Independent Researcher

Retired, doing independent research

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Old Kingdom arithmetic was written in an infinite series system that rounded off unity (1) = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... by throwing away a 1/64 unit. The rounded off binary numeration system was "healed" in hieratic Egyptian fraction mathematics.

Middle Kingdom scribes ciphered rational numbers in 2/n tables in the Kahun Papyrus (1800 BCE) and the RMP (1650 BCE) and solved over 200 unit fraction problems. Scribes applied theoretical unit fraction arithmetic in algebra, geometry and practical wage payments made in hekat (grain) units.

"Healing" was term used by Tanja Pemmerening in 2005 to report a hieroglyphic dja symbol. Middle Kingdom scribes scaled the dja to a 1/64 hekat. The smallest grain unit was 1/320 of a hekat (named ro). Ro units were remainders in inventory control and wage payment problems.

Scribes in the MK wrote

unity (1) = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 10 ro

and corrected OK approximations by exact statements.

At times MK scribes recorded rational numbers 2/64 to 10/320, and n/64 to 5n/320, by LCM 5 in algebra and geometry problems such that:

(8 + 2)/320 = 1/40 + 1/160

recording unity as:

1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/40 + 1/160

and applied the finite arithmetic formula

(64/64)/n = Q/64 + (5R/n)ro

with Q = quotient and R = remainder

five times in the 1950 BCE Akhmim Wooden Tablet and over 60 times in the RMP.