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A portion of the Rhind Papyrus
A portion of the Rhind Papyrus

The Rhind Mathematical Papyrus (RMP) also designated as: papyrus British Museum 10057, and pBM 10058), 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. The British Museum is a Museum of human history and culture in London. for others with the same name see Rhind Alexander Henry Rhind (1833 &ndash 1863 was a Scottish Lawyer and Scotland ( Gaelic: Alba) is a Country in northwest Europethat occupies the northern third of the island of Great Britain. Papyrus (/pəˈpaɪrəs/ (Rhymes -aɪrəs)is a thick paper-like material produced from the Pith of the papyrus plant Cyperus papyrus Year 1858 ( MDCCCLVIII) was a Common year starting on Friday (link will display the full calendar of the Gregorian Calendar (or a Common Luxor (in Arabic: الأقصر al-Uqṣur) is a city in Upper (southern Egypt and the capital of Luxor The Ramesseum is the memorial temple (or mortuary temple of Pharaoh Ramesses II ("Ramesses the Great" also spelt "Ramses" and The British Museum, where the papyrus is now kept, acquired it in 1864 along with the Egyptian Mathematical Leather Roll, also owned by Henry Rhind; there are a few small fragments held by the Brooklyn Museum in New York. The Egyptian Mathematical Leather Roll (also referred to as EMLR) was a 10" x 17" leather roll purchased by Alexander Henry Rhind in 1858. 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 ( is a state in the Mid-Atlantic and Northeastern regions of the United States and is the nation's third most populous It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus. The Moscow Mathematical Papyrus is also called the Golenischev Mathematical Papyrus, after its first owner Egyptologist Vladimir Goleniščev. 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 of Egypt and is the best example of Egyptian mathematics. The Second Intermediate Period marks a period when Ancient Egypt once again fell into disarray between the end of the Middle Kingdom, and the start of the New The History of Ancient Egypt spans the period from the early predynastic settlements of the northern Nile Valley to the Roman conquest in 30 Egyptian mathematics refers to the style and methods of Mathematics performed in Ancient Egypt. It was copied by the scribe Ahmes (i. Ahmes (c 1680 BC-c 1620 BC (more accurately Ahmose) was an Egyptian scribe who lived during the Second Intermediate Period. e. , Ahmose; Ahmes is an older transcription favoured by historians of mathematics), from a now-lost text from the reign of king Amenemhat III (12th dynasty). Transcription is the conversion into written typewritten or printed form of a Spoken language source such as the proceedings of a court hearing Pharaoh is the title given in modern parlance to the ancient Egyptian kings of all periods Amenemhat III, alt Amenemhet III, (c 1860 BC-1814 BC was a Pharaoh of the Twelfth Dynasty of Egypt. The Eleventh (all of Egypt Twelfth, Thirteenth and Fourteenth Dynasties of ancient Egypt are often combined under the group title Middle Kingdom. Written in the hieratic script, this Egyptian manuscript is 33 cm tall and over 5 meters long, and was first translated in the late 19th century. Hieratic is a Cursive writing system used in pharaonic Egypt that developed alongside the hieroglyphic system to which it is intimately A manuscript is any Document that is Written by hand as opposed to being printed or reproduced in some other way The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar The document is dated to Year 33 of the Hyksos king Apophis and also contains a separate later Year 11 on its verso likely from his successor, Khamudi. The Hyksos ( Egyptian heqa khasewet, "foreign rulers" Greek,, Arabic,) were an Asiatic people who invaded the eastern Nile Khamudi (also known as Khamudy) was the last Pharaoh of the Hyksos Fifteenth dynasty of Egypt, who came to power in the northern portion of Egypt [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

The papyrus begins with the RMP 2/n table and follows with 84 problems, worked, written on both sides. The Rhind Mathematical Papyrus contains among other mathematical contents a Table of Egyptian fractions created from 2/ n. Taking up roughly one third of the manuscript is the RMP 2/n table which expresses 2 divided by the odd numbers from 5 to 101 in terms only of unit fractions. The Rhind Mathematical Papyrus contains among other mathematical contents a Table of Egyptian fractions created from 2/ n. There are two basic vulgar fraction methods used, one to convert 2/p and another to convert 2/pq vulgar fractions to Egyptian fractions. An Egyptian fraction is the sum of distinct Unit fractions such as \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16} The 2/p method was noted by F. Hultsch in 1895, and confirmed by E. M. Bruins in 1945, commonly called the H-B method. The 2/pq method consists of three methods, with two based on the Egyptian Mathematical Leather Roll and the third factoring 2/95 into 1/5 x 2/19, with 2/19 converted by the H-B Method.

The RMP's 84 problems begin with six division-by-10 problems, the central subject of the Reisner Papyrus. The Reisner Papyrus is one of the most basic of the hieratic mathematical texts There are 15 problems dealing with addition, and 18 algebra problems. There are 15 algebra problems of the same type. They ask the reader to find x and a fraction of x such that the sum of x and its fraction equals a given integer. Problem #24 is the easiest, and asks the reader to solve this equation, x + 1/7x = 19. Ahmes, the author of the RMP, worked the problem this way:

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

with 133/8 being the initial vulgar fraction find 16 as the quotient and 5/8 as the remainder term. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object 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.

