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An element of a completion of the field of rational numbers with respect to a p-adic ultrametric.
Definition source: English Wiktionary via Wiktextract
Topics: mathematics, number-theory, sciences
Examples
- The expansion (21)2121p is equal to the rational p-adic number #92;textstyle#123;2p#43;1#92;overp2-1#125;.
- 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=3n+1. This closed ball partitions into exactly three smaller closed balls of radius 1/9: x|∃n∈Z.,x=1+9n, x|∃n∈Z.,x=4+9n, and x|∃n∈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|∃n∈Z.,x=1+n/3, which is one out of three closed balls forming a closed ball of radius 9, and so on.
Meaning relationships
Synonyms: none provided
Antonyms: none provided
No etymology was provided for this word.
Across languages
Translations
8 source translations are retained for this English entry.
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Bulgarian:
p-адично́ число́
(p-adičnó čisló)
— element of a completion of the rational numbers with respect to a p-adic ultrametric
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Chinese Mandarin:
p進數 /p进数
(pì-jìnshù)
— element of a completion of the rational numbers with respect to a p-adic ultrametric
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Finnish:
p-adinen luku
— element of a completion of the rational numbers with respect to a p-adic ultrametric
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French:
nombre p-adique
— element of a completion of the rational numbers with respect to a p-adic ultrametric
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German:
p-adische Zahl
— element of a completion of the rational numbers with respect to a p-adic ultrametric
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Italian:
numero p-adico
— element of a completion of the rational numbers with respect to a p-adic ultrametric
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Polish:
liczba p-adyczna
— element of a completion of the rational numbers with respect to a p-adic ultrametric
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Romanian:
număr p-adic
— element of a completion of the rational numbers with respect to a p-adic ultrametric
