## Wednesday, June 22, 2005

### Akhmim Wooden Tablet (AWT)

INTRODUCTION
-----------------------

The Akhmim Wooden Tablet may date to 2,000 BCE, 12th dynasty, or as late as 15th dynasty. The tablet is housed in the Cairo, Egypt Museum. It is 46.5 x 26n cm in size and mentions 27 servant names, an unknown king's name (citing the 8th year of his reign), five division calculations, one of which was repeated four times and five proofs. The document was reported in 1901 and analyzed and published in 1906 by Georges Daressy. Daressy indicated five divisions by 3, 7, 10, 11 and 13, and wrote out exact 1/p unit fraction series, and validated three of the five proofs.

Daressy discussed the AWT in terms of binary fractions and minimized aspects of the five Egyptian fraction series. Typos and other errors muddled the 1/11 and 1/13th multiplications. Exactness was not identified in the scribal proof for the 1/11 and 1/13 cases. Daressy cited the cubit-cubit rather than the hekat, the actual AWT context (Peet's main complaint).

However, Daressy's cubit view was consistent with a binary fraction remainder arithmetic, and exact partitions. Peet did not identify several scribal arithmetic facts. Daressy's approach properly analyzed AWT data that included binary fractions and scaled remainders.

A small number of scholars worked on the 1/10th of a hekat (a volume unit) named hin, hinu, or henu scaled a linear cubit to a cubit-cubit-cubit within a hekat unity (64/64) such that 1/320 of a hekat was named ro

(64/64)/10 hekat = (6/64 + 4/640)hekat =

(4 + 2)/64 hekat + 20/10 ro =

(1/16 + 1/32)hekat + 2ro =

1 hin

Ahmes in Rhind Mathematical Papyrus (RMP) 81 used 29 binary hekat quotients + scaled ro remainders by following the theoretical statement

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

with Q = Quotient, R = Remainder, and

n limited to the range 1/64 < n < 64. The AWT and RMP two-part statements used one part statements. For example: 10/n hin simply meant a hekat was scaled to a 1/10 unit named hin was the limit scholarly discussions. Peet in 1923 incompletely discussed the AWT's binary fractions statements that conflicted with Daressy's earlier work. Peet reported 1/3, 1/7, 1/10, 1/11, 1/13 multiplication aspects of the problems and only stressed the 1/320 ro unit. Peet had not reported the meta (64/64)/n division that reported five binary quotient plus a 1/320 remainder answers and proofs. Peet under reported the scribal context and modern translation of the AWT n = 11 and 13 cases, correctly reported by: a. (64/64)/11 hekat = (5/64 + 9/704)hekat = (4 + 1)/64 hekat + (45/11)ro = (1/16 + 1/64)hekat + (4 + 1/11)ro b. (64/64)/13 hekat = (4/64 + 12/832)hekat = 4/64 hekat + (60/13)ro = 1/16 hekat + (4 + 8/13)ro with 8/13 scaled by LCM 2 to 16/26 = (13 + 2 + 1)/26 = 1/2 + 1/13 + 1/26 recorded as: (64/64)/13 hekat = 1/16 hekat + (4 + 1/2 + 1/13 + 1/26)ro It 80 years for Vymazalova to correctly report the proof context of the AWT arithmetic story. Hana Vymazalova reported the proof side of the five two-part statements that returned (64/64) five times when multiplied by the initial divisors. Vymazalova did not challenge Peet's calculation views of the five AWT binary quotient and 1/320 remainder answers, points that were corrected in 2006.

Daressy's 1906 review of the AWT's data garbled the n = 11 and n = 13 proofs thereby confusing Peet and later researchers. In 2002 by Hana Vymazalova corrected Daressy's two proof errors. Vymazalova's corrections and other meta points were published in 2006 and 2011.
The AWT reported a well-defined system of weights and measures arithmetic from an Old Kingdom inexact system to an exact Middle Kingdom rational number system.

The meta context in which Peet properly identified the 1/320 ro aspect of the AWT was missed until Hana Vyamazalova's 2002 paper. Scholars for almost 100 years did not connect the AWT partition method to RMP 81 and 29 data points, and over 30 additional two-part partitions of a (64/64) hekat unity discussed in the RMP.

CONTENTS OF THE AWT
------------------------------
Translated to our modern base 10, the AWT simply states that unity (64/64th) was divided by 3, 7, 10, 11 and 13 following a general rule of division, as is clearly read by:

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

with Q = quotient, and R = Remainder

writing the initial problem in modern base 10 fractions.

Middle Kingdom scribes used this pattern, by easily writing the quotient term into a Horus-Eye series, for example (64/64)/3 = 21/64 (Q). Several scholars have seen this portion, but become foggy with respect to the remainder, one (1) in the case of n = 3.

The 2nd portion, the handling of the remainder, has been grossly confused by scholars. One reason can be excused since R/(n*64) was 'encoded' by scribes replacing 1/64th with equivalent 5/320 remainders. This allowed student scribes to add Q and R values as one number.

To the average scholar seeing an Egyptian fraction series, actually (5R/n) followed by ro (long known to be 1/320) did not 'feel' like a remainder component. However, Ro, clearly used as a common divisor, can also have been seen by Ahmes as n LCM, or even a GCD. Whatever ro's meaning to Ahmes, its details was left to modern code breakers.

Only a few scholars opnenly 'scratched' their respective heads when seeing AWT and RMP two-part data. Two recent scholars: Robins-Shute saw (6400/64)/70 one of 10 RMP 47 problems. Robins-Shute did not report the two-part Q/64 + (5R/70)ro expression in a 1987 RMP book. Robins-Shute fairly reported products and remainders contained in the data, a form of scratching of their respective heads.

An earlier motivated scholar was Chace. In a 1927 RMP book, RMP 83 reported three data sets for (64/64) divided by n = 6, 20, and 40. The bird-feeding rates made little sense to him, suggesting that Ahmes had garbled the data. Chace mentioned that Ahmes left no clues to on this matter, a personal point that is obviously incorrect when reading the AWT in its broader context.

Factually, Ahmes and the AWT scribes created

Q/64 +(5R/n)(1/320)statements, over 60 times following the same theoretical style.

For none mathematicians reading, it may be best to refer to the modern base 10 version of the 4,000 year old arithmetic notation (shorthand), and then, say a few days later, actually attempt write in the ancient 2-part notation. That is, clearly think in our modern base 10 for a few days, before trying to think and write as a 4,000 year old scribe.

There are four trees being discussed here, each suggesting confusion, unless care is taken. They are: (1) the Horus-Eye notation, followed by a hekat, written in the first half of the expression, (2) Hieratic Egyptian fraction notion, followed by ro in the second half of the expression, (3) the Egyptian fraction series represented only the rational number (5*R/n), noting n in the denominator, allowing it to grow to any size, thereby allowing an exact computation, every time, and (4)the word ro, as 1/320th was factored from the remainder term.

Clearly ro was a minor term, in the expression, possibly only a common divisor factor, used to add the Horus-Eye and Egyptian fraction series together, allowing a proof to be quickly performed, as was listed five times in the AWT (and not at all in the RMP).

It is therefore recommended that novice readers of this blog do not allow one or two of the different types of trees that you run into, to be confusing. Look for the forest of each type of notation, stated as clearly as your education allows. Then and only then try to read and work with the inner workings of the AWT, such as an individual tree. The work on a complete (64/64)/n division problem as the 2,000 BC student scribe was trained.

Have patience, each of the ideas are simple, seen separately. Taking in the set of anjcient ideas at once causes problems for many people. Please avoid shortcuts, especially the ones that Ahmes himself practiced, until you discover the foundations of Ahmes' arithmetic, such as the ancient methods that created 2/n tables.

Peet, Gillings and other may have taken a couple of modern shortcuts, missing an ancient tree or two.

