|Hindu–Arabic numeral system|
|Positional systems by base|
|Non-standard positional numeral systems|
|List of numeral systems|
The duodecimal system (also known as base 12 or dozenal) is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated "2" (2) and the number eleven by a rotated "3" (3). This notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ (Code point 218A) and ↋ (Code point 218B), respectively. Other notations use "A", "T", or "X" for ten and "B" or "E" for eleven. The number twelve (that is, the number written as "12" in the base ten numerical system) is instead written as "10" in duodecimal (meaning "1 dozen and 0 units", instead of "1 ten and 0 units"), whereas the digit string "12" means "1 dozen and 2 units" (i.e. the same number that in decimal is written as "14"). Similarly, in duodecimal "100" means "1 gross", "1000" means "1 great gross", and "0.1" means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth").
The number twelve, a superior highly composite number, is the smallest number with four non-trivial factors (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the subitizing range. As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers (whereas ten has only two non-trivial factors: 2 and 5, and not 3, 4, or 6), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. As a result, duodecimal has been described as the optimal number system. Of its factors, 2 and 3 are prime, which means the reciprocals of all 3-smooth numbers (such as 2, 3, 4, 6, 8, 9...) have a terminating representation in duodecimal. In particular, the five most elementary fractions (1/, 1/, 2/, 1/ and 3/) all have a short terminating representation in duodecimal (0.6, 0.4, 0.8, 0.3 and 0.9, respectively), and twelve is the smallest radix with this feature (because it is the least common multiple of 3 and 4). This all makes it a more convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary, octal and hexadecimal systems. Although the trigesimal and sexagesimal systems (where the reciprocals of all 5-smooth numbers terminate) do even better in this respect, this is at the cost of unwieldy multiplication tables and a much larger number of symbols to memorize.
- 1 Origin
- 2 Places
- 3 Comparison to other numeral systems
- 4 Conversion tables to and from decimal
- 5 Fractions and irrational numbers
- 6 Advocacy and "dozenalism"
- 7 Duodecimal digits on computerized writing systems
- 8 See also
- 9 References
- 10 External links
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; the Chepang language of Nepal and the Mahl language of Minicoy Island in India are known to use duodecimal numerals. In fiction, J. R. R. Tolkien's Elvish languages can express numbers either decimally (in Quenya, maquanótië "hand-counting" or *quaistanótië "tenth-counting") or duodecimally (presumably *rastanótië "dozen-counting").
Germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif (respectively one left and two left), both of which were decimal.
Historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, and the Babylonians had twelve hours in a day (although at some point this was changed to 24). Traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in an imperial foot, 12 troy ounces in a troy pound, 12 old British pence in a shilling, 24 (12×2) hours in a day, and many other items counted by the dozen, gross (144, square of 12) or great gross (1728, cube of 12). The Romans used a fraction system based on 12, including the uncia which became both the English words ounce and inch. 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|
|English unit of mass
at sea level
|12−2||ligne||line||2 scruples||2 scrupulum||slug|
The importance of 12 has been attributed to the number of lunar cycles in a year, and also to the fact that humans have 12 finger bones (phalanges) on one hand (three on each of four fingers). It is possible to count to 12 with the thumb acting as a pointer, touching each finger bone in turn. A traditional finger counting system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. In this system, the one (usually right) hand counts repeatedly to 12, displaying the number of iterations on the other (usually left), until five dozens, i. e. the 60, are full.
In a duodecimal place system, ten can be written as 2, ᘔ, or ↊ (a rotated digit two); eleven can be written as 3, Ɛ, or ↋ (a reversed digit three); and twelve is written as 10. For alternative symbols, see below.
According to this notation, duodecimal 50 expresses the same quantity as decimal 60 (= five times twelve), duodecimal 60 is equivalent to decimal 72 (= six times twelve = half a gross), duodecimal 100 has the same value as decimal 144 (= twelve times twelve = one gross), etc.
Comparison to other numeral systems
The number 12 has six factors, which are 1, 2, 3, 4, 6, and 12, of which 2 and 3 are prime. The decimal system has only four factors, which are 1, 2, 5, and 10; of which 2 and 5 are prime. Vigesimal adds two factors to those of ten, namely 4 and 20, but no additional prime factor. Although twenty has 6 factors, 2 of them prime, similarly to twelve, it is also a much larger base (i.e. the digit set and the multiplication table are much larger). Binary has only two factors, 1 and 2, the latter being prime. Hexadecimal has five factors, adding 4, 8 and 16 to those of 2, but no additional prime. Trigesimal is the smallest system that has three different prime factors (all of the three smallest primes: 2, 3 and 5) and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30). Sexagesimal—which the ancient Sumerians and Babylonians among others actually used—adds the four convenient factors 4, 12, 20, and 60 to this but no new prime factors. The smallest system that has four different prime factors is base 210 and the pattern follows the primorials. In all base systems, there are similarities to the representation of multiples of numbers which are one less than the base.
