Numeral systems 

Hindu–Arabic numeral system 
East Asian 
Alphabetic 
Former 
Positional systems by base 
Nonstandard positional numeral systems 
List of numeral systems 
The duodecimal system (also known as base 12, dozenal, or rarely uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the base ten 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" (i.e. the same number that in decimal is written as "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").
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. As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers (whereas ten has only two nontrivial factors: 2 and 5, and not 3, 4, or 6), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. As a result, duodecimal has been described as the optimal number system.^{[1]} Of its factors, 2 and 3 are prime, which means the reciprocals of all 3smooth numbers (such as 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, ...) have a terminating representation in duodecimal. In particular, the five most elementary fractions ( ^{1}⁄_{2}, ^{1}⁄_{3}, ^{2}⁄_{3}, ^{1}⁄_{4} and ^{3}⁄_{4}) all have a short terminating representation in duodecimal (0.6, 0.4, 0.8, 0.3 and 0.9, respectively), and twelve is the smallest radix with this feature (because it is the least common multiple of 3 and 4). This all makes it a more convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary, octal and hexadecimal systems. Although the trigesimal and sexagesimal systems (where the reciprocals of all 5smooth numbers terminate) do even better in this respect, this is at the cost of unwieldy multiplication tables and a much larger number of symbols to memorize.
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✪ Base 12  Why Counting In Twelves Would Make Life Easier

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✪ Base 12, and Why the Way We Count Sucks

✪ We NEED the Doudecimal (Dozenal) System!
Transcription
Contents
Origin
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;^{[2]} the Chepang language of Nepal^{[3]} and the Maldivian language (Dhivehi) of the people of the Maldives and Minicoy Island in India are known to use duodecimal numerals.
Germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from ProtoGermanic *ainlif and *twalif (respectively one left and two left), both of which were decimal.^{[4]}^{[5]}
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. 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, and many other items 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 
The importance of 12 has been attributed to the number of lunar cycles in a year, and also to the fact that humans have 12 finger bones (phalanges) on one hand (three on each of four fingers).^{[6]}^{[7]} It is possible to count to 12 with the thumb acting as a pointer, touching each finger bone in turn. A traditional finger counting system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. In this system, the one (usually right) hand counts repeatedly to 12, displaying the number of iterations on the other (usually left), until five dozens, i. e. the 60, are full.^{[8]}^{[9]}
Notations and pronunciations
Transdecimal symbols
In a duodecimal place system twelve is written as 10, but there are numerous proposals for how to write ten and eleven.^{[10]}
To allow entry on typewriters, letters such as A and B (as in hexadecimal), T and E (initials of Ten and Eleven), X and E (X from the Roman numeral for ten), or X and Z are used. Some employ Greek letters such as δ (standing for Greek δέκα 'ten') and ε (for Greek ένδεκα 'eleven'), or τ and ε.^{[10]} Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his book New Numbers an X and ℰ (script E, U+2130).^{[11]}
Edna Kramer in her 1951 book The Main Stream of Mathematics used a sixpointed asterisk (sextile) ⚹ and a hash (or octothorpe) #.^{[10]} The symbols were chosen because they are available in typewriters, they also are on pushbutton telephones.^{[10]} This notation was used in publications of the Dozenal Society of America (DSA) in the period 1974–2008.^{[12]}^{[13]}
From 2008 to 2015, the DSA used
andThe Dozenal Society of Great Britain proposed a rotated digit two (2) and a reversed or rotated digit three (3).^{[10]} This notation was introduced by Sir Isaac Pitman.^{[15]}^{[10]}^{[16]}
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.^{[17]} 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 the Unicode 8.0 release in June 2015.^{[18]}^{[19]} They are available in LaTeX as \textturntwo
and \textturnthree
.^{[20]}
After the Pitman digits were added to Unicode the DSA took a vote and then began publishing content using the Pitman digits instead.^{[21]} They still use the letters X and E in ASCII text.
