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The Classic Greek Ladder and Newton's Method

Robert J. Wisner (New Mexico State University)


For many students in early mathematics courses, their familiarity with approximations is limited to \( \sqrt{2}\approx{1.414} \), \( \sqrt{3}\approx{1.732} \), \( \pi\approx{\frac{22}{7}} \), and maybe a few more. But a topic of number theory, Diophantine Approximations (honoring Diophantus, a mathematician of Alexandria who lived circa 207 - 291 AD and wrote books called Arithmetica), involves approximating irrational numbers by ordinary reduced fractions. One of the approximation "tools" of ancient mathematicians is a construct called Greek ladders. Maybe Greek ladders will ignite your interest in approximations by ordinary fractions.

The phrase "classic Greek Ladder" is taken here to mean the infinite array that begins \[ \begin{array}{cc} 1 & 1\\2 & 3\\5 & 7\\12 & 17\\29 & 41\\70 & 99\end{array} \] where each rung \( \langle a \quad b \rangle \) is followed by \( \langle a+b \quad 2a+b \rangle \) and the approximations to \( \sqrt{2} \) are the fractions \( b/a \) .

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