Talk:Balanced ternary

Are there balanced systems for any base other than three? I think there would be a requirement for the base to be odd, but other than that, why not balanced base 5? Balanced base 11 could be counted on the fingers, with one hand representing negative numbers and the other positive. Linguofreak 02:13, 19 March 2006 (UTC)[reply]

Yes, there are ballanced systems with digits other than three. An for the really preverse, there are workable systems with negative and positive digits that aren't balanced for even bases. (But this drifts off into original research, many people have played with them, nobody seems to have officially published.) Nahaj 11:46, 17 April 2006 (UTC)[reply]

Even bases, how does that work? I shall have to play around with this too. Linguofreak 02:28, 18 April 2006 (UTC)[reply]

Actually, I've been doing some OR in this myself (developed balanced ternary on my own about 15 years ago, though I called it "nullcentric trinary" back then, for obvious reasons), and it seems that 0 can occupy any position at all within the choice of digits; e.g., you could have an octal system that starts at -2, and goes up to 5 (though why you'd want to is beyond me). I haven't even ruled out systems that don't include 0, e.g. octal from 3 to (decimal) 10, but only because I haven't considered them yet. --John Owens | (talk) 18:10, 16 February 2007 (UTC)[reply]

+0- Dave Smith: DaveAt168@aol.com -0+

I entered this field many years ago, while toying with the concept of numerical complexity. That is the idea that the complexity of a number is the product of the number of digits required to encode a value, multiplied by the number of states that digit could take. I was struck by the fact that both base2 and base4 had the same complexity, and wondered if there was a reason for this. Eventually, I came up with a formula which proved that the nearer the base was to e, the less was the value of numerical complexity.

Having reached this conclusion, I struggled to devise a logic system based on what we now call Balanced Ternary.

After much struggle, almost coming to the conclusion that no logic was possible, other than negation, I decided to work backwards from arithmetic, and using the Binary half-adder as a model, devised the 'And' and 'Exclusive Or' functions.

Here are my results for And, Exclusive Or, and Negate

       B + 0 -                 B + 0 -      
      A                       A                      A Y
      +  + 0 0                +  - + 0               + -
      0  0 0 0                0  + 0 -               0 0
      -  0 0 -                -  0 - +               - +

I was unable to define any other unambiguous function, and still consider that this is the closed set.

I was concerned also that the operation according to de Morgan, which turned an And gate into an Or gate, actually reflected all three of these functions back on them selves, such that F(And) = And, F(Exclusive Or) = Exclusive Or, and F(Negate)= Negate.

This seems to confirm that the domain is closed.

I have serious doubts about refering to other so called ternary systems as ternary as they seem only to be binary logic being input by binary states, and undefined states, but the logic in the gate is still true binary.

Best regards,

Dave.

Fudge Dice?

I think someone should mention that one Fudge Die can produce one digit in balanced ternary. I'm not sure how to introduce this into either article.

Abhijit Bhattacharjee's work appears to be original research

Abhijit Bhattacharjee's work appears to be original research which has not yet been peer reviewed. It appears to be correct but I am not a mathematician so in line with policy I have added a template

Looking at the web page, I wouldn't exactly call it original. The system he describes for representing non-integer numbers is just the normal one: digits to the right of the decimal point get multiplied by successively smaller powers of the base (in this case, 1/3, 1/9, 1/27...). The description is longwinded, but that's all it boils down to. Carl Muckenhoupt 18:15, 25 October 2006 (UTC)[reply]

No, there is a crucial difference when it comes to fractional numbers as it has been pointed out in that work. For fractional numbers, you have to approach it from two directions, for those below .5 and for those above .5. Hence there is an inventive step. Just carrying on with ternary numbers wouldnt work because even if you add up all the ternary fractions it would not add up to more that .5

Don't we accept self published non peer-reviewed stuff from well published and recognised authors to some degree anyway? See Wikipedia:Reliable_sources#Self-published_sources. I'm not sure of coruse if this stuff was self-published Nil Einne 19:10, 26 October 2006 (UTC)[reply]

For fractional numbers, you have to approach it from two directions, just like balanced ternary, where you have to approach 26 from two directions: +00- = 1*3^3 + -1. There is no inventive step here. This is just carrying on with balanced ternary numbers. I am a high school junior, and this is completely obvious to me. --Zarel 21:03, 24 November 2006 (UTC)[reply]

Let me explain: Just like 9.0/10 is the same as 0.90 in decimal, 11.0/10 is the same as 1.10 in balanced ternary. All you're doing is moving the decimal place one to the left to divide by 10. I've edited the article accordingly. --Zarel 21:27, 24 November 2006 (UTC)[reply]

I am Abhijit Bhattacharjee and it wouldnt be possible for me to restore the page after each vandalisation. Just in case anyone cares, the page can be found here. http://www.abhijit.info/tristate/tristate.html where I have belaboured enough about its need and existence. —The preceding unsigned comment was added by 59.93.246.74 (talk • contribs) 03:10, 3 December 2006 (UTC).[reply]

Use as currency

Has the suggestion to use balanced tertiary for currency considered that it would be difficult to have people willingly keep things with negative value?-- unsigned edit by user:72.144.117.150 at 07:55, 28 October 2006

