Red auxiliary numbers
Encyclopedia
In the study of ancient Egyptian mathematics, red auxiliary numbers were additive numbers that summed to a numerator used in Middle Kingdom arithmetic problems. In the 1650 BCE Rhind Mathematical Papyrus
(RMP), the additive set of divisors were written in red ink. Red auxiliary numbers were used for several purposes The red auxiliary numbers were divisors of a greatest common divisor
(GCD) that was used to generally convert 2/n to optimized, but not optimal, unit fraction series. The main purpose created 2/n tables by hi-lighting scribal skills in converting rational numbers to optimized, but not optimal, unit fraction series. The RMP 2/n table reported Egyptian fraction series by multiplying 2/n by a LCM m, written as a unity (m/m), that obtained 2m/mn. Ahmes
the RMP author, practiced the selection of red auxiliary numbers in RMP 21, 22, and 23 and detailed other aspects of the red number method in RMP 36. The scaled 2m/mn rational number was converted to a unit fraction series by finding the best set of divisors of (GCD) mn, as discussed below.
For example, Ahmes converted 2/43 by LCM 42, thinking:
To parse 84/1806 Ahmes selected the best divisors of 1806 from {43, 42, 21, 14, 7, 6, 4, 3, 2, 1}, denoting the divisors that best summed to numerator 84 in .
Ahmes chose rather than , or ,
such that:
The 2/n table method that was validated in RMP 36 with an explicit conversion of 3/53 scaled to 60/1060. The best divisors of 1060 were summed to 60 within /1060 and 1/20 + 1/265 + 1/530 + 1/1060. The red number method was used over six times in RMP 36 and over seven times in RMP 37.
Scribal red numbers, LCMs and GCDs were known for over 130 years before being rigorously parsed. Math historians failed to parse aspects of red numbers seen in the Rhind Mathematical Papyrus
and the (RMP 2/n table) in the manner recorded by Ahmes
. Math historians recognized red numbers were connected to LCMs, but few explicit details were described in the 2/n table. In the 21st century math historians began to parse Middle Kingdom origins and applications of red numbers. Ahmes practiced finding LCMs and its red divisor factored aspect of GCDs in RMP 21, 22 and 23, a topic introduced by George G. Joseph, "Crest of the Peacock" in 1991:
On page 37, example 3.7 Joseph reports:
Complete 2/3 + 1/4 + 1/28 to 1.
This meant: solve 2/3 + 1/4 + 1/28 + x = 1 (example 3.7)
The lowest common denominator (LCM) is not 28, but rather 42. Modern students would likely multiple 3 times 28 finding an LCM of 84. But 42 was sufficient for Ahmes, and Egyptian scribes, as noted by:
was written out in fractions
with 42 marked in red, and not our modern algebra form that would have written 42x.
The unknown fraction x is found by solving
meant that
Additional RMP problems asked Ahmes
to complete a series of fractions that added up to a given number include:
using Least common multiple
(LCM) 30 to find
such that:
was rewritten as:
and,
using LCM 45 to compute x = 1/9 + 1/40
(certain details have been left for the reader)
It is important to note that the Egyptian Mathematical Leather Roll
(EMLR) and the RMP 2/n table employed LCMs within modern-like multiplication and division operations. Decoding Red numbers arithmetic and its applications have surprisingly decoded modern-like arithmetic operations that were hidden in scribal shorthand notes.
In RMP 37 LCM 72 scaled 1/4 to 72/288 and 1/8 to 72/576. Red numbers parsed 72/88 by /288 and 72/576 by /576, both non-optimal unit fraction series. Ahmes aligned each red number below a unit fraction showing that red denoted an inverse relationship to a unit fraction, even to numerators that were not integers.
The optimized Egyptian fraction series was an implicit topic of the RMP 2/n table. The RMP began with 1/3 of the text reporting 51 2/n optimized, but not always optimal, Egyptian fraction series by implicitly selecting optimized, but not optimal, red auxiliary numbers from the divisors, aliquot parts, of scaled denominators mn.
