Talk:rational numbers

RFD discussion
rfd-sense: "The set of numbers that can be expressed as a ratio of integers (fraction) m/n, where n is not zero. In set-builder notation, it is defined as {m/n|m∈ℤ,n∈ℕ}." Not that I doubt it, but is this a dictionary definition or a mathematical one? Does 'plural of rational number' cover the dictionary aspect? For example we don't define 0.9 recurring (0.9 recurring). Mglovesfun (talk) 20:01, 16 December 2013 (UTC)
 * Added "rfd-sense:". --Dan Polansky (talk) 14:26, 21 December 2013 (UTC)
 * No objections (though I was simply expecting people to read the entry). Mglovesfun (talk) 13:28, 27 December 2013 (UTC)
 * Is this term used outside of mathematics? I doubt it, which is why it has a mathematical definition. Also, I don't see what 0.9 recurring has anything to do with this. --WikiTiki89 20:08, 16 December 2013 (UTC)
 * I agree it's an infelicitous definition. If the first sentence stopped at ‘integers’ it would be just as accurate (wouldn't it?) and less confusing. We need to be defining things like this for people who don't understand mathematics, that's really the whole point of a general-purpose reference work like a dictionary. (I did maths to A-level and the set-builder stuff still means nothing to me: is it necessary? Are we just saying the same thing again in a less accessible way?). Move to RFC, or possibly delete if it's SOP (you can also have rational roots, rational coefficients etc., where ‘rational’ just means ‘expressible as a ratio of two integers’ in all cases). Ƿidsiþ 21:01, 16 December 2013 (UTC)
 * I agree that the definition should be cut off after the word "integers", the rest is unnecessary. But I still don't understand what point Mglovesfun was trying to make with the 0.9 repeating. --WikiTiki89 21:06, 16 December 2013 (UTC)
 * I think it took me so long to find out how to write it, I'd forgotten why I wanted to in the first place. I mean to define 0.9 recurring as a synonym of 1. Mglovesfun (talk) 21:13, 16 December 2013 (UTC)
 * Yes, but what does that have to do with rational numbers? --WikiTiki89 21:15, 16 December 2013 (UTC)
 * Nothing, never claimed it did. It was just a comparison. Mglovesfun (talk) 21:24, 16 December 2013 (UTC)
 * Oh, I see what you mean now. Sorry for not getting it before. --WikiTiki89 21:29, 16 December 2013 (UTC)


 * Keep . There's a difference between the plural of rational number and a reference to the entire set of rational numbers. I agree that the definition should be cut off after "integers" - the rest is a mathematical definition, not a dictionary definition. For comparison, we have definitions for real numbers and natural numbers, but only "plural of" entries for irrational numbers, complex numbers, hyperreal numbers, and imaginary numbers. —Mr. Granger (talk • contribs) 21:34, 16 December 2013 (UTC)
 * Keep, more or less as is. Non-experts will get what they need from the first part of the definition and glaze over the rest; experts (or. at any rate, formal math students) will get what they need from the whole thing. Definitions of complex topics in technical fields should be suitably useful both to the average reader, and to the member of that technical field. bd2412 T 21:56, 16 December 2013 (UTC)
 * Keep. I have edited the definition, but if you disagree, revert me. It should be kept either way. --WikiTiki89 22:00, 16 December 2013 (UTC)
 * I restored the deleted material to a usage note, which should be sufficiently out of the way to avoid intimidating non-math majors, but contains the information that hardcore math fans will be looking for. bd2412 T 22:30, 16 December 2013 (UTC)
 * I think it is unnecessary to have the information there. A link to Wikipedia should be enough. But if you really think it is crucial, I won't object. --WikiTiki89 22:32, 16 December 2013 (UTC)
 * Keep per Mr. Granger. The set Q is a mathematical object frequently referred to and is not simply the plural of rational number, it is the set of all rational numbers.  I do agree, however, that the expression in set notation is not needed.  It adds nothing to the definition, nor is it really a usage note.  It is what it is; a formal expression in the language of sets.  If we want that sort of thing we should put it under a different language head because it ain't English.  Mathematical notation is either a language of its own or else it is translingual.  By the way, for those that can't read gobbledegook, the expression translates into English as "the set of all quotients of m and n where m is an element of the set of integers and n is an element of the set of natural numbers". Pretty much the same as the definition we already have in words except that it is not necessary to explicitly exclude n=0 since zero is not included in N. Spinning Spark  00:55, 17 December 2013 (UTC)
 * Keep. A considerable number of non-native users are math-literate, and for them the mathematical definition given in addition to the verbal one is very clarifying. --Hekaheka (talk) 06:27, 19 December 2013 (UTC)
 * An additional point: most of our math-related definitions include a formula. Deleting this would logically indicate that we would prefer to remove them all. Probably all chemical formulae would have to go as well. --Hekaheka (talk) 06:34, 19 December 2013 (UTC)


