Talk:Rational number

Recurring Decimals?
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)

To clarify this problem I've added...
 * However, the use of recurrence (even infinitely recurring zeroes) can violate the property of injectivity required of a countable set because it enables more than one way to write the same natural number. For example, the convention of ascribing "0.999..." as equal to 1 is not valid in discrete mathematics because "0.999..." both represents a real and can be also represented as "1.000...".

Alexander Bunyip (talk) 05:04, 19 June 2017 (UTC)
 * The characterisation of the rational numbers as the real numbers with a decimal expansion which is either finite or eventually periodic is presently given in the second paragraph of the lead. Thus this section should be considered as closed. D.Lazard (talk) 08:26, 19 June 2017 (UTC)
 * The 2nd sentence in the 2nd paragraph is incorrect. A "repeating decimal", implying "infinitely repeating" or "infinitely recurring" digit, is not a valid concept or representation in a positional numeral system. A number system S is countable set IFF there exists an injective function f from S to the natural numbers N = {0, 1, 2, 3, ...}. "0.999...=1" is a declaration that there are now two ways to write the same number "one" in the otherwise countable decimal positional numeral system. If you say that some numbers can be written in more than one way then the function f is surjective. In general, recurrence is an attempt to represent a rational number that is not completely divisible within the finite precision of the base of the decimal positional numeral system in which it is written. Recurrence denies the clear limitation in precision of decimal representation due to its positional dependecy. Recurrence has the additional effect of voiding ordinality because it is not possible to say what is the previous/next sequential number before/after a number with digits of infinite recurrence. For example the next number after 0.333 is 0.334 but what is the next number after "0.333..." ? Thus the infinitely recurring digit voids the countable number system through loss of injectivity with the set of natural numbers, and loss of ordinality due to having no terminating positional numeral.

The invalidity of infinite recurrence is the elephant in the room of decimal representation. The invalidity of specious ideas like "0.999...=1" needs to be well and truly exposed everywhere it appears. Alexander Bunyip (talk) 16:16, 24 June 2017 (UTC)
 * It's not invalid. As I said at another talk page, there is nothing wrong with having an infinite string of digits - there are countably many repeating decimals. To prove this directly, note that for every integer (so only paying attention to the fractional part), there are finitely many repeating decimals where the repeating pattern is 1 digit long, finitely many 2 digits long, finitely many 3 digits long, and so on. One can easily setup a bijection (which is by definition also an injection) of this set with the natural numbers in a similar way as the rational numbers. Also, there are infinitely many (but countably many) rational numbers between any given pair of rational numbers. So at best, you're not demonstrating that you understand the meaning of cardinality, nor does your comment comply with a basic property of the rational numbers.--Jasper Deng (talk) 19:31, 24 June 2017 (UTC)

Complex Rationals
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
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$$\subseteq$$Q$$\subseteq$$R$$\subseteq$$C... 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
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)

circular
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
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 &#9742; 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)

Lead
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)

Quotients of complex numbers as rational numbers
Does anyone define quotients like (a+bi)/(c+di) as rational, where a, b, c, and d are integers?CountMacula (talk) 01:55, 7 September 2018 (UTC)
 * These are Gaussian rationals and form the field of fractions of Gaussian integers. D.Lazard (talk) 08:28, 7 September 2018 (UTC)

Yes if (c+di) is not equal to 0 Anubhav0708 (talk) 14:50, 14 April 2021 (UTC)

Lead
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)

Lead
In the final paragraph, the expansion method is valid in any base not just base 10. A proof of this theorem of Cantor appears in Irrational Numbers by Ivan Niven. Aliotra (talk) 13:52, 15 September 2019 (UTC)
 * Which final paragraph? The fact that "the expansion method is valid in any base not just base 10" appears explicitely in the second paragraph of the lead. D.Lazard (talk) 14:22, 15 September 2019 (UTC)
 * The paragraph indicating how the reals can be constructed from the rationals. — Preceding unsigned comment added by Aliotra (talk • contribs) 00:14, 16 September 2019 (UTC)
 * I see. This paragraph is not about the main subject of the article. It is in the lead. These two facts make normal to emphasize readibility over mathematical accuracy. In this case, including other bases would need to replace a single word by one or several sentences; this would give too much emphasis on this specific method over the others. Nevertheless, I have added a link where all the methods of construction are detailed. D.Lazard (talk) 07:20, 16 September 2019 (UTC)

