I was wondering is there any way to express $0.999999$ recurring as an actual fraction without equaling $1$? Because I tried to convert it into a fraction following the rules for normal recurring decimals like this:
$$\begin{align}n&=0.999\dot9\\10n&=9.999\dot9\\n&=0.999\dot9\\9n&=9\\\therefore n&=9/9\end{align}$$
But as you can see the result is $9/9$ which ultimately is equal to $1$ . And I've even tried calculating it other ways like this:
$$\begin{align}1/3&=0.333\dot3\\2/3&=0.666\dot6\\\therefore3/3&=0.999\dot9\end{align}$$
But it always ends up telling me that $0.9999999... = 1$. Is there any mistake in my logic? And I also realized this applied to other recurring decimals ending in $9$. E.g: $0.5999999...=5.4/9 = 0.6$ . So is there a way to write $0.999999...$ as a fraction so you can differentiate it from $1$?
Answer
As you have proven by yourself,
$$
0.999999999999999999999\dot9 = 1.
$$
There are no (big) logical mistakes in your post.
Because $0.99\dots$ is equal to $1$, it also cannot be another fraction.
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