Talk:Ternary numeral system

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This article just had an edit (which I undid) which only added the line: "Currently there is a private project underway to develop ternary logic into a usable programming language.". No facts, nothing that could be checked. Without knowing who is allegedly attempting to do this, there is no way to tell if this is a legitimate project, or another scam to get people to support a new startup. I also question the semantics... Has binary logic been built into a usable programming language? What would that mean? I can't think of any language feature (other than the list of possible operators) that cares if the underlying logic is binary or ternary. Nahaj 02:45:39, 2005-09-06 (UTC) [addendum: Fortran back in the 1960's had three way if statements... (Positive, negative, zero). (: What's the ternary version of "foreach" ? :)


From the article:

Similarily, a currency system using balanced ternary would save visits to the bank - customers would be likely to have exact change, or be able to get a small number of coins in change, and sellers would just occasionally need to deposit a large coin or two

My question: Who is going to want to carry around coins that indicate negative amounts of money. I would quickly throw away (or conveniently lose) any such coins.

Not buy something worth, say, ψ17 (with ψ meaning a generic currency), you would give the shopkeeper ψ27 and he would give you coins for ψ9 and ψ1. 23191Pa (chat me!) 04:59, 21 December 2009 (UTC)

It seems to be a common misconception that you need negative money for this. Here's a way to think about it: when you are buying something, taking the positive digits alone gives the amount of money you give to the cashier, and taking the negative digits alone gives the amount of money you receive back as change. Of course the opposite holds true when you are selling something. Double sharp (talk) 07:50, 22 June 2014 (UTC) (Protactinium-231 with a new username)


Update. I see that Pakaran has deleted my sentence in the article "While mathematically appealing, such a system [using balanced ternary as a monetary system] would require the existance of coins representing negative amounts of money, which would be impractical." This was added to a sentence that Pakaran wrote here (see above "Similarily, a currency system using balanced ternary...").

Here is my understanding of a balanced ternary monetary system. In balanced ternary, you have digits that represent postive number, and digits that represent negative numbers. For example, 5 is represented "+--": 9 - 3 - 1. To represent five dollars in a balanced ternary system, one would have a nine dollar bill, a negative three dollar bill, and a negative one dollar bill. It's not balanced ternary unless we have some way of representing negative amounts. Well, if I had five dollars from having a positive nine dollar bill, and two negative bills, well I would just throw out the two negative bills and have nine dollars!

So, yes, either I'm hopelessly confused about this, or we will need to keep the above sentence in the article. Please clarify if I am mistaken somehow. Samboy 14:07, 23 Dec 2004 (UTC)

The point is that with balanced coin sizes, it's convenient for people to pay in exact change, which counts down on the number of bank visits by customers to turn in rolled coin, and by sellers to get more small coins. That reasoning is taken from a website, I forget which. Pakaran (ark a pan) 14:11, 23 Dec 2004 (UTC)
Oh, and you don't need negative coins for this to work, any more than you need a negative mass for the balance to work. Pakaran (ark a pan) 14:13, 23 Dec 2004 (UTC)
OK, I found something here. I can't read ps documents right now (!@#$ xmas vacation and no real internet for my Linux laptop), but the Google text version of the document says this:

A Currency System based on Balanced Ternary All currency systems use a set of tokens, usually called coins and notes. Suppose we choose the value of the tokens (let us call them coins), to be multiples of 3 : 1, 3, 9, 27, 81 and so on. Further suppose that we want to pay the sum of x units. Representing x in ordinary ternary with digits 0, 1 and 2 gives a means of representing and hence of paying x units using no more than 2 coins of each denomination. However, if we express x in balanced ternary with digits

-1, 0 and 1, this represents a transaction where the customer pays x units by exchanging coins with the shopkeeper. Every 1 represents a coin that the customer gives the shopkeeper and every -1 represents a coin that the shopkeeper gives the

customer. Each need have only one coin of each denomination in order to make any transaction possible.

This system has a pleasing symmetry to it. Since -1 and 1 digits are equally likely, the exchanges will tend to balance out so that no particular denomination tends to run out or build up (assuming a uniform distribution of prices). Thus the customer will keep a supply of each denomination, having to withdraw high denomination tokens periodically and only rarely having to either split a coin into 3 of the next denomination down or exchange 3 of one denomination for one of the next denomination up

So it looks like I was mistaken. I will reword a clarification taking this information in to account. Samboy 14:19, 23 Dec 2004 (UTC)
Update: Description fixed. Samboy 14:22, 23 Dec 2004 (UTC)

The argument that all digits (-1,0,-1) are "equally likely" is suspect. The American Mathematical Society has had papers explaining why (back in the days that log tables were still used) the pages with numbers starting with "1" were more worn than those starting with "9". Wouldn't that same idea apply here?

