Talk:Non-standard positional numeral systems

Article creation

I have addressed certain issues by creating the article Non-standard positional numeral systems, and making related changes to Unary numeral system, Golden ratio base, Quater-imaginary base, Positional notation, Base (mathematics), and Category:Positional numeral systems. I suggest further discussion of these issues takes place here.--Niels Ø 14:44, 26 February 2006 (UTC)

I have now made related changes to Negabinary, Negaternary, Mixed radix and Bijective numeration as well. I'm not quite happy with the development at the category page category:Positional numeral systems - can someone help me with the cat sorting tags used for the non-standard systems? They should appear separate from the standard ones, perhaps in this order:

• Non-standard positional numeral systems
• Mixed radix
• Unary numeral system
• Golden ratio base
• Negabinary
• Quater-imaginary base
• Negaternary

--Niels Ø 21:04, 26 February 2006 (UTC)

First paragraph

The first paragraph is erroneous and the article needs some Wikification. As the article positional notation indicates, a positional notation system does not contain a base number of glyphs. For example, the sexagesimal system does not contain 60 glyphs—the number varies from two glyphs in Babylonia to fifteen in the Greek form (there are ten in the modern form). But this article is not about standard systems, so I'm not sure whether to correct it or delete it. Wikipedia requires the article title to appear in bold as close to the beginning of the article as possible. You only have This article. This means that the first paragraph should introduce non-standard systems, it should not discuss standard systems first, maybe later. — Joe Kress 05:57, 27 February 2006 (UTC)

I'm not sure about the terminology; you may be right. There are 60 different somethings in sexagesimal, but those somethings may not be properly denoted glyphs as they are themselves made up of a number of more primitive somethings. Help if you can see a good solution.

Having the title appear in bold close to the beginning seems a bit silly to me in this case, but I will try to rephrase it to satisfy this policy.--Niels Ø 17:04, 27 February 2006 (UTC)

Glyph is unfortunate. Numeral? Sign? Septentrionalis 17:08, 23 March 2006 (UTC)

Symbols? I really don't know - someone, please help!--Niels Ø 21:30, 23 March 2006 (UTC)

Two possibilities, "representations" or "positions", neither of which sound good. An alternative would be to classify the sexagesimal system itself as a non-standard system because it does not use sixty glyphs. Unfortunately, that opens up a Pandora's box because the vigesimal system used by MesoAmerican peoples does not use twenty symbols, rather it uses three: a shell for zero, a dot for one, and a bar for five. I note that the vigesimal article has the same error as this article, assuming that it has twenty glyphs, or at least it only mentions modern respresentations. The title of this article itself is unfortunate. I suspect that whoever invented the term "non-standard positional numeral systems" was only thinking of the decimal, hexagesimal, and other computer systems. It would be quite odd to include both sexagesimal and vigesimal as non-standard when both were invented long before the decimal system. That makes the decimal system itself non-standard because it is so recent. Maybe exclude "classic" numeral systems? Or maybe call these "invented positional numeral systems"? — Joe Kress 19:33, 24 March 2006 (UTC)

Substituting numeral for glyph; if someone sees a better solution, go for it. Septentrionalis 21:55, 24 March 2006 (UTC)

It also appears that there's a problem with the claim that with just numerals, a radix, and the minus sign, all reals can be represented. It doesn't appear to me that any real that's not also a rational can be represented with just those three things, in order to represent a non-rational real, you'd need more symbols. And not even all rationals can be represented, those with repeating digits need a symbol to indicate that they're repeating. — Preceding unsigned comment added by 157.166.167.129 (talk) 12:54, 28 June 2018 (UTC)

Well, it is a statement that needs to be understood or interpreted properly to be true - but when I wrote it I felt it would become too long-winded to make it 100% precise in this context. The understanding needed is that digit strings can be infinitely long. One could add an overbar to the inventory of symbols to indicate a repeating period, and then any rational could be represented in any base with a finite number of symbols. But to represent e.g. pi or squareroot-of-2 in any base, you would still need an infinite number of symbols, or a finite formulation of a rule that would generate those infinitely many symbols. Another thing that could be included (but that I think would be too long-winded) is that, not allowing leading zeros before the radix point or trailing zeros after it, the representation as a digit string is unique except that those numbers that have a finite representation also have an infinite one (like 1. = 0.999... in decimal).

As for editing the article, one possibility is to remove the sentence about signs and radices, limiting the scope to nonnegative integers, but I think it is better to mention this briefly.

