Appreciating p-adics, the Fake Numbers

Abstract

66 p.What does the phrase “Fake Numbers” mean? For me, the origins are in the semantics of food. If you will bear with me for a moment, please consider the term real food and its implied opposite, fake food. Taken literally fake food would refer to a plastic apple. However, that is not the comparison that is being made when real food is brought up. It references healthy food, natural food, or perhaps the food that people have been eating for most of history. Therefore, we know that the implied fake food is the modern, in this case processed and industrialized food. However, we can note that the latter is still edible. It still has calories, and by most definitions would qualify as food. Let this then inform our understanding of “fake numbers”. Much of the early if not full mathematical journey of many of my peers has been to learn about the real numbers (R). I content the p-adic numbers are the implied, albeit often omitted, fake number counterpart to R. A more modern construction, one with quite the overlap and enticing features that fake food has, yet not necessarily a full substitute for the original and all of its properties. With some semantics and analogies over with, we can discuss the layout what this paper and what we wish to accomplish over its duration. On a personal level it has been the culmination of my undergraduate experience with mathematics. A chance for me to appreciate much of what I have learned, apply past concepts to new landscapes, finally get my head around some key ideas, left me with a clearer picture of how little I know, and has been a continued exploration my newly kindled love of seeing math1. Therefore, I have a goal for us to accomplish. We will enjoy looking into the interesting, weirdly shaped, sometimes counterintuitive, yet also very accessible world of the p-adics. Along the way we will make several comparisons with R. We should also not believe in the above analogy so much as to begin viewing the p-adics as unhealthy or as a full replacement for R. To accomplish this goal, we will start at the beginning, what are the p-adics