generalized continued fractions- square root calculations

• Apr 13th 2013, 01:25 AM
mathlover10
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
• Apr 13th 2013, 04:30 PM
Soroban
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:

. . $\displaystyle \sqrt{2} \;=\;1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2+ \cdots}}}$
• Apr 14th 2013, 03:53 AM
mathlover10
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
• Apr 14th 2013, 03:55 AM
mathlover10
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
• May 13th 2013, 05:53 PM
mathlover10
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?