Duodecimal
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List of numeral systems 
The duodecimal system (also known as base 12, or dozenal) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead written as "10" in duodecimal (meaning "1 dozen and 0 units", instead of "1 ten and 0 units"), whereas the digit string "12" means "1 dozen and 2 units" (decimal 14). Similarly, in duodecimal, "100" means "1 gross", "1000" means "1 great gross", and "0.1" means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth", respectively).
Various symbols have been used to stand for ten and eleven in duodecimal notation; this page uses A and B, as in hexadecimal, which make a duodecimal count from zero to twelve read 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 10. The Dozenal Societies of America and Great Britain (organisations promoting the use of duodecimal) use turned digits in their published material: 2 (a turned 2) for ten and 3 (a turned 3) for eleven.
The number twelve, a superior highly composite number, is the smallest number with four nontrivial factors (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the subitizing range, and the smallest abundant number. All multiples of reciprocals of 3smooth numbers (a/2^{b}·3^{c} where a,b,c are integers) have a terminating representation in duodecimal. In particular, +1⁄4 (0.3), +1⁄3 (0.4), +1⁄2 (0.6), +2⁄3 (0.8), and +3⁄4 (0.9) all have a short terminating representation in duodecimal. There is also higher regularity observable in the duodecimal multiplication table. As a result, duodecimal has been described as the optimal number system.^{[1]}
In these respects, duodecimal is considered superior to decimal (which has only 2 and 5 as factors) and other proposed bases like octal or hexadecimal. Sexagesimal does even better in this respect (the reciprocals of all 5smooth numbers terminate), but at the cost of unwieldy multiplication tables and a much larger number of symbols to memorize.
Origin
 In this section, numerals are in decimal. For example, "10" means 9+1, and "12" means 9+3.
The origin of the duodecimal system is typically traced back to a system of finger counting based on the knuckle bones of the four larger fingers. Using the thumb as a pointer, it is possible to count to 12 by touching each finger bone, starting with the farthest bone on the fifth finger, and counting on. In this system, one hand counts repeatedly to 12, while the other displays the number of iterations, until five dozens, i.e. the 60, are full. This system is still in use in many regions of Asia.^{[2]}^{[3]}^{[4]}
Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, GbiriNiragu (GureKahugu), Piti, and the Nimbia dialect of Gwandara;^{[5]} and the Chepang language of Nepal^{[6]} are known to use duodecimal numerals.
Germanic languages have special words for 11 and 12, such as eleven and twelve in English. They come from ProtoGermanic *ainlif and *twalif (meaning, respectively, one left and two left), suggesting a decimal rather than duodecimal origin.^{[7]}^{[8]} However, Old Norse used a hybrid decimal–duodecimal counting system, with its words for "one hundred and eighty" meaning 200 and "two hundred" meaning 240.^{[9]} On the British Isles, this style of counting survived well into the Middle Ages as the long hundred.
Historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, and the Babylonians had twelve hours in a day (although at some point, this was changed to 24). Traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches or 24 (12×2) Solar terms. There are 12 inches in an imperial foot, 12 troy ounces in a troy pound, 12 old British pence in a shilling, 24 (12×2) hours in a day; many other items are counted by the dozen, gross (144, square of 12), or great gross (1728, cube of 12). The Romans used a fraction system based on 12, including the uncia, which became both the English words ounce and inch. Predecimalisation, Ireland and the United Kingdom used a mixed duodecimalvigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the pound sterling or Irish pound), and Charlemagne established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places.
Table of units from a base of 12  

Relative value 
French unit of length 
English unit of length 
English (Troy) unit of weight 
Roman unit of weight 
English unit of mass 
12^{0}  pied  foot  pound  libra  
12^{−1}  pouce  inch  ounce  uncia  slinch 
12^{−2}  ligne  line  2 scruples  2 scrupula  slug 
12^{−3}  point  point  seed  siliqua 
Notations and pronunciations
In a numbering system, the base (twelve for duodecimal) must be written as 10, but there are numerous proposals for how to write the quantities (counting values) "ten" and "eleven".^{[10]}
Notation  

⟨ten, eleven⟩  Background  Note  By keyboard 
By dedicated characters  
⟨A, B⟩  As in hexadecimal  To allow entry on typewriters.  
⟨T, E⟩  Initials of Ten and Eleven  
⟨X, E⟩  X from the Roman numeral for ten; E from English Eleven.  
⟨X, Z⟩  Origin of Z unknown but has been attributed to "D'Alambert [sic] & Buffon" by the Dozenal Society of America (DSA).^{[10]}  
⟨δ, ε⟩  Greek δ, ε, from δέκα "ten" and ένδεκα "eleven"^{[10]}  
⟨τ, ε⟩  Greek τ, ε^{[10]}  
⟨W, ∂⟩  W comes from doubling the Roman numeral for five; ∂ is based on a pendulum  Silvio Ferrari in Calcolo Decidozzinale (1854).^{[11]}  
⟨X, ℰ⟩  X, U+2130 ℰ SCRIPT CAPITAL E 
Frank Emerson Andrews in New Numbers (1935).^{[12]}  
⟨⚹, #⟩  sextile or sixpointed asterisk, hash or octothorpe 
Edna Kramer in The Main Stream of Mathematics (1951). Used in publications of the Dozenal Society of America (DSA) from 1974 to 2008,^{[13]}^{[14]} also on pushbutton telephones.^{[10]} 

