(Letβs imagine that our world is full dozenal!!! e.g. the dozenal number 1,234,567 can be read as βone million two hundred thirty-four thousand five hundred sixty-sevenβ, and the number 10;^20; can also be simply called βone septillionβ) Tag: Visual edit |
(Undo revision 4923 by Xayahrainie43 (talk)) |
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The Harmonic System adopts 'day' as a unit of calendar time corresponding to one mean solar day, and uses the calendar time unit structure shown on the right figure. |
The Harmonic System adopts 'day' as a unit of calendar time corresponding to one mean solar day, and uses the calendar time unit structure shown on the right figure. |
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==5 Number counting== |
==5 Number counting== |
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β | The Universal Unit System recommends [http://www.asahi-net.or.jp/~dd6t-sg/univunit-e/myriad.pdf the |
+ | The Universal Unit System recommends [http://www.asahi-net.or.jp/~dd6t-sg/univunit-e/myriad.pdf the duodecimal myriad system]. See Appendix C of [http://www.asahi-net.or.jp/~dd6t-sg/univunit-e/revised.pdf revised.pdf]. |
+ | When designing number counting system, we must make sure that the same expression does not represent different numbers for decimal and dozenal context. |
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β | When units of the Harmonic System are combined with [[Systematic Dozenal Nomenclature]], only powers divisible with 4 or -1, -2, and -3 should be used. |
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+ | Since [[Systematic Dozenal Nomenclature]] fulfills this requirement, it can be used in combination with the Harmonic System. |
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+ | When units of the Harmonic System are combined with Systematic Dozenal Nomenclature, only powers divisible with 4 or -1, -2, and -3 should be used. |
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In this situation, dozen(10;), gross(100;) and doz gross(1000;) are used in substitution for powers 1, 2, and 3. |
In this situation, dozen(10;), gross(100;) and doz gross(1000;) are used in substitution for powers 1, 2, and 3. |
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β | |||
β | Also, the dozenal short scale system (dozenal thousand system): |
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β | |||
β | (n) (n-illion) (number of zeros: 3n+3) (value: 10;<sup>3n+3</sup>) |
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β | |||
β | 1 million 6 (1,000,000;) |
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β | |||
β | 2 billion 9 (1,000,000,000;) |
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β | |||
β | 3 trillion 10; (1,000,000,000,000;) |
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β | |||
β | 4 quadrillion 13; (1,000,000,000,000,000;) |
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β | |||
β | 5 quintillion 16; (1,000,000,000,000,000,000;) |
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β | |||
β | 6 sextillion 19; (1,000,000,000,000,000,000,000;) |
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β | |||
β | 7 septillion 20; (1,000,000,000,000,000,000,000,000;) |
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β | |||
β | 8 octillion 23; (1,000,000,000,000,000,000,000,000,000;) |
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β | |||
β | 9 nonillion 26; (1,000,000,000,000,000,000,000,000,000,000;) |
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β | |||
β | X dekrillion 29; (1,000,000,000,000,000,000,000,000,000,000,000;) |
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β | |||
β | E elpillion 30; (1,000,000,000,000,000,000,000,000,000,000,000,000;) |
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β | |||
β | 10; dozillion 33; (1,000,000,000,000,000,000,000,000,000,000,000,000,000;) |
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β | |||
β | 11; undozillion 36; (etc...) |
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β | |||
β | 12; duodozillion 39; |
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β | |||
β | 13; tredozillion 40; |
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β | |||
β | 14; quattuordozillion 43; |
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β | |||
β | 15; quindozillion 46; |
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β | |||
β | 16; sexdozillion 49; |
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β | |||
β | 17; septendozillion 50; |
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β | |||
β | 18; octodozillion 53; |
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β | |||
β | 19; novemdozillion 56; |
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β | |||
β | 1X; dekradozillion 59; |
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β | |||
β | 1E; elpindozillion 60; |
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β | |||
β | 20; vigintillion 63; |
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β | |||
β | 21 unvigintillion 66 |
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β | |||
β | 22 duovigintillion 69 |
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β | |||
β | 23 trevigintillion 70 |
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β | |||
β | 24 quattuorvigintillion 73 |
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β | |||
β | 25 quinvigintillion 76 |
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β | |||
β | 26 sexvigintillion 79 |
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β | |||
β | 27 septenvigintillion 80 |
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β | |||
β | 28 octovigintillion 83 |
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β | |||
β | 29 novemvigintillion 86 |
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β | |||
β | 2X dekravigintillion 89 |
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β | |||
β | 2E elpinvigintillion 90 |
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β | |||
β | 30 trigintillion 93 |
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β | |||
β | 31 untrigintillion 96 |
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β | |||
β | 32 duotrigintillion 99 |
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β | |||
β | 33 tretrigintillion X0 |
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β | |||
β | 34 quattuortrigintillion X3 |
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β | |||
β | 35 quintrigintillion X6 |
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β | |||
β | 36 sextrigintillion X9 |
