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Dozenal Usage Reference

Digits

The dozenal digits used here are: 0 1 2 3 4 5 6 7 8 9 X E. Two new, transdecimal numerals are added.

End-Digit Sequences

The final digit resulting from addition or subtraction is given here. The sequences for 5 and 7 are both single, as those numbers share no prime factor with the base.

Sequences for 2:
• 0, 2, 4, 6, 8, X
• 1, 3, 5, 7, 9, E

Sequences for 3:
• 0, 3, 6, 9
• 1, 4, 7, X
• 2, 5, 8, E

Sequences for 4:
• 0, 4, 8
• 1, 5, 9
• 2, 6, X
• 3, 7, E

Sequence for 5:
• 0, 5, X, 3, 8, 1, 6, E, 4, 9, 2, 7

Sequences for 6:
• 0, 6
• 1, 7
• 2, 8
• 3, 9
• 4, X
• 5, E

The sequences for 7, 8, 9 and X are the reverse of those for 5, 4, 3 and 2 respectively.

Multiplication Table

1,2,3,4,5,6,7,8,9,X,E,10
2,4,6,8,X,10,12,14,16,18,1X,20
3,6,9,10,13,16,19,20,23,26,29,30
4,8,10,14,18,20,24,28,30,34,38,40
5,X,13,18,21,26,2E,34,39,42,47,50
6,10,16,20,26,30,36,40,46,50,56,60
7,12,19,24,2E,36,41,48,53,5X,65,70
8,14,20,28,34,40,48,54,60,68,74,80
9,16,23,30,39,46,53,60,69,76,83,90
X,18,26,34,42,50,5X,68,76,84,92,X0
E,1X,29,38,47,56,65,74,83,92,X1,E0
10,20,30,40,50,60,70,80,90,X0,E0,100

In addition to the table, it may be useful to memorise the following multiplication rows as an aid in divisibility tests:

Multiples of 5 less than the gross:
• 5, X, 13, 18, 21, 26, 2E, 34, 39, 42, 47, 50, 55, 5X, 63, 68, 71, 76, 7E, 84, 89, 92, 97, X0, X5, XX, E3, E8

Multiples of 14 up to and including the gross:
• 14, 28, 40, 54, 68, 80, 94, X8, 100

Divisibility Tests

In dozenal, there are trivial tests for divisibility by 2, 3, 4 or 6 (divisors of the base), easy final-digit checks for 8, 9 (semidivisors of the base), 14 and 16 (divisors of the gross), the usual neighbour-related tests for E (the ω-totative) and 11 (the α-totative), and modular arithmetic tests of various degrees of difficulty for 5, 7 (totatives) and X (semitotative).

Divisors:
• 2: the final digit is 0, 2, 4, 6, 8 or X.
• 3: the final digit is 0, 3, 6 or 9.
• 4: the final digit is 0, 4 or 8.
• 6: the final digit is 0 or 6.

Semidivisors:
• 8: the second-last digit is even and the final digit is 0 or 8; or the second-last digit is odd and the final digit is 4.
• 9: the second-last digit is 3n−1 and the final digit is 3; or the second-last digit is 3n+1 and the final digit is 6; or the second-last digit is 3n and the final digit is 0 or 9.
• 14: the last two digits are divisible by 14 (see list above of nine numbers to be memorised).
• 16: the last two digits are divisible by 16 (not listed, because checking for divisibility by 16 is of little practical use).

Neighbours of the base:
• E: the sum of all the digits is a multiple of E. (Digit-sum rule)
• 11: the sum of all the digits in even places and the sum of all the digits in odd places differ by a multiple of 11. (Alternating-digit rule)

For the totatives with no simple rule, 5 and 7, there are two types of test: one that works on a digit-by-digit basis and another that reduces the tested number by more than one digit at a time. In those tests, too, there is more than one option.

