P-adic number

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 to the left rather than to the right.

For example, comparing the expansion of the rational number ${\displaystyle {\tfrac {1}{5}}}$ in base 3 vs. the 3-adic expansion,

${\displaystyle {\begin{alignedat}{3}{\tfrac {1}{5}}&{}=0.01210121\ldots \ ({\text{base }}3)&&{}=0\cdot 3^{0}+0\cdot 3^{-1}+1\cdot 3^{-2}+2\cdot 3^{-3}+\cdots \\[5mu]{\tfrac {1}{5}}&{}=\dots 121012102\ \ ({\text{3-adic}})&&{}=\cdots +2\cdot 3^{3}+1\cdot 3^{2}+0\cdot 3^{1}+2\cdot 3^{0}.\end{alignedat}}}$

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 an 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 ${\displaystyle a_{i}\neq 0}$ (if all ${\displaystyle a_{i}}$ are zero, one has the zero p-adic number, which has 0 as its p-adic absolute value).

Every rational number can be uniquely expressed as the sum of a series as above, with respect to the p-adic absolute value. This allows considering rational numbers as special p-adic numbers, and alternatively defining the p-adic numbers as the completion of the rational numbers for the p-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.

p-adic numbers were first described by Kurt Hensel in 1897,^[1] though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using p-adic numbers.^{[note 1]}

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