To calculate the total weight of all the ants in the world, you can multiply the average weight of an ant by the total number of ants:
\[ \text{Total weight} = (\text{weight of one ant}) \times (\text{number of ants}) = (3 \times 10^{-6} , \text{kg}) \times (1 \times 10^{16}). \]
Now, perform the multiplication:
\[ \text{Total weight} = 3 \times 1 \times 10^{-6} \times 10^{16} = 3 \times 10^{(-6 + 16)} = 3 \times 10^{10} , \text{kg}. \]
Thus, the approximate weight of all the ants in the world is
\[ \boxed{3 \times 10^{10}} , \text{kg}. \]
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