Analysis by Milo Gardner: The Achmim Wooden Tablets (C.G. 25.367 and C.G. 25.368) | |||||||||||

LAST MODIFIED: 3/9/05 | |||||||||||

Middle Kingdom texts exactly computed a hekat, a unit of grain, written partly in the | |||||||||||

` | Horus-Eye system and also partly in the Hieratic system. The myth of Anubis, "monitor of | ||||||||||

the scales of truth" suggests that this modular arithmetic system may date to the Old | |||||||||||

Kingdom, Saqqara. There was a gate** at Saqqara that metaphorically solved the Horus-Eye | |||||||||||

balance beam problem. Today this means that there was a literal scaling of Horus Eye | |||||||||||

binary fractions by a substitution process, that mathematicans may title as parametrics. | |||||||||||

Modular arithmetic statements of radix (quotients) and remainders were written out in | |||||||||||

the Middle Kingdom and later times, following an exact system that may not have been | |||||||||||

mathematically altered after its birth. Names within the exact 1/n divisor system of volume | |||||||||||

measurement may have changed over 2,000 years or more of Egyptian history, however, | |||||||||||

the modular arithmetic that began by replacing 1/64 with 5/320 in the remainder term, did not. | |||||||||||

Radix 10 (Q) | Remainder (R) | ||||||||||

with ro = 1/320, after 1/64 replaced by 5/320 | |||||||||||

Q | 5*R | ||||||||||

(64/64)/n = | --------------- | + ---------- * ro | and (5*R)/n being written as exact Egyptian fractions | ||||||||

64 | n | series, as noted below, maybe for the first time. | |||||||||

Historical Sources: | |||||||||||

AWT: The Achmim Wooden Tablets | |||||||||||

(C.G. 25.367 and C.G. 25.368) Two plastered wood planks inscribed in Hieratic. 12th Dynasty | |||||||||||

RMP: The Rhind Mathematical Papyrus | |||||||||||

(BM 10057 and 10058) Hieratic papyrus originally connected, now torn in two with small fragments in NY | |||||||||||

**Gate at Saqqara: Text reference; gate location | |||||||||||

Unity means that 1 = 64/64 in the hekat case | |||||||||||

remainder divisor | |||||||||||

Source & | hin unit size | Hekat | divisor | Horus-Eye | quasi ro units | quasi ro units | remainder units | was an LCM = (n*320) before the ro factoring | actual ro units | ||

Problem # | Unity | (n) | quotient | factors | (Q) | (R) | (R) | ||||

implied | 10 hin | 64/64 | 64 | 64 | 1 hekat | 320 | |||||

RMP #81 | 8.75 | 64/64 | 1 1/7 | 56 | 32+ 16 + 8 | 320 | 0 | 0 | 365.7143 | 0.875 | 280 |

RMP #81 | 8.125 | 64/64 | 1 3/13 | 52 | 32+ 16 + 4 | 320 | 0 | 0 | 393.8462 | 0.8125 | 260 |

RMP #81 | 7.8125 | 64/64 | 1 7/25 | 50 | 32+ 16 + 2 | 320 | 0 | 0 | 409.6 | 0.78125 | 250 |

RMP #81 | 7.65625 | 64/64 | 1 15/49 | 49 | 32+ 16 + 1 | 320 | 0 | 0 | 417.9592 | 0.765625 | 245 |

RMP #81 | 7.5 | 64/64 | 1 1/3 | 48 | 32 + 16 | 320 | 0 | 0 | 426.6667 | 0.75 | 240 |

RMP #81 | 6 hin | 64/64 | 1 3/5 | 40 | 32 + 8 | 320 | 0 | 0 | 512 | 2/3 hekat | 200 |

