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 ia he mea paʻakikī loa. Akā he aha ka mea e hana ʻia ke noʻonoʻo ʻoe i ka helu aʻe, ka mea i ka nānā mua ʻana he mea ʻano ʻē: $$ ...99999 $$ , kahi e hoʻolōʻihi ʻole ai ka palena ʻole i ka ʻākau, akā i 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, $$A + B = 1 + (-1) = 0 = C$$ .
Nānā: Ua hōʻike ʻia ʻo nā waiwai \(1, -1\) a me \(0\) ʻo ia ka wā i wehewehe ʻia ai \(A, B\) a me \(C\) a ua hāʻawi ʻia kahi waiwai kūpono iā lākou.