Duodecimal

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The number twelve, a 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, with neither 3 nor 4), 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 (Template:Frac, Template:Frac, Template:Frac, Template:Frac and Template:Frac) 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 sexagesimal system (where the reciprocals of all 5-smooth numbers terminate) does better in this respect (but at the cost of an unwieldy multiplication table and a much larger number of symbols to memorize).

Origin

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, 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 use a hybrid decimal–duodecimal system, primarily decimal but with special names for multiples of six.

Germanic languages have special words for 11 and 12, such as eleven and twelve in English, which are often misinterpreted as vestiges of a duodecimal system.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. 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 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.

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 your 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.

Places

In a duodecimal place system, ten can be written as ᘔ, eleven can be written as Ɛ, 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

A dozenal multiplication table

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.Benefits of Residing in a Apartment or Landed property in Singapore Property New Launches & Project Showcase In Singapore Many residential Singapore property sales involve buying property in Singapore at new launches. These are often homes underneath building, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, based on personal property transactions in May Cell Apps FREE Sign Up Log in Property Brokers Feedback

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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 + 0.58Template:Overline333333333... + 0.0Template:Overline5555555555...

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   =          0.58Template:Overline333333333...
        0.08  =          0.0Template:Overline5555555555...
--------------------------------------------
  123,456.78  =    296,130.63Template:Overline888888888...

That is, (duodecimal) 123,456.78 equals (decimal) 296,130.63Template:Overline ≈ 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 + 0.8Template:Overline4972497249724972497... + 0.Template:Overline0Ɛ62...

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   =          0.8Template:Overline4972497249724972497...
        0.08  =          0.Template:Overline0Ɛ62...
--------------------------------------------------------
  123,456.78  =     5Ɛ,540.9Template:Overline43ᘔ...

That is, (decimal) 123,456.78 equals (duodecimal) 5Ɛ,540.9Template:Overline... ≈ 5Ɛ,540.94

Duodecimal to decimal digit conversion

Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.
100,000	248,832	10,000	20,736	1,000	1,728	100	144	10	12	1	1	0.1	0.08Template:Overline	0.01	0.0069Template:Overline
200,000	497,664	20,000	41,472	2,000	3,456	200	288	20	24	2	2	0.2	0.1Template:Overline	0.02	0.013Template:Overline
300,000	746,496	30,000	62,208	3,000	5,184	300	432	30	36	3	3	0.3	0.25	0.03	0.0208Template:Overline
400,000	995,328	40,000	82,944	4,000	6,912	400	576	40	48	4	4	0.4	0.Template:Overline	0.04	0.02Template:Overline
500,000	1,244,160	50,000	103,680	5,000	8,640	500	720	50	60	5	5	0.5	0.41Template:Overline	0.05	0.0347Template:Overline
600,000	1,492,992	60,000	124,416	6,000	10,368	600	864	60	72	6	6	0.6	0.5	0.06	0.041Template:Overline
700,000	1,741,824	70,000	145,152	7,000	12,096	700	1008	70	84	7	7	0.7	0.58Template:Overline	0.07	0.0486Template:Overline
800,000	1,990,656	80,000	165,888	8,000	13,824	800	1152	80	96	8	8	0.8	0.Template:Overline	0.08	0.0Template:Overline
900,000	2,239,488	90,000	186,624	9,000	15,552	900	1,296	90	108	9	9	0.9	0.75	0.09	0.0625
ᘔ00,000	2,488,320	ᘔ0,000	207,360	ᘔ,000	17,280	ᘔ00	1,440	ᘔ0	120	ᘔ	10	0.ᘔ	0.8Template:Overline	0.0ᘔ	0.069Template:Overline
Ɛ00,000	2,737,152	Ɛ0,000	228,096	Ɛ,000	19,008	Ɛ00	1,584	Ɛ0	132	Ɛ	11	0.Ɛ	0.91Template:Overline	0.0Ɛ	0.0763Template:Overline

Decimal to duodecimal digit conversion

Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.
100,000	49,ᘔ54	10,000	5,954	1,000	6Ɛ4	100	84	10	ᘔ	1	1	0.1	0.1Template:Overline	0.01	0.0Template:Overline
200,000	97,8ᘔ8	20,000	Ɛ,6ᘔ8	2,000	1,1ᘔ8	200	148	20	18	2	2	0.2	0.Template:Overline	0.02	0.0Template:Overline
300,000	125,740	30,000	15,440	3,000	1,8ᘔ0	300	210	30	26	3	3	0.3	0.3Template:Overline	0.03	0.0Template:Overline
400,000	173,594	40,000	1Ɛ,194	4,000	2,394	400	294	40	34	4	4	0.4	0.Template:Overline	0.04	0.0Template:Overline
500,000	201,428	50,000	24,Ɛ28	5,000	2,ᘔ88	500	358	50	42	5	5	0.5	0.6	0.05	0.0Template:Overline
600,000	24Ɛ,280	60,000	2ᘔ,880	6,000	3,580	600	420	60	50	6	6	0.6	0.Template:Overline	0.06	0.0Template:Overline
700,000	299,114	70,000	34,614	7,000	4,074	700	4ᘔ4	70	5ᘔ	7	7	0.7	0.8Template:Overline	0.07	0.0Template:Overline
800,000	326,Ɛ68	80,000	3ᘔ,368	8,000	4,768	800	568	80	68	8	8	0.8	0.Template:Overline	0.08	0.Template:Overline
900,000	374,ᘔ00	90,000	44,100	9,000	5,260	900	630	90	76	9	9	0.9	0.ᘔTemplate:Overline	0.09	0.1Template:Overline

