Category talk:Positional numeral systems

[untitled]
I set up this category to avoid various lists of "see also" links in each article, linking the others.

Is there a way of sorting the links on this category page in a more intelligent way? The system I'm using presently is to pipe each categorization as blank-blank-digit for one-digit bases (2-9, e.g. " 2") and as blank-digit-digit for two-digit bases (10-99, e.g. " 16"). No articles on bases higher than 99 seem to exist (so far!). If you think the system will work up to 999 you are wrong - beginning with 100, annoying headings like 1 (for 100-199) etc. would appear on the category page.

There is also an article series defined by the table "Table Numeral Systems" (included on this page), but its focus is on numeral systems used by different cultures, rather than the math of different bases. In fact, it would make sense to remove binary, octal and hexagesimal from this table, as no cultures seem to use these (unless computers or computer specialists are considered a culture).

Non-standard positional numeral systems
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 at talk:Non-standard positional numeral systems.--Niels Ø 14:42, 26 February 2006 (UTC)

Rename this category?
When I created this category, the idea was to list pos.num.systs by base. This idea has repeatedly been violated by other editors, perhaps because the category has a misleading name. My purpose creating the category was to avoid accumulation of "see also" links to other bases at the bottom of each article.

There seems to be two issues involved: One is the sort order of the category (where my intention with this category was to sort by base), the other is that some articles deal have a specific mathematical base as their subject, others have the number notation used in a certain culture as their subject (where my intention was to categorize the articles with a base as their subject).

What to do?

One could create a new subcategory "Positional numeral systems by base", leaving this main category for the culturally inclined articles (alphabetically sorted). But again, the fact that "Rod numerals" belongs in the main category and not in the "by base" sub category would not me absolutely clear. Or one might change the "by base" list from a category into a list.

Ideas? Views?--Noe (talk) 13:08, 23 June 2008 (UTC)


 * Hmm, no answers for quite some time. I still think there is a problem to be solved, but I'm not sure how. The category mechanism in wikipedia is not too good if we want the list sorted under headings like "2", "3", ..., "10", etc. Perhaps we should let the category be sorted by default (Binary under "B" etc.), and instead make a List of positional numeral systems withe a section By base with the desired subheadings, and where the subsection "10" could include both Decimal and Rod numerals. And another section By culture, e.g. with subheadings like "Asia", "Europe" etc. Would that be appropriate?--Noe (talk) 14:03, 29 March 2009 (UTC)

Notability
sligocki has raised questions about the notability of the articles about these bases: I propose to discuss that question below.--Noe (talk) 09:48, 23 October 2009 (UTC)
 * 5 Quinary
 * 7 Septenary
 * 11 Undecimal
 * 13 Base 13
 * 14 Tetradecimal
 * 15 Pentadecimal
 * 18 Octadecimal
 * 26 Hexavigesimal
 * 27 Septemvigesimal
 * 30 Base 30
 * 32 Base 32
 * 62 Base 62
 * 64 Base 64


 * sligocki has allready changed e.g. Hexavigesimal into a redirect to Positional numeral system, but as this article doesn't contain e.g. the word Hexavigesimal, I do not think this is a satisfactory solution. If Hexavigesimal is a word that belongs in wikipedia at all (even if just as a redirect), then looking it up should at least provide the information that it's base 26.--Noe (talk) 09:48, 23 October 2009 (UTC)


 * Right, I'm not exactly familiar with formal procedures on Wikipedia. I decided that Hexavigesimal (base 26) (along with Octadecimal (base 18) and Septemvigesimal (base 27)) are just ridiculous and clearly not notable so I redirected them to positional notation. It is not to say that they are synonyms, but rather unnotable types of positional notation. What do you guys think? I don't have problem with a formal deletion (except that there seems to be no reason to get rid of the history). Cheers, — sligocki (talk) 10:06, 23 October 2009 (UTC)


 * Nevermind, they've been reverted back to articles by User:Robo37. I've added them to my classification of the bases bellow. Cheers, — sligocki (talk) 09:42, 25 October 2009 (UTC)

