The Rhind Mathematical Papyrus (RMP) (also designated as: papyrus British Museum The British Museum is a museum of human history and culture in London. Its collections, which number more than seven million objects, are amongst the largest and most comprehensive in the world and originate from all continents, illustrating and documenting the story of human culture from its beginning to the present.[a] 10057, and pBM 10058), is named after Alexander Henry Rhind, a Scottish Scotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the southwest. In addition to the mainland, Scotland antiquarian, who purchased the papyrus Papyrus is a thick paper-like material produced from the pith of the papyrus plant, Cyperus papyrus, a wetland sedge that was once abundant in the Nile Delta of Egypt in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum The Ramesseum is the memorial temple of Pharaoh Ramesses II ("Ramesses the Great", also spelled "Ramses" and "Rameses"). It is located in the Theban necropolis in Upper Egypt, across the River Nile from the modern city of Luxor. The name – or at least its French form, Rhamesséion – was coined by Jean-François. It dates to around 1650 BC. The British Museum, where the papyrus is now kept, acquired it in 1864 along with the Egyptian Mathematical Leather Roll The Egyptian Mathematical Leather Roll was a 10" x 17" leather roll purchased by Alexander Henry Rhind in 1858. It was sent to the British Museum in 1864, along with the Rhind Mathematical Papyrus but the former was not chemically softened and unrolled until 1927 (Scott, Hall 1927), also owned by Henry Rhind; there are a few small fragments held by the Brooklyn Museum The Brooklyn Museum, located at 200 Eastern Parkway, in the New York City borough of Brooklyn, is the second-largest art museum in New York City, and one of the largest in the United States. Arnold L. Lehman is the museum's Director in New York New York City, which is geographically the largest city in the state and most populous in the United States, is known for its history as a gateway for immigration to the United States and its status as a financial, cultural, transportation, and manufacturing center. According to the U.S. Department of Commerce, it is also a destination of choice[citation needed]. It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus The Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenischev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Goleniščev. It later entered the collection of the Pushkin State Museum of Fine Arts in Moscow, where it remains today. Based on the palaeography of the hieratic text, it. The Rhind Papyrus is larger than the Moscow Mathematical Papyrus, while the latter is older than the former.[1]

