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Counting to 12 on fingers

I am flummoxed by the correct name for this body part, but I know how to count to 144 on my fingers using my thumbs as marker and counter - what do you call the finger-lengths from joint to joint?

Take your hand (R, L, whatever works for you) and touch your thumb to the farthest finger-length on the index finger = 1. then the next length down = 2. the one closest to your palm = 3; then you move your counting to the middle finger (4,5,6), ring finger (7,8,9), little finger (10,11,12). Mark the first length on the OTHER hand with your thumb. Start over = 13-24. Etc. You can tally a gross of whatever you need to count without writing. MOST convenient, and often used as an explanation for the use of duodecimal systems in ancient Mesopotamia - and hence in Astrology, Astronomy, Chronology, etc., since we all adopted their version of that. I learned this at my father's knee, but I think I saw it confirmed in one of Eviatar Zerubavel's books on time. --MichaelTinkler.

• The bones are called phalanges. There are 14 per hand (the 12 you mention, plus 2 for the thumb). --Zundark
  • Aha! Thank you. Yes, one uses the thumb to touch the phalanges.

• I post a question on the talk:Decimal page sometime ago about using the thumb to count the finger joints and finger tips in a hexadecimal system. That may have been mistaken version of the duodecimal system. I think this tallying method should be part of the article.

I have also come across reference to so using the fingers for hexadecimal. Jackiespeel (talk) 16:31, 17 December 2008 (UTC)

The reference for this is a dead link. So I've searched for and read a few other articles that talk about ancient Egyptians counting to twelve by their fingers. I haven't yet seen a primary source referenced however. Does anyone know of a primary source, ancient papyrus, or even an ancient historian, etc., that describes this? — Preceding unsigned comment added by BobEnyart (talk • contribs) 01:51, 28 December 2015 (UTC)

Notation

Isn't the standard base notation for digits in a base greater than 10 is to say 1,2,3,...,8,9,A,B,C...? Instead of the X we have here. Is this notation only for the duodecimal system? Dysprosia 12:53, 23 Aug 2003 (UTC)

• Yes, X is only for the duodecimal system. A,B,C,D,E,F are used for the Hexadecimal system. There is no standard notation for digits in a base greater than 10, that I know of in use. -- Karl

• Script capital E is possible in Unicode: ℰ. You could also use ℇ or ℇ. --Sonjaaa 07:49, Sep 10, 2004 (UTC)
  • There is also Ɛ, LATIN CAPITAL LETTER OPEN E. However, there still seems to be no satisfactory Unicode glyph for the DSGB's suggested "ten" symbol (a rotated 2) . Livajo 07:57, 10 Sep 2004 (UTC)

• It would be fun to see a table that shows the character used for ten and eleven according to various stardards, e.g. DSGB, the American guy who used script X and E, etc. That way you can compare at a glance the various ways people have suggested to write ten and eleven, and maybe decide which standard you prefer, or see their similarities and how they differ, etc.--Sonjaaa 08:09, Sep 10, 2004 (UTC)

• For script x, Unicode character 1D4CD (𝒳) can be used. It's outside the Basic Multilingual Plane, so its usefulness is limited, but Firefox is one browser that displays it by means of substitution (to the normal ASCII x). FF has a substitution file, ./res/entityTables/transliterate.properties, that provides for that. --82.80.19.232 (talk) 14:36, 10 September 2009 (UTC)
  • Assuming it's the script capitals you want, it's U+1D4B3 for the X and U+2130 for the E. An alternative, especially if you want to use Pitman's "rotated 2" glyph, is the Private Use Area. I suggest U+E03A for base-12 b^1−2 and U+E03B for b^1−1, because that way they look like a continuation of the ASCII codes 30–39 (the digits 0–9). There's even a third alternative, though much less comfortable to handle: use decimal multiple digits as duodecimal digits, as customary for sexagesimal, like 1:2:10 doz = 178 dec. --89.138.41.81 (talk) 23:35, 6 October 2009 (UTC)

• The thing about using X and E, or V and E as I do, is that it presupposes decimal numbers still exist. In my case, this is true, because i use base twelfty (120), and V, E are used for teenty and elefty respectively. For those who really want to use a strict duodecimal system, one has to eliminate the underlying decimal substrate, deriving new numbers and runes for 10, 11. Although i use E for writing, the thing is actually a back-to-front nine or large 'e'.

Were one to push for Unicode digits (rather than just picking out letters from assorted fonts), one would be advised for digits from -1 to 19 or something like this. There are other things to consider. For example, one needs to look at the 'lower case' or hanging form, such as one might see in the font Georgia. One might look at assorted diacritics that might be added (eg writing 17 in base 20 as '7. In my script, V is written as x-height, while E is written with a descender.

In some cases, there is need for a second series of number (eg roman digits). This allows things like apposition of numbers without conflict, eg 2009ix9. I don't think enough detail has been given to this. --Wendy.krieger (talk) 07:03, 11 September 2009 (UTC)

Please move to DozensOnline. PiotrGrochowski - 16:29, 17 July 2014

Pronunciation?

How do they intend we pronounce a dozenal number like 14? "A dozen and four"? Would 3E be pronounced "three dozen eleven"? What about higher numbers, in the 3rd column where we normally had hundreds before. Dozenal 100 is decimal 144. Is there already an English word for the decimal number 144?--Sonjaaa 08:14, Sep 10, 2004 (UTC)

• Oh, there's gross for 12*12 and great gross for 12*12*12! How far do such names go? --Sonjaaa 08:15, Sep 10, 2004 (UTC)

There are a few systems for dozenal pronunciation. I only really know of the Pendlebury system, where X = ten, E = elv and 10 = zen. 14 becomes zen four, and 3E becomes threezen elv. N4m3 (talk) 22:01, 16 June 2011 (UTC)

This system of counting in dozenal seems reasonable. — Preceding unsigned comment added by 68.146.23.244 (talk) 02:07, 13 March 2012 (UTC)

This is generally what I've heard before as far as pronunciation. One, do, gro, mo, do-mo, gro-mo, etc. — Preceding unsigned comment added by 76.88.24.190 (talk) 23:40, 28 December 2012 (UTC)

I usually use something like Pitman's scheme, where one to eleven keep their current names, and the multiples of twelve are: onezen, twozen, threezen, fourzen, fivezen, sixzen, sevenzen, eightzen, ninezen, tenzen, elevenzen. Twelve twelves make a gross; twelve grosses make a grand; a grand grands make a zillion (neatly rationalizing the "z" as coming from "dozen"). The multiples of twelve are basically contracted forms of "one dozen", "two dozen", etc.: if communicating in dozenal to someone else, I would use the full forms, and "great gross" instead of "grand". Double sharp (talk) 15:12, 12 April 2015 (UTC)

I can't remember where I got it, maybe http://www.dozenal.org/archive/ManualOfTheDozenSystem1174-web.pdf but I like one, two, ... nine, dec, el, do, one do one, one do two, one do three ... one do el, two do, two do one, two do two ... and so on. it's quick and easy to learn and handy enough to use. Then they used Gro for gross and some other stuff from then on

Polygons

I find it worth mentioning that there also seems to be a relation to polygons: Regular triangles, squares and hexagons (3,4 and 6) will tessellate in the plane with themselves as well as in combination with each other (triangles/squares, triangles/hexagons, triangles/squares/hexagons), whereas regular pentagons (5) will neither tesselate in the plane with themselves nor with other regular polygons. Twelve regular pentagons may however form a pentagonal dodecahedron in three dimensions. Article on Polygons from DSGB (Adobe PDF)

Names

About the special names for 11 and 12 in european Languages:

English: eleven (not one-teen)  twelve (not two-teen)
German:  elf (not eins-zehn)    zwölf (not zwei-zehn)
Dutch:   elf (not een-tien)     twaalf (not twee-tien)
--InsectAttack 14:32, 29 Sep 2004 (UTC)

• And in french onze (not dix-un), douze (not dix-deux) but also treize (not dix-trois), quatorze (not dix-quatre), quinze (not dix-cinq) and seize (not dix-six). So I don't think french is accurate in the article. 82.224.88.105 14:23, 28 Nov 2004 (UTC)

• In Finnish we say "yksitoista", "kaksitoista", "kolmetoista" etc. This literally means "one of the second", "two of the second", "three of the second", etc. This can be extended further, for example "yksikolmatta", meaning "one of the third", means 21. This was in very common use for centuries right until the early 20th century, but is now archaic. Nowadays we only use "-toista" for 11 through 19 and then we use the normal form of concatenating the tens and the ones. — JIP | Talk 09:59, 4 Mar 2005 (UTC)

All of this suggest base 4. (multiples of 4 are 8, 12, 16, 20). Also 8 in the most ancient form is a dual form (ie being 2×4), and nine and new come from the same root (the new group of four). French also uses scores to count in, as in sixty-seventeen and four-twenty-eightteen (98 = 4×20+8+10). Wendy.krieger (talk) 10:46, 9 March 2012 (UTC)

Time?

