&gt; The distance between 0 and 1 is smaller than the distance between 0 and 2.<p>This is correct.  However, the number of real-valued points between 0 and 1 is the same as the number of real-valued points between 0 and 2.<p>&gt; The number of points between 0 and 1 is larger than the number of rational numbers.<p>This is also true because there are uncountably many real-valued points between 0 and 1 and countably many rational numbers.