### Continued Fractions: Part 2

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 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).