The Rhind Mathematical Papyrus dates to the Second Intermediate Period The Second Intermediate Period marks a period when Ancient Egypt fell into disarray for a second time, between the end of the Middle Kingdom and the start of the New Kingdom. It is best known as the period when the Hyksos made their appearance in Egypt and whose reign comprised the fifteenth and sixteenth dynasties of Egypt The History of ancient Egypt spans the period from the early predynastic settlements of the northern Nile Valley to the Roman conquest in 30 BC. The Pharaonic Period is dated from around 3150 BC, when Lower and Upper Egypt became a unified state, until the country fell under Greek rule in 332 BC and is the best example of Egyptian mathematics After 2050 BC, a form of Egyptian multiplication doubled answers to problems. The doubled answers reported the arithmetic correctness of answers. Scholars noted that many initial and intermediate calculations were missing, and searched for the ab initio information. Finding few ciphered hieratic letters were transliterated into modern numbers. It was copied by the scribe Ahmes Ahmes was an ancient Egyptian scribe who lived during the Second Intermediate Period and the beginning of the Eighteenth Dynasty (the first dynasty of the New Kingdom). He is best known for his work in mathematics (i.e., Ahmose; Ahmes is an older transcription Transcription in the linguistic sense is the conversion of a representation of language into another representation of language, usually in the same language but in a different form. A transcriptionist is a person who performs transcription favoured by historians of mathematics), from a now-lost text from the reign of king Pharaoh is a title used in many modern discussions of the ancient Egyptian rulers of all periods. In antiquity this title began to be used for the ruler who was the religious and political leader of united ancient Egypt. This was true only during the New Kingdom, specifically during the middle of the eighteenth dynasty. For simplification, however, Amenemhat III Amenemhat III, also spelled Amenemhet III , was a pharaoh of the Twelfth Dynasty of Egypt. He ruled from ca.1860 BC to ca.1814 BC, the latest known date being found in a papyrus dated to Regnal Year 46, I Akhet 22 of his rule. He is regarded as the greatest monarch of the Middle Kingdom. He may have had a long coregency (of 20 years) with his (12th dynasty The chronology of the Twelfth Dynasty is the most stable of any period before the New Kingdom. Manetho stated that it was based in Thebes, but from contemporary records it is clear that the first king moved its capital to a new city named "Amenemhat-itj-tawy" , more simply called Itjtawy. The location of Itjtaway has not been found, but). Written in the hieratic Hieratic is a cursive writing system that was used in the provenance of the pharaohs in Egypt and Nubia that developed alongside the hieroglyphic system, to which it is intimately related. It was primarily written in ink with a reed brush on papyrus, allowing scribes to write quickly without resorting to the time-consuming hieroglyphs. The word script, this Egyptian manuscript A manuscript or handwrit is a recording of information that has been manually created by someone or some people, such as a hand-written letter, as opposed to being printed or reproduced some other way. The term may also be used for information that is hand-recorded in other ways than writing, for example inscriptions that are chiselled upon a hard is 33 cm tall and over 5 meters long, and began to be transliterated and mathematically translated in the late 19th century. In 2008, the mathematical translation aspect is incomplete in several respects. The document is dated to Year 33 of the Hyksos The Hyksos were an Asiatic people who invaded the eastern Nile Delta, in the Twelfth dynasty of Egypt initiating the Second Intermediate Period of Ancient Egypt. The people are shown below wearing the cloaks of many colors associated with the mercenary Mitanni bowmen and cavalry (ha ibrw) of Northern Canaan, Aram, Kadesh, Sidon and Tyre king Apophis Apepi (also Ipepi; Egyptian language ipp) or Apophis (Greek Άποφις; regnal names Neb-Khepesh-Re, A-Qenen-Re and A-User-Re) was a ruler of Lower Egypt during the fifteenth dynasty and the end of the Second Intermediate Period that was dominated by this foreign dynasty of rulers called the Hyksos. According to the Turin Canon of Kings, he and also contains a separate later Year 11 on its verso likely from his successor, Khamudi.[2]

In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving “Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries...all secrets”.

Contents

Mathematical problems

A portion of the Rhind Papyrus

The papyrus, written on both sides, began with a RMP 2/n table The Rhind Mathematical Papyrus contains, among other mathematical contents, a table of Egyptian fractions created from 2/n. The text reports 51 rational numbers converted to short and concise unit fraction series. The document was written in 1650 BCE by Ahmes. Aspects of the document may have been copied from an unknown 1850 BCE text. Another, followed by 87 problems. The RMP 2/n table The Rhind Mathematical Papyrus contains, among other mathematical contents, a table of Egyptian fractions created from 2/n. The text reports 51 rational numbers converted to short and concise unit fraction series. The document was written in 1650 BCE by Ahmes. Aspects of the document may have been copied from an unknown 1850 BCE text. Another took up one third of the manuscript. The table converted 2 divided by the odd numbers from 3 to 101 by sums of Egyptian fractions An Egyptian fraction is the sum of distinct unit fractions, such as . That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The sum of an expression of this type is a positive rational number a/b; for instance the Egyptian fraction above using an Egyptian multiplication and division method that 19th and 20th century scholars found hard-to-read.

Alternative scribal 2/n table methods were proposed by 19th and 20th century scholars. A German Egyptologist F. Hultsch (1895) proposed an aliquot part aspect of the 2/n table. Hultsch's aliquot part fragments were independently confirmed by E.M. Bruins in 1944. The hard-to-read singular 2/n conversion method apparently consisted of selecting least common multiples In arithmetic and number theory, the least common multiple or lowest common multiple or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both of a and of b. Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive (LCMs). A scribe implicitly selected LCMs in the Egyptian Mathematical Leather Roll The Egyptian Mathematical Leather Roll was a 10" x 17" leather roll purchased by Alexander Henry Rhind in 1858. It was sent to the British Museum in 1864, along with the Rhind Mathematical Papyrus but the former was not chemically softened and unrolled until 1927 (Scott, Hall 1927) by converting 1/p and 1/pq rational numbers to non-optimal unit fraction series. Ahmes' improved 2/n table LCMs was published in 2005 exposed LCM 12 and/or LCM 60 scaled 2/95 and/or 1/5*(2/19) per:

