Search
The Classic Greek Ladder and Newton's Method
Greek ladders for approximating square roots may be more ancient than the ancient Greeks. Students at any level can appreciate their beauty and simplicity. Those who have studied calculus can compare them with Newton's Method for approximating roots.
Random Quotation
Mathematics seems to endow one with something like a new sense.
In N. Rose (ed.), Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Loci: Convergence
show printer friendly
send to a friend
The Classic Greek Ladder and Newton's Method
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
This article uses jsMath, which requires JavaScript, to process the mathematics expressions. If your browser supports JavaScript, be sure it is enabled. Once the jsMath scripts are running, clicking the "jsMath" button in the lower right corner of the browser window brings up a panel with configuration options and links to documentation and download pages, including instructions for installing missing mathematics fonts.
Pages: | 1 | 2 | 3 | 4 | 5 | 6 |
Wisner, Robert J., "The Classic Greek Ladder and Newton's Method," Loci (August 2009), DOI: 10.4169/loci003330
Discuss this article
thread #1:
MY COMMENTS ON THIS ARTICLE
I read with big interest this article of Dr. Robert J. Wisner (also the article on the same subject “A disquisition of the square root of three�). It is very rewarding for a lover of mathematics as I am to find in Convergence article always very clear and very appealing. The only point that was not clear to me in this article was relevant to the “rating system�. I understand from a personal E-mail interchanged with Prof. Wisner that in Diophantine approximation the goal is to get a good approximation with a minimal denominator, but following this line for sqrt(3) we can have these approximations: 5/3 with a rating of (-2,1) and a value 5/3=1,66666666666667 Or 19/11 with a rating of (-2,2) and a value 19/11 = 1,72727272727273 But the real value of sqrt(3) is 1,732051 therefore 5/3 is far from the value Following the explanation “in Diophantine approximation the goal is to get a good approximation with a minimal denominator� there is no need for proceed along the “greek ladder� because the approximation with a minimal denominator is always the first rung I like to share also a my explanation of the formula reported for the “greek ladder�, always said that is not known how was found For me this was the way If we say that b/a = sqrt(N) we have b^2/a^2 = N if we add to both said b/a we have b/a + b^2/a^2 = N + b/a b/a ( 1+b/a) = N + b/a b/a = [N+ b/a] / (1+b/a) and with some manipulation b/a = (a *N +b) /(a+b) from which bn = an-1 * N + bn-1 an = an-1 + bn-1 Best reagards Luigi Rivara