In the short paragraph where it states the rationals are countable, it follows they have measure zero and then almost all reals are irrational. Does this follow from the fact the irrationals are uncountable as is suggested? Thanks Aliotra (talk) 23:58, 17 September 2019 (UTC)
 * Not really, this follows, as said in the article, from the fact that the set of all reals are uncountable. The fact that irrationals are uncountable results ffrom the facts that reals are uncountable and the set complement) of a countable set in a uncountable one is also uncountable (because the union of two countable sets is countable). D.Lazard (talk) 08:28, 18 September 2019 (UTC)
 * Indeed, the logic was broken; now hopefully I've fixed it. Boris Tsirelson (talk) 10:07, 18 September 2019 (UTC)

Is the arithmetic paragraph necessary?
I think it should be moved to the arithmetic at fraction, or if not that, then at least it should be shortened. Fr.dror (talk) 08:01, 16 October 2019 (UTC)
 * I'd opt to keep it here, as the paragraph lists essential operations with the rational numbers, which after all is the subject of the article. - DVdm (talk) 09:08, 16 October 2019 (UTC)
 * Fraction (mathematics) exists already, and is linked at the top of the section. So, you are not asking for a move but for a merge. In any case, I oppose to such a merge for several reasons. Firstly, fractions and rational numbers are not exactly the same thing, as a rational number is a number, and a fraction is a numeral that represents a rational number (if the numerator and the denominator are integers; the numerator and the denominator of a fraction need not to be integers). The equality of fractions is not the same as the equality of rational numbers (2/3 and 4/6 are different fractions that represent the same rational number). Secondly, the arithmetic of rational numbers and its property are fundamental in number theory and in all mathematics. So, a lot of articles link to rational number, and people that follow these links need a complete summary of the subject. :As the section contains nothing more than the definition of the operations on rational numbers, I do not see any other way for shortening the section than removing subsection headings, which would make the article much less clear. On the other hand, section "Properties" deserves to be moved just after or just before the section "Arithmetic". Also this section must be expanded for listing the basic properties that are hidden behind the phrase "Q is the unique ordered field that has no proper subfield". D.Lazard (talk) 09:15, 16 October 2019 (UTC)

Subtraction
In the section about subtracting fractions, it claims that if and only if two fractions have coprime denominators and are in canonical form, their difference will also be in canonical form. However, consider the counterexample 1/2 - 1/4 = 1/4. 2 and 4 are not coprime, as their GCD is 2, but 1/4 is in canonical form. 100.35.122.101 (talk) 10:47, 9 June 2022 (UTC)


 * You misunderstand. The formula quoted
 * $$\frac{a}{b} - \frac{c}{d} = \frac{ad-bc}{bd} = \frac{1\times 4 - 2 \times 1}{2 \times 4} = \frac{2}{8}$$
 * is not in canonical form.  Dr Greg  talk 11:17, 9 June 2022 (UTC)

Decimal expansion
The lead includes the sentence The decimal expansion of a rational number either terminates after a finite number of digits or eventually begins to repeat the same finite sequence of digits over and over.

The lead appears to present this as part of the definition of a rational number. It cites Encyclopaedia Britannica but that source also presents it as part of the definition, and offers no proof or explanation of why this statement can be made.

It is my view that the statement quoted above should not be presented as part of the definition of the rational number, so should not be included in the lead. It should be moved downwards into the body of the page, and should be presented as a characteristic that can be proved or demonstrated for all rational numbers. An improved citation should be used; ideally one that provides a proof or explanation that links the statement to the definition of a rational number. Dolphin ( t ) 08:18, 28 June 2022 (UTC)


 * The definition is in the first paregraph of the lead, and your quotation is in the second one. So I do not understand why you think that your quotation is presented as a part of the definition. It is a property, and the lead is a place for the most important properties of a subject. This property is indeed important, as it is taught to very young people. Nevertheless, the lead deserves to be improved. D.Lazard (talk) 13:14, 28 June 2022 (UTC)


