
The duodecimal system, also known as base twelve 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 (144), "1,000" means twelve cubed (1,728), and "0.1" means a twelfth (.8333...).
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, and finally 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 (dek, pronounced dɛk) and 3 (a turned 3) for eleven (el, pronounced ɛl).
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/2b·3c 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. In 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, 24 (12×2) hours in a day; many other items are counted by the dozen, gross (144, twelve squared), or great gross (1728, twelve cubed). The Romans used a fraction system based on 12, including the uncia, which became both the English words ounce and inch. Historically, many parts of western Europe used a mixed vigesimal–duodecimal currency system of pounds, shillings, and pence, with 20 shillings to a pound and 12 pence to a shilling, originally established by Charlemagne in the 780s.
Relative value | Length | Weight | ||
---|---|---|---|---|
French | English | English (Troy) | Roman | |
120 | pied | foot | pound | libra |
12−1 | pouce | inch | ounce | uncia |
12−2 | ligne | line | 2 scruples | 2 scrupula |
12−3 | point | point | seed | siliqua |
Notations and pronunciations
In a positional numeral system of base n (twelve for duodecimal), each of the first n natural numbers is given a distinct numeral symbol, and then n is denoted "10", meaning 1 times n plus 0 units. For duodecimal, the standard numeral symbols for 0–9 are typically preserved for zero through nine, but there are numerous proposals for how to write the numerals representing "ten" and "eleven". More radical proposals do not use any Arabic numerals under the principle of "separate identity."
Pronunciation of duodecimal numbers also has no standard, but various systems have been proposed.
Transdecimal symbols
2 3 | |
---|---|
duodecimal ⟨ten, eleven⟩ | |
In Unicode |
|
Block Number Forms | |
Note | |
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Several authors have proposed using letters of the alphabet for the transdecimal symbols. Latin letters such as ⟨A, B⟩ (as in hexadecimal) or ⟨T, E⟩ (initials of Ten and Eleven) are convenient because they are widely accessible, and for instance can be typed on typewriters. However, when mixed with ordinary prose, they might be confused for letters. As an alternative, Greek letters such as ⟨τ, ε⟩ could be used instead. Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his 1935 book New Numbers ⟨X, Ɛ⟩ (italic capital X from the Roman numeral for ten 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 PDF content using the Pitman digits instead, but continues to use the letters X and E on its webpage.
Symbols | Background | Note | |
---|---|---|---|
A | B | As in hexadecimal | Allows entry on typewriters. |
T | E | Initials of Ten and Eleven | Used (in lower case) in music set theory |
X | E | X from the Roman numeral; E from Eleven. | |
X | Z | Origin of Z unknown | Attributed to D'Alembert & Buffon by the DSA. |
δ | ε | Greek delta from δέκα "ten"; epsilon from ένδεκα "eleven" | |
τ | ε | Greek tau, epsilon | |
W | ∂ | W from doubling the Roman numeral V; ∂ based on a pendulum | Silvio Ferrari in Calcolo Decidozzinale (1854). |
X | Ɛ | italic X pronounced "dec"; rounded italic Ɛ, pronounced "elf" | Frank Andrews in New Numbers (1935), with italic 0–9 for other duodecimal numerals. |
* | # | sextile or six-pointed asterisk, hash or octothorpe | On push-button telephones; used by Edna Kramer in The Main Stream of Mathematics (1951); used by the DSA 1974–2008 |
2 | 3 |
| Isaac Pitman (1857); used by the DSGB; used by the DSA since 2015; included in Unicode 8.0 (2015) |
Pronounced "dek", "el" |
|
Base notation
There are also varying proposals of how to distinguish a duodecimal number from a decimal one. The most common method used in mainstream mathematics sources comparing various number bases uses a subscript "10" or "12", e.g. "5412 = 6410". To avoid ambiguity about the meaning of the subscript 10, the subscripts might be spelled out, "54twelve = 64ten". In 2015 the Dozenal Society of America adopted the more compact single-letter abbreviation "z" for "dozenal" and "d" for "decimal", "54z = 64d".
Other proposed methods include italicizing duodecimal numbers "54 = 64", adding a "Humphrey point" (a semicolon instead of a decimal point) to duodecimal numbers "54;6 = 64.5", prefixing duodecimal numbers by an asterisk "*54 = 64", or some combination of these. The Dozenal Society of Great Britain uses an asterisk prefix for duodecimal whole numbers, and a Humphrey point for other duodecimal numbers.
Pronunciation
The Dozenal Society of America suggested ten and eleven should be pronounced as "dek" and "el", respectively.
Terms for some powers of twelve already exist in English: The number twelve (1012 or 1210) is also called a dozen. Twelve squared (10012 or 14410) is called a gross. Twelve cubed (100012 or 172810) is called a great gross.
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.
Both the Dozenal Society of America (founded as the Duodecimal Society of America in 1944) and the Dozenal Society of Great Britain (founded 1959) promote adoption of the duodecimal system.
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)
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)
In media
In "Little Twelvetoes," an episode of the American educational television series Schoolhouse Rock!, a farmer encounters an alien being with twelve fingers on each hand and twelve toes on each foot who uses duodecimal arithmetic. The alien uses "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, and 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 9+3.
