In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers that 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 versus the 3-adic expansion,
1 5
= 0. 0121 ¯ ( base 3 )
= 0 ⋅ 3 0 + 0 ⋅ 3 − 1 + 1 ⋅ 3 − 2 + 2 ⋅ 3 − 3 + ⋯ 1 5
= 1210 ¯ 2 ( 3-adic )
= ⋯ + 2 ⋅ 3 3 + 1 ⋅ 3 2 + 0 ⋅ 3 1 + 2 ⋅ 3 0 . {\displaystyle {\begin{alignedat}{3}{\tfrac {1}{5}}&{}=0.{\overline {0121}}\ ({\text{base }}3)&&{}=0\cdot 3^{0}+0\cdot 3^{-1}+1\cdot 3^{-2}+2\cdot 3^{-3}+\cdots \\[5mu]{\tfrac {1}{5}}&{}={\overline {1210}}2\ \ ({\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.}
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