p-adic number

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English[edit]

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Noun[edit]

p-adic number (plural p-adic numbers)

  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 .
    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.

Usage notes[edit]

  • 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.

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