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.
× | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | 10 |
2 | 2 | 4 | 6 | 8 | A | 10 | 12 | 14 | 16 | 18 | 1A | 20 |
3 | 3 | 6 | 9 | 10 | 13 | 16 | 19 | 20 | 23 | 26 | 29 | 30 |
4 | 4 | 8 | 10 | 14 | 18 | 20 | 24 | 28 | 30 | 34 | 38 | 40 |
5 | 5 | A | 13 | 18 | 21 | 26 | 2B | 34 | 39 | 42 | 47 | 50 |
6 | 6 | 10 | 16 | 20 | 26 | 30 | 36 | 40 | 46 | 50 | 56 | 60 |
7 | 7 | 12 | 19 | 24 | 2B | 36 | 41 | 48 | 53 | 5A | 65 | 70 |
8 | 8 | 14 | 20 | 28 | 34 | 40 | 48 | 54 | 60 | 68 | 74 | 80 |
9 | 9 | 16 | 23 | 30 | 39 | 46 | 53 | 60 | 69 | 76 | 83 | 90 |
A | A | 18 | 26 | 34 | 42 | 50 | 5A | 68 | 76 | 84 | 92 | A0 |
B | B | 1A | 29 | 38 | 47 | 56 | 65 | 74 | 83 | 92 | A1 | B0 |
10 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | A0 | B0 | 100 |
Conversion tables to and from decimal
To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under positional notation). Alternatively, one can use digit-conversion tables. The ones provided below can be used to convert any duodecimal number between 0.1 and BB,BBB.B 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 + 1,1A8 + 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 = 1,1A8 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 |
A0,000 | 207,360 | A,000 | 17,280 | A00 | 1,440 | A0 | 120 | A | 10 | 0.A | 0.83 |
B0,000 | 228,096 | B,000 | 19,008 | B00 | 1,584 | B0 | 132 | B | 11 | 0.B | 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 | 6B4 | 100 | 84 | 10 | A | 1 | 1 | 0.1 | 0.12497 |
20,000 | B,6A8 | 2,000 | 1,1A8 | 200 | 148 | 20 | 18 | 2 | 2 | 0.2 | 0.2497 |
30,000 | 15,440 | 3,000 | 1,8A0 | 300 | 210 | 30 | 26 | 3 | 3 | 0.3 | 0.37249 |
40,000 | 1B,194 | 4,000 | 2,394 | 400 | 294 | 40 | 34 | 4 | 4 | 0.4 | 0.4972 |
50,000 | 24,B28 | 5,000 | 2,A88 | 500 | 358 | 50 | 42 | 5 | 5 | 0.5 | 0.6 |
60,000 | 2A,880 | 6,000 | 3,580 | 600 | 420 | 60 | 50 | 6 | 6 | 0.6 | 0.7249 |
70,000 | 34,614 | 7,000 | 4,074 | 700 | 4A4 | 70 | 5A | 7 | 7 | 0.7 | 0.84972 |
80,000 | 3A,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.A9724 |
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/A is one tenth)
- 1/14 = 0.09 (this is one sixteenth, 1/12 is one fourteenth)
while other rational numbers have recurring duodecimal fractions:
- 1/5 = 0.2497
- 1/7 = 0.186A35
- 1/A = 0.12497 (one tenth)
- 1/B = 0.1 (one eleventh)
- 1/11 = 0.0B (one thirteenth)
- 1/12 = 0.0A35186 (one fourteenth)
- 1/13 = 0.09724 (one fifteenth)
Examples in duodecimal | Decimal equivalent |
---|---|
1 × 5/8 = 0.76 | 1 × 5/8 = 0.625 |
100 × 5/8 = 76 | 144 × 5/8 = 90 |
576/9 = 76 | 810/9 = 90 |
400/9 = 54 | 576/9 = 64 |
1A.6 + 7.6 = 26 | 22.5 + 7.5 = 30 |
As explained in recurring decimals, whenever an irreducible fraction is written in radix point notation in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factors of its denominator are also prime factors of the base.
Because in the decimal system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: 1/8 = 1/(2×2×2), 1/20 = 1/(2×2×5), and 1/500 = 1/(2×2×5×5×5) can be expressed exactly as 0.125, 0.05, and 0.002 respectively. 1/3 and 1/7, however, recur (0.333... and 0.142857142857...).
Because in the duodecimal system, 1/8 is exact; 1/20 and 1/500 recur because they include 5 as a factor; 1/3 is exact, and 1/7 recurs, just as it does in decimal.
The number of denominators that give terminating fractions within a given number of digits, n, in a base b is the number of factors (divisors) of , the nth power of the base b (although this includes the divisor 1, which does not produce fractions when used as the denominator). The number of factors of
is given using its prime factorization.
For decimal, . The number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together, so the number of factors of
is
.
For example, the number 8 is a factor of 103 (1000), so and other fractions with a denominator of 8 cannot require more than three fractional decimal digits to terminate.
For duodecimal, . This has
divisors. The sample denominator of 8 is a factor of a gross
(in decimal), so eighths cannot need more than two duodecimal fractional places to terminate.
Because both ten and twelve have two unique prime factors, the number of divisors of for b = 10 or 12 grows quadratically with the exponent n (in other words, of the order of
).
Recurring digits
The Dozenal Society of America argues that factors of 3 are more commonly encountered in 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: B Prime factors of one above the base: 11 (=1310) All other primes: 5, 7, 15 (=1710), 17 (=1910), 1B (=2310), 25 (=2910), 27 (=3110) | ||||
Fraction | Prime factors of the denominator | Positional representation | Positional representation | Prime factors of the denominator | Fraction |
---|---|---|---|---|---|
1/2 | 2 | 0.5 | 0.6 | 2 | 1/2 |
1/3 | 3 | 0.3 | 0.4 | 3 | 1/3 |
1/4 | 2 | 0.25 | 0.3 | 2 | 1/4 |
1/5 | 5 | 0.2 | 0.2497 | 5 | 1/5 |
1/6 | 2, 3 | 0.16 | 0.2 | 2, 3 | 1/6 |
1/7 | 7 | 0.142857 | 0.186A35 | 7 | 1/7 |
1/8 | 2 | 0.125 | 0.16 | 2 | 1/8 |
1/9 | 3 | 0.1 | 0.14 | 3 | 1/9 |
1/10 | 2, 5 | 0.1 | 0.12497 | 2, 5 | 1/A |
1/11 | 11 | 0.09 | 0.1 | B | 1/B |
1/12 | 2, 3 | 0.083 | 0.1 | 2, 3 | 1/10 |
1/13 | 13 | 0.076923 | 0.0B | 11 | 1/11 |
1/14 | 2, 7 | 0.0714285 | 0.0A35186 | 2, 7 | 1/12 |
1/15 | 3, 5 | 0.06 | 0.09724 | 3, 5 | 1/13 |
1/16 | 2 | 0.0625 | 0.09 | 2 | 1/14 |
1/17 | 17 | 0.0588235294117647 | 0.08579214B36429A7 | 15 | 1/15 |
1/18 | 2, 3 | 0.05 | 0.08 | 2, 3 | 1/16 |
1/19 | 19 | 0.052631578947368421 | 0.076B45 | 17 | 1/17 |
1/20 | 2, 5 | 0.05 | 0.07249 | 2, 5 | 1/18 |
1/21 | 3, 7 | 0.047619 | 0.06A3518 | 3, 7 | 1/19 |
1/22 | 2, 11 | 0.045 | 0.06 | 2, B | 1/1A |
1/23 | 23 | 0.0434782608695652173913 | 0.06316948421 | 1B | 1/1B |
1/24 | 2, 3 | 0.0416 | 0.06 | 2, 3 | 1/20 |
1/25 | 5 | 0.04 | 0.05915343A0B62A68781B | 5 | 1/21 |
1/26 | 2, 13 | 0.0384615 | 0.056 | 2, 11 | 1/22 |
1/27 | 3 | 0.037 | 0.054 | 3 | 1/23 |
1/28 | 2, 7 | 0.03571428 | 0.05186A3 | 2, 7 | 1/24 |
1/29 | 29 | 0.0344827586206896551724137931 | 0.04B7 | 25 | 1/25 |
1/30 | 2, 3, 5 | 0.03 | 0.04972 | 2, 3, 5 | 1/26 |
1/31 | 31 | 0.032258064516129 | 0.0478AA093598166B74311B28623A55 | 27 | 1/27 |
1/32 | 2 | 0.03125 | 0.046 | 2 | 1/28 |
1/33 | 3, 11 | 0.03 | 0.04 | 3, B | 1/29 |
1/34 | 2, 17 | 0.02941176470588235 | 0.0429A708579214B36 | 2, 15 | 1/2A |
1/35 | 5, 7 | 0.0285714 | 0.0414559B3931 | 5, 7 | 1/2B |
1/36 | 2, 3 | 0.027 | 0.04 | 2, 3 | 1/30 |
The duodecimal period length of 1/n are (in decimal)
- 0, 0, 0, 0, 4, 0, 6, 0, 0, 4, 1, 0, 2, 6, 4, 0, 16, 0, 6, 4, 6, 1, 11, 0, 20, 2, 0, 6, 4, 4, 30, 0, 1, 16, 12, 0, 9, 6, 2, 4, 40, 6, 42, 1, 4, 11, 23, 0, 42, 20, 16, 2, 52, 0, 4, 6, 6, 4, 29, 4, 15, 30, 6, 0, 4, 1, 66, 16, 11, 12, 35, 0, ... (sequence A246004 in the OEIS)
The duodecimal period length of 1/(nth prime) are (in decimal)
- 0, 0, 4, 6, 1, 2, 16, 6, 11, 4, 30, 9, 40, 42, 23, 52, 29, 15, 66, 35, 36, 26, 41, 8, 16, 100, 102, 53, 54, 112, 126, 65, 136, 138, 148, 150, 3, 162, 83, 172, 89, 90, 95, 24, 196, 66, 14, 222, 113, 114, 8, 119, 120, 125, 256, 131, 268, 54, 138, 280, ... (sequence A246489 in the OEIS)
Smallest prime with duodecimal period n are (in decimal)
- 11, 13, 157, 5, 22621, 7, 659, 89, 37, 19141, 23, 20593, 477517, 211, 61, 17, 2693651, 1657, 29043636306420266077, 85403261, 8177824843189, 57154490053, 47, 193, 303551, 79, 306829, 673, 59, 31, 373, 153953, 886381, 2551, 71, 73, ... (sequence A252170 in the OEIS)
Irrational numbers
The representations of irrational numbers in any positional number system (including decimal and duodecimal) neither terminate nor repeat. The following table gives the first digits for some important algebraic and transcendental numbers in both decimal and duodecimal.