2/95 = (1/5)*(2/19*(12/12) =(1/5)*(24/228) = (19 + 3 + 2 )/1140 = 1/60 + 1/380 + 1/570

and/or,

2/95 = (2/95)*(60/60)= 120/5700 = (95 + 15 + 10)/5700 = 1/60 + 1/380 + 1/570

LCM 12 suggests that Ahmes factored 2/pq to 1/q times 2/p parsing an important scribal 2/n table application.

In 2008 and 2010 an explicit RMP 36 red number method converted 2/53, 3/53, 5/53 + 15/53 + 28/53 by

2/53 = 60/1590 = (53 + 5 + 2)/1590 = 1/30 + 1/318 + 1/795

3/53 = 60/1060 = (53 + 4 + 2 + 1)/1060 = 1/20 + 1/265 + 1/530 + 1/1060

5/53 = 60/636 = (53 + 4 + 2 + 1)/636 = 1/12 + 1/159 + 1/319 + 1/636

15/53 = 60/212 = (53 + 4 + 2 + 1)/212 = 1/4 + 1/53 + 1/106 + 1/212

28/53 = 56/106 = (53 + 2 + 1)/106 = 1/2 + 1/53 + 1/106

and,

30/53 = 28/53 + 2/53

making explicit a second application of RMP 2/n tables The Rhind Mathematical Papyrus contains, among other mathematical contents, a table of Egyptian fractions created from 2/n. The text reports 51 rational numbers converted to short and concise unit fraction series. The document was written in 1650 BCE by Ahmes. Aspects of the document may have been copied from an unknown 1850 BCE text. Another. In RMP 31 Ahmes solved 28/97 by substituting 26/97 + 2/97 unit fraction series. Generally, Ahmes converted difficult n/p rational number conversions by substituting (n -2)/p + 2/p.

Ahmes' hard-to-read shorthand notes included additive numerators in RMP 36 that were written in red. Ahmes omitted initial and intermediate steps, however, Ahmes' method is clear based on his shorthand notes containing informational links to other RMP problems. Today, Ahmes' 2/n table calculation omissions that had confused math historians began to clear up in 2001 by citing the EMLR's 26 lines of data. The EMLR paper set decoding ground work for a 2005 Akhmim Wooden Tablet paper. The final piece of the 2/n table problem puzzle was confirmed in the 2009 RMP 36 paper.

21st century scholars are increasingly reporting the RMP's 87 problems that began with six division-by-10 problems to be amplified by the contents of the Reisner Papyrus. In the RMP there were 15 problems that dealt with addition, and 19 algebra problems. There were 15 algebra problems, RMP 18-23 and RMP 24- 34, of the same type, followed by RMP 35- 38 written in a hekat context. The 19 algebra problems asked Ahmes to find x and a fraction of x such that the sum of x and its fraction equaled an integer. Problem #24 is the easiest that asked Ahmes to solve the equation, x + 1/7x = 19. Ahmes worked the problem this way:

(8/7)x = 19, or x = 133/8 = 16 + 5/8,

with 133/8 being the initial vulgar fraction A fraction is a number that can represent part of a whole find 16 as the quotient and 5/8 as the remainder term. Ahmes converted 5/8 to an Egyptian fraction series by (4 + 1)/8 = 1/2 + 1/8, making his final quotient plus remainder based answer x = 16 + 1/2 + 1/8.

The algebra problems, from RMP 21 -34, produced increasingly difficult vulgar fractions. RMP 38 converted a hekat, written 320 ro, by multiplying by 35/110, 7/22, obtaining 101 9/11. The initial 320 ro was obtained by multiplying 101 9/11 by 22/7. RMP 82 partitioned a hekat written as (64/64). Hekat unity problems limited n to 1/64 < n < 64, obtaining quotient (Q) and remainder (R) two-part numbers: Q/64 + (5R/n)ro. Ro answers were converted to a one-part 1/10 hekat hin unit by writing 10/n (29 times). Vulgar fractions were easily converted to an optimal (short and small last term) Egyptian fraction An Egyptian fraction is the sum of distinct unit fractions, such as . That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The sum of an expression of this type is a positive rational number a/b; for instance the Egyptian fraction above series in all RMP problems.