So, if you have a suggestion for a way to make it more precise without making it too long-winded, be bold and fix it (or suggest it here if you prefer)!

If not, my suggestion is to leave it as it is.--Nø (talk) 14:29, 28 June 2018 (UTC)

Cultural history vs mathematical idea

I'm quite happy with the improvements made by others to "my" article. There is an issue bothering me, relevant to many of the articles on numeral systems: Some have a focus on the abstract mathematical idea of various numeral systems, and others focus on the cultural history of numeral systems. Many mix these two subject areas, and a clearer distinction between them would improve many of the articles. I don't really see how to do it, though. Quoting Joe Kress above, "the sexagesimal system does not contain 60 glyphs—the number varies from two glyphs in Babylonia to fifteen in the Greek form (there are ten in the modern form)." Now, that's cultural history; in the mathematical ideal sexagesimal system, there definitely are sixty somethings, that now are called numerals instead of glyphs in the article. The word glyph that I originally used was a bad choice, as it puts focus on the physical manifestation, and hence on the cultural history, and not on the mathematical idea I intended.--Niels Ø 09:18, 25 March 2006 (UTC)

Neologism

As the originator of this article, I must confess the title was a neologism. However,

1. It seems to have caught on - not only wiki mirrors, but others too now use it.
2. You can see the title of the article as descriptive of the contents, rather than as a generally accepted concept to be described.
3. The little word "here" very early in the intro is meant to indicate that the title is not a generally accepted concept, but only a convenience used "here", i.e. in the present encyclopedia.

Still I wonder, have I sinned here?--Niels Ø (noe) 13:14, 13 June 2007 (UTC)

Japanese article

There's an interwiki link to ja:広義の記数法 - I've no idea what that article is saying, but it clearly has material we don't have.--Niels Ø (noe) (talk) 17:02, 6 December 2007 (UTC)

The comment above that I made nearly nine years ago still holds true. Anyone knowledagble in Japanese who can help?--Nø (talk) 11:53, 27 October 2016 (UTC)

Using Google translate, it seems the Japanese article is about numeral systems in some broader sense, but not at all about what´we cover in this article. I have now removed the interwiki link or claim or whatever those are called today. The Korean one is OK, though.--Nø (talk) 11:54, 13 October 2018 (UTC)

Gray code?

The Gray code does not currently appear in this article, or in the associated category. Should it do so?--Noe (talk) 19:54, 8 August 2008 (UTC)

Citations?

I've just removed a general citations-needed-tag from the article - I don't think it is relevant.

The article is little more than a brief summary of and pointer to uncontroversial information found in other wikipedia articles; if they lack sources, they should be flagged. (To my knowledge, the only thing that some editors feel is controversial is the inclusion of Unary as a Positional system - but I think that is dealt with in a satisfactory way in the article.)

And if anything in particular in this article lacks sources, that should be flagged.

As can be gathered from the posts above, the title of this article is a neologism, or (as I prefer to see it) a description of something that it was convenient to collect in one place as a separate wikipedia article. Perhaps that idea is challenged by the editor asking for sources for the concept "Non-standard positional numeral systems". Such sources do not exist! (Actually they do now - but only in the multitudes of wiki mirrors and in a few scientific articles that also have picked it up from us.)

Think of this article as similar to the many articles called "List of (blah blah)"!--Nø (talk) 13:30, 28 June 2010 (UTC)

So the general citations tag is back; I still think it makes little sense. Possibly this article should be renamed to "List of non-standard positional numeral systems"; maybe detailed citations could be dispensed with then. (As I said above, of course the article on each numeral system must be sourced!). I'm not on top of official wikipedia policies on citations and list, but there are load of lists without citations that haven't been challenged. Or is this article too detailed to be considered a list? - Is that why someone feels it needs citations? Please explain! If it is specific statements in the article that bothers someone, please tag those rather than the whole article. -- There's also a new article called List of numeral systems (clearly still work in progress); perhaps it would be a good idea to merge the present article into that list.--Nø (talk) 07:46, 18 March 2011 (UTC)

Endianness

Are there positional number systems where the least significant digits are on the left? Like decimal number written in reverse. — Preceding unsigned comment added by 195.113.21.109 (talk) 10:55, 24 October 2012 (UTC)

Some computers put the least significant byte first (and arguably they all put the least significant bit first within the byte, i.e. the LSB is "bit 0"). That's all that springs to my mind. —Tamfang (talk) 14:31, 24 October 2012 (UTC)

