. Carl: I am a bit confused by what you write, but as far as I understand it, (1) the "external" real numbers will never be the usual ones familiar from ZFC; (2) the "standard" real numbers are meant to replace what ZF-afficionados think of as ordinary real numbers; (3) since IST is in a suitable sense "isomorphic" to Robinson, it does entail an extension of the reals. The claim of IST being a conservative extension of ZFC I think is meant as a claim that they are simultaneously consistent, but I am out of my depth here. Tkuvho (talk) 21:17, 22 December 2010 (UTC)
Yes, well, see, I disagree with the claim that ZF is the "background set theory". The background set theory is not really any axiomatic theory whatsoever. --Trovatore (talk) 21:50, 22 December 2010 (UTC)

If you don't specify the background set theory then you have considerable latitude as to what your reals could look like. They could for example look like Nelson's reals, assorted with the "standard" predicate, in such a way that some real numbers already behave as infinitesimals. This is of course fabulous news for analysis; you can define a derivative by actually forming the infinitesimal quotient a la ancienne; you can write down a Dirac delta function that's actually a function rather than a distribution. So all in all I am sympathetic to your point of view. Though I am not entirely convinced this is what you meant :) Tkuvho (talk) 21:56, 22 December 2010 (UTC)

The set theory is not unspecified. It's just not axiomatic. --Trovatore (talk) 22:02, 22 December 2010 (UTC)

You seem to want to eat the cake and have it, too. The background theory is there, and presumably it is ZF, but it's not axiomatic ZF. It's real life, platonic, sweaty under the armpits kind of ZF, that certainly does not tolerate any competition from funny standard predicates. Tkuvho (talk) 22:06, 22 December 2010 (UTC)

If you call it "ZF" you're referring to an axiomatization. The "real life etc kind of ZF" is just called "set theory". --Trovatore (talk) 22:11, 22 December 2010 (UTC)
When I say "set theory" I mean ZF(C)... If I mean the standard model, I say that too. — Carl (CBM · talk) 22:13, 22 December 2010 (UTC)
So you don't count even large cardinals? --Trovatore (talk) 22:50, 22 December 2010 (UTC)
It's not necessary to resolve this in order to find an acceptable wording for the article. But I don't see why we need ZFC to see that the real line is a complete Archimedean ordered field; I think this is clear just from our intuition about various properties of idealized geometric lines. — Carl (CBM · talk) 22:13, 22 December 2010 (UTC)
Trovatore: I am still not sure what you are trying to say. Fine, let's call it "set theory". The traditional approach to defining the reals certainly starts with set theory. The question is, which set theory do you start with? If you don't specify the axioms, you don't know which set theory you are in. If you start with ZF, you get one kind of real field. If you start with the IST axiomatisation, you get a real field with greater ability to express intuitively appealing constructions in calculus. Carl: I think I can appreciate that your "idealized geometric line" is a ZF-based real field. On the other hand, quantum theory tells us that certain tiny lengths are inaccessible even theoretically. Similarly, the finite size of the universe tells us that certain huge sizes are inaccessible even theoretically (see the related motivational discussion at IST). My "idealized geometric line" is much closer to Nelson's than to Cantor's. Tkuvho (talk) 10:25, 23 December 2010 (UTC)
No, the traditional approach does not "start" with set theory. It can be expressed in set theory, which is completely different. I think you're falling prey here to the error of foundationalism. --Trovatore (talk) 03:25, 24 December 2010 (UTC)
No, I am perfectly aware of the trap of foundationalism, I am just saying that the traditional approach falls into the said trap, as one can see by consulting any textbook. At any rate, the point seems a bit academic. The real line can be "expressed" in ZF, or it can be expressed in an enriched axiomatisation such as IST. In this formulation, all of my remarks still apply. Tkuvho (talk) 05:36, 24 December 2010 (UTC)
(←) What Nelson means by "conservative extension" is that if you take ZFC augmented with an extra predicate for "standard" and Nelson's extra axioms for IST that give meaning to this predicate, then any sentence of ZFC that does not mention the new "standard" predicate and which is provable in IST is already provable in ZFC. — Carl (CBM · talk) 21:29, 22 December 2010 (UTC)


The following entry seems highly questionable:

'Note importantly, however, that all actual measurements of physical quantities yield rational numbers because the precision of such measurements can only be finite.'

If you measure a long distance by counting how many times you can fit a 1 m^2 square plate diagonally on the way, your measurement will yield an irrational number of meters. Lapasotka (talk) 09:21, 5 January 2011 (UTC)

Yeah, it's problematic, as is the following bit about computers being able to use only rationals. It's true in both cases that the usual representations used to report a measurement, or to represent a value in a computer, denote rationals, but the text as stated is not really justified. I'm not quite ready to just rip the text out, though; this needs a little thought. --Trovatore (talk) 10:32, 5 January 2011 (UTC)
I agree that the sentence is questionable as it is, but it is because of a bad presentation of a relevant fact. I suggest to replace this sentence by something like:
Note importantly, however, that all actual measurements of physical quantities yield approximate values which are usually expressed by finite decimal or binary numbers, which are rational numbers (here finite means that there are only a finite number of digits in the fractional part of the number). In particular, real numbers are approximately represented in computers by floating point numbers, which are also rational numbers. Nevertheless these rational values are not manipulated as rational numbers, but as approximations of unknown real numbers. For example the product of two floating point numbers is different of their product as rational numbers, as the fractional part is truncated to keep a fixed precision.
D.Lazard (talk) 11:07, 5 January 2011 (UTC)
Notice that Lapasotka said "by counting how many times you can fit a 1 m^2 square plate diagonally on the way" (emphasis added). The actual result of your measurement is a count, i.e. a natural number. Only after you multiply it by the assumed length of the plate's diagonal does it become an irrational number. JRSpriggs (talk) 12:10, 5 January 2011 (UTC)
My counter argument is probably too naive, but there is certainly a problem here. In mainstream physics the space and time themselves are modeled using the real numbers, and one can certainly conduct measurements which necessarily produce irrational values for some quantities in the same physical model. If you model an actual cylinder by a mathematical cylinder, the diameter and the circumference of the cylinder will have the ratio π in your physical model, and measuring either of these quantities from the actual cylinder will yield an irrational value for at least one of them in the physical model.
Of course one stores only an appropriate number of digits for different quantities when conducting physical measurements in practice, so taking the opposite point of view one could say that actual measurements only yield integer quantities. But this is only a convention. I really cannot see why I couldn't take an ordinary measuring tape and wrap it around an actual cylinder to see that its circumference is about π m. This is a legitimate physical measurement which yields an irrational number.
BTW, has anyone else heard about a court ruling where the ratio of the diameter and the circumference of a circle was fixed as 3.14? Lapasotka (talk) 21:31, 5 January 2011 (UTC)

As I see it, the valid point being made by the existing text is the following: Real numbers are infinite-precision objects. It is impossible to make a measurement to infinite precision, nor can computers store infinite-precision representations.