Hi guys, here's my roundup of articles
 * There are articles on bases 1–16, 18, 20, 24, 26, 27, 30, 32, 36, 60, 62, 64
 * I'm leaning toward keep on:
 * unary, binary, trinary, octal, decimal, hexadecimal are all commonly used in math. Clear notability.
 * base 20 and base 60 have historical notability as early numeral systems, seems good.
 * base 12 is used for music and has often been suggest because there are many factors of 12, seems like a pretty good article.
 * base 36 seems to be widely used in CS for some reason, probably notable.
 * base 4 claims to be useful for Hilbert curves.
 * Skeptical about:
 * base 24 seems to claim notability because of the 24-hour/day system, but that's not really a positional system, only one 24-digit, I'm skeptical of notability.
 * base 6 and base 30 claim notability because all primes have certain endings, it is not noted whether this is used by anyone favorably. Dubious notability.
 * And leaning towards delete (redirect to positional system) for:
 * base 32 and base 64 seem to be here simply because of Base32 and Base64. I don't see why the have their own articles.
 * base 5, base 15, base 27 seem most notable for being used in a natural language (maybe). Copied info over to Decimal.
 * base 7, base 9, base 11, base 13 seem most notable for the plethora of sci-fi refs (perhaps we could have a general article about non-standard bases)
 * base 26, base 62 seem to claim notability because there are 26 letters in our alphabet ...
 * base 14, base 18 seem completely unnotable.
 * Cheers, — sligocki (talk) 09:59, 23 October 2009 (UTC)

Please let me know what you guys think of my suggestions and appropriate courses of action to take (merge and redirect, delete, etc.) Cheers, — sligocki (talk) 09:42, 25 October 2009 (UTC)


 * My humble opinion is that several of these articles need improvements, and perhaps a handfull of them (but no more) might simply be deleted or changed to redirects, if the article to which they redirect would be helpfull for the reader.--Noe (talk) 17:14, 25 October 2009 (UTC)


 * I generally agree with your assessments of which numeral systems to keep (your first list). I would add base 5 to your list, because the comparison of quinary and biquinary systems in natural languages is relevant information that is particular to base 5, but the other natural-language-based systems (bases 15 and 27) can easily be merged. rspεεr (talk) 00:23, 26 October 2009 (UTC)


 * Mostly agree. base 32 and base 64 can be merged/redirected to Base32 and Base64 without much loss. The non-decimal natural language ones might be ok to merge, but certainly not delete. The sci-fi ones can be merged and have a half-decent article between them. --Cyber cobra (talk) 05:04, 26 October 2009 (UTC)


 * I'd like to see a draft of those merged articles (or of modifications to existing articles that the redirects would lead to), before I agree to this. I think it's hard to do in an elegant way, and I don't see any harm in having a few stub-like articles on various bases. (I absolutely agree about e.g. base 32 and 64, though.)--Noe (talk) 08:16, 26 October 2009 (UTC)


 * Looking over some of the articles it seems to me there are two types of material. There is linguistic material that describes the use of non-decimal numbering systems in various exotic languages. It's a bit of a stretch to put this in articles about positional numerals since they are basically linguistic and not mathematical in nature. It seems to me that this material could be merged into the articles on the respective languages and perhaps summarized in a single article about numbering systems in comparative linguistics. The other type of material is about actual positional notation in various bases. Articles on these should probably be restricted to those in common usage (base 10), with historical significance (e.g. base 60), applications in mathematics or computer science (e.g. binary), or those where a reliable source can be found to show that it's of theoretical interest. I doubt the articles in sligocki's list would survive this test.--RDBury (talk) 17:18, 28 October 2009 (UTC)


 * I agree, I think there could be an interesting article/section on non-decimal numeral systems in exotic languages. Do you think it would be it's own article or if not, which article should it be a part of? Cheers, — sligocki (talk) 18:19, 28 October 2009 (UTC)

This discussion seems to have lost focus. Proposals to merge have to be considered on their individual merits, naturally. I have a question: why assume that no one is interested in a question like "where in the world do people use base 5 counting?" In other words, this material could be merged into the articles on the respective languages with the suggestion that this is an issue in comparative linguistics seems not the meet the research needs of an enquiry that could be homework for a ten-year-old. And that's for counting base-5: which is not particularly alien by the way (shepherds in my part of the world used it till quite recently). By the way, "exotic" applied to languages as if they were some kind of imported fruit is just derogatory. The value of information to the encyclopedia about languages should have nothing to do with considerations of remoteness, since our content policy mentions nothing of the kind. Altogether there have been some sweeping statements and proposals made, and the debate here should be brought back to the common ground of looking at what verifiable, serious material and sources can be found for the articles in question. Charles Matthews (talk) 19:14, 28 October 2009 (UTC)


 * If you would like to post notable content about base 5 as used by shepherds that would be great. However, as a simple example, base 27 claims that it's used by "the Telefol language and the Oksapmin language of Papua New Guinea." I claim that this in-and-of itself do not make the topic notable and the article clearly has not developed into much and is basically an orphan (linked to from base 3, 27 (number) and a two lists). How likely is it that some researcher (be they 10 or 50) is going to search for base 27 and find this page notable? I think it would be someone who was searching for non-standard bases used in natural language. Why not collect all of those facts together into a single article on non-standard bases used in natural language? Maybe we can even combine it with non-standard bases used in science fiction! But right now, there are a smattering of articles which list one natural language fact and then some boilerplate, unnotable mathematical properties. Cheers, — sligocki (talk) 22:32, 28 October 2009 (UTC)


