I don’t know why I’m so fascinated that the **rational numbers** are countable even though they’re a dense subset of the *uncountable* real numbers. A rational number can be arbitrarily close to any real number, making you *think* they’d be infinite like the reals, but in fact, nearly all numbers are * irrational* (and an uncountable subset of the reals).

So, the rational numbers — good old ** p/q** fractions — though still infinite are

*countably*infinite (see this post for details).

More to the point here, a common way of enumerating the rational numbers, when graphed results in some pretty curves and illustrates some fun facts about the rational numbers.