Egyptian multiplication and division
Encyclopedia
Since the 1880s, as formalized in the 1920s, an incomplete view has defined Egyptian multiplication. Springer's on-line encyclopedia summarizes the 1920s view this way (from Planetmath: http://planetmath.org/encyclopedia/EgyptianMultiplicationAndDivision.html):
The 1920s conclusions properly decoded an incomplete additive version of Egyptian multiplication. The 1920s historians had not followed up a 1895 report that suggested a second form of multiplication method was present in Ahmes' RMP 2/n table and RMP 36. The second method included aliquot parts, as Springer suggested. Aliquot part were reported by F. Hultsch in 1895. Hultsch parsed Ahmes' 2/n table revealing aliquot part patterns. Yet, Springer's Egyptian multiplication encyclopedia entry did not specify critical aliquot part operational details that are required to translate the information into modern arithmetic statements. Sadly, 1920s math historians had skipped over several operational details, such as of F. Hultsch's 1895 aliquot part discussion points, thereby improperly concluding that aliquot part patterns had not been seen in Ahmes
' 2/n table.
The aliquot part story line remained an unsolved issue until the 21st century. Shortly after 2002 the Kahun Papyrus
and the RMP 2/n table revealed two aliquot part operational methods: (1) new inverse multiplication and division methods, and (2) a LCM
number method written in red (RMP 38). The multiplication and division methods had been hidden Hultsch's aliquot part operational steps, including red auxiliary numbers steps that selected 'optimized' divisors of the LCM. 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
. This method scaled the conversion of 1/p, 1/pq, 2/p, 2/pq, n/p and n/pq rational numbers by an LCM m, written as m/m.
Ahmes' aliquot part division steps, sensed in the 19th century, not decoded during the 20th century began to release its secrets after 2001, increasingly by 2006 and 2009 (by RMP 36). Two reasons had misdirected 1920s math historians. The first prematurely closed the subject of Egyptian fraction arithmetic operations by concluding Egyptian multiplication contained only additive steps. Second, scribal division was suggested have followed a non-inverse process called 'single false position'.
Moreover, Springer followed the traditional 1920s 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 1920s proposed Egyptian division method 'single false position'. Ironically, 'single false position' was first documented in 800 AD. Later Arab texts improved its root finding 'double false position' method.
Springer's definition of Egyptian division was historically incomplete. To complete a definition of Egyptian division the first six RMP problems, a division by 10 labor rate (defined earlier in the Reisner Papyrus
) set of problems are consulted. In addition, RMP algebra problems and methods are consulted. For example, Ahmes divided 28 by 97, in 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 did not mention 'single false position' in algebra problems, a valid point made by Robins-Shute in 1987. The inaccurate 1920s supposition has been replaced by parsing large vulgar fractions by stripping away the unit fraction notation. For example, 28/97, in RMP 31, and RMP 23 expose Ahmes' LCM multplication method. In RMP 23 where a 45 multiplier was introduced to solve most of the problem. Yet, to read the complete problem LCM 360 was needed as other RMP algebra problems were solved.
In the 21st century, Ahmes is becoming clearly reported by converting vulgar fractions into optimized unit fractions series within a LCM method. The LCM method also applied aliquot parts of the denominator to solve 2/97 in RMP 31, and in 2/n table. Ahmes converted 28/97 into two problems, 2/97 and 26/97, selecting two LCM multipliers such that:
1. To convert 2 by 97: Ahmes' 2/n table wrote 2/n conversions less than 2/101, he 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:
and,
2. To convert 26/97 to a unit fraction series 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:
such that:
and,
3. Ahmes combined steps 2/97 and 26/97 into one Egyptian fraction series by writing:
as RMP 36 converted 30/53 by 2/53 + 28/53 with 2/53 scaled by (30/30) and 28/53 scaled by (2/2).
4. Egyptian multiplication was an inverse operation to the Egyptian division operation, and visa verse. The modern looking multiplication and division operations had been hidden within the Egyptian fraction notation.