Red numbers defined a core method that math historians pondered for over 130 years finally parsing the method in the 21st century. Aspects of the method were exposed in a 2002 AD EMLR paper, 2006 with a Akhmim Wooden Tablet
paper, and an Ebers Papyrus
paper. Non-additive numerical methods connected Egyptian weights and measures, 2/n tables, and other ancient scribal methods to red numbers, with RMP 36 and RMP 37 providing the final abstract aspects of the method.
In summary, after 2002 alternative views of Middle Kingdom red numbers exposed LCMs and GCDs in updated translations of the scribal raw data. Several scribal notes of RMP problems included modern-like multiplication and division operations. Recent journal papers report Ahmes' use of red numbers, one being origin issues related to the factoring of rational numbers into its unique prime factors. Scribal division operations and applications employed modern-like quotients and remainders in a finite arithmetic system. Ahmes divided 2 by n 51 times to create a 2/n table. Ahmes also divided a hekat unity (64/64) and 320 ro, by n in two different volume based weights and measures systems. The traditional Old Kingdom duplation multiplication operationally proved the arithmetic accuracy of unit fraction answers, and was not the primary Middle Kingtdom multiplication operation.
Rhind Mathematical Papyrus
The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...
(RMP), the additive set of divisors were written in red ink. Red auxiliary numbers were used for several purposes The red auxiliary numbers were divisors of a greatest common divisor
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
(GCD) that was used to generally convert 2/n to optimized, but not optimal, unit fraction series. The main purpose created 2/n tables by hi-lighting scribal skills in converting rational numbers to optimized, but not optimal, unit fraction series. The RMP 2/n table reported Egyptian fraction series by multiplying 2/n by a LCM m, written as a unity (m/m), that obtained 2m/mn. 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...
the RMP author, practiced the selection of red auxiliary numbers in RMP 21, 22, and 23 and detailed other aspects of the red number method in RMP 36. The scaled 2m/mn rational number was converted to a unit fraction series by finding the best set of divisors of (GCD) mn, as discussed below.
For example, Ahmes converted 2/43 by LCM 42, thinking:
- 2/43×(42/42) = 84/1806
To parse 84/1806 Ahmes selected the best divisors of 1806 from {43, 42, 21, 14, 7, 6, 4, 3, 2, 1}, denoting the divisors that best summed to numerator 84 in .
Ahmes chose rather than , or ,
such that:
- 2/43 = 84/1806 = /1806 = 1/42 + 1/86 + 1/129 + 1/301
The 2/n table method that was validated in RMP 36 with an explicit conversion of 3/53 scaled to 60/1060. The best divisors of 1060 were summed to 60 within /1060 and 1/20 + 1/265 + 1/530 + 1/1060. The red number method was used over six times in RMP 36 and over seven times in RMP 37.
Scribal red numbers, LCMs and GCDs were known for over 130 years before being rigorously parsed. Math historians failed to parse aspects of red numbers seen in the Rhind Mathematical Papyrus
Rhind Mathematical Papyrus
The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...
and the (RMP 2/n table) in the manner recorded by 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...
. Math historians recognized red numbers were connected to LCMs, but few explicit details were described in the 2/n table. In the 21st century math historians began to parse Middle Kingdom origins and applications of red numbers. Ahmes practiced finding LCMs and its red divisor factored aspect of GCDs in RMP 21, 22 and 23, a topic introduced by George G. Joseph, "Crest of the Peacock" in 1991:
On page 37, example 3.7 Joseph reports:
Complete 2/3 + 1/4 + 1/28 to 1.
This meant: solve 2/3 + 1/4 + 1/28 + x = 1 (example 3.7)
The lowest common denominator (LCM) is not 28, but rather 42. Modern students would likely multiple 3 times 28 finding an LCM of 84. But 42 was sufficient for Ahmes, and Egyptian scribes, as noted by:
- 84/3 + 42/4 + 42/28 + 42x = 42 (example 3.7.1)
was written out in fractions
- 28 + (10 + 1/2) + (1 + 1/2) + 2 = (example 3.7.2)
with 42 marked in red, and not our modern algebra form that would have written 42x.