Other proposals are more creative or aesthetic; for example, many do not use any Arabic numerals under the principle of "separate identity."^{[10]}
As these Unicode symbols are not rendering in many browsers, this article uses U+2D52 ⵒ TIFINAGH LETTER YAP and U+0190 Ɛ LATIN CAPITAL LETTER OPEN E, or physically rotated '2' and '3' glyphs.
Base notation
There are also varying proposals of how to distinguish a duodecimal number from a decimal one.^{[22]} 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)^{[22]} such as "54_{z} = 64_{d}," "54_{12} = 64_{10}" or "doz 54 = dec 64." Programming languages limited to ASCII could use a prefix, similar to 0x used for hexadecimal, perhaps 0z but there is no standard.
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 famous systems.
The dogromo system
In this system, each order has its own name and the prefix e is added for fractions.^{[14]}^{[23]}
Duodecimal  Name  Decimal  Duodecimal fraction  Name 

1;  one  1  
10;  do  12  0;1  edo 
100;  gro  144  0;01  egro 
1,000;  mo  1,728  0;001  emo 
10,000;  domo  20,736  0;000,1  edomo 
100,000;  gromo  248,832  0;000,01  egromo 
1,000,000;  bimo  2,985,984  0;000,001  ebimo 
1,000,000,000;  trimo  5,159,780,352  0;000,000,001  etrimo 
Multiple digits in this are pronounced differently. 12; is "one do two", 30; is "three do", 100; is "one gro", 239; is "el gro dek do nine", 38,652,300; is "el do eight bimo, six gro five do dek mo, three gro", and so on.^{[23]}
Systematic Dozenal Nomenclature (SDN)
This system uses "qua" ending for the positive powers of 12 and "cia" ending for the negative powers of 12.^{[24]}^{[25]}
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 the prevalence in commerce.^{[26]}
The case for the duodecimal system was put forth at length in F. 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.
Both the Dozenal Society of America and the Dozenal Society of Great Britain promote widespread adoption of the basetwelve system. They use the word "dozenal" instead of "duodecimal" to avoid the more overtly baseten terminology. The etymology of 'dozenal' is itself also an expression based on baseten terminology since 'dozen' is a direct derivation of the French word 'douzaine' which is a derivative of the French word for twelve, douze which is related to the old French word 'doze' from Latin 'duodecim'.
Since at least as far back as 1945 some members of the Dozenal Society of America and Duodecimal Society of Great Britain have suggested that a more apt word would be 'uncial'. Uncial is a derivation of the Latin word 'onetwelfth' which is 'uncia' and also the basetwelve analogue of the Latin word 'onetenth' which is 'decima'. In the same manner as decimal comes from the Latin word for onetenth decima, (Latin for ten was decem), the direct analogue for a basetwelve system is uncial.^{[27]}
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)^{[28]}
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)^{[29]}
In Lee Carroll's Kryon: Alchemy of the Human Spirit, a chapter is dedicated to the advantages of the duodecimal system. The duodecimal system is supposedly suggested by Kryon (a fictional entity believed in by New Age circles) for allround use, aiming at better and more natural representation of nature of the Universe through mathematics. An individual article "Mathematica" by James D. Watt (included in the above publication) exposes a few of the unusual symmetry connections between the duodecimal system and the golden ratio, as well as provides numerous number symmetrybased arguments for the universal nature of the base12 number system.^{[30]}
In "Little Twelvetoes", American television series Schoolhouse Rock! portrayed an alien child using basetwelve arithmetic, using "dek", "el" and "doh" as names for ten, eleven and twelve, and Andrews' scriptX and scriptE for the digit symbols.^{[31]}
Duodecimal systems of measurements
Systems of measurement proposed by dozenalists include:
 Tom Pendlebury's TGM system^{[32]}^{[25]}
 Takashi Suga's Universal Unit System^{[33]}^{[25]}
Comparison to other number systems
The number 12 has six factors, which are 1, 2, 3, 4, 6, and 12, of which 2 and 3 are prime. The decimal system has only four factors, which are 1, 2, 5, and 10, of which 2 and 5 are prime. Vigesimal (base 20) adds two factors to those of ten, namely 4 and 20, but no additional prime factor. Although twenty has 6 factors, 2 of them prime, similarly to twelve, it is also a much larger base, and so the digit set and the multiplication table are much larger. Binary has only two factors, 1 and 2, the latter being prime. Hexadecimal (base 16) has five factors, adding 4, 8 and 16 to those of 2, but no additional prime. Trigesimal (base 30) 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). Sexagesimal—which the ancient Sumerians and Babylonians among others actually used—adds the four convenient factors 4, 12, 20, and 60 to this but no new prime factors. The smallest system that has four different prime factors is base 210 and the pattern follows the primorials. In all base systems, there are similarities to the representation of multiples of numbers which are one less than the base.
