Duodecimal

Short description: Base-12 numeral system

The duodecimal system, also known as base 12 or dozenal, is a positional numeral system using twelve as its base. In duodecimal, the number twelve is denoted "10", meaning 1 twelve and 0 units; in the decimal system, this number is instead written as "12" meaning 1 ten and 2 units, and the string "10" means ten. In duodecimal, "100" means twelve squared, "1000" means twelve cubed, and "0.1" means a twelfth.

Various symbols have been used to stand for ten and eleven in duodecimal notation; this page uses Template:D2 and Template:D3, as in hexadecimal, which make a duodecimal count from zero to twelve read 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, Template:D2, Template:D3, 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 non-trivial 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 3-smooth 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.

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 (base sixty) does even better in this respect (the reciprocals of all 5-smooth 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.

Georges Ifrah speculatively traced the origin of the duodecimal system 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.

Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu (Gure-Kahugu), Piti, and the Nimbia dialect of Gwandara; and the Chepang language of Nepal 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 Proto-Germanic *ainlif and *twalif (meaning, respectively, one left and two left), suggesting a decimal rather than duodecimal origin. 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. 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. Pre-decimalisation, Ireland and the United Kingdom used a mixed duodecimal-vigesimal 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
120	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".

Notation
⟨ten, eleven⟩	Background	Note	By keyboard
By dedicated characters
⟨A, B⟩	As in hexadecimal	To allow entry on typewriters.	Y
⟨T, E⟩	Initials of Ten and Eleven		Y
⟨X, E⟩	X from the Roman numeral for ten; E from English Eleven.		Y
⟨X, Z⟩	X from Roman numeral; origin of Z unknown	Attributed to "D'Alambert [sic] & Buffon" by the Dozenal Society of America (DSA).	Y
⟨δ, ε⟩	Greek δ, ε, from δέκα "ten" and ένδεκα "eleven"
⟨τ, ε⟩	Greek τ, ε
⟨W, ∂⟩	W comes from doubling the Roman numeral for five; ∂ is based on a pendulum	Silvio Ferrari in Calcolo Decidozzinale (1854).
⟨X, Ɛ⟩	italic X pronounced "dec"; rounded italic Ɛ, similar in appearance to U+0190 Ɛ LATIN CAPITAL LETTER OPEN E, pronounced "elf"	Frank Emerson Andrews in New Numbers (1935); Andrews used italic 0–9 for the other duodecimal numerals.
⟨*, #⟩	sextile or six-pointed 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, also on push-button telephones.	Y
⟨2, 3⟩	The Arabic digits 2 and 3 rotated 180°: U+218A ↊ TURNED DIGIT TWO, U+218B ↋ TURNED DIGIT THREE	Isaac Pitman (1857). Used by the Dozenal Society of Great Britain (DSGB) Used by DSA 2015–present Included in Unicode 8.0 (2015).
⟨, ⟩	Pronounced "dek", "el"	William Addison Dwiggins (1945/1932?). Used by DSA 1945–1974 and 2008–2015
By base notation
duodecimal ⇔ decimal	Background	Note	By keyboard
54 = 64 54;6 = 64.5	In italics Use semicolon instead of a decimal point	Humphrey point	– Y
*54 = 64 54;6 = 64.5	Asterisked for whole numbers, Humphrey points for others	Used by DSGB.	– Y
54z = 64d	Subscript "z"	From "dozenal". Used by DSA since 2015.
5412 = 6410	Subscript base number	Common usage by mathematicians and mathematics textbooks
54twelve = 64ten	Subscript base spelt out	Variation of the above sometimes found in school textbooks
doz 54 = dec 64			Y

Transdecimal symbols

2Template:NBSP3
duodecimal ⟨ten, eleven⟩
In Unicode	U+218A ↊ TURNED DIGIT TWO U+218B ↋ TURNED DIGIT THREE

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⟩. Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his 1935 book New Numbers ⟨X, Ɛ⟩ (italic capital X and a rounded italic capital E similar to open E), along with italic numerals 0–9.