⟨↊, ↋⟩ 



⟨, ⟩  Pronounced "dek", "el" 


By base notation^{[19]}  
duodecimal ⇔ decimal  Background  Note  By keyboard 
54 = 64 54;6 = 64.5 
In italics Use semicolon instead of a decimal point 
Humphrey point 
– 
*54 = 64 54;6 = 64.5 
Asterisked for whole numbers, Humphrey points for others  Used by DSGB.^{[19]}  – 
54_{z} = 64_{d}  Subscript "z"  From "dozenal". Used by DSA since 2015.^{[19]}  
54_{12} = 64_{10}  Subscript base number  Common usage by mathematicians and mathematics textbooks^{[19]}  
54_{twelve} = 64_{ten}  Subscript base spelt out  Variation of the above sometimes found in school textbooks^{[19]}  
doz 54 = dec 64 
Transdecimal symbols
↊ ↋  

duodecimal ⟨ten, eleven⟩  
In Unicode 

Block Number Forms  
Note  

To allow entry on typewriters, letters such as ⟨A, B⟩ (as in hexadecimal), ⟨T, E⟩ (initials of Ten and Eleven), ⟨X, E⟩, or ⟨X, Z⟩ (X from the Roman numeral for ten) are used. Some employ Greek letters, such as ⟨δ, ε⟩ (from Greek δέκα "ten" and ένδεκα "eleven") or ⟨τ, ε⟩.^{[10]} Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his book New Numbers ⟨X, ℰ⟩ (script capital E, U+2130).^{[12]}
Edna Kramer in her 1951 book The Main Stream of Mathematics used a ⟨⚹, #⟩ (sextile or sixpointed asterisk, hash or octothorpe).^{[10]} The symbols were chosen because they were available on some typewriters; they are also on pushbutton telephones.^{[10]} This notation was used in publications of the Dozenal Society of America (DSA) from 1974 to 2008.^{[21]}^{[22]}
From 2008 to 2015, the DSA used ⟨ , ⟩, the symbols devised by William Addison Dwiggins.^{[10]}^{[18]}
The Dozenal Society of Great Britain (DSGB) proposed symbols ⟨ 2, 3 ⟩.^{[10]} This notation, derived from Arabic digits by 180° rotation, was introduced by Isaac Pitman.^{[23]}^{[10]}^{[15]} In March 2013, a proposal was submitted to include the digit forms for ten and eleven propagated by the Dozenal Societies in the Unicode Standard.^{[24]} Of these, the British/Pitman forms were accepted for encoding as characters at code points U+218A ↊ TURNED DIGIT TWO and U+218B ↋ TURNED DIGIT THREE. They were included in Unicode 8.0 (2015).^{[16]}^{[25]}
After the Pitman digits were added to Unicode, the DSA took a vote and then began publishing content using the Pitman digits instead.^{[26]} They still use the letters X and E in ASCII text. As the Unicode characters are poorly supported, this page uses "A" and "B".
Other proposals are more creative or aesthetic; for example, many do not use any Arabic numerals under the principle of "separate identity."^{[10]}
Base notation
There are also varying proposals of how to distinguish a duodecimal number from a decimal one.^{[19]} They include italicizing duodecimal numbers "54 = 64", adding a "Humphrey point" (a semicolon instead of a decimal point) to duodecimal numbers "54;6 = 64.5", or some combination of the two. Others use subscript or affixed labels to indicate the base, allowing for more than decimal and duodecimal to be represented (for single letters, "z" from "dozenal" is used, as "d" would mean decimal),^{[19]} such as "54_{z} = 64_{d}," "54_{12} = 64_{10}" or "doz 54 = dec 64."
Pronunciation
The Dozenal Society of America suggested the pronunciation of ten and eleven as "dek" and "el". For the names of powers of twelve, there are two prominent systems.
Duodecimal numbers
In this system, the prefix e is added for fractions.^{[18]}^{[27]}
Duodecimal Number  Duodecimal Number Name  Duodecimal Number Fraction  Duodecimal Fraction Name 