×  0  1  2  3  4  5  6  7  8  9  2  3  10 

0  0  0  0  0  0  0  0  0  0  0  0  0  0 
1  0  1  2  3  4  5  6  7  8  9  2  3  10 
2  0  2  4  6  8  2  10  12  14  16  18  12  20 
3  0  3  6  9  10  13  16  19  20  23  26  29  30 
4  0  4  8  10  14  18  20  24  28  30  34  38  40 
5  0  5  2  13  18  21  26  23  34  39  42  47  50 
6  0  6  10  16  20  26  30  36  40  46  50  56  60 
7  0  7  12  19  24  23  36  41  48  53  52  65  70 
8  0  8  14  20  28  34  40  48  54  60  68  74  80 
9  0  9  16  23  30  39  46  53  60  69  76  83  90 
2  0  2  18  26  34  42  50  52  68  76  84  92  20 
3  0  3  12  29  38  47  56  65  74  83  92  21  30 
10  0  10  20  30  40  50  60  70  80  90  20  30  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 ƐƐƐ,ƐƐƐ.ƐƐ 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, of course, 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...
Now, because the summands are already converted to base ten, 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 basically same. Using the digit conversion tables:
(decimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (duodecimal) 49,ⵒ54 + Ɛ,6ⵒ8 + 1,8ⵒ0 + 294 + 42 + 6 + 0.849724972497249724972497... + 0.0Ɛ62ⵒ68781Ɛ05915343ⵒ0Ɛ62...
However, in order to do this sum and recompose the number, now the addition tables for the duodecimal system have to be used, instead of the addition tables for decimal most people are already familiar with, because the summands are now in base twelve and so the arithmetic with them has to be in duodecimal as well. In decimal, 6 + 6 equals 12, but in duodecimal it equals 10; 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,ⵒ54 20,000 = Ɛ,6ⵒ8 3,000 = 1,8ⵒ0 400 = 294 50 = 42 + 6 = + 6 0;7 = 0.849724972497249724972497... 0;08 = 0.0Ɛ62ⵒ68781Ɛ05915343ⵒ0Ɛ62...  123,456.78 = 5Ɛ,540.943ⵒ0Ɛ62ⵒ68781Ɛ05915343ⵒ...
