I've always liked factorial base -- which uses integer coefficients -- the version for nonnegative integers: (*) N = a_1 1! + a_2 2! + . . . + a_k k!, with 0 <= a_j < j for all j and the one for fractions in [0,1): (**) f = c_1 / 2! + c_2 / 3! + . . . + c_k / k! + . . . with 0 <= c_j <= j for all j . If the c_j's are all = j, then the series sums to 1. The nice thing is, this doesn't depend on a specific choice of base, so the factorial representation of a number might be of number-theoretic interest. But I don't know of theorems linking number-theoretic properties of a number to factorial representations. E.g., can one say something about the representation (*) of a prime number? About the representation (**) of an algebraic number as compared to a transcendental one? Etc. --Dan