RMP #81 | 5.625 | 64/64 | 1 7/9 | 36 | 32 + 4 | 320 | 0 | 0 | 568.8889 | 0.5625 | 180 |

RMP #81 | 5.3125 | 64/64 | 1 15/17 | 34 | 32 + 2 | 320 | 0 | 0 | 602.3529 | 0.53125 | 170 |

RMP #81 | 5.15625 | 64/64 | 1 31/33 | 33 | 32 + 1 | 320 | 0 | 0 | 620.6061 | 0.515625 | 165 |

RMP #81 | 5 hin | 64/64 | 2 | 32 | 32 | 320 | 0 | 0 | 640 | 0.5 | 160 |

RMP #81 | 5.15625 | 64/64 | 2 2/3 | 24 | 16 + 8 | 320 | 0 | 0 | 853.3333 | 0.515625 | 165 |

AWT #3 | 3 hin | 64/64 | 3 | 21 | 16 + 4 + 1 | 315 | 5 | 1 | 960 | 0.375 | 120 |

RMP #81 | 3.125 | 64/64 | 3 1/5 | 20 | 16 + 4 | 320 | 0 | 0 | 1024 | 0.3125 | 100 |

RMP #81 | 2.8125 | 64/64 | 3 4/9 | 18 | 16 + 2 | 320 | 0 | 0 | 1102.222 | 0.28125 | 90 |

RMP #81 | 2.65625 | 64/64 | 3 13/17 | 17 | 16 + 1 | 320 | 0 | 0 | 1204.706 | 0.265625 | 85 |

RMP #81 | 2 hin | 64/64 | 4 | 16 | 16 | 320 | 0 | 0 | 1280 | 1/4 hekat | 80 |

RMP #81 | 2 hin | 64/64 | 5 | 12 | 12 | 300 | 20 | 4 | 1600 | 1/5 hekat | 64 |

RMP #81 | 1.5 hin | 64/64 | 6 | 10 | 10 | 300 | 20 | 4 | 1920 | 1/6 hekat | 53.33333 |

AWT #7 | 64/64 | 7 | 9 | 6 + 2 | 315 | 5 | 1 | 2240 | 1/7 hekat | 45.71429 | |

RMP #81 | 1.2 hin | 64/64 | 8 | 8 | 8 | 320 | 0 | 0 | 2560 | 1/8 hekat | 40 |

AWT #10 | 1 hin | 64/64 | 10 | 6 | 6 | 300 | 20 | 4 | 3200 | 1/10 hekat | 32 |

AWT #11 | 0.909090909 | 64/64 | 11 | 5 | 4 + 1 | 275 | 45 | 9 | 3520 | 0.090909 | 29.09091 |

RMP #81 | 0.769230769 | 64/64 | 13 | 4 | 4 | 260 | 60 | 12 | 4160 | 0.076923 | 24.61538 |

RMP #81 | 2/3 hin | 64/64 | 15 | 3 | 2 + 1 | 255 | 65 | 13 | 4800 | 0.066667 | 21.33333 |

RMP #81 | 5/8 hin | 64/64 | 16 | 3 | 2 + 1 | 300 | 20 | 4 | 5120 | 0.0625 | 20 |

RMP #81 | 1/2 hin | 64/64 | 20 | 2 | 2 | 230 | 90 | 18 | 6400 | 0.05 | 16 |

RMP #81 | 1/3 hin | 64/64 | 30 | 2 | 2 | 300 | 20 | 4 | 9600 | 0.033333 | 10.66667 |

RMP #81 | 5/16 hin | 64/64 | 32 | 2 | 2 | 310 | 10 | 2 | 10240 | 0.03125 | 10 |

RMP #81 | 1/4 hin | 64/64 | 40 | 1 | 1 | 185 | 135 | 27 | 12800 | 0.025 | 8 |

RMP #81 | 1/6 hin | 64/64 | 60 | 1 | 1 | 205 | 115 | 23 | 19200 | 0.016667 | 5.333333 |

RMP #81 | 9/32 hin | 64/64 | 64 | 1 | 1 | 225 | 95 | 19 | 20480 | 0.015625 | 5 |

RMP #47 | 0.142857143 | 6400/64 | 70 | 91 | 64+16+8+2+1 | 170 | 150 | 30 | 22400 | 0.014286 | 4.571429 |

implied | 0.03125 | 64/320 | 320 | 0 | 0 | 319 | 1 | 1/5 | 102400 | 0.003125 | 1 |

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