Continued Fractions: Part 2

by landonlehman

So in the first post on continued fractions, I found out about an interesting pattern in the continued fraction of e, but didn’t get to my main point.  Hopefully I will get at least halfway there in this post!  As I said in the first post, this material is by no means original with me, and a large part of the information and motivation for these posts comes from The Princeton Companion to Mathematics.

At the end of the last post I said that continued fractions provide us with approximations to irrational numbers, and that in a certain sense they provide the “best” approximations.  Here is what I meant.  Take any finite continued fraction of an irrational number m.  Through standard algebra this can be rewritten as a “normal” fraction, i.e. written in the form p/q, where p and q are integers.  It can be shown that there are no fractions r/s that are closer to m than p/q such that s is smaller than q.

It is worthwhile to think about what this means.  It is not saying that there are no fractions r/s that are closer to m than p/q.  The fractional (rational) approximation of any irrational number can always be improved by using larger numbers in the numerator and denominator.  In fact, for any finite continued fraction approximating m, we could always just include a few more terms after the original (arbitrary) cutoff and thus obtain a rational number closer to m with a larger numerator and denominator than the original p/q.

What this notion of “best approximation” is saying in a more formal way is the following.  We say a fraction p/q is the “best approximation” to m if 0<s\leq q implies that s|m-\frac{r}{s}|>q|m-\frac{p}{q}|.  This takes into account the value of the denominator of the fraction, so we are saying that “best approximation” means that there does not exist any fraction with a smaller denominator than q that is closer to m.

I can see that I am going to have to move faster and gloss over some things if I want to be able to get to the main interesting result.  So I am not going to try proving any of this (I would probably get stuck pretty quickly if I did, plus I am very busy right now with classes/homework sets).