There seems to be a dozenal-inspired system used in time as well. Days have 24 hours (or two sets of 12), hours have 60 minutes (or 12 sets of 5), there are likewise 60 seconds in a minute, and there are 12 months in the year in most calendars, including the Julian, Gregorian, Hebrew, Hindu, Islamic, and Persian (although admittedly there are practical considerations for this, it could be divided another way). I can't say I know anything about the history of the time measurement, but maybe someone who does could include something on the topic? Sarge Baldy 00:39, July 15, 2005 (UTC)

• The number of months is due primarily to the length of the lunar month. The 60 goes back to the Babylonians, who used a base-60 numerical system.

Incidentally, in East Asia, the day was traditionally divided into 12 units, each, therefore, 2 hours in length, and named after the animals of the Chinese zodiac -- Nik42 15:06, 15 July 2005 (UTC)

• The sexagesimal number system, base 60, comes from the Sumerians and is what led to the Babylonian number system. Many Sumerian creations are with us today. 12 hours in a day, 60 seconds per minute, 60 minutes per hour, and even 360 degrees in a circle originate from the first civilization to bring us both mathematics and writing. Base 60 comes from counting to twelve on one hand with your thumb as a pointer and tallying the number of sets of twelve you have counted on your other hand. Five sets of twelve equals 60. — Preceding unsigned comment added by 24.44.206.248 (talk) 14:43, 3 January 2012 (UTC)

The sumerian day is divided sixty-wise, preserved in India as a day of 60 ghurries, a ghurry of 60 pali, a pali of 60 vipali. The hour is an egyptian decimal day: a day of 10 hours + two twilights + rising of 12 decans at night = 24 hours. The modern time system is a greek composite of egyptian hours divided in the sumerian fashion.

The number of signs in the zodiac is pretty much cultural. It places the sun and planets at points of the sky, but the division into 12, 13, 15, 18 (mayan) or 36 (egyptian) is a matter of calendar. Months are quite used, so 12 is common for that reason (alone). Cetus is usually the next constellation to use.

Base 60 actually comes from three scores, and then multiples of 10 alternated with divisions of 6. Using twelfts in 60 is like using twentieths in decimal. You soon think in terms of tens and halfs of tenths or sixths and halves of sixths.

Twelve in the divisions is a roman process, and used on cultural dependencies of Rome. The dram, for example, was divided into 6 obols, and 100 made the pound (mina). This is pretty much how it would happen when one counts 'drachma' coins. The romans incorporated the unit into a pound, divided into 12 uncia, so the dram is rounded to 1/96 lb, and the obol to 1/576 lb. A scruple of 1/288 lb creeps in elsewhere. It should be noted that the romans only divided duodecimally, counts are always in decimal.

In the north of europe, lies an entirely different metrological tradition based on 120 and its multiples. The pound of king Offa was made by an ounce of 10 dirhems, a pound of 12 ounces. The dirhem yields two silver pennies, which leads to an oz of 20 dwt. Likewise, a hundredweight of 120 lbs is usual in north europe. Wendy.krieger (talk) 10:43, 9 March 2012 (UTC)

Music Notes

Twelve half-steps are present per octave in music. Counting a scale of 8 notes, with the bottom and top note identical, is the etymology of the word "octave," but that is only counting one key's scale. Counting every half-step, without counting two of the same note, makes a piano, keyboard, or guitar frets useful in counting according to the dozenal system.

A, A#/Bb, B, C, C#/Db, D, D#/Eb, E, F, F#/Gb, G, G#/Ab — Preceding unsigned comment added by 99.71.180.153 (talk) 22:38, 15 April 2014 (UTC)

growth

This article has grown really well. Can we nominate it for featured article?--Sonjaaa 05:25, 27 January 2006 (UTC)

Not yet, please. There's still a lot of info I'd like to add before having this article featured, such as about prime number identification, factorials, the patterns in the multiplication table, the relation of twelve to certain elementary angles and geometric shapes, the relevance of twelve in Western music, the system of dozenal fractions used by the Romans, the proposals for filling the gaps in English duodecimal nomenclature and the way some African languages developed a duodecimal nomenclature from a decimal one and viceversa, the existence of a complete proposal for a consistently duodecimal system of measures, the hurdles an eventual dozenalization would have to face, etc. I'm also planning to refurbish the whole article so that the issue is handled in a more structured way. I want this article to describe the duodecimal system in depth, with all its pros and cons, so that readers have all the information to make a fair comparison with the decimal system they are already familiar with, and thus be able make an informed judgement about dozenalism for themselves rather than going with the preconceived idea that decimal is "the natural way for humans to count" and dozenalism merely "a freaky idea no-one should take seriously". Uaxuctum 04:21, 4 February 2006 (UTC)

English duodecimal names

In the article it says that A^5 (49,A54) would be:

four dozen and nine great gross, ten gross five dozen and four

and A^6 (402,854):

four gross and two great gross, eight gross five dozen and four

Are these right? Above it seems to imply that there are no names for 10,000 or 100,000 in duodecimal. And even if there were, wouldn't 49,A54 be something like four great great gross and nine great gross...? --Aceizace 21:53, 8 March 2006 (UTC)

So far there are no standard English names for dozenal 10,000 and 100,000 that I know of (let alone for higher dozenal numbers), save for the straightforward a dozen great-gross and a gross great-gross (maybe great-great-gross and great-great-great-gross have already been used by some people, but I cannot tell for sure). Note that four gross and two great gross is not to be read as "four-gross and two-great-gross", but as "four-gross-and-two great-gross", i.e. analogously to "four hundred and two thousand" meaning 402,000 (the same possible ambiguity with the meaning 400+2000 exists in decimal). There have been a number of proposals to expand and standardize English nomenclature for the big dozenal numbers, though. In the , some dozenists have suggested to substitute a simpler, more manageable name (like grand) for great-gross. Others have proposed naming schemes that would generate a unique name for every dozenal power. I myself have suggested the name zyriad (from dozenal myriad) for 10^4 = 10,000 = 1,0000, as well as similar z- names like zillion for 10^6 = 10^(3+3) = 1,000,000 = 100,0000, zyrion for 10^8 = 10^(4+4) = 100,000,000 = 1,0000,0000, zilliard for 10^9 = 10^(3+3+3) = 1,000,000,000 = 10,0000,0000, and the merely tentative name doogol (based on googol) for 10^10 = 10^(3+3+3+3) = 10^(4+4+4) = 1,000,000,000,000 = 1,0000,0000,0000). But so far, all of these are mere suggestions. Uaxuctum 16:31, 22 April 2006 (UTC)

There is this scheme http://www.dozenal.org/archive/ManualOfTheDozenSystem1174-web.pdf do = 10 gro = 100 mo = 1000 do mo = 10,000 gro mo = 100,000

It would be even funnier in German, "vier gross-gross-Grosse und neun gross-Grosse"... ;) 惑乱 分からん 15:08, 18 November 2006 (UTC)

The names above seem to be as transparent and "standard English" as you'll ever get for high dozenal numbers. Double sharp (talk) 15:14, 12 April 2015 (UTC)

Question on societies

Why isn't there a separate article on the Duodecimal Society of America? PrimeFan 23:02, 24 October 2006 (UTC)

Well, :-), because it seems no one has cared to create it so far. I've found that in general the articles dealing with topics related to other number bases than decimal and the ones commonly used in computing are still poor and lack info on many important points (for example, until I added it the other day, the article on ternary didn't mention anything about using it to represent rational numbers like the basic fraction 1/2, and I had to correct the still-stubby article on sexagesimal where it said that Babylonian sexagesimal was mixed radix just because they represented their digits using a sub-base of ten—which is analogous to how the Maya represented their digits using a sub-base of five and doesn't mean they used mixed radix because of that, although the Maya actually used mixed radix of twenty and eighteen when computing dates). This very article on dozenal still lacks tons of info, without which it is not possible to make a fair judgement about the case for dozenal over decimal that the DSA and DSGB promote. But I myself am already working on expanding it. I'm currently finishing two comparative charts, one showing the effect of decimal, dozenal and hexadecimal in the perception and choice of numbers (which numbers look "rounder" than others in that base; e.g., people using decimal tend to prefer numbers such as 10, 25 and 50 over 12, 24 and 60, even though the latter are inherently more well-suited for many purposes), and the other chart showing how the choice of base affects the representation of rationals and thus the everyday choice of certain fractions and proportions over others according to their ease of representation in that base. Here's an almost finished version of the first one: http://img214.imageshack.us/img214/4301/tabla8oo.png Uaxuctum 18:37, 1 November 2006 (UTC)

Easier to memorize?

The article states:

"As can be seen, it is easier to memorize the first nine digits of pi in base twelve than in base ten, while the opposite happens with the first ten digits of the number e."

How can one prove that this is the case? Unless there is a pattern (which dose not appear to be the case), what might be easier for you to remember, might not be easier for me.

Unless some can give a cite for this, I think it should be removed. —Gary van der Merwe 10:26, 14 December 2006 (UTC)

Sorry - I just noticed the "1828" repartition in decimal e. Still, how is it easier to remember dozenal pi as apposed to decimal pi? —Gary van der Merwe 10:31, 14 December 2006 (UTC)

Can't you really see the patterns?

doz pi        dec pi          dec e         doz e
         vs                            vs
3.            3.              2.7           2.7
  18 48         14 15           18 28         87 52
  09 49         92 65           18 28         36 06

They should be pretty obvious if you just care to look at the numbers for a while: one-eight, four-eight, then oh-nine, four-nine (patterns: 1_4_ / 0_4_ and _8_8 / _9_9); that's a more regular pattern than one-four, one-five, then nine-two, six-five (patterns: 14 / 15 and ___5 / ___5). Uaxuctum 00:01, 18 December 2006 (UTC)

I see the patterns, but as far as I'm concerned that's original research. And I'm also not at all sure what the point is, because it doesn't have anything intrinsic to do with decimal or duodecimal bases. Essentially irrational numbers produce different random sequences when expressed in different bases, and some random sequences are easier to perceive patterns in than others.

