Monday, February 07, 2005

Egyptian Math History

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.

Reading only the RMP problems and answers, and considering no other texts, Chace's view was correct for his time period. However, considering the Akhmim Wooden Tablet, translated by a Charles U, Prague grad student, in 2002 and other texts at later times, Ahmes' hekat 'feeding rates' is clearly6 read and confirmed, thereby updating Chace's excellent 1927 transliteration in major ways.

In other words, traditional language-type transliterations had poorly detailed scribal number facts written when considering all four arithmetic operations. Scribal subtraction, and division operations had not been fully understood by the early researchers, especially 1920's scholars.

Modern decoding keys are needed to convert all four of the scribal arithmetic operations into modern arithmetic operations. Scribal subtraction, and division, decoding keys became available after 2002, facts that are not well known. Beginning in 2002, the new scribal subtraction and division decoding keys has allowed big chunks of once unreadable hieratic data to be translated into modern base 10 fractions (for the first time). These post-2002 two-way translations have made the 1920's and other transliteration views of the RMP and other texts obsolete, in several important respects, exposing additional aspects of Ahmes and other scribe's mathematics.

Translation efforts published before 2002 had been unfair and incomplete in reporting scribal arithmetic and algebra exclusively within additive arithmetic. Traditional scholars had omitted the hard-to-read aspects of the ancient arithmetic operations that had not been additive in scope. Scholarly omissions had introduced additive 'transliterations' of the numerical information, thereby being unfair to the ancient scribal math in serious ways.

Seen as modern translation gaps, post 2002 readings of the ancient arithmetic methods, especially subtraction and division operations had begun anew. Several new decoding keys had been pin-pointed related to scribal subtraction and division. The new decoding keys were not additive. Two-way abstract and practical streets had been built that details rigorous modern arithmetic versions of the newly decoded aspects of the ancient hieratic arithmetic.

One of the subtraction and division keys was the scribal use vulgar fractions. Scholars had suggested that scribal division consisted only of unit fraction terms, thereby eliminating the need for scholars to consider vulgar fractions. However, vulgar fractions appear in almost every RMP problem, hidden away in the often unreported (or under reported) scribal shorthand, sometimes explicitly, and sometimes implicitly.

For example, RMP # 32 states in modern terms

x + (1/2 + 1/4)x = 2 or

x + 7/12x = 2, or

19/12 x = 2, or

x = 24/19 = 1 + 5/19

Ahmes' shorthand notes, often fragmented in reporting its details, worked with the 5/19 vulgar fraction. Noting his 1 + 1/6 + 1/12 + 1/114 + 1/224 answer, it appears generalized, but not optimized. As a Hultsch-Bruin method, Ahmes did the following:

Find 5/19 by first subtracting 1/12 or

5/19 - 1/12 = (41)/(12*19)

= (38 + 2 + 1)/(12*19)

= 1/6 + 1/114 + 1/224, or

the final answer

1 + 1/6 + 1/12 + 1/114 + 1/224

Ahmes' choice of 1/12 followed the 2/19 2/nth table suggestion, where:

2/19 = 1/12 + (3 + 2)/(12*19) = 1/12 + 1/76 + 1/114

as also used in 2/95, where 2/19 x 1/5 = 1/5 (1/12 + 1/76 + 1/114)

as Wilbur Knorr, in 1982, as published in the journal Historia Mathematica, and others at other times.

Had Ahmes 'thought of another number' like 1/4, a more optional series 1/4 + 1/76 would have been found (as the 400 AD Akhmim Papyrus, from the Coptic era includes in its n/19 table).

Gillings and other authors, i.e. like Robins-Shute, had avoided discussing Ahmes' shorthand that revealed the general use of vulgar fractions . Ahmes's shorthand notes are fragmentary, for sure, but facts that point in the remainder direction when intermediate vulgar fractions were converted by Ahmes, and other scribes, to Egyptian fraction series.