That is true and worth saying. What's fishy is the idea that, given that the objects represented are not infinite-precision real numbers, they must be rational numbers. I don't think they are. To me they seem more like "fuzzy real numbers", real numbers represented within a tolerance.

Unfortunately, going into too much detail along those lines is likely to run afoul of WP:OR. So I think it would be better to say that you can't make measurements to infinite precision nor represent them in computers, and leave it at that. --Trovatore (talk) 22:06, 5 January 2011 (UTC)

The redaction I have proposed above for replacing the questionable sentence seems to me to be more or less the same as Trovatore's "fuzzy real numbers". The fact that the product of two floating point numbers is different from their product as rational numbers appears as a key to show that measured values, although represented by rationals, are not rational numbers (we find here a key distinction, which is common in computer science, between a (mathematical) object and its representation.
I am surprised that nobody has yet discussed my proposition for replacing the questionable sentence. D.Lazard (talk) 22:51, 5 January 2011 (UTC)
But I said (or at least meant) that I would not put the "fuzzy reals" idea into the article, on WP:NOR grounds. I think the same about your discussion on floating-point numbers, unless you can source it. Even then the idea should probably be attributed to a particular source rather than stated as a fact, unless you can show that it is the consensus view in the field. --Trovatore (talk) 22:57, 5 January 2011 (UTC)
To Trovatore: I like your proposed alternative wording. JRSpriggs (talk) 06:53, 6 January 2011 (UTC)
To D.Lazard: The alternative wording is correct and nice, but a little lengthy. Perhaps you could merge or replace the following passage with it:
Note importantly, however, that all actual measurements of physical quantities yield rational numbers because the precision of such measurements can only be finite.
Computers cannot directly operate on real numbers, but only on a finite subset of rational numbers, limited by the number of bits used to store them. However, computer algebra systems are able to treat some irrational numbers exactly by storing their algebraic description (such as "sqrt(2)") rather than their rational or their decimal approximation.
Another perplexing passage in this section is
Unlimited precision real numbers in the physical universe are prohibited by the holographic principle and the Bekenstein bound.
The citation leads to ArXiv (not peer reviewed) into an article where this problematics is not even mentioned (at least in a short glance.) Besides, what is an "unlimited precision real number"? Has 1 or 2 less precision than π or e? I propose total discarding of this sentence. Rest of the section seems to be fine. Lapasotka (talk) 09:44, 7 January 2011 (UTC)

On introduction[edit]

I think the introduction should start with a conscise non-jargon description of the concept of real numbers. For instance "continuum" is jargon and does not describe much to anyone who does not understand the concept of real numbers already. Also the dichotomy of reals into rationals and irrationals should be postponed a little bit, because it doesn't describe anything either (unless you already understand what real number AND rational numbers are).

Proposal for introduction:

Symbol often used to denote the set of real numbers

Real numbers are mathematical objects that are used to measure various quantities in mathematics, physics and other sciences. Real numbers include natural numbers, which are used in counting objects (two apples, four cars, three Cadillacs etc.) as well as ordering them (first, second and third place in a contest etc.), but real numbers are also useful in measuring quantities which can "change continuously", such as the temperature inside a refrigerator or the length of a fuse.

The question "What real numbers really are?" is largely philosophical. Instead, in mathematics the real numbers are defined rigorously by how they behave. The real number system consists of the set of all real numbers, the operations of addition and multiplication, and the order relation between two real numbers. The axioms of real numbers describe precisely how this system of real numbers behaves, and they are tailored to encapsulate the geometric idea of an infinitely long line with two special points 0 (the origin) and 1 on it (this particular line is frequently referred to as the number line or the real line). Real numbers correspond to the points on the number line and the order relation x<y means that the points x and y appear on the number line in the same order as 0 and 1. Addition of real numbers x and y corresponds to parallel translation of the corresponding line segments starting from 0, and their product is determined by the proportions of these line segments and the line segment from 0 to 1.

Real numbers as points on an infinitely long number line, and integers as its subset.

The set of natural numbers contains those real numbers 1, 2=1+1, 3=1+1+1, ... that can be reached by adding the number 1 arbitrarily many times to itself, the set of integers contains 0, all natural numbers and their negatives, and the set of rational numbers contains all real numbers , where m is an integer and n is a natural number. Rational numbers are dense in the set of real numbers in the sense that for any real number there is a rational number arbitrarily close to it. However, there are points on the real line that do not correspond to rational numbers, a fact of which the ancient Greek and Indian mathematicians were already aware of. The most common example is the diagonal of the unit square, which is of irrational length .

The completeness property of the real number system encapsulates the idea that there are no holes (or gaps) on the real line, and also that there are no infinitely small or infinitely large real numbers. Informally it states that if one cuts the real line into two halves, then exactly one of the halves will contain an end point. The set of rational numbers does not satisfy this property, since one can cut it into two halves at . There are extensions of the real number system where both infinitely small and infinitely large real numbers exist, and they are studied in non-standard analysis.

A common way to represent real numbers is by decimal expansions such as 523, -4.25, and 4.91648367... where "..." signifies that the decimal expansion continues indefinitely. Integers correspond to decimal expressions without a fractional part (the sequence of numerals after the decimal point) and rational numbers to those decimal expressions for which the tail of the rational part is periodic, such as =2.08333... Irrational numbers always have infinitely long decimal representations without any periodicity, for example, π=3.1415926535...

End of proposal, last edited by Lapasotka (talk) 11:17, 14 January 2011 (UTC), D.Lazard (talk) 16:07, 14 January 2011 (UTC)

Please comment. I will wait for few days before tampering with anything. Maybe some pictures will be helpful in visualizing the meaning of multiplication and division of real numbers. Lapasotka (talk) 15:35, 9 January 2011 (UTC)

The "holes" or more precisely, "gaps", and the completeness property are relevant to this page. Note however that the rest of the material above applies just as well to the rational numbers. I know it may be shocking to realize one does not need the "real" numbers to measure temperature inside the refrigerator, but I don't think we should necessarily impose such an illusion on our readers. Tkuvho (talk) 16:10, 9 January 2011 (UTC)

I added "gaps" by your suggestion, and also something about irrational numbers. The geometric constructions for the multiplication and addition are of course the same for rational numbers, but the crux is that they work equally well for all real numbers, and that they can be explained to people without knowledge of higher mathematics or axiomatic systems.

