0,(9): Versiyalar orasidagi farq

$2, $3 dagi (joriy) koʻrinishi

0,(9) yoki 0,999… („nol va davrda toʻqqiz“) matematikada oʻnli davriy kasr boʻlib, 1ga tengdir. Yaʼni „0,999…“ ham, „1“ ham aynan bir sondir^[1].

Ushbu ayniyatni isbotlovchi bir necha matematik isbot mavjud. Masalan:

\frac{1}{3} = 0{,}333\ldots = 0{,}(3);

1 = 3 \cdot \frac{1}{3} = 0{,}999\ldots = 0{,}(9).

yoki
${\displaystyle \begin{align} x &= 0{,}(9); \\ 10 x &= 9{,}(9); \\ 10 x - x &= 9{,}(9) - 0{,}(9); \\ 9 x &= 9{,}0; \\ x &= 1. \end{align}}$

Manbalar

↑ This definition is equivalent to the definition of decimal numbers as the limits of their summed components, which, in the case of 0.999..., is the limit of the sequence (0.9, 0.99, 0.999, ...). The equivalence is due to bounded increasing sequences having their limit always equal to their least upper bound.

Matematikaga oid uşbu maqola çaladir. Siz uni boyitib, ZiyoSferaga yordam berişingiz mumkin.

[1] This definition is equivalent to the definition of decimal numbers as the limits of their summed components, which, in the case of 0.999..., is the limit of the sequence (0.9, 0.99, 0.999, ...). The equivalence is due to bounded increasing sequences having their limit always equal to their least upper bound.

[1]