Conversion of powers

Exponent	b=2		b=3		b=4		b=5		b=6		b=7
	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.
b6	64	54	729	509	4,096	2454	15,625	9,061	46,656	23,000	117,649	58,101
b5	32	28	243	183	1,024	714	3,125	1,985	7,776	4,600	16,807	9,887
b4	16	14	81	69	256	194	625	441	1,296	900	2,401	1,481
b3	8	8	27	23	64	54	125	ᘔ5	216	160	343	247
b2	4	4	9	9	16	14	25	21	36	30	49	41
b1	2	2	3	3	4	4	5	5	6	6	7	7
b−1	0.5	0.6	0.Template:Overline	0.4	0.25	0.3	0.2	0.Template:Overline	0.1Template:Overline	0.2	0.Template:Overline	0.Template:Overline
b−2	0.25	0.3	0.Template:Overline	0.14	0.0625	0.09	0.04	0.Template:Overline	0.02Template:Overline	0.04	0.Template:Overline	0.Template:Overline

Exponent	b=8		b=9		b=10		b=11		b=12
	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.
b6	262,144	107,854	531,441	217,669	1,000,000	402,854	1,771,561	715,261	2,985,984	1,000,000
b5	32,768	16,Ɛ68	59,049	2ᘔ,209	100,000	49,ᘔ54	161,051	79,24Ɛ	248,832	100,000
b4	4,096	2,454	6,561	3,969	10,000	5,954	14,641	8,581	20,736	10,000
b3	512	368	729	509	1,000	6Ɛ4	1,331	92Ɛ	1,728	1,000
b2	64	54	81	69	100	84	121	ᘔ1	144	100
b1	8	8	9	9	10	ᘔ	11	Ɛ	12	10
b−1	0.125	0.16	0.Template:Overline	0.14	0.1	0.1Template:Overline	0.Template:Overline	0.Template:Overline	0.08Template:Overline	0.1
b−2	0.015625	0.023	0.Template:Overline	0.0194	0.01	0.0Template:Overline	0.Template:Overline	0.Template:Overline	0.0069Template:Overline	0.01

Fractions and irrational numbers

Fractions

Duodecimal fractions may be simple:

• Template:Frac = 0.6
• Template:Frac = 0.4
• Template:Frac = 0.3
• Template:Frac = 0.2
• Template:Frac = 0.16
• Template:Frac = 0.14

or complicated

• Template:Frac = 0.24972497... recurring (easily rounded to 0.25)
• Template:Frac = 0.186ᘔ35186ᘔ35... recurring (easily rounded to 0.187)
• Template:Frac = 0.124972497... recurring (rounded to 0.125)
• Template:Frac = 0.11111... recurring (rounded to 0.11)
• Template:Frac = 0.0Ɛ0Ɛ... recurring (rounded to 0.0Ɛ)

Examples in duodecimal	Decimal equivalent
1 × (Template:Frac) = 0.76	1 × (Template:Frac) = 0.625
100 × (Template:Frac) = 76	144 × (Template:Frac) = 90
Template:Frac = 76	Template:Frac = 90
Template:Frac = 54	Template:Frac = 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: Template:Frac = Template:Frac, Template:Frac = Template:Frac and Template:Frac = Template:Frac can be expressed exactly as 0.125, 0.05 and 0.002 respectively. Template:Frac and Template:Frac, however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, Template:Frac is exact; Template:Frac and Template:Frac recur because they include 5 as a factor; Template:Frac is exact; and Template:Frac recurs, just as it does in decimal.

Recurring digits

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, 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 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.).

Values in bold indicate that value is exact.

