Duodecimal Numbers

The duodecimal numbers system is a base twelve number system. Its basis, as in the case of the decimal system, is rational.

It is mainly used in Anglo-Saxon societies that use magnitudes of measurement on this basis, such as feet, inches, ounces, etc.

In base twelve, we use the ten digits from 0 to 9, and two variable symbols to represent ten and eleven. There are many situations where A represents ten and B represents eleven, but other symbols are also used.

Examples of Duodecimal Numbers

• 12₁₂ = 14₁₀ (actually, 1᛫12¹ + 2᛫12⁰)
• 26₁₂ = 30₁₀ (actually, 2᛫12¹ + 6᛫12⁰)
• 30₁₂ = 36₁₀ = 100₆ (actually, 3᛫12¹ + 0᛫12⁰)
• 50₁₂ = 60₁₀ (actually, 5᛫12¹ + 0᛫12⁰)
• 69₁₂ = 81₁₀ (actually, 6᛫12¹ + 9᛫12⁰)
• 76₁₂ = 90₁₀ (actually, 7᛫12¹ + 6᛫12⁰)
• 85₁₂ = 101₁₀ (actually, 8᛫12¹ + 5᛫12⁰)
• 100₁₂ = 144₁₀ (actually, 1᛫12² + 0᛫12¹ + 0᛫12⁰)
• 160₁₂ = 216₁₀ = 1000₆ (actually, 1᛫12² + 6᛫12¹ + 0᛫12⁰)
• 1A6₁₂ = 270₁₀ (actually, 1᛫12² + 10᛫12¹ + 6᛫12⁰)
• 265₁₂ = 365₁₀ (actually, 2᛫12² + 6᛫12¹ + 5᛫12⁰)
• 294₁₂ = 400₁₀ = 100₂₀ (actually, 2᛫12² + 9᛫12¹ + 4᛫12⁰)
• 400₁₂ = 576₁₀ (actually, 4᛫12² + 0᛫12¹ + 0᛫12⁰)
• 576₁₂ = 810₁₀ (actually, 5᛫12² + 7᛫12¹ + 6᛫12⁰)
• 6B4₁₂ = 1000₁₀ (actually, 6᛫12² + 11᛫12¹ + 4᛫12⁰)
• 900₁₂ = 1296₁₀ = 10000₆ (actually, 9᛫12² + 0᛫12¹ + 0᛫12⁰)
• 1000₁₂ = 1728₁₀ (actually, 1᛫12³ + 0᛫12² + 0᛫12¹ + 0᛫12⁰)
• 11A8₁₂ = 2000₁₀ (en effet, 1᛫12³ + 1᛫12² + 10᛫12¹ + 8᛫12⁰)
• 2454₁₂ = 4096₁₀ = 1000₁₆ (actually, 2᛫12³ + 4᛫12² + 5᛫12¹ + 4᛫12⁰)
• 3969₁₂ = 6561₁₀ = 10000₉ (actually, 3᛫12³ + 9᛫12² + 6᛫12¹ + 9᛫12⁰)
• 4600₁₂ = 7776₁₀ = 100000₆ (en effet, 4᛫12³ + 6᛫12² + 0᛫12¹ + 0᛫12⁰)
• 4768₁₂ = 8000₁₀ = 1000₂₀ (actually, 4᛫12³ + 7᛫12² + 6᛫12¹ + 8᛫12⁰)
• 5000₁₂ = 8640₁₀ (actually, 5᛫12³ + 0᛫12² + 0᛫12¹ + 0᛫12⁰)
• 789A₁₂ = 13366₁₀ (actually, 7᛫12³ + 8᛫12² + 9᛫12¹ + 10᛫12⁰)
• 10000₁₂ = 20736₁₀ (actually, 1᛫12⁴ + 0᛫12³ + 0᛫12² + 0᛫12¹ + 0᛫12⁰)
• 23000₁₂ = 46656₁₀ = 1000000₆ (actually, 2᛫12⁴ + 3᛫12³ + 0᛫12² + 0᛫12¹ + 0᛫12⁰)

Duodecimal counting with the phalanges. Рыцарь поля, CC0, via Wikimedia Commons

Prime numbers

In any base, a prime number is that integer that cannot be divided by any other integer, except 1 or itself, and yield another integer. As the multiplications are different in the base 12 system than in the base 10 (decimal system), the prime numbers in both bases are different. Thus, in base 12 it turns out that prime numbers can only end in 1, 5, 7 or B (except the numbers 2 and 3).

The number of prime numbers is, as in the other infinite bases (including the decimal), infinite. The first prime numbers in the decimal and duodecimal bases are:

In duodecimal base	2	3	5	7	B	11	15	17	1B	25	27	31	35	37	3B	45	4B	51	57	5B	61	67	6B	75	81	85	87	8B	91	95	A7	AB	B5	B7	…
Dalam basis desimal	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97	101	103	107	109	113	127	131	137	139	…

The multiplication table in the duodecimal or base 12 system is as follows:

×	1	2	3	4	5	6	7	8	9	A	B	10
1	1	2	3	4	5	6	7	8	9	A	B	10
2	2	4	6	8	A	10	12	14	16	18	1A	20
3	3	6	9	10	13	16	19	20	23	26	29	30
4	4	8	10	14	18	20	24	28	30	34	38	40
5	5	A	13	18	21	26	2B	34	39	42	47	50
6	6	10	16	20	26	30	36	40	46	50	56	60
7	7	12	19	24	2B	36	41	48	53	5A	65	70
8	8	14	20	28	34	40	48	54	60	68	74	80
9	9	16	23	30	39	46	53	60	69	76	83	90
A	A	18	26	34	42	50	5A	68	76	84	92	A0
B	B	1A	29	38	47	56	65	74	83	92	A1	B0
10	10	20	30	40	50	60	70	80	90	A0	B0	100

Examples of arithmetic operations

Examples of arithmetic operations

Senary (Base 6)	Decimal (Base 10)	Duodecimal (Base 12)	Vigesimal (Base 20)
140 + 50 = 230	60 + 30 = 90	50 + 26 = 76	30 + 1A = 4A
3430 – 213 = 3213	810 – 81 = 729	576 – 69 = 509	20A – 41 = 1G9
13132 – 140 = 12552	2000 – 60 = 1940	11A8 – 50 = 1158	500 – 30 = 4H0
1130 ᛫ 52 = 104000	270 ᛫ 32 = 8640	1A6 ᛫ 28 = 5000	DA ᛫ 1C = 11C0
2400 / 13 = 144	576 / 9 = 64	400 / 9 = 54	18G / 9 = 34
3430 / 13 = 230	810 / 9 = 90	576 / 9 = 76	20A / 9 = 4A
220 = 30544	212 = 4096	210 = 2454	2C = A4G

Fractions and irrational numbers

In any rational base positional numeral system (such as decimal and duodecimal), all those irreducible fractions whose denominator contains prime factors other than those that factor the base, will lack finite representation, obtaining for them an infinite series of digits of fractional value (commonly called “decimals”, although it is absurd to use this term for bases other than decimal). Furthermore, this infinite series of digits will present a recurrence period, giving pure recurrence when there is no prime factor in common with the base, and mixed recurrence (the one in which there are fractional digits at the beginning that are not part of the period) when there are at least one prime factor in common with the base.

Examples in duodecimal and Decimal equivalent

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

So, in a duodecimal base, the representation of all those fractions whose denominator contains prime factors other than 2 and 3 is infinite and recursive; while in decimal base this occurs when they are different from 2 and 5:

