Information about 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.

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.

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.

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 + 0.583333333333... + 0.055555555555...

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 = 0.583333333333... 0.08 = 0.055555555555...

---

123,456.78 = 296,130.638888888888...

That is, (dozenal) 123,456.78 equals (decimal) 296,130.638888888888... ≈ 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 + 0.849724972497249724972497... + 0.0B62A68781B05915343A0B62...

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 = 0.849724972497249724972497... 0.08 = 0.0B62A68781B05915343A0B62...

---

123,456.78 = 5B,540.943A0B62A68781B05915343A...

That is, (decimal) 123,456.78 equals (dozenal) 5B,540.943A0B62A68781B05915343A... ≈ 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	0.083	0.01	0.00694
200,000	497,664	20,000	41,472	2,000	3,456	200	288	20	24	2	2	0.2	0.16	0.02	0.0138
300,000	746,496	30,000	62,208	3,000	5,184	300	432	30	36	3	3	0.3	0.25	0.03	0.02083
400,000	995,328	40,000	82,944	4,000	6,912	400	576	40	48	4	4	0.4	0.3	0.04	0.027
500,000	1,244,160	50,000	103,680	5,000	8,640	500	720	50	60	5	5	0.5	0.416	0.05	0.03472
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	0.0416
700,000	1,741,824	70,000	145,152	7,000	12,096	700	1008	70	84	7	7	0.7	0.583	0.07	0.04861
800,000	1,990,656	80,000	165,888	8,000	13,824	800	1152	80	96	8	8	0.8	0.6	0.08	0.05
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	0.83	0.0A	0.0694
B00,000	2,737,152	B0,000	228,096	B,000	19,008	B00	1,584	B0	132	B	11	0.B	0.916	0.0B	0.07638

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	0.12497	0.01	0.015343A0B62A68781B059
200,000	97,8A8	20,000	B,6A8	2,000	1,1A8	200	148	20	18	2	2	0.2	0.2497	0.02	0.02A68781B05915343A0B6
300,000	125,740	30,000	15,440	3,000	1,8A0	300	210	30	26	3	3	0.3	0.37249	0.03	0.043A0B62A68781B059153
400,000	173,594	40,000	1B,194	4,000	2,394	400	294	40	34	4	4	0.4	0.4972	0.04	0.05915343A0B62A68781B0
500,000	201,428	50,000	24,B28	5,000	2,A88	500	358	50	42	5	5	0.5	0.6	0.05	0.07249
600,000	24B,280	60,000	2A,880	6,000	3,580	600	420	60	50	6	6	0.6	0.7249	0.06	0.08781B05915343A0B62A6
700,000	299,114	70,000	34,614	7,000	4,074	700	4A4	70	5A	7	7	0.7	0.84972	0.07	0.0A0B62A68781B05915343
800,000	326,B68	80,000	3A,368	8,000	4,768	800	568	80	68	8	8	0.8	0.9724	0.08	0.0B62A68781B05915343A
900,000	374,A00	90,000	44,100	9,000	5,260	900	630	90	76	9	9	0.9	0.A9724	0.09	0.10B62A68781B05915343A

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	0.3	0.4	0.25	0.3	0.2	0.2497	0.16	0.2	0.142857	0.186A35
^−2	0.25	0.3	0.1	0.14	0.0625	0.09	0.04	0.05915343A0 B62A68781B	0.027	0.04	0.0204081632653 06122448979591 836734693877551	0.02B322547A05A 644A9380B908996 741B615771283B

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	0.1	0.14	0.1	0.12497	0.09	0.1	0.083	0.1
^−2	0.015625	0.023	0.012345679	0.0194	0.01	0.015343A0B6 2A68781B059	0.00826446280 99173553719	0.0123456789B	0.00694	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) can be rounded to 1.5 in duodecimal. This number is more precise than rounding to 1.41 in the decimal system.

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... = 0.3	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

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 × 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 algebraic and trascendental 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+√5)⁄2)	1.6180339887498... (≈ 1.618)	1.74BB6772802A4... (≈ 1.75)
Trascendental 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, 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 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.

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 25th, 1962 [1]

	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) [2]

See also

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

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