Origin Edit

Notations and pronunciations Edit

Transdecimal symbols Edit In a duodecimal place system twelve is written as 10, but there are numerous proposals for how to write ten and eleven.[10] The simplified notations use only basic and easy to access letters such as A and B (as in the hexadecimal and vigesimal), T and E (initials of Ten and Eleven), X and Z. Some employ Greek letters such as δ (standing for Greek δέκα 'ten') and ε (for Greek ένδεκα 'eleven'), or τ and ε.[10] Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his book New Numbers an X (from the Roman numeral for ten) and a script E (ℰ, U+2130).[11]

The Dozenal Society of Great Britain proposes a rotated digit two 2 for ten and a reversed or rotated digit three 3 for eleven.[10] This notation was introduced by Sir Isaac Pitman.[12][10][13] These digit forms are available as Unicode characters since June 2015[14][15] as (↊, U+218A) and (↋, U+218B) respectively.[16] Until 2015, the Dozenal Society of America (DSA) used and , the symbols devised by William Addison Dwiggins.[10][17] After the Pitman digits (32) were added to Unicode the DSA took a vote and then began publishing content using the Pitman digits instead.[18][19] They still use the letters X and E as the equivalent in ASCII text. Other proposals are more creative or aesthetic, for example, Edna Kramer in her 1951 book The Main Stream of Mathematics used a six-pointed asterisk (sextile) ⚹ for ten and a hash (or octothorpe) # for eleven.[10] The symbols were chosen because they are available in typewriters and already present in telephone dials.[10] This notation was used in publications of the Dozenal Society of America in the period 1974–2008.[20][21] Many don't use any Arabic numerals under the principle of "separate identity."[10] Base notation Edit There are also varying proposals of how to distinguish a duodecimal number from a decimal one, or one in a different base. They include italicizing duodecimal numbers (54 = 64), adding a "Humphrey point" (a semicolon ";" instead of a decimal point ".") to duodecimal numbers (54; = 64.) (54;0 = 64.0), or some combination of the two. More also add extra marking to one or more bases. Others use subscript or affixed labels to indicate the base, allowing for more than decimal and duodecimal to be represented:[19] Common Base Abb. Letter Cardinal Decimal Duodecimal b inary b in b two 2 2 o ctal o ct o eight 8 8 d ecimal d ec d ten 10 ↊ do z enal (duodecimal) do z z twelve 12 10 he x adecimal he x x sixteen 16 14 This allows one to write "54 z = 64 d ," "54 twelve = 64 ten " or "doz 54 = dec 64." In programming, binary, octal, and hexadecimal often use a similar scheme: a binary number starts with 0b , octal with 0o , and hexadecimal with 0x . Pronunciation Edit The Dozenal Society of America suggests the pronunciation of ten and eleven as "dek" and "el", each order has its own name and the prefix e- is added for fractions.[17][22] The symbol corresponding to the decimal point or decimal comma, separating the whole number part from the fractional part, is the semicolon ";". The overall system is:[17] Duodecimal Name Decimal Duodecimal fraction Name 1 one 1 10 do 12 0;1 edo 100 gro 144 0;01 egro 1,000 mo 1,728 0;001 emo 10,000 do-mo 20,736 0;000,1 edo-mo 100,000 gro-mo 248,832 0;000,01 egro-mo 1,000,000 bi-mo 2,985,984 0;000,001 ebi-mo 1,000,000,000 tri-mo 5,159,780,352 0;000,000,001 etri-mo Multiple digits in this are pronounced differently. 12 is "one do two", 30 is "three do", 100 is "one gro", BA9 (ET9) is "el gro dek do nine", B8,65A,300 (E8,65T,300) is "el do eight bi-mo, six gro five do dek mo, three gro", and so on.[22]