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 ƐƐƐ,ƐƐƐ.ƐƐ to decimal, or any decimal number between 0.01 and 999,999.99 to duodecimal. To use them, the given number must first be decomposed into a sum of numbers with only one significant digit each. For example:
123,456.78 = 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08
This decomposition works the same no matter what base the number is expressed in. Just isolate each 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, of course, left out in the digit decomposition (102,304.05 = 100,000 + 2,000 + 300 + 4 + 0.05). Then the digit conversion tables can be used to obtain the equivalent value in the target base for each digit. If the given number is in duodecimal and the target base is decimal, we get:
(duodecimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (decimal) 248,832 + 41,472 + 5,184 + 576 + 60 + 6 + 0.583333333333... + 0.055555555555...
Now, because the summands are already converted to base ten, the usual decimal arithmetic is used to perform the addition and recompose the number, arriving at the conversion result:
Duodecimal -----> Decimal 100,000 = 248,832 20,000 = 41,472 3,000 = 5,184 400 = 576 50 = 60 + 6 = + 6 0.7 = 0.583333333333... 0.08 = 0.055555555555... -------------------------------------------- 123,456.78 = 296,130.638888888888...
That is, (duodecimal) 123,456.78 equals (decimal) 296,130.638 ≈ 296,130.64
If the given number is in decimal and the target base is duodecimal, the method is basically same. Using the digit conversion tables:
(decimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (duodecimal) 49,ᘔ54 + Ɛ,6ᘔ8 + 1,8ᘔ0 + 294 + 42 + 6 + 0.849724972497249724972497... + 0.0Ɛ62ᘔ68781Ɛ05915343ᘔ0Ɛ62...
However, in order to do this sum and recompose the number, now the addition tables for the duodecimal system have to be used, instead of the addition tables for decimal most people are already familiar with, because the summands are now in base twelve and so the arithmetic with them has to be in duodecimal as well. In decimal, 6 + 6 equals 12, but in duodecimal it equals 10; so, if using decimal arithmetic with duodecimal numbers one would arrive at an incorrect result. Doing the arithmetic properly in duodecimal, one gets the result:
Decimal -----> Duodecimal 100,000 = 49,ᘔ54 20,000 = Ɛ,6ᘔ8 3,000 = 1,8ᘔ0 400 = 294 50 = 42 + 6 = + 6 0.7 = 0.849724972497249724972497... 0.08 = 0.0Ɛ62ᘔ68781Ɛ05915343ᘔ0Ɛ62... -------------------------------------------------------- 123,456.78 = 5Ɛ,540.943ᘔ0Ɛ62ᘔ68781Ɛ05915343ᘔ...
That is, (decimal) 123,456.78 equals (duodecimal) 5Ɛ,540.943ᘔ0Ɛ62ᘔ68781Ɛ059153... ≈ 5Ɛ,540.94
Duodecimal to decimal digit conversion
Decimal to duodecimal digit conversion
Conversion of powers
Fractions and irrational numbers
Duodecimal fractions may be simple:
- 1/ = 0.6
- 1/ = 0.4
- 1/ = 0.3
- 1/ = 0.2
- 1/ = 0.16
- 1/ = 0.14
- 1/ = 0.1
- 1/ = 0.24972497... recurring (rounded to 0.24ᘔ)
- 1/ = 0.186ᘔ35186ᘔ35... recurring (rounded to 0.187)
- 1/ = 0.124972497... recurring (rounded to 0.125)
- 1/ = 0.111111... recurring (rounded to 0.111)
- 1/ = 0.0Ɛ0Ɛ0Ɛ... recurring (rounded to 0.0Ɛ1)
- 1/ = 0.0ᘔ35186ᘔ35186... recurring (rounded to 0.0ᘔ3)
|Examples in duodecimal||Decimal equivalent|
|1 × (5/) = 0.76||1 × (5/) = 0.625|
|100 × (5/) = 76||144 × (5/) = 90|
|576/ = 76||810/ = 90|
|400/ = 54||576/ = 64|
|1ᘔ.6 + 7.6 = 26||22.5 + 7.5 = 30|
As explained in recurring decimals, whenever an irreducible fraction is written in radix point notation in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factors of its denominator are also prime factors of the base. Thus, in base-ten (= 2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: 1/ = 1/, 1/ = 1/ and 1/ = 1/ can be expressed exactly as 0.125, 0.05 and 0.002 respectively. 1/ and 1/, however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, 1/ is exact; 1/ and 1/ recur because they include 5 as a factor; 1/ is exact; and 1/ recurs, just as it does in decimal.