1;  one  
10;  do  0;1  edo 
100;  gro  0;01  egro 
1,000;  mo  0;001  emo 
10,000;  domo  0;000,1  edomo 
100,000;  gromo  0;000,01  egromo 
1,000,000;  bimo  0;000,001  ebimo 
10,000,000;  dobimo  0;000,000,1  edobimo 
100,000,000;  grobimo  0;000,000,01  egrobimo 
1,000,000,000;  trimo  0;000,000,001  etrimo 
10,000,000,000;  dotrimo  0;000,000,000,1  edotrimo 
100,000,000,000;  grotrimo  0;000,000,000,01  egrotrimo 
1,000,000,000,000;  quadmo  0;000,000,000,001  equadmo 
10,000,000,000,000;  doquadmo  0;000,000,000,000,1  edoquadmo 
100,000,000,000,000;  groquadmo  0;000,000,000,000,01  egroquadmo 
1,000,000,000,000,000;  pentamo  0;000,000,000,000,001  epentamo 
10,000,000,000,000,000;  dopentamo  0;000,000,000,000,000,1  edopentamo 
100,000,000,000,000,000;  gropentamo  0;000,000,000,000,000,01  egropentamo 
1,000,000,000,000,000,000;  hexamo  0;000,000,000,000,000,001  ehexamo 
Multiple digits in this series are pronounced differently: 12 is "do two"; 30 is "three do"; 100 is "gro"; BA9 is "el gro dek do nine"; B86 is "el gro eight do six"; 8BB,15A is "eight gro el do el, one gro five do dek"; ABA is "dek gro el do dek"; BBB is "el gro el do el"; 0.06 is "six egro"; and so on.^{[27]}
Systematic Dozenal Nomenclature (SDN)
This system uses "qua" ending for the positive powers of 12 and "cia" ending for the negative powers of 12, and an extension of the IUPAC systematic element names (with syllables dec and lev for the two extra digits needed for duodecimal) to express which power is meant.^{[28]}^{[29]}
Duodecimal  Name  Decimal  Duodecimal fraction  Name 

1;  one  1  
10;  unqua  12  0;1  uncia 
100;  biqua  144  0;01  bicia 
1,000;  triqua  1,728  0;001  tricia 
10,000;  quadqua  20,736  0;000,1  quadcia 
100,000;  pentqua  248,832  0;000,01  pentcia 
1,000,000;  hexqua  2,985,984  0;000,001  hexcia 
10,000,000;  septqua  35,831,808  0;000,000,1  septcia 
100,000,000;  octqua  429,981,696  0;000,000,01  octcia 
1,000,000,000;  ennqua  5,159,780,352  0;000,000,001  enncia 
10,000,000,000;  decqua  61,917,364,224  0;000,000,000,1  deccia 
100,000,000,000;  levqua  743,008,370,688  0;000,000,000,01  levcia 
1,000,000,000,000;  unnilqua  8,916,100,448,256  0;000,000,000,001  unnilcia 
10,000,000,000,000;  ununqua  106,993,205,379,072  0;000,000,000,000,1  ununcia 
Advocacy and "dozenalism"
William James Sidis used 12 as the base for his constructed language Vendergood in 1906, noting it being the smallest number with four factors and its prevalence in commerce.^{[30]}
The case for the duodecimal system was put forth at length in Frank Emerson Andrews' 1935 book New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics. Emerson noted that, due to the prevalence of factors of twelve in many traditional units of weight and measure, many of the computational advantages claimed for the metric system could be realized either by the adoption of tenbased weights and measure or by the adoption of the duodecimal number system.^{[12]}
Both the Dozenal Society of America and the Dozenal Society of Great Britain promote widespread adoption of the duodecimal system. They use the word "dozenal" instead of "duodecimal" to avoid the more overtly decimal terminology. However, the etymology of "dozenal" itself is also an expression based on decimal terminology since "dozen" is a direct derivation of the French word douzaine, which is a derivative of the French word for twelve, douze, descended from Latin duodecim.
Mathematician and mental calculator Alexander Craig Aitken was an outspoken advocate of duodecimal:
The duodecimal tables are easy to master, easier than the decimal ones; and in elementary teaching they would be so much more interesting, since young children would find more fascinating things to do with twelve rods or blocks than with ten. Anyone having these tables at command will do these calculations more than oneandahalf times as fast in the duodecimal scale as in the decimal. This is my experience; I am certain that even more so it would be the experience of others.
— A. C. Aitken, "Twelves and Tens" in The Listener (January 25, 1962)^{[31]}
But the final quantitative advantage, in my own experience, is this: in varied and extensive calculations of an ordinary and not unduly complicated kind, carried out over many years, I come to the conclusion that the efficiency of the decimal system might be rated at about 65 or less, if we assign 100 to the duodecimal.
— A. C. Aitken, The Case Against Decimalisation (1962)^{[32]}
In media
In "Little Twelvetoes", American television series Schoolhouse Rock! portrayed an alien being using duodecimal arithmetic, using "dek" and "el" as names for ten and eleven, and Andrews' scriptX and scriptE for the digit symbols.^{[33]}^{[34]}
Duodecimal systems of measurements
Systems of measurement proposed by dozenalists include:
 Tom Pendlebury's TGM system^{[35]}^{[29]}
 Takashi Suga's Universal Unit System^{[36]}^{[29]}
 John Volan's Primel system^{[37]}
Comparison to other number systems
In this section, numerals are in decimal. For example, "10" means 9+1, and "12" means 6×2.
The Dozenal Society of America argues that if a base is too small, significantly longer expansions are needed for numbers; if a base is too large, one must memorise a large multiplication table to perform arithmetic. Thus, it presumes that "a number base will need to be between about 7 or 8 through about 16, possibly including 18 and 20".^{[38]}
The number 12 has six factors, which are 1, 2, 3, 4, 6, and 12, of which 2 and 3 are prime. It is the smallest number to have six factors, the largest number to have at least half of the numbers below it as divisors, and is only slightly larger than 10. (The numbers 18 and 20 also have six factors but are much larger.) Ten, in contrast, only has four factors, which are 1, 2, 5, and 10, of which 2 and 5 are prime.^{[38]} Six shares the prime factors 2 and 3 with twelve; however, like ten, six only has four factors (1, 2, 3, and 6) instead of six; therefore, its corresponding base, senary, is below the DSA's stated threshold.
Eight has four factors (1, 2, 4, and 8) but only has one prime factor (2). Sixteen adds itself as a fifth factor but no additional primes. This is because , and 8 already contains 2 as a factor. Therefore, in octal and hexadecimal, the only repeating numbers are the poweroftwo reciprocals.
Thirty is the smallest number that has three different prime factors (2, 3, and 5, the first three primes), and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30). Sexagesimal was actually used by the ancient Sumerians and Babylonians, among others; its base, sixty, adds the four convenient factors 4, 12, 20, and 60 to 30 but no new prime factors. The smallest number that has four different prime factors is 210; the pattern follows the primorials. However, these numbers would result in very large bases.
In all base systems, there are similarities to the representation of multiples of numbers that are one less than or one more than the base.
In the following multiplication table, numerals are based on duodecimal. For example, "10" means twelve, and "12" means fourteen.
×  1  2  3  4  5  6  7  8  9  A  B  10 