Begin at the beginning of each of the AWT problems, and compare your beginning, middle and end work with the same type of problem written out in the RMP (#47, 81, 83 are the best examples). Then work to the end of each AWT problem, doing your own work, every step of the way.

You will be rewarded. Spend the necessary time to work through more than one ancient problem as scribes solved it, using all of the old tools.

BACKGROUND
-------------------
Thomas E. Peet, 1923, partially repeated an analysis of the AWT by showing several connections between Egyptian math and the practical experiences of an ancient Egyptian scribe that used two numeration systems, Horus-Eye and the Egyptian fractions (cited in the RMP). Peet muddled where one system ended and where the other system began by only detailing additive aspects of the two numeration systems, missing the exact Egyptian division features using 5ro as a partitioning idea, using numerators and denominators = 320 in an interesting way. Peet also prematurely concluded that the Egyptian division was only an inverse of the Egyptian multiplication operation.

Peet did not directly discuss Egyptian division, as a general operation, as confirmed by the AWT examples. However, contrary to Gillings and Robins-Shute, Peet did seem to compute with 5ro, 4ro, 3ro, 2 ro and ro, but only from a limited view of the AWT student. Peet was slightly myopic, asking few meta questions, such as: were all of the student's divisions required to be exact? More importantly, no comparisons of Peet's view of the AWT were made to the RMP and its 84 problems. At least ten RMP problems, 36-43, and 81-82, have been misread with respect to ro, suggesting it was a weights and measures unit. Ro was actually connected to a generalized partitioning role, as closely related to other exact partitioning methods cited in the RMP, and other Middle Kingdom mathematical texts.

Scholars are, of course, free to explore these issues on their own, commenting on the actual mix of Egyptian mathematics that meets a few of the standards that are deduced from the AWT, and its interesting set of division methodologies.

Peet, and two later scholars, Gillings, 1972, and Robin-Shute, 1987, show that all the three scholars prematurely concluded in independent analyses that MK ro data (from the RMP and AWT) only meant 1/320 of a hekat, and no more. Not one of the three scholars grasped Ahmes central fact, that quotients and scaled ro remainders defined a weights and measures unit by beginning with a hekat unity (64/64), and dividing by any rational number less than 64.

Returning to Peet, and his analysis will be shown in the next few paragraphs. He apparently made serious errors with respect to ro and its relationship to Egyptian division, as was vividly declared in the AWT hekat and 1/64 divided by 1/n context. Peet did not see ro's actual association with remainders, though he mentioned remainders from time to time. It is clear that 64 times 5, an early form of mod 5 in the R/3 term, included the use of numerators and denominators, or, by example, let the divisor of 64/64 be n, then

Q + R/3

appeared in two shorthand forms, the first being

(1) (64/64)/n = Q/64 + R/(n*64).

(2) (64/64)/n = Q/64 + (5*R/n)* 1/320, with ro = 1/320

Ahmes used the second form. The first form is the manner in which modern mathematicians read this type of information.

One of the most exciting aspects of the AWT is that the rational number (5*R/n) was easily converted to an Egyptian fraction series. It not known if this was the
first generalized used of Egyptian fractions.

Silverman's 1975 point, that Egyptian fractions were found in the Old Kingdom, may mean that the balance beam problem was solved by an Old Kingdom scribe, noting:

R/(n*64)

with the Egyptian fractions series being either R/n or R/(n*64).

Egyptian remainder arithmetic research continues. Thanks to the ancient scribes for leaving red flags raised by the AWT so that the simple five division problems opens a major door to decoding Egyptian fraction math from any era prior to 800 AD.

Daressy in 1906 discussed Egyptian fraction math as remainder based. Daressy saw the exact divisions of some unit labeled in cubit units. Going beyond Daressy's analysis, a vivid clue to the Old Kingdom manner of partitioning cubit units, the hekat was partitioned in a manner that defined Egyptian fraction math for 2,800 years, ending in 800 AD.