Algebraic irrational number | In decimal | In duodecimal |
---|---|---|
√2, the square root of 2 | 1.414213562373... | 1.4B79170A07B8... |
φ (phi), the golden ratio = | 1.618033988749... | 1.74BB6772802A... |
Transcendental number | In decimal | In duodecimal |
π (pi), the ratio of a circle's circumference to its diameter | 3.141592653589... | 3.184809493B91... |
e, the base of the natural logarithm | 2.718281828459... | 2.875236069821... |
See also
- Vigesimal (base 20)
- Sexagesimal (base 60)
References
- Dvorsky, George (January 18, 2013). "Why We Should Switch To A Base-12 Counting System". Gizmodo. Retrieved December 21, 2013.
- Pittman, Richard (1990). "Origin of Mesopotamian duodecimal and sexagesimal counting systems". Philippine Journal of Linguistics. 21 (1): 97.
- Ifrah, Georges (2000) [1st French ed. 1981]. The Universal History of Numbers: From prehistory to the invention of the computer. Wiley. ISBN 0-471-39340-1. Translated from the French by David Bellos, E. F. Harding, Sophie Wood and Ian Monk.
- Matsushita, Shuji (October 1998). "Decimal vs. Duodecimal: An interaction between two systems of numeration". www3.aa.tufs.ac.jp. Archived from the original on October 5, 2008. Retrieved May 29, 2011.
- Mazaudon, Martine (2002). "Les principes de construction du nombre dans les langues tibéto-birmanes". In François, Jacques (ed.). La Pluralité (PDF). Leuven: Peeters. pp. 91–119. ISBN 90-429-1295-2. Archived from the original (PDF) on 2016-03-28. Retrieved 2014-03-27.
- von Mengden, Ferdinand (2006). "The peculiarities of the Old English numeral system". In Nikolaus Ritt; Herbert Schendl; Christiane Dalton-Puffer; Dieter Kastovsky (eds.). Medieval English and its Heritage: Structure Meaning and Mechanisms of Change. Studies in English Medieval Language and Literature. Vol. 16. Frankfurt: Peter Lang. pp. 125–145.
- von Mengden, Ferdinand (2010). Cardinal Numerals: Old English from a Cross-Linguistic Perspective. Topics in English Linguistics. Vol. 67. Berlin; New York: De Gruyter Mouton. pp. 159–161.
- Gordon, E V (1957). Introduction to Old Norse. Oxford: Clarendon Press. pp. 292–293.
- De Vlieger, Michael (2010). "Symbology Overview" (PDF). The Duodecimal Bulletin. 4X [58] (2).
- Pakin, Scott (2021) [2007]. "The Comprehensive LATEX Symbol List". Comprehensive TEX Archive Network (14.0 ed.). Rei, Fukui (2004) [2002]. "tipa – Fonts and macros for IPA phonetics characters". Comprehensive TEX Archive Network (1.3 ed.). The turned digits 2 and 3 employed in the TIPA package originated in The Principles of the International Phonetic Association, University College London, 1949.
- Andrews, Frank Emerson (1935). New Numbers: How Acceptance of a Duodecimal (12) Base Would Simplify Mathematics. Harcourt, Brace and company. p. 52.
- Note that the symbol displayed is a standard asterisk; for technical reasons, this page cannot display the sextile inline.
- "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).
- "Mo for Megro" (PDF). The Duodecimal Bulletin. 1 (1). 1945.
- Pitman, Isaac (24 November 1857). "A Reckoning Reform". Bedfordshire Independent. Reprinted as "Sir Isaac Pitman on the Dozen System: A Reckoning Reform" (PDF). The Duodecimal Bulletin. 3 (2): 1–5. 1947.
- Pentzlin, Karl (March 30, 2013). "Proposal to encode Duodecimal Digit Forms in the UCS" (PDF). ISO/IEC JTC1/SC2/WG2. Retrieved 2024-06-25.
- "The Unicode Standard, Version 8.0: Number Forms" (PDF). Unicode Consortium. Retrieved 2016-05-30.
- "The Unicode Standard 8.0" (PDF). Retrieved 2014-07-18.
- The Dozenal Society of America (n.d.). "What should the DSA do about transdecimal characters?". Dozenal Society of America. The Dozenal Society of America. Retrieved January 1, 2018.