 * Well, your argument that a numerical system used by native speakers of a certain language is not thereby made notable is about a clear as example of systemic bias as one could wish to see. In other words, you are arguing from a cultural position that may predominate here, but your argument has no merit in encyclopedic terms. The fact that you combine it with the suggestion that references in science fiction rate as highly as the way some human beings live is simply crass. Charles Matthews (talk) 22:49, 28 October 2009 (UTC)


 * It looks a lot less crass once you realise that the combined number of speakers of Telefol and Oksapmin (according to our articles) is less than 15,000. This suggests to me that it makes sense to just follow our usual principles: Base 27 gets its notability from the fact that it is discussed in reliable sources as being used in these languages, not directly from the fact that it is thus being used. Hans Adler 23:02, 28 October 2009 (UTC)

My feeling is that if we can document that a certain base was significant as a positional number system in a historical culture, and the article contains significant content on how that number system was used in that culture, it should stay. If it just says "obscure culture X used a number system that was related to the number B" then that's not really base B arithmetic and it verges on trivia. And if some programmer somewhere found it convenient to use a certain base because of the limitations of some communications channel or whatever, that's probably also not good enough. If some standards committee found it convenient to incorporate modulo-b arithmetic into their standard as a checksum, that's definitely not enough, because it's not about positional number systems. And if some sci-fi author invented seven-fingered aliens with a septenary system, that's definitely not enough because it's just trivia. So: bases 2, 3, 8, 10, 16, 60: obviously deserve separate articles. Base 5: probably does if the claims of historical usage as a positional system can be backed up and expanded. All the rest: marginal at best. Bases 13, 15, 18, 27, and 32 contain claims of historical usage but it's far from obvious whether that usage could reasonably be described as a positional number system, and the rest don't even have that. —David Eppstein (talk) 00:51, 29 October 2009 (UTC)


 * Base 12 is missing from your list. The duodecimal article has a whole lot of notable information, including references to entire books written about it, so I wouldn't call it "marginal at best". rspεεr (talk) 01:26, 29 October 2009 (UTC)
 * Yes, add 12 to the list of clear keeps. I looked through the list of bases posted at the start of this thread, but that didn't include some that were obviously notable. —David Eppstein (talk) 02:44, 29 October 2009 (UTC)

Right, I've listed all of the notable things about each of the bases below. It should help you tell which ones are noticeable enough to have their own articles.

Table

 * Base -10 (negadecimal)
 * Has been studied in pure mathematics.
 * Can accommodate all the same numbers as standard place-value systems.
 * Both positive and negative numbers are represented without the use of a minus sign.


 * Base -3 (negaternary)
 * Has been studied in pure mathematics.
 * Can accommodate all the same numbers as standard place-value systems.
 * Both positive and negative numbers are represented without the use of a minus sign.


 * Base -2 (negabinary)
 * Has been studied in pure mathematics.
 * Can accommodate all the same numbers as standard place-value systems.
 * Both positive and negative numbers are represented without the use of a minus sign.
 * Implemented in the early Polish computer BINEG.


 * Base 1 (unary)
 * The simplest numeral system to represent natural numbers.
 * Widely used.


 * Base 2 (binary)
 * 2 is a primorial.
 * Radix economy is second only to base 3.
 * The "root base" for bases 4, 8, 16, 32 and 64.
 * Widely used.


 * Base 3 (ternary or trinary)
 * Used in comparison logic.
 * Integer base with best radix economy.
 * A balanced ternary variant provides exceptional economy for some applications.
 * Used in ternary computers.
 * Used in Islam to keep track of counting Tasbih to 99 or to 100 on a single hand for counting prayers.
 * Used to denote fractional parts of an inning in baseball.
 * Used to convey self-similar structures like a Sierpinski Triangle or a Cantor set conveniently.
 * The "root base" for bases 9 and 27.
 * Used in a few cultures.
 * Fewest sides on a regular polygon, a triangle.


 * Base 4 (quaternary)
 * 4 = 22
 * Each digit can be represented as two binary digits.
 * Useful in the study of powers of two.
 * Used in the representation of 2D Hilbert curves.
 * Originally used by the Chumashan languages.
 * Used in the Ventureño language.
 * Related to the way the genetic code is represented by DNA.
 * Has been used for the data transmission from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits.
 * Radix economy is second only to base 3.
 * A "sub-base" for bases 16 and 64.
 * Number of sides on a square.
 * Fewest faces, and vertexes, of a regular polyhedron, a tetrahedron.


 * Base 5 (quinary)
 * Can be represented by counting your fingers on one hand.
 * Often used for counting as groups, sets or clusters of 5.
 * Used in the Gumatj language.
 * Used in the Nunggubuyu language.
 * Used in the VKuurn Kopan Noot language.
 * Used in the Saraveca language.
 * The "root base" for base 25.
 * Number of sides on a pentagon.


 * Base 6 (senary or heximal)
 * 6 is a primorial.
 * 6 is a perfect number.
 * More unit fractions terminate than with decimal, and this is the lowest base to have this property.
 * Useful in the study of prime numbers.
 * Useful in the study of perfect numbers.
 * Can be represented with one hand per hexit.
 * Used with Diceware.
 * Used in the Ndom language.
 * Used in the Proto-Uralic language.
 * The "root base" of base 36.
 * Largest number of sides on a regular polygon that can tile a flat surface, a hexagon.
 * Number of faces on a cube; number of vertexes of an octahedron.
 * Fewest edges of a regular polyhedron, a tetrahedron.


 * Base 7 (septenary)
 * Used by the Tau of Sci-fi Table-top battle game Warhammer 40,000.
 * Used by the time lords in Doctor Who.
 * Used for a countdown clock in the Halo 3 Alternate Reality Game "IRIS".
 * Used by the Dwarven miners in the online RPG Kingdom of Loathing.


 * Base 8 (octal)
 * Each digit can be represented as three binary digits.
 * Useful in the study of powers of two.
 * Used in computing.
 * Can be represented by counting the gaps in between your fingers.
 * Used by the Pamean languages.
 * Used in the Yuki language.
 * Used by Charles XII of Sweden.
 * Used by the fictional alien felinoid species Kilrathi of the Wing Commander universe.
 * Used by the Octospider species of Rama Revealed and the computer game RAMA.
 * Used by the Alterans from Stargate SG-1.
 * Referred to by the satirist Tom Lehrer in his song parodying new math.
 * Used in the first-person shooter Prey.
 * Used by the Tau in the Warhammer 40,000 universe.
 * Used in The Beekeeper's Apprentice, Laurie R. King's first Sherlock Holmes pastiche.
 * Used by the Hutts in Star Wars.
 * Used by the pachydermoid Fithp in the Niven/Pournelle novel Footfall.
 * A "sub-base" for base 64.
 * Number of faces on an octahedron; number of vertexes on a cube.


 * Base 9 (nonary)
 * Each digit can be represented as two ternary digits.
 * Useful in the study of powers of three.
 * Useful in the study of triangular numbers.
 * Except for three, no primes in nonary end in 0, 3 or 6.
 * A nonary number is divisible by two, four or eight, if the sum of its digits is also divisible by two, four or eight respectively.
 * Used by the fictional civilization, The Culture, found in Iain M. Banks' books.
 * There are 9 Heegner numbers.


 * Base 10 (decimal)
 * Widely used.


 * Base 11 (undecimal)
 * Use for the check digit for ISBN.
 * Used in Carl Sagan's novel Contact
 * Used by the fictional Psychlos in L. Ron Hubbard's book Battlefield Earth.
 * Used in the show Babylon 5.


 * Base 12 (duodecimal or dozenal)
 * More unit fractions terminate than with decimal.
 * Number of faces on a dodecahedron; number of vertexes on a icosahedron.
 * Number of edges on a cube or octahedron.
 * Widely used.


 * Base 13 (tridecimal or tredecimal)
 * Used in the Maya calendar.
 * In the end of The Restaurant at the End of the Universe by Douglas Adams, a possible question to get the answer "forty-two" is presented "What do you get if you multiply six by nine?", which is correct in Base 13.


 * Base 14 (tetradecimal)
 * Used for programming for the HP 9100A/B calculator.
 * Used for image processing applications.


 * Base 15 (pentadecimal)
 * Used in the Huli language.
 * Used for telephony routing over IP.


 * Base 16 (hexadecimal)
 * 16 = 32
 * Each digit can be represented as four binary digits.
 * Each digit can be represented as two quaternary digits.
 * Useful in the study of powers of two.
 * Useful in the study of powers of four.
 * Widely used.


 * Base 18 (octadecimal)
 * More unit fractions terminate than with decimal.
 * Used in the Mayan calendar.
 * Used for some specialised computer systems where octadecimal notation has been used for high levels of data compression.
 * All prime numbers apart from 2 and 3 will end in 1, 5, 7, B, D or H.


 * Base 20 (vigesimal)
 * Every unit fraction that terminates in decimal also terminates in vigesimal.
 * Number of faces on an icosahedron; number of vertexes of a dodecahedron.
 * Widely used.


 * Base 24 (quadrovigesimal)
 * More unit fractions terminate than with decimal.
 * Used to represent the number of hours in a day.
 * Used in the Umbu-Ungu language.
 * Used in the Kakoli language.


