Talk:Nested radical

(untitled)
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)


 * I've added some language to clarify the situation. --TheMaestro (talk) 20:53, 13 September 2009 (UTC)


 * 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)

Infinitely Nested Radicals
Going by this article, and the convention to use the positive sqaure root, the foollowing expression:


 * $$ x = \sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\cdots}}}} $$

Evaluates to x = 1. This doesn't seem intutive, so is there some other answer(ie. 0) or is 1 the accepted value? 122.162.197.54 (talk) 15:08, 5 November 2008 (UTC)


 * The article expressly says that the formula is to be applied for n&gt;0. It breaks down for n=0.&mdash;208.181.115.4 (talk) 21:28, 11 December 2008 (UTC)

A quadratic equation has two solutions. In the case you cite, one of the two solutions is 1. The other is 0. Michael Hardy (talk) 16:16, 5 October 2009 (UTC)


 * It isn't obvious that it's a quadratic equation. 86.132.222.67 (talk) 16:27, 9 September 2017 (UTC)

It is hardly surprising that commenters here are not following this part of the page, as written. The wording seems to me to be very philosophically muddled:

"Under certain conditions . . . . . represent ....."

More accurate would be:

". . . . . can be interpreted to represent. . . ."

Strings of symbols only have a meaning after we have given them one, and explained it. I am no expert on accepted conventions (which is the main thing we we should actually at this point be talking about) for interpreting infinitely nested radicals but if the following is the case, then it needs to be prominently stated, right at the start of this section.


 * $$ \sqrt{x_0+\sqrt{x_1+\sqrt{x_2+\sqrt{x_3+\cdots}}}} $$

has the conventional interpretation


 * $$ \lim_{k\rightarrow\infty}\sqrt{x_0+\sqrt{x_1+\sqrt{x_2+\sqrt{x_3+\cdots\sqrt{x_k}}}}} $$

Is that interpretation in terms of widely known concepts correct? I won't edit the page myself, in case I am wrong in my guess here; but if my guess is correct then that should be said in the page. — Preceding unsigned comment added by 83.217.170.175 (talk) 19:11, 21 December 2020 (UTC)

Uh?
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)

Clarification needed
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 &minus; x &minus; 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)

External links modified (February 2018)
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An example with the golden ratio
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)


 * Be bold! yoyo (talk) 14:07, 29 September 2018 (UTC)

What is the cube root of 2+sqrt(5)?
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)

Be bold! — Preceding unsigned comment added by 180.183.71.47 (talk) 04:43, 8 November 2019 (UTC)

Herschfeld's Convergence Theorem
The theorem added in april by Svennik is Herschfeld's theorem I, but Herschfeld's theorem III is stronger.

It is stated in Radical imbriqué and proved in Approfondissement sur les suites numériques/Exercices/Convergence.

Anne Bauval (talk) 06:34, 6 August 2020 (UTC)

Ramanujan Infinitely Nested Radical
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)

Two-Stroke Engine Link?
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)


 * . Thanks for pointing to this vandalism. D.Lazard (talk) 09:49, 16 December 2023 (UTC)