The distance between two real numbers, defined as |a-b|, represents the intrinsic separation between them, independent of how the interval is partitioned. This definition holds true even though there are infinitely many numbers between any two distinct real numbers, making counting them impractical. The discussion highlights that the distance remains constant regardless of the chosen unit of measurement, such as integers or fractions. Misunderstandings often arise from intuitive notions of counting elements within intervals, which can lead to errors like the "fence-post" problem. Ultimately, the metric |a-b| serves as a consistent measure of distance in the real number system.