This blog details a range of specific textual ways that scribal shorthand had been hard to read. This researcher has been lucky in several areas. Several others had assisted over the last 20 years.

In summary ad hoc relationships have corrected several scholarly oversights over the last few years. Sadly, gaps still exist in reading several ancient texts. As a consequence the oldest Egyptrian number system, and its Horus-Eye and Egyptian fraction arithmetic methods, will continue to be poorly read by only applying language transliteration rules. Additional ancient arithmetic rules, parsed from the available texts, are needed.

Special care needs to be taken when reading hieratic arithmetic and its fragmented scribal shorthand. One new new method to achieve this goal is documented on this blog. Choose whatever name that fits this new discovered method. One way is to build two-way streets, working within ancient and modern remainder arithmetic. Final translations should not be offered until the ancient and modern remainder arithmetic are doubled checked.

This blog shows that ancient words and numbers are only modern snapshots of ancient arithmetic statements. To fully translate a snapshot of an arithmetic statement, and find the ancient context, fragmented arithmetic shorthand must be read using rigorous methods. Filling in a scribal gap at one's leisure, skipping ones that are not readily understood, had commonly omitted major segments of the scribal methods, one being the use of remainder arithmetic.

To resolve a few of the fragmented aspects of the ancient texts, wider translations of the scribal math shorthand has been developed that covers added aspects of the available data. One technique translates the ancient data into modern base 10 fractions as a bilingual text. The new methods may be seen as comparable to the Rosetta Stone's use of phonetic symbols to link the third language's decoding to the first two. The new method uses quotients and remainders, often written within common divisors, like 1/320 of a hekat.

Math and language scholars had often taken an aspect of a text, and offered interpretations outside of the original context. When attempting to translate MK mathematics and its metrology, translations have been claimed to be accurate and complete. Later translators found that the early translators had 'translated what they could', and had omitted important fragmented facts .

This blog focuses on clearing up a few of the fragmented facts, one being the remainder arithmetic cited in the RMP and other texts.

For example, RMP 62 states that

100/13 = 7 + 2/3 + 1/39

But what does that mean? The Reisner Papyrus will be shown to have written:

n/10 = Q + R

where Q was a quotient and R was a remainder, first stated as vulgar fraction, that was easily converted to an Egyptian fraction series. The Reisner cites:

8/n1 = 1/2 + 1/4 + 1/20 = 8/10

48/n2 = 4 + 1/2 + 1/4 + 1/20 = 4 + 8/10

18/x3 = 1 + 1/2 + 1/10 = 1 + 6/10

64/x4 = 6 + 1/4 + 1/10 + 1/20 = 6 + 4/10

36/x5 = 3 + 1/2 + 1/10 = 3 + 6/10

such that x1 = x2 = x3 = x4 = x5 = 10

That is,

100/13 = 7 + 9/13 or 7 + 2/3 + 1/39

with 7 being a quotient and 9/13 being a remainder that was converted to 2/3 + 1/39.

Additional proof is provided by the word ro, a term often read in metrology as 1/320th of a hekat. The word ro has a broader meaning including rest, remainder and related ideas that connect to the 'healing' of Horus-Eye arithmetic.

Ro was defined in at least two different situations. By reading the hekat division operation, cited in the Akhmim Wooden Tablet and the RMP, ro was used as a remainder scaling factor. Specifically Middle Kingdom scribes wrote (64/64)/n = 64/n + (5R/n)ro in the AWT and RMP, and (320/n)ro in the RMP and the medical texts.

On an arithmetic level the word ro also meant 'common divisor' as well scaling factor. Ro was linked to binary fractions and Egyptian fraction series, points that scholars have been slowly appreciating since 2002.

Stated in other terms, scholars have reported ancient math data in fragmentary ways beginning in the 1880's. By 1923 scholars like Peet began to conclude that their Egyptian fraction decoding work was complete. Peet concluded that the ro idea had only equaled 1/320th of a hekat and, by implication, could not have been re-valued within another context, such as partitioning a cubit,
or other hekat situations.

