The Rhind Mathematical Papyrus (RMP) outlined two classes of hekat (volume) units. The first volume unit replaced one hekat by a unity value (64/64). The second volume unit replaced a hekat by 320 ro, an unscaled hekat unity in RMP 36, and 10 hin and 64 dja, also unscaled hekat unities, in other RMP problems.

The first method is recorded in five Akhmim Wooden Tablet problems, as well in RMP 47, 82, and 83.

The second method converted 2/53, 3/53, 5/53, 15/53, 28/53, and 30/53 to unit fraction series in RMP 36. Initial and intermediate calculations infer the 320 ro substitution for 64/64. In RMP 35-38 and RMP 66, summarized by:

A. RMP 35: Find 3/10 of one hekat in ro units

1. $$320 ro * 3/10 = 96 ro$$

2. The 96 ro data created a unity from

a. $$320*3/10 = 96 ro$$

b. $$320*6/10 = 192 ro$$

c. $$320*1/10 = 32 ro$$

d. Unity sum $$320 ro = 1 hekat$$

B. RMP 36 solved $$3x + (1/3)x + 1/5(x) = 1 hekat$$

with LCM 15 such that:

$$(45x + 5x + 3x)/15 = 1$$

$$(53/15)x = 1$$

$$53x = 15$$ and

$$x = 15/53 hekat$$

as algebra was solved by Greeks, Arabs, the Liber Abaci written in 1202 AD, and the modern era, defining multiplication and division operations near to our modern definitions.

Ahmes converted 2/53, 3/53, 5/53, 15/53, 28/53 and 30/53 using LCMs allow red auxiliary numerators to partition rational numbers in optimized manners. The exception was 30/53. Ahmes converted $$2/53*(30/39) + 28/53*(2/2)$$ as 2/53 was listed in the 2/n table. For example $$2/53*(30/30) = 60/(30*53) = (53 + 5 + 2)/(30*53) = 1/30 + 1/318 + 1/795$$ with 5 + 2 written in red, details the initial, intermediate, and final calculations.

Ahmes combined RMP 18-23, completion lessons; with RMP 24 - 34, algebra lessons to find $$x = 15/53 hekat$$ .

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 also converted 2/53, 3/53, 5/53, 28/53, and 30/53 to unit fraction series by 2/n table red auxiliary numbers within two proofs.

1. The first 2/n tsble proof:

a. $$15/53*(4/4) = 60/212= (53 + 4 + 2 + 1)/212 = 1/4 + 1/53 + 1/106 + 1/212$$

b. $$(15/53)*2 = 30/53 = 2/53 + 28/53= (2/53)*(30/30) + (28/53)*(2/2) = 1/53 + 1/318 + 795 + 1/2 + 1/53 + 1/106$$

c. $$5/53 = (5/53)*(12/12) = (53 + 4 + 2 + 1)/(12*53)= 1/12 + 1/159 + 1/318 + 1/636$$

d. $$3/53 = (5/53)*(20/20) = (53 + 4 + 2 + 1)/(20*53)= 1/20 + 265 + 1/530 + 1/1060$$

e. sum: $$15/53 + 30/53 + 5/53 + 3/53 = 53/53 = one $$ (heket unity)

2. The second proof discussed 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 points, such as:

a. $$(20 + 10 + 5)$$ scaled $$15/53 = (3/53)*5 = 3/53 - 1/20) = (4 + 2 + 1)/212]*5 = (20 + 10 + 5)$$

b. $$(35 + 1/3) + (3 + 1/3) + (1 + 2/3) + 20 + 10$$ scaled $$28/53 + 2/53 = 30/53$$

c. $$(88 + 1/3) + (6 + 2/3) + (3 + 1/3) + (1+ 2/3)$$ scaled $$5/53= (5/3)*[(3/53)(15/15) = (53 + 4 + 2 + 1)] = 3/53$$

d. $$53 + 4 + 2 + 1$$ scaled $$(3/53) = (3/53)*(20/20)= 60/1060 = (53 + 4 + 2 + 1)/1060$$

e. Each part of $$15/53 = (1/4 + 1/53 + 1/106 + 1/212)hekat$$ namely 3/53 and 5/53 were multiples of 15/53.

