Step 1: Understanding oxidation states.
The complex \([Ru(NH_3)_5H_2O]Cl_2\) consists of the \([Ru(NH_3)_5H_2O]^2+\) cation and two chloride anions (Cl\(^-\)). The ammonia (NH\(_3\)) and water (H\(_2\)O) molecules are neutral ligands. The oxidation number of chloride is \(-1\).
Step 2: Calculating the oxidation number of Ru.
The overall charge of the complex is \(+2\), and since the chloride ions are each \(-1\), we have:
\[
x + 5(0) + 1(0) - 2 = 0,
\]
where \(x\) is the oxidation state of Ru. Solving for \(x\), we get \(x = +2\).
Step 3: Conclusion.
Thus, the oxidation number of Ru in the complex is \(\mathbf{+2}\).