Talk:Rational number

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Recurring Decimals?[edit]

Does this article mention anywhere that recurring decimals are rational? I looked pretty hard, but couldn't see it. I know that to someone who has alot of mathematical expierience it would be relatively obvious, but from reading the page on '0.999... = 1' arguments it seems that it is unclear to the majority that recurring decimals are rational. --Cmdr Clarke (talk) 20:32, 18 December 2007 (UTC)

This information is briefly discussed on the Fraction (mathematics) page.Cliff (talk) 22:19, 21 January 2011 (UTC)

Complex Rationals[edit]

Is there an analogous idea of a "complex rational number"? One guess would be

 \left\{ \frac{w}{z} : w,z \in \mathbb{Z}[i], \ z \neq 0 \right\} (i.e. w and z are Gaussian integers, and z is non-zero).

Does anyone know of an exisiting theory?  Δεκλαν Δαφισ   (talk)  09:17, 29 May 2009 (UTC)

Let w = a + ib and z = c + id for integers a, b, c, d (c or d non-zero) then
 \frac{w}{z} = \left(\frac{ac+bd}{c^2+d^2}\right) + i \left(\frac{bc-ad}{c^2+d^2}\right)
 \implies \left\{ \frac{w}{z} : w,z \in \mathbb{Z}[i], \ z \neq 0 \right\} \subseteq \mathbb{Q}[i] .
So the next question is: can we choose a, b, c, d such that Re(w/z) and Im(w/z) can have artitrary rational values? If so then \subseteq should be replaced by = and the complex rationals are just \mathbb{Q}[i]. If not then \subseteq should be replaced by \subset and there is still work to be done...  Δεκλαν Δαφισ   (talk)  09:32, 29 May 2009 (UTC)
 \left\{ \frac{w}{z} : w,z \in \mathbb{Z}[i], \ z \neq 0 \right\} is the field of fractions of \mathbb{Z}[i]. It clearly contains \mathbb{Q} and i, and must therefore contain \mathbb{Q}[i], and so (from what you have shown) is equal to \mathbb{Q}[i] (which is the same as \mathbb{Q}(i), the field of Gaussian rationals). --Zundark (talk) 11:37, 29 May 2009 (UTC)
Thanks Zundark, but that was only a guess at an idea of a "complex rational number". Is this the well held notion?  Δεκλαν Δαφισ   (talk)  11:42, 29 May 2009 (UTC)

Corresponds to V is[edit]

I reversed Jorge Stolfi's edit. Saying that every integer corresponds to a rational number is, IMHO, better than saying that every integer is a rational number. We can look at the rational numbers as being the quotient space \mathbb{Z} \times (\mathbb{Z} - \{0\}) / \sim where (m1,n1) ~ (m2,n2) if, and only if, m1/n1 = m2/n2. In this case the rational numbers can be seen as a quotient space where the rational numbers themselves are equivalence classes. Although saying that every integer corresponds to a rational number is, at a basic level, no better than saying that every integer is a rational number; the former phrase lends itself to more technical investigations. ~~ Dr Dec (Talk) ~~ 10:13, 2 August 2009 (UTC)