- Arnold Whittall, The Cambridge Introduction to Serialism (New York: Cambridge University Press, 2008): 276. ISBN 978-0-521-68200-8 (pbk).
- Ferrari, Silvio (1854). Calcolo Decidozzinale. p. 2.
- "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).
- "The Unicode Standard 8.0" (PDF). Retrieved 2014-07-18.
- Volan, John (July 2015). "Base Annotation Schemes" (PDF). The Duodecimal Bulletin. 62.
- "Definition of GROSS". www.merriam-webster.com. Retrieved 2025-02-17.
- "Definition of GREAT GROSS". www.merriam-webster.com. Retrieved 2025-02-17.
- The Prodigy (Biography of WJS) pg [42]
- A. C. Aitken (January 25, 1962) "Twelves and Tens" The Listener.
- A. C. Aitken (1962) The Case Against Decimalisation. Edinburgh / London: Oliver & Boyd.
- "SchoolhouseRock - Little Twelvetoes". 6 February 2010. Archived from the original on 6 February 2010.
- Bellos, Alex (2011-04-04). Alex's Adventures in Numberland. A&C Black. p. 50. ISBN 978-1-4088-0959-4.
- Pendlebury, Tom; Goodman, Donald (2012). "TGM: A Coherent Dozenal Metrology" (PDF). The Dozenal Society of Great Britain.
- Goodman, Donald (2016). "Manual of the Dozenal System" (PDF). Dozenal Society of America. Retrieved 27 April 2018.
- Suga, Takashi (22 May 2019). "Proposal for the Universal Unit System" (PDF).
- Volan, John. "The Primel Metrology" (PDF). The Duodecimal Bulletin. 63 (1): 38–60.
- De Vlieger, Michael Thomas (30 November 2011). "Dozenal FAQs" (PDF). dozenal.org. The Dozenal Society of America. Retrieved November 20, 2022.
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.
The duodecimal system also known as base twelve 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 144 1 000 means twelve cubed 1 728 and 0 1 means a twelfth 8333 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 and finally 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 dek pronounced dɛk and 3 a turned 3 for eleven el pronounced ɛl 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 2b 3c 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 OriginIn 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 In 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 24 12 2 hours in a day many other items are counted by the dozen gross 144 twelve squared or great gross 1728 twelve cubed The Romans used a fraction system based on 12 including the uncia which became both the English words ounce and inch Historically many parts of western Europe used a mixed vigesimal duodecimal currency system of pounds shillings and pence with 20 shillings to a pound and 12 pence to a shilling originally established by Charlemagne in the 780s Duodecimally divided units Relative value Length WeightFrench English English Troy Roman120 pied foot pound libra12 1 pouce inch ounce uncia12 2 ligne line 2 scruples 2 scrupula12 3 point point seed siliquaNotations and pronunciationsIn a positional numeral system of base n twelve for duodecimal each of the first n natural numbers is given a distinct numeral symbol and then n is denoted 10 meaning 1 times n plus 0 units For duodecimal the standard numeral symbols for 0 9 are typically preserved for zero through nine but there are numerous proposals for how to write the numerals representing ten and eleven More radical proposals do not use any Arabic numerals under the principle of separate identity Pronunciation of duodecimal numbers also has no standard but various systems have been proposed Transdecimal symbols 2 3duodecimal ten eleven In UnicodeU 218A TURNED DIGIT TWOU 218B TURNED DIGIT THREEBlock Number FormsNoteArabic digits with 180 rotation by Isaac PitmanIn LaTeX using the TIPA package textturntwo textturnthree Several authors have proposed using letters of the alphabet for the transdecimal symbols Latin letters such as A B as in hexadecimal or T E initials of Ten and Eleven are convenient because they are widely accessible and for instance can be typed on typewriters However when mixed with ordinary prose they might be confused for letters As an alternative Greek letters such as t e could be used instead Frank Emerson Andrews an early American advocate for duodecimal suggested and used in his 1935 book New Numbers X Ɛ italic capital X from the Roman numeral for ten 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 PDF content using the Pitman digits instead but continues to use the letters X and E on its webpage Symbols Background NoteA B As in hexadecimal Allows entry on typewriters T E Initials of Ten and Eleven Used in lower case in music set theoryX E X from the Roman numeral E from Eleven X Z Origin of Z unknown Attributed to D Alembert amp Buffon by the DSA