Hello, I looked at some examples, as far as I know, for 1 example : sqrt(10), the classical approximation is better : the 4'th convergent of the continued fraction is 771/228 which has an error of 3.04 E-06, his approximation of sqrt(10) is 838/265 and the error is -1.35092E-05. So the fraction is bigger and the error is bigger, it should be the inverse. So for that example : his method FAILS to be a better approximation. Actualy, if you start from 771/228 there are a number of ways to get away from the best possible approximation to a not that bad approximation BUT the knowns theorems about approximations will clearly state that there are NO smaller numerator that will give a better approximation (period). In other words : it is impossible to get a smaller error from sqrt(10) with a smaller numerator. There are numerous references about this fact. Another example could be given by the ratio of successive fibonacci, if you start with 1, 3 instead of 1,2 it will still converge to 1.618033... BUT the numerators will be BIGGER and the approximation of the golden ratio won't be better. There is a classical reference about these facts in the book of Hardy and Wright (An introduction to the Theory of Numbers). Again : in other words (cite) The best possible approximation of the golden ratio with the smallest possible numbers is the one and only succ. Fibonacci numbers. There is no way to get a better example and they give a clear proof of that. That principle can be extended to all other real numbers. The continued fraction algorithm is the best possible rational approximation of a real number. The exercice he is doing is not without interest but it cannot be said that this is a better way, it is plain impossible. Simon Plouffe