Wikipedia:Featured article candidates/Euclidean algorithm/archive1: Difference between revisions
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<math>
a = bq + r </math>
uniquely with a (degree) condition on r (and some conditions on a and b). They are related, of course, but I was wondering if it might be worthwhile saying something to that effect off the bat. [[User:Fowler&fowler|<
:Perhaps you're referring to the "division lemma" or [[division algorithm]]? If so, the article mentions it in the subsection "Calculating the quotients and remainders". But I haven't encountered a source that calls it the "Euclidean algorithm"; could you point me towards one? [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 02:45, 29 April 2009 (UTC)
::I had a vague feeling Fowler had a point there, so I dug through some books I had handy. I found that (as expected) Hardy & Wright's number theory book states that "Euclid's algorithm" is defined by generating the sequence of remainders which terminates (it seems to give no name for the "division algorithm", merely calling it division with remainder at times), Dummit & Foote's abstract algebra book states that the "Euclidean algorithm" comes from the ''division algorithm'', but Herstein's ''Topics in Algebra'' does indeed call the above, ''the'' Euclidean algorithm. Herstein is a pretty well-known algebra book, so I dug a bit more and I found that a source on the [[Euclidean ___domain]] article, which was published in the Bulletin of the American Mathematical Society in 1949, also uses the term Euclidean algorithm for division algorithm (see [http://projecteuclid.org/euclid.bams/1183514381]). I think in number theory books there is a pretty well-established tradition of using "Euclidean algorithm" to mean generating the sequence of remainders from the two initial numbers. In algebra texts that discuss Euclidean domains, my suspicion is that books here and there may use Euclidean algorithm to mean division algorithm but I suspect that modern books generally don't; I find Dummit & Foote is pretty reliable regarding modern terminology, while Herstein is from 1961 and sometimes a bit outdates on terminology. In any case, I think a note or footnote is in order. --[[User:C S|C S]] ([[User talk:C S|talk]]) 04:59, 29 April 2009 (UTC)
:::Yes, I guess you are right: number theory texts do call it the "division algorithm," and in fact, come to think of it, in high-school we called it the "division algorithm" ourselves, perhaps because it was introduced as a part of elementary number theory. Somewhere in college though the name became less certain (as I remember it). As for texts, among the classic algebra texts, both van der Waerden and Birkoff/Mac Lane do call it the "division algorithm," but Herstein doesn't (as you say). The more recent texts seem to be a mixed bag. I don't know Dummit and Foote, but among the books published in the last 20 years that refer to the "division algorithm" as the "Euclidean algorithm" are (the links should take you straight to the page about the "Euclidean Algorithm"): Hilton and Wu's ''[http://books.google.com/books?id=ua5gKZt3R6AC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA10,M1 A Course in Modern Algebra]'' (1989), Rowen's [http://books.google.com/books?id=EmO9ejuMHNUC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA116,M1 Algebra: Groups, Rings, and Fields] (1994), Lang's ''[http://books.google.com/books?id=Fge-BwqhqIYC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA173,M1 Algebra: A Graduate Course]'' (2002), Murty and Esmonde's [http://books.google.com/books?id=YaqVpdrngNYC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA3,M1 Problems in algebraic number theory] (2004), Lang's ''[http://books.google.com/books?id=PdBirmNTwu0C&printsec=frontcover&source=gbs_summary_r&cad=0#PPA113,M1 Undergraduate Algebra]'' (2005), and Lowen's [http://books.google.com/books?id=8svFC09gGeMC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA27,M1 Graduate Algebra: The Noncommutative View] (2008). Lang (2002), in particular, is still widely used, I think. So the footnote will be useful for any others who have questions like mine. :) Thanks! [[User:Fowler&fowler|<
::::I didn't check all those books publication dates, but certainly Lang's well-known classic ''Algebra'' is not really a book "published in the last 20 years". The revised 3rd edition is from 2002, but the original was from 1965, and subsequent versions are essentially the same (but with fixing of errors and so forth). Unlike the others you mention, which I've never heard of, certainly it is still used (mainly by the top graduate programs and more that like to think they are in the same class). --[[User:C S|C S]] ([[User talk:C S|talk]]) 20:47, 29 April 2009 (UTC)
