Are there number systems with fractional or irrational bases?

Question

I'm wondering if there are number systems with bases other than integers? For example, with a fractional, imaginary, irrational, transcendental basis, or with the basis "infinity"? If there are, then how is the translation made in and between them?

Answer 1

I like this example ... in $\mathbb C$, take base $-1+i$ with digit set $D := \{0,1\}$. Numbers of the form $$ \sum_{k=0}^K a_k (-1+i)^k,\qquad a_k \in \{0,1\} $$ are the Gaussian integers . And numbers of the form $$ \sum_{k=-\infty}^K a_k (-1+i)^k,\qquad a_k \in \{0,1\} $$ give us all of $\mathbb{C}$. And the "fractions" for this number system $$ \sum_{k=-\infty}^{-1} a_k (-1+i)^k,\qquad a_k \in \{0,1\} $$ form the fractal known as the twindragon.