 * Delete the nominated sense as redundant to the definition in rational number, and we have "plural form of rational number" at rational numbers. I am unconvinced by the argument by Mr. Granger from 21:34, 16 December 2013; the other arguments above I cannot follow at all. Checked: . By the way, "{m/n|m∈ℤ,n∈ℕ}" is no more precise than rational number's "A real number that can be expressed as the ratio of two integers"; it is just a technical notation. If you want to present this notation to the reader, you can do so by adding "; any member of {m/n|m∈ℤ,n∈ℕ}" to rational number. --Dan Polansky (talk) 14:31, 21 December 2013 (UTC)
 * Since you found it necessary to add "any member of" to the front of the expression, I take it that you understand that the expression is for the set of rational numbers. The term rational numbers when used with the meaning of this set is not the same as the plural of rational number.  That plural meaning is any old collection of rational numbers.  The set meaning is specifically the set ℚ, the set of all rational numbers. Spinning Spark  03:51, 25 December 2013 (UTC)
 * Re: "... the expression is for the set of rational numbers." You see, you yourself do not write "the set of all rational numbers", since all is implied. The plural of a noun can refer to all items, depending on context and use of articles: if I say "cats are animals", I mean "all cats" or "cats in general, with possible exceptions". Sure, I had to write "member of" to allow the use of the set notation in the singular entry for the lovers of the set notation; there is nothing necessary about using the set notation. Put differently, if you use a technical notation that naturally constructs collectives rather than predicates, you have to say "member of". The letter "ℚ" does not work like mantra, neither does the set notation; chanting mathematical symbols and notation as if they were some sort of mantra does not bolster any argument and brings one closer to mysticism. "ℚ" is just a convenience to enable writing things like "x∈ℚ", one that you want to have since you have a short symbol for "∈"--"member of"--while having no such symbol for "is a" AKA "instance of".
 * On another note, let us have a look at what other sources do. Let us first check and . When you follow the latter link and click through Collins, Vocabulary.com, Wikipedia, and "Encyclopedia" (item 5 there), you land on pages that use "rational number" as the leading headword of term. As a second check, have a look at https://www.google.com/#q=define+%22rational+numbers%22, a search that uses plural, and check whether the pages that you find use plural or singular. From what I can see, sources online do not feel the need to define "rational numbers" as something separate that is not sufficiently covered by "rational number". --Dan Polansky (talk) 10:15, 25 December 2013 (UTC)
 * To prove that this sense is not covered by the plural of rational number, consider the term open interval. I have never heard someone say "the open intervals" to refer to the set of all open intervals. —Mr. Granger (talk • contribs) 13:02, 25 December 2013 (UTC)
 * I don't think that common use of "the rational numbers" in contrast to "the open intervals" to refer to the set of all of them proves that we need to have the entry for "rational numbers" defined as anything else but the plural. We certainly do not need two different or redundant definitions at rational number and rational numbers. If this common use of the plural to refer to all of them is considered an oddity to be highlighted, a usage note at rational number or rational numbers should do, IMHO. Somewhat similarly, we do not have a separate definition at computer to cover the use in which "the computer" does not refer to any particular item taking space but rather to the invention in general. --Dan Polansky (talk) 13:27, 25 December 2013 (UTC)
 * A quick Google Books search turns up this and this. Given that there are only 19 hits for "the set of open intervals", the rarity of the expression may have something to do with the nature of open intervals as a class rather than with this construction. Perhaps it's because it consists of relations between pairs of numbers rather than numbers sharing a common property. Chuck Entz (talk) 23:45, 27 December 2013 (UTC)
 * Open intervals are sets, not relations between pairs of numbers. But okay, I'm convinced. Delete. —Mr. Granger (talk • contribs) 00:04, 28 December 2013 (UTC)
 * Reduce it to a plural of "rational number". Delete the set rubbish. Equinox ◑ 10:17, 25 December 2013 (UTC)


