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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)
To clarify this problem I've added...
Alexander Bunyip ( talk) 05:04, 19 June 2017 (UTC)
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)
Is there an analogous idea of a "complex rational number"? One guess would be
Does anyone know of an exisiting theory? Δεκλαν Δαφισ (talk) 09:17, 29 May 2009 (UTC)
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 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 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)
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)
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)
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)
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)
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)
Yes if (c+di) is not equal to 0 Anubhav0708 ( talk) 14:50, 14 April 2021 (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)
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)
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)
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)
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)
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)
This article currently has no discussion of division by zero. Seems like a huge oversight. – jacobolus (t) 19:38, 2 January 2023 (UTC)
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)
This
level-4 vital article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||
|
To-do list for Rational number:
|
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)
To clarify this problem I've added...
Alexander Bunyip ( talk) 05:04, 19 June 2017 (UTC)
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)
Is there an analogous idea of a "complex rational number"? One guess would be
Does anyone know of an exisiting theory? Δεκλαν Δαφισ (talk) 09:17, 29 May 2009 (UTC)
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 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 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)
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)
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)
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)
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)
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)
Yes if (c+di) is not equal to 0 Anubhav0708 ( talk) 14:50, 14 April 2021 (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)
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)
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)
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)
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)
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)
This article currently has no discussion of division by zero. Seems like a huge oversight. – jacobolus (t) 19:38, 2 January 2023 (UTC)
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)