1  1  2  3  4  5  6  7  8  9  A  B  10 
2  2  4  6  8  A  10  12  14  16  18  1A  20 
3  3  6  9  10  13  16  19  20  23  26  29  30 
4  4  8  10  14  18  20  24  28  30  34  38  40 
5  5  A  13  18  21  26  2B  34  39  42  47  50 
6  6  10  16  20  26  30  36  40  46  50  56  60 
7  7  12  19  24  2B  36  41  48  53  5A  65  70 
8  8  14  20  28  34  40  48  54  60  68  74  80 
9  9  16  23  30  39  46  53  60  69  76  83  90 
A  A  18  26  34  42  50  5A  68  76  84  92  A0 
B  B  1A  29  38  47  56  65  74  83  92  A1  B0 
10  10  20  30  40  50  60  70  80  90  A0  B0  100 
Conversion tables to and from decimal
To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under positional notation). Alternatively, one can use digitconversion tables. The ones provided below can be used to convert any duodecimal number between 0;01 and BBB,BBB;BB to decimal, or any decimal number between 0.01 and 999,999.99 to duodecimal. To use them, the given number must first be decomposed into a sum of numbers with only one significant digit each. For example:
 123,456.78 = 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08
This decomposition works the same no matter what base the number is expressed in. Just isolate each nonzero digit, padding them with as many zeros as necessary to preserve their respective place values. If the digits in the given number include zeroes (for example, 102,304.05), these are left out in the digit decomposition (102,304.05 = 100,000 + 2,000 + 300 + 4 + 0.05). Then, the digit conversion tables can be used to obtain the equivalent value in the target base for each digit. If the given number is in duodecimal and the target base is decimal, we get:
 (duodecimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0;7 + 0;08 = (decimal) 248,832 + 41,472 + 5,184 + 576 + 60 + 6 + 0.583333333333... + 0.055555555555...
Because the summands are already converted to decimal, the usual decimal arithmetic is used to perform the addition and recompose the number, arriving at the conversion result:
Duodecimal > Decimal
100,000 = 248,832 20,000 = 41,472 3,000 = 5,184 400 = 576 50 = 60 + 6 = + 6 0;7 = 0.583333333333... 0;08 = 0.055555555555...  123,456;78 = 296,130.638888888888...
That is, (duodecimal) 123,456.78 equals (decimal) 296,130.638 ≈ 296,130.64
If the given number is in decimal and the target base is duodecimal, the method is same. Using the digit conversion tables:
(decimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (duodecimal) 49,A54 + B,6A8 + 1,8A0 + 294 + 42 + 6 + 0;849724972497249724972497... + 0;0B62A68781B05915343A0B62...
However, to do this sum and recompose the number, the addition tables for the duodecimal system have to be used, instead of the addition tables for decimal most people are already familiar with. This is because the summands are now in duodecimal; thus, the arithmetic with them has to be in duodecimal as well. In decimal, ; in duodecimal, . So, if using decimal arithmetic with duodecimal numbers, one would arrive at an incorrect result. Doing the arithmetic properly in duodecimal, one gets the result:
Decimal > Duodecimal
100,000 = 49,A54 20,000 = B,6A8 3,000 = 1,8A0 400 = 294 50 = 42 + 6 = + 6 0.7 = 0;849724972497249724972497... 0.08 = 0;0B62A68781B05915343A0B62...  123,456.78 = 5B,540;943A0B62A68781B05915343A...
That is, (decimal) 123,456.78 equals (duodecimal) 5B,540;943A0B62A68781B059153... ≈ 5B,540;94
Duodecimal to decimal digit conversion
Duod.  Decimal  Duod.  Decimal  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec. 