 * Perhaps our resident mathematician would like to weigh in. Pinging User:msh210... :) - -sche (discuss) 01:57, 26 December 2013 (UTC)
 * I'm unfamiliar with this sense of the term, but that's an RFV issue. (A quotation like "function on the rational numbers" is not for this sense: it means a function on all of the rational numbers (i.e. the usual plural). I don't know of uses with "The rational numbers is", but cites will tell.) As to RFD: certainly it's a separate sense from the plural-of sense: keep. &#x200b;—msh210℠ (talk) 07:27, 26 December 2013 (UTC)
 * @msh210: I don't think the point of the sense "The set of numbers that ..." is to claim that "the rational numbers" is grammatically used in singular just like "the set". The point seems to be that "the rational numbers" means all of them. I could send the sense to RFV, but I do not think I would be able to require that only the likes of "The rational numbers is" count toward attestation. Furthermore, the use of plural or singular with the likes of "the group of ..." seems to be pondian anyway, so the genus "the set" does not unequivocally force the grammatical singular. --Dan Polansky (talk) 15:12, 26 December 2013 (UTC)
 * AFAIK even in Leftpondia "the set of Xes" is singular; certainly in a math context. I understood the nominated definition as referring to, well, the set, which would always take a singular verb (in a math context at least). If it refers to rational numbers generally, then it's redundant to the other, and delete. &#x200b;—msh210℠ (talk) 05:20, 27 December 2013 (UTC)
 * @msh210, two questions:
 * One, when you say "even in Leftpondia 'the set of Xes' is singular", do you mean that you would say something like "integers is those elements of the infinite and numerable set {...,-3,-2,-1,0,1,2,3,...}, and rational numbers is those numbers that [blah blah blah]"? Something about that sounds grammatically "off" to me. Or do you mean that you would say "the set of rational numbers is..."? In that case "is" goes with "set" and makes grammatical sense, but also seems irrelevant, since the entry under discussion is [[rational numbers]], not [[set of rational numbers]]. (We wouldn't add a usage note to [[phonebook]] claiming it is used with a singular verb just because it is possible to say "the set of all phonebooks is...")
 * Two, is there any mathematical concept X for which the term that means "the set of all Xs" is not the same written word as the plural form of X? There are some non-mathemetical things for which there exist different terms, e.g. multiple people = "humans" but the set of all people = "humanity". (Even then, "humans" can also mean "the set of all people": "Humans are mammals that [blah blah blah].") It seems to be a rule that the general plural Xs of a word X can mean either "the set of all Xs" or "multiple Xs". For example, I can speak of the "trees" outside my flat, or of all "trees"...
 * ...and if that is a general rule—that plurals can refer to either multiple Xs or the set of all Xs—then it seems appropriate to explain what it takes to be an X in the entry X, not the entry Xs.
 * (general comment) I think this RFD was ill-formed, and has thus turned into a discussion of whether or not we should give a precise explanation of what it takes to be a rational number. Really, it should be a discussion of whether to give that precise explanation in the singular entry, the plural entry, or both. (Accordingly, I have half a mind to start a RFM once the RFD concludes.) - -sche (discuss) 06:12, 27 December 2013 (UTC)
 * Ad 1: I mean the latter, and it's relevant because the nominated sense is "The set of numbers that can be expressed as a ratio of integers…". See also below.
 * Ad 2: You're not using set the way a mathematician uses set, and I read the entry as using it the way a mathematician uses it. So we're speaking past each other a bit here. Let me try to clarify: By set of Xes I mean a single entity, the collection of all Xes. That phrase (set of Xes) takes a singular verb, always. Humans are mammals refers to all humans, not to the set of humans. The set of humans contains some seven billion members refers to the set. Rational numbers refers to any, possibly all, rational numbers, not to the set (AFAIK; again, that's an RFV issue). The nominated sense "The set…" thus is different from the usual-plural sense, which is why I said to keep it, and it one that I doubt can be cited (not that I've looked). I hope I've made my point more clearly.
 * &#x200b;—msh210℠ (talk) 07:49, 27 December 2013 (UTC)
 * Since this discussion is moving in the direction of RFV, let's start looking for citations. Here's one that seems to meet MSh210's criterion,
 * First, we review his method for showing that the cardinality of the rational numbers is the same as the cardinality of the natural numbers.
 * The set of rational numbers is clearly meant here even though "rational numbers" is not explicitly preceded by "set of". The claim that the cardinality is the same implies this: clearly, taking any old bunch of rational numbers and comparing with any old bunch of natural numbers will not necessarily result in equal cardinality. Spinning Spark  17:44, 27 December 2013 (UTC)
 * You cannot pick "any old bunch of rational numbers" using an expression that contains a definite article: "the rational numbers". Your quotation has some merit, but the idea that "the rational numbers" could possibly refer to any old set containing some rational numbers but not all of them not so. --Dan Polansky (talk) 17:54, 27 December 2013 (UTC)
 * The prefix "the" is relative to context, as is "you". The set of rational numbers given by the number of urinations my dog makes in any day over the number of different trees he uses is not the set in the quotation.  Cantor is not talking about my dog pissing up a tree, he is talking about ℚ. Spinning Spark  19:21, 27 December 2013 (UTC)
 * Delete. Using "the" without qualifiers and without cues to the contrary in the context means that you're talking about about the entirety of that which follows, either as a whole or as a class. The fact that someone might say "the British speak English differently than the Americans do" doesn't mean we're talking about left-handed accountants from Derbyshire vs. people who watch Duck Dynasty from the Midwest. It's a well-known construction that can be applied to any class of numbers: the irrational numbers, the integers, the natural numbers, the imaginary numbers, the powers of two, the even numbers, the odd numbers, etc. Chuck Entz (talk) 23:01, 27 December 2013 (UTC)
 * After following the above links, I can see that we have a real inconsistency problem in analogous terms, with some entries using this concept, and others using simply "plural of". Chuck Entz (talk) 23:17, 27 December 2013 (UTC)