Of course one can measure the temperature of a refrigerator with quite a few different number systems, of even with some suitable refrigeratory Performance Index taking values from blue to red, for that matter, but this article is about real numbers, which is the standard system for measuring physical quantities "taking values in the continuum". I was being careful not to state that one needs real numbers to do physical measurements, but only that they are useful in things like that. Lapasotka (talk) 22:47, 9 January 2011 (UTC)

Not really. People interested in doing physical measurements on the one hand, and people interested in gappiness and completeness on the other, form two largely disjoint groups of individuals. The concern for completeness is a largely theoretical concern having little to do with computation. You can do all of Euclid and not notice there are any gaps, as Dedekind pointed out. Tkuvho (talk) 01:20, 10 January 2011 (UTC)

You cannot do all Euclid and not notice that there are gaps on the rational line (which I suppose you meant). Algebraic numbers will do, however. But back to the point:

We need more opinions on the relevance of the discussion of what real numbers are really needed in physical measurements within the article. I see it as an unneccessary complification, since one can also turn the question upside down and ask what numbers need to be discarded. I think we agree that there should be a section on the subsets of real numbers that can be constructed by certain finite processes, such as floating point numbers, algebraic numbers and geometrically constructible numbers, as well as some others subsets such as definable real numbers. However, my impression is that mainstream physicists (or other scientists) do not care too much about these finesses and perhaphs they should not be mentioned in the introduction even though they are of mathematical and (perhaps) philosophical importance.

If there are no objections, corrections or improvements I will paste the text above into the introduction within a few days. Lapasotka (talk) 07:12, 10 January 2011 (UTC)

I object, the the grounds that the expressions "real measurement" and "real number" do not refer to the same meaning of the word "real". Namely, real numbers are a number system that has been developed with certain theoretical considerations in mind, such as the property of completeness making it a useful system in mathematical arguments. Measurements in the real world, on the other hand, and as Lapasotka acknowledges, "do not care too much about these finesses". Arguing that real numbers are needed to measure the temperature in the refrigerator is therefore merely a play on words. Tkuvho (talk) 10:36, 10 January 2011 (UTC)

To Tkuvho: Please re-read 'but real numbers are also useful in measuring quantities which can "change continuously", such as the temperature inside a refrigerator or the length of a fuse.' I agree on the questionability of the statement 'real numbers are needed to measure the temperature in the refrigerator', but this is a different claim. What do you think of the first (actual) one? Lapasotka (talk) 13:27, 10 January 2011 (UTC)

The subsection Real_number#Uses already contains an appropriately brief discussion of measurement and other physical applications. If you mean to ask literally whether real numbers, including the completeness property, are useful in measuring temperature inside the refrigerator, I would have to express scepticism. Are you confusing the two meanings of the word "real"? Tkuvho (talk) 14:07, 10 January 2011 (UTC)
A clarification: decimal numbers are useful in measurement; a similar claim for real numbers is questionable. Tkuvho (talk) 14:09, 10 January 2011 (UTC)

Alright. Maybe one needs to be more precise: 'but real numbers are also useful in modeling quantities which can "change continuously", such as the temperature inside a refrigerator or the length of a fuse.' Either in Real_number#Uses subsection or in another subsection covering decimal representations and floating point numbers one could underline your point. BTW, I don't think I have confused the meaning of the word "real" as the part of the combined word "real number" into its other possible meanings. Whether any numbers are "real" in the everyday sense of the word is a philosophical question that I will not try to answer. Lapasotka (talk) 14:37, 10 January 2011 (UTC)

I just don't think that the completeness property is useful in anything having to do with refrigerators. Decimal numbers and floating point numbers are useful in dealing with refrigerators. I still think you have the wrong address. Tkuvho (talk) 15:11, 10 January 2011 (UTC)
My objections are somewhat different. Completeness is not used directly in real-world applications, but the reals are the most mathematically natural structure that works, and that naturalness aids you in all sorts of ways.
Basically I think the proposed text is too long, insufficiently geometrically-oriented, and it says a few things I don't agree with. It's too much like an essay — not a lot too much, but a little too much. It is very well written, and I am certainly not married to the existing text, which is also insufficiently geometrical. We can discuss further, but please don't just dump in the text as it is. --Trovatore (talk) 17:43, 10 January 2011 (UTC)
I have a deja vu feeling that we had this discussion somewhere before :) Prithee tell me whereby are the real numbers so natural? They are the unique complete ordered field alright, but by the time you are done explaining completeness as well as uncountability (go tell that to a laboratory assistant) how much more natural have you gotten than saying "roots of all polynomials"? Decimals, not real numbers, are used in measurement. Tkuvho (talk) 17:54, 10 January 2011 (UTC)

To Trovatore: Thank you for the comment. I feel myself that the geometric explanation of the real number system is too lengthy, so it is probably better to cut it down to a few words and move the rest together with some pictures into a short subsection. I somehow feel the same about N,Z,Q-part, but from the previous comments it seems that constrasting real numbers with rational numbers cannot be left out of the introduction. Most importantly, saying only that "real numbers can be defined axiomatically" is not a satisfactory solution. Can you please point out the things you do not agree with. Also some ideas how to switch the bias would be useful, but let us keep in mind that this article is about real numbers (as defined in mathematics), not about "real world" or practices of storing measured data in science laboratories. Lapasotka (talk) 19:58, 10 January 2011 (UTC)

To Trovatore: Adding a more detailed discussion of geometry is a good idea. I am not sure whether it should go in the lede or in one of the sections, but one could mention what is known in the literature as "Cantor's axiom", namely the positing of the identification of the numerical continuum on the one hand, and the geometric line in space, on the other. Tkuvho (talk) 05:58, 11 January 2011 (UTC)
I think that's a little bit backwards. My position is that the geometrical line is the motivation for the reals in the first place. To be sure, there are other applications; they're also the natural way of modeling other physical quantities that appear to vary continuously, such as energy. But geometry, I'm pretty sure, came first, and is still the most natural way of understanding the structure. --Trovatore (talk) 07:56, 11 January 2011 (UTC)
This is one place I agree with Trovatore. The identification of points on the line with numbers has to date back at least as far as Descartes. But the ancient Greeks were able to work with lines, and obtain non-obvious results about real numbers such as the irrationality of the square root of two, starting with the real line and adding a coordinate system rather than starting with a field and adding topology.
A little later I will try editing the proposed text above to shorten it, and I'll post the results here. — Carl (CBM · talk) 12:39, 11 January 2011 (UTC)

