I have a small follow up on rational numbers and real numbers. We've established that rational numbers have holes, and the real numbers fill those holes. The question is, how big are the holes, and how many of them are there?

It's possible to show that there is a rational number between any two real numbers. The metaphorical proof is straightforward. Imagine the real number line as a sidewalk you are walking on. If you want to find a rational number between the real numbers x and y, imagine a gap in the sidewalk between x and y. If you are taking steps which are a constant rational size, then what happens when you get to the gap depends on the size of your steps. If your steps are big enough, you could step right over the gap and keep going. But if your steps are smaller than the gap size, you must step into the gap. The position of the step in the gap is a rational number, and it is between x and y.

A formal version of the stepping in the gap argument proves that between any two real numbers, you can always find a rational number. Unsurprisingly, a similar argument proves that between any two real numbers, you can always find an irrational number. (An irrational number is just any real number which is not a rational number.)

Here's my interpretation: irrational numbers are holes in the set of rational numbers. Since you can find irrational numbers everywhere, there are lots of holes everywhere. On the other hand, since you can find rational numbers everywhere, that proves that the holes are tiny. It's important that the real numbers fill in the holes in the rational numbers, but depending on what you're doing, you may be able to ignore the holes anyway.