d e Greek delta from deka ten epsilon from endeka eleven t e Greek tau epsilonW W from doubling the Roman numeral V based on a pendulum Silvio Ferrari in Calcolo Decidozzinale 1854 X Ɛ italic X pronounced dec rounded italic Ɛ pronounced elf Frank Andrews in New Numbers 1935 with italic 0 9 for other duodecimal numerals sextile or six pointed asterisk hash or octothorpe On push button telephones used by Edna Kramer in The Main Stream of Mathematics 1951 used by the DSA 1974 20082 3 Digits 2 and 3 rotated 180 Isaac Pitman 1857 used by the DSGB used by the DSA since 2015 included in Unicode 8 0 2015 Pronounced dek el William Dwiggins 1945 1932 Used by the DSA 1945 1974 and 2008 2015Base notation There are also varying proposals of how to distinguish a duodecimal number from a decimal one The most common method used in mainstream mathematics sources comparing various number bases uses a subscript 10 or 12 e g 5412 6410 To avoid ambiguity about the meaning of the subscript 10 the subscripts might be spelled out 54twelve 64ten In 2015 the Dozenal Society of America adopted the more compact single letter abbreviation z for dozenal and d for decimal 54z 64d Other proposed methods include italicizing duodecimal numbers 54 64 adding a Humphrey point a semicolon instead of a decimal point to duodecimal numbers 54 6 64 5 prefixing duodecimal numbers by an asterisk 54 64 or some combination of these The Dozenal Society of Great Britain uses an asterisk prefix for duodecimal whole numbers and a Humphrey point for other duodecimal numbers Pronunciation The Dozenal Society of America suggested ten and eleven should be pronounced as dek and el respectively Terms for some powers of twelve already exist in English The number twelve 1012 or 1210 is also called a dozen Twelve squared 10012 or 14410 is called a gross Twelve cubed 100012 or 172810 is called a great gross 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 founded as the Duodecimal Society of America in 1944 and the Dozenal Society of Great Britain founded 1959 promote adoption of the duodecimal system 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 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 In media In Little Twelvetoes an episode of the American educational television series Schoolhouse Rock a farmer encounters an alien being with twelve fingers on each hand and twelve toes on each foot who uses duodecimal arithmetic The alien uses 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 and John Volan s Primel system Comparison to other number systemsIn this section numerals are in decimal For example 10 means 9 1 and 12 means 9 3 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 A B 101 1 2 3 4 5 6 7 8 9 A B 102 2 4 6 8 A 10 12 14 16 18 1A 203 3 6 9 10 13 16 19 20 23 26 29 304 4 8 10 14 18 20 24 28 30 34 38 405 5 A 13 18 21 26 2B 34 39 42 47 506 6 10 16 20 26 30 36 40 46 50 56 607 7 12 19 24 2B 36 41 48 53 5A 65 708 8 14 20 28 34 40 48 54 60 68 74 809 9 16 23 30 39 46 53 60 69 76 83 90A A 18 26 34 42 50 5A 68 76 84 92 A0B B 1A 29 38 47 56 65 74 83 92 A1 B010 10 20 30 40 50 60 70 80 90 A0 B0 100Conversion tables to and from decimalTo 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 BB BBB B 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 gt 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 1A8 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 gt Duodecimal 10 000 5 954 2 000 1 1A8 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 08320 000 41 472 2 000 3 456 200 288 20 24 2 2 0 2 0 1630 000 62 208 3 000 5 184 300 432 30 36 3 3 0 3 0 2540 000 82 944 4 000 6 912 400 576 40 48 4 4 0 4 0 350 000 103 680 5 000 8 640 500 720 50 60 5 5 0 5 0 41660 000 124 416 6 000 10 368 600 864 60 72 6 6 0 6 0 570 000 145 152 7 000 12 096 700 1 008 70 84 7 7 0 7 0 58380 000 165 888 8 000 13 824 800 1 152 80 96 8 8 0 8 0 690 000 186 624 9 000 15 552 900 1 296 90 108 9 9 0 9 0 75A0 000 207 360 A 000 17 280 A00 1 440 A0 120 A 10 0 A 0 83B0 000 228 096 B 000 19 008 B00 1 584 B0 132 B 11 0 B 0 916Decimal to duodecimal digit conversion Dec Duod Dec Duod Dec Duod Dec Duod Dec Duod Dec Duodecimal10 000 5 954 1 000 6B4 100 84 10 A 1 1 0 1 0 1249720 000 B 6A8 2 000 1 1A8 200 148 20 18 2 2 0 2 0 249730 000 15 440 3 000 1 8A0 300 210 30 26 3 3 0 3 0 3724940 000 1B 194 4 000 2 394 400 294 40 34 4 4 0 4 0 497250 000 24 B28 5 000 2 A88 500 358 50 42 5 5 0 5 0 660 000 2A 880 6 000 3 580 600 420 60 50 6 6 0 6 0 724970 000 34 614 7 000 4 074 700 4A4 70 5A 7 7 0 7 0 8497280 000 3A 368 8 000 4 768 800 568 80 68 8 8 0 8 0 972490 000 44 100 9 000 5 260 900 630 90 76 9 9 0 9 0 A9724Fractions