I don't see the point... and even if I did, it is not a description of anything that is a well-known or well-established characteristic of duodecimal numbers. If it were, you would be able to cite a source for it. It's just your own personal observation. It's not even clear whether everyone perceives these patterns the same way. Dpbsmith (talk) 01:12, 18 December 2006 (UTC)

So if the article mentions that 1/35 or the Euler-Mascheroni constant in base twelve equal (blah blah) and those particular numbers happen to not be explicitly published somewhere (I haven't cared to check), but those are simply the digit sequences one gets by doing the well-known base conversion algorithm (which any calculator with a base conversion function can do), then doing those straighforward mathematical conversions to include them in the illustrative charts would be original research too? Would someone place a "citation needed" tag until someone provides some reference asserting explicitly that 1/35 in base twelve is indeed (blah blah)? If in the multiplication table article someone puts an illustrative chart that highlights the objective digit patterns in the tables (e.g. the patterns in the table of 7, which are there, one just has to look at the numbers for a moment to find them) and then mentions "without a source" that the patterns in the table of 5 (5-10-15-20-25-30-35...) make it easier to remember this table in decimal as opposed to the table of 5 in e.g. dozenal (5-A-13-18-21-26-2B...), will you ask them for a "citation" to back up such a straightforward fact saying that "It is not clear that everyone perceives those patterns the same way"? The article is just mentioning that the patterns are there in dozenal 3.18480949... and decimal 2.718281828..., which is an objetive, uncontroversial fact about those digit sequences (it is not a matter of personal perception that 1_4_-0_4_, or _8_8-_9_9, or 18-28-18-28, or 5-10-15-20-25, create patterns, the patterns are there objectively, and they are not at all particularly difficult to see). And the point is, patterns serve as a mnemonic technique (e.g. telephone numbers: a patterned number like 1-800-234-5656 is easier to memorize than some random number like 1-475-823-9465 — would you challenge the assertion of this common-sense fact asking for a citation to back it up?), and the memorization of pi's digits is relevant given that this number is very frequently used. So the more regular patterns in the first dozenal digits of pi (3 - 18 48 - 09 49) as compared to the patterns observable in its first decimal digits (3 - 14 15 - 92 65) make the dozenal representation of pi easier to memorize than the decimal equivalent at least up to the ninth digit (in decimal it is straightforward to memorize up to the fifth digit: 3.1415... or the rounding 3.1416, which are fairly good approximations for everyday purposes but not as good as the 9-digit approximation which you can get in dozenal with about the same memorization effort), and this is a relative practical advantage of dozenal over decimal; while, to balance the matter, decimal allows easier memorization than dozenal for the first ten digits of e, another very frequently used number, and that is a relative practical advantage of decimal over dozenal. I don't see why this simple statement of fact about mathematically objective digit sequences should be any controversial at all. In any case, given that it seems that even simple statements about straightforward mathematical facts need to be "sourced", I will ask the DSGB/DSA people in the DozensOnline forum for "citations" that explicitly mention it (some of the people there like to discover really arcane dozenal patterns, so probably someone will laugh at me when I ask them for a reference about the in-your-face-simple pi thing). I'm sure there must be some explicit reference somewhere, since there are entire books devoted to the many patterns observable in dozenal numbers, and the one in the first digits of pi is one of the most straightforward to see and is of practical use for remembering this everyday number, although it is a "trivial" one in that it is not a consequence of the divisibility properties of twelve the way e.g. the patterns in the multiplication tables are. Uaxuctum 03:43, 18 December 2006 (UTC)

Of pi, e, etc. Use provides a rapid way of recalling these numbers. A four-function calculator does wonders to learning the irrationals to any number of places. These are usually limited by the places of the calculator, so eg 1/2(sqrt(2)+sqrt(6)) is 1.93185165259, and sqrt(2.5+sqrt(1.25) = 1.90211303259, are numbers learnt on a 12-place calculator, while the chords of the heptagon 1.801937736 and 2.2469796037 date from a period when i had a 10-place calculator.

The fraction for e is best thought as 2721/1001. The number is 2.718 281 828, add 718+281=999, so it's 2 719/1001. In duodecimal, we get 2.8 7 5 2 3 6 0 6 9 for e and 2.8 7 5 2 3 5 8 7 5 2 3 5 for 2721/1001. It's much easier to calculate this fraction in any given base.

For pi, the easiest to find is 3 16/113, or say 3 17/120 -1/113. However, another approximation exists as 7^7/8^6, in octal, we have pi = 3.110 375 524 cf 7^7 = 3.110 367, corresponding to a difference of 6 in the sixth octal. In base 56, we get eg that pi is 3.7 52 1 52 21 32, 1/pi = 0. 17 46 12 17 16, while 3 7 51 53 26 20 7 × 17 46 12 42 35 33 8 gives 56^13, the numbers are the 13th powers of 7 and 8 respectively.

Of course, if you want ease of remembering pi, base 22 is hard to go by. pi = 3. 3 2 11 14, 22/7 is 3. 3 3 3 3, and 3.3 3 is 1.17 squared, while sqrt(pi) is 1.16 21 19 1 20 (a difference of 3 in the third place). pi^2 is 9.19 2 19 12, while 227/23 is 9.19 2 19 2 19 .. pi^3 is 1 9.0 3 0 8, pi^4 is 4 9.9 0 0 0 0 14 3. 1/pi is 0.7 0 1 8 , etc. --Wendy.krieger (talk) 11:04, 25 December 2010 (UTC)

i see the pattern . π 10 : 3.14159265 , π 12 : 3.w5x6y5z6 jjbernardiscool (talk) 09:18, 28 July 2013 (UTC)

Conversion from decimal

Is there any known system for quick conversions from decimal to dozenal? Oberiko 21:47, 2 February 2007 (UTC)

What do you mean by "quick conversions"? The fastest way is usually to simply use a calculator that supports that function (there is one available from the DSGB site, although it has some minor bugs). The other two methods are: to use the general base-conversion algorithm (which can be a bit tedious), or to memorize or look up the tables of digit conversions (this method is easier for dozenal-to-decimal conversions, unless you also learn to do simple dozenal arithmetic, because it works by adding the corresponding values in the target base for each digit in the source base, e.g. dozenal 123=100+20+3 corresponds to decimal 144+24+3=171, which is an easy sum to do because we all are already trained to do sums with decimal numbers, but to convert decimal 123 into dozenal one needs to know how to sum the dozenal numbers 84+18+3, which is not really difficult at all but requires that you learn to "switch" from decimal arithmetic thinking to dozenal arithmetic thinking, because in dozenal 4+8+3 equals 13 and not 15 as would be expected in decimal, and then 8+1+1 adds to A and not to 10, so the result of 84+18+3 is A3 in dozenal and not 105 as one would have obtained doing the sum in decimal, and thus decimal 123 corresponds to dozenal A3 (as a side note, in this particular example it would be easier if one just remembers that decimal 120, i.e. twelve tens, equals dozenal A0, i.e. ten dozens). Alternatively, if what you're looking for is merely an estimate of "how large" the number is, rather than its accurate converted value, then it's useful to memorize some easy to remember "landmarks" (e.g., doz 90 = dec 108, doz 300 = dec 432, doz 600 = dec 864, doz 700 = doz 1008, doz 2000 = dec 3456). For your reference, the following are the digit conversion tables in both directions, up to fifth digits:

DOZENAL TO DECIMAL
Doz Dec      Doz  Dec       Doz   Dec        Doz    Dec         Doz     Dec
1   1        10   12        100   144        1000   1728        10000   20736
2   2        20   24        200   288        2000   3456        20000   41472
3   3        30   36        300   432        3000   5184        30000   62208
4   4        40   48        400   576        4000   6912        40000   82944
5   5        50   60        500   720        5000   8640        50000   103680
6   6        60   72        600   864        6000   10368       60000   124416
7   7        70   84        700   1008       7000   12096       70000   145152
8   8        80   96        800   1152       8000   13824       80000   165888
9   9        90   108       900   1296       9000   15552       90000   186624
A   10       A0   120       A00   1440       A000   17280       A0000   207360
B   11       B0   132       B00   1584       B000   19008       B0000   228096
DECIMAL TO DOZENAL
Dec Doz      Dec  Doz       Dec   Doz        Dec    Doz         Dec     Doz
1   1        10   A         100   84         1000   6B4         10000   5954
2   2        20   18        200   148        2000   11A8        20000   B6A8
3   3        30   26        300   210        3000   18A0        30000   15440
4   4        40   34        400   294        4000   2394        40000   1B194
5   5        50   42        500   358        5000   2A88        50000   24B28
6   6        60   50        600   420        6000   3580        60000   2A880
7   7        70   5A        700   4A4        7000   4074        70000   34614
8   8        80   68        800   568        8000   4768        80000   3A368
9   9        90   76        900   630        9000   5260        90000   44100

This tables should be added to the article eventually. Uaxuctum 14:37, 5 February 2007 (UTC)

You could use my number converter that has been on the web since 1995 at http://www.footrule.com/1/conversn/convjvsc.htm?sr=homepage&ac=1&jt=1.1.1 . This allows conversions between all bases up to 36 (the limit of decimal digits plus 26 alphabetic characters). This also allows fractional values. There are some limitations due to the inability of javascript handling that may produce a close approximation which can usually be recognized and rounded yourself. Ruthe (talk) 22:17, 14 June 2016 (UTC)

Dozenal Doesn't? come again?

