believe the non-integer base/radix has appeared more frequently under the name of "beta-expansion", which was introduced first in late 50's, e.g., the golden ratio base has been extensively studied and used in A/D conversion. ExSan should be interested in looking at complex bases, since they are known to generate 2-d fractal geometry (dragon curves). one good thorough reference is Knuth's TAOCP (4.1 in Vol 2).regarding whether the number of representations of e.g., 1 in a certain base q is countable or uncountable, this question has been solved for 1<q<2, where the digits can only be 0 or 1. except for a class of numbers (e.g., Pisot numbers), in almost every base, 1 has a continuum of beta-expansions. but there are quite remarkable exceptions like Komornik-Loreti constant, which is transcendental and 1 has a unique expansion in this base.
Last edited by wileysw
on June 26th, 2015, 10:00 pm, edited 1 time in total.