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Multiplication Tables of Various Bases by Michael Th Omas De Vlieger

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Multiplication Tables of Various Bases by Michael Th Omas De Vlieger Th e Dozenal Society of America Multiplication Tables of Various Bases by Michael Th omas De Vlieger Ever wonder what multiplication tables might look like in alternative bases? Th e dsa Updated 6 February 2011. presents the following tables in an eff ort to thoroughly explore number bases in general. Not long into any study of the multiplication tables, we require new numerals for digits Undecimal (Base 11) greater than 9 (“transdecimal” digits). At bases eleven and twelve, this document uses Numeral Set: the dsa-Classic or Dwiggins transdecimals, where a = digit-ten and b = digit-eleven. decimal equivalent Beginning with base thirteen, we need more digits. Here we use the author’s numeral set 012345678910 known as “arqam” (< Arabic, “numbers”). Th is number 0123456789a Binary (Base 2) set extends to about four hundred numerals, however undecimal digits 1 10 only sixty are currently available in the typeface. Th is 1 23456789a10 10 100 document will be expanded from time to time to in- clude tables for larger bases. 2 4 6 8 a 11 13 15 17 19 20 Ternar y (Base 3) 369 11 14 17 1a 22 25 28 30 1 210 Octal (Base 8) 481115 19 22 26 2a 33 37 40 2 11 20 1 23456710 5 a 14 19 23 28 32 37 41 46 50 10 20 100 2 4 6 1012141620 6 1117222833 39 44 4a 55 60 7 131a26323945 51 58 64 70 Quaternary (Base 4) 3611 14 17 22 25 30 4101420 24 30 34 40 8 15222a37445159 66 73 80 1 2310 5 12172431 36 43 50 9 172533414a586674 82 90 2 10 12 20 6 1422303644 52 60 a 192837465564738291 a0 31221 30 7 162534435261 70 10 20 30 40 50 60 70 80 90 a0 100 10 20 30 100 10 20 30 40 50 60 70 100 Note that there are no standard undecimal numerals. The nu- Quinary (Base 5) merals presented here are the set of Dwiggins duodecimal nu- Nonary (Base 9) merals whose values are less than the base r = decimal 11. The 1 23410 numeral “X” is employed by the International Standard Book 1 234567810 Number (isbn) as a check digit. 2 4 11 13 20 2 4 6 8 11 13 15 17 20 (Base 12) 31114 22 30 Dozenal 3610 13 16 20 23 26 30 Numeral Set: 4132231 40 481317 22 26 31 35 40 decimal equivalent 10 20 30 40 100 0 1 2 3 4 5 6 7 8 9 10 11 5 11162227 33 38 44 50 0123456789ab Senal (Base 6) 6 1320263340 46 53 60 dozenal digits 1 234510 7 152331384654 62 70 1 23456789ab10 2 4 10 12 14 20 8 17263544536271 80 2 4 6 8 a 10 12 14 16 18 1a 20 31013 20 23 30 10 20 30 40 50 60 70 80 100 369 10 13 16 19 20 23 26 29 30 4122024 32 40 Decimal (Base 10) 481014 18 20 24 28 30 34 38 40 5 14233241 50 5 a 13 18 21 26 2b 34 39 42 47 50 1 2345678910 10 20 30 40 50 100 6 1016202630 36 40 46 50 56 60 2 4 6 8 10 12 14 16 18 20 7 1219242b3641 48 53 5a 65 70 Septenary (Base 7) 369 12 15 18 21 24 27 30 8 14202834404854 60 68 74 80 1 2345610 481216 20 24 28 32 36 40 9 1623303946536069 76 83 90 2 4 6 11131520 5 10152025 30 35 40 45 50 a 