I've added a similar discussion to the introduction. ~~ Dr Dec (Talk) ~~ 12:58, 2 August 2009 (UTC)
I still don't understand the argument. Once one chooses a definition, either formal or informal, one is entitled to use the copula "is"; that is what the word was invented for.
In a context where rational numbers are formally defined, it is entirely correct and appropriate to say that a rational number *is* an equivalence class of Z×Z etc. etc.. In an informal definition, too, it is quite correct and appropriate to say that a rat.num. *is* a number that can be expressed as a quotient a/b etc.; and since a/1 is defined as a, it follows that every integer *is* a rational number. Now, Wikipedia is not to be a "Bourbakipedia", where there is only one "holy" definition for each concept. Rather, all sound definitions of a concept should be treated as equally valid. So there is nothing wrong with using "is" in both definitions; the word does not exclude any views.
Or, from another angle: notice that rational numbers, equivalence classes, quotients, and reduced fractions are all equally *abstract* concepts. Trying to figure out whether two equivalent abstract concepts "are the same thing" or merely "correspond to each other" is a sure recipe for insanity. (Trust me, I have been down that path before.)
My advide is to relax. Mathematics is not formalism. Formalism is a resource that, like anything in life, should be used with moderation, and only when it does good. All the best, --Jorge Stolfi (talk) 00:24, 3 August 2009 (UTC)
I didn't say that one is not entitled to use the copula "is". I simply said that saying that every integer corresponds to a rational number is, IMHO, better than saying that every integer is a rational number, after all \mathbb{Z} \subset \mathbb{Q}. As you say a rational number is (and corresponds to) an equivalence class of \mathbb{Z} \times (\mathbb{Z} - \{0\}) / \sim . But, IMHO, there is more mileage to gained using corresponds to. Also, I begain my thread using the British convention of writing V for verses. The American practise is to write Vs. It is bad Wikipedia form to edit something that someone else has written on a talk page. I would ask you not to do that again. I thank you for your advise, and I can assure you that I am perfectly relaxed.
~~ Dr Dec (Talk) ~~ 18:41, 3 August 2009 (UTC)
I agree with Jorge Stolfi that it's better to say "is". It's clearer that way, and even you appear to agree that it's correct. (And by the way, section titles on Talk pages are communal property.) --Zundark (talk) 20:32, 3 August 2009 (UTC)
Zundark, I'm not sure that Jorge Stolfi says that it's better to use "is". His argument seems to be that there is no real difference in meaning, to which I agree! My argument is that the phrasing corresponds to lends itself to the formal setting in a better way. Maybe I'm being a pedant. As for your link section titles on Talk pages are communal property it clearly says "To avoid disputes it is best to discuss changes with the editor who started the thread..." Given that the existing header was not hard to understand, misleading, offensive, etc, there is no need to have changed it. Given Jorge Stolfi's save summary his change seems to be little more than a retaliatory measure.
~~ Dr Dec (Talk) ~~ 20:43, 3 August 2009 (UTC)
It also says "it is generally acceptable to change section headers when a better header is appropriate". Since your section header contained a misspelling of versus, Jorge Stolfi's replacement was clearly better (even if it was American). --Zundark (talk) 21:02, 3 August 2009 (UTC)
Zundark, how do your comments add to this page? Why do you want to focus on the thread header? Your own link say "To avoid disputes it is best to discuss changes with the editor who started the thread..." I have had no such discussion. End of conversation! Your own link also says "It is not necessary to bring talk pages to publishing standards, so there is no need to correct typing/spelling errors, grammar, etc. It tends to irritate the users whose comments you are correcting." Please try to focus your energy on improving this article and not trying to score points. ~~ Dr Dec (Talk) ~~ 21:20, 3 August 2009 (UTC)

I have reverted Zundark's edit. Let me explain why. The original article used the phrase that "...every integer corresponds to a rational number." This was changed, without talk page discussion, by Jorge Stolfi to "...every integer is a rational number." The popular consensus seems to be that there is linguistically no difference; so why make the change in the first place? I reverted Jorge Stolfi's edit so that the article was as it originally was, and added a second paragraph introducing the abstract theory. I believe that the original wording lends itself to the abstract approach better. Zundark reverted my revert and also removed this introduction to the abstract theory, saying that this second paragraph was not necessary. The first paragraph is, as Wikipedia guideline recommend, of a very introductory nature. The second paragraph then takes an introductory approach to the abstraction. This is standard policy: simple, informal introductory paragraph, then down to the real business. As Zundark rightly says: the abstract approach is explained later; but in much more detail, e.g. composition and metrics are discussed. I feel that the article is best as it was before Jorge Stolfi's change and with a more abstract second paragraph, i.e. as it stands now. I would be interested to read the views of editors other than Jorge Stolfi and Zundark although, naturally, their input is most welcome. ~~ Dr Dec (Talk) ~~ 11:23, 4 August 2009 (UTC)

About the is/corresponds issue, here is a simple argument: every math author that I can think of assumes that Z\subseteqQ\subseteqR\subseteqC... In other words, everybody agrees that integers *are* rationals (and also reals, complexes, etc..
If one must distinguish "is" from "corresponds", then it is preferable to say that the equivalence classes of ordered pairs merely *correspond* to (emulate, simulate, represent, encode, ...) the rationals. (Come to think of it, the notion that those contrived objects *are* the rationals is rather bizarre indeed. 8-)
About the thread header: I wasn't familiar with the "British V" notation and it took me a couple seconds to realize what it meant. I "fixed" it only because I thought that it was a typo. (I often fix sectioning, titles, and indentation of talk pages for improved readability; and I regard that as a basic good citizen's duty.) I cannot understand how the save history could possibly indicate "retaliatory intent"(?) on my part. All the best, --Jorge Stolfi (talk) 02:19, 5 August 2009 (UTC)
No-one is saying that one must distinguish "is" from "corresponds"; they are, in this case, indistinguishable. It's just that the latter lends itself to the abstract arguments better. It was your save history comment: "Disagreein on that nit. Oh well, what is life if not endless strife? 8-)" which made me think you were trying to get your own back. If that's not the case, then I apologise. ~~ Dr Dec (Talk) ~~ 22:37, 5 August 2009 (UTC)

Formal vs. (or V) Informal[edit]

I have restored the original informal definition ("quotient of two integers"), which had been deleted without discussion.
While it is informal, it is no less precise and correct than the formal definition. Moreover, it can be clearly understood by any reader who may come to this page; while the formal definition makes no sense unless the reader (a) already knows the informal definition, and (b) is clever enough to recognize that those integer pairs are disguised fractions, and the equivalence "≈" means that the fractions denote the same quantity.
The informal definition has served mankind, including the finest mathematicians in history, for over 4000 years; and is still effectively used by most people, including the finest mathematicians of today. The formal definition — part of a formalistic fad that started in the 19th century, and lost much of its appeal in the 20th — adds nothing to our understanding of rational numbers. It is only useful within formal logic, and its only merit is to show that one does not need to include "rational number" as a separate primitive concept, since it can be emulated with the concepts of "set" and "cartesian product". That is nice, but it does not make it the official mathematical definition, outside of formal algebra. (One can build a chair out of Lego blocks, but no sane person would define a chair as "a bunch of Lego blocks that one can sit upon".)
Besides, there are infinitely many formal constructions that are distinct from that one, but esentially equivalent to it. For instance, one may use an appropriate subset of Z^4 with the encoding (m,n,k,e) <--> (m + n/(10^k - 1))*10^e. Why pick that one in particular?
All the best, --Jorge Stolfi (talk) 11:50, 17 November 2009 (UTC)

  • I've reverted the in-line LaTeX \Q\, to the HTML bold Q. In-line LaTeX can really mess with some people's we browsers. Everything gets messed-up and out of line. ~~ Dr Dec (Talk) ~~ 12:33, 17 November 2009 (UTC)
I'm changing the formal definition to allow integers as the second coordinate. I don't understand why the definition is restricted the way it is. Revert if you choose, I'll source soon. Cliff (talk) 05:12, 28 March 2011 (UTC)


Describing the Archimedean metric on Q as "derived from the reals" is circular. The metric is used to construct the reals, rather than being derived from the reals. Tkuvho (talk) 05:02, 11 August 2010 (UTC)

Merger proposal[edit]

I am unsure why there are separate pages for Fraction and Rational number. These are equivalent terms as currently covered and do not need separate pages. Has this merge been discussed yet? Clifsportland (talk) 21:38, 10 January 2011 (UTC)

These terms are not equivalent at all. Rational numbers are elements of the field of rational numbers, they are numbers. Fractions are expressions comprising two other expressions and a division sign. Every rational number can be represented by a fraction with integer numerator and denominator, but it can also be represented in other ways, and conversely, fractions can represent other objects, such as rational functions.—Emil J. 11:29, 11 January 2011 (UTC)
While this is a decent interpretation of the two terms, don't you think that the entry for Fraction should reflect that understanding? The entire article is used to discuss rational numbers, how they are added, subtracted, multiplied, and divided. The Fraction page does not discuss the use of fractions to describe rational functions. As the pages exist now, they do not describe different terms.Clifsportland (talk) 21:27, 14 January 2011 (UTC)
Further, Rational functions should be covered in the Rational function page, not also on a second page with rational numbers. Cliff (talk) 20:43, 21 January 2011 (UTC)
I'm opposed to the proposed merger. In addition to the reasons given above by EmilJ, these two articles are appropriate for different audiences, with different mathematical backgrounds. Paul August 02:58, 26 January 2011 (UTC)
Generally opposed to the merger as well, there are "See Also" reciprocal links to the pages. EmilJ's explanation strikes a chord. Intersofia (talk) 17:27, 4 February 2011 (UTC)


As in Field, the lead here also needs significant simplification as well as clarification. I move we shuffle most of its present content below in the main text. Inorout (talk) 12:33, 16 March 2015 (UTC)