Duodecimal system

The duodecimal system is a base-twelve numeral system, also called dozenal. This means that each set of 12 units of a level generates a unit of the next higher level. Twelve elements make a dozen; 12 dozen, one thick; 12 gross, a fourth level unit. So 3457_D is equal to 3 gross dozen, 4 gross, 5 dozen, and 7 ones. The basic numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B; they are all less than base twelve = 10 = 1 dozen + 0 ones. (A = 10; B = 11 of the base 10).

There are societies in Britain and the US that promote the use of base-twelve, arguing the following:

• The 12 has four own factors (excluding 1 and 12), which are 2, 3, 4 and 6; while the 10 has only two own factors: 2 and 5. Because of this, the 12-based multiplications and divisions are easier (see below) and therefore the duodecimal system is more efficient than decimal.
• Historically, the 12 has been used by many civilizations. It is believed that the observation of 12 appearances of the Moon over a year is the reason why the 12 is universally employed in all cultures. Examples include the year of 12 months, 12 zodiacal signs, 12 animals in Chinese astrology, etc.
• Because the 12 is an abundant number, it is used with profusion in the measuring units, for example, a foot is 12 inches, a troy pound equals 12 ounces, a dozen items have 12 items, a thick one has 12 dozen, etc.
• It sells dishes, knives, forks, pots, at present, for half a dozen, a quarter of dozen; equally the powder rocket rocket rockets for a thick average, as well as the bengala lights.

In Gran Canaria (Canary Islands), as surely in the rest of the world since apparently the Sumerians used this same method, among the people of the countryside the base twelve is used because it is a human base. Counted with one hand using the thumb as a pointer, the phalanges of the remaining four fingers count twelve, a dozen, 3 phalanges for 4 fingers. Using the other hand as a multiplier 12 x 5 = 60. That is, with both hands they could count up to sixty, or their five dozen, the hands used as an abacus. Hence there are twelve months, apostles, hours of the day, signs of the zodiac, etc. And what is most important is the sexagesimal system (base 60), and also the division of the hour into sixty minutes and these into sixty seconds.

Fractions and Irrational Numbers

In any positional numeral system with a rational base (such as decimal and duodecimal), all those irreducible fractions whose denominator contains prime factors other than those that factorize the base, will not have a 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. Thus, 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:

On the other hand, in any rational-based positional numeral system, 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:

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

Multiply Table

Finding Prime Numbers

In base 12, a prime number can only end in 1, 5, 7, or B (with the only exceptions being the prime numbers 2 and 3). The remaining eight possibilities always generate composite numbers:

Detection of multiples

• Numbers ending in 0, 2, 4, 6, 8 and A are multiples of two
• Of these, those who row in 0, 4 and 8 are also divisible by four
• Numbers finished in 0, 3, 6 and 9 are multiples of three
• Of these, the finished in 0 and 6 are also multiples of six
• Of all the above, those who conclude in 0 are on their side, multiples of twelve.
• Be the number N =xyz...w on base 12, if the encryption sum x+y+z+...+w = 11 or multiple of 11, the number N is divisible by 11 = B (like property of 9, on base 10). Example 542_D
• In the same way, the criterion of the difference of order figures for less than the odd order, to detect if a base number 12 is multiple of 11_D= 13.
• If a base N number 12 ends in 2 zeros or the last two figures as numeral duodecimal is multiple of 8, then N is divisible by 8; e.g.: 3514, 4934.
• When a base 12 K numeral, rows in 2 zeros or the numeral formed by the last 2 duodemic figures is multiple of 9, then K is multiple of 9. Ejm: 4A23, 7B16.
• The numeral duodecimal N is a multiple of 16 if it rows in 2 zeros or the last two figures form a duodecimal multiple of 16. E.g.: 2928, BA30.

Natural cousins in decimal and duodecimal notations

The following is a list of the series of prime numbers (up to those with less than three digits) in duodecimal base: