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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