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.
The 
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