Egyptian Mathematics

Circa 2700 BC Egyptians introduced the earliest fully developed base 10 numeration system. Though it was not a positional system, it allowed the use of large numbers and also fractions in the form of unit fractions and Eye of Horus fractions, or binary fractions.By 2700 BC, Egyptian construction techniques included precision surveying, marking north by the sun's location at noon. Clear records began to appear by 2000 BC citing approximations for π and square roots. Exact statements of number, written arithmetic tables, algebra problems, and practical applications with weights and measures also began to appear around 2000 BC, with several problems solved by abstract arithmetic methods.

For example, the Akhmim Wooden Tablet (AWT) lists five divisions of a unit of volume called a hekat, beginning with one hekat unity valued as 64/64. The hekat unity was divided by 3, 7, 10, 11 and 13, with all answers being exact. The first half of the answers cite a binary quotient, i.e. one hekat (64/64), divided by 3, found a quotient 21 with a remainder of 1. The scribe wrote 21 as (16 + 4 + 1), such that a binary series was obtained by (16 + 4 + 1)/64 = 1/4 + 1/16 + 1/64. The second half of the answer scaled the remainder one (1) to 1/320th (ro) units or 1/(192) = (5/3)*1/320 = (1 + 2/3)*ro.

The scribe combined the quotient and remainder into one statement. The 1/3rd of a hekat answer was written as: 1/4 1/16 1/64 1 2/3 ro. Scribal addition and multiplication signs are not seen. Note that the scribal series was written from right to left. The scribe proved all of his results by multiplying the answers by its initial divisors, finding the initial hekat unity value of(64/64 all five times. The AWT scribe wrote out this exact partitioning method in more detail, a method that was shorteded by Ahmes and other Middle Kingdom scribes. Ahmes' steps did not include the proof aspect, for example. However, Ahmes' partitioning steps, however, did follow the AWT's two-part structure, using it 29 times in Rhind Mathematical Papyrus #81.

Hana Vymazalova published in 2002 a fresh copy of the AWT that showed that all five AWT divisions had been exact, by first parsing the proof steps, returning all five division answers to 64/64. Vymazalova thereby updated Daressy's 1906 incomplete discussion of the subject that had only found 1/3, 1/7 and 1/10 to be exact.
Beyond the fact that (64/64)/n = Q/64 - (5R/n)*ro, with Q = quotient and R = remainder, fairly states the 2,000 BCE scribal form of hekat division, two additional facts reveal early scribal thinking. One fact reveals that whenever the divisor n was between 1/64 and 64 a limit of 64 had been reached. RMP 80 details this two-part limit. Second, to go beyond the divisor n = 64 limit, hin, ro and other sub-units of the hekat were developed.

Gillings summaries the RMP data with 29 examples in an appendix, thereby contrasting the two-part statements to the equivalent one-part hin statements. The medical texts and of a hekat, and 320/n ro for 1/320th of a hekat for prescription ingredients.its 2,000 examples also used the extended one-part formats following: 10/n hin for 1/10th
Ahmes was able to go beyond the 64 divisor limit and its two-part remainder arithmetic in other ways, one being to increase the size of the numerator. The two-part hekat partitioning method was described in problem 35 as 100 hekat divided by n= 70. Ahmes wrote 100*(64/64)/70 = (6400/64)/70 = 91/64 + 30/(70*64). The quotient was written as (64 + 16 + 8 + 2 + 1)/64 =(1 + 1/4 + 1/8 + 1/32+ 1/64). Ahmes then wrote the remainder part as (150/70)*1/320 = (2 + 1/7)ro. Finally, the combined 1 1/4 1/8 1/32 1/64 2 1/7 ro answer was written down following the right to left, using no arithmetic addition or multiplication