One implication is that 'single false position' represented a 20th century supposition that failed to historically read Ahmes' additive numerators written in multiplication
problems. Ahmes division operations, described by aliquot part steps in over 20 algebra problems, embed ancient and modern division methods, as inverse to Egyptian multiplications. Egyptian scribes applied several modern theoretical ideas, mostly arithmetic ones, as recorded in Ahmes math tool box.
A second implication is contained in RMP 38. It details Ahmes multiplying 320 ro, one hekat, by 35/11 times 1/10 = 7/22, obtaining 101 9/11. Ahmes proved that 101 9/11 was correct by multiplying by the inverse of 7/22, or 22/7. Egyptian division generally applied an inverse of Egyptian multiplication in the 1900 BCE Akhmim Wooden Tablet
(AWT) and all other Middle Kingdom mathematical texts. The AWT, for example. divided one hekat, (64/64), by n = 3, 7, 10, 11 and 13. Quotient
and remainder
answers were multiplied by divisor inverses, 1/3, 1/7, 1/10, 1/11 and 1/13, exactly returning the beginning rational number (64/64).
Finally, the red numerator numerators implied by the 2/n table were directly discussed in RMP 36. Ahmes converted, 2/53, 3/53, 5/53, 15/53, 28/53 and 30/53 by two rules. The first rule scaled 2/53*(30/30) = 60/1590, 3/53(20/20) = 60/1060, 5/53*(12/12) = 60/636, 15/53*(4/4) = 60/212, 28/53*(2/2) = 56/106. The second rule converted 30/53 by parsing 30/53 into 2/53 + 28/53. as Ahmes has converted 28/97 by parsing 29/97 into 2/97 + 26/97.
Conclusion: To understand ancient Egyptian multiplication and division, Ahmes' 2/n table aliquot part arithmetic operational steps must be translated into modern arithmetic statements. Ahmes multiplication and division methods were inverse to each other, with RMP 38, and the AWT provided vivid examples of the arithmetic relationships. RMP 36 the details of two rational number conversion methods were detailed, one for n/p, n/pq, 2/p and 2/pq and another for hard to convert n/p rational numbers that were parsed into solvable 2/p and (n-2)/p statements.
Egyptian multiplication contained two aspects, a theoretical side, and a practical side. Egyptian division by a rational number was Egyptian multiplication by an inverse of the rational number. Early Egyptian scholars had not considered the theoretical aspects of the RMP and other Egyptian texts until the 21st century. Theoretical definitions had been hidden in conversion of rational numbers by scaled multipliers applied in an aliquot part rule. RMP 38 multiplied a hekat, stated as 320 ro, by 7/22, and returned 320 ro by multiplying the answer by 22/7. Egyptian division was quotient and remainder based, theoretical aspects that scholars are increasingly studying in terms of aliquot parts, 2/n tables, and other ancient scribal applications after 2005.
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 BCE. Additive hieroglyphic methods representation numbers (cf. Numbers, representations of) in ways that Old Kingdom Egyptians had perform addition and subtraction operations on natural numbers in relatively simple ways. For example, one multiplication method 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 1920s conclusions properly decoded an incomplete additive version of Egyptian multiplication. The 1920s historians had not followed up a 1895 report that suggested a second form of multiplication method was present in Ahmes' RMP 2/n table and RMP 36. The second method included aliquot parts, as Springer suggested. Aliquot part were reported by F. Hultsch in 1895. Hultsch parsed Ahmes' 2/n table revealing aliquot part patterns. Yet, Springer's Egyptian multiplication encyclopedia entry did not specify critical aliquot part operational details that are required to translate the information into modern arithmetic statements. Sadly, 1920s math historians had skipped over several operational details, such as of F. Hultsch's 1895 aliquot part discussion points, thereby improperly concluding that aliquot part patterns had not been seen in Ahmes
Ahmes
Ahmes was an ancient Egyptian scribe who lived during the Second Intermediate Period and the beginning of the Eighteenth Dynasty . He wrote the Rhind Mathematical Papyrus, a work of Ancient Egyptian mathematics that dates to approximately 1650 BC; he is the earliest contributor to mathematics...