The number of denominators which give terminating fractions within a given number of digits, say n, in a base b is the number of factors (divisors) of bn, 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 bn is given using its prime factorization.
For decimal, 10n = 2n * 5n. The number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together. Factors of 10n = (n+1)(n+1) = (n+1)2.
For example, the number 8 is a factor of 103 (1000), so 1/8 and other fractions with a denominator of 8 can not require more than 3 fractional decimal digits to terminate. 5/8 = 0.625ten
For duodecimal, 12n = 22n * 3n. This has (2n+1)(n+1) divisors. The sample denominator of 8 is a factor of a gross (122 = 144), so eighths can not need more than two duodecimal fractional places to terminate. 5/8 = 0.76twelve
Because both ten and twelve have two unique prime factors, the number of divisors of bn for b = 10 or 12 grows quadratically with the exponent n (in other words, of the order of n2).
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 doesn't help the main inconvenience that one does not get a finite representation for such fractions in the given base (so rounding, which introduces inexactitude, is necessary to handle them in calculations), and overall one is more likely to have to deal with infinite recurring digits when fractions are expressed in decimal than in duodecimal, because one out of every three consecutive numbers contains the prime factor 3 in its factorization, whereas only one out of every five contains the prime factor 5. All other prime factors, except 2, are not shared by either ten or twelve, so they do not influence the relative likeliness of encountering recurring digits (any irreducible fraction that contains any of these other factors in its denominator will recur in either base). Also, the prime factor 2 appears twice in the factorization of twelve, whereas only once in the factorization of ten; which means that most fractions whose denominators are powers of two will have a shorter, more convenient terminating representation in duodecimal than in decimal representation (e.g. 1/(22) = 0.25 ten = 0.3 twelve; 1/(23) = 0.125 ten = 0.16 twelve; 1/(24) = 0.0625 ten = 0.09 twelve; 1/(25) = 0.03125 ten = 0.046 twelve; etc.).
Values in bold indicate that value is exact.
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
Prime factors of the base: 2, 3
Prime factors of one below the base: Ɛ
Prime factors of one above the base: 11
All other primes: 7
of the denominator
|Positional representation||Positional representation||Prime factors
of the denominator
|1/6||2, 3||0.16||0.2||2, 3||1/6|
|1/10||2, 5||0.1||0.12497||2, 5||1/ᘔ|
|1/12||2, 3||0.083||0.1||2, 3||1/10|
|1/14||2, 7||0.0714285||0.0ᘔ35186||2, 7||1/12|
|1/15||3, 5||0.06||0.09724||3, 5||1/13|
|1/18||2, 3||0.05||0.08||2, 3||1/16|
|1/20||2, 5||0.05||0.07249||2, 5||1/18|
|1/21||3, 7||0.047619||0.06ᘔ3518||3, 7||1/19|
|1/22||2, 11||0.045||0.06||2, Ɛ||1/1ᘔ|
|1/24||2, 3||0.0416||0.06||2, 3||1/20|
|1/26||2, 13||0.0384615||0.056||2, 11||1/22|
|1/28||2, 7||0.03571428||0.05186ᘔ3||2, 7||1/24|
|1/30||2, 3, 5||0.03||0.04972||2, 3, 5||1/26|
|1/33||3, 11||0.03||0.04||3, Ɛ||1/29|
|1/34||2, 17||0.02941176470588235||0.0429ᘔ708579214Ɛ36||2, 15||1/2ᘔ|
|1/35||5, 7||0.0285714||0.0414559Ɛ3931||5, 7||1/2Ɛ|
|1/36||2, 3||0.027||0.04||2, 3||1/30|
The duodecimal period length of 1/n are
- 0, 0, 0, 0, 4, 0, 6, 0, 0, 4, 1, 0, 2, 6, 4, 0, 16, 0, 6, 4, 6, 1, 11, 0, 20, 2, 0, 6, 4, 4, 30, 0, 1, 16, 12, 0, 9, 6, 2, 4, 40, 6, 42, 1, 4, 11, 23, 0, 42, 20, 16, 2, 52, 0, 4, 6, 6, 4, 29, 4, 15, 30, 6, 0, 4, 1, 66, 16, 11, 12, 35, 0, ... (sequence A246004 in OEIS)
The duodecimal period length of 1/(nth prime) are
- 0, 0, 4, 6, 1, 2, 16, 6, 11, 4, 30, 9, 40, 42, 23, 52, 29, 15, 66, 35, 36, 26, 41, 8, 16, 100, 102, 53, 54, 112, 126, 65, 136, 138, 148, 150, 3, 162, 83, 172, 89, 90, 95, 24, 196, 66, 14, 222, 113, 114, 8, 119, 120, 125, 256, 131, 268, 54, 138, 280, ... (sequence A246489 in OEIS)
