Definitions

Duodecimal

Duodecimal

[doo-uh-des-uh-muhl, dyoo-]
The duodecimal system (also known as base-12 or dozenal) is a numeral system using twelve as its base. In this system, the number ten may be written as 'A', and the number eleven as 'B' (another common notation, introduced by Sir Isaac Pitman, is to use a rotated '2' for ten and a reversed '3' 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 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. 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 (, , and ), 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 (since 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 unwieldily large multiplication table).

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 (Kahugu), 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 used duodecimal.

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. However, they are considered to come from Proto-Germanic *ainlif and *twalif (respectively one left and two left), both of which were decimal. Admittedly, the survival of such apparently unique terms may be connected with duodecimal tendencies, but their origin is not duodecimal.

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

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 mixed duodecimal-vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the pound sterling), and Charlemagne established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places.

Places

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

For alternative symbols, see the section "Advocacy and 'dozenalism'" 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 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
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) 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. 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 and 20 to this but no new prime factors.

Conversion tables to and from decimal

To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under radix). Alternatively, one can use digit-conversion tables. The ones provided below can be used to convert any dozenal number between 0.01 and BBB,BBB.BB to decimal, or any decimal number between 0.01 and 999,999.99 to dozenal. To use them, we first decompose the given number 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 we use the digit conversion tables to obtain the equivalent value in the target base for each digit. If the given number is in dozenal and the target base is decimal, we get:

(dozenal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (decimal) 248,832 + 41,472 + 5,184 + 576 + 60 + 6 + 0.58333333333... + 0.05555555555...

Now, since the summands are already converted to base ten, we use the usual decimal arithmetic to perform the addition and recompose the number, arriving at the conversion result:

  Dozenal  ----->  Decimal


  100,000     =    248,832
   20,000     =     41,472
    3,000     =      5,184
      400     =        576
       50     =         60
 +      6     =   +      6
0.7 = 0.58333333333... 0.08 = 0.05555555555...
--------------------------------------------
123,456.78 = 296,130.63888888888...

That is, (dozenal) 123,456.78 equals (decimal) 296,130.63888888888... ≈ 296,130.64

If the given number is in decimal and the target base is dozenal, the method is basically same. Using the digit conversion tables:

(decimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (dozenal) 49,A54 + B,6A8 + 1,8A0 + 294 + 42 + 6 + 0.84972497249724972497... + 0.0B62...

However, in order to do this sum and recompose the number, we now have to use the addition tables for dozenal, 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 dozenal as well. In decimal, 6 + 6 equals 12, but in dozenal it equals 10; so if we used decimal arithmetic with dozenal numbers we would arrive at an incorrect result. Doing the arithmetic properly in dozenal, we get the result:

  Decimal  ----->  Dozenal


  100,000     =     49,A54
   20,000     =      B,6A8
    3,000     =      1,8A0
      400     =        294
       50     =         42
 +      6     =   +      6
0.7 = 0.84972497249724972497... 0.08 = 0.0B62...
--------------------------------------------------------
123,456.78 = 5B,540.943A...

That is, (decimal) 123,456.78 equals (dozenal) 5B,540.943A... ≈ 5B,540.94

Dozenal to Decimal digit conversion

Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec.
100,000 248,832 10,000 20,736 1,000 1,728 100 144 10 12 1 1 0.1 0.08 0.01 0.0069
200,000 497,664 20,000 41,472 2,000 3,456 200 288 20 24 2 2 0.2 0.1 0.02 0.013
300,000 746,496 30,000 62,208 3,000 5,184 300 432 30 36 3 3 0.3 0.25 0.03 0.0208
400,000 995,328 40,000 82,944 4,000 6,912 400 576 40 48 4 4 0.4 0. 0.04 0.02
500,000 1,244,160 50,000 103,680 5,000 8,640 500 720 50 60 5 5 0.5 0.41 0.05 0.0347
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 0.041
700,000 1,741,824 70,000 145,152 7,000 12,096 700 1008 70 84 7 7 0.7 0.58 0.07 0.0486
800,000 1,990,656 80,000 165,888 8,000 13,824 800 1152 80 96 8 8 0.8 0. 0.08 0.0
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
A00,000 2,488,320 A0,000 207,360 A,000 17,280 A00 1,440 A0 120 A 10 0.A 0.8 0.0A 0.069
B00,000 2,737,152 B0,000 228,096 B,000 19,008 B00 1,584 B0 132 B 11 0.B 0.91 0.0B 0.0763

Decimal to Dozenal digit conversion

Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz.
100,000 49,A54 10,000 5,954 1,000 6B4 100 84 10 A 1 1 0.1 0.1 0.01 0.0
200,000 97,8A8 20,000 B,6A8 2,000 1,1A8 200 148 20 18 2 2 0.2 0. 0.02 0.0
300,000 125,740 30,000 15,440 3,000 1,8A0 300 210 30 26 3 3 0.3 0.3 0.03 0.0
400,000 173,594 40,000 1B,194 4,000 2,394 400 294 40 34 4 4 0.4 0. 0.04 0.0
500,000 201,428 50,000 24,B28 5,000 2,A88 500 358 50 42 5 5 0.5 0.6 0.05 0.0
600,000 24B,280 60,000 2A,880 6,000 3,580 600 420 60 50 6 6 0.6 0. 0.06 0.0
700,000 299,114 70,000 34,614 7,000 4,074 700 4A4 70 5A 7 7 0.7 0.8 0.07 0.0
800,000 326,B68 80,000 3A,368 8,000 4,768 800 568 80 68 8 8 0.8 0. 0.08 0.
900,000 374,A00 90,000 44,100 9,000 5,260 900 630 90 76 9 9 0.9 0.A 0.09 0.1