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β | |||
β | 37 septentrigintillion E0 |
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β | |||
β | 38 octotrigintillion E3 |
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β | |||
β | 39 novemtrigintillion E6 |
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β | |||
β | 3X dekratrigintillion E9 |
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β | |||
β | 3E elpintrigintillion 100 |
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β | |||
β | 40 quadragintillion 103 |
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β | |||
β | 41 unquadragintillion 106 |
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β | |||
β | 42 duoquadragintillion 109 |
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β | |||
β | 43 trequadragintillion 110 |
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β | |||
β | 44 quattuorquadragintillion 113 |
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β | |||
β | 45 quinquadragintillion 116 |
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β | |||
β | 46 sexquadragintillion 119 |
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β | |||
β | 47 septenquadragintillion 120 |
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β | |||
β | 48 octoquadragintillion 123 |
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β | |||
β | 49 novemquadragintillion 126 |
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β | |||
β | 4X dekraquadragintillion 129 |
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β | |||
β | 4E elpinquadragintillion 130 |
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β | |||
β | 50 quinquagintillion 133 |
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β | |||
β | 51 unquinquagintillion 136 |
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β | |||
β | 52 duoquinquagintillion 139 |
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β | |||
β | 53 trequinquagintillion 140 |
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β | |||
β | 54 quattuorquinquagintillion 143 |
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β | |||
β | 55 quinquinquagintillion 146 |
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β | |||
β | 56 sexquinquagintillion 149 |
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β | |||
β | 57 septenquinquagintillion 150 |
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β | |||
β | 58 octoquinquagintillion 153 |
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β | |||
β | 59 novemquinquagintillion 156 |
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β | |||
β | 5X dekraquinquagintillion 159 |
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β | |||
β | 5E elpinquinquagintillion 160 |
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β | |||
β | 60 sexagintillion 163 |
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β | |||
β | 61 unsexagintillion 166 |
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β | |||
β | 62 duosexagintillion 169 |
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β | |||
β | 63 tresexagintillion 170 |
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β | |||
β | 64 quattuorsexagintillion 173 |
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β | |||
β | 65 quinsexagintillion 176 |
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β | |||
β | 66 sexsexagintillion 179 |
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β | |||
β | 67 septensexagintillion 180 |
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β | |||
β | 68 octosexagintillion 183 |
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β | |||
β | 69 novemsexagintillion 186 |
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β | |||
β | 6X dekrasexagintillion 189 |
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β | |||
β | 6E elpinsexagintillion 190 |
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β | |||
β | 70 septuagintillion 193 |
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β | |||
β | 71 unseptuagintillion 196 |
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β | |||
β | 72 duoseptuagintillion 199 |
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β | |||
β | 73 treseptuagintillion 1X0 |
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β | |||
β | 74 quattuorseptuagintillion 1X3 |
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β | |||
β | 75 quinseptuagintillion 1X6 |
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β | |||
β | 76 sexseptuagintillion 1X9 |
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β | |||
β | 77 septenseptuagintillion 1E0 |
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β | |||
β | 78 octoseptuagintillion 1E3 |
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β | |||
β | 79 novemseptuagintillion 1E6 |
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β | |||
β | 7X dekraseptuagintillion 1E9 |
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β | |||
β | 7E elpinseptuagintillion 200 |
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β | |||
β | 80 octogintillion 203 |
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β | |||
β | 81 unoctogintillion 206 |
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β | |||
β | 82 duooctogintillion 209 |
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β | |||
β | 83 treoctogintillion 210 |
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β | |||
β | 84 quattuoroctogintillion 213 |
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β | |||
β | 85 quinoctogintillion 216 |
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β | |||
β | 86 sexoctogintillion 219 |
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β | |||
β | 87 septenoctogintillion 220 |
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β | |||
β | 88 octooctogintillion 223 |
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β | |||
β | 89 novemoctogintillion 226 |
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β | |||
β | 8X dekraoctogintillion 229 |
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β | |||
β | 8E elpinoctogintillion 230 |
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β | |||
β | 90 nonagintillion 233 |
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β | |||
β | 91 unnonagintillion 236 |
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β | |||
β | 92 duononagintillion 239 |
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β | |||
β | 93 trenonagintillion 240 |
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β | |||
β | 94 quattuornonagintillion 243 |
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β | |||
