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Chislo Liuvillya irracionalne chislo x displaystyle x yake mozhna nabliziti racionalnimi chislami tak sho dlya bud yakogo cilogo n displaystyle n isnuye neskinchenno bagato par cilih p q displaystyle p q q gt 1 displaystyle q gt 1 takih sho 0 lt x p q lt 1 q n displaystyle 0 lt left x frac p q right lt frac 1 q n Diofantove chislo irracionalne chislo yake v takij sposib podati ne mozhna tobto pri nablizhenni racionalnim chislom pomilka stanovit ne menshe deyakogo stepenya znamennika C a gt 0 p Z q N x p q C q a displaystyle exists C alpha gt 0 colon quad forall p in mathbb Z q in mathbb N quad left x frac p q right geqslant frac C q alpha Za teoremoyu Liuvillya pro nablizhennya algebrichnih chisel bud yake irracionalne algebrichne chislo ye diofantovim Zokrema bud yake liuvilleve chislo transcendentne sho dozvolyaye yavno buduvati transcendentni chisla yak sumi nadshvidko zbizhnih ryadiv racionalnih chisel Diofantovi chisla ye metrichno tipovimi yih mnozhina maye povnu miru Lebega Chisla Liuvillya navpaki tipovi z topologichnoyi tochki zoru yih mnozhina zalishkova Mira irracionalnosti chisel Liuvillya m x displaystyle mu x infty krim togo yaksho mira irracionalnosti chisla neskinchenna to vono liuvilleve inodi cyu vlastivist prijmayut yak viznachennya chisel Liuvillya Klasichnij priklad liuvillevogo chisla stala Liuvillya yaku viznachayut yak k 1 10 k 0 110 0010000000000000000010000 displaystyle sum k 1 infty 10 k 0 1100010000000000000000010000 ldots PrimitkiMilnor Dzh Golomorfnaya dinamika Vvodnye lekcii Dynamics in One Complex Variable Introductory Lectures Izhevsk NIC Regulyarnaya i haoticheskaya dinamika 2000
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