Duodecimal

A duodecimal multiplication table

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

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

The number 12 has six factors, which are 1, 2, 3, 4, 6, and 12. It is a more convenient number system for computing fractions compared with the decimal or vigesimal systems. The decimal system has only four factors, which are 1, 2, 5, and 10.

Origin

In this section, numerals are based on decimal places. For example, 10 means ten, 12 means twelve.

Languages based on the duodecimal system are uncommon. Languages in the Nigerian Middle Belt such as Janji, Kahugu, the Nimbia dialect of Gwandara, Mahl language of Minicoy 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). 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 a 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 the English word ounce. 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.

Places

In a duodecimal place system, ten is written as A, eleven is written as B, twelve is written as 10.

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 in English we say 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)	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

Fractions and irrational numbers

Fractions

Duodecimal fractions are usually 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.

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 notinfluence 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 thedecimal and duodecimal representation of several of the most important irrational numbers. As can be seen, it is easier to memorize the first nine digits of pi in base twelve than in base ten, while the opposite happens with the first ten digits of the number e:

Irrational number	In decimal	In duodecimal / dozenal
' (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)

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. In contrary to 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,

Scriptx.png

and

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

Scriptx.png

for its resemblance to the Roman numeral X, and

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.

External links

*Decimal vs. Duodecimal: An interaction between two systems of numeration — duodecimal numerals in languages in Nigerian Middle Belt
*The origin of a duodecimal system (Japanese) — explains a possible origin of a duodecimal system in a language
*Dozenal Society of America
*Dozenal Society of Great Britain website