The Classic Greek Ladder and Newton's Method

Editor's note: This article was published in August 2009.

Introduction

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 \( b/a \) .

• Introduction
• Classic Greek Ladders
• Comparison with Newton's Method
• Extending the Connection
• Suggestions for Further Exploration
• Conclusion: A Little Story
• Addendum
• Acknowledgements
• About the Author
• References
• References for Addendum

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