# 0,99999... = 1?1214

Inā ma ke kula a i ʻole ma ke kula nui: ʻO kahi nīnau hoihoi e $$0,99999... = 1$$ a laila e hoʻopili pū ana i ka nīnau inā he ʻoiaʻiʻo ka hoʻohālikelike ʻana: $$0,99999... = 1$$ . ʻOiai $$0,99999... = A$$ infinity i ka $$0,99999... = A$$ hema o ka hoʻohālikelike, hāʻawi mākou i kahi inoa: $$0,99999... = A$$ . Ma hope o ka hoʻonui ʻia e ka mea $$10$$ a me nā hoʻololi algebraic maʻalahi, loaʻa iā mākou kahi ʻike kupaianaha mua.

$$\begin{array}{rcll} 9,99999... & = & 10\cdot A & \Leftrightarrow \\ 9 + 0,99999... & = & 10 \cdot A & \Leftrightarrow \\ 9 + A & = & 10 \cdot A & \Leftrightarrow \\ 9 & = & 9 \cdot A & \Leftrightarrow \\ 1 & = & A & \Leftrightarrow \\ 1 & = & 0,99999... & \end{array}$$

ʻAʻole paʻakikī kēlā. Akā he aha ka hana ke nānā ʻoe i ka helu aʻe o $$...99999$$ , ka mea i ka nānā mua ʻana he ʻano ʻē paha ia, kahi e kau ʻole ai ka palena ʻole ma ka ʻākau akā ma ka hema.

Lawe mākou i nā loli like ma luna a loaʻa:

$$\begin{array}{rcll} ...99999 & = & B & \Leftrightarrow \\ ...999990 & = & 10\cdot B & \Leftrightarrow \\ B - 9 & = & 10 \cdot B & \Leftrightarrow \\ - 9 & = & 9 \cdot B & \Leftrightarrow \\ -1 & = & B & \Leftrightarrow \\ -1 & = & ...99999 & \end{array}$$

ʻO ka hope loa mākou e noʻonoʻo i ka helu $$...99999,99999...$$

a loaʻa iā ʻoe ka mea kupaianaha i ka maka mua

$$\begin{array}{rcll} ...99999,99999... & = & C & \Leftrightarrow \\ ...99999,99999... & = & 10\cdot C & \Leftrightarrow \\ C & = & 10 \cdot C & \Leftrightarrow \\ 0 & = & 9 \cdot C & \Leftrightarrow \\ 0 & = & C & \end{array}$$

Akā kūlike nō kēia, ʻoiai ma kekahi ʻaoʻao $$A + B = 0,99999... + ...99999 = 99999,99999 = C$$ a ma kekahi, $$A + B = 1 + (-1) = 0 = C$$ .

Kahakaha: Hōʻike ʻia inā wehewehe kekahi iā $$A, B$$ a me $$C$$ a hāʻawi i kahi kūpono kūpono iā lākou, a laila ʻo nā kumukūʻai $$1, -1$$ a me $$0$$ .

Hope