p-adic number

English

Noun

1. Template:Lb An element of a completion of the field of rational numbers with respect to a p-adic ultrametric.

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

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Usage notes

• An expanded, constructive definition:
  • For given $p$, the natural numbers are exactly those expressible as some finite sum ${\sum }_{k=0}^n{a}_k{p}^k$, where each $a_k$ is an integer: $0\le a_k<p$ and $n\ge 0$. (To this extent, $p$ acts exactly like a base).
  • The slightly more general sum ${\sum }_{k=N}^n{a}_k{p}^k$ (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 $\mathbb{Q}$) results from permitting infinite sums: ${\sum }_{k=N}^{\mathrm{\infty }}{a}_k{p}^k$.
    • 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 Template:M has limited meaning for p-adic numbers; it is best simply interpreted as Template:M.)
  • 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 $p>1$), but it is only for prime $p$ 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.
• ${\mathbb{Q}}_p$ is not the same as $ℝ$.
  • For example, $\sqrt{p}\notin {\mathbb{Q}}_p$ for any $p$, and, for some values of $p$, $\sqrt{-1}\in {\mathbb{Q}}_p$.

Hyponyms

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  • rational number
    • integer

Related terms

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Translations

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See also

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Further reading

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• p-adic Number on Template:W
• P-adic number on Template:W