and irrational numbersFractions 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 A is one tenth 1 14 0 09 this is one sixteenth 1 12 is one fourteenth while other rational numbers have recurring duodecimal fractions 1 5 0 2497 1 7 0 186A35 1 A 0 12497 one tenth 1 B 0 1 one eleventh 1 11 0 0B one thirteenth 1 12 0 0A35186 one fourteenth 1 13 0 09724 one fifteenth Examples in duodecimal Decimal equivalent1 5 8 0 76 1 5 8 0 625100 5 8 76 144 5 8 90 576 9 76 810 9 90 400 9 54 576 9 641A 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 2 5 10 displaystyle 2 times 5 10 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 2 2 3 12 displaystyle 2 times 2 times 3 12 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 bn displaystyle b n the n th 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 bn displaystyle b n is given using its prime factorization For decimal 10n 2n 5n displaystyle 10 n 2 n times 5 n 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 10n displaystyle 10 n is n 1 n 1 n 1 2 displaystyle n 1 n 1 n 1 2 For example the number 8 is a factor of 103 1000 so 18 textstyle frac 1 8 and other fractions with a denominator of 8 cannot require more than three fractional decimal digits to terminate 58 0 62510 textstyle frac 5 8 0 625 10 For duodecimal 10n 22n 3n displaystyle 10 n 2 2n times 3 n This has 2n 1 n 1 displaystyle 2n 1 n 1 divisors The sample denominator of 8 is a factor of a gross 122 144 textstyle 12 2 144 in decimal so eighths cannot need more than two duodecimal fractional places to terminate 58 0 7612 textstyle frac 5 8 0 76 12 Because both ten and twelve have two unique prime factors the number of divisors of bn displaystyle b n for b 10 or 12 grows quadratically with the exponent n in other words of the order of n2 displaystyle n 2 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 04612Decimal base Prime factors of the base 2 5 Prime factors of one below the base 3 Prime factors of one above the base 11 All other primes 7 13 17 19 23 29 31 Duodecimal base Prime factors of the base 2 3 Prime factors of one below the base B Prime factors of one above the base 11 1310 All other primes 5 7 15 1710 17 1910 1B 2310 25 2910 27 3110 Fraction Prime factors of the denominator Positional representation Positional representation Prime factors of the denominator Fraction1 2 2 0 5 0 6 2 1 21 3 3 0 3 0 4 3 1 31 4 2 0 25 0 3 2 1 41 5 5 0 2 0 2497 5 1 51 6 2 3 0 16 0 2 2 3 1 61 7 7 0 142857 0 186A35 7 1 71 8 2 0 125 0 16 2 1 81 9 3 0 1 0 14 3 1 91 10 2 5 0 1 0 12497 2 5 1 A1 11 11 0 09 0 1 B 1 B1 12 2 3 0 083 0 1 2 3 1 101 13 13 0 076923 0 0B 11 1 111 14 2 7 0 0714285 0 0A35186 2 7 1 121 15 3 5 0 06 0 09724 3 5 1 131 16 2 0 0625 0 09 2 1 141 17 17 0 0588235294117647 0 08579214B36429A7 15 1 151 18 2 3 0 05 0 08 2 3 1 161 19 19 0 052631578947368421 0 076B45 17 1 171 20 2 5 0 05 0 07249 2 5 1 181 21 3 7 0 047619 0 06A3518 3 7 1 191 22 2 11 0 045 0 06 2 B 1 1A1 23 23 0 0434782608695652173913 0 06316948421 1B 1 1B1 24 2 3 0 0416 0 06 2 3 1 201 25 5 0 04 0 05915343A0B62A68781B 5 1 211 26 2 13 0 0384615 0 056 2 11 1 221 27 3 0 037 0 054 3 1 231 28 2 7 0 03571428 0 05186A3 2 7 1 241 29 29 0 0344827586206896551724137931 0 04B7 25 1 251 30 2 3 5 0 03 0 04972 2 3 5 1 261 31 31 0 032258064516129 0 0478AA093598166B74311B28623A55 27 1 271 32 2 0 03125 0 046 2 1 281 33 3 11 0 03 0 04 3 B 1 291 34 2 17 0 02941176470588235 0 0429A708579214B36 2 15 1 2A1 35 5 7 0 0285714 0 0414559B3931 5 7 1 2B1 36 2 3 0 027 0 04 2 3 1 30 The duodecimal period length of 1 n are in decimal 0 0 0 0 4 0 6 0 0 4 1 0 2 6 4 0 16 0 6 4 6 1 11 0 20 2 0 6 4 4 30 0 1 16 12 0 9 6 2 4 40 6 42 1 4 11 23 0 42 20 16 2 52 0 4 6 6 4 29 4 15 30 6 0 4 1 66 16 11 12 35 0 sequence A246004 in the OEIS The duodecimal period length of 1 nth prime are in decimal 0 0 4 6 1 2 16 6 11 4 30 9 40 42 23 52 29 15 66 35 36 26 41 8 16 100 102 53 54 112 126 65 136 138 148 150 3 162 83 172 89 90 95 24 196 66 14 222 113 114 8 119 120 125 256 131 268 54 138 280 sequence A246489 in the OEIS Smallest prime with duodecimal period n are in decimal 11 13 157 5 22621 7 659 89 37 19141 23 20593 477517 211 61 17 2693651 1657 29043636306420266077 85403261 8177824843189 57154490053 47 193 303551 79 306829 673 59 31 373 153953 886381 2551 71 73 sequence A252170 in the OEIS Irrational numbers The representations of irrational numbers in any positional number system including decimal and duodecimal neither terminate nor repeat The following table gives the first digits for some important algebraic and transcendental numbers in both decimal and duodecimal Algebraic irrational number In decimal In duodecimal 