That is, (decimal) 123,456.78 equals (duodecimal) 5Ɛ,540;943ⵒ0Ɛ62ⵒ68781Ɛ059153... ≈ 5Ɛ,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 
ⵒ,000,000  29,859,840  ⵒ00,000  2,488,320  ⵒ0,000  207,360  ⵒ,000  17,280  ⵒ00  1,440  ⵒ0  120  ⵒ  10  0;ⵒ  0.83  0;0ⵒ  0.0694 
Ɛ,000,000  32,845,824  Ɛ00,000  2,737,152  Ɛ0,000  228,096  Ɛ,000  19,008  Ɛ00  1,584  Ɛ0  132  Ɛ  11  0;Ɛ  0.916  0;0Ɛ  0.07638 
Decimal to duodecimal digit conversion
Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duod.  Dec.  Duodecimal 

100,000  49,ⵒ54  10,000  5,954  1,000  6Ɛ4  100  84  10  ⵒ  1  1  0.1  0;12497  0.01  0;015343ⵒ0Ɛ62ⵒ68781Ɛ059 
200,000  97,8ⵒ8  20,000  Ɛ,6ⵒ8  2,000  1,1ⵒ8  200  148  20  18  2  2  0.2  0;2497  0.02  0;02ⵒ68781Ɛ05915343ⵒ0Ɛ6 
300,000  125,740  30,000  15,440  3,000  1,8ⵒ0  300  210  30  26  3  3  0.3  0;37249  0.03  0;043ⵒ0Ɛ62ⵒ68781Ɛ059153 
400,000  173,594  40,000  1Ɛ,194  4,000  2,394  400  294  40  34  4  4  0.4  0;4972  0.04  0;05915343ⵒ0Ɛ62ⵒ68781Ɛ 
500,000  201,428  50,000  24,Ɛ28  5,000  2,ⵒ88  500  358  50  42  5  5  0.5  0;6  0.05  0;07249 
600,000  24Ɛ,280  60,000  2ⵒ,880  6,000  3,580  600  420  60  50  6  6  0.6  0;7249  0.06  0;08781Ɛ05915343ⵒ0Ɛ62ⵒ6 
700,000  299,114  70,000  34,614  7,000  4,074  700  4ⵒ4  70  5ⵒ  7  7  0.7  0;84972  0.07  0;0ⵒ0Ɛ62ⵒ68781Ɛ05915343 
800,000  326,Ɛ68  80,000  3ⵒ,368  8,000  4,768  800  568  80  68  8  8  0.8  0;9724  0.08  0;0Ɛ62ⵒ68781Ɛ05915343ⵒ 
900,000  374,ⵒ00  90,000  44,100  9,000  5,260  900  630  90  76  9  9  0.9  0;ⵒ9724  0.09  0;10Ɛ62ⵒ68781Ɛ05915343ⵒ 
Divisibility rules
(In this section, all numbers are written with duodecimal)
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 ⵒ.
 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(5*5)
Examples:
13 rule => 12*3 = 5 which is divisible by 5.
2Ɛⵒ5 rule => 2Ɛⵒ2*5 = 2Ɛ0(5*70) which is divisible by 5(or apply the rule on 2Ɛ0).
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(5*3)
Examples:
13 rule => 33*1 = 0 which is divisible by 5.
2Ɛⵒ5 rule => 53*2Ɛⵒ = 8Ɛ1(5*195) which is divisible by 5(or apply the rule on 8Ɛ1).
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 101 = 5*25, thus this rule can be also tested for the divisibility by 25.
Example:
97,374,627 => 2746+3797 = 7Ɛ 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 2Ɛ(7*5)
Examples:
12 rule => 3*2+1 = 7 which is divisible by 7.
271Ɛ rule => 3*Ɛ+271 = 29ⵒ(7*4ⵒ) which is divisible by 7(or apply the rule on 29ⵒ).
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(7*2)
Examples:
12 rule => 22*1 = 0 which is divisible by 7.
271Ɛ rule => Ɛ2*271 = 513(7*89) which is divisible by 7(or apply the rule on 513).
OR
To test for divisibility by 7, 4 times 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(7*7)
Examples:
12 rule => 4*21 = 7 which is divisible by 7.
271Ɛ rule => 4*Ɛ271 = 235(7*3Ɛ) 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 1001 = 7*11*17, thus this rule can be also tested for the divisibility by 11 and 17.
Example:
386,967,443 => 443967+386 = 168 which is divisible by 7.
