Now this might be a very dumb question but this has been bothering me from some days.
Imagine I want to create the real number line and for that I start with the rational numbers. So I start to put the numbers on the line (like toppings on a pizza). Now I know that between any two rational numbers I can add another rational number.
So I keep on adding infinitely many rationals between any two rational numbers by adding more and more digits in the decimal expansion.
I can keep on doing this forever but I don't think I will ever reach a point where I can't put a rational and only an irrational number can go there i.e. I don't think there will be any gap between two rationals which can be only fit in by an irrational (or is it ?)
So where do irrationals actually fit in on the line ?
I know this entire argument can be done the other way starting with irrationals too.