Page "Duodecimal" Paragraph 1

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.

As a result, duodecimal is sometimes named the number system with the most optimal radix economy.

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 ( ½, ⅓, ⅔, ¼ and ¾ ) 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 unwieldy multiplication table ).