why is nitrous oxide is n₂o and not n₂o₃

Nitrogen has variable valencies and it depends on which element it is combining with. With oxygen, nitrogen can have oxidation states from 1+ to 5+. Hence it can form six principle oxides. They are:

Nitrous oxide N2O, valency of nitrogen 1

Nitric oxide NO, valency of nitrogen 2

Dinitrogen trioxide N2O3, valency of nitrogen 3

Nitrogen dioxide NO2, valency of nitrogen 4

Dinitrogen tetraoxide N2O4, valency of nitrogen 4 and

Dinitrogen pentoxide N2O5 valency of nitrogen 5

The formula of the oxide depends on the valecy in which nitrogen is combined with oxygen. Hence the formula of nitrous oxide is N2O with nitrogen valency 1 and dinitrogen trioxide is N2O3 with valency of nitrogen 3.