See p-adic number in All languages combined, or Wiktionary
Download JSON data for p-adic number meaning in English (5.9kB)
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{
"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": "quotation"
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"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": "quotation"
},
{
"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.",
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"An element of a completion of the field of rational numbers with respect to a p-adic ultrametric."
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"(number theory) An element of a completion of the field of rational numbers with respect to a p-adic ultrametric."
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"word": "p-adic absolute value"
},
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"word": "p-adic norm"
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{
"word": "p-adic integer"
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{
"word": "p-adic ordinal"
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{
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"word": "n-adic"
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"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",
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"note": "not used in Polish",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric"
},
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"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
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"word": "liczba p-adyczna"
},
{
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"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
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"word": "număr p-adic"
}
],
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"word": "p-adic number"
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{
"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": "quotation"
},
{
"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": "quotation"
},
{
"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": "quotation"
}
],
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"An element of a completion of the field of rational numbers with respect to a p-adic ultrametric."
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"(number theory) An element of a completion of the field of rational numbers with respect to a p-adic ultrametric."
],
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"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"
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"word": "liczba p-adyczna"
},
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"code": "ro",
"lang": "Romanian",
"sense": "element of a completion of the rational numbers with respect to a p-adic ultrametric",
"tags": [
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"word": "număr p-adic"
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"word": "p-adic number"
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This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-05 from the enwiktionary dump dated 2024-05-02 using wiktextract (f4fd8c9 and c9440ce). 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.
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