"They use the word dozenal instead of "duodecimal" because the latter comes from Latin roots that express twelve in base-ten terminology."

And how does dos + zehn not? Is it just something about the Latin that the Gallic is more acceptable? —Preceding unsigned comment added by 75.28.41.130 (talk • contribs)

Uh? What do you mean by that strange mix of Spanish/Latin + German, "dos + zehn"? That's by no means the origin of the word "dozen". The point is, "duodecimal" is clearly analyzable as a compound of the Latinate prefix duo- meaning "two" + the Latinate root dec- meaning "ten" + the Latinate adjectival ending -imal, all of which are identifiable elements to be found in many other English words (duo, duo-poly, duo-logue, duo-dec-illion, dec-illion, dec-imal, dec-imal-ize, tri-dec-imal, hexa-dec-imal, hex-imal, viges-imal, sexages-imal, etc.). That is, in English, the term "duodecimal" is obviously a ten-based way of naming this numeral system, implying the decimal point of view of "twelve = ten + two" (twelve is no more "ten + two" than it is "nine + three" or "eleven + one" or "seven + five" or just "twelve"; viewing it as "ten + two" is a decimal-minded construct). In contrast, "dozen" is an unanalyzable term, not a compound, referring to a group of "::::::" elements directly without implying the decimal-based way of looking at it as ":::::" plus ":". Sure the source of the word "dozen" goes back to the Latin compound duodecim (which in old French became douze and then entered English through the derived word douzaine); that is, ultimately it has the same etymological origin as "duodecimal". However, that decimal-based origin in a different language does not apply to the English language, where "dozen" entered as a single unanalyzable root referring directly to twelve without implying "ten + two". You cannot break up the English word "dozen" into *do- + *-zen the way you can break up "duodecimal" into duo-dec-imal implying "two + ten". English "dozen" is an unbreakable, unanalyzable single morpheme referring to a "group of ::::::" just like "score" means "group of ::::::::::" without the ten-based analysis as "::::: + :" and "::::: + :::::". "Dozen" refers to the number twelve as a counting unit, as an underived quantity, which is precisely the point of view of the dozenal numeral system (in dozenal you count by the dozen, so the name is very well-fitted), unlike "duodecimal" which implies the decimal perspective of "ten" as the counting unit and of "twelve" as a derived quantity composed of "one ten, plus two units" (in dozenal, twelve is "10" not "12"). "Hundred" and "thousand" are two other examples of words which in their remote etymological origin were breakable, analyzable compounds (both ultimately based on the Proto-Indoeuropean root *dekm for "ten") but which by the time English came into existence had become single unanalyzable morphemes. Uaxuctum 17:52, 26 May 2007 (UTC)

I concurr with Uaxuctum that the word 'dozen' is derived from the French word 'douzaine', and it is also true that the French word was derived from the Latin duodecima and is therefore still a reference to the 'two-ten' origin. If it is desirous to provide a word for a 12 base number system, it would be far more acceptable to use a root that is completely disconnected from the base ten system using the Latin basis of 'decima' for the 'decimal' system. Therefore why not use the same form of root as used for the decimal system but based on 12, and there is such a perfectly good basis. I discovered this when researching Roman mathematics and the Roman hand abacus, which led to the Roman names for fractions, which were themselves overwhelmingly 12 based. The Roman word for a tenth was the decima, but the Roman word for a twelfth was 'uncia'. What better word for a base 12 root than the base 12 analogue of the base 10 word? Thus the 'Uncial' or even 'Uncimal' system! Note that although I have suggested such a name for a base 12 number system, I was later to find that it had been suggested many years earlier by Pvt. William S. Crosby in a letter to the DSA and published in the Vol.1 No.2 edition of The Duodecimal Bulletin of June 1945 under the title of "Uncial Jottings of a Harried Infantryman". I quote directly from that article.

"On Nomenclature, Notation, and Numeration: Maybe I am a factionlist, but here are some of the prejudices I stick by.

Duodecimal, (two more than ten) is a derived concept as well as a clumsy word. What is needed is a word expressing "counting by the scale of twelve", but as far as possible not depending on any other concept. "Uncial" is a suitable word to replace "decimal" in naming point-form fractions, and I myself use the word for the whole field of counting by dozens. Its chief drawback is that only a specialized meaning (in the field of paleography) is given in most dictionaries. "Dozenal I consider beneath contempt."

The entire article can be found beginning on page 9 of Vol. 1 No.2 edition of The Duodecimal Dozen.

It seems good ideas will always re-occur!

Ruthe (talk) 20:35, 22 December 2010 (UTC)

If I may pick an etymological nit or two: the classical Latin words for 'ten, twelve' are decem, duodecim; the corresponding ordinal adjectives are decim–um, duodecim–um (–us, –a, etc). The French word douzaine is douze (from a Vulgar Latin form that probably resembled Italian dodici) with a French suffix –aine that forms other "group of N" words (e.g. quarant-aine for a set of 40 days); this suffix is unique to French as far as I know. Nor is –imal a morpheme as Uaxuctum would have it. —Tamfang (talk) 21:41, 22 December 2010 (UTC)

Fractions

Hi everyone!

I have tried to work some fractions out using duodecimal though I am very stuck!

Can someone explain in a very easy way to me (since I am very dumb!) how the duodecimal works and how to get those fractions the article showed?

Thanks a lot!! —Preceding unsigned comment added by Mariekitty (talk • contribs) 13:59, 9 November 2008 (UTC)

Calculations in base 12 are done by carrying or borrowing in multiples of 12. For example, the example that 1/9 = 0.14, goes like this:

  9 goes into 1 0 times remainder 1 :  units = 0   carry  remainder 1  as 12 twelfths
  9 goes into 12 1 times remainder 3:  digit = 1   carry remainder 3 as 36 
  9 goes into 36 4 times remainder 0:  digit = 4   carry remainder 0 as 0

So we have 9 goes into 1 as 0.14

 cf dec
  9 goes into 1 0 remainder 1   write 0,  carry 1 as 1*10 tenths
  9 goes into 10 1 remainder 1  write 1,  carry 1 as 1*10 
      repeat
  1/9 = 0.11111111111 decimal

Books on arithmetic give this calculation as a means of dealing with products of feet, inches and twelfths, where the output is given as cu ft, 144 cu in, 12 cu in, cu in, etc.

--Wendy.krieger (talk) 12:53, 27 August 2009 (UTC)

Germanic and Twelve / Twelfty

There are some commnets about germanic languages and twelve that need fixing.

Indo-European is a decimal system that went as far as 100, but not 1000. The words for 100 are easily seen to be related, but the words for 1000 are later inventions (eg mill vs thousand).

The change from eleven/twelve to thirteen etc, is a base 4 feature, not a base 12 one. Nine and new are etymologically related in all IE languages, nine being the new one in the third four. Also, Finnish changes from names to decimal relatives at 9 (nine = one-removed). French changes from two-lif to -teen forms at 16 to 17 (dix-sept, dix-huit).

There is no germanic tradition of dodecimal multiple or division. The hundred is always 120, divided into twelve tens. The thousand is always 1200. Germanic languages have words for writing numbers in 'long' and 'short' forms, or twelftywise, teentywise.

Latin weights and measures show decimal multiples (eg mile = 5000 paces), while the divisions are largely duodecimal, with factors of 2 and 3 only. Germanic number systems tend to have relatively equal numbers of 3 and 5 on each side (eg cwt = 120 lb, lb = 7680 gr).

The six score hundred was typically used of things with heads (eg people, nails), while the new five-score hundred has elsewere use.

One notes that E Gordon 1956 "Introduction to old norse" gives in section 107 (page 292), as numerals, 100 tiu tigir (teenty), 110 ellifu tigir (elefty), 120 hundrað (hundred), 200 hundrað ok atta tigir (hundred and eighty), 240 twau hundrað (two hundred), 1200 þusand (thousand), all given without comment.

The system of dozens, grosses, and great grosses form a system of super-divisions, ie larger units that are intended to be divided. A grocer is one who deals in grosses, dividing these into dozens for sale, and the buyer divides the dozen to units. Dozens and grosses are only used of things that are itself cheap and undivided: fruit, eggs, etc: the undivided nature suggests that 1 is a division, the cheap-price suggests that one might want to make larger units to divide into smaller ones.

--Wendy.krieger (talk) 12:40, 27 August 2009 (UTC)

Firstly, Finnish does not change naming pattern from 9 (yhdeksän), but from 8 (kahdeksan); the second member of the compounds in both kahdeksan and yhdeksän is an Indo-European loan meaning ‘ten’. All the Uralic languages have invented new words for both 8 and 9; no common denominator can be found or reconstructed—in fact, only 1–5 can be reconstructed for Uralic with any certainty. The rest are later innovations or borrowings in individual branches.

Secondly, I don’t see how Germanic ‘hundred’, being always 120 and divided into twelve equal units, is not a tradition of duodecimal division. To me, that’s exactly what it is: a tradition of mixed decimal and duodecimal division.

Thirdly, while ‘nine’ and ‘new’ are etymologically related, there doesn’t seem to be any evidence that they were transparently connected as such by Germanic times (when the ‘one-left’ and ‘two-left’ terms were coined). Common Germanic forms like *newjaz and *niwun are not that parallel. And any association between ‘eight’ and ‘sharp’ was definitely completely gone by Germanic times (and probably only vaguely relevant even in Proto-Indo-European).

There’s little doubt that *neu̯n̥ is ‘the new one’; but that only points to vestiges of a base-4 system in Indo-European itself. It doesn’t support any theory that new terms coined (to replace the inherited, purely decimal ones) in a much later stage of a daughter language, are base-4 in nature. Nor do terms naming added or removed items from a decimal base lend much credence to claims that they are in fact base-4.

130.225.244.206 (talk) 12:49, 16 October 2012 (UTC)

Unwarranted Emphasis on English Weights & Measures for Duodecimal Usage

The ultimate sentence in the section titled "Origin" attributes the usage of 12 as a major basis for several units of measure and a monetary system mainly to the English system of measures and coinage. Although there is a reference to Charlemagne which does provide a more complete description of the origin of these systems, the emphasis in this section on the 'English' origin gives a skewed impression by ignoring the details in the referenced entry for Charlemagne that clearly explains that the mixed 20/12 based monetary system originated with Charlemagne, was adopted by King Offa in Britain, but was widespread throughout central and western Europe. It was probably the introduction of the metric system (not the S.I.) that mainly involved the adoption of decimal monetary systems. For example, the French system of money used the livre of 20 sous and the sou of 12 deniers up until the adoption of the metric system. Likewise, the monetary systems of many European counties used the same mixed base monetary system, and it was only the fact that the United Kingdom was the last country to abandon that system that made it the "English" system in common experience.

Again the entry for Charlemagne does note the origin of the duodecimal use of fractions by the Romans, and it was this that resulted in the base 12 portion of these monetary systems. It also notes that the English words 'inch' and 'ounce' are both derived from the Roman word for /₁₂ or an 'uncia'. This use of a base of 12 was also widespread across Europe for the use of length measures as exemplified by the words for 'inch' in several European languages. Whereas England (Britain) used the word inch, the main practice in continental Europe was to use a word derived from the Roman usage for that measure which was a 'thumb inch', such that the words for inch and thumb in many languages are the same or closely related. Examples are shown below.

(1)

• duim als lengtemaat
• Engelse duim

I think it would be appropriate to modify this section to broaden the origin of the mixed base monetary system and to give a better understanding of the wider European usage of a base of 12 in many units of measure, not limiting this to the English, but acknowledging the true origins that began with Charlemagne and earlier with the Romans.

Ruthe (talk) 00:45, 1 October 2010 (UTC)

The Romans used a weight-fraction system, where an 'as' is a unit (foot, pound, grain, coin-measure), which was then divided into lesser units by weight (uncia, drachma, scrupule, obolus, calculus, etc). These typically consistent with the duodecimal fractions, and Roman and Greek money traditions. (drachma = 6 obolus, in copper was a handful of 6 spear-coins). Every foot and pound the Romans inherited was divided in the same manner.

For weights, the usual weights were in ounces, the pound becoming an ever larger number of ounces. The divisions of the ounce are still preserved.

Of length, the pre-roman tradition (Greece, Egypt), was to divide the foot into 16 digits. The division into 16 is a clavis (claw, clove, digit, (finger)nail), which becomes a 16th of any other measure. We see dutch nagel of weight, being 1/16 of a larger unit. Likewise, english digit, nail, clove divide the foot, yard and cwt into 16, are all etymologically related to 16 fingernails to the foot.

Most of Europe inherited the Roman tradition, and even though the uncia becomes the thumb (and this divided duodecimally to lines and points), still retain the Roman twelfth divisions, rather than other divisions. When decimalisation came to europe pre-metrically, the same sequence of names (foot, inch, line, point) are applied to various duodecimal/decimal divisions. The duodecimal units in common use (Eng foot, inch; French inch, foot, line), retained their values, and decimals applied elsewhere. A tradition in Germany has decimalisation on the rood (12 ft), foot, inch as the Kette-mass, Geometric and miners units.

A similar process occurs today with 1000 vs. 1024 for powers of kilo. A 1.44 MB diskette, is actually 1.44 × 1000 × 1024.

--Wendy.krieger (talk) 10:26, 25 December 2010 (UTC)

Usage section

Would it be applicable to have a section about practical usage of the system in speaking, writing and such? There is already a section on symbology (which takes up most of the advocacy section), but the article lacks mention of any nomenclature. This is likely to leave the reader confused with saying things like "aye-ty bee", which sounds like "eighty three" (the digit 'A' especially so). Of nomenclatures to suggest, I know about the classic "ten, eleven, dozen, gross", TGM's "ten, elv, zen, duna" and Schoolhouse Rock!'s "dek, el, doh, " (as said in this article).

It would be useful if this section was early on, so that readers know about it before trying to use it.

M1n1f1g on Dozensonline, N4m3 (talk) 16:21, 28 August 2011 (UTC)

Old English had words that become teenty and elefty in ME for the decades after 90. These avoid clashes between tenty/twenty and eleventy/seventy, because there is some word-space between teenty/twenty and elefty/seventy. Most of the duodecimal words i have seen have been based on decimal ten / eleven, which mean 'two hands' and 'one left'.

For a hundred of six-score (which is what OE, old norse etc used), it does not really matter, because there is a decimal step, and teenty, elefty are perfectly appropriate. For a pure twelfty base, one should take cognance that PIE used a four-based system, and that 'nine' is the /new/ one of the third tetrad. One might suppose, 'med' and 'alt' might form the etymological roots for 10 and 11 digits in a pure twelve-system, and higher numbers in the manner of new-welsh, eg one ty six (for 18). This allows one to use simpler numbers for the 144's etc, as one decides whether one should use 144-based or 1728-based steps (cf 100, 1000 based commas).

Much of what is written in 12 is mainly how to calculate in the duodecimal divisions. Older German units had names for 12*cube and 144*cube, along with 12*sq, which allows duodecimal names for something like 144*foot³. See, eg Muret-Sanders English/German Dictionary 1902, in particular the extensive table of weights and measures.

--Wendy.krieger (talk) 08:17, 29 August 2011 (UTC)

is base-30 the smallest system that has three different prime factors?

Article says: "Trigesimal is the smallest system that has three different prime factors (all of the three smallest primes: 2, 3 and 5) and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30)". IMHO it's not true. Base-15 is the smallest system that has three different prime factors — 2, 3, 5 (and has five divisors — 1, 2, 3, 5, 15).
83.26.192.123 (talk) 20:37, 9 February 2012 (UTC)

Base 15 does not have 2. Karl (talk) 13:03, 10 February 2012 (UTC)

Dozenal...

...is the most optimal radix, am I not correct? Avengingbandit 05:36, 18 May 2012 (UTC)

No, twelfty (six-score) is. Wendy.krieger (talk) 07:00, 18 May 2012 (UTC).

Well, if society can memorize 7T34_doz multiplication problems and able to remember T0_doz different symbols, base T0_doz is the most optimal base.

13572 = E3.12? i don't understand. I use 144 = 1.24. 120 symbols? I use 12. Still, i did a fairly extensive survey of number systems and assorted tools, including bases, to determine the most suited method for hand calculations. 120 won hands down. You just need a more advanced notion of bases to handle it. I've been using it as the normal base for a number of calculations (such as symmetries). I will often do calculations in an assortment of other bases, such as 12 and 18 and 30, but the only ones that don't need annotations are 10 and 120, since these were designed to be concurrent. Wendy.krieger (talk) 06:59, 19 May 2012 (UTC)

Yes but if I was to pick a new number base for society, it would be 10_doz. Avengingbandit 15:08, 19 May 2012 (UTC)

twelfty has historical usage. twelve doesn't. so þere ye go. Wendy.krieger (talk) 10:27, 21 May 2012 (UTC)

10 is not as bad as many 12-ists seem to think: while 10 does not support 3 completely, the fact that 9 (adjacent to 10) is a multiple of 3 does partially alleviate this issue. 12 does not have similar solutions for 5. (6, 14, and 15 are also not bad.) I'm aware you can test for 5 in base 12 based on the fact that 12 ≡ −1 (mod 5), but this is not really on the order of ease of finding the digital root. Nevertheless the advantages of 12 are not inconsiderable. Double sharp (talk) 11:19, 5 April 2013 (UTC)

I've kinda thought about a little more. I'm not really a fan of 12. The solutions for 5 are only partial, and while its treatment of its divisors is awesome, its treatment of 5 leaves much to be desired. And I feel 5 is important. (DeVlieger explains perfectly why.) I'm beginning to think that sixty might be the optimal radix. (No, I do not want to be teleported to ancient Babylon. I just want to have their arithmetic back. IMO we should have stayed with sixty.) It solves the problem of 5, has a half-maximal expansion for 1/7 (0.08;34;17;08;34;17...), and anything higher is not really necessary. And the 1/7 expansion is quite recognizable. I can estimate it pretty well as 0.08;30, by just feeding of twelfty's (0.17;17;17...) 120 solves that problem, but I think it is really too large to handle, while sixty isn't quite there yet IMO. (Sexagesimalist here.) 120 has a lower totient ratio, but has twice the digits, utterly removing the benefits and making it about 50% (30 per sixty) more complicated. Double sharp (talk) 15:41, 10 November 2013 (UTC)

Now that I have thought about it a bit more, I now think 12 is actually the best human-scale base, with 10 very close behind, though if 60 were considered human scale I think it might actually win. Double sharp (talk) 13:36, 15 November 2013 (UTC)

Merge with Base 13

While this has been historically labeled as "duodecimal", this article technically describes a base-13 number system, not base-12. This article and Base 13 should be merged and a new page accurately describing a base-12 number system should be created.

Explanation:
Base-2: 0 1
Base-3: 0 1 2
Base-4: 0 1 2 3
Base-5: 0 1 2 3 4
Base-6: 0 1 2 3 4 5
Base-7: 0 1 2 3 4 5 6
Base-8: 0 1 2 3 4 5 6 7
Base-9: 0 1 2 3 4 5 6 7 8
Base-10: 0 1 2 3 4 5 6 7 8 9
Base-11: 0 1 2 3 4 5 6 7 8 9 "10"

Base-12 should be: 0 1 2 3 4 5 6 7 8 9 "10" "11" yet it's described here as what should logically be a base-13 system...
Base-13: 0 1 2 3 4 5 6 7 8 9 "10" "11" "12"

David Lones (talk) 22:03, 17 July 2013 (UTC)

You're wrong. — Arthur Rubin (talk) 00:10, 18 July 2013 (UTC)

Duodecimal / Dozenal uses 12 "digits"; whether they are 0 1 2 3 4 5 6 7 8 9 10/Z/T/A 11/E/B or 1 2 3 4 5 6 7 8 9 "10" "11" "12", there are 12 digits. — Arthur Rubin (talk) 00:20, 18 July 2013 (UTC)

I have reverted the addition of the merge tags as they are based on completely bogus reasoning. As for counting systems that use digits 1...base instead of 0...base–1, see bijective numeration. —David Eppstein (talk) 00:39, 18 July 2013 (UTC)

Then I recommend the wording of this article be a bit more clear in that regard (perhaps in including more direct examples of the system) and that the inaccurate multiplication table included in this page be corrected, re-labeled, or removed. — David Lones (talk) 01:07, 18 July 2013 (UTC)

I don't see anything wrong the multiplication table. — Arthur Rubin (talk) 01:27, 18 July 2013 (UTC)

The table is base-13! Also, in reading the article as it is currently, I should also point out that its Places section describes the number system as including characters for both zero and twelve as well.

In a duodecimal place system, ten can be written as A, eleven can be written as B, and twelve is written as 10.

If twelve can be written as "10" and the multiplication table includes a character for both zero and twelve, then (at least at first glance) what is being described here is a base-13 system. — David Lones (talk) 01:54, 18 July 2013 (UTC)

If you think that "10" is a single digit, in base 12 or any other base, then you have no business editing or discussing articles on number bases. It is two digits. And the multiplication table is fine; it includes 13 columns, but that is no different than a decimal multiplication table that includes multiplication by ten (with eleven columns, zero through ten). —David Eppstein (talk) 01:58, 18 July 2013 (UTC)

Sorry. I just counted columns, apparently. I didn't realize I was making a complete fool of myself until after I typed that. — David Lones (talk) 02:11, 18 July 2013 (UTC)

Oh, well, no harm done I think except maybe to some dignity. I'm sorry if my language earlier came off as a bit harsh. —David Eppstein (talk) 04:59, 18 July 2013 (UTC)

Fractions

How do you convert fractions from base 10 to base 12? — Preceding unsigned comment added by Alhadialika (talk • contribs) 21:10, 26 September 2013 (UTC)

Fractions like decimals and dozenals, actally are continued numerators, so 0.538 = 5.38/10, eg 5 dm and then 0.38 is 3.8/10 (cm), and so forth. The conversion is to replace these 10's to 12's. 0.538 * 12/12 = 6.456/12, so the first digit is 6. 0.456 * 12/12 gives 5.472/12, this is the second place, and 12*0.472/12 gives 5.664, gives 5 as the third place. So 0.538 decimal = 0.6558 dozenal.

The same method can be used of changing numerators, so 0,538 * 6/6 = 3.228, 0.228*10/10 = 2.28/10; .28*6/6 gives 1.68/6, and 0.68 *10/10 gives 6.8/10, so this gives 0.538 hours = 0.3217 (sixtywise) hours = 32' 17.

Likewies, you can use it of units: 0.538 acres * 4/4 = 2.152 roods, 0.152 * 40/40 gives 6.08 perches, so 0.538 acres = 2 roods, 6 perches. Wendy.krieger (talk) 10:36, 27 September 2013 (UTC)

(I am aware that this section is nearly a month old.)
There is another way to convert dozenal fractions to decimal, though you have to be proficient with dozenal arithmetic since that is used instead of decimal. For example, I'll convert decimal 0.375 to dozenal. (ᘔ is 10 in decimal form.) Take the least significant digit, 5, and divide it by ᘔ, yielding 0.6. Take that result and add it by the next least significant digit, 7, to get 7.6. Divide that by ᘔ to get 0.9. Take that and add by the next least significant digit, 3, to get 3.9. Divide that by ᘔ to arrive at 0.46. Avengingbandit 21:36, 26 October 2013 (UTC)

Thanks to both of you, and I know how to so dozenal math. — Preceding unsigned comment added by Alhadialika (talk • contribs) 02:58, 6 November 2013 (UTC)

you can also do this. for example if you want to make 0.375 decimal to dozenal all you do is convert 375 into dozenal and 1000 into dozenal and divide them — Preceding unsigned comment added by Xirtyan (talk • contribs) 04:28, 22 November 2018 (UTC)

Article Title

Has it been discussed yet whether this article should be titled "Dozenal" rather than "Duodecimal"? There are Dozenal Societies of America and Great Britain, not Duodecimal Societies. Most people I hear speaking of the topic use "dozenal" and avoid "duodecimal." Thank you, startswithj (talk) 02:11, 18 January 2014 (UTC)

I support changing the article's title to "Dozenal". It's been getting more and more attention in recent years and I almost never see people refer to it as "duodecimal". Avengingbandit 01:10, 21 January 2014 (UTC)

Oppose. Duodecimal is the logical word and title in line with decimal. --JorisvS (talk) 13:54, 21 January 2014 (UTC)

Please elaborate. Avengingbandit 06:52, 26 January 2014 (UTC)

Yes, please elaborate a little more on what makes it logical, why being in line with decimal is preferable. Thank you, startswithj (talk) 19:27, 26 January 2014 (UTC)

"Duodecimal" is derived from Latin duodecem analogous to "decimal" from Latin decem. Dozenal, however, is derived from English "dozen" + "-al". --JorisvS (talk) 07:38, 27 January 2014 (UTC)

You still haven't given a real reason why the rename shouldn't occur. Base twelve has been getting famous in recent years (partially due to numberphile's video Base 12) and it's better known as Dozenal than Duodecimal. I'm seeing more and more people on the net treat it as Dozenal rather than Duodecimal. As the OP said, there's no "Duodecimal" society of America or Britian, but dozenal. Imo I feel that base twelve and the term dozenal has gained enough prominence to rename the article to Dozenal. Avengingbandit 20:20, 2 February 2014 (UTC)

Haven't I? It apparently just isn't anything you deem good enough (from what I can tell, only strict "most common usage" matters to you). --JorisvS (talk) 08:12, 3 February 2014 (UTC)

From what I read above, the only argument presented in favor of keeping the title "duodecimal" is that it more closely correlates to the word "decimal." But why would that matter—shouldn't this article title be most relevant to the numbering system it discusses, rather than being named in contrast or comparison to another system? I believe most-common-usage would better guide here.startswithj (talk) 18:02, 3 February 2014 (UTC)

Not just decimal, though, but all base-number articles. Decimal is just the best-known one. Other examples can be found in the box on the right in this article's lead. Note especially hexadecimal, among several similar ones. --JorisvS (talk) 18:22, 3 February 2014 (UTC)

There is little consistency between the articles, and there are many exceptions. Of course Binary, Octal, and all other bases below ten have no "‑decimal" in their names. Many articles are named simply "Base‑X" (Base-13, Base-32). And there are others above base ten without "‑decimal" (Vigesimal and Hexavigesimal, from the Latin vigesimus for "twenty"). startswithj (talk) 20:43, 4 February 2014 (UTC)

"Vigesimal" and the others are actually consistent. The only ones that are not consistent are the "Base XX" titles, and even their inconsistency is quite distinct from that of "dozenal". --JorisvS (talk) 21:13, 4 February 2014 (UTC)

