Talk:Square root of 5

Existence of the square root


A creation of square root of 5 must come before a property of $$\,\sqrt{5},\, $$ like its irrationality exposed now in the first section. If $$\,k\,$$ denotes the ratio of a given similarity,   or its coefficient,  then this transformation multiplies areas by $$k^{\,2}.\;$$  Therefore here are two direct similarities  that multiply areas by 5. Arthur Baelde (talk) 13:19, 20 November 2021 (UTC)

No longer any section “Irrationality”, now the article begins by defining $$\,\sqrt{5}\,$$ as a limit. The beginning would be easier with a first section titled “Geometry”, in which $$\,\sqrt{5}\,$$ would be the dimension $$\,h\,$$ of a square that has an area of 5. Square in red dashed lines on the left. This new image also illustrates the Pythagorean theorem. Arthur Baelde (talk) 15:02, 13 December 2021 (UTC)

In the current article,  the image named Pinwheel_1.svg  shows a right triangle divided into five congruent parts,  each one is similar to the initial triangle. Nothing in the text corresponds to this image,  a neophyte cannot understand anything. Why this dividing of triangle in the article?  Neither on the image nor in its caption,  nothing about the coefficient of this similarity,  $$\text{equal to }\,\tfrac{\sqrt{\,5\,}}{5}.$$

Here on the left,  the Pythagorean tiling could be a good beginning to present $$\,\sqrt{5}\;$$ in a first section titled “Geometry”. Arthur Baelde (talk) 11:00, 21 December 2021 (UTC)


 * I agree that the decomposition of one triangle into five is not necessary to illustrate $$\sqrt{5}$$, but not that the busy image shown here would be better. What would really be nice would be just a simple 1, 2, sqrt(5) right triangle, but I don't see one on commons. Also note that new sections should be placed at the bottom of talk pages.  Cheers. Danstronger (talk) 13:42, 21 December 2021 (UTC)


 * I have not written on Wikipedia for a long time,  excuse me for the wrong position of this section on the talk page. I will propose soon a better image.    Arthur Baelde (talk) 09:51, 23 December 2021 (UTC)

The So-called Proof
Quite possibly the most difficult to follow, most unorganized and most back-and-forth nonsense I have ever read. How about a proof that actually constitutes a proof? — Preceding unsigned comment added by 75.172.60.221 (talk) 10:59, 13 July 2012 (UTC)

Fair enough...
I removed the refimprove tag because I don't think it applies to articles that don't have uncited statements. So then someone added a handful of citation needed tags. Fair enough... Anton Mravcek 23:42, 10 August 2007 (UTC)


 * I would consider it fair enough if not because he started putting "citation needed" tags in the most improbable places. The purpose of citations is to let readers verify the information. When this information is readily verifiable without any need of citations (as in the case, for example, of simple geometric facts such as that the length of the diagonal of a unit cube is the square root of 3, which to be verified the only thing one needs is to know about the Pythagorean theorem, i.e. to be acquainted with the most elemental geometry), the addition of such tags I can only qualify as ludicrous. I would even say he is adding them in bad faith, because so far it seems clear he is not really interested in improving the article, but only in having it deleted which is what he has been trying to do since the beginning (in the summary of an edit to the article on the golden ratio he even stated that this article "I hope and expect will be speedy deleted", which proves he was not acting in good faith), and his "contributions" to this article have dealt mainly with deleting useful information, such as most of the formulae that clarify the relations between sqrt(5), phi and its conjugate (which are very useful to let readers see their geometric relationships very clearly), and with disputing simple mathematical and geometric facts which no one else thinks are dubious. Uaxuctum 19:26, 12 August 2007 (UTC)


 * I apologize if I'm gone to far in calling for citations. The intent was to discourage and remove trivia that anyone can derive by a bit of original research, in favor of saying things that matter, as evidenced by being from reliable sources.  Please do continue to assume good faith, and let's discuss them on a case-by-case basis, or remove items for which no source is available.