 8
If the 2digit number formed by the last 2 digits of the given number is divisible by 8 then the given number is divisible by 8.
Example: 1Ɛ48, 4120
rule => since 48(8*7) divisible by 8, then 1Ɛ48 is divisible by 8. rule => since 20(8*3) divisible by 8, then 4120 is divisible by 8.
 9
If the 2digit number formed by the last 2 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.
 ⵒ
If the number is divisible by 2 and 5 then the number is divisible by ⵒ.
 Ɛ
If the sum of the digits of a number is divisible by Ɛ then the number is divisible by Ɛ (the equivalent of casting out nines in decimal).
Example: 29, 61Ɛ13
rule => 2+9 = Ɛ which is divisible by Ɛ, then 29 is divisible by Ɛ. rule => 6+1+Ɛ+1+3 = 1ⵒ which is divisible by Ɛ, then 61Ɛ13 is divisible by Ɛ.
 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) = ⵒⵒ = 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 2digit number formed by the last 2 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 may be simple:
 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 (note that this is a twelfth, 1/ⵒ is a tenth)
 1/14 = 0.09 (note that this is a sixteenth, 1/12 is a fourteenth)
or complicated:
 1/5 = 0.249724972497... recurring (rounded to 0.24ⵒ)
 1/7 = 0.186ⵒ35186ⵒ35... recurring (rounded to 0.187)
 1/ⵒ = 0.1249724972497... recurring (rounded to 0.125)
 1/Ɛ = 0.111111111111... recurring (rounded to 0.111)
 1/11 = 0.0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ... recurring (rounded to 0.0Ɛ1)
 1/12 = 0.0ⵒ35186ⵒ35186... recurring (rounded to 0.0ⵒ3)
 1/13 = 0.0972497249724... recurring (rounded to 0.097)
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 
1ⵒ.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. Thus, in baseten (= 2×5) 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...). In the duodecimal (= 2×2×3) 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 which give terminating fractions within a given number of digits, say n, in a base b is the number of factors (divisors) of b^{n}, 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 b^{n} is given using its prime factorization.
For decimal, 10^{n} = 2^{n} * 5^{n}. The number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together. Factors of 10^{n} = (n+1)(n+1) = (n+1)^{2}.
For example, the number 8 is a factor of 10^{3} (1000), so 1/8 and other fractions with a denominator of 8 can not require more than 3 fractional decimal digits to terminate. 5/8 = 0.625_{ten}
For duodecimal, 12^{n} = 2^{2n} * 3^{n}. This has (2n+1)(n+1) divisors. The sample denominator of 8 is a factor of a gross (12^{2} = 144), so eighths can not need more than two duodecimal fractional places to terminate. 5/8 = 0.76_{twelve}
Because both ten and twelve have two unique prime factors, the number of divisors of b^{n} for b = 10 or 12 grows quadratically with the exponent n (in other words, of the order of n^{2}).
Recurring digits
The Dozenal Society of America argues that factors of 3 are more commonly encountered in reallife division problems than factors of 5.^{[34]} 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 doesn't 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 representation (e.g. 1/(2^{2}) = 0.25 _{ten} = 0.3 _{twelve}; 1/(2^{3}) = 0.125 _{ten} = 0.16 _{twelve}; 1/(2^{4}) = 0.0625 _{ten} = 0.09 _{twelve}; 1/(2^{5}) = 0.03125 _{ten} = 0.046 _{twelve}; etc.).
Values in bold indicate that value is exact.