Edna Kramer in her 1951 book The Main Stream of Mathematics used a ⟨*, #⟩ (sextile or six-pointed asterisk, hash or octothorpe). The symbols were chosen because they were available on some typewriters; they are also on push-button telephones. This notation was used in publications of the Dozenal Society of America (DSA) from 1974 to 2008.

From 2008 to 2015, the DSA used ⟨ ,  ⟩, the symbols devised by William Addison Dwiggins.

The Dozenal Society of Great Britain (DSGB) proposed symbols ⟨ 2, 3 ⟩. This notation, derived from Arabic digits by 180° rotation, was introduced by Isaac Pitman in 1857. 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. 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).

After the Pitman digits were added to Unicode, the DSA took a vote and then began publishing content using the Pitman digits instead. They still use the letters X and E in ASCII text. As the Unicode characters are poorly supported, this page uses "Template:D2" and "Template:D3".

Other proposals are more creative or aesthetic; for example, many do not use any Arabic numerals under the principle of "separate identity."

Base notation

There are also varying proposals of how to distinguish a duodecimal number from a decimal one. 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), such as "54_z = 64_d," "54₁₂ = 64₁₀" 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.

Duodecimal Number	Number Name	Duodecimal Number Fraction	Fraction Name
1;	one
10;	do	0;1	edo
100;	gro	0;01	egro
1,000;	mo	0;001	emo
10,000;	do-mo	0;000,1	edo-mo
100,000;	gro-mo	0;000,01	egro-mo
1,000,000;	bi-mo	0;000,001	ebi-mo
10,000,000;	do-bi-mo	0;000,000,1	edo-bi-mo
100,000,000;	gro-bi-mo	0;000,000,01	egro-bi-mo
1,000,000,000;	tri-mo	0;000,000,001	etri-mo
10,000,000,000;	do-tri-mo	0;000,000,000,1	edo-tri-mo
100,000,000,000;	gro-tri-mo	0;000,000,000,01	egro-tri-mo
1,000,000,000,000;	quad-mo	0;000,000,000,001	equad-mo
10,000,000,000,000;	do-quad-mo	0;000,000,000,000,1	edo-quad-mo
100,000,000,000,000;	gro-quad-mo	0;000,000,000,000,01	egro-quad-mo
1,000,000,000,000,000;	penta-mo	0;000,000,000,000,001	epenta-mo
10,000,000,000,000,000;	do-penta-mo	0;000,000,000,000,000,1	edo-penta-mo
100,000,000,000,000,000;	gro-penta-mo	0;000,000,000,000,000,01	egro-penta-mo
1,000,000,000,000,000,000;	hexa-mo	0;000,000,000,000,000,001	ehexa-mo

Multiple digits in this series are pronounced differently: 12 is "do two"; 30 is "three do"; 100 is "gro"; Template:D3Template:D29 is "el gro dek do nine"; Template:D386 is "el gro eight do six"; 8Template:D3Template:D3,15Template:D2 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.

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.

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.

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 ten-based weights and measure or by the adoption of the duodecimal number system.

A duodecimal clockface as in the logo of the Dozenal Society of America, here used to denote musical keys

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 one-and-a-half 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.

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.

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' script-X and script-E for the digit symbols.

Duodecimal systems of measurements

Systems of measurement proposed by dozenalists include:

• Tom Pendlebury's TGM system
• Takashi Suga's Universal Unit System
• John Volan's Primel system

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".

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. 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. Its corresponding base, senary, is below the DSA's stated threshold.

Eight and Sixteen only have 2 as a prime factor. Therefore, in octal and hexadecimal, the only terminating fractions are those whose denominator is a power of two.

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 are quite large to use as bases, and are far beyond the DSA's stated threshold.

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 written in duodecimal. For example, "10" means twelve, and "12" means fourteen.