 * Base 25 (pentavigesimal)
 * Each digit can be represented as two quinary digits.
 * Useful in the study of powers of five.
 * Can be represented with your hands.


 * Base 26 (hexavigesimal)
 * Can represent numbers using nothing but the 26 letters used in the Latin alphabet.
 * Can use this to turn numbers into letters.
 * Can use this to turn letters into numbers.
 * Used in nominal numbers.
 * Used in serial numbers.


 * Base 27 (septemvigesimal)
 * 27 = 23
 * Each digit can be represented as three ternary digits.
 * Useful in the study of powers of three.
 * Used in the Telefol language.
 * Used in the Oksapmin language.
 * Occasionally used in puzzles to transform ternary triplets into words or messages.


 * Base 30 (trigesimal)
 * 30 is a primorial.
 * More unit fractions terminate than with decimal.
 * More unit fractions terminate than with any other base lower than 60.
 * Every unit fraction that terminates in decimal also terminates in trigesimal.
 * Every unit fraction from 1/1 up to 1/6 terminates.
 * Platonic solid can't have more than 30 edges.


 * Base 32 (duotrigesimal)
 * Each digit can be represented as five binary digits.
 * Useful in the study of powers of two.
 * Base32
 * Used in the Ngiti language.


 * Base 36 (hexatridecimal, sexatrigesimal or hexatrigesimal)
 * Each digit can be represented as two senary digits.
 * Useful in the study of powers of six.
 * More unit fractions terminate than with decimal.
 * The digits can be represented using the Arabic numerals 0-9 and the Latin letters A-Z
 * Used for transmitting coordinates in a compact form by the Remote Imaging Protocol for bulletin board systems.
 * Used by many URL redirection systems like TinyURL or SnipURL/Snipr.
 * Used as a compact representation of Gregorian dates in file names by various systems such as RickDate.
 * Used by Dell as a compact version of their Express Service Codes.
 * Used by the software package SalesLogix as part of its database identifiers.
 * Used as labels for purchases by TreasuryDirect.
 * Used by the E-mail client program PMMail to encode the UNIX time of an email.
 * Used to represent uploaded files stored in MediaWiki.


 * Base 42 (duoquadecimal)
 * More unit fractions terminate than with decimal.


 * Base 48 (octaquadecimal)
 * More unit fractions terminate than with decimal.


 * Base 49 (nonaquadecimal)
 * Each digit can be represented as two septenary digits.
 * Useful in the study of powers of seven.


 * Base 52 (duopentadecimal)
 * Can represent numbers using nothing but the 26 lowercase and 26 uppercase letters used in the Latin alphabet.
 * Can use this to turn numbers into letters.
 * Can use this to turn letters into numbers.


 * Base 54 (quadropentadecimal)
 * More unit fractions terminate than with decimal.


 * Base 60 (sexagesimal)
 * More unit fractions terminate than with decimal.
 * Every unit fraction that terminates in decimal also terminates in sexagesimal.
 * Every unit fraction from 1/1 up to 1/6 terminates.
 * Widely used.


 * Base 62 (duosexagesimal)
 * The digits can be represented using the numerals 0-9, uppercase letters A-Z and lowercase letters a-z.
 * Used for UNIX time stamps.
 * Can be included in a URL without encoding any characters.


 * Base 64 (quadrosexagesimal)
 * Each digit can be represented as six binary digits.
 * Each digit can be represented as three quaternary digits.
 * Each digit can be represented as two octal digits.
 * Useful in the study of powers of two.
 * Useful in the study of powers of four.
 * Useful in the study of powers of eight.
 * Base64


 * Base 66 (hexasexagesimal)
 * More unit fractions terminate than with decimal.


 * Base 70 (septagesimal)
 * More unit fractions terminate than with decimal.
 * Every unit fraction that terminates in decimal also terminates in septagesimal.


 * Base 72 (duoseptagesimal)
 * More unit fractions terminate than with decimal.


 * Base 78 (octaseptagesimal)
 * More unit fractions terminate than with decimal.


 * Base 81 (unoctagesimal)
 * Each digit can be represented as two nonary digits.
 * Useful in the study of powers of nine.


 * Base 84 (quadroctagesimal)
 * More unit fractions terminate than with decimal.


 * Base 85 (pentaoctagesimal)
 * Base85


 * Base 210 (duocentidecimal)
 * 210 is a primorial.
 * More unit fractions terminate than with decimal.
 * More unit fractions terminate than with any other base lower than 420.
 * Every unit fraction that terminates in decimal also terminates in duocentidecimal.
 * Every unit fraction from 1/1 up to 1/10 terminates.
 * There can be up to 210 elements. (see extension of the periodic table beyond the seventh period)


 * Base 256 (duocentihexapentadecimal)
 * 256 = 24
 * Each digit can be represented as eight binary digits.
 * Each digit can be represented as four quinary digits.
 * Each digit can be represented as two hexadecimal digits.
 * Useful in the study of powers of two.
 * Useful in the study of powers of four.
 * Useful in the study of powers of sixteen.


