There is a certain pattern to learning mathematics that I got used to in primary and secondary school. It starts like this: first, there are only positive numbers. We have 3 apples, or 2 apples, or maybe 0 apples, and that’s that. Sometime after realizing that 100 apples is a lot of apples (I’m sure that’s how my 6 year old self would have thought of it), we learn that we might have a negative number. That’s how I learned that they don’t always tell us everything, and that sometimes the things that they do tell us have silly names.
We know how the story goes – for a while, there aren’t remainders in division. Imaginary numbers don’t exist. Under no circumstance can we divide or multiply by infinity, or divide by zero. And this doesn’t go away: in my calculus courses (and the ones I’ve helped instruct), almost every function is continuous (at least mostly) and continuity is equivalent to ‘being able to draw it without lifting a pencil.’ It would be absolutely impossible to conceive of a function that’s continuous precisely at the irrationals, for instance (and let’s not talk about 
So when I hit my first class where I learned and used the pigeon-hole principle regularly (which I think was my combinatorics class? Michelle – if you’re reading this, perhaps you remember), I thought the name “pigeon-hole” was another one of those names that will get tossed. And I was wrong.
I was in a seminar today, listening to someone talk about improving results related to equidistribution theorems, approximating reals by rationals, and… the Dirichlet Box Principle. And there was much talking of pigeons and their holes (albeit a bit stranger, and far more ergodic-sounding than what I first learned on).
Not knowing much ergodic theory (or any at all, really), I began to think about a related problem. A standard application of pigeonholing is to show that any real number can be approximated to arbitrary accuracy by a rational 
More after the fold -