Smallest prime with duodecimal period n are
- 11, 13, 157, 5, 22621, 7, 659, 89, 37, 19141, 23, 20593, 477517, 211, 61, 17, 2693651, 1657, 29043636306420266077, 85403261, 8177824843189, 57154490053, 47, 193, 303551, 79, 306829, 673, 59, 31, 373, 153953, 886381, 2551, 71, 73, ... (sequence A252170 in OEIS)
As for irrational numbers, none of them have a finite representation in any of the rational-based positional number systems (such as the decimal and duodecimal ones); this is because a rational-based positional number system is essentially nothing but a way of expressing quantities as a sum of fractions whose denominators are powers of the base, and by definition no finite sum of rational numbers can ever result in an irrational number. For example, 123.456 = 1 × 102 + 2 × 101 + 3 × 100 + 4 × 1/101 + 5 × 1/102 + 6 × 1/103 (this is also the reason why fractions that contain prime factors in their denominator not in common with those of the base do not have a terminating representation in that base). Moreover, the infinite series of digits of an irrational number does not exhibit a pattern of repetition; instead, the different digits succeed in a seemingly random fashion. The following chart compares the first few digits of the decimal and duodecimal representation of several of the most important algebraic and transcendental irrational numbers. Some of these numbers may be perceived as having fortuitous patterns, making them easier to memorize, when represented in one base or the other.
|Algebraic irrational number||In decimal||In duodecimal|
|√2 (the length of the diagonal of a unit square)||1.41421356237309... (≈ 1.4142)||1.4Ɛ79170ᘔ07Ɛ857... (≈ 1.5)|
|√3 (the length of the diagonal of a unit cube, or twice the height of an equilateral triangle of unit side)||1.73205080756887... (≈ 1.732)||1.894Ɛ97ƐƐ968704... (≈ 1.895)|
|√5 (the length of the diagonal of a 1×2 rectangle)||2.2360679774997... (≈ 2.236)||2.29ƐƐ132540589... (≈ 2.2ᘔ)|
|φ (phi, the golden ratio = )||1.6180339887498... (≈ 1.618)||1.74ƐƐ6772802ᘔ4... (≈ 1.75)|
|Transcendental irrational number||In decimal||In duodecimal|
|π (pi, the ratio of circumference to diameter)||3.1415926535897932384626433
|e (the base of the natural logarithm)||2.718281828459045... (≈ 2.718)||2.8752360698219Ɛ8... (≈ 2.875)|
The first few digits of the decimal and duodecimal representation of another important number, the Euler–Mascheroni constant (the status of which as a rational or irrational number is not yet known), are:
|Number||In decimal||In duodecimal|
|γ (the limiting difference between the harmonic series and the natural logarithm)||0.57721566490153... (≈ 0.577)||0.6Ɛ15188ᘔ6760Ɛ3... (≈ 0.7)|
Advocacy and "dozenalism"
The case for the duodecimal system was put forth at length in F. Emerson Andrews' 1935 book New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics. Emerson noted that, due to the prevalence of factors of twelve in many traditional units of weight and measure, many of the computational advantages claimed for the metric system could be realized either by the adoption of ten-based weights and measure or by the adoption of the duodecimal number system.
Rather than the symbols "A" for ten and "B" for eleven as used in hexadecimal notation and vigesimal notation (or "T" and "E" for ten and eleven), he suggested in his book and used a script X and a script E, (U+1D4B3) and (U+2130), to represent the digits ten and eleven respectively, because, at least on a page of Roman script, these characters were distinct from any existing letters or numerals, yet were readily available in printers' fonts. He chose for its resemblance to the Roman numeral X, and as the first letter of the word "eleven".