 * Base φ
 * Has been studied in pure mathematics.
 * 11φ = 100φ


 * Base plastic number
 * Has been studied in pure mathematics.
 * 11p = 1000p.


 * Base e
 * Has been studied in pure mathematics.
 * The natural logarithm serves the same purpose as the common logarithm.
 * The area under the line y = 1/x can be expressed more easily.
 * Best radix economy of any base.


 * Base π
 * Has been studied in pure mathematics.
 * Easier to show the relationship between the diameter of a circle to its circumference.
 * Easier to show the relationship between the diameter of a circle to its area.


 * Base √2
 * Has been studied in pure mathematics.
 * Can accommodate all the same numbers as standard place-value systems.
 * Easier to show the relationship between the side of a square to its diagonal.
 * Easier to show the silver ratio.


 * Base 2i
 * Has been studied in pure mathematics.
 * Can accommodate all the same numbers as standard place-value systems.
 * Both positive and negative numbers are represented without the use of a minus sign.
 * Complex numbers can be represented by the digits 0, 1, 2, and 3.
 * All Gaussian integers can be represented without sign.


 * Base −1±i
 * Has been studied in pure mathematics.
 * Creates the twindragon shape.
 * All Gaussian integers can be represented without sign.


 * Mixed radix
 * Used for representing units that are equivalent to each other, but not by the same ratio.
 * Widely used.

Robo37 (talk) 20:17, 29 October 2009 (UTC)


 * I find this list useless without a little more care and detail in the justifications. E.g. the individual negative bases "studied in pure mathematics": really? Very little pure mathematics is specific to any single base. I'd like to see multiple nontrivial publications specifically devoted to a single base to believe this claim. I'm happy to have a separate article as we do now on negative bases but that's not what this discussion is about. Base 13 "used in Maya calendars": clearly no. The Maya used a calendar that may have had 13 as one of its multipliers, but that's not base 13, at best that's a mixed-radix system in which 13 is used for only one of the positions. "More unit rationals terminate than decimal": true of infinitely many bases, not acceptable as a jsutification. Etc. Basically I see this long WP:TL;DR list as derailing the discussion by repeating all the inadequate justifications already present in the articles and by scrolling all the past discussion of thiss issue off the screen, and I am very tempted to hide it within an archive box for that reason. —David Eppstein (talk) 02:25, 30 October 2009 (UTC)
 * I agree. This list has almost nothing to do with notability. I'm going to be bold and do that. rspεεr (talk) 07:33, 30 October 2009 (UTC)


 * Okay guys feel free to edit the list if you want, I only created it as a rough guideline. If you think that there should be 'a little more care and detail in the justifications' then if you're allowed to edit other peoples lists I have no problem with you adding this content.


 * About there being an infinite number of bases that have more terminating unit fractions than with decimal - that is correct, but there's only a finite number of these bases can be easily represented using numbers and letters. Out of these Base 6 is of interest as its the only base with this property that uses less digits than decimal, and Base 30 + Base 60 are also of interest as they both are able to show more terminating fractions and show the same terminating fractions as decimal, and Base 30 is the lowest base that this is applicable to. I have also included Base 210 in the list as this also has this property but is also the lowest base that is able to show the first 10 unit fractions as terminating ones. I know I seem to be mainly focusing on these fraction related properties... but, in my opinion it's the only property that is of any actual use... though I haven’t just listed the first load of bases that are primorials because that would, well, be silly. I am not here to say which articles should or shouldn't exist, I'm just pointing out the facts. Robo37 (talk) 09:23, 30 October 2009 (UTC)


 * My first thought is that the Non-integer representation article that many of the entries link to has no references and based on my studies of the subject is full of OR. There is a paper on Base φ out there and Knuth has some general material, but there is nothing to signify Base π as notable. The section on Base √2 is wrong, at least up to interpretation of the definition given since it's vague. I get 3 base √2 as 1000.000001001000000001... not 101 as the article implies.


