Egyptian multiplication and division

Egyptian multiplication and division

Since the 1880s an incomplete and muddled view defined ancient Egyptian multiplication and

division

operations

.

http://eom.springer.de/A/a013260.htm

Springer’s on-line encyclopedia summarizes a 1920s view on

http://en.

wikipedia

.org/wiki/Egyptian_multiplication_and_division

Wikpedia:

The art of

computation

arose and developed before the time of the oldest written records. The oldest advanced mathematical records include the Kahun (Cahoon) papyri and the famous

Rhind papyrus

, which, among other papyri, date to 2050 BCE. Earlier

additive

hieroglyphic

arithmetic

methods represented numbers in ways that Old Kingdom Egyptians performed

addition

and

subtraction

operations in relatively simple ways. For example,

multiplication

was carried out by doubling, i.e. the factors were decomposed into sums of powers of two, the

individual

summands were multiplied, and the components added. Operations on

fractions

were

reduced

in Ancient Egypt to operations on aliquot fractions. More complicated fraction operations were decomposed with the aid of tables as the sums of aliquot fractions.

In 1862 the first Egyptian text, the Berlin Papyrus, was published. Formal scholarly solutions appeared in 1900 with the work of Schack-Shackenberg. European scholars did not consider the Berlin Papyrus when 1920s transliterations of the RMP were published in incomplete additive versions of Egyptian multiplication. The 1920s historians had also not followed up a 1895 report (by F. Hultsch) that suggested a second form of multiplication method was present in Ahmes’ 2/n table, and other RMP problems (i.e. RMP 38). The second method included

aliquot parts

, as Springer suggests above.

Aliquot part were reported by

http://

planetmath

.org/encyclopedia/HultschBruinsMethodEgyptianFractions2.html

F. Hultsch in 1895. Hultsch parsed Ahmes’ 2/n table revealing scribal patterns of

divisors

of scaled

rational number

denominators that parsed 2/n table data. Springer’s Egyptian multiplication encyclopedia entry did not specify scribal aliquot part details a topic that was not resolved until the 21th century.

Ahmes’ aliquot part division steps, sensed in the 19th century, were not decoded during the 20th century for several reasons. Two reasons misdirected 1920s math historians by considering transliterations rather than translations. The first class of reason prematurely closed the subject of

Egyptian fraction

arithmetic operations by concluding Egyptian multiplication

contained

only additive steps. The Second class of reason scribal division was suggested have followed an non-inverse process called ’single false position’, or ’trial and error’ reported by scholars related to RMP 24-35, Ahmes’

algebra

problems.

The aliquot part story line remained unsolved until 2008. Shortly after 2002 the

http://planetmath.org/encyclopedia/KahunPapyrusAndArithmeticProgressions.html

Kahun Papyrus and the

http://rmprectotable.blogspot.com/

Rhind Papyrus 2/n table were decoded revealing two aliquot part operational methods: (1) new

inverse

multiplication and division methods, and (2) a LCM number method written in red. The multiplication and division methods had been hidden in same set of aliquot part operational steps, including

http://planetmath.org/encyclopedia/FirstLCMMethodRedAuxiliaryNumbers.html

red auxiliary numbers steps. In 2006, the 1895

Hultsch-Bruins method

was confirmed from a second direction, detailing a common aliquot method used in the RMP and

Egyptian Mathematical Leather Roll

, and published on-line in 2008.

Springer’s summary of

Egyptian multiplication and division

followed the misleading 1920s

definition

of Egyptian division suggesting: ”Division was carried out by subtracting from the number to be divided the numbers obtained by successive doubling of the divisor.” Math historians from 1920 to 1990 (i.e.; Howard Eves) called the proposed Egyptian division method ’single false position’, and 1990 scholars (i.e. Spalinger) called the scribal information ’trial and error’. Ironically, ’single false position’ was first documented in 800 AD, and at no earlier data. Later Arab texts improved up its root finding ’double

http://en.wikipedia.org/wiki/False_position

false position ’, method. ’Trial and error’ in Ahmes’ algebra problems selected LCM m to scale rational numbers n/p to mn/pm, and therefore can not be a scribal division fragment.

Springer’s definition of Egyptian division was historically incomplete on several levels. A

complete

definition of Egyptian scribal division should include discussion of the first six RMP problems. RMP 1-6 reported division by 10 labor rates defined in the

http://planetmath.org/encyclopedia/ReisnerPapyrus.html

Reisner Papyrus

. In addition, RMP algebra problems and methods are consulted. For example, Ahmes divided 28 by 97, in

http://mathforum.org/kb/thread.jspa?threadID=1768698&tstart=0

RMP 31(confirmed in RMP 34) by solving: x + (2/3 + 1/2 + 1/7)x = 33 and x + (2/3 + 1/2 + 1/7)x = 37 as other vulgar fraction problems were solved in the Kahun Papyrus and Rhind Papyrus 2/n tables. Aliquot part steps were hidden in theoretical multiplication and division operations for over 100 years.