Advocacy and "dozenalism" Edit

Comparison to other numeral systems Edit

The number 12 has six factors, which are 1, 2, 3, 4, 6, and 12, of which 2 and 3 are prime. The decimal system has only four factors, which are 1, 2, 5, and 10, of which 2 and 5 are prime. Vigesimal (base 20) adds two factors to those of ten, namely 4 and 20, but no additional prime factor. Although twenty has 6 factors, 2 of them prime, similarly to twelve, it is also a much larger base, and so the digit set and the multiplication table are much larger. Binary has only two factors, 1 and 2, the latter being prime. Hexadecimal (base 16) has five factors, adding 4, 8 and 16 to those of 2, but no additional prime. Trigesimal (base 30) is the smallest system that has three different prime factors (all of the three smallest primes: 2, 3 and 5) and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30). Sexagesimal—which the ancient Sumerians and Babylonians among others actually used—adds the four convenient factors 4, 12, 20, and 60 to this but no new prime factors. The smallest system that has four different prime factors is base 210 and the pattern follows the primorials. In all base systems, there are similarities to the representation of multiples of numbers which are one less than the base. Duodecimal multiplication table × 0 1 2 3 4 5 6 7 8 9 ᘔ Ɛ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 ᘔ Ɛ 2 0 2 4 6 8 ᘔ 10 12 14 16 18 1ᘔ 3 0 3 6 9 10 13 16 19 20 23 26 29 4 0 4 8 10 14 18 20 24 28 30 34 38 5 0 5 ᘔ 13 18 21 26 2Ɛ 34 39 42 47 6 0 6 10 16 20 26 30 36 40 46 50 56 7 0 7 12 19 24 2Ɛ 36 41 48 53 5ᘔ 65 8 0 8 14 20 28 34 40 48 54 60 68 74 9 0 9 16 23 30 39 46 53 60 69 76 83 ᘔ 0 ᘔ 18 26 34 42 50 5ᘔ 68 76 84 92 Ɛ 0 Ɛ 1ᘔ 29 38 47 56 65 74 83 92 ᘔ1

