Now this question might sound a bit weird to some people, but the situation is this:
Say I have the number $0.999..$ where there is an infinite number of 9's (much like $0.3333..$ with $\frac{1}{3}$). For convenience, let's put a line above the 9's to indicate there is an infinite number of 9's: $0.\overline{99}$ (probably not official notation, but what the heck..).
Now if I see a statement like this $1=0.\overline{99}$, I would think this is not true, at first, through more or less the same logic that $1\neq2$.
But the following confuses me: $\frac{1}{9}=0.\overline{11}$, so If I do this: $$ \frac{1}{9}\cdot9=\frac{9}{9}=1 $$ And also: $$ 0.\overline{11}\cdot9=0.\overline{99} $$
That means that $1=0.\overline{99}$, but that doesn't make sense to me, because $1$ and $0.\overline{99}$ look like totally different numbers to me... Am I missing something here or is my above logic just false?
Any clarification would be great :)
