Talk:Duodecimal

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)

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) [1]. 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)

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)

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 men-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)

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)

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 [<link removed - in blacklist> DozensOnline forum], 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)

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)

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 ^(Talk) 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 ^(Talk) 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)

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)

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)

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 ¹/₁₂ 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.

Language	Inch	Thumb
Catalan	polzad	polze
Danish	tomme	tommelfinger
Dutch	inch(1)	duim
French	pouce	pouce
Finnish	tuuma	peukalo
Galician	polgada	polgar
Icelandic	tomma	thumb
Irish	orlach	ordóg
Italian	pollice	pollice
Portuguese	polegada	polegar
Spanish	pulgada	pulgar
Swedish	tum	tumme

(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, [don't know]" (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)