' 2/n table.
The aliquot part story line remained an unsolved issue until the 21st century. Shortly after 2002 the Kahun Papyrus
Kahun Papyrus
The Kahun Papyri are a collection of ancient Egyptian texts discussing administrative, mathematical and medical topics. Its many fragments were discovered by Flinders Petrie in 1889 and are kept at the University College London. This collection of papyri is one of the largest ever found. Most of...
and the RMP 2/n table revealed two aliquot part operational methods: (1) new inverse multiplication and division methods, and (2) a LCM
LCM
LCM is an abbreviation used in:* mathematics for least common multiple* naval warfare for Landing Craft Mechanized* accounting for Lower of Cost or Market* computing for Lipikar Custom Map file formatLCM may also refer to:...
number method written in red (RMP 38). The multiplication and division methods had been hidden Hultsch's aliquot part operational steps, including red auxiliary numbers steps that selected 'optimized' divisors of the LCM. 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
Egyptian Mathematical Leather Roll
The Egyptian Mathematical Leather Roll was a 10 × 17 in leather roll purchased by Alexander Henry Rhind in 1858...
. This method scaled the conversion of 1/p, 1/pq, 2/p, 2/pq, n/p and n/pq rational numbers by an LCM m, written as m/m.
Ahmes' aliquot part division steps, sensed in the 19th century, not decoded during the 20th century began to release its secrets after 2001, increasingly by 2006 and 2009 (by RMP 36). Two reasons had misdirected 1920s math historians. The first prematurely closed the subject of Egyptian fraction arithmetic operations by concluding Egyptian multiplication contained only additive steps. Second, scribal division was suggested have followed a non-inverse process called 'single false position'.
Moreover, Springer followed the traditional 1920s 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 1920s proposed Egyptian division method 'single false position'. Ironically, 'single false position' was first documented in 800 AD. Later Arab texts improved its root finding 'double false position' method.
Springer's definition of Egyptian division was historically incomplete. To complete a definition of Egyptian division the first six RMP problems, a division by 10 labor rate (defined earlier in the Reisner Papyrus
Reisner Papyrus
The Reisner Papyri date to the reign of Senusret I, who was king of Ancient Egypt in the 19th century BCE. The documents were discovered by Dr. G.A. Reisner during excavations in 1901-04 in Naga ed-Deir in southern Egypt. A total of four papyrusrolls were found in a wooden coffin in a tomb. * The...
) set of problems are consulted. In addition, RMP algebra problems and methods are consulted. For example, Ahmes divided 28 by 97, in 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 did not mention 'single false position' in algebra problems, a valid point made by Robins-Shute in 1987. The inaccurate 1920s supposition has been replaced by parsing large vulgar fractions by stripping away the unit fraction notation. For example, 28/97, in RMP 31, and RMP 23 expose Ahmes' LCM multplication method. In RMP 23 where a 45 multiplier was introduced to solve most of the problem. Yet, to read the complete problem LCM 360 was needed as other RMP algebra problems were solved.
In the 21st century, Ahmes is becoming clearly reported by converting vulgar fractions into optimized unit fractions series within a LCM method. The LCM method also applied aliquot parts of the denominator to solve 2/97 in RMP 31, and in 2/n table. Ahmes converted 28/97 into two problems, 2/97 and 26/97, selecting two LCM multipliers such that:
1. To convert 2 by 97: Ahmes' 2/n table wrote 2/n conversions less than 2/101, he 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 to a unit fraction series 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
as RMP 36 converted 30/53 by 2/53 + 28/53 with 2/53 scaled by (30/30) and 28/53 scaled by (2/2).
4. Egyptian multiplication was an inverse operation to the Egyptian division operation, and visa verse. The modern looking multiplication and division operations had been hidden within the Egyptian fraction notation.
One implication is that 'single false position' represented a 20th century supposition that failed to historically read Ahmes' additive numerators written in multiplication
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
problems. Ahmes division operations, described by aliquot part steps in over 20 algebra problems, embed ancient and modern division methods, as inverse to Egyptian multiplications. Egyptian scribes applied several modern theoretical ideas, mostly arithmetic ones, as recorded in Ahmes math tool box.
