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RMP 36, 37 and the EMLR
Posted:
May 5, 2010 3:39 PM
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RMP 36 and RMP 37 validate that Ahmes used EMLR-type non-optimal unit fraction series, created from red auxiliary numbers, within proofs.
RMP 36 solved
3x+(13)x+15(x)=1hekat
(45x+5x+3x)15=1
(53/15)x=1
53x=15
and
x=15/53 hekat
very much like algebra problems were solved Greeks, Arabs, and medieval scribes. Ahmes used a form of modern multiplication and division operations. To Ahmes the division of a rational number by another rational number was obtained by inverting the divisor, and multiplying. Ahmes' multiplication operation mentally calculated many answers. But with division and multiplication answers in-hand, Ahmes applied the traditional duplation multiplication operation as a proof.
In the conversion of 2/53, 3/53, 5/53, and 15/53, all were scaled to the same numerator 60, by different LCMs m, that obtained different denominators 53m. A fifth rational number, 28/53, obtained a numerator 56. Considering all five conversions Ahmes scaled n/53 to mn/53m, with numerator mn additively parsed by selecting the best divisors of m. The selected divisors of m were denoted in red ink. A required conversion of 30/53 to a unit fraction series was obtained by the substitution of
28/53+2/53
.
In RMP 18-23, the scribe worked "completion to 1 algebra problems" that practiced the selection of LCMs. In RMP 24 - 34, algebra lessons were worked that led up to scribe
obtained in RMP 36
x=15/53 hekat
Ahmes converted 15/53 to a unit fraction series by considering:
(15/53)(4/4)= 60/212=(53+4+2+1)212=
(1/4+1/53+1/106+1.212)hekat
.
Ahmes converted 2/53, 3/53, 5/53, 28/53, and 30/53 to unit fraction series by following 2/n table red auxiliary rules within two proofs.
1. The first 2/n table proof considered:
a. 15/53(4/4)=60/212=(53+4+2+1)212=1/4+1/53+1/106+1/212
b.30/53=2/53+28/53=(2/53)(3030)+(28/53*(2/2) =
1/30+1/318+1/795+ 1/2+1/53+1/106
c.5/53=(5/53)(12/12)=(53+4+2+1)636=
1/12+1/159+1/318+/1636
d.3/53=(3/53)*(20/20)=(53+4+2+1)1060)=
1/20+1/265+1/530+1/1060
e. sum: 15/53+30/53+5/53+3/53=53/53= one(heket unity)
2. The second proof considered 2/53, 3/53, 5/53, 15/53, 28/53 and 30/53) as parts of a hekat in terms of red auxiliary numbers, and other issued per:
a. (20+10+5) scaled 15/53= (4+2+1)212 =(20+10+5)
with 20 + 10 + 5 recorded in red ink.
b. 28/53+2/53=30/53
c. (88 + 1/3) + (6 + 2/3) + (3 + 1/3) + (1+ 2/3)= 100
scaled 3/53 = 60/1060 = (53 + 4 + 2 + 1)/1060
d. red numbers (53+4+2+1) had scaled
(3/53)=(3/53)*(2/020)=60/1060=(53+4+2+1)/1060
e. Each part of
15/53=(1/4+1/53+1/106+1/212)hekat
3/53 and 5/53 are multiples of 15/53.
Conclusion: Proofs converted 2/53, 3/53, 15/5, 28/53, and 30/53, with 30/53 = 2/53+28/53) and
3/53+5/53+15/53+30/53=53/53= one hekat(unity)
Unity aspects were mentioned by Peet as
45/53+5/53+3/53 = 1hekat
.
Ahmes' proofs contained proto-number theory that were not mentioned by Peet, Chace or Marshall Clagett, Ancient Egyptian Science, Vol III have 1999.
For example, red auxiliary numbers were incorrectly parsed by Peet, Chace and Clagett in RMP 36. RMP 36 generally converted one difficult n/p, 30/53, by solving for (n?2/)p +2/ p, a method that was used in RMP 31 to convert 28/97 = 2/97+26/97
.
A second set of RMP 36 facts showed that Ahmes' used multiplication and division as inverse operations (as discussed in RMP 24-34) as well as converting 7/212, 7/106, 7/53 and 3/53 by red auxiliary proofs that summed to 265, showing the parts of 1060 as 530 + 265 + 265.
C. RMP 37: Found 1/90 of a hekat in ro units from:
1. 320 ro (190)= 3 + 1/2 +1/18 = 64/18
2. Ahmes playfully reported four unity sum methods. Three inverse red number calculations included EMLR-like conversion of 1/4 = 72/288 = (9 + 18 + 24 + 3 + 8 + 1 + 8 + 1)/288, with additive numerators recorded in red ink.
Ahmes aligned red numbers (9 + 18 + 24 + 3 + 8 + 1 + 8 + 1) below a non-optimal(1/32 + 1/16 + 1/12 + 1/96 + 1/36 + 1/288 + 1/36 + 1/288) series. The paired lines meant that red integers were inverses of unit fractions. Ahmes recorded 1/8 as 72/576 with (8 + 36 + 18 + 9 + 1) recorded in red, again below (1/72 + 1/16 + 1/32 + 1/64 + 1/576), another EMLR-like non-optimal series, ending a playful problem.
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