 * This really needs to be moved to RFV. If it can be shown to be used with set-related terms, then it should be kept. For example "member of the rational numbers", would support this if it is cited. --WikiTiki89 00:25, 28 December 2013 (UTC)
 * It wouldn't support it, any more than "one of my children" would support "children" being anything more than the plural of "child". Equinox ◑ 00:28, 28 December 2013 (UTC)
 * You may have misunderstood me: "one" is not a particularly set-related term, while "member" is. You would never say "this child is a member of my children" unless you were thinking of "my children" as a mathematical set. --WikiTiki89 00:31, 28 December 2013 (UTC)
 * It makes no difference. If you think we need separate senses at prime numbers (plural), odd numbers (plural), numbers!!! (plural) etc. merely because they can be sets, then perhaps you're right in some arcane branch of mathematics but you are not right in English. This dictionary is English. Equinox ◑ 02:13, 28 December 2013 (UTC)
 * And to attempt to put it in slightly more mathematical terms: take any two elements x, y from the set of rationals. I ask you, "Are x and y rational numbers?" Yes, obviously. Each one belongs to the set. It doesn't matter that they don't make up the entire set. It's the same as Fido and Rover being dogs, even though there are other dogs in the world. There is no precedent or lexicographical reason for trying to restrict the plural term to the entire set. Equinox ◑ 02:33, 28 December 2013 (UTC)
 * We're not "restricting" the definition; we are adding a second definition. There is a linguistic distinction between using "rational numbers" as the plural of "rational number" and using it as a set. You are right that "prime numbers" and "odd numbers" can be used this way too, as can the plural of any type of anything ("regular polygons", "unit vectors", etc.), which is why I'm starting to think that it would be impossible to include the set definition on each of these terms. I guess it could be considered SOP (or something like that) to use a plural as the name of a set. --WikiTiki89 03:01, 28 December 2013 (UTC)
 * The best proof of a set-only sense would be: "The rational numbers are made up of rational numbers". Do you think anyone would ever say such a thing? Chuck Entz (talk) 06:05, 28 December 2013 (UTC)
 * The "best" proof is not always available, so we'll have to settle for the second best. --WikiTiki89 17:21, 28 December 2013 (UTC)