Read again... that's not what is suggested. But I guess it could be stated more clearly. The idea is really to use regular ternary coin sizes (1, 3, 9, 27), as that would minimize the number of coins to be exchanged, assuming both parties in the deal have a good supply of different coins.--Niels Ø 13:14, 28 October 2006 (UTC)[reply]

I have just reverted an addition along the same lines.--Niels Ø 16:48, 22 November 2006 (UTC)[reply]

I remember encountering coins with values of 3 and 15 in the Soviet Union, shortly before it ceased to exist. Does anyone have accurate info on these coins; is it covered anywhere in the wikipedia; is it (at least marginally) relevant here; is there a good way to include it?--Niels Ø 16:48, 22 November 2006 (UTC)[reply]

rounding

From the Article (text in italics recently added):

Donald Knuth has pointed out that truncation and rounding are the same operation in balanced ternary—they produce exactly the same result. Moreover, there is no ambiguity in rounding (a property shared with other odd bases) since the number half is not representable.

Isn't it because there is no representation for 0.5 (mid point) that truncation is the same as rounding. -- Chris Q 15:22, 8 December 2006 (UTC)[reply]

No. Rounding and truncating is not the same for normal (unbalanced) ternary, even though it doesn't have a representation for 0.5 either. With unbalanced ternary the number 1.2 would get rounded to 2 but truncated to 1. -- Thowllly 01:27, 14 August 2007 (UTC)[reply]

Evenness test

The article states, "The quick test for even is the analog of the base ten divide-by-nine test: add up all the digits and repeat until you have a one-digit number; the number is even if the final sum is zero."

It can be stated much more simply than that. A number is even if it has an even number of nonzero digits. (In general, in any odd base, balanced or ordinary, a number is even if it has an even number of odd digits.)

I'm not sure whether this should be given as an alternative method or whether it should replace the existing one. The general "odd base evenness test" and the "one less than base" division test coincide, making them both reasonable ways to approach the problem. However, I don't feel that the evenness test deserves too much text devoted to it, so if nobody says anything then (provided I don't forget!) I will replace the complicated method with the simple one. MarkC77 02:09, 9 December 2006 (UTC)[reply]

LeRoy Eide's algorithm

Does anyone know what LeRoy Eide's algorithm (mentioned in main article) is? Ian S 14:19, 27 March 2007 (UTC)[reply]

On the web: http://www.dyalog.com/dfnsdws/n_JitSub.htm for example, Possibly the reference should be removed as original research. While it seems to be well known in a number of circles, there doesn't seem to have ever been a formal paper. For balanced ternary it is a slick way of doing divides by 2 and 10. 155.101.224.65 (talk) 17:17, 27 March 2008 (UTC)[reply]

It is certainly a slick way of dividing, but it relies on prior knowledge that the dividend is in fact divisible by the divisor (try following the algorithm when it is not). When the quotient is 2 this can be cheaply ascertained (see main article), and if the dividend is not even then a remainder of 1 can be added (or subtracted) to make it so. Can this be generalized to other (3ⁿ ± 1)? Ian S (talk) 18:01, 10 April 2008 (UTC)[reply]

Yes, there is a generalization (but not as easily stated.). But then you are really and truely off into original research with no published paper or even web page. 155.101.244.53 (talk) 18:30, 13 April 2008 (UTC)[reply]

In the light of this I have removed the reference to divisors other than 2. I think it would be a pity to remove any more as the correctness of the rest is obvious from the description of the evenness test (in the article) and of the algorithm (in the www.dyalog.dk page referred to above), but strictly there is no accepted source. Ian S (talk) 18:15, 17 April 2008 (UTC)[reply]

Note that the companies domain has changed now from www.dyalog.dk to www.dyalog.com. 155.101.224.65 (talk) 13:50, 2 September 2008 (UTC)[reply]

Notation

Using overline for repeating digits and underline for minus makes it rather nerdy to read the examples. I think the overline is a good choice, but could we change the examples to use "+" for +1 and "-" for -1, or is there in the literature precedence for using some other set of symbols like "1" for +1 and "m", say, for -1?--Niels Ø (noe) 07:25, 20 September 2007 (UTC)[reply]

I have added a suggested notation ("bop") which is visually symmetric. Unfortunately it's from a colleague who has not chosen to publish it elsewhere. I find it quite appealing, and I think that putting it in Wikipedia (rather than letting it vanish) is a reasonable approach. Snezzy (talk) 23:09, 19 January 2008 (UTC)[reply]

If it is unpublished from a single colleague it is OR, I am afraid. I think there is a wiki for this type of thing, but not Wikipedia. -- Q Chris (talk) 22:25, 20 January 2008 (UTC)[reply]

Noe, I have asked where this can be published, as I agree that it is a neat notation, and it would be a shame for it to vanish. It has been suggested that a Wikiversity article would be a good place if you want to publish it. -- Q Chris (talk) 08:18, 22 January 2008 (UTC)[reply]

Kepler?

I vaguely recall seeing this system attributed to the astronomer Johannes Kepler. Can anyone confirm or deny anything? Michael Hardy (talk) 17:12, 6 August 2008 (UTC)[reply]