A required meta view of the AWT included Egyptian economy and absentee landlords and Pharaohs that used the Egyptian fraction math for practical applications.

author: Milo Gardner

## Sunday, June 19, 2005

### Forward and Reverse Engineering AWT & RMP data

Nine data points will be forward and reverse engineered as ancient
scribes would have recognized. Five data points are found in the
Akhmim Wooden Tablet, 1/3, 1/7, 1/10, 1/11 and 1/13, and four
are found in the RMP, 1/6, 1/20, 1/40 and 100/70, following the
modern remainder arithmetic form:

(64/64)/n = Q/64 + R/(n*64)

with Q = quotient and R = remainder.

For example, AWT 1 divides 1/3 of a hekat as:

(64/64)/3 = 21/64 + 1/(3*64),

then the scribe converted the remainder data (often mentally) into
a fixed common divisor 1/320, a number that scribes named ro.
To better explain the scibal method of compulation, which may
have looked like this to the scribe:

(64/64)/3 = (4' 16' 64')hekat + ( 1 3")ro, consider the following,

A. Forward computation: follow a remainder arithmetic structure,
computing a quotient (Q) and a remainder (R), a two-part statement:

1. AWT 1:

A Foward Engineeriing

(64/64)/3 = 21/64 (Q) + 1/(3*64)(R)
= (16 + 4 + 1)/64) + (5/5)*1/192)
= (4' 16' 64') hekat + 5/960
= (4' 16' 64') hekat + (5/3) ro
= (4' 16' 64') hekat + (1 3") ro

Note, the 2nd remainder step multiplied 5/5 * 1/(3*64) = 5/(3*320)
thereby creating one common divisor multiple, 1/320, for all five
AWT solutions.

B. Reverse engineering computation: begin with (4' 16' 64') hekat
+ (1 2")ro and find the original problem. First, it must be noted that
2' = 32/64. 4' = 16/64, 8' = 8/64, 16' = 4/64, 32' = 2/64 and
64' = 1/64, or:

(1) Quotient: (4' 16' 64') = (16 + 4 + 1)/64 = 21/64

(2) Remaidner, Egyptian fraction: ( 1 2/3) = 5/3

(3) Remainder, including ro: 5/3 ro = 5/(3*320)
= 5/960
= 1/192

C: Double check the result consists of taking a Q value like 21/64
and an R values, such as 1/192 and adding them:

21/64 + 1/192 = (63 + 1)/192 = 64/192 = 1/3.

2. RMP 83 - 1

A. Forward Engineering

(64/64)/6 = 10/64 + 4/(6*64)
= (8 + 2)/64 + (20/6 * 1/320)
= (8' 32') hekat + ( 3 3') ro

B. Reverse Engineering

(1) Quotient part (8' 32') = (8 + 2)/64 = 10/64

(2) Remainder part (3 3') ro = 10/3 * 1/320 = 10/960 = 1/96

C. Proof ( Q + R):

10/64 + 1/96 = (30 + 2)/192 = 32/192 = 1/6

which means that B (2) actually should read:
(3 3') ro = 20/6 * 1/320 = 20/1920 = 1/96

3. AWT 2

A. Forward Engineering

(64/64)/7 = 9/64 + 1/(7*64)
= (8 + 1)/64 + (5/7)* 1/320
= (8' 64') hekat + (2' 7' 14') ro

B. Reverse Engineereing

(1) quotient part: (8' 64') = (8 + 1)/64 = 9/64

(2) remainder part: convert Egyptian fraction

2' 14' 28' = (7 + 2 + 1)/14 = 10/14 = 5/7, or

(3) remainder part including ro : 5/7 ro = 5/(2240) = 1/448

C.Proof: (Q + R):

9/64 + 1/448 = (63 + 1)/448 = 64/448 = 1/7

4. AWT 3

A. Forward Engineering

(64/64)/10 = 6/64 + 4/(10*64)
= (4 + 2)/64 + 20/(10)* 1/320
= (16' 32') hekat + 2 ro