 * Keep: Another one of these "shoot-first-ask-questions-later" RfDs. This is a valid definition and I see no reason to arbitrarily limit this project  Purplebackpack89  (Notes Taken) (Locker) 02:10, 28 December 2013 (UTC)
 * LOL, if anyone ever "shoots fist" (sic) it's you, Purple. I doubt you'd even want to delete "man wearing a blue hat", because hey, it's words that have a meaning. Equinox ◑ 02:16, 28 December 2013 (UTC)
 * We're a dictionary of words, not a dictionary of words...except for a bunch of words a few editors who vote in RfDs arbitralily decide to delete. And, no, I don't shoot first, or fist, because I don't nominate a lot of entries for deletion.  Also, my proclivities for voting are hardly germane here because this isn't a SOP RfD.  TBH, the only rationale I'm seeing is that the nominator doesn't understand the definition.  Purplebackpack89  (Notes Taken) (Locker) 15:26, 28 December 2013 (UTC)


 * As suggested by several users above, including some who have voted "keep" in this forum, I have opened a Request for Verification. (That does not, IMO, prevent this RFD discussion from continuing, though it should probably not be closed until the RFV is resolved.) - -sche (discuss) 03:11, 28 December 2013 (UTC)


 * Keep Isn't the challenged sense a proper noun sense? DCDuring TALK 14:22, 28 December 2013 (UTC)


 * mathworld has an entry only for the singular. In that article it says The set of all rational numbers is referred to as the "rationals," and forms a field that is denoted Q. (my bold) SemperBlotto (talk) 15:44, 28 December 2013 (UTC)
 * Well, "rationals" is short for "rational numbers" in both the plural sense and the set sense. --WikiTiki89 17:21, 28 December 2013 (UTC)
 * I will note that statements like "rational numbers form a field" are true only if by "rational numbers" you mean the set of all of rational numbers (equipped with the appropriately defined addition and multiplication). If you choose to interpret this as "given some unspecified set of rational numbers, it forms a field", you will get an obvious falsehood: a single number can never be a field. No, field with one element does not count.
 * On the other hand, I would expect most languages conflate the two interpretations just like English does. So from a practical standpoint, splitting translation tables would only create editorial burden for little benefit. Keφr 19:09, 28 December 2013 (UTC)

Kept. <i style="background:lightgreen">bd2412</i> T 18:15, 16 April 2014 (UTC)

