Motivation
We all know that:
$$ .\bar{9} =.999 \dots= 1$$
I was wondering if the following (obviously not rigorous) statement could be defined on the same footing?
Question
$$ x = \bar{9} $$
$$ \implies x/10 = \bar{9}.9 $$
Subtracting the above equations:
$$ .9 x = -.9 $$ $$ \implies x = -1 $$
Hence, $ x=\bar{9} = -1 $
Does this already exist? Does this representation of negative numbers lead to any apparent contradictions? What are the operations one can preserve with this kind of representation which still make sense?
P.S: I know this bit of a wild idea ...