Wikipedia:Featured article candidates/Euclidean algorithm/archive1: Difference between revisions

Revision as of 20:18, 27 April 2009

Revision as of 20:25, 27 April 2009

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::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)

:"Euclid's:::If "algorithm" canis bedefined appliedonly toas reala numbersset of numerical rules (without requiring proof that the rules always work or understanding why they work), asthen describedsurely bythere Euclidwere inalgorithms Bookfor 10thousands of hisyears Elements"before looksEuclid. like anI'll anachronism.review Didthe Euclidliterature knowand reallist numbersthe opinions of other people besides Knuth. ? --[[User:El CaroProteins|El CaroProteins]] ([[User talk:El CaroProteins|talk]]) 1820:1725, 27 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)

:::The ancient Greek certainly knew about irrational numbers, at least as [[nth root|surds]], even if they hadn't defined [[Dedekind cut]]s. Book X of Euclid's ''Elements'' is devoted entirely to questions of [[Commensurability (mathematics)|incommensurability]], and this real-number version of Euclid's algorithm begins that exposition. Knuth states elsewhere that the Greeks treated real numbers by infinite continued fractions, but he doesn't explain his remark further; I took him to be referring to this version of Euclid's algorithm. One could argue, I suppose, that the modern concept of real numbers embraces more than just "the set of all rational and irrational numbers", but that seems beyond the level of this article. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 20:18, 27 April 2009 (UTC)

:::The ancient Greek certainly knew about irrational numbers, at least as [[nth root|surds]], even if they hadn't defined [[Dedekind cut]]s. Book X of Euclid's ''Elements'' is devoted entirely to questions of [[Commensurability (mathematics)|incommensurability]], and this real-number version of Euclid's algorithm begins that exposition. Knuth states elsewhere that the Greeks treated real numbers by infinite continued fractions, but he doesn't explain his remark further; I took him to be referring to this version of Euclid's algorithm. One could argue, I suppose, that the modern concept of real numbers embraces more than just "the set of all rational and irrational numbers", but that seems beyond the level of this article. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 20:18, 27 April 2009 (UTC)