P-adic numbers

P-adic numbers are a system of numbers introduced by the mathematician Kurt Hensel in 1897, which extends the concept of the usual rational numbers. They are constructed in a way that allows for a different notion of "closeness" between numbers, based on a chosen prime number \( p \). The core idea of p-adic numbers is to define a distance between numbers that is based on divisibility by a prime \( p \).

The \( p \)-adic exponential function is an important concept in \( p \)-adic analysis, which is a branch of mathematics that deals with \( p \)-adic numbers. The \( p \)-adic numbers are a system of numbers that extend the rational numbers and provide a different perspective on number theory and algebra.

P-adic modular forms are a generalization of classical modular forms, extending the notion into the realm of p-adic analysis. They arise in number theory and algebraic geometry and are particularly important for studying p-adic representations and for the Langlands program. ### Definition and Context 1.

P-adic quantum mechanics is an approach to quantum mechanics that is based on p-adic numbers instead of the usual real or complex numbers. P-adic numbers are a system of numbers used in number theory, defined with respect to a prime number \( p \). Unlike real and complex numbers, which extend indefinitely in both directions, p-adic numbers allow for expansions that are focused around a prime base, leading to a different structure that can have unique properties.