What exactly are all those real numbers? This reminds me of one of my favorite parts of studying math.  There are wayyyy more real numbers than rational numbers. It’s really hard to get a handle on what those “extra” numbers are, and yet you need all of them to do analysis. (Here “analysis” is a specific term meaning, very roughly, “the stuff you need to make calculus work.”)

For a bit of background, the Hilbert Hotel, a classic thought exepriment.  What the hell does it mean for one infinite thing to be “bigger” than some other infinite thing?