Benford's law doesn't really apply to the initial digits here, since every positive number starts with 1 and every negative number with -1. For second and later digits, however, we should see the exact same sort of bias: for the second digit of positive numbers, for instance, -1 is commoner than 0 which is commoner than 1 (in the ratios .4649735206 : .3062702279 : .2287562514).
I'm not sure what consequences this has for the original point about the symmetry of a balanced ternary system. I don't think the poster was talking about a scenario in which one's equally likely to be on the buying as the selling end of a transaction, for instance... 4pq1injbok 03:44, 2 August 2005 (UTC)

The balanced ternary section badly needs to be wikified and cleaned up, I think. 4pq1injbok 03:24, 2 August 2005 (UTC)

Balanced Ternary math[edit]

I have a short tutorial on balanced ternary (and the math tables) here:

I was going to add the tables in where the current multiplication one is, but I don't have the time right now to learn the Wiki table format. If anyone would like to convert the tables over for use here, feel free.

FYI the link is dead, the user's directory no longer exists. pity, sounded like what I was looking for. -- (talk) 16:09, 14 January 2010 (UTC)

Balanced Ternary math algorithms?[edit]

Should there be a section on ternary or balanced ternary math tricks? (Such as the "'repeatedly add up the digits until you get a one digit number' trick to test for even in BT?" [If the resulting digit is zero, it is even))

Should there be a section of random ternary facts (Such as "Multiply by 3^n+1 in balanced ternary leaves the last n digits intact")

Should there be a section on ternary math algorithms? (Is BT shift and add or shift and exponentiate enough different from the binary versions to be worth showing? (Or even mentioning?)

Should there be some actual examples of arithmetic in the system?


There is a pointer to the Russian machines, but not to Thomas Fowler's 1840 ternary calculator. ("The Ternary Calculating Machine of Thomas Fowler", in the IEEE journal "Annals of the History of Computing, Vol. 27, No. 3, p 4-22 at IEEE Explore) This leaves one with the misconception (promoted by the Russian pages) that there was nothing in ternary before the Russian machines. Nahaj 14:57, 8 November 2005 (UTC) Addendum: There is now a link in the article. Nahaj 02:15, 25 May 2006 (UTC)


I don't recognise this system as a computing student, is it really that commonly used?

It is not that commonly used. But it has interesting theoretical advantages (Such as [as Knuth points out] rounding and truncation being the same operation. And there have been machines built using it. (Since the 1800's!!!) I encountered it as a computer science student, so some "computing students" do. Nahaj 03:01, 3 January 2006 (UTC)

The earliest mention of a base three counting / number system is the Tai Hsuan Jing by Yang Tse Yun (Yang Hsiung), approx -10 to +10. Yang's work has been translated (mostly) but the passage detailing the construction of the Shou of the Tai Hsuan (0-80) are clearly a base 3 counting system, was not included! This was outlined in my paper to Dr. Needham.

VR Pat Kirol —Preceding unsigned comment added by Sunwukongmonkeygod (talkcontribs) 21:44, 25 October 2008 (UTC)


I moved the article to Ternary numeral system, for consistency and also because "ternary" can mean ternary operation, which is used frequently in several programming languages. Ternary has been turned into a disambiguation page. æle 19:13, 16 January 2006 (UTC)

Cantor set[edit]

Should the article mention the Cantor set which consists of the elements of [0,1] whose ternary expansion does not contain any 1s? —Tobias Bergemann 10:45, 19 January 2006 (UTC)

The article says: "(i.e. you do not have non-unique representations for the same number such as .999.. and 1.00...)". This is untrue. The ternary repeating number 0.22 is exactly equal to 1. Ternary is useful for describing the Cantor set because of exactly what Tobias Bergemann said above. In fact, the ternary representation of the Cantor set as ternary numbers not containing the digit 1 relies on the equivalency of 0.1 (which is in the set) and 0.02, which fits the construction. —jellyvista 01:59, 21 July 2009 (UTC)

The size of a tryte[edit]

From the article:

"Ternary also has a unit similar to a byte, the tryte, which is six ternary digits."

