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Loci: Convergence
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A Disquisition on the Square Root of Three
Continued Fractions
A "standard" sequence of continued fractions for approximating \(\sqrt{3}\) follows.
\[1+\frac{2}{2}=2=2.000000\]
\[1+\frac{2}{2+\frac{2}{2}}=\frac{5}{3}\approx 1.666667\]
\[1+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}=\frac{7}{4}=1.750000\]
\[1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}=\frac{19}{11}\approx 1.727273\]
\[1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}=\frac{26}{15}\approx 1.733333\]
\[1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}}=\frac{71}{41}\approx 1.731707\]
\[1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}}}=\frac{97}{56}\approx 1.732143\]
\[1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}}}}=\frac{265}{153}\approx 1.732026\]
\[1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}}}}}=\frac{362}{209}\approx 1.732057\]
\[1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}}}}}}=\frac{989}{571}\approx 1.732049\]
\[1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}}}}}}}=\frac{1351}{780}\approx 1.732051\]
These approximations give the same fractions as the Greek ladder table. Thus, it seems fair to declare that the continued fraction method of estimating \(\sqrt 3\) ties the Greek ladder method.
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Wisner, Robert J., "A Disquisition on the Square Root of Three," Loci (June 2010), DOI: 10.4169/loci003514