## English

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### Noun

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 ${2 p + 1 \over p^2 - 1}$.
In the set of 3-adic numbers, the closed ball of radius 1/3 "centered" at 1, call it B, is the set $\{x | \exists n \in \mathbb{Z} . \, x = 3 n + 1 \}$. This closed ball partitions into exactly three smaller closed balls of radius 1/9: $\{x | \exists n \in \mathbb{Z} . \, x = 1 + 9 n \}$, $\{x | \exists n \in \mathbb{Z} . \, x = 4 + 9 n \}$, and $\{x | \exists n \in \mathbb{Z} . \, x = 7 + 9 n \}$. 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, $\{x| \exists n \in \mathbb{Z} . \, x = 1 + {n\over 3} \}$, which is one out of three closed balls forming a closed ball of radius 9, and so on.

#### Usage notes

• 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 $\mathbb{Q}_p$ 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.