Continued Fractions: Part 2
So in the first post on continued fractions, I found out about an interesting pattern in the continued fraction of , 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 . Through standard algebra this can be rewritten as a “normal” fraction, i.e. written in the form , where and are integers. It can be shown that there are no fractions that are closer to than such that is smaller than .
It is worthwhile to think about what this means. It is not saying that there are no fractions that are closer to than . 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 , we could always just include a few more terms after the original (arbitrary) cutoff and thus obtain a rational number closer to with a larger numerator and denominator than the original .
What this notion of “best approximation” is saying in a more formal way is the following. We say a fraction is the “best approximation” to if implies that . 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 that is closer to .
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).