B. Reverse Engineering

(1) quotient: (16' 32') = (4 + 2)/64 = 6/64

(2) remainder: 2 ro = 2/320 = 1/160

C. Proof (Q + R):

6/64 + 1/160 = (30 + 2)/320 = 32/320 = 1/10

5. AWT 4

A. Forward Engineering

(64/64)/11 = 5/64 + 9/(11*64)

= (4 + 1)/64 + (45/11)* 1/320

= (16' 64') hekat + (4 11') ro

B. Reverse Engineering

(1) quotient: (16' 64') = (4 + 1)/64 = 5/64

(2) remainder, Egyptian fraction: (4 11') = ( 44 + 1)/11 = 45/11

(3) remainder including ro: 45/11 ro = 45/11 * 1/320 =
45/(11*320) = 9/(11*64) = 9/704

C. Proof: (Q + R):

5/64 + 9/704 = (55 + 9)/704= 64/704 = 1/11

6. AWT 5

A. Forward Engineering

(64/64)/13 = 4/64 + 12/(13*64)
= 16' + (60/13)* 1/320
= 16' hekat + (4 8/13 ) ro
= 16' hekat + (4 2' 13' 26') ro

B. Reverse Engineering

(1) quotient: 16' = 4/64

(2) remainder, Egyptian fraction:

4 2' 13' 26' = (104 + 13 + 2 + 1)/26
= 120/26 = 60/13
(3) remainder including ro

(60/13)*ro = 60/(13*320) = 12/(13*64)

C. Proof: (Q + R):

4/64 + 12/(13*64)= (52 + 12)/(13*64)
= 64/(13*64) = 1/13

7. AWT 83 - 2

A. Forward Engineering

(64/64)/20 = 3/64 + 4/(20*64)
= (2 + 1)/64 + (20/20)*1/320
= (32' 64') hekat + 1 ro

B. Reverse Engineering

(1) quotient: (32' 64') = (2 + 1)/64 = 3/64

(2) remainder, Egyptian fraction 1, insufficient data

(3) remainder including ro : 1 ro = 1/320

C. Proof: (Q + R):

3/64 + 1/320= (15 + 1)/320
= 16/320 = 1/20

which means the Egyptian fraction step B (2) should be 20/20,
or 4/(20*64)

8. RMP 83 - 3

A. Forward Engineering

(64/64)/40 = 1/64 + 24/(40*64)
= 64' + (120/40)*1/320
= 64' hekat + 3 ro

B. Reverse Engineering

(1) quotient: 64' = 1/64

(2) remainder, Egyptian fraction 3, insufficient data

(3) remainder including ro: 3 ro = 3/320

C. Proof (Q + R):

1/64 + 3/320 = (5 + 3)/320= 8/320 = 1/40

which means that the Egyptian fraction was 120/40, or
24/(40*64)

9 . RMP 47

A. Forward Engineering

(6400/64)/70 = 91/64 + 30/(70*64)
= (64 + 16 + 8 + 2 + 1)/64
+ (150/70)*1/320

= (1 4' 8' 32' 64')hekat
+ (2 7') ro

B. Reverse Engineering:

(1) Quotient: (1 4' 8' 32' 64') = (64 + 16 + 8 + 2 + 1)/64
= 91/64

(2) Remainder, Egyptian fraction: (2 7') = 15/7

(3) Remainder including ro: (15/7) ro = 15/(7*320)
= 3/(7*64)

C. Proof: (Q + R)

91/64 + 3/(7*64) = (637 + 3)/(7*64)
= 640/(7*64)
= 10/7

The 100/70 fact seems hidden. However, the point that is found
in the scribal narrative, meaning the initial problem was 100/70.

In summary the forward and reverse processes work equally well
for the 33 additional data points found in RMP 69 and 81. Oddly
scholars reporting RMP 81 only cited hin data. A hin was 1/10th
of a hekat, and implementing forward and reverse engineering takes
the hin fact into account, while exactly scaling to 1/3200th in the
remainder term as scribes first calculated the data.

