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

Revision as of 13:59, 4 May 2009

Revision as of 17:25, 4 May 2009

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* '''Comments''' from '''[[User:Cryptic C62|Cryptic C62]] · [[User talk: Cryptic C62|Talk]]'''

**<s>The lead does not adequately summarize all of the main sections of the article. Unless I am misreading, it appears that ''Other number systems'' is not represented in the lead.</s>

::**:Apparently you missed the sentence "In the 19th century, the algorithm was generalized to other types of numbers, which led to modern [[abstract algebra]]ic notions such as [[Euclidean ___domain]]s." in the lead. But you're right, that's too terse, so I expanded it to "The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as [[Gaussian integer]]s and [[polynomial]]s of one variable. This led to modern [[abstract algebra]]ic notions such as [[Euclidean ___domain]]s." [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 09:59, 4 May 2009 (UTC)

**<s>Caption: "The greatest common divisor of ''a'' and ''b'' is the largest square tile that covers an ''a''-by-''b'' rectangle exactly. Here, a 24-by-60 rectangle is covered with 12-by-12 square tiles." In the first sentence, it needs to be made clear that it is not one single square tile that covers the rectangle, but multiple iterations of that square tile. "exactly" is somewhat ambiguous, consider expanding. It would also be helpful to say "ten 12-by-12 square tiles".</s> Addendum: upon reading the relevant paragraph, it might be helpful to make this into an animation which demonstrates the various ways in which a 60-by-24 rectangle can be divided.

**:Reworded caption, thanks. The animation might be helpful, but that would require someone to create and position precisely 1440 1-by-1 squares. It's possible — are you volunteering, by any chance? [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 09:59, 4 May 2009 (UTC)

**Caption: "The greatest common divisor of ''a'' and ''b'' is the largest square tile that covers an ''a''-by-''b'' rectangle exactly. Here, a 24-by-60 rectangle is covered with 12-by-12 square tiles." In the first sentence, it needs to be made clear that it is not one single square tile that covers the rectangle, but multiple iterations of that square tile. "exactly" is somewhat ambiguous, consider expanding. It would also be helpful to say "ten 12-by-12 square tiles". Addendum: upon reading the relevant paragraph, it might be helpful to make this into an animation which demonstrates the various ways in which a 60-by-24 rectangle can be divided.

**:::RewordedI'm caption,not thanks.familiar with Thehow animationto mightconvert bea helpfulseries of images into an animation, but thatI'd would requirebe someonewilling to createmake andthe positionimages precisely(or 1440at 1-by-1least squarestry). If It'sI possiblemake —them, arecan you volunteering,make bythe any chanceanimation? --'''[[User:ProteinsCryptic C62|ProteinsCryptic C62]] (· [[User talk:Proteins Cryptic C62|talkTalk]])''' 0917:5925, 4 May 2009 (UTC)

**"The greatest common divisor is often written as GCD(a, b) or, more simply, as (a, b)." Yes, the second version is simpler, but that notation is also used for ''lots'' of other things in mathematics. What (a, b) represents depends on the context of the problem, and I think it would be wise to mention this so as not to mislead our less mathematically-inclined readers.

**"neither 6 = 2×3 nor 35 = 5×7 is a prime number, since they both have two prime factors" I think it may be a tad confusing to include the prime factorization at first; perhaps this should be added later: "neither 6 nor 35 is a prime number, since they both have two prime factors: 6 = 2x3 and 35 = 5x7." or something like that. Also, shouldn't it be "neither 6 nor 35 '''are''' prime number'''s'''" ?

::**:Excellent suggestion for the rewording. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 09:59, 4 May 2009 (UTC)

***::[http://www.bartleby.com/68/47/4047.html Neither/nor] - it depends if you think 6 and 35 are singular or plural. Are they singular because they are individual numerals or are they plural because they abstractly represent "more than one"? Tricky. [[User:Awadewit|Awadewit]] ([[User talk:Awadewit|talk]]) 20:05, 3 May 2009 (UTC)

****:::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&amp;fowler|<font color="#B8860B">Fowler&amp;fowler</font>]][[User talk:Fowler&amp;fowler|<font color="#708090">«Talk»</font>]] 22:40, 3 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>

**:I presume that you are not objecting to the [[imperative mood]] (a staple of mathematics: "Let x be..."), just the verb "imagine". I re-worded this to use "consider" for both: "ImagineConsider a rectangular area a by b, and consider any common divisor c that divides both a and b exactly." Wikipedia Since this ''is'' an encyclopedia, notwe anshould episodeboth ofstrive Spongebobto Squarepantskeep our comments less colorful. No sentence[[User:Proteins|Proteins]] in([[User antalk:Proteins|talk]]) encyclopedia10:32, should4 startMay with2009 "imagine."(UTC)

**<s>"the GCD(462, 1071) = 3×7" In all other instances thus far, you have chosen not to use an article before GCD. Did you mean to say "the'''n''' GCD(462, 1071) = 3×7"?</s>

:::I presume that you are not objecting to the [[imperative mood]] (a staple of mathematics: "Let x be..."), just the verb "imagine". I re-worded this to use "consider" for both: "Consider a rectangular area a by b, and any common divisor c that divides both a and b exactly." Since this ''is'' an encyclopedia, we should both strive to keep our comments less colorful. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 10:32, 4 May 2009 (UTC)

::**:Thank you for catching that inconsistency, which I've fixed. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 10:32, 4 May 2009 (UTC)

**"the GCD(462, 1071) = 3×7" In all other instances thus far, you have chosen not to use an article before GCD. Did you mean to say "the'''n''' GCD(462, 1071) = 3×7"?