Duodecimal multiplication table

×	1	2	3	4	5	6	7	8	9	Template:D2	Template:D3	10
1	1	2	3	4	5	6	7	8	9	Template:D2	Template:D3	10
2	2	4	6	8	Template:D2	10	12	14	16	18	1Template:D2	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	Template:D2	13	18	21	26	2Template:D3	34	39	42	47	50
6	6	10	16	20	26	30	36	40	46	50	56	60
7	7	12	19	24	2Template:D3	36	41	48	53	5Template:D2	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
Template:D2	Template:D2	18	26	34	42	50	5Template:D2	68	76	84	92	A0
Template:D3	Template:D3	1Template:D2	29	38	47	56	65	74	83	92	Template:D21	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 digit-conversion tables. The ones provided below can be used to convert any duodecimal number between 0;1 and Template:D3Template:D3,Template:D3Template:D3Template:D3;Template:D3 to decimal, or any decimal number between 0.1 and 99,999.9 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:

12,345.6 = 10,000 + 2,000 + 300 + 40 + 5 + 0.6

This decomposition works the same no matter what base the number is expressed in. Just isolate each non-zero 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, 7,080.9), these are left out in the digit decomposition (7,080.9 = 7,000 + 80 + 0.9). 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) 10,000 + 2,000 + 300 + 40 + 5 + 0;6
= (decimal) 20,736 + 3,456 + 432 + 48 + 5 + 0.5

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
  
  10,000    =   20,736
   2,000    =    3,456 
     300    =      432
      40    =       48
       5    =        5
 +     0;6  =  +     0.5
--------------------------------------------
  12,345;6  =   24,677.5

That is, (duodecimal) 12,345;6 equals (decimal) 24,677.5

If the given number is in decimal and the target base is duodecimal, the method is same. Using the digit conversion tables:

(decimal) 10,000 + 2,000 + 300 + 40 + 5 + 0.6
= (duodecimal) 5,954 + 1,1Template:D28 + 210 + 34 + 5 + 0;7249

To sum these partial products and recompose the number, the addition must be done with duodecimal rather than decimal arithmetic:

  Decimal --> Duodecimal
  
  10,000    =   5,954
   2,000    =   1,1Template:D28
     300    =     210
      40    =      34
       5    =       5
 +     0.6  =  +    0;7249
--------------------------------------------------------
  12,345.6  =   7,189;7249

That is, (decimal) 12,345.6 equals (duodecimal) 7,189;7249

Duodecimal to decimal digit conversion

Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.
10,000	20,736	1,000	1,728	100	144	10	12	1	1	0;1	0.083
20,000	41,472	2,000	3,456	200	288	20	24	2	2	0;2	0.16
30,000	62,208	3,000	5,184	300	432	30	36	3	3	0;3	0.25
40,000	82,944	4,000	6,912	400	576	40	48	4	4	0;4	0.3
50,000	103,680	5,000	8,640	500	720	50	60	5	5	0;5	0.416
60,000	124,416	6,000	10,368	600	864	60	72	6	6	0;6	0.5
70,000	145,152	7,000	12,096	700	1,008	70	84	7	7	0;7	0.583
80,000	165,888	8,000	13,824	800	1,152	80	96	8	8	0;8	0.6
90,000	186,624	9,000	15,552	900	1,296	90	108	9	9	0;9	0.75
Template:D20,000	207,360	Template:D2,000	17,280	Template:D200	1,440	Template:D20	120	Template:D2	10	0;Template:D2	0.83
Template:D30,000	228,096	Template:D3,000	19,008	Template:D300	1,584	Template:D30	132	Template:D3	11	0;Template:D3	0.916

Decimal to duodecimal digit conversion

Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duodecimal
10,000	5,954	1,000	100	84	10	1	1	0.1	0;12497
20,000	Template:D3,6Template:D28	2,000	200	148	20	18	2	2	0.2	0;2497
30,000	15,440	3,000	300	210	30	26	3	3	0.3	0;37249
40,000	1Template:D3,194	4,000	2,394	400	294	40	34	4	4	0.4	0;4972
50,000	5,000	500	358	50	42	5	5	0.5	0;6
60,000	2Template:D2,880	6,000	3,580	600	420	60	50	6	6	0.6	0;7249
70,000	34,614	7,000	4,074	700	70	7	7	0.7	0;84972
80,000	3Template:D2,368	8,000	4,768	800	568	80	68	8	8	0.8	0;9724
90,000	44,100	9,000	5,260	900	630	90	76	9	9	0.9	0;Template:D29724

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 Template:D2.

