> I believe real numbers to be completely natural
You can teach middle school children how to define complex numbers, given real numbers as a starting point. You can't necessarily even teach college students or adults how to define real numbers, given rational numbers as a starting point.
well it's hard to formally define them, but it's not hard to say "imagine that all these decimals go on forever" and not worry about the technicalities.
An infinite decimal expansion isn't enough. It has to be an infinite expansion that does not contain a repeating pattern. Naively, this would require an infinite amount of information to specify a single real number in that manner, and so it's not obvious that this is a meaningful or well-founded concept at all.
The way I think of it is this:
Imagine you have a ruler. You want to cut it exactly at 10 cm mark.
Maybe you were able to cut at 10.000, but if you go more precise you'll start seeing other digits, and they will not be repeating. You just picked a real number.
Also, my intuition for why almost all numbers are irrational: if you break a ruler at any random part, and then measure it, the probability is zero that as you look at the decimal digits they are all zero or have a repeating pattern. They will basically be random digits.