p-2

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




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.

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