Conversion tables to and from decimal Edit

To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under positional notation). Alternatively, one can use digit-conversion tables. The ones provided below can be used to convert any duodecimal number between 0.01 and ƐƐƐ,ƐƐƐ.ƐƐ to decimal, or any decimal number between 0.01 and 999,999.99 to duodecimal. To use them, the given number must first be decomposed into a sum of numbers with only one significant digit each. For example: 123,456.78 = 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 This decomposition works the same no matter what base the number is expressed in. Just isolate each non-zero digit, padding them with as many zeros as necessary to preserve their respective place values. If the digits in the given number include zeroes (for example, 102,304.05), these are, of course, left out in the digit decomposition (102,304.05 = 100,000 + 2,000 + 300 + 4 + 0.05). Then the digit conversion tables can be used to obtain the equivalent value in the target base for each digit. If the given number is in duodecimal and the target base is decimal, we get: (duodecimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (decimal) 248,832 + 41,472 + 5,184 + 576 + 60 + 6 + 0.58 3 333333333... + 0.0 5 5555555555... Now, because the summands are already converted to base ten, the usual decimal arithmetic is used to perform the addition and recompose the number, arriving at the conversion result: Duodecimal -----> Decimal 100,000 = 248,832 20,000 = 41,472 3,000 = 5,184 400 = 576 50 = 60 + 6 = + 6 0.7 = 0.58 3 333333333... 0.08 = 0.0 5 5555555555... -------------------------------------------- 123,456.78 = 296,130.63 8 888888888... That is, (duodecimal) 123,456.78 equals (decimal) 296,130.638 ≈ 296,130.64 If the given number is in decimal and the target base is duodecimal, the method is basically same. Using the digit conversion tables: (decimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (duodecimal) 49,ᘔ54 + Ɛ,6ᘔ8 + 1,8ᘔ0 + 294 + 42 + 6 + 0.849724972497249724972497... + 0.0Ɛ62ᘔ68781Ɛ05915343ᘔ0Ɛ62... However, in order to do this sum and recompose the number, now the addition tables for the duodecimal system have to be used, instead of the addition tables for decimal most people are already familiar with, because the summands are now in base twelve and so the arithmetic with them has to be in duodecimal as well. In decimal, 6 + 6 equals 12, but in duodecimal it equals 10; so, if using decimal arithmetic with duodecimal numbers one would arrive at an incorrect result. Doing the arithmetic properly in duodecimal, one gets the result: Decimal -----> Duodecimal 100,000 = 49,ᘔ54 20,000 = Ɛ,6ᘔ8 3,000 = 1,8ᘔ0 400 = 294 50 = 42 + 6 = + 6 0.7 = 0.8 4972 4972497249724972497... 0.08 = 0. 0Ɛ62ᘔ68781Ɛ05915343ᘔ 0Ɛ62... -------------------------------------------------------- 123,456.78 = 5Ɛ,540.9 43ᘔ0Ɛ62ᘔ68781Ɛ059153 43ᘔ... That is, (decimal) 123,456.78 equals (duodecimal) 5Ɛ,540.943ᘔ0Ɛ62ᘔ68781Ɛ059153... ≈ 5Ɛ,540.94 Duodecimal to decimal digit conversion Edit Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. 1,000,000 2,985,984 100,000 248,832 10,000 20,736 1,000 1,728 100 144 10 12 1 1 0.1 0.08 3 0.01 0.0069 4 2,000,000 5,971,968 200,000 497,664 20,000 41,472 2,000 3,456 200 288 20 24 2 2 0.2 0.1 6 0.02 0.013 8 3,000,000 8,957,952 300,000 746,496 30,000 62,208 3,000 5,184 300 432 30 36 3 3 0.3 0.25 0.03 0.0208 3 4,000,000 11,943,936 400,000 995,328 40,000 82,944 4,000 6,912 400 576 40 48 4 4 0.4 0. 3 0.04 0.02 7 5,000,000 14,929,920 500,000 1,244,160 50,000 103,680 5,000 8,640 500 720 50 60 5 5 0.5 0.41 6 0.05 0.0347 2 6,000,000 17,915,904 600,000 1,492,992 60,000 124,416 6,000 10,368 600 864 60 72 6 6 0.6 0.5 0.06 0.041 6 7,000,000 20,901,888 700,000 1,741,824 70,000 145,152 7,000 12,096 700 1,008 70 84 7 7 0.7 0.58 3 0.07 0.0486 1 8,000,000 23,887,872 800,000 1,990,656 80,000 165,888 8,000 13,824 800 1,152 80 96 8 8 0.8 0. 6 0.08 0.0 5 9,000,000 26,873,856 900,000 2,239,488 90,000 186,624 9,000 15,552 900 1,296 90 108 9 9 0.9 0.75 0.09 0.0625 ᘔ,000,000 29,859,840 ᘔ00,000 2,488,320 ᘔ0,000 207,360 ᘔ,000 17,280 ᘔ00 1,440 ᘔ0 120 ᘔ 10 0.ᘔ 0.8 3 0.0ᘔ 0.069 4 Ɛ,000,000 32,845,824 Ɛ00,000 2,737,152 Ɛ0,000 228,096 Ɛ,000 19,008 Ɛ00 1,584 Ɛ0 132 Ɛ 11 0.Ɛ 0.91 6 0.0Ɛ 0.0763 8 Decimal to duodecimal digit conversion Edit Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. 100,000 49,ᘔ54 10,000 5,954 1,000 6Ɛ4 100 84 10 ᘔ 1 1 0.1 0.1 2497 0.01 0.0 15343ᘔ0Ɛ62ᘔ68781Ɛ059 200,000 97,8ᘔ8 20,000 Ɛ,6ᘔ8 2,000 1,1ᘔ8 200 148 20 18 2 2 0.2 0. 2497 0.02 0.0 2ᘔ68781Ɛ05915343ᘔ0Ɛ6 300,000 125,740 30,000 15,440 3,000 1,8ᘔ0 300 210 30 26 3 3 0.3 0.3 7249 0.03 0.0 43ᘔ0Ɛ62ᘔ68781Ɛ059153 400,000 173,594 40,000 1Ɛ,194 4,000 2,394 400 294 40 34 4 4 0.4 0. 4972 0.04 0. 05915343ᘔ0Ɛ62ᘔ68781Ɛ 500,000 201,428 50,000 24,Ɛ28 5,000 2,ᘔ88 500 358 50 42 5 5 0.5 0.6 0.05 0.0 7249 600,000 24Ɛ,280 60,000 2ᘔ,880 6,000 3,580 600 420 60 50 6 6 0.6 0. 7249 0.06 0.0 8781Ɛ05915343ᘔ0Ɛ62ᘔ6 700,000 299,114 70,000 34,614 7,000 4,074 700 4ᘔ4 70 5ᘔ 7 7 0.7 0.8 4972 0.07 0.0 ᘔ0Ɛ62ᘔ68781Ɛ05915343 800,000 326,Ɛ68 80,000 3ᘔ,368 8,000 4,768 800 568 80 68 8 8 0.8 0. 9724 0.08 0. 0Ɛ62ᘔ68781Ɛ05915343ᘔ 900,000 374,ᘔ00 90,000 44,100 9,000 5,260 900 630 90 76 9 9 0.9 0.ᘔ 9724 0.09 0.1 0Ɛ62ᘔ68781Ɛ05915343ᘔ

