Eric Angelini wrote:
Is the introduction of a third symbol "economi- cally" efficient? (I think we will use less symbols to represent all first 1000 natural integers with this "ternary notation" than with the usual binary system -- the price might be to high, though)...
You're coming up with a variation on what's called "balanced ternary" notation, base 3 with digits {1,0,-1}. There are too many possible conventions for how to write this using the digits {0,1,2} instead; the EIS entry A072998 does it by adding 1 to each digit. Yes, base-3 computers (built on "trits", not "bits") were indeed pursued before binary ones became universal. Christoph is quite right in his recollection that "base e" is optimal, and base 3 is the best integer choice, if it weren't for the physical simplicity of 2-state storage. American Scientist had an article recently with some base-3 history: http://www.americanscientist.org/template/AssetDetail/assetid/14405/page/1 Two particularly nice properties of trits are that (1) a single trit can hold a determination of <=>, and (2) there's a tritwise operation analogous to xor, well-known now as the rule of the award-winning children's card game "Set": if a=b, then a#b is a, and if a!=b, then a#b is the third state, c. (Or, a#b is -a-b mod 3, if you prefer.) --Michael Kleber (I'm suppressing an odd urge to include a middle name in my signature, to go with the rest of the 3-not-2 theme...) -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.