Two arithmetical progressions (A.P.) were solved, one being RMP 64. The method of solution followed the method defined in the Kahun Papyrus The Kahun Papyrus is an ancient Egyptian text discussing mathematical and medical topics. Its many fragments were discovered by Flinders Petrie in 1889 and are kept at the University College London. Most of the texts are dated to ca 1825 BC, to the reign of Amenemhat III. One of its fragments, referred to as the Kahun Gynaecological Papyrus, deals. The problem solved sharing 10 hekats of barley, between 10 men, by a difference of 1/8th of a hekat finding 1 7/16 as the largest term.

The second A.P. was RMP 40, the problem divided 100 loaves of bread between five men such that the smallest two shares (12 1/2) were 1/7 of the largest three shares' sum (87 1/2). The problem asked Ahmes to find the shares for each man, which he did without finding the difference (9 1/6) or the largest term (38 1/3). All five shares 38 1/3, 29 1/6, 20, 10 2/3 1/6, and 1 1/3) were calculated by first finding the five terms from a proportional A.P. that summed to 60. The median and the smallest term, x1, were used to find the differential and each term. Ahmes then multiplied each term by 1 2/3 to obtain the sum to 100 A.P. terms. In reproducing the problem in modern algebra, Ahmes also found the sum of the first two terms by solving x + 7x = 60.

The RMP continues with 5 hekat division problems from the Akhmim Wooden Tablet, 15 problems similar to ones from the Moscow Mathematical Papyrus The Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenischev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Goleniščev. It later entered the collection of the Pushkin State Museum of Fine Arts in Moscow, where it remains today. Based on the palaeography of the hieratic text, it, 23 problems from practical weights and measures, especially the hekat, and three problems from recreational diversion subjects, the last the famous multiple of 7 riddle, written in the Medieval era as, "Going to St. Ives".

The Rhind Mathematical Papyrus also contains the following problem related to trigonometry Trigonometry is a branch of mathematics that studies triangles, particularly right triangles. Trigonometry deals with relationships between the sides and the angles of triangles, and with trigonometric functions, which describe those relationships and angles in general, and the motion of waves such as sound and light waves:[3]

"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?"

The solution to the problem is given as the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face.[3]

Mathematical knowledge

Upon closer inspection, modern-day mathematical analyses of Ahmes' problem-solving strategies reveal a basic awareness of composite A composite number is a positive integer which has a positive divisor other than one or itself. In other words, if n > 0 is an integer and there are integers 1 < a, b < n such that n = a × b then n is composite. By definition, every integer greater than one is either a prime number or a composite number. The number one is a unit - it is and prime numbers In mathematics, a prime number is a natural number that has exactly two distinct natural number divisors: 1 and itself. The smallest twenty-five prime numbers (all the prime numbers under 100) are:;[4] arithmetic In mathematics and statistics, the arithmetic mean of a list of numbers is the sum of all of the list divided by the number of items in the list. If the list is a statistical population, then the mean of that population is called a population mean. If the list is a statistical sample, we call the resulting statistic a sample mean, geometric The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, which is what most people think of with the word "average", except that the numbers are multiplied and then the nth root of the resulting product is taken and harmonic means In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired;[4] a simplistic understanding of the Sieve of Eratosthenes In mathematics, the Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer. It works efficiently for the smaller primes (below 10 million). It was created by Eratosthenes, an ancient Greek mathematician. However, none of his mathematical works survived - the sieve was described and attributed[4], and perfect numbers In mathematics, a perfect number is a positive integer that is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors , or σ(n) = 2n.[4][5]

The papyrus also demonstrates knowledge of solving first order linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable[5] and summing arithmetic In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 3, 5, 7, 9, 11, 13, … is an arithmetic progression with common difference 2 and geometric series In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series.[5]

The papyrus calculates π π is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.141593 in the usual decimal notation. Many formulae from mathematics, science, and engineering involve π, which as (a margin of error of less than 1%). In addition 255/81 was considered (3.1481481...) and 22/7. In RMP 38, Ahmes multiplied a hekat, 320 ro, by 7/22 obtaining 101 9/11. The divisor 7/22 was inverted to 22/7 and multiplied by 101 9/11 obtaining 320 ro as a proof. Ahmes' use of 22/7 may have corrected the hekat's built-in loss based on using 256/81 as pi.

