The present course of mathematical education in school and college introduces a student rather casually to the various properties of integers, rational numbers, and real numbers as they are needed for arithmetic, algebra, geometry, and calculus. Sometimes the relations among these mathematical structures are sketched as indications of things to come in mathematics. When the student begins graduate study he finds that he is expected to be familiar with the structure of the real number HyHtem. If his graduate courses deal explicitly with this system at all, they usually do so in terms of a brief summary. It is then assumed that the student has a usable knowledge of the intricately interwoven properties of set theory, algebra and topology which characterize the Hystcm of real numbers. Actually, this assumption is frequently false and a gap is left in the student's knowledge which, if not filled, hinders his development.