Divisibility rules Edit

(In this section, all numbers are written with duodecimal) This section is about the divisibility rules in duodecimal. 1 Any integer is divisible by 1. 2 If a number is divisible by 2 then the unit digit of that number will be 0, 2, 4, 6, 8 or ᘔ. 3 If a number is divisible by 3 then the unit digit of that number will be 0, 3, 6 or 9. 4 If a number is divisible by 4 then the unit digit of that number will be 0, 4 or 8. 5 To test for divisibility by 5, double the units digit and subtract the result from the number formed by the rest of the digits. If the result is divisible by 5 then the given number is divisible by 5. This rule comes from 21(5*5) Examples:

13 rule => |1-2*3| = 5 which is divisible by 5.

2Ɛᘔ5 rule => |2Ɛᘔ-2*5| = 2Ɛ0(5*70) which is divisible by 5(or apply the rule on 2Ɛ0). OR To test for divisibility by 5, subtract the units digit and triple of the result to the number formed by the rest of the digits. If the result is divisible by 5 then the given number is divisible by 5. This rule comes from 13(5*3) Examples:

13 rule => |3-3*1| = 0 which is divisible by 5.

2Ɛᘔ5 rule => |5-3*2Ɛᘔ| = 8Ɛ1(5*195) which is divisible by 5(or apply the rule on 8Ɛ1). OR Form the alternating sum of blocks of two from right to left. If the result is divisible by 5 then the given number is divisible by 5. This rule comes from 101, since 101 = 5*25, thus this rule can be also tested for the divisibility by 25. Example:

97,374,627 => 27-46+37-97 = -7Ɛ which is divisible by 5. 6 If a number is divisible by 6 then the unit digit of that number will be 0 or 6. 7 To test for divisibility by 7, triple the units digit and add the result to the number formed by the rest of the digits. If the result is divisible by 7 then the given number is divisible by 7. This rule comes from 2Ɛ(7*5) Examples:

12 rule => |3*2+1| = 7 which is divisible by 7.

271Ɛ rule => |3*Ɛ+271| = 29ᘔ(7*4ᘔ) which is divisible by 7(or apply the rule on 29ᘔ).

OR To test for divisibility by 7, subtract the units digit and double the result from the number formed by the rest of the digits. If the result is divisible by 7 then the given number is divisible by 7. This rule comes from 12(7*2) Examples:

12 rule => |2-2*1| = 0 which is divisible by 7.

271Ɛ rule => |Ɛ-2*271| = 513(7*89) which is divisible by 7(or apply the rule on 513).

OR To test for divisibility by 7, 4 times the units digit and subtract the result from the number formed by the rest of the digits. If the result is divisible by 7 then the given number is divisible by 7. This rule comes from 41(7*7) Examples:

12 rule => |4*2-1| = 7 which is divisible by 7.

271Ɛ rule => |4*Ɛ-271| = 235(7*3Ɛ) which is divisible by 7(or apply the rule on 235).

OR Form the alternating sum of blocks of three from right to left. If the result is divisible by 7 then the given number is divisible by 7. This rule comes from 1001, since 1001 = 7*11*17, thus this rule can be also tested for the divisibility by 11 and 17. Example:

