generalized continued fractions- square root calculations

Lagrange used continued fractions to solve the thousand year old Pell's equation $\displaystyle x^{2}-ny^{2}=1$. square roots can be calculate without calculators with continued fractions sqrt(2+sqrt(-2+sqrt(20))) - sqrt(2-sqrt(-2+sqrt(20))) Generalized continued fraction - Wikipedia, the free encyclopedia

Re: generalized continued fractions- square root calculations

Hello, mathlover10!

A rather disorganized explanation . . .

Quote:

Lagrange used continued fractions to solve the thousand year old Pell's equation $\displaystyle x^{2}-ny^{2}=1$.

Square roots can be calculated without calculators with continued fractions:

. . $\displaystyle \sqrt{2+\sqrt{-2+\sqrt{20}}} - \sqrt{2-\sqrt{-2+\sqrt{20}}}$ . . I don't see a continued fraction.

The above expression equals $\displaystyle \sqrt{5}-1$ . . . but so what?

This is a continued fraction:

. . $\displaystyle \sqrt{2} \;=\;1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2+ \cdots}}} $

Re: generalized continued fractions- square root calculations

Thanks for your reply! so calculation of square roots with continued fractions are used in complex analysis.... I guess 1000 year old problems are not necessarily important ones to spend time on

$\displaystyle \sqrt{20}~4.5$ can be estimated using a few continued fractions

Re: generalized continued fractions- square root calculations

does anyone know some good complex analysis texts or videos? it's interesting you can calculate even decimal square roots and other roots using continued fractions. Equal Temperament's perfect 5th can be expressed this way from $\displaystyle \sqrt[{12}]{2^{7}}$. Gauss's formula can be used to express elementary functions and the Bessel functions

Re: generalized continued fractions- square root calculations

this equation is really interesting for the coupling of the n does it arise from celestial mechanics? is Lagrange's proof available online?