Noun
p-adic number (plural p-adic numbers)
(number theory) An element of a completion of the field of rational numbers with respect to a p-adic ultrametric.
The expansion (21)2121p is equal to the rational p-adic number
2
p
+
1
p
2
−
1
.
{\displaystyle \textstyle {2p+1 \over p^{2}-1}.}
In the set of 3-adic numbers, the closed ball of radius 1/3 "centered" at 1, call it B, is the set
{
x
|
∃
n
∈
Z
.
x
=
3
n
+
1
}
.
{\displaystyle \textstyle \{x|\exists n\in \mathbb {Z} .\,x=3n+1\}.}
This closed ball partitions into exactly three smaller closed balls of radius 1/9:
{
x
|
∃
n
∈
Z
.
x
=
1
+
9
n
}
,
{\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=1+9n\},}
{
x
|
∃
n
∈
Z
.
x
=
4
+
9
n
}
,
{\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=4+9n\},}
and
{
x
|
∃
n
∈
Z
.
x
=
7
+
9
n
}
.
{\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=7+9n\}.}
Then each of those balls partitions into exactly 3 smaller closed balls of radius 1/27, and the sub-partitioning can be continued indefinitely, in a fractal manner.Likewise, going upwards in the hierarchy, B is part of the closed ball of radius 1 centered at 1, namely, the set of integers. Two other closed balls of radius 1 are "centered" at 1/3 and 2/3, and all three closed balls of radius 1 form a closed ball of radius 3,
{
x
|
∃
n
∈
Z
.
x
=
1
+
n
3
}
,
{\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=1+{n \over 3}\},}
which is one out of three closed balls forming a closed ball of radius 9, and so on.