1,000,000  2,985,984  100,000  248,832  10,000  20,736  1,000  1,728  100  144  10  12  1  1  0;1  0.083  0;01  0.00694 
2,000,000  5,971,968  200,000  497,664  20,000  41,472  2,000  3,456  200  288  20  24  2  2  0;2  0.16  0;02  0.0138 
3,000,000  8,957,952  300,000  746,496  30,000  62,208  3,000  5,184  300  432  30  36  3  3  0;3  0.25  0;03  0.02083 
4,000,000  11,943,936  400,000  995,328  40,000  82,944  4,000  6,912  400  576  40  48  4  4  0;4  0.3  0;04  0.027 
5,000,000  14,929,920  500,000  1,244,160  50,000  103,680  5,000  8,640  500  720  50  60  5  5  0;5  0.416  0;05  0.03472 
6,000,000  17,915,904  600,000  1,492,992  60,000  124,416  6,000  10,368  600  864  60  72  6  6  0;6  0.5  0;06  0.0416 
7,000,000  20,901,888  700,000  1,741,824  70,000  145,152  7,000  12,096  700  1,008  70  84  7  7  0;7  0.583  0;07  0.04861 
8,000,000  23,887,872  800,000  1,990,656  80,000  165,888  8,000  13,824  800  1,152  80  96  8  8  0;8  0.6  0;08  0.05 
9,000,000  26,873,856  900,000  2,239,488  90,000  186,624  9,000  15,552  900  1,296  90  108  9  9  0;9  0.75  0;09  0.0625 
A,000,000  29,859,840  A00,000  2,488,320  A0,000  207,360  A,000  17,280  A00  1,440  A0  120  A  10  0;A  0.83  0;0A  0.0694 
B,000,000  32,845,824  B00,000  2,737,152  B0,000  228,096  B,000  19,008  B00  1,584  B0  132  B  11  0;B  0.916  0;0B  0.07638 
Decimal to duodecimal digit conversion
Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duodecimal  Dec.  Duodecimal 

1,000,000  402,854  100,000  49,A54  10,000  5,954  1,000  6B4  100  84  10  A  1  1  0.1  0;12497  0.01  0;015343A0B62A68781B059 
2,000,000  805,4A8  200,000  97,8A8  20,000  B,6A8  2,000  1,1A8  200  148  20  18  2  2  0.2  0;2497  0.02  0;02A68781B05915343A0B6 
3,000,000  1,008,140  300,000  125,740  30,000  15,440  3,000  1,8A0  300  210  30  26  3  3  0.3  0;37249  0.03  0;043A0B62A68781B059153 
4,000,000  1,40A,994  400,000  173,594  40,000  1B,194  4,000  2,394  400  294  40  34  4  4  0.4  0;4972  0.04  0;05915343A0B62A68781B 
5,000,000  1,811,628  500,000  201,428  50,000  24,B28  5,000  2,A88  500  358  50  42  5  5  0.5  0;6  0.05  0;07249 
6,000,000  2,014,280  600,000  24B,280  60,000  2A,880  6,000  3,580  600  420  60  50  6  6  0.6  0;7249  0.06  0;08781B05915343A0B62A6 
7,000,000  2,416,B14  700,000  299,114  70,000  34,614  7,000  4,074  700  4A4  70  5A  7  7  0.7  0;84972  0.07  0;0A0B62A68781B05915343 
8,000,000  2,819,768  800,000  326,B68  80,000  3A,368  8,000  4,768  800  568  80  68  8  8  0.8  0;9724  0.08  0;0B62A68781B05915343A 
9,000,000  3,020,400  900,000  374,A00  90,000  44,100  9,000  5,260  900  630  90  76  9  9  0.9  0;A9724  0.09  0;10B62A68781B05915343A 
Divisibility rules
In this section, numerals are in duodecimal. For example, "10" means 6×2, and "12" means 7×2.
This section is about the divisibility rules in duodecimal.
 1
Any integer is divisible by 1.
 2
If a number is divisible by 2, then the unit digit of that number will be 0, 2, 4, 6, 8, or A.
 3
If a number is divisible by 3, then the unit digit of that number will be 0, 3, 6, or 9.
 4
If a number is divisible by 4, then the unit digit of that number will be 0, 4, or 8.
 5
To test for divisibility by 5, double the units digit and subtract the result from the number formed by the rest of the digits. If the result is divisible by 5, then the given number is divisible by 5.
This rule comes from 21 ().
Examples:
13 rule → , which is divisible by 5.
2BA5 rule → , which is divisible by 5 (or apply the rule on 2B0).
OR
To test for divisibility by 5, subtract the units digit and triple of the result to the number formed by the rest of the digits. If the result is divisible by 5, then the given number is divisible by 5.
This rule comes from 13 ().
Examples:
13 rule → , which is divisible by 5.
2BA5 rule → , which is divisible by 5 (or apply the rule on 8B1).
OR
Form the alternating sum of blocks of two from right to left. If the result is divisible by 5, then the given number is divisible by 5.
This rule comes from 101, since ; thus, this rule can be also tested for the divisibility by 25.
Example:
97,374,627 → , which is divisible by 5.
 6
If a number is divisible by 6, then the unit digit of that number will be 0 or 6.
 7
To test for divisibility by 7, triple the units digit and add the result to the number formed by the rest of the digits. If the result is divisible by 7, then the given number is divisible by 7.
This rule comes from 2B ()
Examples:
12 rule → , which is divisible by 7.
271B rule → , which is divisible by 7 (or apply the rule on 29A).
OR
To test for divisibility by 7, subtract the units digit and double the result from the number formed by the rest of the digits. If the result is divisible by 7, then the given number is divisible by 7.
This rule comes from 12 ().
Examples:
12 rule → , which is divisible by 7.
271B rule → , which is divisible by 7 (or apply the rule on 513).
OR
To test for divisibility by 7, quadruple the units digit and subtract the result from the number formed by the rest of the digits. If the result is divisible by 7, then the given number is divisible by 7.
This rule comes from 41 ().
Examples:
12 rule → , which is divisible by 7.
271B rule → , which is divisible by 7 (or apply the rule on 235).
OR
Form the alternating sum of blocks of three from right to left. If the result is divisible by 7, then the given number is divisible by 7.
This rule comes from 1001, since ; thus, this rule can be also tested for the divisibility by 11 and 17.
Example:
386,967,443 → , which is divisible by 7.
 8
If the twodigit number formed by the last two digits of the given number is divisible by 8, then the given number is divisible by 8.
Example: 1B48, 4120
rule => since 48(8*7) divisible by 8, then 1B48 is divisible by 8. rule => since 20(8*3) divisible by 8, then 4120 is divisible by 8.
 9
If the twodigit number formed by the last two digits of the given number is divisible by 9, then the given number is divisible by 9.
Example: 7423, 8330
rule => since 23(9*3) divisible by 9, then 7423 is divisible by 9. rule => since 30(9*4) divisible by 9, then 8330 is divisible by 9.
 A
If the number is divisible by 2 and 5, then the number is divisible by A.
 B
If the sum of the digits of a number is divisible by B, then the number is divisible by B (the equivalent of casting out nines in decimal).
Example: 29, 61B13
rule => 2+9 = B, which is divisible by B, then 29 is divisible by B. rule => 6+1+B+1+3 = 1A, which is divisible by B, then 61B13 is divisible by B.
 10
If a number is divisible by 10, then the unit digit of that number will be 0.
 11
Sum the alternate digits and subtract the sums. If the result is divisible by 11, the number is divisible by 11 (the equivalent of divisibility by eleven in decimal).
Example: 66, 9427
rule => 66 = 0, which is divisible by 11, then 66 is divisible by 11. rule => (9+2)(4+7) = AA = 0, which is divisible by 11, then 9427 is divisible by 11.
 12
If the number is divisible by 2 and 7, then the number is divisible by 12.
 13
If the number is divisible by 3 and 5, then the number is divisible by 13.
 14
If the twodigit number formed by the last two digits of the given number is divisible by 14, then the given number is divisible by 14.
Example: 1468, 7394
rule => since 68(14*5) divisible by 14, then 1468 is divisible by 14. rule => since 94(14*7) divisible by 14, then 7394 is divisible by 14.
Fractions and irrational numbers
Fractions
Duodecimal fractions for rational numbers with 3smooth denominators terminate:
 1/2 = 0;6
 1/3 = 0;4
 1/4 = 0;3
 1/6 = 0;2
 1/8 = 0;16
 1/9 = 0;14
 1/10 = 0;1 (this is one twelfth, 1/A is one tenth)
 1/14 = 0;09 (this is one sixteenth, 1/12 is one fourteenth)
while other rational numbers have recurring duodecimal fractions:
 1/5 = 0;2497
 1/7 = 0;186A35
 1/A = 0;12497 (one tenth)
 1/B = 0;1 (one eleventh)
 1/11 = 0;0B (one thirteenth)
 1/12 = 0;0A35186 (one fourteenth)
 1/13 = 0;09724 (one fifteenth)
Examples in duodecimal  Decimal equivalent 

