See p-adic number in All languages combined, or Wiktionary
{
"forms": [
{
"form": "p-adic numbers",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {
"1": "p-adic numbers",
"head": "p-adic number"
},
"expansion": "p-adic number (plural p-adic numbers)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
{
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "English links with redundant target parameters",
"parents": [
"Links with redundant target parameters",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "English links with redundant wikilinks",
"parents": [
"Links with redundant wikilinks",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "English terms containing italics",
"parents": [
"Terms containing italics",
"Terms by orthographic property",
"Terms by lexical property"
],
"source": "w"
},
{
"kind": "other",
"name": "Entries with translation boxes",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with 1 entry",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with DISPLAYTITLE conflicts",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Finnish translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with French translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with German translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Italian translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Mandarin translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Polish translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Romanian translations",
"parents": [],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Number theory",
"orig": "en:Number theory",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
}
],
"examples": [
{
"text": "The expansion (21)2121ₚ is equal to the rational p-adic number #x5C;textstyle#x7B;2p#x2B;1#x5C;overp²-1#x7D;.",
"type": "example"
},
{
"text": "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.\nLikewise, 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."
},
{
"ref": "1914, Bulletin of the American Mathematical Society, page 452:",
"text": "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#x5C;alpha#x5C;omega#x5C;betae#x5C;gamma.",
"type": "quote"
},
{
"ref": "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:",
"text": "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.",
"type": "quote"
},
{
"ref": "2000, Kazuya Kato, Nobushige Kurokawa, Takeshi Saitō, Takeshi Saito, translated by Masato Kuwata, Number Theory: Fermat's dream, American Mathematical Society, page 58:",
"text": "#x5C;Q#x5F;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.\nThe p'''-adic numbers were originally introduced by Hensel around 1900.",
"type": "quote"
}
],
"glosses": [
"An element of a completion of the field of rational numbers with respect to a p-adic ultrametric."
],
"hyponyms": [
{
"sense": "rational number",
"word": "integer"
}
],
"id": "en-p-adic_number-en-noun-o6MOE761",
"links": [
[
"number theory",
"number theory"
],
[
"completion",
"completion"
],
[
"rational number",
"rational number"
],
[
"p-adic ultrametric",
"p-adic ultrametric"
]
],
"raw_glosses": [
"(number theory) An element of a completion of the field of rational numbers with respect to a p-adic ultrametric."
],
"related": [
{
"word": "p-adic"
},
{
"word": "p-adic absolute value"
},
{
"word": "p-adic norm"
},
{
"word": "p-adic integer"
},
{
"word": "p-adic ordinal"
},
{
"word": "p-adic ultrametric"
},
{
"word": "n-adic"
}
],
"topics": [
"mathematics",
"number-theory",
"sciences"
],
"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"word": "p進數"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "pì-jìnshù",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"word": "p进数"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"word": "p-adinen luku"
},
{
"code": "fr",
"lang": "French",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"tags": [
"masculine"
],
"word": "nombre p-adique"
},
{
"code": "de",
"lang": "German",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"tags": [
"feminine"
],
"word": "p-adische Zahl"
},
{
"code": "it",
"lang": "Italian",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"tags": [
"masculine"
],
"word": "numero p-adico"
},
{
"code": "pl",
"lang": "Polish",
"note": "not used in Polish",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric"
},
{
"code": "pl",
"lang": "Polish",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"tags": [
"feminine"
],
"word": "liczba p-adyczna"
},
{
"code": "ro",
"lang": "Romanian",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"tags": [
"neuter"
],
"word": "număr p-adic"
}
],
"wikipedia": [
"p-adic number"
]
}
],
"word": "p-adic number"
}
{
"forms": [
{
"form": "p-adic numbers",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {
"1": "p-adic numbers",
"head": "p-adic number"
},
"expansion": "p-adic number (plural p-adic numbers)",
"name": "en-noun"
}
],
"hyponyms": [
{
"sense": "rational number",
"word": "integer"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "p-adic"
},
{
"word": "p-adic absolute value"
},
{
"word": "p-adic norm"
},
{
"word": "p-adic integer"
},
{
"word": "p-adic ordinal"
},
{
"word": "p-adic ultrametric"
},
{
"word": "n-adic"
}
],
"senses": [
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English lemmas",
"English links with redundant target parameters",
"English links with redundant wikilinks",
"English multiword terms",
"English nouns",
"English terms containing italics",
"English terms with quotations",
"English terms with usage examples",
"Entries with translation boxes",
"Pages with 1 entry",
"Pages with DISPLAYTITLE conflicts",
"Pages with entries",
"Terms with Finnish translations",
"Terms with French translations",
"Terms with German translations",
"Terms with Italian translations",
"Terms with Mandarin translations",
"Terms with Polish translations",
"Terms with Romanian translations",
"en:Number theory"
],
"examples": [
{
"text": "The expansion (21)2121ₚ is equal to the rational p-adic number #x5C;textstyle#x7B;2p#x2B;1#x5C;overp²-1#x7D;.",
"type": "example"
},
{
"text": "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.\nLikewise, 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."
},
{
"ref": "1914, Bulletin of the American Mathematical Society, page 452:",
"text": "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#x5C;alpha#x5C;omega#x5C;betae#x5C;gamma.",
"type": "quote"
},
{
"ref": "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:",
"text": "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.",
"type": "quote"
},
{
"ref": "2000, Kazuya Kato, Nobushige Kurokawa, Takeshi Saitō, Takeshi Saito, translated by Masato Kuwata, Number Theory: Fermat's dream, American Mathematical Society, page 58:",
"text": "#x5C;Q#x5F;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.\nThe p'''-adic numbers were originally introduced by Hensel around 1900.",
"type": "quote"
}
],
"glosses": [
"An element of a completion of the field of rational numbers with respect to a p-adic ultrametric."
],
"links": [
[
"number theory",
"number theory"
],
[
"completion",
"completion"
],
[
"rational number",
"rational number"
],
[
"p-adic ultrametric",
"p-adic ultrametric"
]
],
"raw_glosses": [
"(number theory) An element of a completion of the field of rational numbers with respect to a p-adic ultrametric."
],
"topics": [
"mathematics",
"number-theory",
"sciences"
],
"wikipedia": [
"p-adic number"
]
}
],
"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"word": "p進數"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "pì-jìnshù",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"word": "p进数"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"word": "p-adinen luku"
},
{
"code": "fr",
"lang": "French",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"tags": [
"masculine"
],
"word": "nombre p-adique"
},
{
"code": "de",
"lang": "German",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"tags": [
"feminine"
],
"word": "p-adische Zahl"
},
{
"code": "it",
"lang": "Italian",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"tags": [
"masculine"
],
"word": "numero p-adico"
},
{
"code": "pl",
"lang": "Polish",
"note": "not used in Polish",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric"
},
{
"code": "pl",
"lang": "Polish",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"tags": [
"feminine"
],
"word": "liczba p-adyczna"
},
{
"code": "ro",
"lang": "Romanian",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"tags": [
"neuter"
],
"word": "număr p-adic"
}
],
"word": "p-adic number"
}
Download raw JSONL data for p-adic number meaning in English (5.7kB)
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-12-21 from the enwiktionary dump dated 2024-12-04 using wiktextract (d8cb2f3 and 4e554ae). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.