View source for 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.

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

==References==
* 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.


[[Category: Mathematics]]