β | 95 quinnonagintillion 246 |
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β | |||
β | 96 sexnonagintillion 249 |
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β | |||
β | 97 septennonagintillion 250 |
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β | |||
β | 98 octononagintillion 253 |
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β | |||
β | 99 novemnonagintillion 256 |
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β | |||
β | 9X dekranonagintillion 259 |
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β | |||
β | 9E elpinnonagintillion 260 |
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β | |||
β | X0 dekragintillion 263 |
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β | |||
β | X1 undekragintillion 266 |
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β | |||
β | X2 duodekragintillion 269 |
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β | |||
β | X3 tredekragintillion 270 |
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β | |||
β | X4 quattuordekragintillion 273 |
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β | |||
β | X5 quindekragintillion 276 |
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β | |||
β | X6 sexdekragintillion 279 |
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β | |||
β | X7 septendekragintillion 280 |
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β | |||
β | X8 octodekragintillion 283 |
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β | |||
β | X9 novemdekragintillion 286 |
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β | |||
β | XX dekradekragintillion 289 |
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β | |||
β | XE elpindekragintillion 290 |
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β | |||
β | E0 elpagintillion 293 |
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β | |||
β | E1 unelpagintillion 296 |
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β | |||
β | E2 duoelpagintillion 299 |
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β | |||
β | E3 treelpagintillion 2X0 |
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β | |||
β | E4 quattuorelpagintillion 2X3 |
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β | |||
β | E5 quinelpagintillion 2X6 |
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β | |||
β | E6 sexelpagintillion 2X9 |
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β | |||
β | E7 septenelpagintillion 2E0 |
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β | |||
β | E8 octoelpagintillion 2E3 |
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β | |||
β | E9 novemelpagintillion 2E6 |
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β | |||
β | EX dekraelpagintillion 2E9 |
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β | |||
β | EE elpinelpagintillion 300 |
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β | |||
β | 100 centillion 303 |
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β | |||
β | (there is no consistent and widely accepted way to extend cardinals beyondΒ centillion) |
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β | |||
β | For the dozenal long scale system (dozenal million system), "n" should be corresponding to 6n (10<sup>6n</sup>) instead of 3n+3 (10<sup>3n+3</sup>), and 6n+3 can be either "thousand n-illion" or "n-illiard". |
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[[Category:Pages]] |
[[Category:Pages]] |
Revision as of 16:19, 22 February 2019
The Harmonic System is one of variations of the Universal Unit System.
1 The Universal Unit System
By surprising coincidences, if the dozenal number system is used to express the speed of light in vacuum and the quantum of action as the defining constants such that these constants are strictly multiples of integer powers of twelve of the unit quantities, it is possible to construct a coherent unit system in which not only the constant that was used in the definition but also the Rydberg constant, the Bohr radius, the unified atomic mass unit, and half the value of the Planck length can be approximated to about or within an error of 2 per gross (11/2%) by a multiple of integer powers of twelve of the unit quantities.
In that case, many other physical constants, including the charge and mass of an electron, the fine structure constant, the molar volume of an ideal gas under standard conditions, the black-body radiation at the ice point, the density and surface tension of water, and others, can be approximated by multiples of integer powers of twelve of the unit quantities. Moreover, by adding the Boltzmann constant and using it in the definition of thermodynamic temperature, the gas constant of an ideal gas can be approximated by a multiple of an integer power of twelve of the unit quantity and the Stefan-Boltzmann constant, and the specific heat of water can be approximated by multiples of integer powers of twelve of the unit quantities with a factor 2 remaining.
For putting these coincidences to use, the dozenal number system is the only choice.
We define the Universal Unit System as βthe unit system that is constructed by using the dozenal number system and the speed of light in vacuum, the quantum of action, and the Boltzmann constant as the defining constants in such a way that these constants become strict multiples of integer powers of twelve of the unit quantities, and the Rydberg constant, the unified atomic mass unit, the Bohr radius, and half the value of the Planck length can be approximated by multiples of integer powers of twelve of the unit quantitiesβ.
(Please refer to dozenal.com for the overall Universal Unit System. The notation of this article is in principle according to footnote 1 of revised.pdf.)
2 Definition of the Harmonic System
The Harmonic System is conceptually designed and strictly defined as following:
2.1 Conceptual design
'Conceptual design' is the system design that prescribes the rough quantity of the units to constitute a unit system conceptually.
The Harmonic System concept is '35G=c0=ħ=kB=ZP=1' .(See also here(Wikipedia Talk:Planck_units).)
The Harmonic System is conceptually designed to express the following quantities as multiples of integer powers of twelve of the corresponding unit quantities.
- 2E;(35.) times Newtonian gravitational constant (35G)
- speed of light in a vacuum (c0)
- quantum of action (ħ)
- Boltzmann constant (kB)
- Planck impedance (ZP)
2.2 Strict definition
'Strict definition' is the system definition that corresponds strictly to the theoretical equations that a unit system is based on.
