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In case you were wondering: "As in ordinary ternary numbers, the digits of a balanced ternary numeral are coefficients of powers of 3, but instead of coming fro

In case you were wondering: "As in ordinary ternary numbers, the digits of a balanced ternary numeral are coefficients of powers of 3, but instead of coming from the set {0, 1, 2}, the digits are –1, 0 and 1." e.g. 1 x 3^3 - 1 x 3^2 + 0 x 3^1 + 1 x 3^0, Imagine, trits instead of bits. I like the idea of base e, if unrealistic. (BTW: I have an old answer on Math.stackexchange that cheats by using base 3: http://math.stackexchange.com/a/734723/108109 http://math.stackexchange.com/a/734723/108109 )

mathattack 12y ago
We'd all be more attuned to power laws if we thought in base 3. :-) I like the -1, 0 and 1 concept, but is there any way to pull it off from an electrical engineering concept? Or would it all have to be overlayed as software?
- nitrogen 12y ago
  https://en.wikipedia.org/wiki/Ternary_computer https://en.wikipedia.org/wiki/Ternary_computer -- it has been done before.
- darkmighty 12y ago
  If you assume a current I=0 represents 0 and I=i represents 1, you could instead choose I={-i,0,i} to represent {-1,0,1} using just 1/6 more power.
plesner 12y ago
If you like the idea of e as a base I wrote a post about how you can do that, at least asymptotically, which might be of interest. http://hummusandmagnets.tumblr.com/post/48829755849/how-to-use-e-as-a-radix http://hummusandmagnets.tumblr.com/post/48829755849/how-to-u...
pmalynin 12y ago
Galois field theory says there is no difference in using {0, 1, 2} or {-1, 0, 1}
- sophacles 12y ago
  Is that "no difference" in the sense that all turing complete languages are "the same", or in some more practical sense?
  - sp332 12y ago
    For representing a number, it makes no difference at all. For binary you can use on/off, up/down, positive/negative etc. For trinary you can have on-mid-off, up-mid-down, positive-neutral-negative, and so on. Any symbols will do as long as you define the math in terms of the symbols you pick.
- darkmighty 12y ago
  Erm not under multiplication, it seems: {a,b,c} => {0,1,2}-> A | {-1,0,1}-> B b * c * c = 1 * 2 * 2 (A) != 0 * 1 * 1 (B)
  - pmalynin 12y ago
    Well, the problem is notation which is quite misleading here. Firstly, sets don't have order so {-1,0,1} == {0,1,-1} == {-1,1,0}. And the symbols "1", "-1" and "0" are quite misleading as well, especially if you are not familiar with finite fields. In reality the symbols "1", "-1" and "0" represent (infinite) sets and mathematics can be done using any "representative" of that set. For this particular example "-1" and "2" are part of the same set, however mathematics has this notion of "well-defined" so if you choose to do your calculations in say {0, 1, 2} and I do it in {3, 4, 5}, we will have the same answer.
    - darkmighty 12y ago
      You get the same answer if you only use '+', but you may get a different answer if you use 'x', that was my point. In other words, once you have a "canonical" addition table for the elements of a set, there are many multiplications tables to choose from which "extend" the addition in a straightforward way. Sorry not very knowledgeable on this stuff so it took a long detour.

mathattack 12y ago

We'd all be more attuned to power laws if we thought in base 3. :-) I like the -1, 0 and 1 concept, but is there any way to pull it off from an electrical engineering concept? Or would it all have to be overlayed as software?

nitrogen 12y ago
https://en.wikipedia.org/wiki/Ternary_computer https://en.wikipedia.org/wiki/Ternary_computer -- it has been done before.
darkmighty 12y ago
If you assume a current I=0 represents 0 and I=i represents 1, you could instead choose I={-i,0,i} to represent {-1,0,1} using just 1/6 more power.

plesner 12y ago

If you like the idea of e as a base I wrote a post about how you can do that, at least asymptotically, which might be of interest. http://hummusandmagnets.tumblr.com/post/48829755849/how-to-use-e-as-a-radix http://hummusandmagnets.tumblr.com/post/48829755849/how-to-u...

pmalynin 12y ago

Galois field theory says there is no difference in using {0, 1, 2} or {-1, 0, 1}

sophacles 12y ago
Is that "no difference" in the sense that all turing complete languages are "the same", or in some more practical sense?
- sp332 12y ago
  For representing a number, it makes no difference at all. For binary you can use on/off, up/down, positive/negative etc. For trinary you can have on-mid-off, up-mid-down, positive-neutral-negative, and so on. Any symbols will do as long as you define the math in terms of the symbols you pick.
darkmighty 12y ago
Erm not under multiplication, it seems: {a,b,c} => {0,1,2}-> A | {-1,0,1}-> B b * c * c = 1 * 2 * 2 (A) != 0 * 1 * 1 (B)
- pmalynin 12y ago
  Well, the problem is notation which is quite misleading here. Firstly, sets don't have order so {-1,0,1} == {0,1,-1} == {-1,1,0}. And the symbols "1", "-1" and "0" are quite misleading as well, especially if you are not familiar with finite fields. In reality the symbols "1", "-1" and "0" represent (infinite) sets and mathematics can be done using any "representative" of that set. For this particular example "-1" and "2" are part of the same set, however mathematics has this notion of "well-defined" so if you choose to do your calculations in say {0, 1, 2} and I do it in {3, 4, 5}, we will have the same answer.
  - darkmighty 12y ago
    You get the same answer if you only use '+', but you may get a different answer if you use 'x', that was my point. In other words, once you have a "canonical" addition table for the elements of a set, there are many multiplications tables to choose from which "extend" the addition in a straightforward way. Sorry not very knowledgeable on this stuff so it took a long detour.