I am told that spoken Arabic is little-endian, so our way of writing digits matches their way of speaking numbers. —Tamfang (talk) 21:27, 27 June 2023 (UTC)

redundancy of "denoted"

If "x may be a y", it is implied the x may also not be a y. To say that "x may be denoted a y" adds nothing. Bhny (talk) 05:31, 28 February 2013 (UTC)

E.g., unary is arguably a postional notation, arguably not, so it may be denoted... Hence, "denoted" is not redundant.--Nø (talk) 07:20, 28 February 2013 (UTC)

that makes no sense Bhny (talk) 08:20, 28 February 2013 (UTC)

What doesn't make sense? Unary matches the description of standard positional systems on most points, but with a twist, and that happens to be a twist that makes it easy to argue that it is not "positional". There is no established truth in the literature to appeal to in order to settle the question whether unary is a "positional system" or not.

Anyway, I've now tried another compromise - quote:

Non-standard positional numeral systems here designates numeral systems that are in some sense positional systems, but that deviate from the following description of standard positional systems:

Personally I don't think it is better than what we had a week ago - I think both version are OK.--Nø (talk) 14:27, 28 February 2013 (UTC)

Alternative suggestion:

Non-standard positional numeral systems here designates numeral systems that may loosely be described as positional systems, but that do not comply with the following description of standard positional systems:

What do you think?--Nø (talk) 10:19, 13 April 2013 (UTC)

Another type of multi-base numeral system?

In Frederik Pohl's 1984 novel Starbust, one of the characters has to transmit a vast amount of information across space, and uses a process she calls "Gödelization": After encoding the information into an integer, she somehow reduces the integer to a short summation of integer powers (e.g. 11^4+31^3+43^17). This is somewhat different from Gödel numbering and I wonder if it's really something Gödel descrbed. But it seems to be a candidate to put in this article.

There are 4 Wikipedia articles that use the term "Gödelization" in passing : Counter machine, Register machine, Turing's proof, and Algorithm characterizations

Casu Marzu (talk) 18:53, 31 October 2013 (UTC)

Systems without radix point

There are no article or information about numeration systems for real (or more general sets) which do not require the use of a radix point symbol. That's important because the radix point is an underutilized symbol, since it can appear only once, so it is wasteful from a coding efficiency perspective. — Preceding unsigned comment added by 181.20.129.191 (talk) 02:51, 28 February 2015 (UTC)

More systems?

The category with the same name as this article includes the articles

• Asymmetric Numeral Systems
• Counting rods
• Ostrowski numeration
• Redundant binary representation
• Skew binary number system

that are not currently linked in the article. I haven't checked each to see if it makes sense to include them - but it would be woth looking into.--Nø (talk) 11:56, 3 November 2016 (UTC)

Asymmetric numeral systems

A subsection on "Asymmetric numeral systems" has been added by an IP user to the section "Bases that are not positive integers". But as far as I can tell, asymmetric numeral systems do not have a single well-defined base, so I think it belongs as a separate section following "Mixed bases" (or perhaps as a subsection of "Mixed bases"). But I do not understand asymmetric numeral systems well enough to be sure.--Nø (talk) 09:23, 31 January 2020 (UTC)

It seems to me it should probably be a subsection of "Mixed bases," as a read-through of the article details a number system to encode data where each place-value may represent a different amount. However, it may be more useful to incorporate it into the prose of the mixed bases section rather than give it its own section at all. Given the subsections are so small, I've also been considering doing this with the subsections of "Bases that are not positive integers." _{Integral Python} ^{click here to argue with me} 14:13, 31 January 2020 (UTC)

Standard numeral systems can be obtained as special cases of ANS, but in general case it works as stack - we can practically only put/push the last digit. Using some base b suggests that all digits have 1/b probability - asymmetry of ANS allows to modify this probability distribution to any chosen, like each digit having an individual basis. 188.147.49.134 (talk) 15:03, 31 January 2020 (UTC)

Though I cannot claim I fully understand the reply from User:188.147.49.134, or the article, I think what it is asying is that, although inspired by mixed base systems, asymmetric systems are (in the most general case) something far more fancy that would not fit into the section on mixed case systems, but that would natrurally follow it.--Nø (talk) 18:38, 31 January 2020 (UTC)

Too technical

I am someone who isn't completely stupid about math, and this article is too technical for most people. I added the template at the top to help with the cleanup.

Language Boi (talk) 21:32, 24 November 2023 (UTC)