Prove that the irrational numbers are dense in R Ie prove th

Prove that the irrational numbers are dense in R. I.e., prove that if

Solution

The property of being dense just means that, essentially, given two distinct
rational numbers a and b such that a < b there exists an irrational number x
such that a < x < b. So the set of rationals is dense in R, the set of real numbers,
but this doesn\'t put a measure on how many rational numbers there are compared
to the measure of irrational numbers; that is a different question.

As you know, the set of rational numbers is countably infinite and the set of irrational
numbers is uncountably infinite, hence \'almost all\' real numbers are irrational. So the
\'proofs\' that you mention are not mutually inconsistent because they deal with
two distinct concepts, namely density and measure.