386,967,443 => 443-967+386 = -168 which is divisible by 7. 8 If the 2-digit number formed by the last 2 digits of the given number is divisible by 8 then the given number is divisible by 8. Example: 1Ɛ48, 4120 rule => since 48(8*7) divisible by 8, then 1Ɛ48 is divisible by 8. rule => since 20(8*3) divisible by 8, then 4120 is divisible by 8. 9 If the 2-digit number formed by the last 2 digits of the given number is divisible by 9 then the given number is divisible by 9. Example: 7423, 8330 rule => since 23(9*3) divisible by 9, then 7423 is divisible by 9. rule => since 30(9*4) divisible by 9, then 8330 is divisible by 9. ᘔ If the number is divisible by 2 and 5 then the number is divisible by ᘔ. Ɛ If the sum of the digits of a number is divisible by Ɛ then the number is divisible by Ɛ (the equivalent of casting out nines in decimal). Example: 29, 61Ɛ13 rule => 2+9 = Ɛ which is divisible by Ɛ, then 29 is divisible by Ɛ. rule => 6+1+Ɛ+1+3 = 1ᘔ which is divisible by Ɛ, then 61Ɛ13 is divisible by Ɛ. 10 If a number is divisible by 10 then the unit digit of that number will be 0. 11 Sum the alternate digits and subtract the sums. If the result is divisible by 11 the number is divisible by 11 (the equivalent of divisibility by eleven in decimal). Example: 66, 9427 rule => |6-6| = 0 which is divisible by 11, then 66 is divisible by 11. rule => |(9+2)-(4+7)| = |ᘔ-ᘔ| = 0 which is divisible by 11, then 9427 is divisible by 11. 12 If the number is divisible by 2 and 7 then the number is divisible by 12. 13 If the number is divisible by 3 and 5 then the number is divisible by 13. 14 If the 2-digit number formed by the last 2 digits of the given number is divisible by 14 then the given number is divisible by 14. Example: 1468, 7394 rule => since 68(14*5) divisible by 14, then 1468 is divisible by 14. rule => since 94(14*7) divisible by 14, then 7394 is divisible by 14.

Fractions and irrational numbers Edit

Fractions Edit Duodecimal fractions may be simple: 1 / 2 = 0.6

= 0.6 1 / 3 = 0.4

= 0.4 1 / 4 = 0.3

= 0.3 1 / 6 = 0.2

= 0.2 1 / 8 = 0.16

= 0.16 1 / 9 = 0.14

= 0.14 1 / 10 = 0.1 (note that this is a twelfth, 1 / ᘔ is a tenth)

= 0.1 (note that this is a twelfth, is a tenth) 1 / 14 = 0.09 (note that this is a sixteenth, 1 / 12 is a fourteenth) or complicated: 1 / 5 = 0.249724972497... recurring (rounded to 0.24ᘔ)

= 0.249724972497... recurring (rounded to 0.24ᘔ) 1 / 7 = 0.186ᘔ35186ᘔ35... recurring (rounded to 0.187)

= 0.186ᘔ35186ᘔ35... recurring (rounded to 0.187) 1 / ᘔ = 0.1249724972497... recurring (rounded to 0.125)

= 0.1249724972497... recurring (rounded to 0.125) 1 / Ɛ = 0.111111111111... recurring (rounded to 0.111)

= 0.111111111111... recurring (rounded to 0.111) 1 / 11 = 0.0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ... recurring (rounded to 0.0Ɛ1)

= 0.0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ... recurring (rounded to 0.0Ɛ1) 1 / 12 = 0.0ᘔ35186ᘔ35186... recurring (rounded to 0.0ᘔ3)

