Introduction to the $p$-adic Space

TL;DR

This paper provides an accessible introduction to $p$-adic numbers, covering their derivation, properties, and applications in fields like quantum mechanics and computer science, aimed at newcomers to number theory.

Contribution

It offers a clear, comprehensive overview of $p$-adic numbers, including foundational definitions, key properties, and interdisciplinary applications, suitable for beginners.

Findings

Proves the Strong Triangle Inequality for $p$-adic spaces

Derives the Product Formula and Ostrowski's Theorem

Discusses applications in quantum mechanics and computer science

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.