I did a little streamlining on my original proposal, deleted the mentions on proportionality and re-explained multiplication by comparison of areas. I really think there should be a section where the relationship between real numbers and plane geometry is explained more throughoutly in elementary (pun intented) terms. The fact that real numbers represent abstract proportions should be presented as well, but I don't have a very good idea how. Lapasotka (talk) 16:43, 11 January 2011 (UTC)

I copied here my major change on the introduction that Trovatore just reverted. I am happy to discuss more, but nothing has happened in the talk page recently. Please comment so we can reach some resolution. The current vesion of the introduction is worse than B-class. Lapasotka (talk) 09:23, 14 January 2011 (UTC)

I'll comment more later. But one thing is it's too long. A lot too long, maybe 2x. Another is the bit about mathematicians talking about how reals behave rather than what they are — that's OK if it means structuralism, not OK if it means formalism. Those are the two big things. I'll come back and analyze more closely. --Trovatore (talk) 09:35, 14 January 2011 (UTC)

Structuralism vs. constructivism is a good point, and writing down a short description taking into account both points is a challenging task. My understanding is that nowadays most mathematicians tend to be somewhere between platonists and structuralists, and constructivists are in the minority, so I didn't consider this as a major fault. I compared the length of the introduction to those of the featured math articles and it seemed to be about 50% longer on average. I added a comment on structuralism and shortened the proposition a little bit. Actually it is not much longer than the introductions of some featured math articles any more. To shorten it further I think the discussion of N,Z and Q could be almost entirely discarded. Lapasotka (talk) 10:27, 14 January 2011 (UTC)

To Trovatore: I think I misunderstood your earlier comment. Indeed, geometric intuition should be emphasized either in the intro (which is hard if we want to keep it short) or in the first section about basic properties. Lapasotka (talk) 09:56, 16 January 2011 (UTC)

Incremental change[edit]

I am not sure if this is encoded in some formal wiki regulations somewhere, but it seems to me that the appropriate way to proceed with a lede is incremental: one generally avoids making sweeping changes, and adds items one at a time. This is surely the best way of avoiding reverts. Tkuvho (talk) 09:33, 14 January 2011 (UTC)

I have read again both introductions. It appears to me that the proposed new one is better but too technical: the fundamental operations need to be enumerated, but their description should be left to the body of the article. Also, in the line of Tkuvho's suggestion, it should be carefully checked that no idea of the old introduction is omitted.
Another thing: the proposed introduction refers to the geometric idea of "the real line". Therefore it should be linked to the article line, where, among other definitions, lines are defined as subsets of Rn.Thus care is needed to avoid circular definitions, even if they are informal. Note that the introduction of line has been recently changed, because it suffered of the same issues (but worse) as the one of this article.
D.Lazard (talk) 10:11, 14 January 2011 (UTC)

I copied the proposition on the talk page for the reason that it can be incrementally changed before replacing the original one. I felt that the original introduction needed so many structural changes that it was better to do them at once to avoid unreadable versions of the page. Here is a list of (seemingly) unmentioned things in the original intro:

  1. Complex numbers (better only to mention only inside the article)
  2. "—indeed, the realization that a better definition was needed—was one of the most important developments of 19th century mathematics." (This remark fits better into the History section)
  3. equivalence classes, Cauchy sequences, complete Archimedean ordered field (too technical for an introduction of such a central topic, which should be readable for a wide audience.)
  4. Dedekind cuts (This is actually described, but "Dedekind cut" is not mentioned)

Now the description of fundamental operations (multiplication and addition) is mentioned but not explained in the proposal. Lapasotka (talk) 10:53, 14 January 2011 (UTC)

let's discuss your suggestions one at a time. Concerning (1): the dichotomy of real versus complex is an important one and should certainly be mentioned in the introduction. Tkuvho (talk) 11:26, 14 January 2011 (UTC)
Perhaps it is good to mention that real numbers can be seen as special cases of complex numbers. Or put it differently, the real line can be seen as a part of the complex plane. Brevity is the only problem. Lapasotka (talk) 12:07, 14 January 2011 (UTC)

Note - Just a thought as a reminder, in order to avoid going through this (or a similar) process every time a new contributor arrives, wouldn't it be a good idea to provide a solid standard textbook source with each change? - DVdm (talk) 11:39, 14 January 2011 (UTC)

It is recommendable when something comes up which is not already mentioned in the references. This topic is standard to the bone at the level of current discussion, and the controversies are primarily on style Lapasotka (talk) 12:07, 14 January 2011 (UTC)
Yes, it was just a reminder. But replace recommendable with mandatory, and you're in business :-) - DVdm (talk) 12:19, 14 January 2011 (UTC)
Getting back to your list, your item (2) is a valid suggestion. Perhaps the whole phrase should be deleted, but obviously other editors would have to express themselves in support. As far as your item (3) is concerned, I think these key properties should be mentioned in the introduction. Keep in mind that this is not a page on decimal numbers, which are a key tool in practical measurement. The remarks on temperature in refrigerators could be more appropriate at decimal, decimal representation, etc., or at Simon Stevin who is largely responsible for the prevalence of decimals today. Tkuvho (talk) 19:19, 15 January 2011 (UTC)
In regards the above list, does anyone know the source for (2)? —Preceding unsigned comment added by (talk) 15:02, 30 March 2011 (UTC)
The introduction of Dedekind essay cited in article Dedekind cut shows clearly that Dedekind realized "that a better definition was needed". (Why this essay is not cited here?). The second part of the sentence is an opinion which could be qualified as original research, but it seems to be a consensus among mathematicians about it. Such a unanimous opinion is difficult to source? D.Lazard (talk) 15:24, 30 March 2011 (UTC)
D.Lazard, If you found a source that would be useful here at the Real number article, add it. I looked at the introduction of the article you mentioned however, and did not find it clear that Dedekind realized "that a better definition was needed". Can you point us to a specific passage in the essay that indicates what you say it does? Cliff (talk) 20:09, 30 March 2011 (UTC)

Structure of the article[edit]

How about dividing the material in the article as follows:

