## p-adic-number

Noun

(*plural* p-adic numbers)

- (mathematics) An element of a completion of the field of rational numbers which has a
*p*-adic ultrametric as its metric.^{ }- The expansion (21)2121
_{p}is equal to the rational*p*-adic number . - In the set of
**3-adic numbers**, the closed ball of radius 1/3 "centered" at 1, call it*B*, is the set . This closed ball partitions into exactly three smaller closed balls of radius 1/9: , and . 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, which is one out of three closed balls forming a closed ball of radius 9, and so on.

- The expansion (21)2121

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 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.

*p*-adic absolute value,*p*-adic norm*p*-adic ordinal*p*-adic ultrametric