Duodecimal

The duodecimal system, also known as base dozenal, is a positional numeral system using a dozen as its base. In duodecimal, the the quantity of one dozen is denoted "10", meaning one dozen and 0 units; in the decimal system, this number is instead written as "12" meaning one ten and two units, and the string "10" means ten_d. In duodecimal, "100" means a dozen squared, "1000" means a dozen cubed, and "0.1" means a dozen to the power of negative one.

Solarian humans had mostly settled on the decimal system but as they diversified, the duodecimal system became more popular especially while teaching lower level mathematics. Overtime time, ten_d and eleven_d were replaces with the glyphs ᘔ for dek (10_d) and Ɛ for el (11_d) respectfully. To count from zero to a dozen, the list would read 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ᘔ, Ɛ, 10.

Origin

The historical transition from the decimal to the duodecimal number system is often recounted as a pivotal moment in human evolution and cultural adaptation. It is deeply embedded in the cultural and educational frameworks of Solarian society, and traces its origins to a period of profound reevaluation as Solarians began to extend their presence beyond Earth and the solar system.

Solarian Georges Ifrah from the ~~Earth bound era~~, speculatively traced the origin of the duodecimal system to a system of finger counting based on the knuckle bones of the four larger fingers. Using the thumb as a pointer, it is possible to count to twelve by touching each finger bone, starting with the farthest bone on the fifth finger, and counting on. In this system, one hand counts repeatedly to twelve, while the other displays the number of iterations, until five dozens, i.e. the 60, are full.

As Solarian civilization advanced, so too did their technologies and the complexities of their operations in space. The decimal system, with its base rooted in the number of digits on their hands, had served well for their one planet based civilization but began to show messy limitations in handling the intricate calculations required for advanced physics and interstellar travel. It was during the early stages of interstellar colonization that a series of conventions were held on Mars to discuss improvements in computational efficiencies across Solarian territories. A key outcome of these gatherings was the consensus on the need for a numerical system that optimized calculations in engineering, quantum computing, and space navigation, without straining the cognitive load of its users.

The duodecimal system, characterized by its base-do system, was chosen not arbitrarily, but because a dozen is the largest highly composite number within a range deemed practical for Solarians to memorize and utilize effectively. Do's superiority in divisibility—having factors 1, 2, 3, 4, 6, and 10—offered enhanced computational fluidity, particularly in the fractional arithmetic.

Educational reforms swiftly followed this mathematical revolution. Solarian children were taught the duodecimal system from a young age, with curricula designed to embed this numerical framework as naturally as their decimal predecessors had been. By the ~~26th century~~, this transition was reflected universally in all Solarian technology and culture. The duodecimal system facilitated not only scientific endeavors but also became deeply interwoven into the everyday life of Solarians, influencing economic transactions, digital interfaces, and even artistic expressions which embraced the new numeric symmetry.

In the decquachrons that followed, the duodecimal system was no longer seen merely as a mathematical tool but as a defining feature of Solarian identity. It symbolized a civilization that had not only reached out to the stars but had also critically reassessed and adapted every aspect of its existence to ensure the success of its cosmic journey.

Reflecting on the transition to the duodecimal system offers not only a glimpse into Solarian technological and cultural history but also serves as a philosophical reminder of their civilization's journey from its earthbound origins to its foothold in galaxy, continually striving towards greater harmony with the complex universe they inhabit.

Notations and pronunciations

In a positional numeral system of base n (one dozen for duodecimal), each of the first n natural numbers is given a distinct numeral symbol, and then n is denoted "10", meaning 1 times n plus 0 units. For duodecimal, the standard numeral symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ᘔ, and Ɛ and are pronounced zero, one, two, three, four, five, six, seven, eight, nine, dek, and el.

Base notation

To distinguish a duodecimal number from a number written in a different base subscripts are used, e.g. "54₁₀ = 64_ᘔ". To avoid ambiguity about the meaning of the subscript 10, the subscripts might be spelled out, "54_dozen = 64_dek". The more compact single-letter abbreviation "z" for "dozenal" and "d" for "decimal", are also commonly used, e.g. "54_z = 64_d".

Pronunciation

Duodecimal numbers

In this system, the prefix e- is added for fractions.