But note that base-36 has been called both "hexatridecimal" (six-and-three-tens) and "sexatrigesimal" (six-twenties). Numbering systems are commonly named in relation to base-10 ("-decimal") when used in reference to base-10, but sometimes they are named after other bases—such as base-20 ("-esimal")—when that base is more relevant or primal. startswithj (talk) 18:59, 5 February 2014 (UTC)

So people have tried to come up with a term based on Latin roots, but have come up with different terms for various reasons. It seems to me that that is why the article is located at base 36. Here, however, it is not about competing terms that are both based on Latin, but one that is and one that is based on an English word to make it seem like it were based on Latin roots. --JorisvS (talk) 08:22, 6 February 2014 (UTC)

The word "dozen" is derived from Middle English, from Old French, from Latin, from Indo-European. The suffix "-al" is applied to words based in Latin (such as "infernal"), Greek (e.g. "comical"), and English ("tidal"). How is the etymology relevant and inhibitory? startswithj (talk) 00:40, 8 February 2014 (UTC)

I know that. The difference is that that word changed a lot in the process, so that "dozen" cannot be considered Latin in origin, only etymologically traceable to a different Latin word. Moving this article to "dozenal" breaks the symmetry with those related articles located at proper titles (thus excluding "Base XXX" articles. --JorisvS (talk) 11:09, 9 February 2014 (UTC)

1. Why should the article title be kept how it is just because other bases' articles are X-decimal?
2. What's wrong with going by "most common usage"?

Avengingbandit 21:54, 8 February 2014 (UTC)

Most common usage breaks consistency and introduces unnecessary irregularities. A redirect here is appropriate and exists, so anyone searching for 'dozenal' will find what they are looking for. --JorisvS (talk) 11:09, 9 February 2014 (UTC)

I don't see where in WP:TITLE that consistency trumps common name. And we don't have consistency anyway (viz. the Base-XX articles). startswithj (talk) 01:11, 13 February 2014 (UTC)

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A problem in base 12

The problem of OEIS: A088622 and OEIS: A088623 in base 12:

Form: n (written in base 12), f(n)=(smallest k>0 such that concatenation of n^k and 1 (in base 12) is prime, or 0 if no such number exists,written in base 12),g(n)=(smallest k>0 such that concatenation of 1 and n^k is prime, or 0 if no such number exists,written in base 12)

n,f(n),g(n)

if n mod 13=1 (n>1) or n mod 143=142, then f(n)=0.

if n is divisible by 2 or 3, then g(n)=0.

1,1,1 2,3,0 3,1,0 4,2,0 5,1,1 6,1,0 7,4,1 8,1,0 9,1,0 ᘔ,2,0 Ɛ,2,1 10,(GFN),0 11,1,1 12,0,0 13,1,0 14,1,0 15,2,2 16,2,0 17,1,1 18,1,0 19,ᘔ,0 1ᘔ,4,0 1Ɛ,1,1 20,36,0 21,9,4 22,1,0 23,0,0 24,1,0 25,1,1 26,713,0 27,1,2 28,2,0 29,1,0 2ᘔ,1,0 2Ɛ,1,1 30,1,0 31,3,1 32,1,0 33,2,0 34,0,0 35,2,7 36,2,0 37,2,6 38,3,0 39,1,0 3ᘔ,6,0 3Ɛ,3,1 40,1,0 41,2,1 42,1,0 43,1,0 44,3,0 45,0,1 46,4,0 47,1,1 48,1,0 49,3,0 4ᘔ,4,0 4Ɛ,1,3 50,2,0 51,1,92 52,23,0 53,1,0 54,1,0 55,490,14 56,0,0 57,Ɛ3,1 58,(>100000),0 59,1,0 5ᘔ,7,0 5Ɛ,1,4 60,24,0 61,1,13 62,2,0 63,49,0 64,2,0 65,3,(?) 66,1,0 67,0,1 68,2,0 69,42,0 6ᘔ,8Ɛ,0 6Ɛ,1,1 70,1,0 71,1,1 72,1,0 73,852,0 74,9,0 75,1,1 76,3,0 77,1,2 78,0,0 79,1,0 7ᘔ,1,0 7Ɛ,2,1 80,1,0 81,2,1 82,2,0 83,26,0 84,1,0 85,1,2 86,1703,0 87,1,11 88,1,0 89,0,0 8ᘔ,Ɛ6,0 8Ɛ,4,1 90,1,0 91,2,4 92,1,0 93,2,0 94,ᘔ,0 95,4,1 96,13,0 97,1,2 98,3Ɛ,0 99,6,0 9ᘔ,0,0 9Ɛ,1,1 ᘔ0,2,0 ᘔ1,1,81 ᘔ2,2,0 ᘔ3,2,0 ᘔ4,1,0 ᘔ5,6,1 ᘔ6,5,0 ᘔ7,2,1 ᘔ8,4,0 ᘔ9,1,0 ᘔᘔ,6,0 ᘔƐ,0,2 Ɛ0,2,0 Ɛ1,1,1 Ɛ2,1,0 Ɛ3,1,0 Ɛ4,2,0 Ɛ5,3,1 Ɛ6,1,0 Ɛ7,1,1 Ɛ8,3,0 Ɛ9,1,0 Ɛᘔ,0,0 ƐƐ,2,(?) 100,0,0 101,1,1 102,1,0 103,2,0 104,1,0 105,1,1 106,1,0 107,2,1 108,4,0 109,2,0 10ᘔ,2,0 10Ɛ,1,3 110,1,0 111,0,(?) 112,2,0 113,ᘔ1,0 114,2,0 115,1,1 116,18,0 117,(>46000),4 118,1776,0 119,2,0 11ᘔ,1,0 11Ɛ,27,2 120,1,0 121,1,14 122,0,0 123,1,0 124,2,0 125,4,1 126,1,0 127,2,(?) 128,1,0 ...

Is there a prime of these forms?

(58^n)1, 1(65^n), 1(ƐƐ^n), 1(111^n), (117^n)1, 1(127^n), ... — Preceding unsigned comment added by 101.14.115.111 (talk • contribs)

This is a talk page for the article "duodecimal". If you are proposing including this in the article, I say "why?" If you asking for information, Misplaced Pages is not the place. — Arthur Rubin (talk) 15:17, 17 August 2015 (UTC)

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Multiplication table

I consider the 30₁₂×30₁₂ multiplication table excessively wide for a Misplaced Pages article. — Arthur Rubin (talk) 08:28, 5 September 2015 (UTC)

And I agree. What does it show that the 12×12 one doesn't? You only need a 12×12 table to multiply two arbitrary numbers in duodecimal, after all. Double sharp (talk) 17:25, 10 September 2015 (UTC)

Ordinals

How should one use ordinal suffixes for duodecimal in English? For now I am using superscript ordinal º for prime factors and ª for everything else. — Preceding unsigned comment added by Repletewithfish (talk • contribs) 13:39, 9 December 2015 (UTC)

Dozenal digits

Pardon me if this isn't the correct place to post this, but my computer (albeit going through many updates) still doesn't have the digits for ten and eleven. How can I get these digits? Avengingbandit 09:27, 12 December 2015 (UTC)

Download a font with them? Try Symbola. Double sharp (talk) 05:28, 13 December 2015 (UTC)

Irrational Numbers not having a pattern?

The article says "Moreover, the infinite series of digits of an irrational number does not exhibit a pattern of repetition; instead, the different digits succeed in a seemingly random fashion." This doesn't seem right to me - that no irrational numbers exhibit a pattern of repetition. For example, the decimal number 0.121121112111121111121111112... exhibits a pattern, and is irrational, right? 174.102.27.76 (talk) 10:20, 16 December 2015 (UTC)

Presumably the article is referring to the fact that a number is rational if and only if the sequence of its digits (in any non-mixing radix) is eventually periodic. Minimalrho (talk) 04:52, 18 December 2015 (UTC)

174.102.27.76 - your number certainly exhibits a pattern, but not a pattern of repetition, as its decimal expansion never repeats itself exactly. Double sharp (talk) 07:39, 18 December 2015 (UTC)

As far as I know, the definition of being rational is whether the number can be represented by the division of two integers (in any base). I do not believe that any pattern being present is addressed. It just so happens that rational numbers either terminate or repeat at some point. 107.182.198.107 (talk) 22:51, 3 June 2016 (UTC)Alan B

I've adjusted the wording to cover your concerns (I hope), with a link to Repeating decimal for anyone who is not clear about the meaning. Please adjust the sentence further if you think it is still ambiguous. Dbfirs 11:15, 4 June 2016 (UTC)

Base identification subscripts (dec and duod)

In the section Fractions and irrational numbers, there are formulas such as 1/(2) = 0.25 _dec = 0.3 _duod; shouldn't a more standard notation be used, such as 1/(2) = 0.25 ₁₀ = 0.3 ₁₂? — Arthur Rubin (talk) 10:42, 19 January 2016 (UTC)Ī

... or 1/(2) = 0.25 _ten = 0.3 _twelve? — Arthur Rubin (talk) 17:19, 20 January 2016 (UTC)

I think that 0.25 _ten = 0.3 _twelve is better, since the former favors the decimal numerical system. Minimalrho (talk) 05:31, 21 January 2016 (UTC)

Well, that may not be a bad thing. Outside the context of these articles about bases, and the few places where you would directly work with binary, octal, or hexadecimal, don't we all interpret all numbers as decimal? Still, I suppose the "ten/twelve" subscripts do save on unnecessary confusion. Double sharp (talk) 13:41, 4 June 2016 (UTC)

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Notation problems

There are some problems with the notation:

1) The "open E" U+0190 and the Canadian syllabary U+1614 may look like a turned two and a turned three, but such a usage is definitely wrong. Moreover, the latter symbol is present only in some specialized fonts.

