# A Disquisition on the Square Root of Three - Continued Fractions

Author(s):
Robert J. Wisner (New Mexico State University)

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.

Robert J. Wisner (New Mexico State University), "A Disquisition on the Square Root of Three - Continued Fractions," Convergence (December 2010), DOI:10.4169/loci003514