p-adic numbers

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 1 5 {\displaystyle {\tfrac {1}{5}}} in base 3 vs. the 3-adic expansion, 1 5 = 0.01210121 … ( base 3 ) = 0 ⋅ 3 0 + 0 ⋅ 3 − 1 + 1 ⋅ 3 − 2 + 2 ⋅ 3 − 3 + ⋯ 1 5 = … 121012102 ( 3-adic ) = ⋯ + 2 ⋅ 3 3 + 1 ⋅ 3 2 + 0 ⋅ 3 1 + 2 ⋅ 3 0 . {\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 s = ∑ i = k ∞ a i p i = a k p k + a k + 1 p k + 1 + a k + 2 p k + 2 + ⋯ {\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 a i {\displaystyle a_{i}} is an integer such that 0 ≤ a i < p . {\displaystyle 0\leq a_{i}<p.} A p-adic integer is a p-adic number such that k ≥ 0. {\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 | s | p = p − k , {\displaystyle |s|_{p}=p^{-k},} where k is the least integer i such that a i ≠ 0 {\displaystyle a_{i}\neq 0} (if all a i {\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, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using p-adic numbers.