# Stop being so irrational

04 Oct 2015I am convinced if you study mathematics long enough, you will eventually discover something unsettling. For me, this moment came in real analysis. This morning, a friend posted on Facebook, that her five-year old son asked, “Why do numbers never end?” I gave the stock answer about countable infinity. And someone else posted a video about Hilbert’s infinite hotel:

I never found the hotel terribly troubling, but it reminds me of something else. And down the rabbit whole we go. Let’s start by establishing 4 propositions:

- We understand that the rational numbers, those numbers constructed using [latex]m / n[/latex], are countably infinite. And that’s cool. Here are several proofs.
- We also understand the irrational numbers are uncountable. Here’s a proof from Theorem of the Week.
- Also, we know that between any two rational numbers, there is (at least) one irrational numbers. Here’s a proof from Khan Academy.

- Finally, we know that between any two irrational numbers, there is a rational number. Here is a proof of that.

This always bothered me. I was never able to reconcile, in my head, these four points. I kind of felt like if the density of rationals matched the density of irrationals, then if one is countable so is the other. Ahh, I eventually gave up and did statistics, instead.

*Image by Phrod / Wikimedia.*