Wikipedia:Featured article candidates/Euclidean algorithm/archive1: Difference between revisions
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I am nominating this mathematical article because I believe it meets the Featured Article criteria. In its simplest form, the Euclidean algorithm is often taught to 10-year-old children; for many, it is the only algorithm they encounter in school. It has several important applications, such as the [[RSA algorithm]] (often used in electronic commerce) and solving [[Diophantine equation]]s. Although the oldest known algorithm (23 centuries), it continues to play a role in developing new mathematics. It would be helpful for Wikipedia to have an excellent article on this topic, both for itself and for the introduction it provides to advanced mathematics such as [[abstract algebra]]. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 16:22, 27 April 2009 (UTC)
:I'm a bit concerned about the inclusion of the "Game of Euclid" in the historical development section. It doesn't seem to be very important as a research topic or achievement, and it certainly doesn't even come close to the other developments in that section. What was the reason for including this? --[[User:C S|C S]] ([[User talk:C S|talk]]) 17:19, 27 April 2009 (UTC)
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:::Well, I spy less than half a dozen "mathematical journals and textbooks" that mention the game. Surely there are many topics with more coverage that have not been included in this article. So I can't see how omitting this game would violate criterion 1b. Not to mention, these math journals you talk of are mainly math education related ones, except the journal INTEGERS, which seems like an ok journal but not particularly well-known. Of course, I don't mean to disparage journals whose primary audience may be math educators, but in terms of using such journals as a justification for including a mathematical topic in this article, I don't think it is sufficient. One has to separate a topic which is primarily used as an educational device from a topic which is considered an important development in understanding of the Euclidean algorithm. --[[User:C S|C S]] ([[User talk:C S|talk]]) 19:23, 27 April 2009 (UTC)
▲==Oldest algorithm?==
I think that the sentence "The Euclidean algorithm is the oldest algorithm in the historical record" is wrong because of Old Babylonian algorithms used to solve problems. --[[User:El Caro|El Caro]] ([[User talk:El Caro|talk]]) 18:04, 27 April 2009 (UTC)
::This is a good point. The passage stated is uncited, but there is further down a box with a quote by Knuth which makes the nuanced observation that it is the "oldest ''nontrivial'' algorithm" that has survived to the present day. Since Knuth actually wrote an article in 1972 on ancient Babylonian algorithms [http://doi.acm.org/10.1145/361454.361514] where he examined written records of their algorithms, presumably he was aware that the "nontrivial" is an important and necessary modifier. "Oldest nontrivial...", of course, is his opinion. --[[User:C S|C S]] ([[User talk:C S|talk]]) 19:21, 27 April 2009 (UTC)
:::I've read the quote in the box. Is this "only" Knuth's opinion or a statement on whom most specialists agree ? --[[User:El Caro|El Caro]] ([[User talk:El Caro|talk]]) 19:39, 27 April 2009 (UTC)
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::::::After consulting more than a dozen mathematical sources, I concede that you are entirely correct. An algorithm is any well-defined (alpha)numerical procedure, and can well be incorrect. Thus, pre-Pythagorean carpenters constructing a right angle by a 3-4-5 triangle were using an algorithm, and so were carpenters who used a 2-3-4 triangle. I accepted the proof requirement (which I read elsewhere) because Knuth's argument for the EA's priority seemed incompatible with the obvious prior existence of many algorithms, such as calendars, money changing, tax and inheritance systems, measurement of area, architecture, multiplication and division, etc. I have also not found a reliable source besides Knuth that identifies an oldest nontrivial algorithm. The solution for this FAC may well be to leave the quote box from Knuth, but to change the assertion in the article. How about "oldest numerical algorithm still in common use", or "one of the oldest algorithms in the historical record"? Would either of those wordings be acceptable? [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 02:45, 29 April 2009 (UTC)
"Euclid's algorithm can be applied to real numbers, as described by Euclid in Book 10 of his Elements" looks like an anachronism. Did Euclid know real numbers ? --[[User:El Caro|El Caro]] ([[User talk:El Caro|talk]]) 18:17, 27 April 2009 (UTC)
:Well, now we're getting into metaphysical matters that I think are tangential. For example, when Euclid considered arithmetical operations on whole numbers, it's not the same in a sense as what we consider such arithmetic, nor is probably what the ancient Greeks considered whole numbers the same as what we do now. So strictly speaking Euclid did not know real numbers, but he didn't know whole numbers, addition, or subtraction either. So that makes your point kind of moot. --[[User:C S|C S]] ([[User talk:C S|talk]]) 19:21, 27 April 2009 (UTC)
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:::Ancient Greeks considered numerical concepts mainly in terms of geometric constructions ala ruler and compass and that is how Euclid's treatment of commensurability goes. This is certainly not the way modern mathematicians think of them. Sure the ancient Greeks knew of some irrational numbers, but they certainly didn't know "e" or many other irrational numbers that are not constructible. So their concept of irrational number was far more limited than our modern understanding, even when one limits the concept of real number to mean "set of rational and irrational numbers". Even on the math where modern and ancient understanding would seem to overlap, it's clear the ancient Greeks just had a different way of thinking about it, so in a metaphysical sense, you could argue that the objects are really different. --[[User:C S|C S]] ([[User talk:C S|talk]]) 20:40, 27 April 2009 (UTC)
*[[:File:Gabriel-Lamé.jpeg]] - There is no source, date, or author for this image that would lead us to believe that it is in the PD. We need to be able to verify that it is in the PD. More research on this image needs to be done.
*[[:File:Dedekind.jpeg]] - The website for this image does not indicate the 1870 date and we have no name or death date for the photographer, so we cannot verify the PD license listed. More research on this image needs to be done.
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:::Ah, good news! We can use the Lame picture from 1897 then. It doesn't matter if Ecole Polytechnique claims copyright; US copyright law is what Wikipedia requires us to follow and that means the picture is considered in the public ___domain in the US (see [[Wikipedia:Public_domain]]). --[[User:C S|C S]] ([[User talk:C S|talk]]) 04:42, 29 April 2009 (UTC)
* Problem - many of the paragraphs are lacking citations or, if having them, there are no citations covering many sentences. See the end of the section "Greatest common divisor" for just one example. This needs to be fixed before it can pass FAC. [[User:Ottava Rima|Ottava Rima]] ([[User talk:Ottava Rima|talk]]) 23:45, 27 April 2009 (UTC)
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:As a conciliatory gesture to a numerically-minded classicist, I'll be glad to add some more references to the 130 already there. When I've finished, please reconsider your strong oppose. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 02:45, 29 April 2009 (UTC)
▲==terminology==
Often in math books, the Euclidean algorithm is not the actual procedure for finding the GCD, but rather the statement that for two a, b (natural numbers, integers, residue classes of integers, polynomials over commutative rings, power series over complete local rings, and so forth) we can solve the equation:
<math>
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::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)
*'''Conditional Support'''. Overall, the article is excellent and it certainly meets criteria 1 and 2. In fact, arguments could be made that the article is ''too comprehensive'' i.e. includes too much on related topics (criterion 4). However, I would not support that argument. The image copyright issues raised by Awadewit should be corrected. Unless I have missed something major, in my view, the article should be featured. [[User:Lwnf360|Lwnf360]] ([[User talk:Lwnf360|talk]]) 06:10, 28 April 2009 (UTC)
:Thank you, and I hope that the copyright issues will now be resolved. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 02:45, 29 April 2009 (UTC)
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