Section 3.12, due Mon Oct 14

Man I need to get better at remembering to do this...

This section was on continued fractions--approximating real numbers by successively closer rational ones.

1. The most difficult portion was the random theorem inserted into the middle. It mentioned that any fraction that was close to the real number being approximated ("close to" was in terms of the denominator of that fraction) had to be in the list of rational approximations. I thought about it for a while, but haven't figured it out quite yet.
   
   I've come up with reasoning why the recursive relationships of p_i and q_i work out, which makes me happy. Suppose that the relationship holds for p_i:
   
   The next step is found by replacing the a_i with (a_i + 1/(a_i+1)), like the following:
   
   However, the q_i term looks similar, and both are non-integral. We need these both to be integers, so we multiply by a_i+1:
   
   Proving our recursion.
   
2. I guess the most interesting part, personally, was that we could find good guesses of a rational number that generated a certain decimal expansion. Like in the text, we found that 3.764705882... probably came from 64/17, a relatively low denominator.