The Dozenal Society of America Dozenal FAQs Michael Thomas De Vlieger 30 November 2011 The following ten questions were sent in recently by a stu- (12 inches to the foot, 12 original ounces to a pound, ‘inch’ dent. These lucid questions are some of those most frequently and ‘ounce’ both deriving from Roman uncia.) asked. The Dozenal Society of America is furnishing the full The integer twelve has six divisors {1, 2, 3, 4, 6, 12} while responses to these questions complete with illustrations and the number ten only has four {1, 2, 5, 10}. Divisors are im- references. Thanks to Mr. Israeli for questions 1–10! portant because they exhibit relatively brief and predict- 0. What is the number one reason why one believes base 12 is able patterns in arithmetic in a number base they divide. superior to decimal? We can see this when we look at the decimal multiplica- A. Briefly, twelve is a highly divisible yet compact num- tion products for the number 5. Decimal products of the ber; it has more divisors than ten. This facilitates learn- number 5 and an integer x have a least significant digit that ing and using arithmetic, and simplifies the natural is one of {0, 5}. Because of this, we understand any integer fractions [1, 2, 3]. This observation dates as far back as the mid- ending with -0 or -5 is some multiple x of 5. Decimal is also dle ages, involving luminaries of mathematics such as Blaise an even base: because of this, the multiplication products Pascal [4, 5] and Laplace [6]. Dozens have been used since an- of 2 end in one of {0, 2, 4, 6, 8}. In decimal, we can test for cient times; the Roman usage as “unciæ”[7, 8, 9] strongly affect- evenness and divisibility by 5 simply by examining the last ing systems of currency (12 pence to a shilling) and measure digit of an integer. The number twelve has two additional divisors than ten. 1 2 3 4 5 6 7 8 9 10 In duodecimal, the number 6 behaves much like 5 in the 2 4 6 8 10 12 14 16 18 20 multiplication table. In duodecimal, any product of 6 and an integer x will have a least significant digit that is one of 3 6 9 12 15 18 21 24 27 30 {0, 6}. Like decimal, duodecimal is even, so any product of 4 8 12 16 20 24 28 32 36 40 2 and an integer x will end with one of {0, 2, 4, 6, 8, a}. 5 10 15 20 25 30 35 40 45 50 Because twelve has two more divisors, users of duo- 6 12 18 24 30 36 42 48 54 60 decimal enjoy two other divisor product lines in the multiplication table. Any product 3x will end in one of 7 14 21 28 35 42 49 56 63 70 {0, 3, 6, 9}, and any product 4x will end in one of {0, 4, 8}. 8 16 24 32 40 48 56 64 72 80 Because twelve has six divisors, with the smallest four con- 9 18 27 36 45 54 63 72 81 90 secutive, it presents a multiplication table featuring brief 10 20 30 40 50 60 70 80 90 100 patterns in the product lines of many numbers. (See page 4 for a study of patterns in the decimal and duodecimal mul- Figure 0.1. The decimal multiplication table. tiplication tables and Appendix A for bases 8 ≤ r ≤ 16.) 1 2 3 4 5 6 7 8 9 a b 10 The divisors of a base have a second important effect. Ev- ery divisor d of base r has a complement d′ [10] such that 2 4 6 8 a 10 12 14 16 18 1a 20 3 6 9 10 13 16 19 20 23 26 29 30 (0.1) r = d × d′ 4 8 10 14 18 20 24 28 30 34 38 40 This relationship between a divisor and its complement 5 a 13 18 21 26 2b 34 39 42 47 50 can be seen in the multiplication table: the decimal prod- ucts of five have 2 possible end digits (10 = 5 × 2), the 6 10 16 20 26 30 36 40 46 50 56 60 duodecimal products of four have three possible end digits 7 12 19 24 2b 36 41 48 53 5a 65 70 (12 = 4 × 3). The relationship also appears in digital ex- 8 14 20 28 34 40 48 54 60 68 74 80 pansions of fractions. Generally, 9 16 23 30 39 46 53 60 69 76 83 90 (0.2) 1∕d = d′∕r a 18 26 34 42 50 5a 68 76 84 92 a0 thus we will see the digit d′ following the radix point. In b 1a 29 38 47 56 65 74 83 92 a1 b0 base ten we have 1∕2 = 0.5 and 1∕5 = 0.2. In base twelve we 10 20 30 40 50 60 70 80 90 a0 b0 100 will have 1∕2 = 0.6, 1∕3 = 0.4, 1∕4 = 0.3 and 1∕6 = 0.2. Thus duo- decimal features single-place terminating fractions for all [11, 12] Figure 0.2. The duodecimal multiplication table. the ‘natural fractions’ : Dozenal FAQs · De Vlieger The Dozenal Society of America Page 1 ½ ⅓ ⅔ ¼ ¾ Twelve gives us the ability to access the power of 6 divisors Decimal .5 .333… .666… .25 .75 with a magnitude of number base that is not much larger than decimal. The dozen gives us clean natural fractions, Duodecimal .6 .4 .8 .3 .9 incorporating these into its digital representation. Duo- Figure 0.3. The natural fractions in decimal and duodecimal notation. decimal divisors are by far the very most important reason One might raise the question, “granted, twelve has more divi- why duodecimal is the optimum number base for general sors than ten, but what about the other numbers?” First, let’s human intuitive computation. acknowledge that number bases fit for general human compu- 1. Other then having more factors than 10 does, what other ma- tation must lie in a certain range. If a number base is too small, jor properties make 12 a better base for a number system? we’ll need many digits to express even a relatively small quan- A. Briefly, twelve has proportionally fewer totatives (digits tity. Imagine a binary speed limit sign said “1000000 kph”! A that have 1 as the greatest common divisor with the num- driver would need to slow down and peruse the sign to be sure ber base) than ten, minimizing resistance to human intu- not to be speeding at double or bumbling at half the posted ition in computation and manipulation of numbers. Fewer limit! The seven decimal-digit telephone number would be- totatives means fewer “difficult” product lines in the mul- come nearly a two-dozen-bit string of 1’s and 0’s. It would be tiplication table, fewer occasions of recurrent fractions, a easy to miss out on a hot date if one made a transcription er- more organized arrangement of primes, and a smaller pro- ror somewhere. If a number base is too large, then one would portion of other digits and numbers that have recurrent need to memorize a larger multiplication table, for example, reciprocals. Dozenal has a denser set of regular numbers, in order to perform arithmetic. If the Mayans multiplied us- aiding human intuition in computation. Regular numbers ing modern techniques, would they need to spend more than have terminating digital fractions and divisibility tests akin double the time a decimal society would in mastering their to that of divisors. multiplication tables? There are 20 digits in the vigesimal (base-20) system, each with 20 combinations. There are 55 These features are not as easily summarized as the fact unique products in the decimal multiplication table vs. 210 in that twelve has more divisors than ten, but are just as im- the vigesimal and 78 in the duodecimal. portant. The following pages attempt to explain the brief synopsis above, and prove it using mathematics. The ex- Let’s presume a number base will need to be between about 7 planation can get technical. The terms used in the above or 8 through about 16, possibly including 18 and 20. Here is a [13, 14] synopsis are explained in the following text. Let’s begin by list of divisor counts and ratios for bases 2 through 20: examining the digits of a number base, since many of the r σ (r) σ (r)/r properties can be summarized by such an examination. r σ (r) σ (r)/r 0 0 0 0 11 2 18 2∕11 Five Universal Types of Digits 2 2 100 12 6 50 The digits of a number base are important because a num- 3 2 66 ⅔ 13 2 15 5∕13 ber base “sees” the world through its digits. Let the integer 4 3 75 [19, 14 4 28 4∕7 r ≥ 2 be a number base. A digit is an integer 0 ≤ n < r 5 2 40 20]. Using digits, we can express the arbitrary integer x as a 15 4 26 ⅔ 6 4 66 ⅔ product of the integers q and r plus the digit n [21, 22, 23]: 16 5 31 ¼ 7 2 28 4∕7 17 2 11 15∕17 x = qr + n 8 4 50 18 6 33 ⅓ The digit “0” signifies congruence with the number base; 9 3 33 ⅓ [24] 19 2 10 10∕19 it will be interpreted as signifying r itself as a digit .