Lancyugo vij drib abo neperervnij drib ce matematichnij viraz vidu a 0 a 1 a 2 a 3 a 0 1 a 1 1 a 2 1 a 3 displaystyle a 0 a 1 a 2 a 3 cdots a 0 cfrac 1 a 1 cfrac 1 a 2 cfrac 1 a 3 ldots de a0 ye cile chislo a vsi inshi an ye naturalnimi chislami Uzagalnenimi lancyugovimi drobami nazivayut virazi vidu a 0 b 1 a 1 b 2 a 2 b 3 a 3 displaystyle a 0 cfrac b 1 a 1 cfrac b 2 a 2 frac b 3 a 3 dots Bud yake dijsne chislo mozhe buti predstavlene lancyugovim drobom Chislo predstavlyayetsya skinchennim lancyugovim drobom todi j lishe todi koli vono racionalne Zmist 1 Rozklad v lancyugovij drib 2 Prikladi rozkladu 3 Vlastivosti 4 Zastosuvannya 5 Div takozh 6 DzherelaRozklad v lancyugovij drib RedaguvatiBud yake dijsne chislo x displaystyle x nbsp mozhe buti predstavlene lancyugovim drobom a 0 a 1 a 2 a 3 displaystyle a 0 a 1 a 2 a 3 cdots nbsp de a 0 x x 0 x a 0 displaystyle a 0 lfloor x rfloor x 0 x a 0 nbsp a 1 1 x 0 x 1 1 x 0 a 1 displaystyle a 1 left lfloor frac 1 x 0 right rfloor x 1 frac 1 x 0 a 1 nbsp displaystyle dots nbsp a n 1 x n 1 x n 1 x n 1 a n displaystyle a n left lfloor frac 1 x n 1 right rfloor x n frac 1 x n 1 a n nbsp displaystyle dots nbsp de x displaystyle lfloor x rfloor nbsp poznachaye cilu chastinu chisla x displaystyle x nbsp Dlya racionalnogo chisla x displaystyle x nbsp cej rozklad zavershitsya pislya oderzhannya nulovogo x n displaystyle x n nbsp dlya deyakogo n V comu vipadku x displaystyle x nbsp predstavlyayetsya skinchennim lancyugovim drobom x a 0 a 1 a n displaystyle x a 0 a 1 cdots a n nbsp Dlya irracionalnogo x displaystyle x nbsp vsi velichini x n displaystyle x n nbsp budut nenulovimi i proces rozkladu mozhna prodovzhuvati neskinchenno Priklad obchislennya lancyugovogo drobu dlya chisla 3 245 podano v tablici Obchislennya lancyugovogo drobu dlya chisla 3 2453 displaystyle 3 nbsp 3 245 3 49 200 3 displaystyle 3 245 left 3 tfrac 49 200 right 3 nbsp 0 245 49 200 displaystyle 0 245 left tfrac 49 200 right nbsp 1 0 245 200 49 displaystyle 1 0 245 left tfrac 200 49 right nbsp 4 082 4 4 49 displaystyle 4 082 left 4 tfrac 4 49 right nbsp 4 displaystyle 4 nbsp 4 082 4 4 49 4 displaystyle 4 082 left 4 tfrac 4 49 right 4 nbsp 0 082 4 49 displaystyle 0 082 left tfrac 4 49 right nbsp 1 0 082 49 4 displaystyle 1 0 082 left tfrac 49 4 right nbsp 12 250 12 1 4 displaystyle 12 250 left 12 tfrac 1 4 right nbsp 12 displaystyle 12 nbsp 12 250 12 1 4 12 displaystyle 12 250 left 12 tfrac 1 4 right 12 nbsp 0 250 1 4 displaystyle 0 250 left tfrac 1 4 right nbsp 1 0 250 4 1 displaystyle 1 0 250 left tfrac 4 1 right nbsp 4 000 displaystyle 4 000 nbsp 4 displaystyle 4 nbsp 4 000 4 displaystyle 4 000 4 nbsp 0 000 displaystyle 0 000 nbsp STOPlancyugovij drib dlya chisla 3 245 rivnij 3 4 12 4 3 245 3 1 4 1 12 1 4 displaystyle 3 245 3 cfrac 1 4 cfrac 1 12 cfrac 