Fracción continua de sqrt(114)

Example, square root of 114 as a continued fraction

Begin with m0 = 0; d0 = 1; and a0 = 10 (102 = 100 and 112 = 121 > 114 so 10 chosen).

<br /><br /><br /> \begin{align}<br /><br /><br /> \sqrt{114} & = \frac{\sqrt{114}+0}{1} = 10+\frac{\sqrt{114}-10}{1} = 10+\frac{(\sqrt{114}-10)(\sqrt{114}+10)}{\sqrt{114}+10} \\<br /><br /><br /> & = 10+\frac{114-100}{\sqrt{114}+10} = 10+\frac{1}{\frac{\sqrt{114}+10}{14}}.<br /><br /><br /> \end{align}<br /><br /><br /> ” /></strong></dd> </dl> <dl> <dd><strong><img src=
d_{1} = \frac{S-m_{1}^2}{d_0} = \frac{114-10^2}{1} = 14 \,.
a_{1} = \left\lfloor \frac{a_0+m_{1}}{d_{1}} \right\rfloor = \left\lfloor \frac{10+10}{14} \right\rfloor = \left\lfloor \frac{20}{14} \right\rfloor = 1 \,.

So, m1 = 10; d1 = 14; and a1 = 1.

<br /><br /><br /> \frac{\sqrt{114}+10}{14} = 1+\frac{\sqrt{114}-4}{14} = 1+\frac{114-16}{14(\sqrt{114}+4)} = 1+\frac{1}{\frac{\sqrt{114}+4}{7}}.<br /><br /><br /> ” /></strong></dd> </dl> <p><strong>Next, <em>m</em><sub>2</sub> = 4; <em>d</em><sub>2</sub> = 7; and <em>a</em><sub>2</sub> = 2.</strong></p> <dl> <dd><strong><img src=
\frac{\sqrt{114}+10}{7}=2+\frac{\sqrt{114}-4}{7}=2+\frac{98}{7(\sqrt{114}+4)} = 2+\frac{1}{\frac{\sqrt{114}+4}{14}}.
\frac{\sqrt{114}+4}{14}=1+\frac{\sqrt{114}-10}{14}=1+\frac{14}{14(\sqrt{114}+10)} = 1+\frac{1}{\frac{\sqrt{114}+10}{1}}.
\frac{\sqrt{114}+10}{1}=20+\frac{\sqrt{114}-10}{1}=20+\frac{14}{\sqrt{114}+10} = 20+\frac{1}{\frac{\sqrt{114}+10}{14}}.

Now, loop back to the second equation above.

Consequently, the simple continued fraction for the square root of 114 is

\sqrt{114} = [10;1,2,10,2,1,20,1,2,10,2,1,20,1,2,10,2,1,20,\dots].\,

Its actual value is approximately 10.67707 82520 31311 21….

En esta página como no esta de lo mas claro:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html#sqrtcf

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~ por leonsotelo en diciembre 4, 2011.

Escrito en Aritmética, Discreta, Number Theory, Temas olímpicos

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