The unknown fraction x is found by solving
- 42x = 2, or x = 2/42 = 1/21 (example 3.7.3)
meant that
- 2/3 + 1/4 + 1/28 + 1/21 = 1 (example 3.7.4)
Additional RMP problems asked 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...
to complete a series of fractions that added up to a given number include:
- RMP 21: Complete 2/3 + 1/15 + x = 1
using Least common multiple
Least common multiple
In arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b...
(LCM) 30 to find
- 60/3 + 30/15 + 30x = 30
- 20 + 2 + 8 = 30
- 30x = 8
- x = 8/30 = 4/15 = (3 + 1)/15 = 1/5 + 1/15
such that:
- 2/3 + 1/5 + 2/15 = 1
was rewritten as:
- 2/3 + 1/5 + 1/10 + 1/30 = 1
and,
- RMP 23: Complete 1/4 + 1/8 + 1/10 + 1/35 + 1/45 + x = 3
using LCM 45 to compute x = 1/9 + 1/40
(certain details have been left for the reader)
It is important to note that the 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...
(EMLR) and the RMP 2/n table employed LCMs within modern-like multiplication and division operations. Decoding Red numbers arithmetic and its applications have surprisingly decoded modern-like arithmetic operations that were hidden in scribal shorthand notes.
In RMP 37 LCM 72 scaled 1/4 to 72/288 and 1/8 to 72/576. Red numbers parsed 72/88 by /288 and 72/576 by /576, both non-optimal unit fraction series. Ahmes aligned each red number below a unit fraction showing that red denoted an inverse relationship to a unit fraction, even to numerators that were not integers.
The optimized Egyptian fraction series was an implicit topic of the RMP 2/n table. The RMP began with 1/3 of the text reporting 51 2/n optimized, but not always optimal, Egyptian fraction series by implicitly selecting optimized, but not optimal, red auxiliary numbers from the divisors, aliquot parts, of scaled denominators mn.
Red numbers defined a core method that math historians pondered for over 130 years finally parsing the method in the 21st century. Aspects of the method were exposed in a 2002 AD EMLR paper, 2006 with a 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...
paper, and an Ebers Papyrus
Ebers papyrus
The Ebers Papyrus, also known as Papyrus Ebers, is an Egyptian medical papyrus dating to circa 1550 BC. Among the oldest and most important medical papyri of ancient Egypt, it was purchased at Luxor, in the winter of 1873–74 by Georg Ebers...
paper. Non-additive numerical methods connected Egyptian weights and measures, 2/n tables, and other ancient scribal methods to red numbers, with RMP 36 and RMP 37 providing the final abstract aspects of the method.
In summary, after 2002 alternative views of Middle Kingdom red numbers exposed LCMs and GCDs in updated translations of the scribal raw data. Several scribal notes of RMP problems included modern-like multiplication and division operations. Recent journal papers report Ahmes' use of red numbers, one being origin issues related to the factoring of rational numbers into its unique prime factors. Scribal division operations and applications employed modern-like quotients and remainders in a finite arithmetic system. Ahmes divided 2 by n 51 times to create a 2/n table. Ahmes also divided a hekat unity (64/64) and 320 ro, by n in two different volume based weights and measures systems. The traditional Old Kingdom duplation multiplication operationally proved the arithmetic accuracy of unit fraction answers, and was not the primary Middle Kingtdom multiplication operation.
External links
- http://planetmath.org/encyclopedia/RMP36AndThe2nTable.html RMP 36 and the 2/n table
- http://planetmath.org/encyclopedia/RMP35To38PlusRMP66.html RMP 35–38, plus RMP 66
- http://emlr.blogspot.com Egyptian Mathematical Leather Roll
- http://planetmath.org/encyclopedia/EgyptianMathematicalLeatherRoll2.html Planetmath/EMLR
- http://rmprectotable.blogspot.com/ RMP 2/n table and LCMs
- http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html New and Old Ahmes Papyrus classifications
- http://planetmath.org/encyclopedia/EgyptianMath3.html Hultsch–Bruins Method and aliquot parts
- http://mathforum.org/kb/message.jspa?messageID=6492421&tstart=0 Math forum /ancient and modern LCMs
- http://mathforum.org/kb/message.jspa?messageID=6579539&tstart=0 Math forum and two ways to calculate 2/7