::::Actually I expect the Hilton and Wu book must be from the 60s too, since "Hilton" is Peter Hilton and he wrote several books in that period, all of which are well regarded but never really caught on like their competitors. Indeed, Hilton is rather infamous (in a humorous way) for writing a book on homology/cohomology (with Wylie) and using the terms homology and cohomology to refer the opposite way as everyone else used them. That never changed in subsequent editions even though by then it became clear they had lost the terminology reformation attempt. Herstein is also a bit weird in that he composes linear transformations from left to right instead of right to left. That never changed in recent printings. So anyway, if your point (which I think it may be) is that even though these books are old, but they were recently republished and so must reflect more modern terminology, I'm afraid I don't buy that. In my experience, republished classic texts often retain their classic (read: outdated) terminology, and the reader is supposed to be on guard for it. --[[User:C S|C S]] ([[User talk:C S|talk]]) 21:01, 29 April 2009 (UTC)
:::::My point was simply that these books, regardless of their provenance, are still being used by students (as both books by Lang are, by your own admission), so it doesn't hurt to have the note. I have no idea if the terminology is outdated. Certainly Lowen's [http://books.google.com/books?id=8svFC09gGeMC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA27,M1 Graduate Algebra: The Noncommutative View] (2008), [http://www.ams.org/bookstore-getitem/item=GSM/91 published by the AMS] does look recent. Anyway, this is not a biggie for me. Regards, [[User:Fowler&fowler|<
I understand and agree wholeheartedly. Perhaps I should have forestalled your comments by mentioning that Herstein is still a widely used book. --[[User:C S|C S]] ([[User talk:C S|talk]]) 23:02, 29 April 2009 (UTC)
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* '''Strong Oppose''' - lack of citations for over 50% of the page means that it fails the citation requirements. [[User:Ottava Rima|Ottava Rima]] ([[User talk:Ottava Rima|talk]]) 13:20, 28 April 2009 (UTC)
*'''Looking to support'''. I haven't read it yet, but it looks pretty good. Will make vote more definite upon reading. [[User:Fowler&fowler|<
*'''Support'''. I have (inofficially) reviewed the article recently and found it very good (see the article talk page), and think it has even improved since. It is very comprehensive, accessible, provides pictures where useful. I only have one suggestion, which is easy to fix: please consider adding reference(s) for the section "Induction, recursion and infinite descent". I don't agree with Ottava Rima's point above, which is exaggerating verifiability, but that section could do with a brief reference for each of the three methods, just in the sense of a "further reading", if readers are interested in learning more about induction etc. (A reference mentioning these techniques in correlation to the EA would be ideal.) [[User:Jakob.scholbach|Jakob.scholbach]] ([[User talk:Jakob.scholbach|talk]]) 21:47, 29 April 2009 (UTC)
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**:::Hrm. Well, I'm not particularly sure about it myself, so use your best judgment. I just wanted to bring it to your attention. --'''[[User:Cryptic C62|Cryptic C62]] · [[User talk: Cryptic C62|Talk]]''' 22:23, 3 May 2009 (UTC)
**::::In my way of thinking, it depends how we are using "6" in the context. Since we say "6 ''is'' composite" (just as we say "7 ''is'' prime"), it seems to me we are thinking of it as singular.
**::::"Neither," the converse of "both," usually has singular verb concord (as the Bartleby reference suggests as well). So, "Neither 6 nor 35 is prime" sounds correct to me. The adjective "prime" will not apply to instances of plural occurrences of 6; in other words, you can't apply "prime" to "six sheep," although you ''can'' say, "The number of sheep (''viz.'' 6) ''is'' prime." That is as far as ''prescriptive grammar'' goes. If you look at usage on the web, "Neither * nor * is" has [http://www.google.com/#hl=en&q=%22Neither+*+nor+*+is%22&btnG=Google+Search&aq=f&oq=%22Neither+*+nor+*+is%22&fp=OlAWEoQSgPM approximately 7 million hits], whereas "Neither * nor * are" has [http://www.google.com/#hl=en&q=%22Neither+*+nor+*+are%22&fp=OlAWEoQSgPM 16.4 million hits] (some are using "are" for plurals, but not all). So, even though most prescriptive grammar books don't look kindly upon plural verb agreement for "neither" in the case of third person singular nouns, as in "Neither Hamas nor Hizebollah ''are'' ...," if people, by a margin of two to one, ''are'' making such constructions, sooner or later the descriptive grammar books will take notice. [[User:Fowler&fowler|<
**:::::I agree that "6" and "35" (in the sense used here) are each singular, as in "7 is a prime number" or "12 is a composite number". Per Bartleby's and other references, I feel that "neither...nor" should take a singular verb if both nouns are singular, as in "Neither Clara nor John was absent from class". [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 10:06, 5 May 2009 (UTC)
**<s>"Imagine a rectangular area a by b, and consider any common divisor c that divides both a and b exactly." Wikipedia is an encyclopedia, not an episode of Spongebob Squarepants. No sentence in an encyclopedia should start with "imagine."</s>
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