# Wiktionaryβ

## EnglishEdit

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### NounEdit

1. (mathematics) An element of a completion of the field of rational numbers which has a p-adic ultrametric as its metric.[1]
The expansion (21)2121p is equal to the rational p-adic number ${\displaystyle {2p+1 \over p^{2}-1}}$ .
In the set of 3-adic numbers, the closed ball of radius 1/3 "centered" at 1, call it B, is the set ${\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=3n+1\}}$ . This closed ball partitions into exactly three smaller closed balls of radius 1/9: ${\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=1+9n\}}$ , ${\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=4+9n\}}$ , and ${\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=7+9n\}}$ . 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, ${\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=1+{n \over 3}\}}$ , which is one out of three closed balls forming a closed ball of radius 9, and so on.

#### Usage notesEdit

• The 'p' in "p-adic" is a parameter which stands for a positive integer, preferably a prime number.
• For a fixed prime value of p, a p-adic number is a member of the field ${\displaystyle \mathbb {Q} _{p}}$  which is a completion of the set of rational numbers.
• For a composite value of p, a p-adic number is a member of a ring which is an extension of the field of rational numbers.