 * As for the other entries it would be interesting to know where all this is coming from but it's almost all trivia. For example, I've read Contact and Base 11 is used in a single paragraph at the end, so listing it here is unduly magnifying its importance.--RDBury (talk) 18:15, 30 October 2009 (UTC)


 * Base √2 is written correctly, 101√2 = (1×√22) + (0×√21) + (1×√20) = (1×2) + (0×√2) + (1×1) = (2) + (0) + (1) = 3, however, 1000.000001001000000001...√2 = (1×√23) + (0×√22) + (0×√21) + (0×√20) + (0×√2-1) + (0×√2-2) + (0×√2-3) + (0×√2-4) + (0×√2-5) + (1×√2-6) + (0×√2-7) + (0×√2-8) + (1×√2-9) + (0×√2-10) + (0×√2-11) + (0×√2-12) + (0×√2-13) + (0×√2-14) + (0×√2-15) + (0×√2-16) + (0×√2-17) + (1×√2-18) +... = √23 + √2-6 + √2-9 + √2-18 +... = 2.826327125... + 0.125 + 0.044194173... + 0.001953125 +... = 3, so you are also correct, and that should perhaps be mentioned somewhere. And although it probably doesn't deserve its own article Base π still arguably deserves a small mention in the non-integer representation article due to its usefulness for representing the proportions of circles, though I doubt it's ever used. And finally on the other entries, all of the information that I got is from Wikipedia itself. Robo37 (talk) 18:24, 31 October 2009 (UTC)
 * Re "deserves a small mention ... due to its usefulness": only if it can be sourced. Please, let's not be committing original research here. —David Eppstein (talk) 18:35, 31 October 2009 (UTC)


 * I'm not a mathematician, (just an interested layperson) (and I probably joined the discussion a bit late) but isn't base-π what radians are? 122.107.15.145 (talk) 12:37, 31 March 2010 (UTC)
 * No. Radians are a unit of measurement whose scale happens to run from 0 to 2π (in most cases, although e.g. for winding number it makes sense to talk about larger angles) but they are conventionally written using standard decimal notation. π is not used as a radix for them. —David Eppstein (talk) 17:00, 31 March 2010 (UTC)

Action
Alright, after letting this simmer for a little while, I've decided to be WP:Bold and take action. I've redirected base 64 to Base64, since that was the only thing that it talked about. The rest of my plans are on user:sligocki/bases. Specifically:
 * Delete: Base 18, 26, 30 and 62
 * Merge into an article on non-standard bases: Base 6, 7, 9, 11, 13, 14, 15, 24, 27, 32 (the content for this article is in user:sligocki/bases)

Let me know what you think. If I don't hear any objections, I'll nominate the first set for deletion and write up a non-standard bases article (any suggestions for better article name welcome). Thanks, — sligocki (talk) 05:03, 4 December 2009 (UTC)


 * Sounds good to me. Maybe you can merge into Non-standard positional numeral systems instead of creating a new article.--RDBury (talk) 13:20, 4 December 2009 (UTC)


 * Non-standard positional numeral systems currently focuses on non-standard numeral systems, rather than non-standard bases for the canonical system. It talks about bijective numeration, mixed radix, negative base, etc. and seems to focus on the mathematical aspects. On the other hand, non-standard bases would be about non-decimal bases used in natural language, literature and technology. Cheers, — sligocki (talk) 22:30, 4 December 2009 (UTC)


 * I don't agree with deletion of any of these pages, but if you're adamant on deleting them can you at least spare Base 30 as I'd say that is one of the 5 most notable bases and is by far the most useful. Robo37 (talk) 15:44, 10 January 2010 (UTC)


 * Can you supply a more compelling reason for keeping than WP:ILIKEIT? —David Eppstein (talk) 16:45, 10 January 2010 (UTC)


 * 30 is divisible by the three smallest prime numbers, so if you look at the table on its article you'll notice that the table cannot be constructed out of any other single digit base; that is, every simple fraction in decimal remains simple, yet a large number of remaining fractions get simplified. It's because of these fraction related properties that make primorial bases of particular interest, but base 2, base 6 and base 30 are the only ones that can represented under single digits. To clarify, base 2 is the lowest base to make every fraction between 1/1 and 1/2 terminate, base 6 is the lowest base to make every fraction between 1/1 and 1/4 terminate, and base 30 is the lowest base to make every fraction between 1/1 and 1/6 terminate. Robo37 (talk) 17:54, 10 January 2010 (UTC)


 * The point is not "would it make a good base to use" but rather "have people actually historically done anything interesting with this base"? Otherwise we're committing original research. —David Eppstein (talk) 18:22, 10 January 2010 (UTC)


 * I would keep base 6, and would merge base 36 into it. I would merge base 25 into base 5, and would merge base 9 and 27 into base 3.  Here i argue that squares and cubes of a base are really just alternate expressions of the lower base.
 * Base 30 could be merged into base 60, as it is of interest for the same reasons base 60 has been used. Similarly, i would merge base 24 into base 12.