= 0.0ᘔ35186ᘔ35186... recurring (rounded to 0.0ᘔ3) 1 / 13 = 0.0972497249724... recurring (rounded to 0.097) Examples in duodecimal Decimal equivalent 1 × ( 5 / 8 ) = 0.76 1 × ( 5 / 8 ) = 0.625 100 × ( 5 / 8 ) = 76 144 × ( 5 / 8 ) = 90 576 / 9 = 76 810 / 9 = 90 400 / 9 = 54 576 / 9 = 64 1ᘔ.6 + 7.6 = 26 22.5 + 7.5 = 30 As explained in recurring decimals, whenever an irreducible fraction is written in radix point notation in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factors of its denominator are also prime factors of the base. Thus, in base-ten (= 2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: 1/8 = 1/(2×2×2), 1/20 = 1/(2×2×5) and 1/500 = 1/(2×2×5×5×5) can be expressed exactly as 0.125, 0.05 and 0.002 respectively. 1/3 and 1/7, however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, 1/8 is exact; 1/20 and 1/500 recur because they include 5 as a factor; 1/3 is exact; and 1/7 recurs, just as it does in decimal. The number of denominators which give terminating fractions within a given number of digits, say n, in a base b is the number of factors (divisors) of bn, the nth power of the base b (although this includes the divisor 1, which does not produce fractions when used as the denominator). The number of factors of bn is given using its prime factorization. For decimal, 10n = 2n * 5n. The number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together. Factors of 10n = (n+1)(n+1) = (n+1)2. For example, the number 8 is a factor of 103 (1000), so 1/8 and other fractions with a denominator of 8 can not require more than 3 fractional decimal digits to terminate. 5/8 = 0.625 ten For duodecimal, 12n = 22n * 3n. This has (2n+1)(n+1) divisors. The sample denominator of 8 is a factor of a gross (122 = 144), so eighths can not need more than two duodecimal fractional places to terminate. 5/8 = 0.76 twelve Because both ten and twelve have two unique prime factors, the number of divisors of bn for b = 10 or 12 grows quadratically with the exponent n (in other words, of the order of n2). Recurring digits Edit The Dozenal Society of America argues that factors of 3 are more commonly encountered in real-life division problems than factors of 5.[34] Thus, in practical applications, the nuisance of repeating decimals is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations. However, when recurring fractions do occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because 12 (twelve) is between two prime numbers, 11 (eleven) and 13 (thirteen), whereas ten is adjacent to the composite number 9. Nonetheless, having a shorter or longer period doesn't help the main inconvenience that one does not get a finite representation for such fractions in the given base (so rounding, which introduces inexactitude, is necessary to handle them in calculations), and overall one is more likely to have to deal with infinite recurring digits when fractions are expressed in decimal than in duodecimal, because one out of every three consecutive numbers contains the prime factor 3 in its factorization, whereas only one out of every five contains the prime factor 5. All other prime factors, except 2, are not shared by either ten or twelve, so they do not influence the relative likeliness of encountering recurring digits (any irreducible fraction that contains any of these other factors in its denominator will recur in either base). Also, the prime factor 2 appears twice in the factorization of twelve, whereas only once in the factorization of ten; which means that most fractions whose denominators are powers of two will have a shorter, more convenient terminating representation in duodecimal than in decimal representation (e.g. 1/(22) = 0.25 ten = 0.3 twelve ; 1/(23) = 0.125 ten = 0.16 twelve ; 1/(24) = 0.0625 ten = 0.09 twelve ; 1/(25) = 0.03125 ten = 0.046 twelve ; etc.). Values in bold indicate that value is exact. Decimal base

Prime factors of the base: 2 , 5

Prime factors of one below the base: 3

Prime factors of one above the base: 11

All other primes: 7 , 13 , 17 , 19 , 23 , 29 , 31 Duodecimal base

Prime factors of the base: 2 , 3

Prime factors of one below the base: Ɛ

Prime factors of one above the base: 11

All other primes: 5 , 7 , 15 , 17 , 1Ɛ , 25 , 27 Fraction Prime factors

of the denominator Positional representation Positional representation Prime factors