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 ([math]\displaystyle{ 5^2 }[/math]).

Examples:
13     rule Template:Right-arrow [math]\displaystyle{ |1-2\times3|=5 }[/math], which is divisible by 5.
2Template:D3Template:D25   rule Template:Right-arrow [math]\displaystyle{ |2\texttt B\texttt A-2\times5| = 2\texttt B0(5\times70) }[/math], which is divisible by 5 (or apply the rule on 2Template:D30).

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 ([math]\displaystyle{ 5\times3 }[/math]).

Examples:
13     rule → [math]\displaystyle{ |3-3\times1|=0 }[/math], which is divisible by 5.
2Template:D3Template:D25   rule → [math]\displaystyle{ |5-3\times2\texttt B\texttt A|=8\texttt B1(5\times195) }[/math], which is divisible by 5 (or apply the rule on 8Template:D31).

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 [math]\displaystyle{ 101=5\times25 }[/math]; thus, this rule can be also tested for the divisibility by 25.

Example:

97,374,627 → [math]\displaystyle{ 27-46+37-97=-7\texttt B }[/math], 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 2Template:D3 ([math]\displaystyle{ 7\times5 }[/math])

Examples:
12     rule → [math]\displaystyle{ |3\times2+1|=7 }[/math], which is divisible by 7.
271Template:D3    rule → [math]\displaystyle{ |3\times\texttt B+271|=29\texttt A(7\times4\texttt A) }[/math], which is divisible by 7 (or apply the rule on 29Template:D2).

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 ([math]\displaystyle{ 7\times2 }[/math]).

Examples:
12     rule → [math]\displaystyle{ |2-2\times1|=0 }[/math], which is divisible by 7.
271Template:D3    rule → [math]\displaystyle{ |\texttt B-2\times271|=513 (7\times89) }[/math], 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 ([math]\displaystyle{ 7^2 }[/math]).

Examples:
12     rule → [math]\displaystyle{ |4\times2-1|=7 }[/math], which is divisible by 7.
271Template:D3    rule → [math]\displaystyle{ |4\times\texttt B-271|=235(7\times3\texttt B) }[/math], 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 [math]\displaystyle{ 1001=7\times11\times17 }[/math]; thus, this rule can be also tested for the divisibility by 11 and 17.

Example:

386,967,443 → [math]\displaystyle{ 443-967+386=-168 }[/math], which is divisible by 7.

8

If the two-digit number formed by the last two digits of the given number is divisible by 8, then the given number is divisible by 8.

Example: 1Template:D348, 4120

     rule => since 48(8*7) divisible by 8, then 1Template:D348 is divisible by 8.
     rule => since 20(8*3) divisible by 8, then 4120 is divisible by 8.

9

If the two-digit 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.

If the number is divisible by 2 and 5, then the number is divisible by Template:D2.

If the sum of the digits of a number is divisible by Template:D3, then the number is divisible by Template:D3 (the equivalent of casting out nines in decimal).

     rule => 2+9 = Template:D3, which is divisible by Template:D3, then 29 is divisible by Template:D3.
     rule => 6+1+Template:D3+1+3 = 1Template:D2, which is divisible by Template:D3, then 61Template:D313 is divisible by Template:D3.

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 => |6-6| = 0, which is divisible by 11, then 66 is divisible by 11.
     rule => |(9+2)-(4+7)| = |Template:D2-Template:D2| = 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 two-digit 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 3-smooth 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/Template:D2 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;186Template:D235
• 1/Template:D2 = 0;12497 (one tenth)
• 1/Template:D3 = 0;1 (one eleventh)
• 1/11 = 0;0Template:D3 (one thirteenth)
• 1/12 = 0;0Template:D235186 (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
1Template:D2.6 + 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 [math]\displaystyle{ 2\times5=10 }[/math] 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 [math]\displaystyle{ 2\times2\times3=12 }[/math] 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 [math]\displaystyle{ b^n }[/math], 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 [math]\displaystyle{ b^n }[/math] is given using its prime factorization.