Other problems in the Rhind papyrus demonstrate knowledge of arithmetic progressions (Kahun Papyrus The Kahun Papyrus is an ancient Egyptian text discussing mathematical and medical topics. Its many fragments were discovered by Flinders Petrie in 1889 and are kept at the University College London. Most of the texts are dated to ca 1825 BC, to the reign of Amenemhat III. One of its fragments, referred to as the Kahun Gynaecological Papyrus, deals), algebra Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure and geometry Geometry "Earth-measuring" is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by.

The papyrus demonstrates knowledge of weights and measures, business distributions of money (paid out in arithmetic progressions, with one group proportionally being paid more than another), and several recreational diversions.

Influence of the RMP

This section does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be and removed. (September 2008)

The Egyptian use of arithmetic proportions in the Rhind Papyrus, problems 40 and 64, and the Kahun Papyrus The Kahun Papyrus is an ancient Egyptian text discussing mathematical and medical topics. Its many fragments were discovered by Flinders Petrie in 1889 and are kept at the University College London. Most of the texts are dated to ca 1825 BC, to the reign of Amenemhat III. One of its fragments, referred to as the Kahun Gynaecological Papyrus, deals, are briefly discussed by Gillings. In particular the use of the Remen, which has two values, is reflected in the foot which has two values, (the second being the nibw or ell Several national forms existed, with different lengths, including the Scottish ell , the Flemish ell (approx. 27 in or 69 cm), the Polish ell (approx. 31 in or 79 cm) and the Danish ell (divided into a length just under 25 inches, approximately 63 cm) which is two feet), and the cubit which has two values. Doubling is also seen in the subdivisions such as fingers and palms. Since doubling seems to have been the basis of most of the unit fraction calculations, which it was not (multiples were) up to and including the calculations of circles with dimensions given in khet (see Ancient Egyptian units of measurement), looking at how the remen and seked were used provided many insights to Greek and Roman geometers A geometer is a mathematician whose area of study is geometry. Some important geometers and their main fields of work are: and architects An architect is a person trained in the planning, design and oversight of the construction of buildings, and is licensed to practice architecture. To practice architecture means to offer or render services in connection with the design and construction of a building, or group of buildings and the space within the site surrounding the buildings,. The actual and proposed readings/decodings of the RMP and Kahun 2/n tables is required to be fairly interjected.

In the Rhind Papyrus we first encounter the remen which is defined as the proportion of the diagonal of a rectangle to its sides when its other sides are whole units. Yet, a singular arithmetic proportion formula reported in the RMP and Kahun Papyrus offer an additional example beyond the remen's diagonal A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- ("through", "across") and gonia ("angle", related to gony "knee"); it was used by both of a square, with its sides a cubit. We also find problems using the seked or unit rise to run proportion. Typical of the Classical orders of the Greeks and Romans, it was built upon the canon of proportions derived from the inscription grids of the Egyptians.

This document is one of the main sources of our knowledge of Egyptian mathematics.

See also

References

  1. ^ Great Soviet Encyclopedia, 3rd edition, entry on "Папирусы математические", available online here
  2. ^ cf. Thomas Schneider's paper 'The Relative Chronology of the Middle Kingdom and the Hyksos Period (Dyns. 12-17)' in Erik Hornung, Rolf Krauss & David Warburton (editors), Ancient Egyptian Chronology (Handbook of Oriental Studies), Brill: 2006, p.194-195
  3. ^ a b Maor, Eli (1998). Trigonometric Delights. Princeton University Press. p. 20. ISBN 0691095418.
  4. ^ a b c d [1] MathPages - Egyptian Unit Fractions.
  5. ^ a b c [2] Scott W. Williams, The Mathematics Department of The State University of New York at Buffalo.

External links

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