Talk:Nested radical

message: The page Exact_trigonometric_constants has a bit of a longer discussion on this, but I don't quite get the math.
I don't understand how the quadradic formula can be used to solve for d and e. I've looked everywhere on the Internet and nowhere dose it actually show the deverivation of the formula for denesting radicals.
:Quick explanation: if we assume that there exist rational d and e such that \sqrt{a+b\sqrt{c}}=\sqrt{d}+\sqrt{e}, then by squaring we get a+b\sqrt{c}=d+e+2\sqrt{de} as the article says. Then, we equate the rational and irrational parts, giving us a=d+e\, and b\sqrt{c}=2\sqrt{de}\rightarrow de=b^2c/2 and you can rearrange the last two to give quadratic formulae for d and e in terms of a, b and c. Of course, it may turn out that your quadratics don't resolve into nice rational expressions, in which case you're screwed. Confusing Manifestation 01:59, 6 June 2007 (UTC)
replies: text: I've added some language to clarify the situation. --TheMaestro (talk) 20:53, 13 September 2009 (UTC)

text: Look at the italian article (haven't to understand the language) and think at Vieta's formulas.  — Preceding unsigned comment added by 80.219.229.120 (talk) 01:52, 16 November 2011 (UTC)
thread_title:  (untitled) 

message: To a layperson, this article is almost worthless as no effort is made to explain any of it which makes any concession to the non-specialist. What, for instance, is a 'radical' in this context and what does it mean to be 'nested'? Can someone do some work on it to make it clearer, please? Thanking you in advance. ThePeg (talk) 14:21, 30 May 2011 (UTC)
thread_title: Uh?

message: The section Cube roots [of infinitely nested radicals] says
:: \sqrt[3]{n+\sqrt[3]{n+\sqrt[3]{n+\sqrt[3]{n+\cdots}}}}
:is the real root of the equation x3 − x − n = 0 for all n > 0.
Is this supposed to say "for all integer n>0"? The reason I ask is that if n^2<4/27, the equation has three real roots, not just one. Or does the statement remain true for one of the real roots? Loraof (talk) 21:24, 13 January 2015 (UTC)
thread_title:  Clarification needed 

message: Hello fellow Wikipedians,
I have just modified one external link on Nested radical. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
*Added archive https://web.archive.org/web/20060312035635/http://www.almaden.ibm.com/cs/people/fagin/symb85.pdf to http://www.almaden.ibm.com/cs/people/fagin/symb85.pdf
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
Cheers.—InternetArchiveBot (Report bug) 07:20, 16 February 2018 (UTC)
thread_title:  External links modified (February 2018) 

message: I wonder if examples of a radical denesting with non-rational constants are worth including in the article as well, like this nice and non-obvious one with the golden ratio:
\sqrt{\phi+\sqrt\phi}=\frac\phi{\sqrt2}+\frac1{\sqrt{2\,\phi}}. VladimirReshetnikov (talk) 20:35, 15 June 2018 (UTC)
replies: text: Be bold!  yoyo (talk) 14:07, 29 September 2018 (UTC)
thread_title:  An example with the golden ratio 

message: What the value of the cube root of 2+\sqrt{5}:
\sqrt[3]{2+\sqrt{5}}? 180.183.71.47 (talk) 04:43, 8 November 2019 (UTC)
thread_title:  What is the cube root of 2+sqrt(5)? 

message: The theorem added in april by Svennik is Herschfeld's theorem I, but Herschfeld's theorem III is stronger.
It is stated in fr:Radical imbriqué#Radicaux infinis de Ramanujan and proved in v:fr:Approfondissement sur les suites numériques/Exercices/Convergence#Exercice 7.
Anne Bauval (talk) 06:34, 6 August 2020 (UTC)
thread_title:  Herschfeld's Convergence Theorem 

message: I believe there is an issue in reasoning for this section. The functional equation F(x)^{2}=1+xF(x+1), i.e. setting n=1 and a=0, can be shown to have infinitely many solutions other than F(x)=x+1. In fact, there are solutions which have an arbitrary number of continuous derivatives which solve the functional equation but aren't equal to x+1. Any function f on the interval [0,1] can be extended to a solution on further intervals simply by taking
:  f(x) = \frac{f(x-1)^{2}-1}{x-1} 
to inductively define the function on (n, n+1] when it is already defined on (n-1, n]. In this Desmos project, I give an example solution which has a continuous third derivative (although it is not always positive). Even restricting to positive functions, a similar solution could be found in a fifth-degree polynomial. 
The solution F(x)=x+1 seems to only be the unique solution if there is an assumption that the function is analytical or at least that it is infinitely continuously differentiable. However, this assumption is not immediately obvious from the definition as a limit of those nested radical functions. I believe the proof in this paper would actually be better suited for this page to give a proper argument as to why that condition is met, although it may be too intensive for the page.  — Preceding unsigned comment added by 104.230.10.122 (talk) 22:56, 18 May 2022 (UTC)
thread_title:  Ramanujan Infinitely Nested Radical 

message: Is there a reason for the "two" in "two nested square roots" to link to the page for two-stroke engine? It doesn't seem relevant to me, but I'm only an amateur editor so I thought I'd bring it up here. Tempestate (talk) 23:52, 15 December 2023 (UTC)
replies: text: . Thanks for pointing to this vandalism. D.Lazard (talk) 09:49, 16 December 2023 (UTC)
thread_title:  Two-Stroke Engine Link?