I have the following problem.
Say I have a finite set of irrational numbers $(z_1,...,z_n)$ that are linearly independent (in the sense that I can not multiply by a rational number to get one of the $z_j$ from another). Consider then a number $z$ in the $\mathbb Z$-span with respect to this set, i.e.
$$
z = \sum_j a_j z_j,
$$
where $a_j \in \mathbb Z$.
In reality I obtain the number $z$ numerically and hence only to some finite digit precission (~5).
Is there any hope (uniqueness) in obtaining the coefficients $a_j$ (i.e. $z$ itself) from this approximation?
If yes, what could be suitable algorithm to determine them?