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: Ɛ Prime factors of one above the base: 11 All other primes: 5, 7, 15, 17, 1Ɛ, 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  0.186ⵒ35  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/ⵒ 
1/11  11  0.09  0.1  Ɛ  1/Ɛ 
1/12  2, 3  0.083  0.1  2, 3  1/10 
1/13  13  0.076923  0.0Ɛ  11  1/11 
1/14  2, 7  0.0714285  0.0ⵒ35186  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  0.08579214Ɛ36429ⵒ7  15  1/15 
1/18  2, 3  0.05  0.08  2, 3  1/16 
1/19  19  0.052631578947368421  0.076Ɛ45  17  1/17 
1/20  2, 5  0.05  0.07249  2, 5  1/18 
1/21  3, 7  0.047619  0.06ⵒ3518  3, 7  1/19 
1/22  2, 11  0.045  0.06  2, Ɛ  1/1ⵒ 
1/23  23  0.0434782608695652173913  0.06316948421  1Ɛ  1/1Ɛ 
1/24  2, 3  0.0416  0.06  2, 3  1/20 
1/25  5  0.04  0.05915343ⵒ0Ɛ62ⵒ68781Ɛ  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  0.05186ⵒ3  2, 7  1/24 
1/29  29  0.0344827586206896551724137931  0.04Ɛ7  25  1/25 
1/30  2, 3, 5  0.03  0.04972  2, 3, 5  1/26 
1/31  31  0.032258064516129  0.0478ⵒⵒ093598166Ɛ74311Ɛ28623ⵒ55  27  1/27 
1/32  2  0.03125  0.046  2  1/28 
1/33  3, 11  0.03  0.04  3, Ɛ  1/29 
1/34  2, 17  0.02941176470588235  0.0429ⵒ708579214Ɛ36  2, 15  1/2ⵒ 
1/35  5, 7  0.0285714  0.0414559Ɛ3931  5, 7  1/2Ɛ 
1/36  2, 3  0.027  0.04  2, 3  1/30 
The duodecimal period length of 1/n are
 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
 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
 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...  1.4Ɛ79170ⵒ07Ɛ8... 
φ (phi), the golden ratio =  1.618033988749...  1.74ƐƐ6772802ⵒ... 
Transcendental number  In decimal  In duodecimal 
π (pi), the ratio of a circle's circumference to its diameter  3.141592653589...  3.184809493Ɛ91... 
e, the base of the natural logarithm  2.718281828459...  2.875236069821... 
See also
 Senary (base 6)
 Decimal (base 10)
 Hexadecimal (base 16)
 Sexagesimal (base 60)
References
 ^ George Dvorsky (20130118). "Why We Should Switch To A Base12 Counting System". Archived from the original on 20130121. Retrieved 20131221.
 ^ Matsushita, Shuji (1998). Decimal vs. Duodecimal: An interaction between two systems of numeration. 2nd Meeting of the AFLANG, October 1998, Tokyo. Archived from the original on 20081005. Retrieved 20110529.
 ^ Mazaudon, Martine (2002). "Les principes de construction du nombre dans les langues tibétobirmanes". In François, Jacques (ed.). La Pluralité (PDF). Leuven: Peeters. pp. 91–119. ISBN 9042912952.
 ^ von Mengden, Ferdinand (2006). "The peculiarities of the Old English numeral system". In Nikolaus Ritt; Herbert Schendl; Christiane DaltonPuffer; Dieter Kastovsky (eds.). Medieval English and its Heritage: Structure Meaning and Mechanisms of Change. Studies in English Medieval Language and Literature. 16. Frankfurt: Peter Lang. pp. 125–145.
 ^ von Mengden, Ferdinand (2010). Cardinal Numerals: Old English from a CrossLinguistic Perspective. Topics in English Linguistics. 67. Berlin; New York: De Gruyter Mouton. pp. 159–161.
 ^ Pittman, Richard (1990). "Origin of Mesopotamian duodecimal and sexagesimal counting systems". Philippine Journal of Linguistics. 21 (1): 97.
 ^ Nishikawa, Yoshiaki (2002). "ヒマラヤの満月と十二進法" [The Full Moon in the Himalayas and the Duodecimal System] (in Japanese). Archived from the original on March 29, 2008. Retrieved 20080324.
 ^ 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.
