Talk:Euclidean algorithm
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Integers: ordinary, normal, usual, real, Gaussian
[edit]In the § Gaussian integers section, we find these expressions:
- "ordinary integers"
- "normal integers".
I found these terms confusing, since
- I don't recall seeing them used, in decades of reading maths, before this; and
- The phrase "normal integers" occurs in close proximity to a discussion of a norm.
Perhaps both of the above phrases were meant to specify the usual or everyday integers, namely those in (or isomorphic to those in) the real numbers? In any case, the sense of the section wouldn't suffer, and accuracy would improve, if we were to replace both "ordinary integers" and "normal integers" by "real integers".
Before I make such a change, I'd like to hear other opinions.
yoyo (talk) 11:55, 6 November 2018 (UTC)
- The phrase "ordinary integer" is commonly used for distinguishing usual integers from Gaussian integers, and more generally from algebraic integers. The phrase "normal integer" is less common and must changed into "ordinary integer". "Real integer" is not a good idea, because it would be WP:OR, and also because (for example) is an algebraic integer and a real number. I'll change "normal" to "ordinary", and add an explanatory footnote after the first use of "ordinary". D.Lazard (talk) 12:35, 6 November 2018 (UTC)
- I agree with D.Lazard. "Ordinary integers" is not at all confusing, "normal integers" may be slightly confusing, but "real integers" is definitely confusing. Maproom (talk) 14:28, 6 November 2018 (UTC)
Inconsistent history
[edit]The present text of the article says that the Euclidean algorithm was first described in Europe by Bachet in 1624. This can hardly be true if it was already described in Euclid's Elements, which was known in Europe in various editions and translations long before Bachet.109.149.2.98 (talk) 13:49, 7 April 2019 (UTC)
- Point well taken. The source only says that Bachet gave the first numerical description of the algorithm in Europe. I'll edit this. --Bill Cherowitzo (talk) 19:02, 7 April 2019 (UTC)
Section "Non commutative rings"
[edit]It seems that the only known example is the ring of Hurwitz quaternions. This must be clarified. If is true, the section must be renamed "Hurwitz quaternions". Otherwise, it must be named "Non commutative Euclidean rings" and moved to Euclidean domain. D.Lazard (talk) 08:47, 18 June 2019 (UTC)
Inaccurate implementations
[edit]I feel the implementations given in the Implementations sections are all inaccurate. For example, gcd(-6, 0) is 6, but the implementations return -6. This is wrong because GCDs are always non-negative. Hexagonalpedia (talk) 12:18, 23 May 2021 (UTC)
Fixed D.Lazard (talk) 16:39, 24 May 2021 (UTC)
Flowcharts
[edit]At least one flowchart that represents an Euclidean algorithm could be expected. Now no flowchart in the current article. So here are flowcharts.
As though there is no instruction modulo in computing, neither in Python nor in JavaScript for example, the first algorithm computes a remainder through successive subtractions. The second algorithm uses successive modulo operations — % in Python or in JavaScript —, in order to yield the GCD of two natural numbers. A little more complicated, the third algorith is conceived for the easiest mathematical proof of the existence of this “great” divisor, which is multiple of every common divisor of a given pair of natural numbers. This last algorithm keeps unchanged the common divisors of such a pair at every step, by replacing the greatest integer with the positive difference between the two.
calculation of n modulo d for n and d
members of ℕ, when d is not zero. Example:
if n = 85 and d = 20, then variable r passes
through the values: 85 ↠ 65 ↠ 45 ↠ 25 ↠ 5,
along four courses of the loop. Therefore,
according to the algorithm, divide 85 by 20
yields the remainder
85 – 4 × 20 = 85 modulo 20 = 5.
Euclidean algorithm, calculation of the GCD
of natural numbers d and g. For example,
if their starting values are 20 and 85, then
dividend, divisor and remainder of the unique
division of the loop are, at every step,
three successive terms of this sequence:
20, 85, 20, 5, 0 (5 being a divisor of 20).
Result: gcd(20, 85) = 5.
is the other number, because zero is the only multiple
of every integer, the “greatest” for the divisibility.
These three algorithms could be inserted in introduction just after the text, while the antique diagram,
less comprehensible than a flowchart, could be transfered in section
“Background: greatest common divisor” on the right,
the rectangle being placed on the left.
All captions of this multiple image are
well presented in the first two available font‑sizes.
Arthur Baelde (talk) 14:28, 5 September 2024 (UTC)
- Flowcharts were a popular method for describing algorithms whent the main control instruction of programming languages was the "go to". Presently, flowcharts are generally replaced by pseudo-code, which is more concise, easier to understand because of its linear structure, and more powerful (it describes programs that cannot be described with flowcharts). The article Flowchart contains
The flowchart became a popular tool for describing computer algorithms, but its popularity decreased in the 1970s, when interactive computer terminals and third-generation programming languages became common tools for computer programming, since algorithms can be expressed more concisely as source code in such languages. Often pseudo-code is used, which uses the common idioms of such languages without strictly adhering to the details of a particular one. Also, flowcharts are not well-suited for new programming techniques such as recursive programming.
- Also, the manual of style of Wikipedia MOS:MATH#Algorithms says
An article about an algorithm may include pseudocode or in some cases source code in some programming language.
, and does not mention flowcharts at all. Effectively, there are many Wikipedia articles that contain pseudocode, and very few that contain flowcharts. - Presently, the section § Implementations contains the pseudo-code of three versions of the Euclidean algorithm. The most complicate one consists of 7 short lines, and the third one can definitively not be converted into a flow chart. Also, the second one is a much better implementation than your thirs flowcharts.
- So, there is no reason for introducing flowcharts in this article, and, in any case, their place is not in the lead. D.Lazard (talk) 16:26, 5 September 2024 (UTC)
- I think it would be okay in principle to have a flowchart somewhere in this article, though an algorithm this simple can also be described as a paragraph or written as pseudocode.
- I find these particular flow charts pretty hard to read and visually distracting. The most important feature of flow charts is organizing information spatially so it can be easily visually scanned. –jacobolus (t) 17:16, 5 September 2024 (UTC)
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