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			Duodecimal / Dozenal base Prime factors of the base: 2, 3 Prime factors of one below the base: Ɛ Prime factors of one above the base: 11
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.Template:Overline	0.4	3	1/3
1/4	2	0.25	0.3	2	1/4
1/5	5	0.2	0.24972497... = 0.Template:Overline	5	1/5
1/6	2, 3	0.1Template:Overline	0.2	2, 3	1/6
1/7	7	0.Template:Overline	0.Template:Overline	7	1/7
1/8	2	0.125	0.16	2	1/8
1/9	3	0.Template:Overline	0.14	3	1/9
1/10	2, 5	0.1	0.1Template:Overline	2, 5	1/ᘔ
1/11	11	0.Template:Overline	0.Template:Overline	Ɛ	1/Ɛ
1/12	2, 3	0.08Template:Overline	0.1	2, 3	1/10
1/13	13	0.Template:Overline	0.Template:Overline	11	1/11
1/14	2, 7	0.0Template:Overline	0.0Template:Overline	2, 7	1/12
1/15	3, 5	0.0Template:Overline	0.0Template:Overline	3, 5	1/13
1/16	2	0.0625	0.09	2	1/14
1/17	17	0.Template:Overline	0.Template:Overline	15	1/15
1/18	2, 3	0.0Template:Overline	0.08	2, 3	1/16
1/19	19	0.Template:Overline	0.Template:Overline	17	1/17
1/20	2, 5	0.05	0.0Template:Overline	2, 5	1/18
1/21	3, 7	0.Template:Overline	0.0Template:Overline	3, 7	1/19
1/22	2, 11	0.0Template:Overline	0.0Template:Overline	2, Ɛ	1/1ᘔ
1/23	23	0.Template:Overline	0.Template:Overline	1Ɛ	1/1Ɛ
1/24	2, 3	0.041Template:Overline	0.06	2, 3	1/20
1/25	5	0.04	0.Template:Overline	5	1/21
1/26	2, 13	0.0Template:Overline	0.0Template:Overline	2, 11	1/22
1/27	3	0.Template:Overline	0.054	3	1/23
1/28	2, 7	0.03Template:Overline	0.0Template:Overline	2, 7	1/24
1/29	29	0.Template:Overline	0.Template:Overline	25	1/25
1/30	2, 3, 5	0.0Template:Overline	0.0Template:Overline	2, 3, 5	1/26
1/31	31	0.Template:Overline	0.Template:Overline	27	1/27
1/32	2	0.03125	0.046	2	1/28
1/33	3, 11	0.Template:Overline	0.0Template:Overline	3, Ɛ	1/29
1/34	2, 17	0.0Template:Overline	0.0Template:Overline	2, 15	1/2ᘔ
1/35	5, 7	0.0Template:Overline	0.Template:Overline	5, 7	1/2Ɛ
1/36	2, 3	0.02Template:Overline	0.04	2, 3	1/30

Irrational numbers

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 × 1/10⁻² + 2 × 1/10⁻¹ + 3 × 1/10⁰ + 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 / dozenal
√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 = $\scriptstyle {\frac {1+{\sqrt {5}}}{2}}$)	1.6180339887498... (≈ 1.618)	1.74ƐƐ6772802ᘔ4... (≈ 1.75)
Transcendental irrational number	In decimal	In duodecimal / dozenal
π (pi, the ratio of circumference to diameter)	3.1415926535897932384626433 8327950288419716939937510... (≈ 3.1416)	3.184809493Ɛ918664573ᘔ6211Ɛ Ɛ151551ᘔ05729290ᘔ7809ᘔ492... (≈ 3.1848)
e (the base of the natural logarithm)	2.718281828459045... (≈ 2.718)	2.8752360698219Ɛ8... (≈ 2.875)

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)	0.57721566490153... (~ 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, $x\!$ (U+1D4B3) and File:Scripte.png (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 $x\!$ for its resemblance to the Roman numeral X, and File:Scripte.png 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) (which again resembles ε) 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 an asterisk * 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 can not be represented in the seven-segment display of most calculator displays (File:Scripte.png being an exception, although "E" is used on calculators to indicate an error message). 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.

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: 31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape.

31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape.

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 (one of the widely popular New Age channeled entities) 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.

Dozenal clock

• Dozenal Clock by Joshua Harkey
• Dozenal Clock with four hands in several variants by Paul Rapoport
• Dozenal Clock by Bill Hall

Dozenal metric systems

Systems of measurement proposed by dozenalists include:

• Tom Pendlebury's TGM system
• Takashi Suga's Universal Unit System

Duodecimal digits on computerized writing systems

Template:Symb In March 2013, a proposal was submitted to include the digits for ten and eleven propagated by the Dozenal Societies of Great Britain and America in Unicode. In June 2013, this was partially accepted, advising for the British digits the provisional code points U+218A 28 year-old Aircraft Maintenance Manufacture (Avionics) Cameron Lester from Port Coquitlam, usually spends time with hobbies which include mountain biking, property developers in singapore and train collecting. Loves to discover unknown towns and spots like Athens.

Here is my web page; test.wbxonline.com and U+218B 28 year-old Aircraft Maintenance Manufacture (Avionics) Cameron Lester from Port Coquitlam, usually spends time with hobbies which include mountain biking, property developers in singapore and train collecting. Loves to discover unknown towns and spots like Athens.

Here is my web page; test.wbxonline.com. By this, the actual availability as Unicode characters can be expected for 2016 (1200_doz.).

Also, the turned digits two and three are available in LaTeX as \textturntwo and \textturnthree.

See also

• Senary (base 6)
• Quadrovigesimal (base 24)
• Sexatrigesimal (base 36)
• Sexagesimal (base 60)
• Babylonian numerals

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• Dozenal Society of America
• Dozenal Society of Great Britain website
• Template:Cite web