2 the square root of 2 1 414213562373 1 4B79170A07B8 f phi the golden ratio 1 52 displaystyle tfrac 1 sqrt 5 2 1 618033988749 1 74BB6772802A Transcendental number In decimal In duodecimalp 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 alsoVigesimal base 20 Sexagesimal base 60 ReferencesDvorsky George January 18 2013 Why We Should Switch To A Base 12 Counting System Gizmodo Retrieved December 21 2013 Pittman Richard 1990 Origin of Mesopotamian duodecimal and sexagesimal counting systems Philippine Journal of Linguistics 21 1 97 Ifrah Georges 2000 1st French ed 1981 The Universal History of Numbers From prehistory to the invention of the computer Wiley ISBN 0 471 39340 1 Translated from the French by David Bellos E F Harding Sophie Wood and Ian Monk Matsushita Shuji October 1998 Decimal vs Duodecimal An interaction between two systems of numeration www3 aa tufs ac jp Archived from the original on October 5 2008 Retrieved May 29 2011 Mazaudon Martine 2002 Les principes de construction du nombre dans les langues tibeto birmanes In Francois Jacques ed La Pluralite PDF Leuven Peeters pp 91 119 ISBN 90 429 1295 2 Archived from the original PDF on 2016 03 28 Retrieved 2014 03 27 von Mengden Ferdinand 2006 The peculiarities of the Old English numeral system In Nikolaus Ritt Herbert Schendl Christiane Dalton Puffer Dieter Kastovsky eds Medieval English and its Heritage Structure Meaning and Mechanisms of Change Studies in English Medieval Language and Literature Vol 16 Frankfurt Peter Lang pp 125 145 von Mengden Ferdinand 2010 Cardinal Numerals Old English from a Cross Linguistic Perspective Topics in English Linguistics Vol 67 Berlin New York De Gruyter Mouton pp 159 161 Gordon E V 1957 Introduction to Old Norse Oxford Clarendon Press pp 292 293 De Vlieger Michael 2010 Symbology Overview PDF The Duodecimal Bulletin 4X 58 2 Pakin Scott 2021 2007 The Comprehensive LATEX Symbol List Comprehensive TEX Archive Network 14 0 ed Rei Fukui 2004 2002 tipa Fonts and macros for IPA phonetics characters Comprehensive TEX Archive Network 1 3 ed The turned digits 2 and 3 employed in the TIPA package originated in The Principles of the International Phonetic Association University College London 1949 Andrews Frank Emerson 1935 New Numbers How Acceptance of a Duodecimal 12 Base Would Simplify Mathematics Harcourt Brace and company p 52 Note that the symbol displayed is a standard asterisk for technical reasons this page cannot display the sextile inline 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 Mo for Megro PDF The Duodecimal Bulletin 1 1 1945 Pitman Isaac 24 November 1857 A Reckoning Reform Bedfordshire Independent Reprinted as Sir Isaac Pitman on the Dozen System A Reckoning Reform PDF The Duodecimal Bulletin 3 2 1 5 1947 Pentzlin Karl March 30 2013 Proposal to encode Duodecimal Digit Forms in the UCS PDF ISO IEC JTC1 SC2 WG2 Retrieved 2024 06 25 The Unicode Standard Version 8 0 Number Forms PDF Unicode Consortium Retrieved 2016 05 30 The Unicode Standard 8 0 PDF Retrieved 2014 07 18 The Dozenal Society of America n d What should the DSA do about transdecimal characters Dozenal Society of America The Dozenal Society of America Retrieved January 1 2018 Arnold Whittall The Cambridge Introduction to Serialism New York Cambridge University Press 2008 276 ISBN 978 0 521 68200 8 pbk Ferrari Silvio 1854 Calcolo Decidozzinale p 2 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 The Unicode Standard 8 0 PDF Retrieved 2014 07 18 Volan John July 2015 Base Annotation Schemes PDF The Duodecimal Bulletin 62 Definition of GROSS www merriam webster com Retrieved 2025 02 17 Definition of GREAT GROSS www merriam webster com Retrieved 2025 02 17 The Prodigy Biography of WJS pg 42 A C Aitken January 25 1962 Twelves and Tens The Listener A C Aitken 1962 The Case Against Decimalisation Edinburgh London Oliver amp Boyd SchoolhouseRock Little Twelvetoes 6 February 2010 Archived from the original on 6 February 2010 Bellos Alex 2011 04 04 Alex s Adventures in Numberland A amp C Black p 50 ISBN 978 1 4088 0959 4 Pendlebury Tom Goodman Donald 2012 TGM A Coherent Dozenal Metrology PDF The Dozenal Society of Great Britain Goodman Donald 2016 Manual of the Dozenal System PDF Dozenal Society of America Retrieved 27 April 2018 Suga Takashi 22 May 2019 Proposal for the Universal Unit System PDF Volan John The Primel Metrology PDF The Duodecimal Bulletin 63 1 38 60 De Vlieger Michael Thomas 30 November 2011 Dozenal FAQs PDF dozenal org The Dozenal Society of America Retrieved November 20 2022 External linksDozenal 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