=asimovd@aol.com

I recently began wondering if anything special happens when one uses the scheme of decimal, binary, etc. representation -- as usual, via a sequence of integer coefficients no greater than the base -- but with an irrational base like e. Does anyone know of research on representing numbers via real irrational bases?

(Perhaps "special" as in "Special Ed.", but...<;-) These base e representations exemplify what I like to call "numbrals" (roughly, "they look like numerals but have different semantics"), such as the "TinkerToy" base sqrt2 (which I've played with more extensively). Since I've raved about this before I'll just mention a few things: Each integer n corresponds to a polynomial in e with coefficients in {0,1,2}, which I notate as 3[n]e (or just [n] when the system is understood from context): n Ternary 3[n]e 0 0 0 1 1 1 2 2 2 3 10 e 4 11 e + 1 5 12 e + 2 6 20 2e 7 21 2e + 1 8 22 2e + 2 9 100 e^2 10 101 e^2 + 1 11 102 e^2 + 2 12 110 e^2 + e 13 111 e^2 + e + 1 ... Typically, the sequence [n] in magnitude order is NOT the lexical order, the first glitch here being at [8] > [9] 2e+2 > e^2 7.43656 > 7.38906 giving (with apologies for neglecting the OEIS) 0 1 2 3 4 5 6 7 9 8 10 11 12 13 14 15 16 18 17 19 20 21 22 23 27 24 28 25 29 26 30 31 32 33 34 36 35 37 38 39 40 41 42 43 45 44 46 47 48 49 50 54 51 55 52 56 53 57 58 59 60 61 63 62 64 65 66 67 68 69 70 81 72 71 82 73 83 74 84 75 85 76 for these particular numbrals (When does a permutation of Z correspond to a numbral base in this way? When uniquely?) Unfortunately (unlike 2[n]sqrt2 and similar systems) sums and products of finite-length 3[n]e aren't generally finite (eg [1]+[2] or [4]*[5]) so the corresponding "numbral theory" ("primes" etc) is generally messier, although some things are doable (What is the number of finite partitions of 3[n]x into 3[k]x with 0<k<n?) (By the way, you can say "x has a finite representation in base b" just when "b[x]1 is finite in magnitude").

Of course, the representation has much more leeway for nonuniqueness than for an integer base, unless an algorithm is specified making it unique.

Interesting topic. Are there really "more", and in what sense, and how does it vary with the bases? Define y(x) := 3[x]e as a function of x. The number of representations of a given value y0 is the multiplicity of the inverse function x(y) at y0. (What is this at x(3)?). Note that the domain of x is ternary strings, and that the non-unique strings in base 3, such as 1.111... (= 2.000...), uniquely represent values in base e (ie e/(e-1) = 1.5819767068693264244...). I guess when the base is positive the greedy algorithm will produce the lexically greatest...