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AWT pg 2
To read the 2,000 BC hieratic version of the above five operations, four considerations (A, B, C, D) must be brought into clear focus. The first two are A & B below.
A. A general rule was available for all n < 65, (64/64)/n = Q/64 + R/(n*64). For n > 64, as the RMP suggestions by n = 70, a hekat unity was increased to mod 10 or mod 100, thereby retaining a value for the Q term.
B. The second consideration is the Q term, stated nearly as a Horus-Eye series, using only the divisors of 64 (1, 2, 4, 8, 16, 32 ). For example 21 = 16 + 4 + 1 stated that (16 + 4 + 1)/64 = (1/4 + 1/16 + 1/64)
Using A & B, the above five partitons are
| 1. 1/3 = (1/4 + 1/16 + 1/64)hekat + 1/(3*64) | 2. 1/7 = (1/8 + 1/64)hekat + 1/(7*64) |
| 3. 1/10 = (1/16 + 1/32)hekat + 4/(10*64) | 4. 1/11 = (1/16 + 1/64)hekat + 9/(11*64) |
| 5. 1/13 = (1/16)hekat + 12/(13*64) | |
C. A third consideration involved the remainder term where two substitutions took place: one replaced 1/64 with 5/320; and a second replaced 1/320 with the word ro, inching closer to the historical data, as given by:
| 1. 1/3 = (1/4 + 1/16 + 1/64)hekat + 5/3 ro | 2. 1/7 = (1/8 + 1/64)hekat + 5/7 ro |
| 3. 1/10 = (1/16 + 1/32)hekat + 2 ro | 4. 1/11 = (1/16 + 1/64)hekat + 45/11 ro |
| 5. 1/13 = (1/16)hekat + 60/13 ro | |
D. Finally, a fourth consideration completed the conversion of the ro term, taking up several lines in the AWT, preparing a standardized Egyptian fraction series, those being:
| 1. 5/3 = (1 + 2/3) ro | 2.
5/7 = 1/2 + (10 - 7)/14 = 1/2 + (2 + 1)/14 = (1/2 + 1/7 + 1/14) ro |
| 3. 2 ro was not changed | 4. 45/11 = (4 + 1/11)ro |
| 5. 60/13 = 4 + 8/13 = 4 + (1/2 + (2 + 1)/26) = (4 + 1/2 + 1/13 + 1/26)ro | |
references: correspondences with Milo Gardner; AWT blog; Math Forum; 46 Lessons in Early Geometry;
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