This kind of leads to an interesting question which I have been vaguely pondering for a few months. What happens if you choose an irrational base for your numb

This kind of leads to an interesting question which I have been vaguely pondering for a few months. What happens if you choose an irrational base for your number system, rather than an integer or even a rational base. For example, if you chose to express a (positive) number as a_1 e + a_2 e^2 + a_3 e^3 + ... where 0 < a_i < e Obviously you could chose e, or pi, or whatever as your base. Can anyone of with more mathematical chops comment?