Duodecimal

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The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base.

The number ten may be written as 'A', and the number eleven as 'B'. The number twelve is written as '10'.

The number 12 has six factors, which are 1, 2, 3, 4, 6, and 12, of which 2 and 3 are prime. It is a more convenient number system for computing fractions than other common number systems such as the decimal, vigesimal, binary and hexadecimal systems. 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) and prime factor 5, being less common in the prime factorization of numbers, is arguably less useful than prime factor 3. 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.

Contents

[edit] Origin

A duodecimal multiplication table
1 2 3 4 5 6 7 8 9 A B 10
2 4 6 8 A 10 12 14 16 18 1A 20
3 6 9 10 13 16 19 20 23 26 29 30
4 8 10 14 18 20 24 28 30 34 38 40
5 A 13 18 21 26 2B 34 39 42 47 50
6 10 16 20 26 30 36 40 46 50 56 60
7 12 19 24 2B 36 41 48 53 5A 65 70
8 14 20 28 34 40 48 54 60 68 74 80
9 16 23 30 39 46 53 60 69 76 83 90
A 18 26 34 42 50 5A 68 76 84 92 A0
B 1A 29 38 47 56 65 74 83 92 A1 B0
10 20 30 40 50 60 70 80 90 A0 B0 100
In this section, numerals are based on decimal places. For example, 10 means ten, 12 means twelve.

Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Kahugu, the Nimbia dialect of Gwandara; the Mahl language of Minicoy Island in India and the Chepang language of Nepal are known to use duodecimal numerals. In fiction, J. R. R. Tolkien's Elvish languages used the duodecimal.

Historically, units of time in many civilizations are duodecimal, which may come as a generalization of the use for months. There are twelve signs of the zodiac. There are twelve European hours in a day or night. Traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches.

Many European languages have special words for 11 and 12 (and sometimes into the teens), which are often misinterpreted as vestiges of a base-twelve system. However, in actuality, most if not all of these terms have been eroded from decimal roots. For example, in Latin, the teens were formed by suffixing -decem (ten) to the respective words. In the modern Romance languages, this is often obscured by sound changes. For example, undecem and duodecem became, in Spanish, once and doce (likewise trece, catorce, quince) and in French, onze and douze (likewise treize, quatorze, quinze, seize from 13 until 16). English “eleven” and “twelve” are believed to come from Proto-Germanic *ainlif and *twalif (respectively “one left” and “two left”), also related to base-ten. Admittedly, the survival of such apparently unique terms may be connected with duodecimal tendencies, but their origin is not duodecimal.

Being a versatile denominator in fractions may explain why we have 12 inches in an imperial foot, 12 ounces in a troy pound, 12 old British pence in a shilling, 12 items in a dozen, 12 dozens in a gross (144, square of 12), 12 gross in a great gross (1728, cube of 12), 24 (12 * 2) hours in a day, etc. The Romans used a fraction system based on 12, including the uncia which became both the English words ounce and inch. Pre-decimalisation, Great Britain used a duodecimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the pound sterling), and Charlemagne established a monetary system that had a base of twelve and twenty, the remnants of which persist in many places.

[edit] Places

In a duodecimal place system, ten is written as A, eleven is written as B, twelve is written as 10. An alternate system has ten written as X and eleven written as E.

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.
Note that the correct term is, a gross of apples, and not a gross apples. In a hypothetical duodecimal system, the term per gross (¹⁄144) might replace per cent (¹⁄100).

