Decimal expansion of rational numbers

Just to correct my answer to a question in the lecture of Monday 7/10: any real number is rational if and only if its decimal expansion is ultimately periodic, see for instance https://en.wikipedia.org/wiki/Repeating_decimal. So indeed, it follows that the set of eventually periodic points of the map $x\to 10 x$ mod 1 on [0,1) is exactly equal to the set of rational points, which is obviously dense in [0,1).