1 4 nbsp Prikladi rozkladu Redaguvatip 3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2 1 84 displaystyle pi 3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2 1 84 cdots nbsp yaksho prote vikoristovuvati uzagalneni lancyugovi drobi to otrimayemo pevni zakonomirnosti p 3 1 2 6 3 2 6 5 2 6 7 2 6 9 2 6 11 2 6 13 2 6 15 2 6 4 1 1 2 3 2 2 5 3 2 7 4 2 9 5 2 11 6 2 13 7 2 15 displaystyle pi 3 cfrac 1 2 6 cfrac 3 2 6 cfrac 5 2 6 cfrac 7 2 6 cfrac 9 2 6 cfrac 11 2 6 cfrac 13 2 6 cfrac 15 2 6 ddots cfrac 4 1 cfrac 1 2 3 cfrac 2 2 5 cfrac 3 2 7 cfrac 4 2 9 cfrac 5 2 11 cfrac 6 2 13 cfrac 7 2 15 ddots nbsp e exp 1 2 1 2 1 1 4 1 1 6 1 1 8 1 1 10 1 1 12 1 1 displaystyle e exp 1 2 1 2 1 1 4 1 1 6 1 1 8 1 1 10 1 1 12 1 1 dots nbsp Yakshon cile chislo bilshe odinici exp 1 n 1 n 1 1 1 3 n 1 1 1 5 n 1 1 1 7 n 1 1 1 displaystyle exp 1 n 1 n 1 1 1 3n 1 1 1 5n 1 1 1 7n 1 1 1 dots nbsp Yaksho takozh n parne exp 2 n 1 n 1 2 6 n 5 n 1 2 1 1 3 n k n 1 2 6 n 2 k 1 3 n k 5 n 1 2 1 1 displaystyle exp 2 n 1 n 1 2 6n 5n 1 2 1 1 dots 3nk n 1 2 6n 2k 1 3nk 5n 1 2 1 1 dots nbsp pri n 1 e 2 exp 2 7 2 1 1 3 18 5 1 1 6 30 8 1 1 9 42 11 1 1 3 k 12 k 6 3 k 2 1 1 displaystyle e 2 exp 2 7 2 1 1 3 18 5 1 1 6 30 8 1 1 9 42 11 1 1 dots 3k 12k 6 3k 2 1 1 dots nbsp tanh 1 n 0 n 3 n 5 n 7 n 9 n 11 n 13 n 15 n 17 n 19 n displaystyle tanh 1 n 0 n 3n 5n 7n 9n 11n 13n 15n 17n 19n dots nbsp yaksho n dodatne chislo takozh tan 1 1 1 1 3 1 5 1 7 1 9 1 11 1 13 1 15 1 displaystyle tan 1 1 1 1 3 1 5 1 7 1 9 1 11 1 13 1 15 1 dots nbsp yaksho n gt 1 tan 1 n 0 n 1 1 3 n 2 1 5 n 2 1 7 n 2 1 displaystyle tan 1 n 0 n 1 1 3n 2 1 5n 2 1 7n 2 1 dots nbsp Vlastivosti RedaguvatiBud yake racionalne chislo mozhe buti predstavlene v vidi skinchennogo lancyugovogo drobu dvoma sposobami bilsh dovgij z yakih zavzhdi zakinchuyetsya odiniceyu a korotshij vidriznyayetsya vid nogo tim sho ostannoyi odinici nemaye a element pered odiniceyu na 1 bilshij Napriklad 9 4 2 3 1 2 4 displaystyle 9 4 2 3 1 2 4 nbsp dd Teorema Lagranzha Chislo mozhna podati u viglyadi neskinchennogo periodichnogo linijnogo drobu todi j lishe todi koli vono ye irracionalnim rozv yazkom kvadratnogo rivnyannya z cilimi koeficiyentami Napriklad 2 1 2 2 2 2 displaystyle sqrt 2 1 2 2 2 2 dots nbsp zolotij podil ϕ 1 1 1 1 displaystyle phi 1 1 1 1 dots nbsp dd Dlya inshih ne kvadratichnih algebrayichnih chisel harakter rozkladu ne vidomij Dlya majzhe vsih dijsnih chisel x serednye geometrichne koeficiyentiv rozkladu chisla v lancyugovij drib rivnij konstanti Hinchina K 2 6854520010 n im nablizhenim drobom dlya lancyugovogo drobu x a 0 a 1 a 2 a 3 displaystyle x a 0 a 1 a 2 a 3 cdots nbsp nazivayetsya skinchennij lancyugovij drib a 0 a 1 a n displaystyle a 0 a 1 cdots a n nbsp znachennya yakogo mozhna podati p n q n displaystyle frac p n q n nbsp Parni