Examples in duodecimal English duodecimal name Decimal equivalent
26 two dozen and six = two and a half dozen 30
50 five dozen 60
76 seven dozen and six = seven and a half dozen 90
A0 ten dozen = one small gross 120
100 one gross 144
1A6 one gross ten dozen and six = twenty-two and a half dozen 270
260 two gross and six dozen = two and a half gross = thirty dozen 360
500 five gross = sixty dozen 720
600 six gross = half a great gross 864
700 seven gross 1 008
B29 eleven gross two dozen and nine 1 617
BBB eleven gross eleven dozen and eleven = one less than a great gross 1 727
1 1B1 one great gross one gross eleven dozen and one 2 005
36 A17 three dozen and six great gross, ten gross one dozen and seven 74 035
Powers of twelve in duodecimal English duodecimal name Decimal equivalent
10 = 101 twelve = one dozen 12 = 121
100 = 102 one gross = twelve dozen 144 = 122
1 000 = 103 one great gross = twelve gross 1 728 = 123
10 000 = 104 = 1002 one dozen great gross = twelve great gross 20 736 = 124 = 1442
100 000 = 105  ? (twelve to the fifth power)=twelve dozen great gross 248 832 = 125
1 000 000 = 106 = 1003 = 1 0002  ? (twelve to the sixth power) 2 985 984 = 126 = 1443 = 1 7282
10 000 000 = 107  ? (twelve to the seventh power) 35 831 808 = 127
100 000 000 = 108 = 1004  ? (twelve to the eighth power) 429 981 696 = 128 = 1444
1 000 000 000 = 109 = 1 0003  ? (twelve to the ninth power) 5 159 780 352 = 129 = 1 7283
10 000 000 000 = 10A = 1005  ? (twelve to the tenth power) 61 917 364 224 = 1210 = 1445
100 000 000 000 = 10B  ? (twelve to the eleventh power) 743 008 370 688 = 1211
1 000 000 000 000 = 1010 = 10^^2 = 1006 = 1 0004 = 10 0003 = 1 000 0002  ? (twelve to the twelfth power = twelve tetrated to the second hyperpower) 8 916 100 448 256 = 1212 = 12^^2 = 1446 = 1 7284 = 20 7363 = 2 985 9842
Powers of ten in duodecimal English duodecimal name Decimal equivalent
A = A1 ten 10 = 101
84 = A2 eight dozen and four 100 = 102
6B4 = A3 six gross eleven dozen and four 1 000 = 103
5 954 = A4 = 842 five great gross nine gross five dozen and four 10 000 = 104 = 1002
49 A54 = A5 four dozen and nine great gross, ten gross five dozen and four 100 000 = 105
402 854 = A6 = 843 = 6B42 four gross and two great gross, eight gross five dozen and four 1 000 000 = 106 = 1003 = 1 0002
3 423 054 = A7  ? (ten to the seventh power) 10 000 000 = 107
29 5A6 454 = A8 = 844 = 59542  ? (ten to the eighth power) 100 000 000 = 108 = 1004 = 10 0002
23A A93 854 = A9 = 6B43  ? (ten to the ninth power) 1 000 000 000 = 109 = 1 0003
1 B30 B91 054 = AA = A^^2 = 845  ? (ten to the tenth power = ten tetrated to the second hyperpower) 10 000 000 000 = 1010 = 10^^2 = 1005
17 469 96A 454 = AB  ? (ten to the eleventh power) 100 000 000 000 = 1011
141 981 B87 854 = A10 = 846 = 6B44 = 5 9543 = 402 8542  ? (ten to the twelfth power) 1 000 000 000 000 = 1012 = 1006 = 1 0004 = 10 0003 = 1 000 0002
Powers of two in duodecimal English duodecimal name Decimal equivalent
2 = 21 two 2 = 21
4 = 22 = 2^^2 four 4 = 22 = 2^^2
8 = 23 eight 8 = 23
14 = 24 = 2^^3 = 42 one dozen and four 16 = 24 = 2^^3 = 42
28 = 25 two dozen and eight 32 = 25
54 = 26 = 43 = 82 five dozen and four 64 = 26 = 43 = 82
A8 = 27 ten dozen and eight = one small gross and eight 128 = 27
194 = 28 = 44 = 4^^2 = 142 one gross nine dozen and four 256 = 28 = 44 = 4^^2 = 162
368 = 29 = 83 three gross six dozen and eight 512 = 29 = 83
714 = 2A = 45 seven gross one dozen and four 1 024 = 210 = 45
1 228 = 2B one great gross two gross two dozen and eight 2 048 = 211
2 454 = 210 = 46 = 84 = 143 = 542 two great gross four gross five dozen and four 4 096 = 212 = 46 = 84 = 163 = 642