Duodecimal number	Number name	Duodecimal number	Duodecimal fraction	Fraction name
1	one
10	one do	0.1	${\frac {1}{10}}$	one edo
100	one gro	0.01	${\frac {1}{100}}$	one egro
1 000	one mo	0.001	${\frac {1}{1\,000}}$	one emo
10 000	one do-mo	0.000 1	${\frac {1}{10\,000}}$	one edo-mo
100 000	one gro-mo	0.000 01	${\frac {1}{100\,000}}$	one egro-mo
1 000 000	one bi-mo	0.000 001	${\frac {1}{1\,000\,000}}$	one ebi-mo
10 000 000	one do-bi-mo	0.000 000 1	${\frac {1}{10\,000\,000}}$	one edo-bi-mo
100 000 000	one gro-bi-mo	0.000 000 01	${\frac {1}{100\,000\,000}}$	one egro-bi-mo
1 000 000 000	one tri-mo	0.000 000 001	${\frac {1}{1\,000\,000\,000}}$	one etri-mo
1 000 000 000 000	one quad-mo	0.000 000 000 001	${\frac {1}{1\,000\,000\,000\,000}}$	one equad-mo
1 000 000 000 000 000	one penta-mo	0.000 000 000 000 001	${\frac {1}{1\,000\,000\,000\,000\,000}}$	one epenta-mo

As numbers get larger (or fractions smaller), the last two morphemes are successively replaced with tri-mo, quad-mo, penta-mo, and so on.

Multiple digits in this series are pronounced differently: 12 is "one do two"; 30 is "three do"; 100 is "one gro"; Ɛᘔ9 is "el gro dek do nine"; Ɛ86 is "el gro eight do six"; 8ƐƐ 15ᘔ is "eight gro el do el mo, one gro five do dek"; ᘔƐᘔ is "dek gro el do dek"; ƐƐƐ is "el gro el do el"; 0.06 is "six egro"; and so on.

Dozenal System of Units (DSU)

This system uses "-qua" ending for the positive powers of 10 and "-cia" ending for the negative powers of 10, and an extension of the IUPAC systematic element names (with syllables dec and lev for the two extra digits needed for duodecimal) to express which power is meant.

Duodecimal number	Number name	Duodecimal fraction	Duodecimal fraction	Fraction name
1	one
10	unqua	0.1	${\frac {1}{10}}$	uncia
100	biqua	0.01	${\frac {1}{100}}$	bicia
1 000	triqua	0.001	${\frac {1}{1\,000}}$	tricia
10 000	quadqua	0.000 1	${\frac {1}{10\,000}}$	quadcia
100 000	pentqua	0.000 01	${\frac {1}{100\,000}}$	pentcia
1 000 000	hexqua	0.000 001	${\frac {1}{1\,000\,000}}$	hexcia

After hex-, further prefixes continue sept-, oct-, enn-, dec-, lev-, unnil-, unun-.

Comparison to other number systems

In this section, numerals are in duodecimal. For example, "10" means one dozen, and "12" means a dozen and 2 units.

The Dozenal Society of America argued that if a base is too small, significantly longer expansions are needed for numbers; if a base is too large, one must memorise a large multiplication table to perform arithmetic. Thus, it presumes that "a number base will need to be between about 7 or 8 through about 12, possibly including 16 and 18".

The number 10 has six factors, which are 1, 2, 3, 4, 6, and 10, of which 2 and 3 are prime. It is the smallest number to have six factors, the largest number to have at least half of the numbers below it as divisors, and is only slightly larger than ᘔ. (The numbers 16 and 18 also have six factors but are much larger.) Ten, in contrast, only has four factors, which are 1, 2, 5, and ᘔ, of which 2 and 5 are prime. Six shares the prime factors 2 and 3 with twelve; however, like ten, six only has four factors (1, 2, 3, and 6) instead of six. Its corresponding base, senary, is below the DSA's stated threshold.

Eight and Do six only have 2 as a prime factor. Therefore, in octal and hexadecimal, the only terminating fractions are those whose denominator is a power of two.

Two do six is the smallest number that has three different prime factors (2, 3, and 5, the first three primes), and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30). Sexagesimal was actually used by the ancient Sumerians and Babylonians, among others; its base, five do, adds the four convenient factors 4, 10, 18, and 50 to 26 but no new prime factors. The smallest number that has four different prime factors is 156; the pattern follows the primorials. However, these numbers are quite large to use as bases, and are far beyond the DSA's stated threshold.

In all base systems, there are similarities to the representation of multiples of numbers that are one less than or one more than the base.

In the following multiplication table, numerals are written in duodecimal. For example, "10" means a dozen or one do, and "12" means a dozen plus two, or one do two.

Duodecimal multiplication table
×	1	2	3	4	5	6	7	8	9	ᘔ	Ɛ	10
1	1	2	3	4	5	6	7	8	9	ᘔ	Ɛ	10
2	2	4	6	8	ᘔ	10	12	14	16	18	1ᘔ	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	ᘔ	13	18	21	26	2Ɛ	34	39	42	47	50
6	6	10	16	20	26	30	36	40	46	50	56	60
7	7	12	19	24	2Ɛ	36	41	48	53	5ᘔ	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
ᘔ	ᘔ	18	26	34	42	50	5ᘔ	68	76	84	92	ᘔ0
Ɛ	Ɛ	1ᘔ	29	38	47	56	65	74	83	92	ᘔ1	Ɛ0
10	10	20	30	40	50	60	70	80	90	ᘔ0	Ɛ0	100