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 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.^[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 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.

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, Gbiri-Niragu (Gure-Kahugu), 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 Proto-Germanic *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. 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.

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.	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).[10]	Y
⟨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, 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.[12]
⟨*, #⟩	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,[13][14] also on push-button telephones.[10]	Y
⟨2, 3⟩	The Arabic digits 2 and 3 rotated 180°: U+218A ↊ TURNED DIGIT TWO, U+218B ↋ TURNED DIGIT THREE	Isaac Pitman (1857).[15] Used by the Dozenal Society of Great Britain (DSGB) Used by DSA 2015–present Included in Unicode 8.0 (2015).[16][17]
⟨, ⟩	Pronounced "dek", "el"	William Addison Dwiggins (1945/1932?).[10][18] Used by DSA 1945–1974 and 2008–2015[13][14]
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	– Y
*54 = 64 54;6 = 64.5	Asterisked for whole numbers, Humphrey points for others	Used by DSGB.[19]	– Y
54z = 64d	Subscript "z"	From "dozenal". Used by DSA since 2015.[19]
5412 = 6410	Subscript base number	Common usage by mathematicians and mathematics textbooks[19]
54twelve = 64ten	Subscript base spelt out	Variation of the above sometimes found in school textbooks[19]
doz 54 = dec 64			Y

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.^[29]

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.^[12]

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.

— A. C. Aitken, "Twelves and Tens" in The Listener (January 25, 1962)^[30]

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)^[31]

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".^[37]

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.^[37] 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.

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;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 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, 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, ${\displaystyle 6+6=12}$; in duodecimal, ${\displaystyle 6+6=10}$. 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,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

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 (${\displaystyle 5^{2}}$).

Examples:
13     rule → ${\displaystyle |1-2\times 3|=5}$, which is divisible by 5.
2BA5   rule → ${\displaystyle |2{\texttt {B}}{\texttt {A}}-2\times 5|=2{\texttt {B}}0(5\times 70)}$, 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 (${\displaystyle 5\times 3}$).

Examples:
13     rule → ${\displaystyle |3-3\times 1|=0}$, which is divisible by 5.
2BA5   rule → ${\displaystyle |5-3\times 2{\texttt {B}}{\texttt {A}}|=8{\texttt {B}}1(5\times 195)}$, 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 ${\displaystyle 101=5\times 25}$; thus, this rule can be also tested for the divisibility by 25.

Example:

97,374,627 → ${\displaystyle 27-46+37-97=-7{\texttt {B}}}$, 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 (${\displaystyle 7\times 5}$)

Examples:
12     rule → ${\displaystyle |3\times 2+1|=7}$, which is divisible by 7.
271B    rule → ${\displaystyle |3\times {\texttt {B}}+271|=29{\texttt {A}}(7\times 4{\texttt {A}})}$, 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 (${\displaystyle 7\times 2}$).

Examples:
12     rule → ${\displaystyle |2-2\times 1|=0}$, which is divisible by 7.
271B    rule → ${\displaystyle |{\texttt {B}}-2\times 271|=513(7\times 89)}$, 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 (${\displaystyle 7^{2}}$).

Examples:
12     rule → ${\displaystyle |4\times 2-1|=7}$, which is divisible by 7.
271B    rule → ${\displaystyle |4\times {\texttt {B}}-271|=235(7\times 3{\texttt {B}})}$, 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 ${\displaystyle 1001=7\times 11\times 17}$; thus, this rule can be also tested for the divisibility by 11 and 17.

Example:

386,967,443 → ${\displaystyle 443-967+386=-168}$, 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: 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 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.

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 => |6-6| = 0, which is divisible by 11, then 66 is divisible by 11.
     rule => |(9+2)-(4+7)| = |A-A| = 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

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 = ${\displaystyle {\tfrac {1+{\sqrt {5}}}{2}}}$	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

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 2018-07-17.