The unit sum number, $u(R)$, of a ring 𝑅 is the least 𝑘 such that every element is the sum of 𝑘 units; if there is no such 𝑘 then $u(R)$ is 𝜔 or $\infty$ depending on whether the units generate 𝑅 additively or not. Here we introduce a finer classification for the unit sum number of a ring and in this new classification we completely determine the unit sum number of the ring of integers of a quadratic field. Further we obtain some results on cubic complex fields which one can decide whether the unit sum number is 𝜔 or $\infty$. Then we present some examples showing that all possibilities can occur.
