This page does not describe how the continued fraction expansion of

was obtained. (Ah, I see your newest post mentions this.) Best rational approximation is a separate topic, interesting in its own right. The best rational approximation of an irrational number for some upper limit

of denominator will always be either a convergent or a semiconvergent of the continued fraction expansion; the choice between convergent and semiconvergent can be determined using the so-called

half rule.

Computing square root continued fractions can be tricky to grasp, but if you review the sources and play around with all the numbers you will eventually see how it works. Post any specific questions.

The description of algorithm in the link I gave you is ideal for computer programmers since it clearly specifies all the variables and how they change from one iteration to the next. On the other hand, it is not immediately clear where the algorithm came from.

You can look at the example worked out in the problem statement of this

Project Euler problem (#64) and see if it's any easier to follow.