 ^ Macey, Samuel L. (1989). The Dynamics of Progress: Time, Method, and Measure. Atlanta, Georgia: University of Georgia Press. p. 92. ISBN 9780820337968.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} De Vlieger, Michael (2010). "Symbology Overview" (PDF). The Duodecimal Bulletin. 4X [58] (2).
 ^ Andrews, Frank Emerson (1935). New Numbers: How Acceptance of a Duodecimal (12) Base Would Simplify Mathematics. p. 52.
 ^ "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).
 ^ ^{a} ^{b} "Mo for Megro" (PDF). The Duodecimal Bulletin. 1 (1). 1945.
 ^ Pitman, Isaac (ed.): A triple (twelve gross) Gems of Wisdom. London 1860
 ^ Pitman, Isaac (1947). "A Reckoning Reform [reprint from 1857]" (PDF). The Duodecimal Bulletin. 3 (2).
 ^ Karl Pentzlin (20130330). "Proposal to encode Duodecimal Digit Forms in the UCS" (PDF). ISO/IEC JTC1/SC2/WG2, Document N4399. Retrieved 20160530.
 ^ "The Unicode Standard, Version 8.0: Number Forms" (PDF). Unicode Consortium. Retrieved 20160530.
 ^ "The Unicode Standard 8.0" (PDF). Retrieved 20140718.
 ^ Scott Pakin (2009). "The Comprehensive LATEX Symbol List" (PDF). Retrieved 20160530.
 ^ "What should the DSA do about transdecimal characters?". The Dozenal Society of America. Retrieved 20180101.
 ^ ^{a} ^{b} Volan, John (July 2015). "Base Annotation Schemes" (PDF). Duodecomal Bulletin. 62.
 ^ ^{a} ^{b} Zirkel, Gene (2010). "How Do You Pronounce Dozenals?" (PDF). The Duodecimal Bulletin. 4E [59] (2).
 ^ "Systematic Dozenal Nomenclature and other nomenclature systems" (PDF). The Duodecimal Bulletin. Retrieved 20190728.
 ^ ^{a} ^{b} ^{c} Goodman, Donald (2016). "Manual of the Dozenal System" (PDF). Dozenal Society of America. Retrieved 27 April 2018.
 ^ The Prodigy (Biography of WJS) pg [42]
 ^ William S. Crosby; "The Uncial Jottings of a Harried Infantryman", The Duodecimal Bulletin, Vol 1 Issue 2, June 1945, Page 9.
 ^ A. C. Aitken (January 25, 1962) "Twelves and Tens" The Listener.
 ^ A. C. Aitken (1962) The Case Against Decimalisation. Edinburgh / London: Oliver & Boyd.
 ^ Carroll, Lee (1995). Kryon—Alchemy of the Human Spirit. The Kryon Writings, Inc. ISBN 0963630482.
 ^ "SchoolhouseRock  Little Twelvetoes". web.archive.org. 6 February 2010.
 ^ Pendlebury, Tom (2012). "TGM: A Coherent Dozenal Metrology" (PDF). The Dozenal Society of Great Britain.
 ^ Suga, Takashi (22 May 2019). "Proposal for the Universal Unit System" (PDF).
 ^ Michael Thomas De Vlieger (30 November 2011). "Dozenal FAQs" (PDF). The Dozenal Society of America.
Further reading
 Savard, John J. G. (2018) [2016]. "Changing the Base". quadibloc. Archived from the original on 20180717. Retrieved 20180717.
 Savard, John J. G. (2018) [2005]. "Computer Arithmetic". quadibloc. The Early Days of Hexadecimal. Archived from the original on 20180716. Retrieved 20180716. (NB. Also has information on duodecimal representations.)
External links
 Dozenal Society of America
 Dozenal Society of Great Britain
 Duodecimal calculator
 Comprehensive Synopsis of Dozenal and Transdecimal Symbologies