Section 3.12, due Monday October 11

1. (Difficult) I understand the theorem, but I am not exactly sure why it is needed in the discussion of continued fractions. Does it have to do with the face that p_k/q_k gives a better approximation to x than any of the preceding rational numbers? Also, is the recurrence relationships related to, or actually, the Euclidean Algorithm? The example demonstrates that the quotients from the Euclidean algorithms are the floored functions after ever step.

2. (Reflective) I have never learned about continued fractions, but this is a really good method. The approximations are surprisingly accurate once you have gone out enough. I remember doing various approximations for numbers, specifically pi, but I never have used this method. I imagine that continued fractions not only could be used in application for good approximation, but could be easily be a research. I think that would be an interesting thing to study and learn more about.