RFV discussion: December 2013–June 2014
In the (currently still ongoing) RFD discussion, several people—including our resident mathematician, msh210—suggested that RFV was more appropriate than RFD. So: is rational numbers attested with any meaning other than, where rational number is defined as "a number (a member of the set of numbers) that can be expressed as a ratio of integers"? See RFD for some discussion of what kind of citations might verify a sense other than the sense. - -sche (discuss) 03:05, 28 December 2013 (UTC)
 * Let me post the same citation that I already provided on the AFD page,
 * First, we review his method for showing that the cardinality of the rational numbers is the same as the cardinality of the natural numbers.
 * Because the cite is talking about cardinality it must be referring to a set in the mathematical sense rather than simply the plural of rational number since cardinality is only meaningful with regard to a set. The claim that the cardinality is the same as that of the natural numbers can only be interpreted as the cardinality of the set of all rational numbers is the same as the set of all natural numbers since subsets of these numbers will, in general, not have the same cardinality.  Anyone who knows the first thing about this subject will instantly recognise that the "he" in this cite is Georg Cantor and that this is his famous theorem on infinite sets - stunningly counter-inuitive.
 * There are more cites on the citations page. <font style="background:#fafad2;color:#C08000">Spinning <font style="color:#4840a0">Spark  09:43, 28 December 2013 (UTC)
 * Here's another cite that is even clearer,
 * Since the rational numbers have been shown to be denumerable, one might suspect that any infinite set is denumerable, and that this is the ultimate result of the analysis of the infinite.
 * The term rational numbers is here clearly intended to be taken as a member of infinite sets. <font style="background:#fafad2;color:#C08000">Spinning <font style="color:#4840a0">Spark  10:06, 28 December 2013 (UTC)
 * This one is interesting,
 * We shall, however, show that the property of completeness does not hold good for the ordered field of rational numbers, i.e., the ordered field ℚ of rationals is not order complete.
 * The author uses rational numbers with the meaning of a set, and then immediately rewords using the mathematical symbol ℚ which makes it unambiguous that the set is meant in the initial wording. <font style="background:#fafad2;color:#C08000">Spinning <font style="color:#4840a0">Spark  12:31, 28 December 2013 (UTC)
 * Widespread use in the relevant mathematical literature (textbooks, etc.). DCDuring TALK 14:15, 28 December 2013 (UTC)
 * Agreed, I don't understand this nomination. Mglovesfun (talk) 18:15, 28 December 2013 (UTC)
 * I believe that the reason it is here is that there is an argument at RFD that rational numbers cannot be unambiguously cited with the meaning of "set of all rational numbers", as opposed to simply the plural of rational number (which might include all of them). The issue was not settled after I provided the first cite, so I don't know if I'm wasting my time on this one. <font style="background:#fafad2;color:#C08000">Spinning <font style="color:#4840a0">Spark  09:21, 29 December 2013 (UTC)
 * Many of the cites on the citations page seem ambiguous to me as to whether they use the term to refer to the set or to its constituents. I've added some in the main entry page that satisfy my sense of rigour, and seem to use the term in the singular. — Pingkudimmi 11:14, 29 December 2013 (UTC)
 * Well, I've given a rationale above for three of those cites not being ambiguous (which is half of them). I can make an at least arguable case for the rest of them as well. <font style="background:#fafad2;color:#C08000">Spinning <font style="color:#4840a0">Spark  13:24, 29 December 2013 (UTC)
 * Perhaps my "many" was only four after all, my standard being ambiguity - that they are compatible with the other sense.
 * In your third citation above, the author repeats (with minor change) the formulation "the ordered field of rational numbers," which I read as the field (a fancy kind of set) comprising the rational numbers (plural sense). The symbol ℚ, to my ears, is an item under discussion that is defined as "the ordered field of rationals." To me, the citation seems perfectly compatible with the plural sense of "rational numbers."
 * That said, it was fairly easy to find supporting citations I was happy with. — Pingkudimmi 23:23, 29 December 2013 (UTC)


 * Am I missing something, or is it blindingly obvious that if there is such a thing as a rational number then the set of all such things would naturally be called "rational numbers"? Why does this require a separate definition?? Ƿidsiþ 08:42, 30 December 2013 (UTC)
 * In the RFD, they mentioned something about distinguishing between the rational numbers "class" versus the rational numbers "set", though I don't particularly see a linguistic distinction myself even in a mathematical context. TeleComNasSprVen (talk) 08:48, 30 December 2013 (UTC)
 * See class (set theory). In this context the distinction between classes and sets is not significant; every respectable set theory can incorporate rational numbers as a set. Keφr 13:21, 30 December 2013 (UTC)


 * What else, then? The booleans, set (or class, if you will) of all absolute truth values? DAVilla 11:04, 31 December 2013 (UTC)
 * The question here is whether or not the term can be cited with the claimed meaning. Whether or not we should have an entry is a different question and a matter for RFD.  It seems to me that a second debate has been pointlessly opened here for something it was already fairly clear could be cited. <font style="background:#fafad2;color:#C08000">Spinning <font style="color:#4840a0">Spark  14:45, 31 December 2013 (UTC)


 * RFV passed: some quotations provided at rational numbers. I think it questionable that they attest the sense as distinct from the plural, but the editors who decided to keep the sense in Talk:rational numbers would probably think otherwise. We are stuck with this sense, I think. --Dan Polansky (talk) 18:44, 20 June 2014 (UTC)