 * Also note that I brought up the proliferation of golden ratio trivia in my recent comments below. It is generally considered best to continue such a discussion once started in talk, rather than just revert with an edit summary that says you don't see why; the object is to find a consensus, and that's best done in the talk page rather than by starting a revert war. Dicklyon 19:59, 12 August 2007 (UTC)

Do we need this article?
There doesn't seem to be anything here that's not already covered at square root or golden ratio, with the exception of the root 5 rectangle, which is just the fifth in an infinite sequence of root n rectangles, with no apparent reason to think it's notable in itself. I'd recommend an article on root rectangles, since there seems to be a lot written about those. Dicklyon 06:08, 5 August 2007 (UTC)
 * I think so, it just needs to be carefully sourced and properly integrated. WP:NUMBER is the relevant guideline for notability and it seems to me it lacks only the third of the three main criteria there. I might be persuaded to support a merge or redirect to root rectangles if such an article existed, but it doesn't. This one does, and it seems to me to be (almost) notable enough. mikaultalk 11:19, 5 August 2007 (UTC)
 * I would have thought the first criterion there would be plenty (Have professional mathematicians published papers on this topic?). So how come nobody gives us such a citation? Dicklyon 15:46, 5 August 2007 (UTC)
 * We're looking... I'm looking in the Fibonacci Quarterly. Please give me a couple of days. Knotslip12 21:26, 6 August 2007 (UTC)


 * 1) Have professional mathematicians published papers on this topic? I've looked on scholar.google.com and found no results. I don't know how to look this up in Math ArXiV. Knodeltheory 21:52, 5 August 2007 (UTC)
 * 2) Is the sequence listed in the On-Line Encyclopedia of Integer Sequences? (In the case of sequences of rational numbers, does the OEIS have the sequences of numerators and denominators of the relevant fractions?)  I've proposed amending this one to ask for irrational numbers if the OEIS lists the decimal expansion AND the continued fraction. If my proposal is accepted, then the answer to this question is YES. Knodeltheory 21:53, 5 August 2007 (UTC)
 * 3) Do MathWorld and PlanetMath have articles on this topic? NO. MathWorld has articles on the square roots of 2 and 3, but not 5. PlanetMath does not have an article on this number but does list it at . Knodeltheory 21:52, 5 August 2007 (UTC) UPDATE: PlanetMath now has an entry on this number. Anton Mravcek 21:45, 11 August 2007 (UTC)


 * That new article by mathnerd at PlanetMath (his only contribution, it appears) does make it clear way paying attention to a wiki as a source is unworkable. It's really unfortunate that this math cite has now been crapped up with the statistics nonsense that someone found where the number 5 in a problem (for 5 work days in a week) leads to sqrt(5) in the answer; this is unbelievably lame nonsense; anyone mind if I take it out again?  Dicklyon 22:02, 11 August 2007 (UTC)

This article is important in terms of sqrt5's relation to compass and ruler constructibility. besides the replication of square (sqrt2) and hexagon (sqrt3) throughout the integers (construction of figures built on multiples thereof) the fact that there is a curious gap from the hexagon to the heptadecagon in terms of original constructible figures is highly notable ito of the relation of sqrts to Fermat primes. As a matter of fact, it makes sqrts 2,3,5,17,257,... unique among sqrts in the field of geometry and as such earns each of them full rights to their own pages apart form other pages on root rectangles, square roots, golden ratio, irrationals, incommensurables, etc. i have not contributed to this page, so don't take this as an intent to preserve my own work here. i do find it deserving of its own place based on this argument. --Euanthes (talk) 20:19, 26 February 2008 (UTC)


 * The problem is that we need sources for that. If you know any, please suggest or be bold.  Brusegadi (talk) 05:18, 27 February 2008 (UTC)

Should this article be deleted?
I do not think the page should be deleted. At least not speedly, I think it deserves some discussion since the number appears in many things.Brusegadi 22:37, 3 August 2007 (UTC)