of the denominator Fraction 1/2 2 0.5 0.6 2 1/2 1/3 3 0. 3 0.4 3 1/3 1/4 2 0.25 0.3 2 1/4 1/5 5 0.2 0. 2497 5 1/5 1/6 2 , 3 0.1 6 0.2 2 , 3 1/6 1/7 7 0. 142857 0. 186ᘔ35 7 1/7 1/8 2 0.125 0.16 2 1/8 1/9 3 0. 1 0.14 3 1/9 1/10 2 , 5 0.1 0.1 2497 2 , 5 1/ᘔ 1/11 11 0. 09 0. 1 Ɛ 1/Ɛ 1/12 2 , 3 0.08 3 0.1 2 , 3 1/10 1/13 13 0. 076923 0. 0Ɛ 11 1/11 1/14 2 , 7 0.0 714285 0.0 ᘔ35186 2 , 7 1/12 1/15 3 , 5 0.0 6 0.0 9724 3 , 5 1/13 1/16 2 0.0625 0.09 2 1/14 1/17 17 0. 0588235294117647 0. 08579214Ɛ36429ᘔ7 15 1/15 1/18 2 , 3 0.0 5 0.08 2 , 3 1/16 1/19 19 0. 052631578947368421 0. 076Ɛ45 17 1/17 1/20 2 , 5 0.05 0.0 7249 2 , 5 1/18 1/21 3 , 7 0. 047619 0.0 6ᘔ3518 3 , 7 1/19 1/22 2 , 11 0.0 45 0.0 6 2 , Ɛ 1/1ᘔ 1/23 23 0. 0434782608695652173913 0. 06316948421 1Ɛ 1/1Ɛ 1/24 2 , 3 0.041 6 0.06 2 , 3 1/20 1/25 5 0.04 0. 05915343ᘔ0Ɛ62ᘔ68781Ɛ 5 1/21 1/26 2 , 13 0.0 384615 0.0 56 2 , 11 1/22 1/27 3 0. 037 0.054 3 1/23 1/28 2 , 7 0.03 571428 0.0 5186ᘔ3 2 , 7 1/24 1/29 29 0. 0344827586206896551724137931 0. 04Ɛ7 25 1/25 1/30 2 , 3 , 5 0.0 3 0.0 4972 2 , 3 , 5 1/26 1/31 31 0. 032258064516129 0. 0478ᘔᘔ093598166Ɛ74311Ɛ28623ᘔ55 27 1/27 1/32 2 0.03125 0.046 2 1/28 1/33 3 , 11 0. 03 0.0 4 3 , Ɛ 1/29 1/34 2 , 17 0.0 2941176470588235 0.0 429ᘔ708579214Ɛ36 2 , 15 1/2ᘔ 1/35 5 , 7 0.0 285714 0. 0414559Ɛ3931 5 , 7 1/2Ɛ 1/36 2 , 3 0.02 7 0.04 2 , 3 1/30 The duodecimal period length of 1/n are 0, 0, 0, 0, 4, 0, 6, 0, 0, 4, 1, 0, 2, 6, 4, 0, 16, 0, 6, 4, 6, 1, 11, 0, 20, 2, 0, 6, 4, 4, 30, 0, 1, 16, 12, 0, 9, 6, 2, 4, 40, 6, 42, 1, 4, 11, 23, 0, 42, 20, 16, 2, 52, 0, 4, 6, 6, 4, 29, 4, 15, 30, 6, 0, 4, 1, 66, 16, 11, 12, 35, 0, ... (sequence A246004 OEIS) The duodecimal period length of 1/(nth prime) are 0, 0, 4, 6, 1, 2, 16, 6, 11, 4, 30, 9, 40, 42, 23, 52, 29, 15, 66, 35, 36, 26, 41, 8, 16, 100, 102, 53, 54, 112, 126, 65, 136, 138, 148, 150, 3, 162, 83, 172, 89, 90, 95, 24, 196, 66, 14, 222, 113, 114, 8, 119, 120, 125, 256, 131, 268, 54, 138, 280, ... (sequence A246489 OEIS) Smallest prime with duodecimal period n are 11, 13, 157, 5, 22621, 7, 659, 89, 37, 19141, 23, 20593, 477517, 211, 61, 17, 2693651, 1657, 29043636306420266077, 85403261, 8177824843189, 57154490053, 47, 193, 303551, 79, 306829, 673, 59, 31, 373, 153953, 886381, 2551, 71, 73, ... (sequence A252170 OEIS) Irrational numbers Edit The representations of irrational numbers in any positional number system (including decimal and duodecimal) neither terminate nor repeat. The following table gives the first digits for some important algebraic and transcendental numbers in both decimal and duodecimal. Algebraic irrational number In decimal In duodecimal √ 2 , the square root of 2 1.414213562373... 1.4Ɛ79170ᘔ07Ɛ8... φ (phi), the golden ratio = 1 + 5 2 {\displaystyle {\tfrac {1+{\sqrt {5}}}{2}}} 1.618033988749... 1.74ƐƐ6772802ᘔ... Transcendental number In decimal In duodecimal π (pi), the ratio of a circle's circumference to its diameter 3.141592653589... 3.184809493Ɛ91... e , the base of the natural logarithm 2.718281828459... 2.875236069821...

See also Edit

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