MY COMMENTS ON THIS ARTICLE
by LUIGI RIVARA (posted 10/17/10)

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