You might wonder why mathematicians want to use such a complicated definition. Wouldn't it be easier to simply define the real numbers to be the Dedekind cuts, or define the real numbers to be the decimal expansions, or something like that? That is the approach taken in some elementary textbooks, but ultimately it is less productive. When we actually use the real number system in proofs, the properties that we need are not specifically the properties of (for instance) Dedekind cuts or of decimal expansions. Rather, the properties we need are the axioms of a Dedekind complete ordered field. It is much simpler to think in terms of those axioms. To think of "numbers" as being cuts or expansions would just encumber us with extra baggage. The cuts or expansions are models -- they are useful for the job proving Theorem 1, but they are useful for little else. Once they've done that job, we can discard and forget them.

If you wish, you can now think of the points on a line as representing the members of a Dedekind-complete ordered field. It is then correct to say that the real numbers are the points on a line.

Information provided by Eric Schechter