Another popular notation, introduced by Sir Isaac Pitman, is to use a rotated 2 (ᘔ) (resembling a script τ for "ten") to represent ten and a rotated or horizontally flipped 3 (Ɛ) to represent eleven. This is the convention commonly employed by the Dozenal Society of Great Britain and has the advantage of being easily recognizable as digits because of their resemblance in shape to existing digits. On the other hand, the Dozenal Society of America adopted for some years the convention of using a sextile ⚹ for ten and a hash ⌗ for eleven. The reason was that the symbol ⚹ resembles a struck-through X, whereas the symbol ⌗ resembles a doubly struck-through 11, and both symbols are already present in telephone dials. However, critics pointed out these symbols do not look anything like digits. Some other systems write 10 as Φ (a combination of 1 and 0) and eleven as a cross of two lines (+, x, or † for example).
Problems with these symbols are evident, most notably that most of them cannot be represented in the seven-segment display of most calculator displays ( and being an exception, although "E" is used on calculators to indicate an error message, and requiring some distortion). However, 10 and 11 do fit, both within a single digit (11 fits as is, whereas the 10 has to be tilted sideways, resulting in a character that resembles an O with a macron, ō or o). A and B also fit (although B must be represented as lowercase "b" and as such, 6 must have a bar over it to distinguish the two figures) and are used on calculators for bases higher than ten.
Other problems relate to the current usage of most of the proposed symbols as variables or quantities in physics and mathematics. Of particular concern to mathematicians is which has ubiquitous usage as an unknown quantity in algebra.
In "Little Twelvetoes", American television series Schoolhouse Rock! portrayed an alien child using base-twelve arithmetic, using "dek", "el" and "doh" as names for ten, eleven and twelve, and Andrews' script-X and script-E for the digit symbols. ("Dek" is from the prefix "deca", "el" being short for "eleven" and "doh" an apparent shortening of "dozen".)
The Dozenal Society of America and the Dozenal Society of Great Britain promote widespread adoption of the base-twelve system. They use the word "dozenal" instead of "duodecimal" because the latter comes from Latin roots that express twelve in base-ten terminology.
The renowned mathematician and mental calculator Alexander Craig Aitken was an outspoken advocate of the advantages and superiority of duodecimal over decimal:
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, 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 (Edinburgh / London: Oliver & Boyd, 1962)
In Leo Frankowski's Conrad Stargard novels, Conrad introduces a duodecimal system of arithmetic at the suggestion of a merchant, who is accustomed to buying and selling goods in dozens and grosses, rather than tens or hundreds. He then invents an entire system of weights and measures in base twelve, including a clock with twelve hours in a day, rather than twenty-four hours.
In Lee Carroll's Kryon: Alchemy of the Human Spirit, a chapter is dedicated to the advantages of the duodecimal system. The duodecimal system is supposedly suggested by Kryon (a fictional entity believed in by New Age circles) for all-round use, aiming at better and more natural representation of nature of the Universe through mathematics. An individual article "Mathematica" by James D. Watt (included in the above publication) exposes a few of the unusual symmetry connections between the duodecimal system and the golden ratio, as well as provides numerous number symmetry-based arguments for the universal nature of the base-12 number system.
Duodecimal metric systems
Systems of measurement proposed by dozenalists include:
Duodecimal digits on computerized writing systems
In March 2013, a proposal was submitted to include the digit forms for ten and eleven propagated by the Dozenal Societies of Great Britain and America in the Unicode Standard. Of these, the British forms were accepted for encoding as characters at code points U+218A turned digit two (↊) and U+218B turned digit three (↋) They have been included in the Unicode 8.0 release in June 2015.
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von Mengden, Ferdinand (2010). Cardinal Numerals: Old English from a Cross-Linguistic Perspective. Topics in English Linguistics 67. Berlin; New York: De Gruyter Mouton. pp. 159–161.
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- Macey, Samuel L. (1989). The Dynamics of Progress: Time, Method, and Measure. Atlanta, Georgia: University of Georgia Press. p. 92. ISBN 978-0-8203-3796-8
- "Little Twelvetoes" Archived June 18, 2013 at the Wayback Machine
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- Kryon—Alchemy of the Human Spirit, ISBN 0-9636304-8-2
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