• 5: subtract the final two digits from all the other digits until you have a number you know to be a multiple of 5. (Works because 101 is a multiple of 5. The list of multiples of 5 above can make this test very useful.)
• 5: multiply the final digit by 3, then add it to all the other digits; repeat as needed. (Works because 2E, a multiple of 5, is one less than 3 times the base.)
• 5: multiply the final digit by 2, then subtract it from all the other digits; repeat as needed. (Works because 21, a multiple of 5, is one more than 2 times the base.)

• 7: subtract the final three digits from all the other digits. (Works because 1001 is a multiple of 7. As there are too many 3-digit multiples of 7, this test is rarely sufficient.)
• 7: multiply the final digit by 3, then add it to all the other digits; repeat as needed. (Works because 2E, a multiple of 7, is one less than 3 times the base.)
• 7: multiply the final digit by 4, then subtract it from all the other digits; repeat as needed. (Works because 41, a multiple of 7, is one more than 4 times the base.)

The semitotative X has a compound test: a multiple of X is an even number that passes the divisibility test for 5.

Fractions

Dozenal fractions cover all the most common, practical fractions neatly. Vulgar fractions, used in many other bases mainly to represent certain common fractions without recurring digits (for example, thirds in decimal), would be rarely needed in dozenal. They would be needed mainly for fractions of the totatives: fifths, sevenths and tenths; these would not be of much use in a dozenal world. It is customary to use a semicolon rather than a dot as the radix point, probably for distinguishing from decimal.

The single-digit terminating fractions are:

Dozenal,0;1,0;2,0;3,0;4,0;5,0;6,0;7,0;8,0;9,0;X,0;E
Vulgar,1/10,1/6,1/4,1/3,5/10,1/2,7/10,2/3,3/4,5/6,E/10

Of the multiple-digit terminating fractions (those of semidivisors), the fractions of 8, 14 and 9 are of most practical value.

Irreducible fractions of 8:

Vulgar,1/8,3/8,5/8,7/8
Dozenal,0;16,0;46,0;76,0;X6

Irreducible fractions of 14:

Vulgar,1/14,3/14,5/14,7/14,9/14,E/14,11/14,13/14
Dozenal,0;09,0;23,0;39,0;53,0;69,0;83,0;99,0;E3

Irreducible fractions of 9:

Vulgar,1/9,2/9,4/9,5/9,7/9,8/9
Dozenal,0;14,0;28,0;54,0;68,0;94,0;X8

The fractions of 5 and 7 are cyclical as well as recurring, with recurrence and circulation of the digits 2497 for n/5 and 186X35 for n/7. Compare 142857 for n/7 in decimal. The fractions of the semitotative X have a single non-recurring digit before the cyclical sequence. For completeness, all the fractions of X (including their reductions to fifths and the half) are given here, with brackets denoting the recurrent digits:

Vulgar,1/X,1/5,3/X,2/5,1/2,3/5,7/X,4/5,9/X
Dozenal,0;1(2497),0;(2497),0;3(7249),0;(4972),0;6,0;(7249),0;8(4972),0;(9724),0;X(9724)

Prime Numbers

All primes in dozenal except 2 and 3 end in the digit 1, 5, 7 or E. The table below gives all the numbers less than the gross in their prime composition. (0 and 1 are neither composite nor prime, hence denoted in brackets.)