 * I think that a separate article for the remaining bases in question, with a title that is not confused with Non-standard positional numeral systems. Perhaps, uncommon integer bases.  This would include bases 7, 11, 13, 14, 15 and 26. Bcharles (talk) 02:11, 7 February 2010 (UTC)


 * Please consider that 9 & 27 are used in Computer Science for Ternary computers like 8 & 16 are in Binary Computers if deletion of 9 & 27 is to take place merging with 3 makes more sense than complete removal. Historical Ternary computer systems have existed and there continues to be research relating to ternary computer systems see Category:Computer_arithmetic Ternary_numeral_system Tryte


 * Though I am a computer person and not an anthropologist there are some notes indicating use of base 3 in Islam and 27 in Telefol & Oksapmin languages of Papua New Guinea 216.99.201.19 (talk) 08:05, 18 January 2011 (UTC)

I have proposed the following merges: Please add your perspectives to the discussions. Bcharles (talk) 04:41, 12 July 2013 (UTC)
 * Merge from "Nonary" and "Base 27" to Ternary
 * Merge from "Base 32" to Binary
 * Merge from "Base 36" to Senary

Proposed deletions


The articles Base 62, Octadecimal, Hexavigesimal and Base 30 have been proposed for deletion. The proposed-deletion notice added to the article should explain why.

While all contributions to Wikipedia are appreciated, content or articles may be deleted for any of several reasons.

You may prevent the proposed deletion by removing the  notice, but please explain why in your edit summary or here.

Please consider improving the article to address the issues raised. Removing  will stop the proposed deletion process, but other deletion processes exist. The speedy deletion process can result in deletion without discussion, and articles for deletion allows discussion to reach consensus for deletion. — sligocki (talk) 17:00, 7 January 2010 (UTC)

List of positional systems by base
This category page used to list the articles ordered by base, but user:Meco has removed all the sort tags so that is no longer the case. Either we need to put them back, or we need to create such a list (or table) somewhere - which probably is a better idea. It could include the names of systems that are not notable enough to have an article of their own; they could be redirected to such a list. (Several such articles were deleted or changed to redirects by User:sligocki some time ago.)

So I will soon create such a table as a new section in the article Positional notation - if not someone else beats me to it or talks me out of it by suggesting a better solution.--Nø (talk) 10:53, 24 October 2010 (UTC)


 * I propose a navigational template which could then go on the bottom of all relevant pages and also on the category page. __meco (talk) 12:58, 24 October 2010 (UTC)


 * That might be a good idea, but I'm not sure. There is a Template:Numeral systems (appearing top right in the relevant articles) that has a selection of these links in it, but it would be too cluttered having them all, so a separate template at the bottom would make sense. However, you can't redirect the names of less notable numeral systems to a template. Ideally, it should be a table connecting bases (like "2") and names ("binary") - it might be too much to include that on every page involved. (Of course, the previous solution with the sort tags did not provide such a connection.) Perhaps a template with an "expand/hide" button would be good; I don't know how to make those. (Might we even add such a button to the existing template instead?)--Nø (talk) 13:28, 24 October 2010 (UTC)


 * Just a tip on creating a template is to find a navigational template in some unrelated article and simply change the links in it, keeping the code. __meco (talk) 17:12, 24 October 2010 (UTC)

I just happened to create List of numeral systems, and then found this talk page. Please come and help out! I suggest we start a discussion on that article's talk page. I'm a bit confused over how to list the numeral systems by culture... --Beao 19:44, 17 March 2011 (UTC)

Based on the above concerns over notability
I've redirected all the articles to List of numeral systems, save the following: All of these are clearly notable, except 4 and 6, which I'm not sure about. They are near the border between notability and non-notability, I think. Double sharp (talk) 16:16, 29 March 2015 (UTC)
 * Unary numeral system (1)
 * Binary number (2)
 * Ternary numeral system (3)
 * Quaternary numeral system (4)
 * Quinary (5)
 * Senary (6)
 * Octal (8)
 * Decimal (10)
 * Duodecimal (12)
 * Hexadecimal (16)
 * Vigesimal (20)
 * Sexagesimal (60)

Add decimal numbers in multiplication tables
Can somebody please divide the rows of the multiplication tables into two and add the corresponding decimal numbers? --Backinstadiums (talk) 15:25, 13 July 2019 (UTC)
 * This would be pointless – anybody having IQ ≥ 100 can superimpose (in mind, or otherwise) the decimal multiplication table up to 12 × 12 with the given one. Or is 12 not enough? Incnis Mrsi (talk) 16:18, 13 July 2019 (UTC)
 * Currently the ternary numeral system has a table showing both binary and ternary numbers for up to the decimal 27, as well as powers up to the ninth and a Fractions table. I propose for the positional systems in this category to add decimal numbers in multiplication tables, and also add the exponential and fractionary ones. --Backinstadiums (talk) 16:59, 13 July 2019 (UTC)

Ternary clock
Hi, could you create a ternary clock similar to the binary one so that it can be added to the respective page? https://commons.wikimedia.org/wiki/File:Binary_clock.svg --Backinstadiums (talk) 16:07, 13 July 2019 (UTC)