 * Let me clarify. When we define something, it doesn’t require proof. (We state that all triangles have three sides, but we never attempt to prove that all triangles have three sides.) When a statement is made with no attempt at proof, or comment about proof, readers are entitled to interpret it as either a definition or very badly written. My quotation is presented without so much as “It can be proved that”. Therefore the inexperienced reader is entitled to assume it must be part of the definition, but the inexperienced reader would have been misled. Perhaps the problem could be mitigated by prefixing the sentence with “It can be proved that ...” or alternatively “An important property of rational numbers is that ...” Such a prefix would call for a suitable in-line citation. Dolphin ( t ) 13:43, 28 June 2022 (UTC)
 * I think that everybody who is able to understand your quotation should be able to understand that it does not belong to the definition given in the preceding paragraph. In any case this does not matter, as the quotation could be an alternative definition, if defining the rational numbers from real numbers would not be a circular definition (reals are defined from rationals).
 * Nevertheless, I have modified the lead for less pedantry and less repetitions. I have added the fundamental property that rational numbers are real numbers, and that the property of repeated decimals is a characterization of rational numbers inside real numbers. I hope this will be clearer to you. D.Lazard (talk) 14:27, 28 June 2022 (UTC)


 * You have written “I have added ... that the property of repeated decimals is a characterisation of rational numbers ...” I disagree; you have added no such thing. The lead remains ambiguous about whether repeated decimals is a characteristic of the rationals, or a part of their definition. Please refresh your memory of WP:Make technical articles understandable. Dolphin ( t ) 01:54, 29 June 2022 (UTC)

Division by zero
This article currently has no discussion of division by zero. Seems like a huge oversight. –jacobolus (t) 19:38, 2 January 2023 (UTC)
 * The opening sentence already introduces rational numbers p/q by means of an explicit non-zero denominator q, so division by zero is automatically and naturally exluded from any possible discussion. So, indeed it is nowhere mentioned in the article. I don't think we really need to mention it, but perhaps we can simply include a pointer to Division by zero in the See also section. - DVdm (talk) 20:27, 2 January 2023 (UTC)
 * Being a field (specifically, the lack of division by zero) is the primary thing distinguishing a "rational number" from an arbitrary ratio of integers. –jacobolus (t)

Error in the Venn Diagram
The Venn diagram that shows the relationships between the sets of Real Numbers, Rational Numbers, Integers, and the Natural numbers is wrong. Here is how the actual relationships exist:

1. The set of Real Numbers contains the sets of Irrational Numbers and the Rational Numbers.

2. Then the set of Rational Numbers contains the sets of Integers and the set of Non-Integer [Positive and Negative] Fractions. The set of non-integer fractions is defined as decimal representations of a fraction which repeats or terminates.

3. Then the set of Integers contains the set of Negative Integers and the set of Whole Numbers.

4. And then finally, the set of Whole Numbers contains the set of Natural Numbers and the Singleton Set of Zero. Thomas Foxcroft (talk) 00:30, 17 July 2023 (UTC)


 * The diagram at the top of the page seems fine, if somewhat simple and unnecessary. Are you talking about the template from the See also section? –jacobolus (t) 01:32, 17 July 2023 (UTC)
 * Jacoblus:
 * There is only one Venn diagram in the article. It's wrong.
 * The members which make the set of rational numbers are the set of integers and the set of non-integer fractions. The members which make the set of integers are the set of negative integers and the set of whole numbers. The members which make the set of whole numbers are the set of natural numbers and the singleton set of Zero.
 * Such a Venn diagram is very important because it provides context about how specific subsets of numbers construct the set of rational numbers, and how the set of rational numbers is a subset that participates in constructing the entire real number system. It properly orientates the reader through proper context, and context is the vital information that forms knowledge which is used for making proper decisions. Thomas Foxcroft (talk) 03:12, 17 July 2023 (UTC)
 * The diagram at the top of the page is not "wrong". It correctly shows the inclusion relationships between the sets of natural numbers, integers, rational numbers, and real numbers. It just does not include all the categories you personally would prefer to show. But there is a downside to adding more categories, which is that such a diagram becomes much more cluttered and confusing. So you have to consider what the intended purpose of the diagram is and who the intended viewer is, before you can properly decide how much information to include. –jacobolus (t) 21:03, 17 July 2023 (UTC)
 * Indeed, the Venn diagram is correct. It is even sufficient, given the importance of the sets R, Q, Z and N, all represented by a single letter. This is not the case for the other named sets in comment. Following that comment we might also feel that we should include the sets of even integers and odd integers, then splitting the former in even multiples of 3 and even non-multiples of 3, and the latter in odd multiples of 3 and odd non-multiples of 3. Etcetera. Etcetera. No need for all this here. - DVdm (talk) 15:48, 18 July 2023 (UTC)