Each of the RMP's other 14 algebra problems produced increasingly difficult vulgar fractions. Yet, all were easily converted to an optimal (short and small last term) Egyptian fraction series. An Egyptian fraction is the sum of distinct Unit fractions such as \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16}

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 (KP is as an ancient Egyptian text discussing mathematical and medical topics 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, 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 Akhmim wooden tablet, is an Ancient Egyptian artifact that has been dated to 2000 BC, near to the beginning of the Egyptian Middle Kingdom. The Moscow Mathematical Papyrus is also called the Golenischev Mathematical Papyrus, after its first owner Egyptologist Vladimir Goleniščev. The hekat or heqat (transcribed HqAt) was an ancient Egyptian volume unit used to measure grain bread and beer " As I was going to St Ives " is a traditional Nursery rhyme which is generally thought to be a Riddle.

Mathematical knowledge

Upon closer inspection, modern-day mathematical analyses of Ahmes' problem-solving strategies reveal a basic awareness of composite and prime numbers;[3] arithmetic, geometric and harmonic means;[3] a simplistic understanding of the Sieve of Eratosthenes[3], and perfect numbers. A composite number is a positive Integer which has a positive Divisor other than one or itself In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided 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 In Mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of Average. In Mathematics, the Sieve of Eratosthenes is a simple ancient Algorithm for finding all Prime numbers up to a specified integer In mathematics a perfect number is defined as a positive integer which is the sum of its proper positive Divisors that is the sum of the positive divisors excluding [3][4]

The papyrus also demonstrates knowledge of solving first order linear equations[4] and summing arithmetic and geometric series. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members In Mathematics, a geometric series is a series with a constant ratio between successive terms. [4]

The papyrus calculates π as (8/9)^2*4 \simeq 3.1605 (a margin of error of less than 1%). IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems [5] Two viable theories offering some insight into a possible motivation for such an accurate derivation have been proposed:[5]

  1. African crafts demonstrating snake curves and sets of equidistant concentric circles,[6] and
  2. Boardgame resembling Mancala,[6] found in the Mortuary Temple of Seti I (both boardgames utilize small and large circles; see Mancala). A board game is a Game in which counters or pieces that are placed on removed from or moved across a "board" (a premarked surface usually specific to that game Mancala is a family of board games played around the world sometimes called " Sowing " games or "count-and-capture" games which describes the The Mortuary Temple of Seti I is the Memorial temple (or mortuary temple of Pharaoh Seti I. Mancala is a family of board games played around the world sometimes called " Sowing " games or "count-and-capture" games which describes the

Other problems in the Rhind papyrus demonstrate knowledge of arithmetic progressions, algebra and geometry. In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position

The papyrus also demonstrates knowledge of weights and measures, business, and recreational diversions.

Influence of the RMP

The Egyptian use of proportion in calculation In the Rhind Papyrus is briefly discussed in 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 which is two feet), and the cubit which has two values. An ell (from Proto-Indo-European *el- "elbow forearm" is a unit of measurement approximating the distance from the elbow to the wrist Doubling is also seen in the subdivisions such as fingers and palms. Since doubling is the basis of most of the unit fraction calculations, 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 and architects. The Ancient Egyptian unit of linear measurement was known as the Royal Cubit, was maintained as 523 A geometer is a Mathematician whose area of study is Geometry. An architect is a licensed individual who leads a design team in the Planning and Design of buildings and participates in oversight of Building Construction

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. In its earliest form it is the diagonal of a square, with its sides a cubit. A diagonal can refer to a line joining two nonconsecutive vertices of a Polygon or Polyhedron, or in contexts any upward or downward sloping line 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. A classical order is one of the ancient styles of building design in the classical tradition, distinguished by their proportions and their characteristic profiles and details Epigraphy (ἐπιγραφολογία from Greek ἐπιγραφή — "inscription" is the study of inscriptions or epigraphs engraved 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. Papyrus Harris I is also known as the Great Harris Papyrus and (less accurately simply the Harris Papyrus (though there are a number of other papyri in the Harris The Great Soviet Encyclopedia ( Большая Советская Энциклопедия, or БСЭ; transliterated Bolshaya Sovetskaya Entsiklopediya 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 c d [1] MathPages - Egyptian Unit Fractions.
  4. ^ a b c [2] Scott W. Williams, The Mathematics Department of The State University of New York at Buffalo.
  5. ^ a b [3] J. J. O'Connor and E. F. Robertson, School of Mathematics and Statistics, University of St Andrews, Scotland.
  6. ^ a b Gerdes: "Three Alternate Methods of Obtaining the Ancient Egyptian Formula for the Area of a Circle," in Historia Math (12,3), 1985, pp. 261-268. Article referenced by J. J. O'Connor and E. F. Robertson; see reference above.
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