A good basis for computing with complex numbers (Q1339058)

The ring of integers in \(Q(\sqrt{-3})\) is generated by \(\xi= (1+ i\sqrt 3)/2\) and the elements can be represented in the form \(a_ 0+ a_ 1b+\cdots+ a_ n b^ n\), where \(b= i\sqrt 3\) and \(a_ i\in S= \{0,1,\xi\}\). Considering the index set as the set of negative integers, the series \(a_{-1} b^{-1}+ a_{-2} b^{-2}+\cdots\) converge absolutely in \(\mathbb{C}\) and we obtain a continuous map into a compact subset \(F\) of \(\mathbb{C}\). The set \(F\) and its subsets are investigated in the paper, computer images of these are also presented.