2) The proper symbols have been added to Unicode only a year ago, but as well there are only a few fonts with them. There is a big chance that it will not be seen by an average reader.

3) The info about notation is split between three chapters, and the main chapter about it is definitely one-sided. There is no authority which would require the usage of exactly turned two and three.

So I suggest: a) some other symbols; b) the merger of three chapters into one (with three sub-chapters if needed).--Lüboslóv Yęzýkin (talk) 11:55, 30 May 2016 (UTC)

4) While a less mathematical problem, the new symbols X wouldn't be able to clearly appear on a seven-segment-display. — Preceding unsigned comment added by 89.100.168.185 (talk) 23:35, 29 June 2016 (UTC)

I've made the merger, however we are left to decide what exact symbols to use. I suggest T and E as the most intuitive, though it would be better to actually employ the Greek capital letters tau and epsilon to avoid confusion and for an easy find-replace.--Lüboslóv Yęzýkin (talk) 13:35, 30 May 2016 (UTC)

I would support capital tau and epsilon for these reasons. Double sharp (talk) 16:20, 26 September 2016 (UTC)

10 and 11

Why use those names, when you can recycle 10 for X and 11 for E, then you can call 10 "onety". 108.71.122.168 (talk) 18:06, 12 July 2016 (UTC)

Requested move 4 January 2017

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: not moved per WP:SNOW. This RM is clearly not going to pass. (non-admin closure) GeoffreyT2000 (talk, contribs) 18:51, 5 January 2017 (UTC)

Duodecimal → Dozenal – more direct/clear and probably more common (dozenal societies etc.) St.nerol (talk) 17:13, 4 January 2017 (UTC)

This is a contested technical request (permalink). Deelay58 (talk) 17:49, 4 January 2017 (UTC)

• Oppose - A reputable mathematics source (Wolfram) titles it as duodecimal, and duodecimal is the version that i have encountered in all (UK) textbooks. — Preceding unsigned comment added by Deelay58 (talk • contribs)
• Oppose. Google Books, while not the only useful metric of course, overwhelmingly supports "Duodecimal", and many of the hits for "Dozenal" pertain to the Dozenal Society of America. Drmies (talk) 17:53, 4 January 2017 (UTC)
• Support per St.nerol. PlanetStar 23:16, 4 January 2017 (UTC)
  • The problem, PlantStar (I can barely read your name--please see WP:COLOR), is that the nominator presents no solid evidence at all, and so "per nom" isn't very strong. Drmies (talk) 04:45, 5 January 2017 (UTC)
• Oppose — Dozenal is not even a word in British English (per OED, though we apparently do have a dozenal society). No evidence has been presented that it is used anywhere. Dbfirs 09:50, 5 January 2017 (UTC)
• Oppose. Duodecimal is the standard term. The only reason why dozenal is ever used is to dissociate twelve from the decimal concept of it being two more than ten, which makes sense for organisations like the Dozenal Societies (which advocate a change to base twelve numeration), but not for a reference work like Misplaced Pages (which should use the common term and not advocate anything). (Besides, dozenal is still decimally constructed, if a bit less clearly. How would one apply such logic to hexadecimal?) Double sharp (talk) 13:00, 5 January 2017 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Compare with music?

Has anyone thought the base 12 number can be correspond with musical scale? 0=C, 1=C#, 2=D, 3=Eb, 4=E, ..., 10=Bb, 11=B. — Preceding unsigned comment added by 101.15.147.75 (talk) 13:16, 31 August 2017 (UTC)

Music is more like mod 12 than base 12, since octave displacements, while not irrelevant, can be considered to be equivalent. Double sharp (talk) 13:19, 31 August 2017 (UTC)

silly divergent series

The list of divergent series seems silly to me. I see 1 + 1 + 1 + 1 + 1 + 1 + ... = −0.6, which seems so weird that it looks like this is some kind of special property of duodecimal notation. But of course 1+1+1+... diverges whether we're doing decimal or duodecimal, and all we're saying is that 0.6 duodecimal is 1/2 in decimal. Is there any point to these series in this article? Staecker (talk) 00:19, 15 May 2018 (UTC)

In my opinion, this and similar divergent series have no place in this article. Dbfirs 00:58, 15 May 2018 (UTC)

I agree with Dbfirs. Double sharp (talk) 01:05, 15 May 2018 (UTC)

As such, I have boldly removed these series. Double sharp (talk) 01:09, 15 May 2018 (UTC)

See the article 1 + 1 + 1 + 1 + ⋯, it says that 1 + 1 + 1 + 1 + ⋯ = -1/2, because it is ${\displaystyle \zeta (0)}$ (see Riemann zeta function)

${\displaystyle \zeta (x)=\sum _{n=1}^{\infty }{\frac {1}{n^{x}}}}$, and when n=0, this sequence become 1 + 1 + 1 + 1 + ⋯, when n=-1, this sequence become 1 + 2 + 3 + 4 + ⋯. Besides, we know that ${\displaystyle \zeta (0)=-{\frac {1}{2}}=-0.6}$ and ${\displaystyle \zeta (-1)=-{\frac {1}{10}}=-0.1}$ (all numbers are written with duodecimal)

Also see the article 1 + 2 + 4 + 8 + ⋯ for the reason that 1 + 2 + 4 + 8 + ⋯ = -1

Those are not the sums of those divergent series, except under some generalisations of the word "sum". Double sharp (talk) 10:51, 22 May 2018 (UTC)

This list continues to get bigger and bigger with more examples that are ultimately WP:OR. I have restored the 15 May version. D4iNa4 (talk) 21:03, 5 July 2018 (UTC)

These things I added are all true. — Preceding unsigned comment added by 175.97.48.79 (talk) 21:21, 5 July 2018 (UTC)

WP:NOTTRUTH. D4iNa4 (talk) 08:10, 6 July 2018 (UTC)

Some properties

I think the entire "some properties" section is junk. Can someone give me a good reason for not reinstating my removal of it? —David Eppstein (talk) 20:55, 22 August 2018 (UTC)

Why remove? Some of them are specific to this base (duodecimal), e.g. 12 (twelve) is the largest base such that both “all squares end with square digits” and “all primes end with prime digits or 1” are true. — Preceding unsigned comment added by 49.215.177.200 (talk) 06:35, 23 August 2018 (UTC)

Repeating another junk property is not a reason to keep. And why remove: because it's primarily original research. —David Eppstein (talk) 06:45, 23 August 2018 (UTC)

This article has been taken over by duodecimal hobbyists. The "some properties" section belongs on someone's personal web page or blog, but not on Misplaced Pages. I support removing it all. BabelStone (talk) 09:33, 23 August 2018 (UTC)

Interesting facts that are unique to duodecimal, or where duodecimal is either the largest or smallest base to have that property, are perhaps interesting. But many pages of stuff about palidromes, especially with no indication whatsoever whether any of this is true or false or different in other bases, is a waste of time. So your one sentence about 12 (twelve) is the largest base such that both “all squares end with square digits” and “all primes end with prime digits or 1” are true is interesting (in fact I am wondering if it is the *only* base where this is true, except for 4 but that is only because all the non-zero digits are prime).Spitzak (talk) 18:23, 23 August 2018 (UTC)

Personally, I can't help but finding some properties worth being reported here. The wholesale removal appears to me as excessive, even when I agree with it to an overwhelming part (primes in certain intervals!). Maybe, from an economic view, it should be the task of the Dozenal Society's fans to suggest selective, noteworthy content for re-enclosing it with the article (no wholesale revert!). I also cannot help but comparing this content with other "junk" in WP. I know I am not allowed to derive anything here from this. Purgy (talk) 08:29, 24 August 2018 (UTC)

To Spitzak: The bases which both “all squares end with square digits” and “all primes end with prime digits or 1” are true are 2, 3, 4, 8 and 12, and 12 is the largest of them. The former is true for bases 2, 3, 4, 5, 8, 12 and 16 (and possible no more, see OEIS: A254328), and the latter is true for bases 2, 3, 4, 6, 8, 12, 18, 24 and 30 (and possible no more, see OEIS: A048597).

Well, why is this section, removed on 2018 Aug 22 considered as junk ? It is a more extended description of several arithmetic facts like primes, recurrent fractions, palindromes, but particularly in Base 12, rather than Base 10, which is decribed in the pages on the subject. S k a t e b i k e r (talk) 14:42, 21 October 2018 (UTC)

Another issue, why are you using these weird characters for the numerals ten and eleven ? No keyboard kan enter these digits, and the convention for base 16 is using the letters A and B for ten and eleven, so why not in Base 12 ? Only in the (very unlikely) case that mankind switches to base 12, these nonletter digits make sense. S k a t e b i k e r (talk) 14:42, 21 October 2018 (UTC)

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