A second implication is contained in RMP 38. It details Ahmes multiplying 320 ro, one hekat, by 35/11 times 1/10 = 7/22, obtaining 101 9/11. Ahmes proved that 101 9/11 was correct by multiplying by the inverse of 7/22, or 22/7. Egyptian division generally applied an inverse of Egyptian multiplication in the 1900 BCE Akhmim Wooden Tablet
Akhmim wooden tablet
The Akhmim wooden tablets or Cairo wooden tablets are two ancient Egyptian wooden writing tablets. They each measure about 18 by 10 inches and are covered with plaster. The tablets are inscribed on both sides. The inscriptions on the first tablet includes a list of servants, which is followed...
(AWT) and all other Middle Kingdom mathematical texts. The AWT, for example. divided one hekat, (64/64), by n = 3, 7, 10, 11 and 13. Quotient
Quotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
and remainder
Remainder
In arithmetic, the remainder is the amount "left over" after the division of two integers which cannot be expressed with an integer quotient....
answers were multiplied by divisor inverses, 1/3, 1/7, 1/10, 1/11 and 1/13, exactly returning the beginning rational number (64/64).
Finally, the red numerator numerators implied by the 2/n table were directly discussed in RMP 36. Ahmes converted, 2/53, 3/53, 5/53, 15/53, 28/53 and 30/53 by two rules. The first rule scaled 2/53*(30/30) = 60/1590, 3/53(20/20) = 60/1060, 5/53*(12/12) = 60/636, 15/53*(4/4) = 60/212, 28/53*(2/2) = 56/106. The second rule converted 30/53 by parsing 30/53 into 2/53 + 28/53. as Ahmes has converted 28/97 by parsing 29/97 into 2/97 + 26/97.
Conclusion: To understand ancient Egyptian multiplication and division, Ahmes' 2/n table aliquot part arithmetic operational steps must be translated into modern arithmetic statements. Ahmes multiplication and division methods were inverse to each other, with RMP 38, and the AWT provided vivid examples of the arithmetic relationships. RMP 36 the details of two rational number conversion methods were detailed, one for n/p, n/pq, 2/p and 2/pq and another for hard to convert n/p rational numbers that were parsed into solvable 2/p and (n-2)/p statements.
Egyptian multiplication contained two aspects, a theoretical side, and a practical side. Egyptian division by a rational number was Egyptian multiplication by an inverse of the rational number. Early Egyptian scholars had not considered the theoretical aspects of the RMP and other Egyptian texts until the 21st century. Theoretical definitions had been hidden in conversion of rational numbers by scaled multipliers applied in an aliquot part rule. RMP 38 multiplied a hekat, stated as 320 ro, by 7/22, and returned 320 ro by multiplying the answer by 22/7. Egyptian division was quotient and remainder based, theoretical aspects that scholars are increasingly studying in terms of aliquot parts, 2/n tables, and other ancient scribal applications after 2005.
External links
- http://planetmath.org/encyclopedia/RMP36AndThe2nTable.html RMP 36 and the 2/n table
- http://rmprectotable.blogspot.com/ RMP 2/n table
- http://planetmath.org/encyclopedia/EgyptianMath3.html
- http://weekly.ahram.org.eg/2007/844/heritage.htm
- http://planetmath.org/encyclopedia/EgyptianMathematicalLeatherRoll2.html
- http://emlr.blogspot.com Egyptian Mathematical Leather Roll
- http://planetmath.org/encyclopedia/FirstLCMMethodRedAuxiliaryNumbers.html
- http://planetmath.org/encyclopedia/AhmesBirdFeedingRateMethod.html Theoretical (expected) economic control numbers, RMP 83
- http://planetmath.org/encyclopedia/RationalNumbers.html
- http://mathforum.org/kb/message.jspa?messageID=6579539&tstart=0 Math forum and two ways to calculate 2/7
- http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html New and Old Ahmes Papyrus classifications