Future study is required to detail the scribal division methods that were comparable, if any, to two forms of hekat remainder arithmetic (scribal division).

Peet's analysis stressed the idea of Egyptian division, as a limiting factor on how and why the Akhmim Wooden Tablet scribe partitioned a hekat (unity) by 3, 7, 10, 11 and 13. Peet did not go into detail to show how the AWT scribe proved his answers or seek comparative cubit or hekat partitioning methods, as reported by Masse and Gewiche.

Peet was not the only scholar that introduced misconceptions by skipping over otherwise 'unreadable' fragments of texts. Peet created a set of unproven additive conclusions concerning the scope and content of Egyptian mathematics. Scribal subtraction and scribal division was not well reported by Peet, especially related to Daressy's 1906 reading of the Akhmim Wooden Tablet.

Considering original texts, often assumed to only additive, Egyptian fraction based arithmetic several texts had often been unfairly read in trivial ways. It will be shown that arithmetic texts containi9ng closely related Horus-Eye and Egyptian fraction data had not been fairly read by Peet and other early scholars.

Over 40 RMP examples of binary and Egyptian fraction data had been ignored, or grossly misread, by early scholars. The following data sets seemed not fit the standard scholarly additive paradigm:

Only recently, 2002 to be precise, and increasingly for the next five nears, these 40 data points been parsed and reviewed. The data was not additive as Peet had concluded in 1923.

This blog analyzes 40 data points and updates an ancient Egyptian division operation. One modern name for a 4,000 year old division method is remainder arithmetic.

This blog also offers a rigorous method that may allow experts and amateurs to double check their steps involved in translating Egyptian mathematical texts to modern base 10 fractions.

Continuing to step back, scholars, had often worked alone. Scholars had unfairly thought that a rigorous ancient text translation to modern base 10 fractions was not necessary. The step is necessary,

Frances I. Griffith in 1891 sensed that ro meant 'greatest common measure' within the RMP. Others scholars working in other texts, Daressy in 1906 with the Akhmim Wooden Tablet sensed other interesting arithmetic features. Daressy sensed exactness in all but the 1/11 and 1/13 cases, where scribal typographical errors had hindered his analysis. Yet, Daressy cited three examples that showed three divisions of a hekat unity and three proofs that found (64/64) , clues that were not followed up by Peet.

Interestingly, by 1923 Peet tried to close off debate related to the highest form of Egyptian division operations by attempting to refute Daressy's 1906 analysis. Sadly, the majority of 1920's scholars had tried to stay within a singular view of Middle Kingdom arithmetical texts, an additive one. Early scholars, therefore, had picked out readable aspects from certain scribal shorthand statements, skipping over otherwise 'unreadable aspects'.

Oddly, by 1923 scholars agreed on the additive aspects of the mathematical texts. But were all the Egyptian mathematical texts only additive or subtractive in scope, as David Silverman (and others) continue to suggest in 1994?

It will be shown that math historians had not completely reported Egyptian mathematical facts. This blog will show that Egyptologists and math historians had created only additive methodologies that had not generally linked the original scribal base 10 data to modern base 10 equivalents. Fair 'decoding' systems are needed.

Corrective oversights continue to be difficult. Formal corrective methodologies have only recently been published. One corrective paper was published in India in 2006. The paper decodes hekat units that involved divisors of 64 or smaller.

Therefore, these newly published decoding keys confirm that additive scribal arithmetic had not, in fact, dominated Middle Kingdom mathematics as had been assumed for over 100 years.

Proof that non-additive arithmetic had dominated scribal arithmetic lies in several examples, beginning with the Rhind Mathematical Papyrus. The RMP lists several beginning type problems that our modern 4-6th graders would recognize, data that appears to have been additive, but actually was not. One problem is #83, where three classes of birds were fed three different amounts, 1/6th of a hekat of grain for three birds, 1/20th for one bird, and 1/40th for each of three birds. Ahmes, the RMP scribe, then asks: how much grain did all seven birds eat in one day?