Conversion of powers

Exponent Powers of 2 Powers of 3 Powers of 4 Powers of 5 Powers of 6 Powers of 7
Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz.
^6 64 54 729 509 4,096 2454 15,625 9,061 46,656 23,000 117,649 58,101
^5 32 28 243 183 1,024 714 3,125 1,985 7,776 4,600 16,807 9,887
^4 16 14 81 69 256 194 625 441 1,296 900 2,401 1,481
^3 8 8 27 23 64 54 125 A5 216 160 343 247
^2 4 4 9 9 16 14 25 21 36 30 49 41
^1 2 2 3 3 4 4 5 5 6 6 7 7
^−1 0.5 0.6 0. 0.4 0.25 0.3 0.2 0. 0.1 0.2 0. 0.
^−2 0.25 0.3 0. 0.14 0.0625 0.09 0.04 0. 0.02 0.04 0. 0.

Exponent Powers of 8 Powers of 9 Powers of 10 Powers of 11 Powers of 12
Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz.
^6 262,144 107,854 531,441 217,669 1,000,000 402,854 1,771,561 715,261 2,985,984 1,000,000
^5 32,768 16,B68 59,049 2A,209 100,000 49,A54 161,051 79,24B 248,832 100,000
^4 4,096 2,454 6,561 3,969 10,000 5,954 14,641 8,581 20,736 10,000
^3 512 368 729 509 1,000 6B4 1,331 92B 1,728 1,000
^2 64 54 81 69 100 84 121 A1 144 100
^1 8 8 9 9 10 A 11 B 12 10
^−1 0.125 0.16 0. 0.14 0.1 0.1 0. 0. 0.08 0.1
^−2 0.015625 0.023 0. 0.0194 0.01 0.0 0. 0. 0.0069 0.01

Fractions and irrational numbers

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.

Because each place is more precise in the duodecimal system, "decimals" can be written with greater accuracy. For example, the square root of 2 (1.4142135... in decimal, 1.4B79170A07B86... in duodecimal) can be rounded to 1.5 in duodecimal. This number is more precise than rounding to 1.41 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, 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. 0.4 3 1/3
1/4 2 0.25 0.3 2 1/4
1/5 5 0.2 0.24972497... = 0. 5 1/5
1/6 2, 3 0.1 0.2 2, 3 1/6
1/7 7 0. 0. 7 1/7
1/8 2 0.125 0.16 2 1/8
1/9 3 0. 0.14 3 1/9
1/10 2, 5 0.1 0.1 2, 5 1/A
1/11 11 0. 0. B 1/B
1/12 2, 3 0.08 0.1 2, 3 1/10
1/13 13 0. 0. 11 1/11
1/14 2, 7 0.0 0.0 2, 7 1/12
1/15 3, 5 0.0 0.0 3, 5 1/13
1/16 2 0.0625 0.09 2 1/14
1/17 17 0. 0. 15 1/15
1/18 2, 3 0.0 0.08 2, 3 1/16
1/19 19 0. 0. 17 1/17
1/20 2, 5 0.05 0.0 2, 5 1/18
1/21 3, 7 0. 0.0 3, 7 1/19
1/22 2, 11 0.0 0.0 2, B 1/1A
1/23 23 0. 0. 1B 1/1B
1/24 2, 3 0.041 0.06 2, 3 1/20
1/25 5 0.04 0. 5 1/21
1/26 2, 13 0.0 0.0 2, 11 1/22
1/27 3 0. 0.054 3 1/23
1/28 2, 7 0.03 0.0 2, 7 1/24
1/29 29 0. 0. 25 1/25
1/30 2, 3, 5 0.0 0.0 2, 3, 5 1/26
1/31 31 0. 0. 27 1/27
1/32 2 0.03125 0.046 2 1/28
1/33 3, 11 0. 0.0 3, B 1/29
1/34 2, 17 0.0 0.0 2, 15 1/2A
1/35 5, 7 0.0 0. 5, 7 1/2B
1/36 2, 3 0.02 0.04 2, 3 1/30

Irrational numbers

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 × 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 doesn't 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... (≈ 1.414) 1.4B79170A07B857... (≈ 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) 1.894B97BB968704... (≈ 1.895)
√5 (the length of the diagonal of a 1×2 rectangle) 2.2360679774997... (≈ 2.236) 2.29BB132540589... (≈ 2.2A)
φ (phi, the golden ratio = ) 1.6180339887498... (≈ 1.618) 1.74BB6772802A4... (≈ 1.75)
Transcendental irrational number In decimal In duodecimal / dozenal
π (pi, the ratio of circumference to diameter) 3.1415926535897932384626433
8327950288419716939937510...
(≈ 3.1416)
3.184809493B918664573A6211B
B151551A05729290A7809A492...
(≈ 3.1848)
e (the base of the natural logarithm) 2.718281828459045... (≈ 2.718) 2.8752360698219B8... (≈ 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) 0.6B15188A6760B3... (~ 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! and , 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 as the first letter of the word "eleven".

Another popular notation, introduced by Sir Isaac Pitman, is to use a rotated 2 to represent ten and a rotated or horizontally flipped 3 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 the symbol * resembles a struck-through X while # 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 ɸ (a combination of 1 and 0) and eleven as a cross of two lines (+, x, or † for example).

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.

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:

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.)

See also

References

External links

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