## Monday, October 18, 2004

### Egyptian Arithmetic (Appendix)

INTRODUCTION: Defining any arithmetic system, at any time,
or anywhere, requires the use of numbers. In the Egyptian
case, Old Kingdom hieroglyphic numbers were many to one,
such as listing the symbol one(1) four times to show that the
number four (4) was being listed. This numeral system was
superceded by ciphered numerals at the beginning of the Middle
Kingdom, about 2200 BC. The ciphered Hieratic numerals began
with one (1), and proceeded to over one million, as was the case
forthe its predecessor. Hieratic numbers were represented within a
base 10 system, mapping each number onto an alphabetical symbol.
To save space the actual listing of numbers versus letters
will not be cited here.

Zero itself was known and used for many situations, in the Old and
New Kingdom. However, zero did not appear in either of the
Egyptian numeration systems, nor any Ancient Near East numeration
system.The first use of a numeration based zero, outside of Europe
and its base 10 application, was in ancient Mexico, and its fully
positional base 20 numeration system (the later being bases 4 and
5, not a pure base 20 system, counting only 0-19). More on zero
later. It is a topic worthy of study, noting that many books have

But, getting back to Western numeration, after number itself was
defined, following the Greek tradition, Europeans began to accept
Vedic- Arabic base 10 numerals, around 900 AD, though zero was
not formalized within a numeration system in that wave of diffusion.
Around 1200 AD, zero did arrive in Europe, first in Germany, and
then it began to formalized in our base 10 decimal positional
system and its four functions of arithmetic, addition, subtraction,
multiplication and division, as we know them today. Given this
incomplete history, other math subjects need to be brought into
the discussion.

THE BASE 10 DECIMAL EXAMPLE

Stevins wrote two books in 1585 AD, one for business and one
for science, that may have first formalized base 10 decimals, using
zero as a place holder. Addition began with the use of rational
numbers, usually a counting number, or integer, a, b, c, and d,
all unequal such that:

1. addition: a + b = c

Addition was straight forward, when all the numbers involved
were > 1. However, Greeks and many cultures had trouble
accepting the general use of a negative number, so even here
problems had arisen in certain cultures. Stevin's work resolved
a range of issues with his definition of our base 10 decimal system.

2. subtraction: a - b = d

Subtraction was apparently straight forward when a > b. However,
when a remainder involved a fraction, a translation to base 10
decimals took place, defining a method of round off was sometimes
required. It should be recalled that Arabic-Vedic history had been
closely linked to the Babylonian infinite series numeration, with
several of base 60 thinking ending up in Stevin's analysis and
final product. Citing an extreme Babylonian case, subtraction base
remainders, stated as fractions, were rounded off when certain
situations took place. For Babylon, one of their cases cited
d = 1/91. Round off too place such that only multiples of 2, 3 and 5
were used as denominators, usually rounding off to 1/90. Stevins
view of base 10 round off was not than complicated. We round off
today based on a predetermined number of significant digits, so
accuracy has been left to the user, to the businessman and to the
scientist.

3. Multiplication: a x b = ab = e

Multiplication was seen as repeated addition, such as
5 x 7 = 35 as, 5 x 1 = 5, added seven time, or 7 x 1 = 7,
added together 5 times. Tables have been memorized
over the years, the easiest ones being based on prime
numbers.

4. Division: a divided by n meant = q + r/n (but from this point
a general notation of A/n = Q + R/n will be used, following the
shorthand notation presented in three Middle Kingdom
mathematical texts, Akhmim Wooden Tablet, 2000 BCE:
Reisner Papyri, 1800 BCE and the RMP or Ahmes
Papyrus, 1650 BCE.
These texts defined R = remainder,
somtimes allowing 0, an integer or a fraction, preceded
by Q = quotient.

A rigorous definition of the factor and remainder theorem, a method
that is reported only in terms of algebra, will not be offered at this
time. At some future point the pure arithmetic arithmetic aspects
will be compared to the ancient versions of Egyptian two-part
numbers. However, to fill a logical hole, I offer Oystein Ore's
analysis on this general history of arithmetic (number theory)
subject, as his history of number theory developed in Greece and
elsewhere, based on LCM's, GCD's, aliquot parts and a
range of other ideas.

