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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.
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Loci: Convergence
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The Classic Greek Ladder and Newton's Method
Acknowledgements
I am indebted to Professor Joseph Zund for introducing me to Greek ladders several years ago, and I thank Professors David Pengelley and Fred Richman for their very useful comments. The material in this article was presented at the MAA Southwestern Annual Conference, Western New Mexico University, Silver City, NM, April 3-4, 2009.
About the Author
Robert J. Wisner is Professor Emeritus in the Department of Mathematical Sciences at New Mexico State University. He was founding editor of SIAM Review, the only one of the 14 journals of the Society for Industrial and Applied Mathematics (SIAM) designed to appeal to all of its members. Wisner also has authored or co-authored numerous K-12 textbooks for Scott Foresman and Company; was Consulting Editor for Mathematics for Brooks/Cole for over 25 years; and recently co-authored a series of interactive pre-calculus textbooks, available on CD from Hardy Calculus.
References
[1] Leonard Eugene Dickson, History of the Theory of Numbers, vol. II, Chelsea Publishing Company, New York, 1919, p. 341.
[2] David Fowler, The Mathematics of Plato's Academy, A New Reconstruction (2nd ed.), Oxford University Press, 2003, pp. 57-58.
[3] Shaun Gilberson and Thomas J. Osler, "Extending Theon's Ladder to Any Square Root," College Math. J. 35 (2004) pp. 222-226.
[4] Russell Jay Hendel, "Approaches to the Formula for the \(n\)-th Fibonacci Number," College Math. J. 25 (1994) pp. 139-142.
[5] H. W. Turnbull, The Great Mathematicians, Simon and Shuster, New York, 1962, pp. 28-30.
[6] G. C. Vedova, "Notes on Theon of Smyrna," Amer. Math. Monthly 58 (1951) pp. 675-683.
[7] Robert J. Wisner, "Square Roots: Greek Ladders, Farey Fractions, and Pascal's Triangle," The AMATYC Rev. 23 (2002) no. 2, pp. 53-60.
References for Addendum
[8] David Fowler, The Mathematics of Plato's Academy, A New Reconstruction (2nd ed.), Oxford University Press, 2003.
[9] John J. O'Connor and Edmund F. Robertson, “Roger Cotes,” MacTutor History of Mathematics Archive, http://www-history.mcs.st-andrews.ac.uk/Biographies/Cotes.html
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Wisner, Robert J., "The Classic Greek Ladder and Newton's Method," Loci (August 2009), DOI: 10.4169/loci003330
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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