Wednesday, November 12, 2008

Least Common Multiples, modern and ancient

Ciphered Egyptian numerals were used within hieratic script after 2050 BCE, with zero written as sfr for accounting and other purposes. One of the oldest texts was the Akhmim Wooden Tablet (AWT). To translate hieratic Egyptian mathematics and hieratic unit fraction arithmetic to modern base 10 fractions missing mental steps must be added back. In the AWT missing steps summed to an initial (64/64) hekat unity. To prove that the correct missing scribal steps are outlined a doubling check method that follows scribal shorthand proof steps must be introduced. The five AWT division of (64/64) by 3, 7, 10, 11 and 13 answers were multiplied by initial divisors and returned (64/64) five times. Hana Vymazalova published the (64/64) hekat unity in 2002.

Incomplete translations were offered by 20th century math historians that did not report the (64/64) hekat unity, and subtle scribal math facts. Clearly 20th century math historians stressed language observations transliterations correctly reporting ancient arithmetic symbols without reporting subtle mathematical subjects beginning with hard-to-read Reisner Papyrus , EMLR and Ahmes 2/n table construction methods.

For example, A.B Chace in 1927 concluded that a translation of the RMP was complete. Chace indirectly discussed Vymazalova's hekat unity approach in Ahmes' bird (hekat) feeding problem (RMP 87). Ahmes' data was garbled so that no modern scholar can decipher the intended facts, per Chace's view.

One RMP entry suggests that a least common multiple (LCM) unity was common as Ahmes thought and wrote out

3/11 (6/6) - 18/66 = (11 + 6 + 1)/66 = 1/6 + 1/11 + 1/66

with (6/6) being a unity that scaled 3/11 to a solvable vulgar fraction, recorded (11 + 6 + 1) in red, and omitted the order initial details.

As readers of Egyptian math history are aware Ahmes' red auxiliary numbers was an idea associated with LCMs. The precise nature of scribal LCMs has come to light.

There are four ways to discuss Egyptian proofs, taken from ,

that 20th century scholars had not considered are summarized by:

1. Read/translate RMP 38, that solved the problem:

a. 10 hekat equals 3200 ro times 7/22 equals 101 9/11

b. proving 101 9/11 times 22/7 equals 3200 ro

c. with a footnote that 35/11 times 1/10 = 35/110 = 7/22

2. Read/translate RMP 47, that solved the problem

a. 100 hekat equals 6400/64 hekat divided by 70

b. proving (91/64)hekat + 150/70 ro was the answer

3. Read/translate RMP 66 problem

a. 10 hekat times 32o ro equals 3200 ro, divided by 365, the number of days per year

b. 3200/365 equals 8 + 280/365 with

c. a duplation proof for the quotient 8, by writing

365 1
730 2
1460 4
2920 8

and the remainder (3200 - 2920) equals 280 by

(243 1/3 + 36 1/2 + 1/6)/365 = 280/365

4. RMP 82 an began with a hekat unity, (64/64), divided by 29 divisors n in the range 1/64 < n < 64, as RMP 83 divided (64/64), divided by 6, 20 , and 40. A final Russian entry generally used an ancient aliquot fraction or ratio idea (Russian terms for red auxiliary numbers) without identifying a hypothetical use of ancient LCMs per:

Note the straight forward modern and ancient arithmetic that easily fills in Ahmes use of red auxiliary LCMs per: