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

Revision as of 10:10, 5 May 2009

Revision as of 16:31, 5 May 2009

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**::I'm not familiar with how to convert a series of images into an animation, but I'd be willing to make the images (or at least try). If I make them, can you make the animation? --'''[[User:Cryptic C62|Cryptic C62]] · [[User talk: Cryptic C62|Talk]]''' 18:52, 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.

::**:Mentioned ambiguity of "(''a'', ''b'')" notation, and swapped order of last two sentences in paragraph for better flow. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 10:10, 5 May 2009 (UTC)

**::Eh, better, but instead of "although the simpler notation is also used for unrelated mathematical objects, such as two-dimensional vectors." how about "although the latter notation is also used for various other mathematical concepts, such as two-dimensional [[vector]]s."

**<s>"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.</s> 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)

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**: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)

**::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]]''' 18:52, 4 May 2009 (UTC)

**<s>"This approach begins by showing that, if the theorem holds for n, it also holds for n + 1." I just learned about induction last semester, and this doesn't seem to be quite right, specifically the last two clauses. My understanding of induction is that it is a two-step process. The first step is proving the basis case (usually n=0 or n=1), and the second step is proving that it holds for n+1. The sentence in question is written as though the first step proves the second step, which is not the case.</s>

::**:For induction, it doesn't matter whether you prove the basis case first and the (''n'' implies ''n''+1) step second, or the reverse. I chose to present the method in the reverse order because (1) I thought it would be easier for lay-people to follow, (2) it emphasizes the (''n'' implies ''n''+1) step, which I feel is more important; and (3) it de-emphasizes the basis case and clarifies that a proof could start with any basis case, e.g., ''n''=7. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 09:36, 5 May 2009 (UTC)

**::Hmm. I've reread that section, and I'm not sure why I had a problem with it the first time, as it makes perfect sense to me now. Perhaps I should have read the entire section before commenting on individual sentences... --'''[[User:Cryptic C62|Cryptic C62]] · [[User talk: Cryptic C62|Talk]]''' 16:31, 5 May 2009 (UTC)

**<s>"A recursion is an equation relating numbers that form a series a1, a2, a3, etc." This is a very poor definition of a recursion, as it does not adequately explain the concept to a reader with no prior familiarity to it. How about "A recursion is an equation in which ''a_n'', an arbitrary term in a [[series]], is defined by the values of previous terms in the series, such as ''a_n-1'' or ''a₀''". This will also help the reader understand the Fibonacci example a bit more clearly.</s>

::**:That's a good suggestion. I've re-worded the recursion along these lines. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 09:36, 5 May 2009 (UTC)

**<s>"Several equations associated with the Euclidean algorithm are recursive, such as ''r''_''k''&nbsp;=&nbsp;''r''_''k''−2&nbsp;−&nbsp;''q''_''k''''r''_''k''−1." This example is essentially useless, as neither the meaning of the equation nor the terms used therein have been defined yet.</s>

::**:Eliminated foreshadowing. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 09:36, 5 May 2009 (UTC)

**"Several equations associated with the Euclidean algorithm are recursive, such as ''r''_''k''&nbsp;=&nbsp;''r''_''k''−2&nbsp;−&nbsp;''q''_''k''''r''_''k''−1." This example is essentially useless, as neither the meaning of the equation nor the terms used therein have been defined yet.

**<s>"Finally, in infinite descent, a given solution is used to construct a smaller solution." I read this sentence and thought I understood the concept being explained. Then I read [[infinite descent]]. Then I reread this sentence, which I now realize does a fairly poor job of explaining infinite descent. My familiarity with the concept is limited to that which I have just read, so I have no suggestion as to how to concisely summarize it, but I strongly urge you to rework the current explanation.</s>

:::Eliminated foreshadowing. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 09:36, 5 May 2009 (UTC)

::**: I hadn't wanted to talk about the (more common) use of infinite descent in impossibility proofs such as [[Fermat's Last Theorem]]. Rather, my goal was to prepare the reader to follow the logic of why the EA must stop eventually. Nevertheless, I've re-written those sentences to to give a broader understanding of the argument. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 09:36, 5 May 2009 (UTC)

**"The validity of the Euclidean algorithm can be shown by two-step argument." Since the title of the section is ''proof of validity'', perhaps this sentence should include the word 'proof': "The validity of the Euclidean algorithm can be proven with a two-step argument."

**"Finally, in infinite descent, a given solution is used to construct a smaller solution." I read this sentence and thought I understood the concept being explained. Then I read [[infinite descent]]. Then I reread this sentence, which I now realize does a fairly poor job of explaining infinite descent. My familiarity with the concept is limited to that which I have just read, so I have no suggestion as to how to concisely summarize it, but I strongly urge you to rework the current explanation.

**"This agrees with the GCD(1071, 462) found by prime factorization above." Erm, there is no mentions of 1071 or 462 in the ''Background'' section. Why not just give the prime factorization here?

::: I hadn't wanted to talk about the (more common) use of infinite descent in impossibility proofs such as [[Fermat's Last Theorem]]. Rather, my goal was to prepare the reader to follow the logic of why the EA must stop eventually. Nevertheless, I've re-written those sentences to to give a broader understanding of the argument. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 09:36, 5 May 2009 (UTC)

**"The sequence ends when there is no residual rectangle, i.e., when the square tiles cover the previous residual rectangle exactly." This paragraph desperately needs to end with: "The length of the sides of the smallest square tile is the GCD of the dimensions of the original rectangle." or something like that.

**"where the magnitude of rk is strictly less than that of rk−1" The use of 'magnitude' here strikes me as being a bit odd. Why not just write a simple inequality? r_k < r_k-1

**"Euclid finds the quotient and remainder by repeated subtraction" Last time I checked, Euclid is dead. Past tense, perhaps?

**"''r''_''k'' &equiv; ''r''_''k''−2 mod ''r''_''k''−1" Is there some article to which we can link '&equiv;'? I'm not sure I know what it means.

* 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)