Decimal base Prime factors of the base: 2 , 5			Duodecimal / dozen 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	two	0.5	0.6	two	1/2
1/3	3	0, 3 3333333…	0.4	3	1/3
1/4	two	0.25	0.3	two	1/4
1/5	5	0.2	0, 2497 24972497…	5	1/5
1/6	2 , 3	0.1 6 6666666…	0.2	2 , 3	1/6
1/7	7	0, 142857 142857142857…	0.186A35 186A35186A35 …	7	1/7
1/8	two	0.125	0.16	two	1/8
1/9	3	0, 1 1111111…	0.14	3	1/9
1/10	2 , 5	0.1	0.1 2497 24972497…	2 , 5	1/Y
11/1	eleven	0, 09 09090909…	0, 1 1111111…	B.	1 B
1/12	2 , 3	0.08 3 3333333…	0.1	2 , 3	1/10
1/13	13	0, 076923 076923076923…	0.0B 0B0B0B0B …	eleven	11/1
1/14	2 , 7	0.0 714285 714285714285…	0.0 A35186 A35186A35186…	2 , 7	1/12
1/15	3 , 5	0.0 6 6666666…	0.0 9724 97249724…	3 , 5	1/13
1/16	two	0.0625	0.09	two	1/14
1/17	17	0, 0588235294117647 0588235294117647…	0, 08579214B36429A7 08579214B36429A7…	fifteen	1/15
1/18	2 , 3	0.0 5 5555555…	0.08	2 , 3	1/16
1/19	19	0, 052631578947368421 052631578947368421…	0.076B45 076B45076B45 …	17	1/17
1/20	2 , 5	0.05	0.0 7249 72497249…	2 , 5	1/18
1/21	3 , 7	0, 047619 047619047619…	0.0 6A3518 6A35186A3518…	3 , 7	1/19
1/22	2 , 11	0.0 45 45454545…	0.0 6 6666666…	2 , B	1/1A
1/23	23	0, 0434782608695652173913 043478260869565…	0, 06316948421 06316948421…	1 B	1/1B
1/24	2 , 3	0.041 6 6666666…	0.06	2 , 3	1/20
1/25	5	0.04	0, 05915343A0B6 05915343A0B6…	5	1/21
1/26	2 , 13	0.0 384615 384615384615…	0.0 56 565656565…	2 , 11	1/22
1/27	3	0, 037 037037037…	0.054	3	1/23
1/28	2 , 7	0.03 571428 571428571428…	0.0 5186A3 5186A35186A3…	2 , 7	1/24
1/29	29	0, 0344827586206896551724137931 0344827586…	0.04B7 04B704B7 …	25	1/25
1/30	2 , 3 , 5	0.0 3 3333333…	0.0 4972 49724972…	2 , 3 , 5	1/26
1/31	31	0, 032258064516129 032258064516129…	0.0478AA093598166B74311B28623A55 0478AA …	27	1/27
1/32	two	0.03125	0.046	two	1/28
1/33	3 , 11	0, 03 03030303…	0.0 4 4444444…	3 , B	1/29
1/34	2 , 17	0.0 2941176470588235 2941176470588235…	0.0 429A708579214B36 429A708579214B36…	2 , 15	1/2A
1/35	5 , 7	0.0 285714 285714285714…	0.0414559B3931 0414559B3931 …	5 , 7	1/2B
1/36	2 , 3	0.02 7 7777777…	0.04	2 , 3	1/30

On the other hand, in any positional numeral system with a rational base, every irrational number not only lacks a finite representation, but also its infinite series of digits lacks a recurrence period. The first few digits of the base duodecimal representation of several of the most important irrational numbers are given below:

irrational number	in decimal base	in duodecimal base
π (pi, the ratio of circumference to diameter )	3.141592653589793238462643… (~ 3.1416)	3.184809493B918664573A6211… (~ 3.1848)
e (the base of the natural or natural logarithm )	2.718281828459… (~ 2.718)	2.875236069821… (~ 2.875)
φ (fi, the golden number or 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…(~1.4B8)
√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)

Irrational numbers

The representations of irrational numbers in any positional number system (including decimal and duodecimal) neither terminate nor repeat. The following table gives the first digits for some important algebraic and transcendental numbers in both decimal and duodecimal.

Algebraic irrational number	In decimal	In duodecimal
√2, the square root of 2	1.414213562373…	1;4B79170A07B8…
φ (phi), the golden ratio = {\displaystyle {\tfrac {1+{\sqrt {5}}}{2}}}	1.618033988749…	1;74BB6772802A…
Transcendental number	In decimal	In duodecimal
π (pi), the ratio of a circle’s circumference to its diameter	3.141592653589…	3;184809493B91…
e, the base of the natural logarithm	2.718281828459…	2;875236069821…

The first digits in duodecimal base of another notable number, the Euler-Mascheroni constant , but for which at the moment it is unknown if it is rational or irrational:

Number	in decimal base	in duodecimal base
γ (the limiting difference between the harmonic series and the natural logarithm)	0.577215664901… (~ 0.577)	0.6B15188A6758… (~0.7)

Sources: PinterPandai, MathIsFun