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:

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.