For decimal, [math]\displaystyle{ 10^n=2^n\times 5^n }[/math]. 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 [math]\displaystyle{ 10^n }[/math] is [math]\displaystyle{ (n+1)(n+1)=(n+1)^2 }[/math].

For example, the number 8 is a factor of 10³ (1000), so [math]\displaystyle{ \frac{1}{8} }[/math] and other fractions with a denominator of 8 cannot require more than three fractional decimal digits to terminate. [math]\displaystyle{ \frac{5}{8}=0.625_{10}. }[/math]

For duodecimal, [math]\displaystyle{ 10^n=2^{2n}\times 3^n }[/math]. This has [math]\displaystyle{ (2n+1)(n+1) }[/math] divisors. The sample denominator of 8 is a factor of a gross [math]\displaystyle{ 12^2=144 }[/math] in decimal), so eighths cannot need more than two duodecimal fractional places to terminate. [math]\displaystyle{ \frac{5}{8}=0.76_{12}. }[/math]

Because both ten and twelve have two unique prime factors, the number of divisors of [math]\displaystyle{ b^n }[/math] for b = 10 or 12 grows quadratically with the exponent n (in other words, of the order of [math]\displaystyle{ n^2 }[/math]).

Recurring digits

The Dozenal Society of America argues that factors of 3 are more commonly encountered in real-life division problems than factors of 5. 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/(2²) = 0.25₁₀ = 0.3₁₂
• 1/(2³) = 0.125₁₀ = 0.16₁₂
• 1/(2⁴) = 0.0625₁₀ = 0.09₁₂
• 1/(2⁵) = 0.03125₁₀ = 0.046₁₂

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: Template:D3 Prime factors of one above the base: 11 (=1310) All other primes: 5, 7, 15, 17, 1Template:D3, 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;186Template:D235	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/Template:D2
1/11	11	0.09	0;1	Template:D3	1/Template:D3
1/12	2, 3	0.083	0;1	2, 3	1/10
1/13	13	0.076923	0;0Template:D3	11	1/11
1/14	2, 7	0.0714285	0;0Template:D235186	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;08579214Template:D336429Template:D27	15	1/15
1/18	2, 3	0.05	0;08	2, 3	1/16
1/19	19	0.052631578947368421	0;076Template:D345	17	1/17
1/20	2, 5	0.05	0;07249	2, 5	1/18
1/21	3, 7	0.047619	0;06Template:D23518	3, 7	1/19
1/22	2, 11	0.045	0;06	2, Template:D3	1/1Template:D2
1/23	23	0.0434782608695652173913	0;06316948421	1Template:D3	1/1Template:D3
1/24	2, 3	0.0416	0;06	2, 3	1/20
1/25	5	0.04	0;05915343Template:D20Template:D362Template:D268781Template:D3	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;05186Template:D23	2, 7	1/24
1/29	29	0.0344827586206896551724137931	0;04Template:D37	25	1/25
1/30	2, 3, 5	0.03	0;04972	2, 3, 5	1/26
1/31	31	0.032258064516129	0;0478Template:D2Template:D2093598166Template:D374311Template:D328623Template:D255	27	1/27
1/32	2	0.03125	0;046	2	1/28
1/33	3, 11	0.03	0;04	3, Template:D3	1/29
1/34	2, 17	0.02941176470588235	0;0429Template:D2708579214Template:D336	2, 15	1/2Template:D2
1/35	5, 7	0.0285714	0;0414559Template:D33931	5, 7	1/2Template:D3
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;4Template:D379170Template:D207Template:D38...
φ (phi), the golden ratio = [math]\displaystyle{ \tfrac{1+\sqrt{5}}{2} }[/math]	1.618033988749...	1;74Template:D3Template:D36772802Template:D2...
Transcendental number	In decimal	In duodecimal
π (pi), the ratio of a circle's circumference to its diameter	3.141592653589...	3;184809493Template:D391...
e, the base of the natural logarithm	2.718281828459...	2;875236069821...

See also

• Vigesimal (base 20)
• Sexagesimal (base 60)

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. http://www.earth360.com/new_number.html. 
• Savard, John J. G. (2018). "Changing the Base". quadibloc. http://www.quadibloc.com/math/baseint.htm.