A modern discussion of this trivial problem requires the addition of 1/6, 1/6, 1/6, 1/20, 1/40, 1/40 and 1/40, Note that an easy to reach common multiple (1/120) is required to solve this problem if no modification in procedure is introduced. Today, kids might improve the ease of working the ancient problem by first adding the three 1/6 fractions, obtaining 1/2. Then they could add 1/20 and 3/40. In this manner only (1/40) is required to be used as a common divisor. This is a basic form of modern logic overlay (a bilingual text, to refer to a cryptanalysis standard) can be restated several ways.

A closely related arithmetic form states, that by adding: 1/2, 1/20 and 3/40 using the common divisor 1/40, or

(20 + 2 + 3)/40 = 5/8th of a hekat of grain, as Ahmes

the requested to know, how much did seven birds eat in one day?

Ahmes performed this same arithmetical task, but chose another interesting value for his common divisor, 1/320. Ahmes wrote his fractional divisors of a hekat, his feeding rates, in terms of a larger one that he named ro, one that was commonly seen in several additional RMP problems, #35-38, 47, and 81.

To understand Ahmes' reason for selected 1/320th for a class of problems, any one of which could have used a smaller common divisor, let us examine Ahmes used of the word ro.

Stated in terms of ro (common divisor) units, Ahmes first added up 53 1/3ro, 53 1/3 ro, 53 1/3 ro, reaching 160 ro. Ahmes then added 16 ro, 8 ro, 8 ro and 8 ro, reaching 200 ro, or 200/320 = 5/8th of hekat. Trivial right? I think not.

At this point several non-trivial points should be hi-lighted. Why did Ahmes apparently chose only one common divisor ro, 1/320th of a hekat, when divisors of a hekat unity was smaller than 64? One clue is given by the idea of common divisor. A common divisor was needed for to find a remainder related to a divisor n, where n less than or equal to 64. A second clue is given by ro being used in another situation 32o/n ro, a statement that appears in the bird feeding problem.

The common divisors used to solve the seven bird eating grain problem makes sense in ancient and modern base 10 fractions. In Ahmes' use, his common divisors were large, all multiples of 1/320. The multiple of 1/320 common divisor was repeated in several other hekat division problems. One other is the Akhmim Wooden Tablet and five division problem. There are an additional 29 cases in the RMP. The 29 RMP data points were also written out in terms of a hin unit, so Ahmes was well versed in using common divisors and other forms of fractions.

This blog reports that Ahmes used remainder arithmetic to exactly partition a hekat a volume unit. Vulgar fractions can be seen as converted into quotients and exact Egyptian fraction remainders. The remainders, scaled to ro units, can be seen as representing 'healed' Horus-Eye units.

The 'healed' Horus-Eye quotients were given Egyptian fraction remainders that included 1/320 ro common divisors. That is, whenever a hekat unity, 64/64, was divided by any prime number, like 3, 7, 11, and 13, or a none-prime number, a remainder term was created that included the possible use of ro.

Ahmes and a mentor writing in the Akhmim Wooden Tablets substituted the fraction 5/320 for the 1/64th fraction that appeared in the remainder arithmetic term, defined by:

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

such that, Ahmes' remainder (R/n)(1/64) was multiplied by 5/5 such that

(64/64)/n = Q/64 + (5R/n)(1/320)

and replaced 1/320 with ro

a final scaled quotient and scaled Egyptian fraction remainder was written as

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

Further, it will be shown that when n > 64, a second form of remainder arithmetic was used. Ro was not often used in the Papyrus Ebers and other medical texts., most often in the form

320/n ro

and generally,

m/n 'unit' wrote quotient and unscaled remainders, such as

m = 10 = hin, was written as 10/n hin

m = 64 = dja, was written as 64/n dja

and so forth.