 * I do not understand the nomination either, and I actually think the claim that "notability is not asserted" is plainly false, since the article does mention several algebraic and geometric aspects that make this number notable. Together with √2, √3, φ, π and e, this is one of the most elemental and important irrationals, being intimately related to the golden ratio and to several of the most geometrically simple shapes (square, cube), as well as appearing in lots of important formulae such as those for the exact trigonometric constants. If this number is not considered notable, then on what grounds are the square root of 2 and the square root of 3 considered so? Besides, an "unreferenced" tag has also been placed, but everything that is currently stated in the article can be verified immediately simply from knowledge of elemental geometry and using well-known mathematical formulae such as the Pythagorean theorem. Sure the addition of more info and references would be very nice, but the "unreferenced" tag makes it seem as if what the article currently says is dubious or disputed, which most definitely is not but is the mere pointing out of very elemental algebraic and geometric relations. Uaxuctum 22:56, 3 August 2007 (UTC)


 * What's hard to understand? The article doesn't include evidence of notability, and such articles are subject to deletion.  Please read WP:Notability.  The edit summary "just the fact that it appears in the golden ration makes it notable enough. I recall this number is important" completely misses the point; it's not your opinon or recollection that matter, but what evidence is cited in the article.  Mere frequent appearance is not relevant.  You need to cite reliable secondary sources that talk about the square root of five as an important topic of its own. Dicklyon 23:20, 3 August 2007 (UTC)


 * I was not using my recollection of the importance of the number as a reason for it to stay, I was using it as a reason for it to not be speedly deleted; which is different. The difference is that one of them gives us time to discuss without just throwing a potentially useful article away.  Basically, I thought that the article should be discussed, like we are doing now.  When I get back home I will look for my number theory books and see if this number is any more important than say, the sqrt of 223 (both irrational by the way...) Brusegadi 23:55, 3 August 2007 (UTC)

I've gone ahead and nominated this article for deletion, since nobody seems to be able to come up with any secondary sources supporting the idea that this particular square root is "notable" in the wikipedia sense. Dicklyon 22:29, 11 August 2007 (UTC)


 * It doesn't look like WP:NOTE is going to be taken seriously, so I guess the article stays; so I've been working on improving it.  For example, the stuff on golden ratio, one of the original reasons for the article, I think, had grown to horrendous proportions, as golden ratio stuff tends to do.  So I pruned it.  And I fixed the ref styles into one consistent style.  And a few other things.  Dicklyon 17:42, 12 August 2007 (UTC)

Golden ratio trivia
Please see immediately above where I said "the stuff on golden ratio, one of the original reasons for the article, I think, had grown to horrendous proportions, as golden ratio stuff tends to do. So I pruned it." If you disagree with a topic that active in a talk page, bring up your objections here, instead of just reverting and objecting in the edit summary. That way, we can work toward a consensus. Dicklyon 20:14, 12 August 2007 (UTC)


 * I don't see why letting readers clearly see the interrelations between this quantity and the golden ration and its conjugate qualifies as unnecessary trivia, and I fail to see where this section has "grown to horrendous proportions" when in fact it was almost telegraphic and your edits have further reduced it to the absolute minimal expression. Those formulae are useful, especially to understand their geometric relations (such as the very clarifying one that sqrt(5) = phi + Phi); sure, the readers could work them out for themselves from the only one provided in your version, but the purpose of an encyclopedia article is to be informative, not to conceal or leave potentially useful or clarifying information unstated. Uaxuctum 20:39, 12 August 2007 (UTC)


 * Thanks for responding. My objection was of course not to "letting readers clearly see" something, but rather to what appeared to be a typical golden ratio algebra fest.  The golden ratio has a ton of fun relationships, which are pretty much discussed in hundreds of articles about the golden ratio, and hence are repeated in the wikipedia article on it.  I don't think those same manipulations are needed in an article on the square root of 5, but if you can find a source for saying that the square root of 5 can be written in terms of the golden ratio in all those ways, then by all means include them and reference it.  The present version is just too much original research, that is, algebra that one can do oneself but has not been pubished, as far as I can tell. Dicklyon 22:32, 12 August 2007 (UTC)

\frac versus \cfrac
Compare and contrast:



4\int_0^\infty\frac{xe^{-x\sqrt{5}}}{\cosh x}\,dx = {1 \over 1 + {1^2 \over {1 + {1^2 \over {1 + {2^2 \over 1 + {2^2 \over {1 + {3^2 \over {1 + {3^2 \over{1 + \cdots \quad}}}}}}}}}}}} $$



4\int_0^\infty\frac{xe^{-x\sqrt{5}}}{\cosh x}\,dx = \cfrac{1}{{}\quad 1 + \cfrac{1^2}{1 + \cfrac{1^2}{1 + \cfrac{2^2}{1 + \cfrac{2^2}{1 + \cfrac{3^2}{1 + \cfrac{3^2}{1 + \cdots \qquad\qquad{}}}}}}} \quad{}} $$
 * Michael Hardy 22:44, 13 August 2007 (UTC)


 * For Ramanujan's identities, cfrac is nicer because I can see the exponents more clearly at every level. For the Docuan table, I think frac is better because it takes up less space, and when the continued fraction just keeps repeating the same denominator, it's OK if I can't see the smaller levels so good. But that's just my opinion. PrimeFan 22:38, 15 August 2007 (UTC)

Here's a more compact form of the above, also using \frac. Comments?



4\int_0^\infty\frac{xe^{-x\sqrt{5}}}{\cosh x}\,dx = \frac{1}{1 +} \frac{1^2}{1 +} \frac{1^2}{1 +} \frac{2^2}{1 +} \frac{2^2}{1 +} \frac{3^2}{1 +} \frac{3^2}{1 +} \cdots $$
 * Glenn L (talk) 19:29, 25 May 2009 (UTC)

I've never liked that form. Compactness is its one virtue, and that seems like something for use in circumstances of poverty of space to write in. The less compact form conveys the meaning to people who've never seen continued fractions before, and I think that's needed here. Michael Hardy (talk) 20:07, 25 May 2009 (UTC)

Unique factorization domains
Is it the case that 5 is the smallest integer n such that the integral domain Z[&radic;-n] fails to be a unique factorization domain? Certainly Z[&radic;-5] is a common example of a domain for which unique factorization fails to hold, and this might be worth mentioning in the article. -- Dominus 23:12, 14 August 2007 (UTC)
 * No. Z[&radic;-3] is not a UFD: 2*2=(1+&radic;-3)(1-&radic;-3). Algebraist 19:44, 24 May 2008 (UTC)

Music
Could someone please explain, or better yet clarify in the article, the following: If someone can send me a copy of the review, I can try to make some sense of it. Thanks. Finell (Talk) 00:46, 15 August 2007 (UTC)
 * "He then applied the square root of five within the golden ratio ... as a way to select the relative length of the two parts of his piece." More specifically, what does "the square root of five within the golden ratio" mean? How did Wuorinen "apply" it to proportion the length(s) of "the two parts of his piece"? What "two parts of his piece"? (E.g., 1st and 2nd movement of a 2-movement work? Theme A and theme B of a sonata form?)
 * "In doing so, he took the bite and harsh dissonance normally accompanying of 12-tone music." Does took in this statement mean he eliminated or reduced "the bite and harsh dissonance"? How did applying some proportion to the lengths of 2 subdivisions of the piece affect its harmonic character?
 * Since the article is discussing a particular composition by Wuorinen, it is essential to name the piece.


 * I had asked here for the guy who posted that to let me know what it says; or just leave it out. He didn't respond, so I take it that leave it out is OK for now.  I'll  take it out and we can see if he cares. Dicklyon 00:56, 15 August 2007 (UTC)
 * From the context, it sounds like a review of a performance of a composition by Wuorinen. There is probably something worthwhile there, but it was not well summarized in the article. If there is something, it probably relates as much or more to φ than to √5. If I had easy access to the source, I would get it myself--but I don't. Finell (Talk) 02:26, 15 August 2007 (UTC)

Only "one million digits"?
Since the Golden ratio is known to billions of digits (at least 17 billion per that article though the Mathematical constant article suggests it's actually 100 billion), shouldn't the Square root of 5 also be known to the same billions of digits? After all:

$$ \sqrt 5 = 2 \varphi - 1 $$

Glenn L (talk) 19:46, 25 May 2009 (UTC)


