Oxidation number is the charge left on the central atom when all the bonding pairs of electrons are removed with the charge assigned to the most electronegative atom.
The sum of the oxidation numbers equals the charge on the ion or molecule or salt...and here this sum must be ZERO...
We gots neutral #Mn_2O_3#...and thus we have #2xxMn^(3+) + 3xxO^(2-)#...
I include the following rules since I have them to hand...
#1.# #"The oxidation number of a free element is always 0."#
#2.# #"The oxidation number of a mono-atomic ion is equal"# #"to the charge of the ion."#
#3.# #"For a given bond, X-Y, the bond is split to give "X^+# #"and"# #Y^-#, #"where Y is more electronegative than X."#
#4.# #"The oxidation number of H is +1, but it is -1 in when"# #"combined with less electronegative elements."#
#5.# #"The oxidation number of O in its"# compounds #"is usually -2, but it is -1 in peroxides."#
#6.# #"The oxidation number of a Group 1 element"# #"in a compound is +1."#
#7.# #"The oxidation number of a Group 2 element in"# #"a compound is +2."#
#8.# #"The oxidation number of a Group 17 element in a binary compound is -1."#
#9.# #"The sum of the oxidation numbers of all of the atoms"# #"in a neutral compound is 0."#
#10.# #"The sum of the oxidation numbers in a polyatomic ion"# #"is equal to the charge of the ion."#
