# awtA

 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 Visit BLOG website http://akhmimwoodentablet.blogspot.com for further details: Visit MATHORIGINS.COM for further details: