Thanks for the correction of my careless error. What I should have written is that uniqueness holds if the greedy algorithm is used. (For each n = 1,2,3,... choose c_n, 0 <= c_n <=n, to be as large as possible such that the partial sum c_1/2! + c_2/3! + . . . + c_n/(n+1)! does not exceed the real number x being represented.) Which brings up the question: Is the greedy algorithm the most efficient? (Say in terms of minimizing the number of nonzero coefficients when x is rational.) It's well-known that this is not the case for Egyptian fractions. But for factorial denominators this seems more likely. --Dan Joerg wrote: << I wrote: << Since high school I've thought factorial base is an interesting way to express real numbers in the unit interval (0,1). That is, since 1/2! + 2/3! + 3/4! + ... = (d/dx((e^x-1)/x))(1) = 1, every x in (0,1) has a unique representation of the form x = Sum_{n=1..oo} c_n/(n+1)! if the integers c_n satisfy 0 <= c_n <= n.

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I do not think so: 1/2 = 0.1 = sum( n>=2, n/(n+1)! ) The problem (same as 1.0 = 0.999.. ) stays with all mixed radix representations. For uniqueness I'd should check non-adjacent forms (no two adjacent digits nonzero), e.g., digits 0,+1,-1 in binary.

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