Introduction to the $p$-adic Space

Abstract

In this paper, we offer a brief introduction to the $p$-adic numbers and operations in the metric space defined under the $p$-adic norm. Specifically, we provide a clear description of the derivation of the $p$-adic number via the completion of the rationals. This work provides definitions of all required background knowledge. We discuss salient features of $p$-adic algebra and explore various properties of the $p$-adic space, proving the Strong Triangle Inequality, the Product Formula, and Ostrowski's Theorem. Finally, we discuss interdisciplinary applications of $p$-adic analysis outside of number theory to quantum mechanics and computer science. This paper is a highly accessible introduction to $p$-adic numbers, ideal for individuals with little to no background in number theory.