I forget who mentioned that e was found to be the "best" number base. I would like to know in what sense it is best -- or is this just some kind of offhand, non-rigorous comment? I chose to look at e simply because it seems to be the math constant (other than 2) whose powers (not necessary integer) arise most commonly. There is the old chestnut that (modulo details) if we start with a real number K >> 0 and wish to break K up into the sum of positive pieces with maximum product, this is attained by making each piece = e. (This is exactly true if K = ne, but needs some modification otherwise, of course.) Is this relevant here? ------------------------------------------------------------------------------ ---------------------------------------- ALSO: There's something unappealing about the asymmetry inherent in using the greedy algorithm to represent a number to (say) base e, since 0 and 1 play a role different from the other digit, 2. One way to avoid this would be as follows. Suppose we want to represent K > 0 to the base e, and suppose that e^r is the largest integer power of e that does not exceed K. Hence K = c (e^r) where 0 <= c < e. Then we use as highest-order digit d = 0, 1, 2 according as 0 <= c < e/3; e/3 <= c < 2e/3; 2e/3 <= c < e, or in brief, d = [3c/e]. (Of course, the calculation of digits continues by applying the above to the quantity K - d(e^r), now using e^(r-1) in place of e^r, ad infinitum.) I wonder if anything nifty comes from treating the digits in this symmetrical fashion. --Dan