This is heavily (or at least somewhat) disputed: some would consider it to be nine trits, and I would go so far as to include 27 as a possibility. --Ihope127 20:34, 19 January 2006 (UTC)
Three years later, this has not been changed. Anyone? (talk) 04:59, 5 March 2009 (UTC)
Removed for lack of verifiability. --beefyt (talk) 20:59, 20 January 2010 (UTC)
Even better: found a reference for the Setun 70, added under tryte. --beefyt (talk) 21:29, 20 January 2010 (UTC)

Balanced Ternary move.[edit]

Balanced ternary was moved to its own separate article, but all the balanced ternary talk was left here.... Anyone object to a move of the talk to follow the article? (: I wouldn't mind, of course, if someone just moved it. :) Nahaj 02:13, 25 May 2006 (UTC)

Strange fonts[edit]

On my machine, the symbols ⅓ and ⅔ appear as thirds when editing, but as halves when reading. A glyph for two halves--how strange! I would think then it would be better to avoid them anywhere. Davilla 22:49, 22 August 2006 (UTC)

More (not quite so) practical uses of trinary[edit]

Trinary has appeared in a few esoteric programming languages, like Malbolge and TriINTERCAL. 09:39, 15 October 2007 (UTC)

Another example[edit]

In electronics there are three possible states: no voltage (0 V), positive voltage (e.g. +2 V), negative voltage (e.g. −3 V). So these states are trits. (talk) 02:19, 30 May 2009 (UTC)

Gideon Frieder's replacement ternary numeral system is not explained[edit]

Gideon Frieder's replacement ternary numeral system called Ternac (different from the one used in the Setun) is not explained.

Fictional Uses and Popular Culture[edit]

As stated in The Klingon Dictionary, the fictional Klingon race from Star Trek once used a trinary number system. Completely random, and I don't know if it would be significant enough to add to the main article, but it's interesting to note a fictional reference to this system. Steamboat28 (talk) 17:30, 27 February 2010 (UTC)

Appositive in lead[edit]

This is minor, but there's no way for me to back and change my edit summary. diff 434244014

Rather, it is an appositive, but it's a restrictive one, and thus commas are erroneous. -- C. A. Russell (talk) 14:53, 14 June 2011 (UTC)

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Merge from "Nonary" and "Base 27"[edit]

As nonary and base 27 are based on powers of three, they are essentially compact notations for two or three digits of ternary. They would best be covered as sections under the "ternary numeral system" article.Bcharles (talk) 21:50, 11 July 2013 (UTC)

This may be a good idea, however there are specific parts of nonary base system which are due to the similar base number as decimal, which should be included. --Reububble (talk) 08:35, 3 August 2013 (UTC)
Though nonary and base 27 are based on powers of three, its better to have separate articles for them. Even octal and hexadecimal are based on powers of two but they are not merged with binary. The Compact ternary representation: base 9 and 27 section in Ternary_numeral_system is good enough and merging will only add to confusion for people wanting information on ternary number system. SmackoVector (talk) 06:16, 1 September 2013 (UTC)
Yup, there're some special properties of these two system, just like that many special properties of octal and hexadecimal. The only difference is their rare usage, which shouldn't be a reason why they'd be merged. Chenxiaoqino (talk) 12:24, 1 January 2014 (UTC)
Given the obscurity of these number-systems, a merge seems appropriate to me. --Yoderj (talk) 19:22, 2 October 2013 (UTC)

I think the merge from nonary would be okay. The nonary article has very little in it that's specific to that base. But base 27 seems a different case to me. There are three reasons to be interested in base 27 (given by the three sections of its article):

  1. its easy conversion to and from ternary,
  2. the closeness of its number of digits to the number of letters in the Latin alphabet, which gives rise to some applications in character encoding and gematria, and
  3. its apparent use in certain southwest Pacific cultures.

The second and third of these have little or nothing to do with ternary, and are properly sourced in the base 27 article. So I think it has enough independent notability to stand on its own (in my opinion at least as much notability as say balanced ternary, which is also a separate article now). Based on this reasoning I have removed the (stale) merge tags from the base 27 article, but left in place the nonary ones. —David Eppstein (talk) 02:01, 12 January 2014 (UTC)

  • They should remain separate articles and not be merged. They aren't obscure, they are quite basic stuff we mathematicians study and occasionally work with. To say these articles should be merged is like saying we should merge articles on subatomic particles because they're "obscure". Sofia Koutsouveli (talk) 13:33, 10 March 2014 (UTC)