$\{10^z|z\in \mathbb Z\}$ looks like a basis of $\mathbb R$ over some field. However, it is definitely not a Hamel basis over $\mathbb Q$ due to
1.It only has countable elements compared with any Hamel basis actually has uncountable many.
2.Most real number needs to be represented as an infinite sum.
3.The most lethal one, itself is not independent over $\mathbb Q$.
So my question is is it another type of basis?