The Harmonic System is strictly defined to express the following quantities as the corresponding unit quantities.
- 100,1700; cycle over Rydberg constant
- 10;-8 times speed of light in a vacuum
- 10;+26; times quantum of action
- 10;+20; times Boltzmann constant
- Planck impedance
3 Units
All the units which have special names peculiar to the Harmonic System are listed as following:
3.1 Length
harmon [hm] - 272.352206 mm, 10.7225278 inches (8 / 9 feet, difference 0.52%)
This is slightly shorter than 1/1000,0000; of the quadrant meridian length of the Earth(=279.136507 mm, difference 2.5%).
Volume of a cube whose side is half harmon (i.e. cubic half harmon) is 2/3 U.S. liquid gallon.(difference 0.06%) Water in this volume includes about 100;10; H2O molecules.(see section 3.3)
100. yards equal 240; harmons.(difference 0.08%)
1 megalithic yard equals about 3 harmons.
3.2 Physical time
nic [nc] - 390.625115 ms (25./64. SI seconds, almost exact)
This is almost 1/1000; of the difference of the length of one Julian year and one mean tropical year.(difference 0.14% at the beginning of year 1900.)
100,000. SI seconds(difference 2.8%), 10,0000; nics(almost exact), and 1 ⅛ days are nearly equal.
1 harmon / nic (=atol [al]) equals 2.51 km / hour. (almost exact)
See also chapter 4.
3.3 Mass
looloh [ββ] - 131.829287 g, 4.65014126 ounces
This is slightly lighter than 1/100; of the mass of water in one cubic harmon capacity.(difference 6.4%)
On the other hand, this is slightly heavier than 1/100; of the mass of ice in the same capacity.(difference 2.4%)
'looloh' is named for the fact that it is 100;10; times unified atomic mass unit.(almost exact) It means that Avogadro constant is expressed by 100;10; times unit quantity(=reciprocal universal mol) of the Harmonic System.
The universal gravitational force between 2 mass points 5 loolohs and 7 loolohs put unit length apart is 10;-8 times force unit.
When we use the Harmonic System, temperature rises twice the unit temperature if we give water of the unit mass 'looloh' heat of the unit energy.(almost exact)
3.4 Impedance
nohm [nh, Ωn] - 29.9792458 Ω(exact)
This is the Planck impedance itself. The unit name 'nohm' is an abbreviation of 'natural ohm'.
The coulomb force between unit electric charges put unit length apart is 10;8 times force unit.
Relations of the electric and magnetic quantities in the Universal Unit System are shown as follows, where Ω2 is total solid angle of the spherical surface(see this ref):
Supplemental comments
These difference percentages show us that the Harmonic System is a kind of dozenalized metric system.
(See also '[Addition]' part of this article.)
Using the Harmonic System, many constants can be approximated by 2n×10;m times the unit quantity, where n = 0 or ±1 and m ∈ integer. The quadrant meridian length and gravitational acceleration of the Earth, the density and specific heat of water, which are used in the past for definition of units, are also included in these constants.
Please refer to this table and online converter for other units.
4 Time unit concepts
The Harmonic System distinguishes 'calendar time' from 'physical time' as a different concept.
4.1 Physical time
This quantity is time which flows uniformly from a viewpoint of physics.
'nic' is the physical time unit of the Harmonic System, and is on the natural time scale ladder of 'powers of twelve'.
The greatest common divisor of one tropical year and one day is 2-7 days, which is also equivalent to about 1000; nics.
4.2 Calendar time
This quantity is the coordinated mean rotation angle of the Earth derived by using the direction of the Sun as a coordinate origin, and its dimension is not time but plain angle.
The dimension of ratio of physical time and calendar time is [time / plain angle], which means the period length of one mean solar day. Because of tidal friction, this length is not constant and becomes longer little by little.
The unit that should be dozenal divisions of mean solar day is calendar time rather than physical time.
Because physical length of mean solar day is not constant but variable, the relation of mean solar day and any unit of physical time will collapse by all means in the future.
'day', as a unit, is equivalent to plain angle of total circumference.
The Harmonic System adopts 'day' as a unit of calendar time corresponding to one mean solar day, and uses the calendar time unit structure shown on the right figure.
5 Number counting
The Universal Unit System recommends the duodecimal myriad system. See Appendix C of revised.pdf.
When designing number counting system, we must make sure that the same expression does not represent different numbers for decimal and dozenal context. Since Systematic Dozenal Nomenclature fulfills this requirement, it can be used in combination with the Harmonic System. When units of the Harmonic System are combined with Systematic Dozenal Nomenclature, only powers divisible with 4 or -1, -2, and -3 should be used. In this situation, dozen(10;), gross(100;) and doz gross(1000;) are used in substitution for powers 1, 2, and 3.