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What is the oxidation number of 1/2 O2 and why?

First let's define the term oxidation number.  It is a way of assigning charges to each atom of a chemical compound or species.  In other words, if you pretended that each atom of a compound or species was an ion, the oxidation number for each element would be the charge assigned to that particular element as a part of the whole.  For an ion with a single element, the oxidation number of the element is the charge of the ion.  For a neutral chemical compound, the oxidation numbers of the individual atoms must add up to a total of zero.  O2 is molecular oxygen and is composed of two oxygen atoms bonded together with a double bond.  Since it has no overall charge and is composed of a single type of element, both of the oxygen atoms in O2 must have an oxidation number of 0 since they both have to be the same and add up to 0.  Now, 1/2 O2 is another way of saying just plain O, and the same rules apply here.  O has a total charge of zero and therefore the oxidation number of O would have to be zero as well.

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