  • Introduction (not a section)
  • 1.Elementary description
  • 2.Historical development
  • 3.Formal definition
  • 4.Uses in mathematics and sciences
  • 5.Extensions and generalizations

I could write an elementary geometric decription with pictures into the first section. That is also a good place to explain the decimal expansions and perhaps uncountability of real numbers, which can be done in relatively simple terms. Lapasotka (talk) 12:33, 14 January 2011 (UTC)

"Incremental change" is the key, as I already mentioned above. I don't see much support among the key editors at this page (many of whom are professional mathematicians) to sweeping changes to the page. Some of your ideas are good, but their implementation requires patience. Tkuvho (talk) 19:15, 15 January 2011 (UTC)

I assume that reordering or merging stubbish sections is an incremental change if their content is not changed. The real proposition here is to divide the definitions/descriptions (1 and 3 in this list) into two parts. The "Elementary description" should be accessible to a wide audience and a good reading for precalculus students in US system. It should also follow the historical development of real numbers, explaining the topics in the order

Geometric considerations -> Algebra -> Decimal representations -> Completeness (no gaps in the real line)

The "Formal definitions", I think, is an appropriate place to go Bourbaki, and to include the current Properties section in it. Also the wordings could be a little bit more precise. For example, real numbers "is not a complete metric space", but rather it "has the standard metric d(x,y)=|x-y|, which is complete." Lapasotka (talk) 20:53, 15 January 2011 (UTC)

I am not a big supporter of Bourbaki, either. On the other hand, this page happens to be about the unique Archimedean complete ordered field, not about measuring temperature in the refrigerator. What's the last time you used an equivalence class of Cauchy sequences to measure temperature? Tkuvho (talk) 21:07, 15 January 2011 (UTC)

Constructive attitude needed![edit]

The page is a pain to elaborate. Almost everything is reverted instead of modified and no-one seems to be able to get anything done with it. It is a shame, since real numbers is one of the most important topics in mathematics, and the page as it is right now is a mess. Before I started on it I took a look on featured articles in mathematics and noticed that an elementary description, accessible at least to high school students, is an essential part of them. Here are some comments:

  • Only stating the definitions and properties using math jargon leads to an article which is readable only to people who are familiar with the content already. (Therefore, to a useless article.)
  • The geometric intuition of an infinitely long number line with addition corresponding to concatenation and multiplication corresponding to the proportionality with respect to the unit [0,1] is the intented model for all axiomatic definitions of real numbers.
  • Claiming that the real line is only a geometric interpretation of the real numbers is backwards, extremely POV and historically inaccurate.
  • Collaborative effort should mean modifying inaccurate passages, not reverting them altogether.

And a kind request:

  • Take a look at the featured articles in mathematics and see how they are written.
  • Elaborate this article. It needs it. Don't destroy everything that is added to it.

Lapasotka (talk) 08:09, 20 January 2011 (UTC)

Recent change may have unintended meaning.[edit]

A recent change by a user made subtle modifications to a sentence that drastically altered the meaning of the statement. Here is the link to the change in question. No sources were added. Is the new meaning or the old one correct? Cliff (talk) 18:12, 18 February 2011 (UTC)

I guess that the new formulation is correct. It is sure that Richard Dedekind has given a rigorous definition of the reals, well known as Dedekind cuts. What has to be checked is the date of this work. In fact the article Dedekind cut refers without date to the essay where Dedekind gave this proof and links to the English translation dated of 1901. On Scholar Google I have found a citation giving the date of 1872 to the English title of this essay. Moreover, in the preface of this essay, Dedekind says that he has communicated this construction to some colleagues a long time before publishing the essay.
In summary the edit seems correct, but the date should be checked and source is needed. Moreover "were given by Richard Dedekind (1856) and then by Georg Cantor ..." would better be replaced by "were given independently by Richard Dedekind (1856) and by Georg Cantor ..."
D.Lazard (talk) 20:04, 18 February 2011 (UTC)

My mistake, I was not clear enough. I don't challenge the information about Dedekind, that's easy enough to verify. My question about the modifications were actually to the second paragraph in the History section. The change implies that the the acceptance of these types of numbers was local to Europe, but that Chinese and Arab scholars had accepted these ideas earlier. The previous incarnation of the paragraph indicated that these three regions accepted these notions at about the same time. Cliff (talk) 20:20, 18 February 2011 (UTC)

The changes have now been reverted, but which is correct? since there is no source for either contribution, how do we know which is correct? reverted article. ---Cliff (talk) 20:24, 18 February 2011 (UTC)

Questionable matters in Uses[edit]

Why is the following metaphysical "deduction" still there?

"Note importantly, however, that all actual measurements of physical quantities yield rational numbers because the precision of such measurements can only be finite."

And what about the other even more questionable statement, which is certainly not widely agreed upon and only backed up by a non-peer-reviewed citation?

"Unlimited precision real numbers in the physical universe are prohibited by the holographic principle and the Bekenstein bound.[3]"

I skimmed through the citation and even if it was reliable and peer-reviewed (which ArXiv is not), it did not imply such a strong and controversial conclusion. Besides, the existence of any sort of numbers in the physical universe is a philosophical question, and should be left out of mathematics articles. At most, there could be a section called "Real numbers and metaphysics" where one can include statements like these. In the way how they are stated now they might give a false impression on the importance and definiteness of real number system at the heart of modern day mathematics and its applications. Lapasotka (talk) 09:11, 27 May 2011 (UTC)

"Actual measurements" actually yield finite decimals, not rational numbers, this should be corrected. As far as unlimited precision, quantum mechanics certainly tells us that infinite divisibility breaks down at sufficiently small scales, so the numbers introduced by Simon Stevin do not model reality accurately at that scale, and we don't need to go to an arxiv post for that. Tkuvho (talk) 11:01, 27 May 2011 (UTC)
The "arxiv post" was published in ACM SIGACT News, Vol. 36, No. 1. (March 2005), pp. 30–52, though I have no idea what the reputation of this publication is, as it is not my field. Does anyone know? Tkuvho (talk) 11:03, 27 May 2011 (UTC)

I elaborated the "physical sciences" part into a form which most working mathematicians and physicists can probably agree with. I am still waiting for more comments on the "infinite divisibility". As far as I understand, the space-time is actually modeled by a Minkowski-space (effectively R4) in relativistic quantum mechanics, or by some higher dimesional fibre bundle in super string theories. I don't see how in such framework "infinite precision real numbers" do not exist in the "physical universe". After all, statements as above can only be made with respect to some physical model of the reality. Lapasotka (talk) 13:23, 27 May 2011 (UTC)

I edited the part on the existence of infinite precision real numbers in the physical universe in accordance to the Wikipedia links. I am not an expert on these matters, but I tried to be true to the referred articles. Somekind of elaboration was needed, because the previous statement was simply too radical to be stated only "as matter of fact". Next I would like to draw attention to the P=NP -part. There are no citations. What might that piece of text actually try to achieve? Lapasotka (talk) 13:57, 27 May 2011 (UTC)

unclear definition in the lead[edit]

"Any real number can be determined by a possibly infinite decimal representation (such as that of π above), where the consecutive digits indicate the tenth of an interval given by the previous digits to which the real number belongs."

