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Noun Concept
Categories: P-adic numbers, Articles with short description, Field theory, Number theory
EN
p-adic number  P-adic Division Algorithm  ...999  ...9999  10-adic
EN
In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to decimals, but with digits based on a prime number p rather than ten, and extending to the left rather than to the right. Wikipedia
Definitions
Examples
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EN
In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to decimals, but with digits based on a prime number p rather than ten, and extending to the left rather than to the right. 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
IS A
number
element
HAS KIND
rational number
p-adic integer
DIFFERENT FROM
modulo
p-adic integer
HAS QUALITY
p-adic valuation
p-adic absolute value
STUDIED BY
p-adic analysis
Wikipedia
EN
p-adic number
Wikidata
EN
p-adic number
Wiktionary
EN
p-adic number
Wikipedia Redirections
EN
...999, ...9999, 10-adic, 2-adic integers, 3-adic, 5-adic, Adic number, Dyadic integers, Dyadic number, Field of p-adic numbers, l-adic, L-adic integers, L-adic number, P-addic number, p-adic, p-adic absolute value, P-adic digit, P-adic division algorithm, P-adic Division Algorithm, p-adic field, p-adic group, P-adic groups, p-adic integer, p-adic integers, p-adic methods, p-adic metric, P-adic Number, p-adic numbers, P-adics, P-ness, Padic number, Padic numbers, Quote notation, ring of p-adic integers
Wikidata Alias
EN
p-adic rational number

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