[edit] Fractions and irrational numbers

[edit] Fractions

Duodecimal fractions may be simple:

  • 1/2 = 0.6
  • 1/3 = 0.4
  • 1/4 = 0.3
  • 1/6 = 0.2
  • 1/8 = 0.16
  • 1/9 = 0.14

or complicated

  • 1/5 = 0.24972497... recurring (easily rounded to 0.25)
  • 1/7 = 0.186A35186A35... recurring (easily rounded to 0.187)
  • 1/A = 0.124972497... recurring (rounded to 0.125)
  • 1/B = 0.11111... recurring (rounded to 0.11)
  • 1/11 = 0.0B0B... recurring (rounded to 0.0B)
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 “decimal” 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: ¹⁄8 = ¹⁄(2×2×2), ¹⁄20 = ¹⁄(2×2×5), and ¹⁄500 = ¹⁄(2×2×5×5×5) can be expressed exactly as 0.125, 0.05, and 0.002 respectively. ¹⁄3 and ¹⁄7, however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, ¹⁄8 is exact; ¹⁄20 and ¹⁄500 recur because they include 5 as a factor; ¹⁄3 is exact; and ¹⁄7 recurs, just as it does in decimal.

[edit] Recurring decimal

Arguably, factors of 3 are more commonly encountered in real-life division problems than factors of 5 (or would be, were it not for the decimal system having influenced most cultures). Thus, in practical applications, the nuisance of recurring 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 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, while 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, while 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 dozenal than in decimal (e.g., 1/(22) = 0.25 dec = 0.3 doz; 1/(23) = 0.125 dec = 0.16 doz; 1/(24) = 0.0625 dec = 0.09 doz; 1/(25) = 0.03125 dec = 0.046 doz; etc.).

Decimal base
Prime factors of the base: 2, 5		 Duodecimal / Dozenal base
Prime factors of the base: 2, 3
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 0.3333... = 0.3 0.4 3 1/3
1/4 2 0.25 0.3 2 1/4
1/5 5 0.2 0.24972497... = 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 0.05915343A0B6 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

As for irrational numbers, none of them has 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 × 103/10 + 2 × 102/10 + 3 × 10/10 + 4 × 1/10 + 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 doesn't exhibit a pattern of recursion; 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 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.

Irrational number In decimal In duodecimal / dozenal
π (pi, the ratio of circumference to diameter) 3.141592653589793238462643... (~ 3.1416) 3.184809493B918664573A6211... (~ 3.1848)
e (the base of the natural logarithm) 2.718281828459... (~ 2.718) 2.875236069821... (~ 2.875)
φ (phi, the golden ratio) 1.618033988749... (~ 1.618) 1.74BB67728022... (~ 1.75)
√2 (the length of the diagonal of a unit square) 1.414213562373... (~ 1.414) 1.4B79170A07B7... (~ 1.5)
√3 (the length of the diagonal of a unit cube, or twice the height of an equilateral triangle) 1.732050807568... (~ 1.732) 1.894B97BB967B... (~ 1.895)
√5 (the length of the diagonal of a 1×2 rectangle) 2.236067977499... (~ 2.236) 2.29BB13254051... (~ 2.2A)

The first few digits of the decimal and dozenal 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 / dozenal
γ (the limiting difference between the harmonic series and the natural logarithm) 0.577215664901... (~ 0.577) 0.6B15188A6758... (~ 0.7)

[edit] 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, image:Scriptx.png and image:Scripte.png, 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 image:Scriptx.png for its resemblance to the Roman numeral X, and image:Scripte.png as the first letter of the word "eleven". Another popular notation, introduced by Isaac Pitman, is to use an inverted 2 to represent ten and an inverted 3 to represent eleven; this is the system commonly employed by the Dozenal Society of Great Britain and has the advantage of being easily recognizable as digits. On the other hand, the Dozenal Society of America adopted for some years the convention of using an asterisk * for ten and a hash # for eleven (the reason was they are present in telephone dials); however, critics pointed out these symbols do not look anything like digits.

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.

[edit] See also

[edit] External links

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