p displaystyle p adichne chislo v matematici ye popovnennyam polya racionalnih chisel vidminnim vid dijsnih chisel Popovnennya vidbuvayetsya ne shodo zvichajnoyi evklidovoyi normi yak u vipadku dijsnih chisel a shodo tak zvanoyi p displaystyle p adichnoyi normi p displaystyle p adichni chisla osoblivo shiroko zastosovuyutsya v teoriyi chisel Zmist 1 Elementarne oznachennya 2 Analitichna pobudova 2 1 p adichna norma 2 2 Fundamentalni poslidovnosti i nul poslidovnosti 2 3 Pobudova chisel 3 Vlastivosti 4 LiteraturaElementarne oznachennya RedaguvatiNehaj p displaystyle p nbsp deyake proste chislo Todi yak vidomo kozhne cile chislo mozhe buti zapisano i 0 n a i p i displaystyle sum i 0 n a i p i nbsp de chisla a i displaystyle a i nbsp nalezhat do mnozhini 0 1 p 1 displaystyle 0 1 dots p 1 nbsp Zagalnovidomim ye rozshirennya danih chisel do mnozhini dijsnih chisel kozhne z yakih mozhe buti zapisane tak i n a i p i displaystyle pm sum i infty n a i p i nbsp de n displaystyle n nbsp deyake cile chislo p displaystyle p nbsp adichni chisla natomist mozhut buti zapisani u vidi i k a i p i displaystyle sum i k infty a i p i nbsp de k displaystyle k nbsp deyake cile chislo Napriklad vzyavshi p 5 displaystyle p 5 nbsp mi matimemo 1 444444444 5 displaystyle 1 dots 444444444 5 nbsp 1 3 131313132 5 displaystyle frac 1 3 dots 131313132 5 nbsp Obchislennya vidbuvayutsya za zvichajnimi pravilami dlya chisel z osnovoyu 5 displaystyle 5 nbsp Chisla dlya yakih a i 0 displaystyle a i 0 nbsp dlya i lt 0 displaystyle i lt 0 nbsp nazivayutsya p displaystyle p nbsp adichnimi cilimi chislami Analitichna pobudova Redaguvatip adichna norma Redaguvati Nehaj mayemo deyake x Z displaystyle x in mathbb Z nbsp cile chislo Nazvemo ordinalom cogo chisla po vidnoshennyu shodo deyakogo prostogo p displaystyle p nbsp ord p x max r p r x displaystyle operatorname ord p x max r colon p r x nbsp Dali dlya a b Q displaystyle frac a b in mathbb Q nbsp viznachimo ord p a b ord p a ord p b displaystyle operatorname ord p frac a b operatorname ord p a operatorname ord p b nbsp Ekvivalentno yaksho x p n a b displaystyle x p n frac a b nbsp de a displaystyle a nbsp b displaystyle b nbsp ne dilyatsya na p displaystyle p nbsp to ord p x n displaystyle operatorname ord p x n nbsp Vvazhatimemo takozh sho ordinal nulya rivnij bezmezhnosti Viznachimo p displaystyle p nbsp adichnu normu dlya x Z displaystyle x in mathbb Z nbsp takim chinom x p p ord p x a 0 p a 0 displaystyle x p begin cases p operatorname ord p x amp mbox a neq 0 p infty amp mbox a 0 end cases nbsp Viznachena podibnim chinom funkciya spravdi ye normoyu oskilki x p 0 displaystyle x p 0 nbsp todi j lishe todi koli x 0 displaystyle x 0 nbsp Spravdi 0 displaystyle 0 nbsp yedine chislo ordinal yakogo rivnij neskinchennosti i vidpovidno yedine dlya yakogo vikonuyetsya dana rivnist x y p x p y p displaystyle xy p x p y p nbsp Spravdi nehaj x p n a b displaystyle x p n frac a b nbsp a y p n c d displaystyle y p n frac c d nbsp de zhodne z chisel a displaystyle a nbsp b displaystyle b nbsp c displaystyle c nbsp d displaystyle d nbsp ne dilitsya na p Todi x y p n m a c b d displaystyle xy p n m frac ac bd nbsp i a c displaystyle ac nbsp b d displaystyle bd nbsp ne dilyatsya na p displaystyle p nbsp Za oznachennyami mayemo x p 1 p n displaystyle x p frac 1 p n nbsp y p 1 p m displaystyle y p frac 1 p m nbsp x y p 1 p n m displaystyle xy p frac 1 p n m nbsp sho j dovodit nashe tverdzhennya x y p max x p y p displaystyle x y p leq max x p y p nbsp Nehaj znovu x p n a b displaystyle x p n frac a b nbsp a y p n c d displaystyle y p n frac c d nbsp de zhodne z chisel a displaystyle a nbsp b displaystyle b nbsp c displaystyle c nbsp d displaystyle d nbsp ne dilitsya na p displaystyle p nbsp Nehaj takozh n m displaystyle