18263442505a687684 92 a0 3612 15 21 24 30 6 1218243036 42 48 54 60 b 1a2938475665748392a1 b0 4111522 26 33 40 7 142128354249 56 63 70 10 20 30 40 50 60 70 80 90 a0 b0 100 5 13212634 42 50 8 16243240485664 72 80 Note that there are no standard dozenal numerals. The numerals 6 1524334251 60 9 1827364554637281 90 presented here are the set of Dwiggins duodecimal numerals, used 10 20 30 40 50 60 100 10 20 30 40 50 60 70 80 90 100 by the dsa between 1945 and 1974, then restored in 2008. Other numerals are proposed by other organizations and individuals. Multiplication Tables of Various Bases · De Vlieger The Dozenal Society of America Page 1 Tridecimal (Base 13) Pentadecimal (Base 15) Numeral Set: Numeral Set: decimal equivalent decimal equivalent 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0123456789abc 0123456789abcde tridecimal digits pentadecimal digits 1 23456789abc10 1 23456789abcde10 2 4 68ac11131517191b20 2 4 68ace11131517191b1d20 369 c 1215181b2124272a30 369 c 101316191c202326292c30 48c13 17 1b 22 26 2a 31 35 39 40 48c11 15 19 1e 22 26 2a 2e 33 37 3b 40 5 a 12 17 1c 24 29 31 36 3b 43 48 50 5 a 10 15 1a 20 25 2a 30 35 3a 40 45 4a 50 6 c 15 1b 24 2a 33 39 42 48 51 57 60 6 c 13 19 20 26 2c 33 39 40 46 4c 53 59 60 7 11182229333a 44 4b 55 5c 66 70 7 e 16 1e 25 2c 34 3b 43 4a 52 59 61 68 70 8 131b263139444c 57 62 6a 75 80 8 1119222a333b44 4c 55 5d 66 6e 77 80 9 15212a36424b5763 6c 78 84 90 9 131c263039434c56 60 69 73 7c 86 90 a 1724313b4855626c79 86 93 a0 a 15202a35404a55606a 75 80 8a 95 a0 b 19273543515c6a788694 a2 b0 b 17232e3a46525d697581 8c 98 a4 b0 c 1b2a39485766758493a2b1 c0 c 192633404c596673808c99 a6 b3 c0 10 20 30 40 50 60 70 80 90 a0 b0 c0 100 d 1b29374553616e7c8a98a6b4 c2 d0 Note that there are no standard tridecimal numerals. The numerals pre- e 1d2c4b4a5968778695a4b3c2d1 e0 sented here are the set of “arqam” numerals invented by Michael Thomas 10 20 30 40 50 60 70 80 90 a0 b0 c0 d0 e0 100 De Vlieger for transdecimal bases, between 1992–2007. There may be other proposals for transdecimal numerals (numerals symbolizing digits greater Note that there are no standard pentadecimal numerals. The numerals pre- than 9, used for bases greater than decimal.) sented here are the set of “arqam” numerals. There may be other numeral proposals for base 15. Quadrodecimal (Base 14) Numeral Set: Hexadecimal (Base 16) decimal equivalent Numeral Set: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 decimal equivalent 0123456789abcd 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 quadrodecimal digits 0123456789abcdef hexadecimal digits 1 23456789abcd10 2 4 68ac10121416181a1c20 1 23456789abcdef10 369 c 1114171a1d2225282b30 2 4 68ace10121416181a1c1e20 48c12 16 1a 20 24 28 2c 32 36 3a 40 369 c f 12 15 18 1b 1e 21 24 27 2a 2d 30 5 a 11 16 1b 22 27 2c 33 38 3d 44 49 50 48c10 14 18 1c 20 24 28 2c 30 34 38 3c 40 6 c 14 1a 22 28 30 36 3c 44 4a 52 58 60 5af1419 1e 23 28 2d 32 37 3c 41 46 4b 50 7 101720273037 40 47 50 57 60 67 70 6 c 12 18 1e 24 2a 30 36 3c 42 48 4e 54 5a 60 8 