 * Seems plausible, since multiplying by 2 seems pretty simple, and the subtraction of 1 effects only the one digit before the decimal point. Michael Hardy (talk) 20:09, 25 May 2009 (UTC)


 * Indeed, it would be trivial to take 17 billion digits of phi and make 17 billion digits of root 5. And if someone would report doing so, in a reliable source, there's no reason we wouldn't include that fact. Dicklyon (talk) 04:21, 26 May 2009 (UTC)

Removal of infobox
Based upon a discussion at Wikipedia talk:WikiProject Mathematics, I've removed the infobox from the article. If anyone disagrees, could you please join the discussion there. Thanks, Paul August &#9742; 14:58, 18 October 2009 (UTC)


 * I have suggested centralizing this discussion to Wikipedia_talk:WikiProject_Mathematics and Wikipedia_talk:WikiProject_Mathematics as it refers to an infobox occurring in several articles. Please go there to build consensus on this edit. RobHar (talk) 19:34, 18 October 2009 (UTC)

Geometry to begin
Here is the beginning of the current section titled [https://en.wikipedia.org/w/index.php?title=Square_root_of_5&oldid=1058260480#Geometry "Geometry". ] That would be obvious faced with a sketch of rectangle divided into two congruent triangles. I propose to insert in the article such an image that will illustrate also the Pythagorean theorem. Admittedly you don’t see my new image yet. Thus you can assess the situation of a reader of the current article. Arthur Baelde (talk) 09:52, 4 January 2022 (UTC)

As well as √5,  from an historical point of view,  began to exist in geometry,  neophytes also conceive √5  first in geometry,  more easily than as a limit like in the current first section,  [https://en.wikipedia.org/w/index.php?title=Square_root_of_5&oldid=1063837830#Continued_fraction titled “Continued fraction”. ]   So I propose to move the current section titled “Geometry” at the beginning of article,  before improving this new first section. Do you agree?  Arthur Baelde (talk) 12:56, 5 January 2022 (UTC)

A rather old version of the section titled “Geometry”,  as edited at 15:38, 27 December 2012,   far in the article like today,  included already the incomprehensible image named Pinwheel_1.svg. '''Far in order [https://en.wikipedia.org/w/index.php?title=Square_root_of_5&oldid=1063837830#Geometry to be hidden? ]'''  Arthur Baelde (talk) 09:47, 6 January 2022 (UTC)
 * I compressed the continued fraction stuff a bit, but I think the current order is logical enough. At least the connection to the golden ratio (which is a big part of the justification for this page's existence) should come before geometry.  It doesn't matter if that pinwheel image has been on the page a long time.  If you have a better one, we should replace it. Danstronger (talk) 13:51, 7 January 2022 (UTC)


 * “Rational approximations” is a better title than “Continued fraction”.  But the word “convergent” is too difficult for the beginning of section, and the expression “continued fraction” should be a link immediatly at its first occurrence.


 * For me the relation between the golden ratio and √5 has to be exposed in geometry, before any concept of limit.  But why would we have to justify the existence of article?  The current second and fourth sections should be two subsections of a first section titled “Geometry”.  Arthur Baelde (talk) 16:08, 7 January 2022 (UTC)


 * Pardon me,  your first occurrence of “Continued fraction” is indeed a link.  Arthur Baelde (talk) 10:12, 8 January 2022 (UTC)
 * The golden ratio is not really a subtopic of geometry, as the ratio has many properties that are not geometric. For whatever reason, trigonometry is not generally considered a subfield of geometry. See sections above on this talk page that discuss whether or not this page should even exist.  Does sqrt(5) really have significant converage in reliable sources?  If everything here could basically be said about the square root of any number, we are wading into "indiscriminate collection of information" territory. I'm not really saying we should delete this page (I tend to lean towards delete, and I think it's borderline, so I don't think a deletion discussion would likely succeed), just explaining why the connection to golden ratio stuff is pretty important to this wikipedia page. Danstronger (talk) 17:01, 8 January 2022 (UTC)