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Topic: Egyptian multiplication and division
Replies: 4   Last Post: Mar 23, 2009 4:28 PM

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 Milo Gardner Posts: 1,105 Registered: 12/3/04
Egyptian multiplication and division
Posted: Oct 8, 2008 11:28 AM

From PlanetMath.com, consider:

Since the 1880s, as formalized in the 1920s, an incomplete view has defined Egyptian multiplication. Springer's on-line encyclopedia

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

summarizes the 1920s view this way:

"The art of computation arose and developed long before the times of the oldest written records extant. The oldest mathematical records are the Cahoon papyri and the famous Rhind papyrus, which is believed to date back to about 2000 B.C.. An additive hieroglyphic system for the representation of numbers (cf. Numbers, representations of) enabled the ancient Egyptians to perform addition and subtraction operations on natural numbers in a relatively simple manner. 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 (cf. Fraction) were reduced in Ancient Egypt to operations on aliquot fractions, i.e. on fractions of the type. More complicated fractions were decomposed with the aid of tables into a sum of aliquot fractions."

The 1920's view properly decoded an additive form of Egyptian multiplication. Yet, 1920's historians could have followed up an incomplete 1895 report that a second form of multiplication created Ahmes' 2/n table. The second multiplication method included a use of aliquot parts. Aliquot parts were first reported in Hultsch-Bruins Method). Springer's Egyptian multiplication entry properly includes hints that aliquot parts were known to Ahmes. However, Springer's description did not specify the details of the aliquot part multiplication method. Sadly, 1920s math historians had skipped over F. Hultsch's 1895 aliquot part discussion.

The aliquot part story line remained unsolved until the 21st century. After 2002 the Kahun Papyrus and the Rhind Papyrus 2/n table were jointly decoded revealing aliquot part multiplication and division. methods. The combined multiplication and division methods had been included within the same set of aliquot parts steps. In 2006, the 1895 Hultsch-Bruins method was confirmed from a second direction, a common method used in the RMP and Egyptian Mathematical Leather Roll.

It is clear that Ahmes' aliquot part division steps, sensed in the 19th century, had not been decoded during the 20th centuries for two reasons. Math historians in the 1920's had prematurely closed the subject of Egyptian fraction arithmetic operations by concluding that multiplication contained only additive steps, and that division only followed an inverse process (called ' single false position').

Moreover, Springer followed the traditional 1920's definition of Egyptian division by suggesting: "Division was carried out by subtracting from the number to be divided the numbers obtained by successive doubling of the divisor." Math historians call the 1920's proposed Egyptian division method 'single false position'. Ironically, 'single false position' has only been documented in 800 AD and later Arab texts in relationship to Arab uses of 'double false position ', a root finding method.

Springer's definition of Egyptian division is therefore historically incorrect. To parse a valid definition of Egyptian division the first six RMP problems, a labor rate context (defined earlier in the Reisner Papyrus) needs to be consulted. In addition, RMP algebra problems need to be consulted. For example, a division of 28 by 97 problem was reported in RMP 31 by solving:

x + (2/3 + 1/2 + 1/7)x = 33

as other vulgar fraction problems were solved in the Kahun Papyrus and Rhind Papyrus 2/n tables within aliquot parts steps that had hid theoretical multiplication and division definitions for over 100 years.

Ahmes did not solve 28/97 by single false position, a 1920s proposal associated with vulgar fraction conversions. Ahmes wrote out a unit fractions series representing 28/97 by breaking the problem into two parts, 2/97 and 26/97, such that:

1. To divide 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 that would become the first partition 1/m, a 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 divide 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 + 1/776

That is, 'single false position' was not used by Ahmes. Ahmes used an aliquot part set of steps. Parsing the aliquot part steps, theoretical multiplication and division steps are visible. Therefore, theoretical multiplication and quotients and remainders operations must be added to Ahmes' arithmetic tool box.

Conclusion: Egyptian multiplication contained additive and theoretical sides. Scholars had overlooked the theoretical side until the 21th century. The theoretical definition had been hidden in an aliquot part 2/n table definition. Egyptian division was quotient and remainder based and was also hidden in the same aliquot part definition.

To directly understand Egyptian multiplication and division Ahmes' mental arithmetic is required to be fully parsed. One example parses 28/97 as 'single false position' has been unable to be solved by math historians. The solution of the 28/97 problem in a historical contexts opens a valid door in which theoretical definitions of Egyptian multiplication and division have cracked two of Ahmes' mental arithmetic codes.

"Egyptian multiplication and division" is owned by milogardner.

Date Subject Author
10/8/08 Milo Gardner
10/12/08 Milo Gardner
12/20/08 Milo Gardner
12/20/08 Milo Gardner
3/23/09 Milo Gardner