–,□0,□1,□2,□3,□4,□5,□6,□7,□8,□9,□X,□E
0□,(0),(1),2,3,2↑2,5,2·3,7,2↑3,3↑2,2·5,E
1□,2↑2·3,11,2·7,3·5,2↑4,15,2·3↑2,17,2↑2·5,3·7,2·E,1E
2□,2↑3·3,5↑2,2·11,3↑3,2↑2·7,25,2·3·5,27,2↑5,3·E,2·15,5·7
3□,2↑2·3↑2,31,2·17,3·11,2↑3·5,35,2·3·7,37,2↑2·E,3↑2·5,2·1E,3E
4□,2↑4·3,7↑2,2·5↑2,3·15,2↑2·11,45,2·3↑3,5·E,2↑3·7,3·17,2·25,4E
5□,2↑2·3·5,51,2·27,3↑2·7,2↑6,5·11,2·3·E,57,2↑2·15,3·1E,2·5·7,5E
6□,2↑3·3↑2,61,2·31,3·5↑2,2↑2·17,7·E,2·3·11,67,2↑4·5,3↑4,2·35,6E
7□,2↑2·3·7,5·15,2·37,3·25,2↑3·E,75,2·3↑2·5,7·11,2↑2·1E,3·27,2·3E,5·17
8□,2↑5·3,81,2·7↑2,3↑2·E,2↑2·5↑2,85,2·3·15,87,2↑3·11,3·5·7,2·45,8E
9□,2↑2·3↑3,91,2·5·E,3·31,2↑4·7,95,2·3·17,5·1E,2↑2·25,3↑2·11,2·4E,7·15
X□,2↑3·3·5,E↑2,2·51,3·35,2↑2·27,5↑3,2·3↑2·7,X7,2↑7,3·37,2·5·11,XE
E□,2↑2·3·E,7·17,2·57,3↑3·5,2↑3·15,E5,2·3·1E,E7,2↑2·5·7,3·3E,2·5E,E·11

Power Series

A few power series are given here for reference. Some are more useful than others, for example the powers of 2 are frequent in the computing world, and the powers of X give an interesting view of our familiar multiples of ten in a totally alien dressing. Other series may not be of much practical use, but it might be a good exercise to try out the divisibility tests on them (especially for the powers of 5 and 7). The squares and cubes of up to 20 are also given.

• 2↑n: 2, 4, 8, 14, 28, 54, X8, 194, 368, 714, 1228, 2454, 48X8, 9594, 16E68, 31E14
• 3↑n: 3, 9, 23, 69, 183, 509, 1323, 3969, E483, 2X209, 86623, 217669
• 5↑n: 5, 21, X5, 441, 1985, 9061, 39265, 16X081, 7X2345
• 7↑n: 7, 41, 247, 1481, 9887, 58101
• X↑n: X, 84, 6E4, 5954, 49X54, 402854, 3423054, 295X6454, 23XX93854
• E↑n: E, X1, 92E, 8581, 7924E, 715261
• 11↑n: 11, 121, 1331, 14641, 15XX51, 1749361
• n↑2: 1, 4, 9, 14, 21, 30, 41, 54, 69, 84, X1, 100, 121, 144, 169, 194, 201, 230, 261, 294, 309, 344, 381, 400
• n↑3: 1, 8, 23, 54, X5, 160, 247, 368, 509, 6E4, 92E, 1000, 1331, 1708, 1E53, 2454, 2X15, 3460, 3E77, 4768, 5439, 61E4, 705E, 8000

Irrational Numbers

The most practical irrational numbers are presented. These are the fundamental square roots of trigonometry, the cube root of 2, the twelfth root of 2 (the constant of musical pitch), the golden ratio (phi), the ratio of circular circumference to diameter (pi) and the natural logarithm base (e).

Note that the degree to radian conversion factor (130/π) is not included here, because a dozenal world would use radians directly, instead of base-50 (decimal base-60, sexagesimal) degrees; the latter is an ancient workaround for base ten’s factor deficiency (especially the inability to divide by 3), while in dozenal it would be simple to represent fractions of π for radians using dozenal fractions.

Number,Dozenal Form (to 10 fractionals),Approximation,Notes
√2,1;4E79170X07E8,1;5,
√3,1;894E97EE9687,1;895,tan 0;4π
√5,2;29EE13254058,2;2X,
√0;6,0;859X696503EX,0;85X,sin 0;3π
√0;9,0;X485X9EEX943,0;X486,sin 0;4π
√0;4,0;6E17E27EE22X,0;7,tan 0;2π
2↑0;4,1;31518811X39X,1;315,
2↑0;1,1;086903X21E3E,1;0869,
φ,1;74EE6772802X,1;75,0;6·(1+√5)
π,3;184809493E91,3;1848,
e,2;875236069821,2;875,

Submitted on
May 10, 2012