1 × (5/8) = 0.76  1 × (5/8) = 0.625 
100 × (5/8) = 76  144 × (5/8) = 90 
576/9 = 76  810/9 = 90 
400/9 = 54  576/9 = 64 
1A.6 + 7.6 = 26  22.5 + 7.5 = 30 
As explained in recurring decimals, whenever an irreducible fraction is written in radix point notation in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factors of its denominator are also prime factors of the base.
Because in the decimal system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: 1/8 = 1/(2×2×2), 1/20 = 1/(2×2×5), and 1/500 = 1/(2×2×5×5×5) can be expressed exactly as 0.125, 0.05, and 0.002 respectively. 1/3 and 1/7, however, recur (0.333... and 0.142857142857...).
Because in the duodecimal system, 1/8 is exact; 1/20 and 1/500 recur because they include 5 as a factor; 1/3 is exact, and 1/7 recurs, just as it does in decimal.
The number of denominators that give terminating fractions within a given number of digits, n, in a base b is the number of factors (divisors) of , the nth power of the base b (although this includes the divisor 1, which does not produce fractions when used as the denominator). The number of factors of is given using its prime factorization.
For decimal, . The number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together, so the number of factors of is .
For example, the number 8 is a factor of 10^{3} (1000), so and other fractions with a denominator of 8 cannot require more than three fractional decimal digits to terminate.
For duodecimal, . This has divisors. The sample denominator of 8 is a factor of a gross in decimal), so eighths cannot need more than two duodecimal fractional places to terminate.
Because both ten and twelve have two unique prime factors, the number of divisors of for b = 10 or 12 grows quadratically with the exponent n (in other words, of the order of ).
Recurring digits
The Dozenal Society of America argues that factors of 3 are more commonly encountered in reallife division problems than factors of 5.^{[38]} Thus, in practical applications, the nuisance of repeating decimals is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.
However, when recurring fractions do occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because 12 (twelve) is between two prime numbers, 11 (eleven) and 13 (thirteen), whereas ten is adjacent to the composite number 9. Nonetheless, having a shorter or longer period does not help the main inconvenience that one does not get a finite representation for such fractions in the given base (so rounding, which introduces inexactitude, is necessary to handle them in calculations), and overall one is more likely to have to deal with infinite recurring digits when fractions are expressed in decimal than in duodecimal, because one out of every three consecutive numbers contains the prime factor 3 in its factorization, whereas only one out of every five contains the prime factor 5. All other prime factors, except 2, are not shared by either ten or twelve, so they do not influence the relative likeliness of encountering recurring digits (any irreducible fraction that contains any of these other factors in its denominator will recur in either base).
Also, the prime factor 2 appears twice in the factorization of twelve, whereas only once in the factorization of ten; which means that most fractions whose denominators are powers of two will have a shorter, more convenient terminating representation in duodecimal than in decimal:
 1/(2^{2}) = 0.25_{10} = 0.3_{12}
 1/(2^{3}) = 0.125_{10} = 0.16_{12}
 1/(2^{4}) = 0.0625_{10} = 0.09_{12}
 1/(2^{5}) = 0.03125_{10} = 0.046_{12}
Decimal base Prime factors of the base: 2, 5 Prime factors of one below the base: 3 Prime factors of one above the base: 11 All other primes: 7, 13, 17, 19, 23, 29, 31 
Duodecimal base Prime factors of the base: 2, 3 Prime factors of one below the base: B Prime factors of one above the base: 11 (=13_{10}) All other primes: 5, 7, 15, 17, 1B, 25, 27  
Fraction  Prime factors of the denominator 
Positional representation  Positional representation  Prime factors of the denominator 
Fraction 

1/2  2  0.5  0;6  2  1/2 
1/3  3  0.3  0;4  3  1/3 
1/4  2  0.25  0;3  2  1/4 
1/5  5  0.2  0;2497  5  1/5 
1/6  2, 3  0.16  0;2  2, 3  1/6 
1/7  7  0.142857  0;186A35  7  1/7 
1/8  2  0.125  0;16  2  1/8 
1/9  3  0.1  0;14  3  1/9 
1/10  2, 5  0.1  0;12497  2, 5  1/A 
1/11  11  0.09  0;1  B  1/B 
1/12  2, 3  0.083  0;1  2, 3  1/10 
1/13  13  0.076923  0;0B  11  1/11 
1/14  2, 7  0.0714285  0;0A35186  2, 7  1/12 
1/15  3, 5  0.06  0;09724  3, 5  1/13 
1/16  2  0.0625  0;09  2  1/14 
1/17  17  0.0588235294117647  0;08579214B36429A7  15  1/15 
1/18  2, 3  0.05  0;08  2, 3  1/16 
1/19  19  0.052631578947368421  0;076B45  17  1/17 
1/20  2, 5  0.05  0;07249  2, 5  1/18 
1/21  3, 7  0.047619  0;06A3518  3, 7  1/19 
1/22  2, 11  0.045  0;06  2, B  1/1A 
1/23  23  0.0434782608695652173913  0;06316948421  1B  1/1B 
1/24  2, 3  0.0416  0;06  2, 3  1/20 
1/25  5  0.04  0;05915343A0B62A68781B  5  1/21 
1/26  2, 13  0.0384615  0;056  2, 11  1/22 
1/27  3  0.037  0;054  3  1/23 
1/28  2, 7  0.03571428  0;05186A3  2, 7  1/24 
1/29  29  0.0344827586206896551724137931  0;04B7  25  1/25 
1/30  2, 3, 5  0.03  0;04972  2, 3, 5  1/26 
1/31  31  0.032258064516129  0;0478AA093598166B74311B28623A55  27  1/27 
1/32  2  0.03125  0;046  2  1/28 
1/33  3, 11  0.03  0;04  3, B  1/29 
1/34  2, 17  0.02941176470588235  0;0429A708579214B36  2, 15  1/2A 
1/35  5, 7  0.0285714  0;0414559B3931  5, 7  1/2B 
1/36  2, 3  0.027  0;04  2, 3  1/30 
The duodecimal period length of 1/n are (in decimal)
 0, 0, 0, 0, 4, 0, 6, 0, 0, 4, 1, 0, 2, 6, 4, 0, 16, 0, 6, 4, 6, 1, 11, 0, 20, 2, 0, 6, 4, 4, 30, 0, 1, 16, 12, 0, 9, 6, 2, 4, 40, 6, 42, 1, 4, 11, 23, 0, 42, 20, 16, 2, 52, 0, 4, 6, 6, 4, 29, 4, 15, 30, 6, 0, 4, 1, 66, 16, 11, 12, 35, 0, ... (sequence A246004 in the OEIS)
The duodecimal period length of 1/(nth prime) are (in decimal)
 0, 0, 4, 6, 1, 2, 16, 6, 11, 4, 30, 9, 40, 42, 23, 52, 29, 15, 66, 35, 36, 26, 41, 8, 16, 100, 102, 53, 54, 112, 126, 65, 136, 138, 148, 150, 3, 162, 83, 172, 89, 90, 95, 24, 196, 66, 14, 222, 113, 114, 8, 119, 120, 125, 256, 131, 268, 54, 138, 280, ... (sequence A246489 in the OEIS)
Smallest prime with duodecimal period n are (in decimal)
 11, 13, 157, 5, 22621, 7, 659, 89, 37, 19141, 23, 20593, 477517, 211, 61, 17, 2693651, 1657, 29043636306420266077, 85403261, 8177824843189, 57154490053, 47, 193, 303551, 79, 306829, 673, 59, 31, 373, 153953, 886381, 2551, 71, 73, ... (sequence A252170 in the OEIS)
Irrational numbers
The representations of irrational numbers in any positional number system (including decimal and duodecimal) neither terminate nor repeat. The following table gives the first digits for some important algebraic and transcendental numbers in both decimal and duodecimal.
Algebraic irrational number  In decimal  In duodecimal 