So what does it mean? Number: 3.1415926535. The last consecutive digits 535 indicate the tenth of an interval given by 3.1415926 to which the real number belongs (the whole π)? What the hell? What's an interval given by "previous digits"? And how do I tell which numbers are consecutive and which are previous. inb4 according to definition it's up to you. Well the one who wrote the definition in the lead should've provided with an example, I really can't understand what is written here. (talk) 22:44, 27 November 2011 (UTC)

I agree that the wording is less than clear. It's clear to me what it means, but that's because I'm used to the notion of a nested sequence of closed intervals converging on a point; to a reader not familiar with that, the phrasing may be less than helpful.
Let me explain what it means and then maybe someone can come up with better wording. The idea is that any initial segment of the decimal representation, say 3.14159, represents an interval that you could round off or truncate to that value, for example the interval [3.141590000...,3.141599999...], where 3.141599999... is an alternative notation for 3.141600000..., I'm using truncation rather than rounding, and I'm giving the interval as closed for a reason that may not matter too much to you (see Heine–Borel property if you're curious; the intersection of a nested sequence of compact sets is always nonempty).
Then the next digit, the 2 in 3.141592, means we have now cut the interval down to a tenth of its previous size, to [3.141592000...,3.141592999...].
Continuing forever, we get a nested collection of closed intervals with exactly one point in all of them, namely π
Is the explanation clear, and does it suggest to anyone how better to phrase things in the lead? --Trovatore (talk) 23:52, 27 November 2011 (UTC)
I've been having a look at that lead and been trying to think to myself what would a middle school child make of it since they are taught about real numbers. There are a lots of unnecessary things there which they mightn't have come across yet. Continuum sounds too highfalutin. They probably have never heard of a transcendental or algebraic number. I see no reason to drag in complex lines. Dmcq (talk) 00:04, 26 March 2012 (UTC)

Everywhere dense?[edit]

In the Advanced Properties section the article claims that the real numbers are everywhere dense. I am familiar with the concept "dense" and "nowhere dense", but not "everywhere dense". Surely when claiming that a set is dense one has to specify the set in which it is dense. The complex numbers are an example of a set in which the real numbers are nowhere dense. Can it still be possible that they are everywhere dense? (talk) 03:21, 21 December 2011 (UTC)

Resolved: by replacing "everywhere dense" by "complete". D.Lazard (talk) 05:57, 21 December 2011 (UTC)
Well, "complete" is not the same as "order-dense", which I think was the intent (strictly between any two points lies a third point). --Trovatore (talk) 06:08, 21 December 2011 (UTC)
... or for that matter, not the same as the topological notion of "dense in itself" (i.e. has no isolated points). --Trovatore (talk) 06:10, 21 December 2011 (UTC)
OK. "Complete" is important here and was lacking here. If another important property remains lacking you may add it. D.Lazard (talk) 08:44, 21 December 2011 (UTC)


I recently added information to the Advanced Properties section regarding the cardinality of the real numbers; specifically that the cardinality of the reals is and that of the natural numbers is . This information was integrated in the first couple of sentences which deals with the cardinality of the reals and how it is strictly larger than that of the natural numbers. However, my edits were reversed. Why? Is this not relevant? — Preceding unsigned comment added by NereusAJ (talkcontribs) 05:44, 21 December 2011 (UTC)

Relevant, sure. Unfortunately it's not clear that it's true. --Trovatore (talk) 05:46, 21 December 2011 (UTC)
What do you mean? Of course it is true. represents the cardinality of the natural numbers. represents the cardinality of the power set of the natural numbers. The cardinality of the continuum, , is the same as by the Continuum Hypothesis. Perhaps you should consult the Wikipedia page on Cardinality. NereusAJ (talk) 06:36, 21 December 2011 (UTC)
This is exactly the point — what is not clear is precisely whether the continuum hypothesis is true. --Trovatore (talk) 06:51, 21 December 2011 (UTC)
(To summarize: The continuum hypothesis is known to be independent of ZFC. ZFC can neither prove that it's true, nor that it's false. That does not by itself close the issue; realists believe that it's either really true or really false, whether or not we can find out which is the case, and some propose various ideas by which we might hope to find out.) --Trovatore (talk) 06:54, 21 December 2011 (UTC)
That is not just a consequence of the Continuum Hypothesis. It was proven by Cantor in his 1874 uncountability proof. See Cardinality of the continuum.NereusAJ (talk) 07:17, 21 December 2011 (UTC)
No, sorry, you're mistaken on this. Take a look at the continuum hypothesis article and search for "independence" or "independent". --Trovatore (talk) 07:20, 21 December 2011 (UTC)
Sorry. Your right. I should wait until we resolve this matter before editing the page again. I don't understand your objection. I believe we can agree than the cardinality of the reals is . The article already states that the cardinality of the reals is equal to the cardinality of the power set of the natural numbers. I assume you don't have a problem with this. Now the cardinality of the natural numbers is (right?). Furthermore, the cardinality of the power set of the natural numbers is . Surely, this means ? Or am I missing something obvious? NereusAJ (talk) 07:34, 21 December 2011 (UTC)
Oh, I think you may have misunderstood a previous post of mine. When I spoke of Cantor's 1874 proof, I did not mean to imply that Cantor proved the Continuum Hypothesis. I meant that he proved the equivalence without using the Continuum Hypothesis. NereusAJ (talk) 08:04, 21 December 2011 (UTC)
  • The cardinality of the reals is , yes. In fact I think that's the definition of .
  • The cardinality of the reals is also , and
  • The cardinality of the powerset of the naturals is
  • None of those points is controversial, and none involves CH
  • But now when you claim that the cardinality of the powerset of the naturals is , or that , well, those are both equivalent to CH.
  • Finally, no, Cantor did not prove in 1874 that . The statement is equivalent to CH, so if he had proved that, he would ipso facto have proved CH. --Trovatore (talk) 08:22, 21 December 2011 (UTC)