Ahmes, therefore, did not use ’single false position’ in any arithmetic operation, a point made by Robins-Shute in 1987. The 1920s ’false position’ and 1990 ’trial and error’ scholarly ideas were false suppositions. For example, 28/97, in RMP 31, and data from RMP 21- 23 expose Ahmes’ LCM method. In RMP 23 where 45 was introduced solving most of the problem, but 360 was needed to Ahmes to complete the problem as all other algebra problems were solved.

In the 21st century, Ahmes is being reported converting vulgar fractions into optimized unit fractions series within a LCM m multiplication method. The LCM method replaced the aliquot parts of the denominator in the numerator. To convert 2/97 in RMP 31, and the 2/n table. Ahmes converted 28/97 into two problems, 2/97 and 26/97, such that:

1. To convert 2 by 97: As Ahmes’ 2/n table wrote for all 2/n conversions less than 2/101, he first selected a highly

divisible

number m as an optimizing multiplier m/m. In the 2/97 case 56 was selected, creating a multiplier 56/56 such that the aliquot parts of 56 (28, 14, 8, 7, 4, 2, 1) were introduced into the solution by writing:

2/97*(56/56) = 112/(56*97) = (97 + 8 + 7)/56*97)

and,

2/97 = 1/56 + 1/679 + 1/776

2. To convert 26/97 Ahmes looked for a multiplier m/m that would increase the numerator to greater than 97. Ahmes found 4/4. By considering the aliquot parts of 4 (4 , 2, 1) Ahmes wrote out:

26/97*(4/4) = 104/(4*97)= (97 + 4 + 2 + 1)/(4*97)

such that:

26/97 = 1/4 + 1/97 + 1/194 + 1/388

and,

3. Ahmes combined steps 2/97 and 26/97 into one Egyptian fraction series by writing:

28/97 = 1/4 + 1/56 + 1/97 + 1/194 + 1/388 + 1/679 + 1/77

’Single false position’ was false 20th century supposition that failed to parse Ahmes’ actual division operation. Ahmes division operation is correctly parsed as inverse to Egyptian multiplication. Egyptian scribes applied theoretical ideas in

http://planetmath.org/encyclopedia/AnOverViewOfAhmesPapyrus.html

Ahmes math tool box to convert rational numbers to Egyptian fractions.

In,

http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html

RMP 35-RMP 38 and RMP 66 a hekat was replaced by its 1/320 unit

equivalent

, 320 ro. In RMP. 10 hekat, 3200 ro, was divided by 365, the number of civil days in the year obtaining:

8 + 280/365

the expected daily use of fat.

The

quotient

8 was proven by the binary steps 1 - 365, 2 - 730, 4 - 1460, and 8 - 2920. The

remainder

280 (3200 - 2920 = 280). Translated to modern notation Ahmes’ proof multiplied the initial divisor 365 by (8 + 2/3 + 1/10 + 2190)

summing

(2920 + 243 1/3 + 36 1/2 + 1/6) obtaining 3200 ro, the exact initial value of 10 hekat of fat.

RMP 38 reported Ahmes multiplying 320 ro, one hekat, by 7/22, obtaining 101 9/11. The 101 9/11 answer was proven by multiplying 101 9/11 by 22/7, with the initial divisor 7/22 parsed by binary steps 34/11 times 1/10 = 7/22 showing that inverse multipliers define division, one of several modern arithmetic ideas that were anticipated by Ahmes.

Egyptian division was an inverse of Egyptian multiplication, reported in the RMP and the 1900 BCE

http://en.wikipedia.org/wiki/Akhmim_Wooden_Tablet

Akhmim Wooden Tablet

(AWT) and other Middle Kingdom texts.

Conclusion

: To parse ancient Egyptian multiplication and division written with a weights and measures

context

, Ahmes’ 2/n table and other arithmetic operations must be stripped away to reveal rational numbers. Ahmes multiplication and division of rational numbers were inverse to each other.

Egyptian multiplication contained two aspects, a theoretical side, and a practical side. Egyptian division was an inverse of Egyptian multiplication, and visa verse. Prior to the 21st century AD

Egyptian math

scholars had not considered theoretical aspects of the RMP and other Egyptian texts. Theoretical definitions hid an aliquot part and other arithmetic definitions, two being a hekat unity stated as (64/64), and a hekat stated as 320 ro. Egyptian division was quotient and exact remainder based, aspects that scholars are increasingly studying, linked to aliquot parts, 2/n tables, and other ancient scribal applications, such as weights and meaures, after 2005.

Title

Egyptian multiplication and division

Canonical

name

EgyptianMultiplicationAndDivision

Date of creation

2013-03-22 18:28:42

Last modified on

2013-03-22 18:28:42

Owner

milogardner (13112)

Last modified by

milogardner (13112)

Numerical id

67

Author

milogardner (13112)

Entry type

Definition

Classification

msc 01A16

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