nablizheni drobi utvoryuyut zrostayuchu poslidovnist a neparni spadnu Obidvi poslidovnosti zbigayutsya do x Vikonuyutsya nastupni rekurentni spivvidnoshennya p 1 1 p 0 a 0 p n a n p n 1 p n 2 displaystyle p 1 1 quad p 0 a 0 quad p n a n p n 1 p n 2 nbsp q 1 0 q 0 1 q n a n q n 1 q n 2 displaystyle q 1 0 quad q 0 1 quad q n a n q n 1 q n 2 nbsp p n q n 1 q n p n 1 1 n 1 displaystyle p n q n 1 q n p n 1 1 n 1 nbsp x p n q n lt 1 q n 2 displaystyle left x frac p n q n right lt frac 1 q n 2 nbsp Zvidsi viplivaye nastupne tverdzhennya nablizhenij drib p n q n displaystyle frac p n q n nbsp ye najkrashim nablizhennyam dlya x displaystyle x nbsp sered vsih drobiv znamennik yakih ne perevishuye q n displaystyle q n nbsp Zastosuvannya Redaguvatipri rozrobci sonyachnogo kalendarya neobhidno znajti racionalne nablizhennya dlya chisla 365 2421988 Za dopomogoyu lancyugovih drobiv oderzhuyetsya poslidovnist 1 4 7 29 8 33 31 128 132 545 displaystyle frac 1 4 frac 7 29 frac 8 33 frac 31 128 frac 132 545 cdots nbsp Pershij z cih drobiv ye osnovoyu yulianskogo kalendarya Dovedennya irracionalnosti chisel Napriklad za dopomogoyu lancyugovih drobiv dovedena irracionalnist dzeta funkciyi Rimana z 3 displaystyle zeta 3 nbsp chisla pi Algoritmi faktorizaciyi SQUFOF i CFRAC Harakteristika ortogonalnih mnogochleniv Harakteristika stabilnih mnogochleniv Algoritm Lancosha vikoristovuye lancyugovi drobi dlya obchislennya vlasnih znachen velikih rozridzhenih matric Div takozh Gipoteza ZarembiDiv takozh RedaguvatiObmezheni nepovni chastkiDzherela RedaguvatiDrozd Yu A 1997 Teoriya algebrichnih chisel Kiyiv RVC Kiyivskij universitet s 82 ISBN 966 594 019 8 ukr V I Arnold Cepnye drobi M MCNMO 2000 T 14 40 s Biblioteka Matematicheskoe prosveshenie A A Buhshtab Teoriya chisel Prosveshenie 1966 384 s I M Vinogradov Osnovy teorii chisel Gos izd tehniko teoreticheskoj literatury 1952 180 s S N Gladkovskij Analiz uslovno periodicheskih cepnyh drobej ch 1 2009 138 s I Ya Depman Istoriya arifmetiki Posobie dlya uchitelej Prosveshenie 1965 S 253 254 G Devenport Vysshaya Arifmetika M Nauka 1965 S V Sizyj Lekcii po teorii chisel Ekaterinburg Uralskij gosudarstvennyj universitet im A M Gorkogo 1999 A Ya Hinchin Cepnye drobi M GIFML 1960 Claude Brezinski History of continued fractions and Pade approximants Arhivovano 22 chervnya 2019 u Wayback Machine Berlin Springer Verlag 1991 VIII 551 pages Chapter 4 Arhivovano 22 chervnya 2019 u Wayback Machine Golden Age Pages 97 140 G Blanch Numerical Evaluation of Continued Fractions SIAM Review Vol 6 No 4 Oct 1964 pp 383 421 J Widz From the History of Continued Fractions Arhivovano 22 chervnya 2019 u Wayback Machine WDS 09 Proceedings of Contributed Papers Part I 176 181 2009 Otrimano z https uk wikipedia org w index php title Lancyugovij drib amp oldid 40290865
