p-adic number

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p-adic number (plural p-adic numbers)

  1. (number theory) An element of a completion of the field of rational numbers with respect to a p-adic ultrametric.[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.
    • 1914, Bulletin of the American Mathematical Society, page 452,
      3. In his recent book Professor Hensel has developed a theory of logarithms of the rational p-adic numbers, and from this he has shown how all such numbers can be written in the form .
    • 1991, M. D. Missarov, Renormalization Group and Renormalization Theory in p-Adic and Adelic Scalar Models, Ya. G. Sinaĭ (editor), Dynamical Systems and Statistical Mechanics: From the Seminar on Statistical Physics held at Moscow State University, American Mathematical Society, page 143,
      p-Adic numbers were introduced in mathematics by K. Hensel, and this invention led to substantial developments in number theory, where p-adic numbers are now as natural as ordinary real numbers. [] Bleher noticed in [19] that the set of purely fractional p-adic numbers is an example of hierarchical lattice.
    • 2000, Masato Kuwata (translator), Kazuya Kato, Nobushige Kurokawa, Takeshi Saitō, Takeshi Saito, Number Theory: Fermat's dream, American Mathematical Society, page 58,
      is called the p-adic number field, and its elements are called p-adic numbers. In this section we introduce the p-adic number fields, which are very important objects in number theory.
      The p-adic numbers were originally introduced by Hensel around 1900.

Usage notes[edit]

  • An expanded, constructive definition:
    • For given , the natural numbers are exactly those expressible as some finite sum , where each is an integer: and . (To this extent, acts exactly like a base).
    • The slightly more general sum (where can be negative) expresses a class of fractions: natural numbers divided by a power of .
    • Much more expressiveness (to encompass all of ) results from permitting infinite sums: .
      • The p-adic ultrametric and the limitation on coefficients together ensure convergence, meaning that infinite sums can be manipulated to produce valid results that at times seem paradoxical. (For example, a sum with positive coefficients can represent a negative rational number. In fact, the concept negative has limited meaning for p-adic numbers; it is best simply interpreted as additive inverse.)
    • Forming the completion of with respect to the ultrametric means augmenting it with the limit points of all such infinite sums.
  • The augmented set is denoted .
  • The construction works generally (for any integer ), but it is only for prime that it becomes of significant mathematical interest.
    • For the power of some prime number, is still a field. For other composite , is a ring, but not a field.
  • is not the same as .
    • For example, for any , and, for some values of , .


  • (element of a completion of the rational numbers with respect to a p-adic ultrametric):

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