Conclusion: Both proofs converted 2/53, 3/53, 15/5, 28/53, and 30/53, with $$30/53 = 2/53 + 28/53)$$ and

$$3/53 + 5/53 + 15/53 + 30/53 = 53/53 = one hekat (unity)$$

The unity aspect was mentioned by Peet citing $$45/53 + 5/53 + 3/53 = 1 hekat$$ (mentioned in the algebra problem). Ahmes' two proofs contained proto-number theory facts not mentioned by Peet, Chace or Clagett. For example, red auxiliary numbers show that proto-number theory was a central RMP 36 method. The proto-number method allowed n/pq to be generally converted by solving for $$2/pq + (n -2)/pq$$ when needed (as also discussed in RMP 31 converting $$28/97 = 2/97 + 26/97$$ . A second RMP 36 fact showed that Ahmes' used multiplication and division as inverse operations (as discussed in RMP 24-34).

C. RMP 37: Find 1/90 of a hekat in ro units, taking 1/3, 1/13 and so forth.

1. $$320 ro*(1/90) = 3 + 1/2 + 1/18 = 64/18$$

2. Ahmes playfully reported four unity sum methods, the first being:

a. $$320*(1/180) = 64/36$$

b. $$320*(1/360) = 64/72$$

c. $$320*(1/720) = 64/144$$

d. $$320*(1/1440) = 64/288$$

e. $$320*(1/2880) = 64/576$$

f. unity sum $$(b + e) = 64/72 + 64/576 = 1$$

The three additional unity sums are in draft form. Each unit sum method will be finalized in the coming weeks.

D. RMP 38 multiply one hekat by 7/22, written in ro units.

1. $$320*(35/11)*(1/10) = 320*(7/22)= (101 + 9/11)ro$$

2. Three implications of the proof are:

a. $$(101 + 9/11)*(22/7) = 320 ro = 1 hekat$$

b. A complete hekat was returned as Ahmes stressed by inverting the divisor 7/22 to 22/7. This meant that 22/7 was a better approximation for pi than 256/81 had been in the Old Kindom. A second aspect of the proof revealed scribal multiplication and division as inverse to one another, a property of modern arithmetic overlooked by Peet, Chace and Clagett.

c. A geometry implication considered the traditional hekat that used $$pi = 256/81$$ . The traditional pi approximation overstated inventory volumes. To attempt to correct for inventory losses a practical $$22/7$$ approximation was implemented by Ahmes.

E. RMP 66: divide 10 hekats of fat by 365 days by reporting a daily rate

1. $$3200/365 = 8 + 2/3 + 1/10 + 1/2190 = 8 + 280/365$$

2. The proof multiplied each unit fraction by 365, the inverted divisor.

a. $$365*8 = 2920 ro$$ b. $$365*(2/3) = (243 + 1/3)ro$$ c. $$365*(1/10)= (36 + 1/2)ro$$ d. $$365*(1/2190) = (1/6)ro$$

d. unity sum $$(a + b + c + d + e) = 320 ro = 1 hekat$$

F. Reference (1): A.B. Chace, Bull, L., Manning, H.P. and Archibald, R.C., The Rhind Mathematical Papyrus, Mathematical Association of America, Vol 1, 1927, vol 2, 1929, and reprint 1979 (NCTM) transliterated the scribal unit fraction shorthand. To translate Ahmes' shorthand to modern arithmetic statements initial calculations and other arithmetic steps are parsed and added back, as noted above. Milo Gardner and Bruce Friedman collaborated on this project.

G. Reference (2) Marshall Clagett, 1999, Egyptian Science and Mathematics (Volume III) includes Chace's 1927 views of the RMP, as well as transliterations of the Kahun Papyrus, the Moscow Mathematical Papyrus. The valid transliterations should not be considered complete translations. Missing initial and intermediate arithmetic steps were not parsed, and inserted, as well as the other mathematics, i.e. the attested arithmetic operations used by all Middle Kingdom scribes, info required to prepare complete translations.

H. Reference (3) Joran Friberg, 2005, "Unexpected links to Egyptian and Babylonian Mathematics" includes the outdated Gillings, Chace, and Peet 1920's transliterations contrasted to Babylonian mathematics. Again, Friberg did not add back the missing initial and intermediate Egyptian fraction statements, or discuss the Akhmim Wooden Tablet in a serious way to create valid translations to modern mathematics.