It is a consequence of countability, but I like to think about it this way: a real number is, roughly, an infinite sequence of "random" digits. What is the prob

It is a consequence of countability, but I like to think about it this way: a real number is, roughly, an infinite sequence of "random" digits. What is the probability that your infinite sequence of random digits will randomly settle into an infinite repeating pattern? That's obviously, zero, right? No way that a random process will produce an infinitely repeating pattern. But just because it's "probabilistically impossible" to pick a rational number doesn't mean that they don't exist.