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number > number theory > p-adic number

Preferred term

p-adic number

Definition(s)

In number theory, given a prime number $p$ , the $p$ -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; $p$ -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number $p$ rather than ten, and extending (possibly infinitely) to the left rather than to the right. Formally, given a prime number $p$ , a $p$ -adic number can be defined as a series

${\\displaystyle s=\\sum _{i=k}^{\\infty }a_{i}p^{i}=a_{k}p^{k}+a_{k+1}p^{k+1}+a_{k+2}p^{k+2}+\\cdots }$

where $k$ is an integer (possibly negative), and each ${\\displaystyle a_{i}}$ is a integer such that ${\\displaystyle 0\\leq a_{i}<p.}$ A $p$ -adic integer is a $p$ -adic number such that ${\\displaystyle k\\geq 0.}$
In general the series that represents a $p$ -adic number is not convergent in the usual sense, but it is convergent for the $p$ -adic absolute value ${\\displaystyle |s|_{p}=p^{-k},}$ where $k$ is the least integer $i$ such that ${\\displaystyle a_{i}\ eq 0}$ (if all ${\\displaystyle a_{i}}$ are zero, one has the zero $p$ -adic number, which has $0$ as its $p$ -adic absolute value).
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/P-adic_number)

Broader concept(s)

number theory

Narrower concept(s)

automorphic number

Related concept(s)

In other languages

nombre p-adique

French

URI

http://data.loterre.fr/ark:/67375/PSR-WG63ZQK3-T

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