In any case this point is treated in section "real numbers and logic" and there is no need to consider it twice. It it another question to know if the organization of the article has to be changed for considering the cardinality question only once. D.Lazard (talk) 08:36, 21 December 2011 (UTC)

My apologies Trovatore. I see now that I am wrong. I was under the illusion that is defined to be . NereusAJ (talk) 09:42, 21 December 2011 (UTC)

Well, you're not alone. Unfortunately the popularizers frequently make this mistake, and even many people who go into mathematics have read the popularizers, and if they never take a set theory course (which most don't) they may never be disabused of the misimpression. --Trovatore (talk) 09:47, 21 December 2011 (UTC)

(Fortunately, kids will grow up reading Wikipedia, so hopefully they will learn a correct deviation.) Also, why is "choice" true? and "continuum hypothesis" is neither true or false? -- Taku (talk)

Choice is normally assumed to be true because most mathematicians take it as an axiom as it makes life much easier, whereas the continuum hypothesis doesn't affect most normal maths and under some defensible axioms it would be wrong so for instance could also be quite a reasonable conclusion. Dmcq (talk) 16:20, 14 June 2012 (UTC)
A little clarification: Choice is usually assumed to be true by pure mathematicians. But, in computational mathematics and in particular for automated theorem proving choice is usually not assumed. D.Lazard (talk) 16:45, 14 June 2012 (UTC)
No, I really think these both miss the point. Choice is assumed to be true because it's just obviously true. Once you've understood the motivating picture, you can accept the picture or not, but if you do accept it, you have to really go out of your way to avoid giving assent to choice.
AC is probably the most complicated axiom that is accepted on the basis of self-evidence. For other axioms (say large-cardinal axioms) there are other forms of evidence and justification, but not self-evidence.
As to the last question, well, it's just not so that CH is "neither true nor false". From the POV of a mathematical realist, CH is either true or false, but we probably don't yet know which one. We may or may not ever know in the future. But in any case, certainly neither CH nor ~CH has the kind of self-evidence that AC does. --Trovatore (talk) 19:47, 14 June 2012 (UTC)
This is not the place for this discussion. But Choice axiom is exactly as "obviously true" and as "self-evident" as Euclides parallel postulate is. The proofs that they are independent axioms are very similar: constructing a model of the theory with the negation of the axiom inside the theory with the axiom. By the way, I did not know "self-evidence" and "obviously true" as mathematical notions. D.Lazard (talk) 21:08, 14 June 2012 (UTC)
The comparison with Euclidean/non-Euclidean geometry is complicated; I agree in some ways and disagree in others, but obviously disagree with your ultimate conclusion. As you say it's not the place. If you like I'll explain on your talk page. --Trovatore (talk) 21:27, 14 June 2012 (UTC)

Misuse of sources[edit]

This article has been edited by a user who is known to have misused sources to unduly promote certain views (see WP:Jagged 85 cleanup). Examination of the sources used by this editor often reveals that the sources have been selectively interpreted or blatantly misrepresented, going beyond any reasonable interpretation of the authors' intent.

Please help by viewing the entry for this article shown at the page, and check the edits to ensure that any claims are valid, and that any references do in fact verify what is claimed.

I searched the page history, and found 3 edits by Jagged 85. Tobby72 (talk) 22:38, 20 January 2012 (UTC)

Real numbers as a Belgian invention[edit]

Simon Stevin seems to be being used as the basis for categorizing real numbers as a Belgian invention. However what did he really invent? As a separate question, was it also novel?

There's some evidence for a claim that he invented a decimal notation for real non-integers. This may have some novelty to it.

He also seems to have worked on quadratic solutions. It's not clear if these were for rational real solutions, or for complex number solutions.

It's claimed here that he invented real numbers. Is that based on his notation, or his work with quadratics? I'm finding it hard (I'm not a mathematician) to see if this justifies a claim for novel invention with rational real numbers (not merely real numbers) from the notation or else for complex numbers from the quadratics. Either way, I'm finding it hard to see his work as being particularly relevant to real numbers specifically. Andy Dingley (talk) 16:49, 13 June 2012 (UTC)

There's always people going around claiming things for different countries. This is just silly, the Babylonians if anyone should be credited with the way they were able to add extra places to the end of the approximation of the square root of two. Dmcq (talk) 17:05, 13 June 2012 (UTC)
I read in Simon Stevin: According to van der Waerden (1985, p. 69), Stevin's "general notion of a real number was accepted, tacitly or explicitly, by all later scientists". I have not checked this citation. One may not credit the Babilonian when they did not know the negative real numbers, but they had some idea of this notion. However, although theorems are usually easy to attribute to someone, notions, like the notion of real numbers, evolve progressively. As the first definition of the real numbers, which is correct from the modern point of view, is due to Kronecker, much later than Stevin, one could also say that real numbers was invented by Kronecker. Conclusion: It is wrong to say that the real numbers have been invented by someone. D.Lazard (talk) 17:39, 13 June 2012 (UTC)
The "Belgian invention" thing is just a ludicrous claim; we don't need elaborate arguments about it. Just revert it any time it gets added. --Trovatore (talk) 19:16, 13 June 2012 (UTC)
I'm less interested here in refuting the "invention" claim than simply trying to find out precisely what it was that Stevin discovered. Andy Dingley (talk) 02:07, 14 June 2012 (UTC)


I wanted, and have included a note on notation as I was looking for R++. I am not sure what the best way to include this is. We can I think have:

The set of positive numbers is often denoted by R+, or R+ or R≥0.
But I haven't seen R≥0.
The set of negative numbers is often denoted by R-, or R≤0
But I haven't seen R≤0.
The set of strictly positive numbers is often denoted by R++, R++ or R>0.
But I haven't seen R>0.
The set of strictly negative numbers is often denoted by R-- or R<0.
But I haven't seen R-- or R<0.

