Is the 10's complement operation defined for a negative decimal number? Two decimal numbers $X$ and $Y$ are considered to be 10's complements of each other if:
$$X + Y = 10^n$$
Where $n$ is the number of digits in $X$ and $Y$ .
For e.g $               $
10's complement of 
$417 = 1000 -417 = 583$ 
DOUBT : Is the 10's complement operation defined for negative numbers?
Since from the above definition, $Y$ for $X=-417$ would be $1417$. But then $n$ would become 4 and it would be violation. 
Note : I am talking about 10's complement as an operation and not as a representation for signed numbers.   
EDIT : The "same no. of digits " property in the definition seems to be wrong . Please provide the correct definition of 10's Complement and answer the question on the basis of that .Thanks.

By your definition, negative numbers do not have complements. This is because if $10^{k-1}<n<10^k$, then the complement of $-n$ would be the number $10^k - (-n) = 10^k + n$ (since $n$ has $k$ digits), but $10^k - n$ has $k+1$ digits, leading to a contradiction.