121a242c364048 52 5a 64 6c 76 80 7 e 15 1c 23 2a 31 38 3f 46 4d 54 5b 62 69 70 9 141d28333c47525b 66 71 7a 85 90 8 10182028303840 48 50 58 60 68 70 78 80 a 16222c3844505a6672 7c 88 94 a0 9 121b242d363f4851 5a 63 6c 75 7e 87 90 b 1825323d4a5764717c89 96 a3 b0 a 141e28323c46505a64 6e 78 82 8c 96 a0 c 1a28364452606c7a8896a4 b2 c0 b 16212c37424d58636e79 84 8f 9a a5 b0 d 1c2b3a495867768594a3b2c1 d0 c 1824303c4854606c788490 9c a8 b4 c0 10 20 30 40 50 60 70 80 90 a0 b0 c0 d0 100 d 1a2734414e5b6875828f9ca9 b6 c3 d0 Note that there are no standard quadrodecimal numerals. The numerals e 1c2a38465462707e8c9aa8b6c4 d2 e0 presented here are the set of “arqam” numerals. There may be other numeral f 1e2d3c4b5a69788796a5b4c3d2e1 f0 proposals for base 14. 10 20 30 40 50 60 70 80 90 a0 b0 c0 d0 e0 f0 100 The standard hexadecimal digits feature {A, B, C, D, E, F} with respective Th is document may be freely shared under the terms of the Creative Commons decimal values of {10, 11, 12, 13, 14, 15}. “Arqam” numerals are used here Att ribution License, Version 3.0 or greater. See htt p://creativecommons.org/li- censes/by/3.0/legalcode regarding the Creative Commons Att ribution License. for consistency. Page 2 The Dozenal Society of America Multiplication Tables of Various Bases · De Vlieger Hexadecimal (Base 16) Octodecimal (Base 18) Numeral Set: Numeral Set: decimal equivalent decimal equivalent 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0123456789abcdef 0123456789abcdefgh hexadecimal digits octodecimal digits 1 23456789abcde f10 1 23456789abcdefgh10 2 4 6 8 a c e 10121416181a1c1e20 2 4 68aceg10121416181a1c1e1g20 369 c f 10 13 16 19 1c 1f 20 23 26 29 2c 2f 30 369 c f 12 15 18 1b 1e 21 24 27 2a 2d 30 48cg 12 16 1a 1e 20 24 28 2c 2g 32 36 3a 3e 40 48c10 14 18 1c 20 24 28 2c 30 34 38 3c 40 5af1217 1c 1h 24 29 2e 31 36 3b 3g 43 48 4d 50 5af1419 1e 23 28 2d 32 37 3c 41 46 4b 50 6 c 10 16 1c 20 26 2c 30 36 3c 40 46 4c 50 56 5c 60 6 c 12 18 1e 24 2a 30 36 3c 42 48 4e 54 5a 60 7 e 13 1a 1h 26 2d 32 39 3g 45 4c 51 58 5f 64 6b 70 7 e 15 1c 23 2a 31 38 3f 46 4d 54 5b 62 69 70 8 g 16 1e 24 2c 32 3a 40 48 4g 56 5e 64 6c 72 7a 80 8 10182028303840 48 50 58 60 68 70 78 80 9 1019202930394049 50 59 60 69 70 79 80 89 90 9 121b242d363f4851 5a 63 6c 75 7e 87 90 a 121c242e363g48505a 62 6c 74 7e 86 8g 98 a0 b 141f28313c454g59626d 76 7h 8a 93 9e a7 b0 a 141e28323c46505a64 6e 78 82 8c 96 a0 c 16202c36404c56606c7680 8c 96 a0 ac b6 c0 b 16212c37424d58636e79 84 8f 9a a5 b0 d 18232g3b46515e69747h8c97 a2 af ba c5 d0 c 1824303c4854606c788490 9c a8 b4 c0 e 1a26323g4c5864707e8a96a2ag bc c8 d4 e0 d 1a2734414e5b6875828f9ca9 b6 c3 d0 f 1c293643505f6c798693a0afbcc9 d6 e3 f0 e 1c2a38465462707e8c9aa8b6c4 d2 e0 g 1e2c3a48566472808g9eacbac8d6e4 f2 g0 f 1e2d3c4b5a69788796a5b4c3d2e1 f0 h 1g2f3e4d5c6b7a8998a7b6c5d4e3f2g1 h0 10 20 30 40 50 60 70 80 90 a0 b0 c0 d0 e0 f0 100 10 20 30 40 50 60 70 80 90 a0 b0 c0 d0 e0 f0 g0 h0 100 Note that there are no standard octodecimal numerals.
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