Is what is on at the moment OK or does anyone have any suggestions? (Msrasnw (talk) 14:46, 4 February 2014 (UTC))

For the strictly positive numbers, one can also see , deriving from the fact that the positive reals are the nonzero and nonnegative ones. This notation is rather common in Number theory where is standard for the nonzero reals. D.Lazard (talk) 15:20, 4 February 2014 (UTC)
Zero is not a positive number (see sign (mathematics)) so R≥0 is incorrect for the positive numbers. What distinction are you making between "positive" and "strictly positive"? R-- can be found in the reference I gave. While the "++" notation makes sense, I have never seen it used and so would like to see a reference for it. In my experience, the superscripted notation (of single symbols) predominates and subscripts are resorted to when other conventions clash (as D.Lazard has pointed out above or when a power notation is to be used significantly). However, there is no single standard notation and there may be areas in mathematics which use conventions that I am not familiar with - hence the need for citations. Bill Cherowitzo (talk) 18:56, 4 February 2014 (UTC)
We can't possibly include every notation so my take is we should just include the most common ones. R , R+, and Rn seem sufficient to me. Mathematical objects such as R-, R, R[x], etc. are probably beyond the scope of this introductory article. Mr. Swordfish (talk) 19:34, 4 February 2014 (UTC)
Sorry about the fuss and my confusion. I came accross this in the conext of convex optimisation in economics but looking here [2] there seem quite a few refs for R+ and R++ (the later which I was looking for here.) One of the sources Anandalingam, G., S. Raghavan, Subramanian Raghavan eds. (2003)Telecommunications Network Design and Management Springer has the line
"We will use standard notations R and R+ for the sets of real and real non- negative numbers, respectively; and a not quite standard notation R++ for the set of strictly positive real numbers."
Best wishes (Msrasnw (talk) 23:37, 4 February 2014 (UTC))

In my experience, none of these notations is used much and I don't think it's clear that they are standardized. If you come across R+ in a paper, and it matters whether it includes 0 or not, then you'd better check what the author said about it. If the author didn't say, well, that's his fault; he should have, because there is not actually a standard meaning.
Most authors avoid the problem by writing explicitly [0,∞) or (0,∞) or something like that. Personally I would prefer to remove all reference to these R+-type notations. They aren't used enough to be worth the trouble of trying to track down whether they're standard or not. --Trovatore (talk) 23:46, 4 February 2014 (UTC)
I agree with this. Do we keep Rn ? I think this is pretty standard notation, but perhaps out-of-place here? Mr. Swordfish (talk) 17:06, 5 February 2014 (UTC)

Do real numbers include the rational numbers or not?[edit]

The introductory paragraph reads "The real numbers include all the rational numbers..."

The first section "Basic Properties" reads "More formally, real numbers have ... the least upper bound property." and then "hence the rational numbers do not satisfy the least upper bound property."

Is this a contradiction or am I even dumber than I realize? Drienstra (talk) 00:09, 8 October 2014 (UTC)

If a set has the least upper bound property, a proper subset of that set doesn't have to have that property (we say that a proper subset doesn't inherit the property in this case). You are pointing to the classical example of this phenomenon. The set of rational numbers less than √2 has lots of rational upper bounds, but no rational least upper bound. The same set, thought of as a set of real numbers has the real number √2 as the least upper bound. To put it another way, given a set with the least upper bound property, if you toss out some elements to get a proper subset, some of the things that you toss could be least upper bounds of some subsets of the proper subset you are left with. This proper subset would then not have the least upper bound property. I hope this helps. Bill Cherowitzo (talk) 02:47, 8 October 2014 (UTC)

misleading article, unhelpful to those who want to learn[edit]

in modern usage, a real number is a contradistinction to an imaginary number. CorvetteZ51 (talk) 09:12, 9 June 2015 (UTC)

Maybe true, if "contradistinction" would be defined in mathematics (this is an article about mathematics). In any case, this cannot define the mathematical concept of "real number", as imaginary numbers need real numbers to be defined. D.Lazard (talk) 09:47, 9 June 2015 (UTC)
In the history section we say that Descartes distinguished between real roots of a polynomial and imaginary roots. Is it worth mentioning this in the lede as a way of explaining the adjective? Tkuvho (talk) 17:23, 9 June 2015 (UTC)
I agree, and I have introduced such a sentence in the lead. This has the advantage to make the lead less WP:TECHNICAL by providing an informal description of what is a real number. D.Lazard (talk) 09:49, 10 June 2015 (UTC)
Thanks but I think we should stick to roots of polynomials, as Descartes did, rather than putting words in his mouth. When you claim that real numbers are encountered in the real world, do you include the almost all of them that are not definable? Tkuvho (talk) 12:01, 10 June 2015 (UTC)

Refs and notes[edit]

  • The first Note could be split into a separate footnotes
  • There's inconsistent mix of inline refs and full refs (tagged)
  • Reference columns User:D.Lazard, User:CBM "Pls. don't add columns for no reason; columns aren't mandatory in any way, esp. not for full refs, and not really an improvement. There is no lack of vertical space in a browser". Sorry adding twice was an edit conflict, so hadn't seen it had been undone. There is a reason we use them, and after about 10 refs is common. Here we're at 15 refs. As you say it's optional. There is a whitespace issue, I selected 37em so it was quite wide columns, hardly no reason, but as in a vocal minority I'll leave it for you guys. Widefox; talk 09:49, 11 April 2016 (UTC)
Note that this is the current style (both fixed numbers or use of colwidth= are both deprecated). Thanks Izno for fixing the ref style. If they're all consistent now, pls remove the maintenance template. Regards Widefox; talk 09:26, 15 April 2016 (UTC)
@Widefox: colwidth == no parameter name per Template:Reflist. Colwidth happens to be more explicit in intent. To call it deprecated is incorrect also; Reflist's documentation says nothing to that effect. Since it's the same output, I have no issue with it currently though. Feel free also to remove the maintenance template yourself. --Izno (talk) 11:14, 15 April 2016 (UTC)
Can't remember where it says it's deprecated, but it's there somewhere (maybe the talk, or in a MOS, or in communication with me). Widefox; talk 11:56, 15 April 2016 (UTC)

Zeno's paradoxes[edit]

A recent discussion I saw on a user talk page on my watchlist reminded me of this: It's very odd that Zeno's paradoxes are not mentioned on this page. In some sense they are a central part of the reason that the notion of real numbers is important in the first place. Once you have internalized the reals, it can be hard to understand why anyone would have ever thought they were paradoxical — but that's because you have the notion of the reals, and the notion of infinitely many points in an interval of finite length is clearly just true, not a paradox.

Of course, by itself, that doesn't differentiate the reals from (say) the rationals, but the paradoxes lead naturally to the notion of a limit point, and from there, the reals are the next natural stop.

I don't know offhand where to find a good source, but surely there must be one. I would think this should be treated fairly centrally in the exposition of the motivation for the concept. --Trovatore (talk) 19:57, 5 July 2016 (UTC)