p-adic number

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p-adic number

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Noun

p-adic number (plural p-adic numbers)

(number theory) An element of a completion of the field of rational numbers with respect to a p-adic ultrametric.^[1]
The expansion (21)2121_p is equal to the rational p-adic number $\textstyle {2p+1 \over p^{2}-1}.$

In the set of 3-adic numbers, the closed ball of radius 1/3 "centered" at 1, call it B, is the set $\textstyle \{x|\exists n\in \mathbb {Z} .\,x=3n+1\}.$ This closed ball partitions into exactly three smaller closed balls of radius 1/9: $\{x|\exists n\in \mathbb {Z} .\,x=1+9n\},$ $\{x|\exists n\in \mathbb {Z} .\,x=4+9n\},$ and $\{x|\exists n\in \mathbb {Z} .\,x=7+9n\}.$ 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, $\{x|\exists n\in \mathbb {Z} .\,x=1+{n \over 3}\},$ which is one out of three closed balls forming a closed ball of radius 9, and so on.
- 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 $p^{\alpha }\omega ^{\beta }e^{\gamma }$ .
- 1991, M. D. Missarov, “Renormalization Group and Renormalization Theory in p-Adic and Adelic Scalar Models”, in 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, Kazuya Kato, Nobushige Kurokawa, Takeshi Saitō, Takeshi Saito, translated by Masato Kuwata, Number Theory: Fermat's dream, American Mathematical Society, page 58:
  $\mathbb {Q} _{p}$ 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

An expanded, constructive definition:
- For given $p$ , the natural numbers are exactly those expressible as some finite sum $\textstyle \sum _{k=0}^{n}a_{k}p^{k}$ , where each $a_{k}$ is an integer: $0\leq a_{k}<p$ and $n\geq 0$ . (To this extent, $p$ acts exactly like a base).
- The slightly more general sum $\textstyle \sum _{k=N}^{n}a_{k}p^{k}$ (where $N$ can be negative) expresses a class of fractions: natural numbers divided by a power of $p$ .
- Much more expressiveness (to encompass all of $\mathbb {Q}$ $\mathbb {Q}$ ) results from permitting infinite sums: $\textstyle \sum _{k=N}^{\infty }a_{k}p^{k}$ $\textstyle \sum _{k=N}^{\infty }a_{k}p^{k}$ .
  - 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 $\mathbb {Q}$ with respect to the ultrametric means augmenting it with the limit points of all such infinite sums.
The augmented set is denoted $\mathbb {Q} _{p}$ .
The construction works generally (for any integer $p>1$ $p>1$ ), but it is only for prime $p$ $p$ that it becomes of significant mathematical interest.
- For $p$ the power of some prime number, $\mathbb {Q} _{p}$ is still a field. For other composite $p$ , $\mathbb {Q} _{p}$ is a ring, but not a field.
$\mathbb {Q} _{p}$ $\mathbb {Q} _{p}$ is not the same as $\mathbb {R}$ $\mathbb {R}$ .
- For example, ${\sqrt {p}}\notin \mathbb {Q} _{p}$ for any $p$ , and, for some values of $p$ , ${\sqrt {-1}}\in \mathbb {Q} _{p}$ .