This chapter begins with a discussion of the Archimedean property and then discusses the fact that we can always find a rational number between two distinct real numbers. We have now reached the point at which we can give rigorous proofs of several facts you have known for some time. We show that there is a nonempty bounded subset of \(\mathbb {Q}\) that does not have a supremum in \(\mathbb {Q}\) . In this sense, \(\mathbb {Q}\) is not complete. Since \(\mathbb {R}\) is complete, there must be numbers in \(\mathbb {R}\) that are not in \(\mathbb {Q}\) . We know what these numbers are, of course; they are the irrational numbers. Thus far we haven’t proven that a particular real number is irrational. But now we can! We complete this chapter with a discussion of the division algorithm. In a set of tips entitled You Solved It. Now What? we ask you to reflect on your work, to learn from the problems that you solved, and to put all parts into context.