√2, the square root of 2  1.414213562373...  1;4B79170A07B8... 
φ (phi), the golden ratio =  1.618033988749...  1;74BB6772802A... 
Transcendental number  In decimal  In duodecimal 
π (pi), the ratio of a circle's circumference to its diameter  3.141592653589...  3;184809493B91... 
e, the base of the natural logarithm  2.718281828459...  2;875236069821... 
See also
 Vigesimal (base 20)
 Sexagesimal (base 60)
References
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 ^ Ifrah, Georges (2000). The Universal History of Numbers: From prehistory to the invention of the computer. John Wiley and Sons. ISBN 0471393401. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk.
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 ^ Ferrari, Silvio (1854). Calcolo Decidozzinale. p. 2.
 ^ ^{a} ^{b} ^{c} Andrews, Frank Emerson (1935). New Numbers: How Acceptance of a Duodecimal (12) Base Would Simplify Mathematics. p. 52.
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 ^ ^{a} ^{b} De Vlieger, Michael (2008). "Going Classic" (PDF). The Duodecimal Bulletin. 49 [57] (2).
 ^ ^{a} ^{b} Pitman, Isaac (1947). "A Reckoning Reform [reprint from 1857]" (PDF). The Duodecimal Bulletin. 3 (2). Archived (PDF) from the original on 20160701.
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 ^ "The Unicode Standard 8.0" (PDF). Retrieved 20140718.
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 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} Volan, John (July 2015). "Base Annotation Schemes" (PDF). The Duodecimal Bulletin. 62. Archived (PDF) from the original on 20180102.
 ^ Scott Pakin (2009). "The Comprehensive LATEX Symbol List" (PDF). Archived (PDF) from the original on 20150905. Retrieved 20160530.
 ^ "Annual Meeting of 1973 and Meeting of the Board" (PDF). The Duodecimal Bulletin. 25 [29] (1). 1974.
 ^ De Vlieger, Michael (2008). "Going Classic" (PDF). The Duodecimal Bulletin. 49 [57] (2).
 ^ Pitman, Isaac (ed.): A triple (twelve gross) Gems of Wisdom. London 1860
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 ^ "The Unicode Standard 8.0" (PDF). Retrieved 20140718.
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 ^ ^{a} ^{b} Zirkel, Gene (2010). "How Do You Pronounce Dozenals?" (PDF). The Duodecimal Bulletin. 4E [59] (2). Archived (PDF) from the original on 20160701.
 ^ "Systematic Dozenal Nomenclature and other nomenclature systems" (PDF). The Duodecimal Bulletin. Archived (PDF) from the original on 20160701. Retrieved 20190728.
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 ^ The Prodigy (Biography of WJS) pg [42]
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External links
 Dozenal Society of America
 "The DSA Symbology Synopsis"
 "Resources", the DSA website's page of external links to third party tools
 Dozenal Society of Great Britain
 Lauritzen, Bill (1994). "Nature's Numbers". Earth360.
 Savard, John J. G. (2018) [2016]. "Changing the Base". quadibloc. Retrieved 20180717.