Concerning the sophistication of Egypt arithmetic or any cultural
view of arithmetic, the manner in which remainders were handed
offers significant insight into the depth their mathematical thinking.
To understand Egypt's Middle Kingdom arithmetic, a close look
at its small set of mathematical texts need to take place.

MIDDLE KINGDOM ARITHMETIC

The four arithmetic operations will be defined in reverse order,
division, multiplication, subtraction, and addition. In this
manner the interesting aspects of division can be high-lighted,
revealing a few of the complexities of Egyptian thinking. As a
basic point, Egyptian math tended to be limited to the domain of
positive rational numbers, with only a small number of exceptions.
For the purposes of this discussion, only positive rational numbers
will be considered.

1. Division

The AWT lits five problems of the division of unity (64/64) by 3, 7, 10,
11 and 13. A shorthand notation was developed that we can recognize
as following our modern base 10 structure, as given by:

(64/64)/n = Q/64 + R/64.

A simplier Reisner and RMP version says

p/n or pq/n = Q + R/n

with p and q being primes, without the additional hekat
unity substitution.

Divisors n were easily read in exact ways for both forms of
arithmetic when remainders were converted to vulgar fractions
and finally to an Egyptian fraction series.

This shorthand notations is explained by one division example, let n = 3,
then Q = 21, R= 1 such that a final answer in our modern base 10 would
have been:

64/64 divided by 3 = 21/64 + 1/(3*64) = 320/960

This form of thinking lies at the center of the discussion of
the AWT and its five division problems. The most complicated
division was dividing a fraction by a fraction, in these cases
1/64 divided by 1/n (with n < 64). The Akhmim Wooden Tablet
lists five of these divisions. The division process was simplified
by a mental process that first divided 64 by n, producing a two
part answer, and quotient (Q) and a remainder(R). Suspecting
that the history of Egyptian numeration was wished to be closely
associated with the Old Kingdom, the Q portion was stated as
a Horus-Eye series. R was stated in the Middle Kingdom mode
of numeration, Hieratic fractions.

In the case of 64/3, Q = 21, such that the question, which
divisors of 64 (1, 2, 4, 8, 16, 32, 64) added up to 21 was asked?
To an ancient scribe 16 + 4 + 1 achieved 21, meaning that 1/64th
fractional inverses of 16, 4 and 1, were easily written out as
(1/4 + 1/16 + 1/64) hekat.

Concerning the 64/3 remainder, R = 1, meant that 1/3 our simple
answer of 1/(3*64) = 1/192 was not written. What was written
can be explained by 1/64 being replaced by 5/320, such that
1/3 x 5/320 = 5/960, or (1 + 2/3) ro was written as the remainder
portion, or totally,

(1/4 + 1/16 + 1/64) hekat (1 + 2/3)ro.

Continuing with . the 64/3 case, 1/192 was written out as 5/3 ro or
(1 + 2/3)ro using Hieratic script, since ro = 320, which also meant
that:

(5*Q*n + 5*R)

was the numerator, which always added up to 320, or generally

320/(n*320) = 1/n

In that way 1/n was multiplied by n, as the AWT did with its
five division problems, thereby finding and confirming that
a 100% accurate hekat was found in each case.

The AWT data is confirmed in the Rhind Mathematical Papyrus,
or the Ahmes Papyrus, if you prefer, by ro being used several
times, always mistranlated by historians, for one myopic reason or
another. Strange but true.

2. Multiplication: repeated additions can be used to describe the
operation, as an introduction. Duplation tables were listed for one
of the numbers being multiplied such that the final answer was picked
from the table. At this time, a rigorous discussion of multiplication
will not be offered.

The AWT proved each of its 1/3, 1/7, 1/10, 1/11, and 1/13 valuations
by multiplying by 3, 7, 10, 11 and 13, respectively. Examples taken
from the AWT will be listed at a later time.

