Stop being so irrational

I 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.