Abstract / Synopsis
Students of real analysis and calculus find that the completeness property of the real numbers is difficult to understand, especially, its importance. The word numbers denote real numbers throughout this piece. After all, the real numbers are not any less imaginary than the so-called imaginary numbers. Although, sometimes, it does gain some mention in calculus courses, its presence as a topic is a mere will-o’-the-wisp of bygone days when teachers would often reach deep into the big ideas of calculus as a mainstay of their courses. For the sake of cultural literacy and the development of mathematical maturity, we believe it is a topic that should not be so slighted. In a fanciful way, we present this topic by way of two fictional characters who were introduced to this topic in their calculus course, contrary to current practice. On the journey to understanding this important idea, other related issues arise along the way. We believe that this and other mathematical concepts are not made concrete enough in today’s calculus classes and, even, real analysis classes, if covered at all. Proofs and explanations are not made alive and direct. Thus, we attempt to make the relationship between connectedness and the S1S2 property more intuitive and more concrete. Moreover, as secondary schools eschew teaching abstract thinking, the need arises to ground mathematical concepts in some sort of reality, which is what Scottish mathematicians of the 18th and 19th centuries called having “sensory referents.”[?] Thus, in this vein, empty sets, rational numbers, and irrational numbers become objects of debate. The grandson of A. Square, a character in Flatland: A Romance of Many Dimensions, makes an appearance.1 He helps our heroes attempt a solution to the mystery of some inexplicable disappearances of people and objects, although with partial success. At the same time, we hope to motivate the completeness axiom. We hope this story aids in the communication of the importance of this significant topic and other issues.
Recommended Citation
Charles A. Coppin, "Nightmare in the Library," Journal of Humanistic Mathematics, Volume 16 Issue 1 (January 2026), pages 414-443. . Available at: https://scholarship.claremont.edu/jhm/vol16/iss1/32
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