How to take one's complement of a positive integer?

Question

Of course we can do that by converting the number to binary and then converting it back to decimal, but to do that directly in decimal?

Answer 1

You need to fix a maximum number of bits $b$ and then the one's complement of a positive integer $x$ with at most $b$ bits is given by $x'=2^b-1-x$, because by definition the sum $x+x'$ has all bits set and hence is of the form $2^b-1$.

If you do not fix a maximum number of bits then you can still flip all significant bits, but then the map $x\mapsto x'$ is no longer injective (and hence not invertible) because for instance $0$ and $2$ are mapped to $1$.