P-adic Numbers

Full Title or Meme

One of number of alternates to the Rational Numbers, which have been used as the basis for metrics used in Dimensions in Physics here-to-fore.

A p-adic number is an extension of the field of rationals such that congruences modulo powers of a fixed prime p are related to proximity in the so called "p-adic metric." https://mathworld.wolfram.com/p-adicNumber.html

Number Theory and Algebraic Geometry:
1. P-adic numbers are essential in number theory, especially in the study of Diophantine equations and congruences.
2. They provide a natural framework for understanding algebraic varieties over finite fields.
Analysis and Topology:
1. P-adic analysis is an alternative to real analysis. It defines a different metric space based on p-adic norms.
2. The p-adic numbers form a complete metric space, which is useful for solving certain types of equations.
3. Ultrametric spaces (such as p-adic spaces) have applications in functional analysis and harmonic analysis.
Representation Theory:
1. P-adic groups (such as p-adic Lie groups) play a crucial role in representation theory.
2. They are used to study representations of algebraic groups and their associated Lie algebras.
Number Fields and Class Field Theory:
1. P-adic fields are important in the study of number fields and their extensions.
2. Class field theory relates p-adic fields to abelian extensions of number fields.
Cryptography:
1. P-adic numbers have been explored for cryptographic purposes due to their unique properties.
2. Some cryptographic protocols use p-adic arithmetic to enhance security.
Physics and Quantum Mechanics:
1. P-adic numbers have been proposed as a way to model quantum mechanics.
2. They provide an alternative approach to understanding wave functions and probabilities.

See also the wiki page on Dimensions in Physics
See also the wiki page on Complex Dimensions for another example of an alternate to Rational Numbers.