bn:03301326n
Noun Concept
Categories: Articles with short description, P-adic numbers, All articles needing additional references, Number theory
EN
P-adic number
EN
In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. Wikipedia
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EN
In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. Wikipedia
A number system for a prime p which extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems Wikidata
An element of a completion of the field of rational numbers with respect to a p-adic ultrametric. Wiktionary
Element of a completion of the rational numbers with respect to a p-adic ultrametric. Wiktionary (translation)
EN
The expansion (21)2121p is equal to the rational p-adic number 2 p + 1 p 2 − 1 . Wiktionary
In the set of 3-adic numbers, the closed ball of radius 1/3 "centered" at 1, call it B , is the set { x | ∃ n ∈ Z . x = 3 n + 1 } . This closed ball partitions into exactly three smaller closed balls of radius 1/9: { x | ∃ n ∈ Z . x = 1 + 9 n } , { x | ∃ n ∈ Z . x = 4 + 9 n } , and { x | ∃ n ∈ Z . x = 7 + 9 n } . 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 | ∃ n ∈ Z . x = 1 + n 3 } , which is one out of three closed balls forming a closed ball of radius 9, and so on. Wiktionary