3. Subtraction: This operation can be seen as a reminder, much as
the division operation calculates a remainder. Seen as a remainder,
Egytians did not allow round off, except when irrational numbers like
pi were being discussed, no matter how small the fractions that were
involved. The exact remainder for all rational number substractions
sets Egyptian method at a higher level than its Babylonian counter
parts. Five examples follow, as variations of the AWT theme,
beginning with unity (1), and subtracting: 1/3, 1/7, 1/10, 1/11 and 1/13.

The Hultsch-Bruins 2/p method will be used, as first re-discovered in
1895, as easily generalized to the n/p case:

1. 1 - 1/3 = 2/3

(The EMLR lists 1/3 + 1/3, but usually 2/3 was considered prime,
and not reduced to lower partitions)

2. 1 - 1/7 = 6/7

= 1/2 + (12 - 7)/14, since 5 can not be found amidst the divisors of 14,

= 1/2 + 1/3 + (15 - 14)/(3*14)

= 1/2 + 1/3 + 1/42

(check 2/3 for the first partition, an interest series)

3. 1 - 1/10 = 9/10

= 1/2 + (18 - 10)/20, with 8 being found amidst the divisors of 20

= 1/2 + (5 + 2 + 1)/20

= 1/2 + 1/4 + 1/10 + 1/20

( try 2/3 as the first partition, it is also interesting)

4. 1 - 1/11 = 10/11
= 1/2 + (20- 11)/22, with 20-11 = 9 not amidst the divsors of 22, therefore
= 1/2 + 1/3 + (27 - 22)/66, with 27 - 22 = 5 admidst divisors of 66, such that
= 1/2 + 1/3 + (3 + 2)/66
= 1/2 + 1/3 + 1/22 + 1/33

5. 1 - 1/13 = 12/13
= 1/2 + (24 - 13)/26, with 11 not being amidst the divisors of 26, therefore
= 1/2 + 1/3 + (33 - 26)/72, with 7 being amidst the divisors of 75, such that
= 1/3 + 1/3 + 1/13 + 1/39

(of course 2/3 can be a first partition, revealing 2/3 + 1/4 + 1/156, as an
alternative)

4. Addition: a + b stated as unit fraction series were added, with
ease, allowing for the understanding of the general conversion
of n/p and n/pq tables of unit fraction series, all providing
guidance to each scribe, to know the range of the expected

Conclusion: Completely reading Egyptian mathematical texts has
always been a difficult task. The primary reason is that scribes left
only brief outlines of their work, and therefore modern scholar,
until very recently, have not fully translate Middle Kingdom data
fully into modern base 10 fractions. This ancient hieratic shorthand
has been called many things, including only a form of hieroglyphc
math from an older era. Clearly, hieratic shorthand contained at
least two forms of remainder arithmetic, as modern researchers
have been detailing in 2005. This researcher has followed faint
clues by fully translating data into modern base 10 fractions, by
finding and explaining the scribal conversion used in the missing
vulgar fraction step. This extra work has been necessary since
scribes usually thought of in a form of mental arithmetic, and
when they did not, scribal notes were very brief.

In addition it had been shown that the word ro, 1/320, also meant
common divisor. Common divisors is a concept long over looked,
one that sheds new light on this 80- 120 year old question.

Irrespective of anyone's particular view of a particular ancient
problem, a clear translation to modern base 10 decimal fractions must
take place at some point. Fuzziness aside, to achieve that goal, as
best as anyone is able, all of the knowable arithmetic steps
mentioned and omitted in the ancient texts must be discussed, and
compared to the modern base 10 decimal fraction version of the
problem. 'No stone or text should be left unturned or unread',
is one way to summarize the operational aspect of this task.
Skipping over one ancient or modern arithmetic issue, or
another, has caused the great confusion that still exists in
the modern reading of the ancient mathematical texts. However,
given a little patience and humility, a better reading will be
appearing over the next few years. Adding a vulgar fraction step
to both forms of remainder arithmetic has greatly assisted in
translating scribal arithmetic into our modern base 10 arithmetic.