# A Disquisition on the Square Root of Three - The Classical Greek Ladder

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

The classical Greek ladder for $\sqrt{3}$ to six-place accuracy begins like this:

 $1$ $1$ $\frac{1}{1}=1.000000$ $1$ $2$ $\frac{2}{1}=2.000000$ $3$ $5$ $\frac{5}{3}\approx 1.666667$ $4$ $7$ $\frac{7}{4}=1.750000$ $11$ $19$ $\frac{19}{11}\approx 1.272727$ where each rung $\langle a\quad b\rangle$ is $15$ $26$ $\frac{26}{15}\approx 1.733333$ followed by $\langle a+b\quad 3a+b\rangle,$ $41$ $71$ $\frac{71}{41}\approx 1.731707$ written in reduced form, $56$ $97$ $\frac{97}{56}\approx 1.732143$ with $\sqrt{3}$ approximated by $\frac{b}{a}.$ $153$ $265$ $\frac{265}{153}\approx 1.732026$ $209$ $362$ $\frac{362}{209}\approx 1.732057$ $571$ $989$ $\frac{989}{571}\approx 1.732049$ $780$ $1351$ $\frac{1351}{780}\approx 1.732051$

While the ladder could begin with any pair of nonnegative integers, not both zero, the rung $\left\langle 1\quad 1\right\rangle$ was used here because it yields the “classical” Greek ladder. The ladder stops where it did because that's where it yields the six-place accuracy that was presented at the outset of this paper. The seven-place denominator of $1000000$ has been beaten by the three-place $780$ – quite an improvement.

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