ABSTRACT
Any two numbers can be ordered on the number line. If point A locates to the left of point B, then A is smaller than B (i.e., A < B). If A < B and B < C, then A < C. Therefore, ordering among real numbers is transitive. Any set with this property is called a totally ordered set. The set of integers, the set of rational numbers, and the set of real numbers are all totally ordered. Points representing consecutive integers on the number line locate discretely, in equally spaced intervals. They look like stepping stones in a garden. There is no integer between any pair of consecutive integers. On the other hand, between any pair of distinct rational numbers a < b, there exists a rational number r such that a < r < b. The density of rational numbers signifies this property, as does the set of real numbers. The set of integers is, by definition, not dense.
