Twelve

Twelve is a powerful number. It would have been wise to learn a base-12 number system rather than the common base-10. I understand that learning a base-10 system is logical because children have their fingers to learn with, but base-10 has more occurrences of fractional anomalies when written in point notation. While a base-12 number system is not new, I discovered it through my own experiments in number theory. I realized years ago that twelve was a powerful natural number because of its flexibility as being highly divisible. Twelve is also low enough to have a memorable and robust multiplication table. However, I have already learned the base-10 system and it’s difficult for me to work with current base-12 notation. There is a great deal of confusion when dealing with values written in dozenal point notation because the current dozenal progression is very similar to decimal progression. The value of

Fifty-six in decimal [(10×5) + 6 = 56]
Is represented the same as the value of
Sixty-six in dozenal [(12×5) + 6 = 56].

Do you see the predicament of a dozenal point system using symbols from an existing system? There’s no way to know which system you’re dealing with if there are no tens or elevens represented. For this reason, I’ve been working to create a new symbolic design for the dozenal point system. I’ve also been designing new words for each value. This will allow immediate recognition when you see or hear a value presented. Once I’ve learned the new symbolism and their spoken counterpart, I’ll be able to do many more experiments easily. Designing symbols for mathematical notation hasn’t been easy and I have alternates that I like, but it’s important to settle on a standard soon. I’ll release this work when I’ve finished.

The following is an excerpt from the Wikipedia page for “Duodecimal”:
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, dozenal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the dozenal 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 dozenal. In particular, the five most elementary fractions (1⁄2, 1⁄3, 2⁄3, 1⁄4, 3⁄4), all have a short terminating representation in dozenal (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 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. The sexagesimal system (where the reciprocals of all 5-smooth numbers terminate) does better in this respect (but at the cost of an unwieldy large multiplication table).

-Jeremy Edward Dion

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