:::Thank you for catching that inconsistency, which I've fixed. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 10:32, 4 May 2009 (UTC)

**"Integer factorization is thought to be a difficult problem for large numbers." A bit weaselly, and it's not particularly difficult if you have a calculator handy. Perhaps "can be" instead of "is thought to be" ?

::**:I clarified the sentence, although perhaps I should have been more clear about "large numbers". A pocket calculator might help in factoring numbers up to 20,000 (5 digits), but it won't be useful in factoring numbers with 500 digits, the rough size of number used in modern cryptography. Nevertheless, the Euclidean algorithm can quickly find the greatest common divisor of two 500-digit numbers. That was the point I was trying to convey. Should I spell that out in the article, do you think? [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 10:32, 4 May 2009 (UTC)

::**::I corrected the statement the first time from asserting "is a difficult problem" to "is thought to be..." since it is "only" thought to be difficult, not proven. The new version which states "The computational difficulty of integer factorization grows exponentially with the size of the number being factored" is a step backwards in that regards. The computational difficulty of integer factorization is in fact unknown. --[[User:C S|C S]] ([[User talk:C S|talk]]) 11:36, 4 May 2009 (UTC)

:**:::You are right that I was inferring too much about the specific scaling of factorization. I've re-worded this sentence to "Factorization of large integers is believed to be such a difficult problem that many modern cryptography systems are based upon it.", which is supported by the Schroeder reference. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 13:59, 4 May 2009 (UTC)

**<s>"A more subtle definition of the GCD is helpful in advanced mathematics, particularly ring theory." This statement should probably be accompanied by a ref.</s>

::**:OK. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 10:32, 4 May 2009 (UTC)

**"A more subtle definition of the GCD is helpful in advanced mathematics, particularly ring theory." This statement should probably be accompanied by a ref.

**<s>"GCD(a, b, c) = GCD(a, GCD(b, c)) = GCD(GCD(a, b), c)" Shouldn't this also include " = GCD(GCD(a, c), b)"?</s>

:::OK. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 10:32, 4 May 2009 (UTC)

**:If only for symmetry. Good catch! [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 10:32, 4 May 2009 (UTC)

**<s>"Thus, Euclid's algorithm, which computes the GCD of two numbers, suffices to calculate the GCD of arbitrarily many numbers." Odd wording at the end. Suggest switching to "integers" to allow the following rewrite: "Thus, Euclid's algorithm, which computes the GCD of two integers, suffices to calculate the GCD of any number of integers."</s>

**"GCD(a, b, c) = GCD(a, GCD(b, c)) = GCD(GCD(a, b), c)" Shouldn't this also include " = GCD(GCD(a, c), b)"?

**:::IfThat's a good point and a good rewording. By using the word "number", I was trying to be general, since this result applies not only to integers, but to any number system for symmetry.which the GoodEA catch!works, such as real numbers or Gaussian integers. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 10:32, 4 May 2009 (UTC)

**::Well, if you'd still like to stick with "numbers" rather than "integers", how about this: "Thus, Euclid's algorithm, which directly computes the GCD of two numbers, can be used to calculate the GCD of any group of numbers, regardless of the size of the group." Or something? As long as we avoid phrases like "number of numbers", it should be fine. --'''[[User:Cryptic C62|Cryptic C62]] · [[User talk: Cryptic C62|Talk]]''' 17:25, 4 May 2009 (UTC)

**"Thus, Euclid's algorithm, which computes the GCD of two numbers, suffices to calculate the GCD of arbitrarily many numbers." Odd wording at the end. Suggest switching to "integers" to allow the following rewrite: "Thus, Euclid's algorithm, which computes the GCD of two integers, suffices to calculate the GCD of any number of integers."

:::That's a good point and a good rewording. By using the word "number", I was trying to be general, since this result applies not only to integers, but to any number system for which the EA works, such as real numbers or Gaussian integers. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 10:32, 4 May 2009 (UTC)

* More to come. Good work thus far. --'''[[User:Cryptic C62|Cryptic C62]] · [[User talk: Cryptic C62|Talk]]''' 19:59, 3 May 2009 (UTC)

:::Thank you very much for your careful reviewing! The article is definitely improving. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 10:32, 4 May 2009 (UTC)