n leq m nbsp Todi x y p p n a d p m n b c b d displaystyle x y p p n left frac ad p m n bc bd right nbsp Tozh ochevidno ordinal x y displaystyle x y nbsp ne mozhe buti menshim n displaystyle n nbsp Okrim togo u vipadku koli n displaystyle n nbsp strogo menshe m displaystyle m nbsp ordinal ye rivnim n displaystyle n nbsp adzhe v takomu vipadku chiselnik i znamennik u rozpisi sumi ochevidno ne dilyatsya na p displaystyle p nbsp Takim chinom p displaystyle cdot p nbsp ye nearhimedovoyu normoyu na poli racionalnih chisel Napriklad dlya chisla x 63 550 2 1 3 2 5 2 7 11 1 displaystyle x 63 550 2 1 cdot 3 2 cdot 5 2 cdot 7 cdot 11 1 nbsp x 2 2 displaystyle displaystyle x 2 2 nbsp x 3 1 9 displaystyle displaystyle x 3 1 9 nbsp x 5 25 displaystyle x 5 25 nbsp x 7 1 7 displaystyle displaystyle x 7 1 7 nbsp x 11 11 displaystyle x 11 11 nbsp x p 1 displaystyle x p 1 nbsp dlya inshih prostih chisel Fundamentalni poslidovnosti i nul poslidovnosti Redaguvati Poslidovnist a i displaystyle a i nbsp nazivayetsya zbizhnoyu do a Q displaystyle a in mathbb Q nbsp za normoyu p displaystyle cdot p nbsp yaksho lim n a i a p 0 displaystyle lim n to infty a i a p 0 nbsp Yaksho a 0 displaystyle a 0 nbsp to taka poslidovnist nazivayetsya nul poslidovnistyu Poslidovnist a i displaystyle a i nbsp nazivayetsya fundamentalnoyu yaksho e gt 0 M Z displaystyle forall varepsilon gt 0 exists M in mathbb Z nbsp take sho m n gt M a m a n p lt e displaystyle m n gt M Rightarrow a m a n p lt varepsilon nbsp Iz zbizhnosti poslidovnosti viplivaye yiyi fundamentalnist Zvorotne tverdzhennya u mnozhini racionalnih chisel ye nevirnim Pobudova chisel Redaguvati Vvedemo na mnozhini fundamentalnih poslidovnostej racionalnih chisel shodo p adichnoyi normi vidnoshennya ekvivalentnosti fundamentalni poslidovnosti a i displaystyle a i nbsp i b i displaystyle b i nbsp ye ekvivalentni todi j lishe todi koli yih riznicya ye nul poslidovnisyu Poznachatimemo klas ekvivalentnosti poslidovnosti a i displaystyle a i nbsp cherez a i displaystyle a i nbsp Na mnozhini klasiv ekvivalentnosti viznachimo arifmetichni operaciyi a n b n a n b n displaystyle a n b n a n b n nbsp a n b n a n b n displaystyle a n b n a n b n nbsp Dani oznachennya ye nesuperechlivimi oskilki suma dvoh nul poslidovnostej ye nul poslidovnistyu i dobutok fundamentalnoyi poslidovnosti na nul poslidovnist ye nul poslidovnistyu Viznachimo takozh zagalnu p displaystyle p nbsp adichnu normu a i p lim n a i p displaystyle a i p lim n to infty a i p nbsp Takim chinom skonstrujovano pole sho ye povnim vidnosno p adichnoyi normi Vono i nazivayetsya polem p displaystyle p nbsp adichnih chisel Racionalni chisla ye shilnim pidpolem danogo polya Chisla x dlya yakih x p 1 displaystyle x p leq 1 nbsp nazivayutsya p adichnimi cilimi chislami Vlastivosti RedaguvatiKozhne p adichne chislo mozhna yedinim chinom podati u vidi i k a i p i displaystyle sum i k infty a i p i nbsp Cim dani chisla vidriznyayutsya vid dijsnih dlya yakih mozhe buti kilka variantiv zapisu cherez sumu stepeniv Napriklad 1 0 999999999 displaystyle 1 0 999999999 dots nbsp Suma i k a i displaystyle sum i k infty a i nbsp p displaystyle p nbsp adichnih chisel zbizhna todi j lishe todi koli a i displaystyle a i nbsp ye nul poslidovnistyu Topologichnij prostir p displaystyle p nbsp adichnih cilih chisel z metrichnoyu topologiyeyu gomeomorfnij mnozhini Kantora a prostir p displaystyle p nbsp adichnih chisel z metrichnoyu topologiyeyu gomeomorfnij mnozhini Kantora z virizanoyu tochkoyu Literatura RedaguvatiBorevich Z I Shafarevich I R Teoriya chisel M Nauka 1985 Koblic N r adicheskie chisla r adicheskij analiz i dzeta funkcii M Mir 1982 Serr Zh P Kurs arifmetiki M Mir 1972 Otrimano z https uk wikipedia org w index php title P adichne chislo amp oldid 39993209
