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| | Andera is what you can call her but she by no means truly favored that name. Doing ballet is some thing she would never give up. Invoicing is what I do. For a while I've been in Alaska but I will have to transfer in a yr or two.<br><br>my website; [http://appin.Co.kr/board_Zqtv22/688025 real psychic readings] |
| <!----------Name---------->
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| | name = Moscow Mathematical Papyrus
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| | location = [[Pushkin State Museum of Fine Arts]] in Moscow
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| <!----------Image---------->
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| | image = File:Moskou-papyrus.jpg
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| | width = 300px
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| | caption = 14th problem of the Moscow Mathematical Papyrus (V. Struve, 1930)
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| <!----------General---------->
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| | Date = [[Thirteenth dynasty of Egypt|13th dynasty]], [[Second Intermediate Period of Egypt]]
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| | Place of origin = [[Thebes, Egypt|Thebes]]
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| | Language(s) = [[Hieratic]]
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| The '''Moscow Mathematical Papyrus''' is an ancient [[Egyptian mathematics|Egyptian mathematical]] papyrus, also called the '''Golenishchev Mathematical Papyrus''', after its first owner, [[Egyptologist]] [[Vladimir Golenishchev]]. Golenishchev bought the papyrus in 1892 or 1893 in [[Thebes, Egypt|Thebes]]. It later entered the collection of the [[Pushkin State Museum of Fine Arts]] in Moscow, where it remains today.
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| Based on the [[palaeography]] and orthography of the [[hieratic]] text, the text was most likely written down in the [[Thirteenth dynasty of Egypt|13th dynasty]] and based on older material probably dating to the [[Twelfth dynasty of Egypt]], roughly 1850 BC.<ref name="Clagett">Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society. ISBN 0-87169-232-5</ref> Approximately 18 feet long and varying between 1½ and 3 inches wide, its format was divided into 25 problems with solutions by the [[Soviet Union|Soviet]] [[oriental studies|Orientalist]] [[Vasily Vasilievich Struve]]<ref>[http://www.encspb.ru/en/article.php?kod=2804014273 Struve V.V., (1889–1965), orientalist :: ENCYCLOPAEDIA OF SAINT PETERSBURG<!-- Bot generated title -->]</ref> in 1930.<ref>Struve, Vasilij Vasil'evič, and [[Boris Turaev]]. 1930. ''Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau''. Quellen und Studien zur Geschichte der Mathematik; Abteilung A: Quellen 1. Berlin: J. Springer</ref> It is a well-known mathematical papyrus along with the ''[[Rhind Mathematical Papyrus]].'' The ''Moscow Mathematical Papyrus'' is older than the ''Rhind Mathematical Papyrus,'' while the latter is the larger of the two.<ref>''[[Great Soviet Encyclopedia]]'', 3rd edition, entry on "Папирусы математические", available online [http://slovari.yandex.ru/art.xml?art=bse/00057/24900.htm here]</ref>
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| ==Exercises contained in the Moscow Papyrus==
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| The problems in the Moscow Papyrus follow no particular order, and the solutions of the problems provide much less detail than those in the [[Rhind Mathematical Papyrus]]. The papyrus is well known for some of its geometry problems. Problems 10 and 14 compute a surface area and the volume of a [[frustum]] respectively. The remaining problems are more common in nature.<ref name="Clagett"/>
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| ===Ship's part problems===
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| Problems 2 and 3 are ship's part problems. One of the problems calculates the length of a ship's rudder and the other computes the length of a ship's mast given that it is 1/3 + 1/5 of the length of a cedar log originally 30 cubits long.<ref name="Clagett"/>
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| ===Aha problems===
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| {{Hiero|Aha|<hiero>P6-a:M35</hiero>|align=left|era=nk}}
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| Aha problems involve finding unknown quantities (referred to as Aha) if the sum of the quantity and part(s) of it are given. The [[Rhind Mathematical Papyrus]] also contains four of these type of problems. Problems 1, 19, and 25 of the Moscow Papyrus are Aha problems. For instance problem 19 asks one to calculate a quantity taken 1 and ½ times and added to 4 to make 10.<ref name="Clagett"/> In other words, in modern mathematical notation one is asked to solve <math>3/2 \times x + 4 = 10</math>
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| ===Pefsu problems===
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| Most of the problems are pefsu problems: 10 of the 25 problems. A pefsu measures the strength of the beer made from a heqat of grain
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| : <math> \mbox{pefsu} = \frac{\mbox{number loaves of bread (or jugs of beer)}}{\mbox{number of heqats of grain}}</math>
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| A higher pefsu number means weaker bread or beer. The pefsu number is mentioned in many offering lists. For example problem 8 translates as:
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| : (1) Example of calculating 100 loaves of bread of pefsu 20
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| : (2) If someone says to you: "You have 100 loaves of bread of pefsu 20
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| : (3) to be exchanged for beer of pefsu 4
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| : (4) like 1/2 1/4 malt-date beer"
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| : (5) First calculate the grain required for the 100 loaves of the bread of pefsu 20
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| : (6) The result is 5 heqat. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer
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| : (7) The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain.
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| : (8) Calculate 1/2 of 5 heqat, the result will be 2 1/2
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| : (9) Take this 2 1/2 four times
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| : (10) The result is 10. Then you say to him:
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| : (11) "Behold! The beer quantity is found to be correct."<ref name="Clagett"/>
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| ===Baku problems===
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| Problems 11 and 23 are Baku problems. These calculate the output of workers. Problem 11 asks if someone brings in 100 logs measuring 5 by 5, then how many logs measuring 4 by 4 does this correspond to? Problem 23 finds the output of a shoemaker given that he has to cut and decorate sandals.<ref name="Clagett"/>
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| ===Geometry problems===
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| Seven of the twenty-five problems are geometry problems and range from computing areas of triangles, to finding the surface area of a hemisphere (problem 10) and finding the volume of a frustum (a truncated pyramid).<ref name="Clagett"/>
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| ==Two Interesting Geometry Problems==
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| ===Problem 10===
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| The 10th problem of the Moscow Mathematical Papyrus asks for a calculation of the surface area of a [[Sphere|hemisphere]] (Struve, Gillings) or possibly the area of a semi-cylinder (Peet). Below we assume that the problem refers to the area of a hemisphere.
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| The text of problem 10 runs like this: "Example of calculating a basket. You are given a basket with a mouth of 4 1/2. What is its surface? Take 1/9 of 9 (since) the basket is half an egg-shell. You get 1. Calculate the remainder which is 8. Calculate 1/9 of 8. You get 2/3 + 1/6 + 1/18. Find the remainder of this 8 after subtracting 2/3 + 1/6 + 1/18. You get 7 + 1/9. Multiply 7 + 1/9 by 4 + 1/2. You get 32. Behold this is its area. You have found it correctly."<ref name="Clagett"/><ref>Williams, Scott W. [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_geometry.html#moscow10 Egyptian Mathematical Papyri]</ref>
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| The solution amounts to computing the area as
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| : <math> \text{Area} = 2 \times \left(\frac{8}{9}\right)^2 \times (\text{diameter})^2 = 2 \times \frac{256}{81} (\text{radius})^2</math>
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| This means the scribe of the Moscow Papyrus used <math> \frac{256}{81} \approx 3.16049</math> to approximate pi.
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| ===Problem 14: Volume of frustum of square pyramid===
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| [[Image:Pyramide-tronquée-papyrus-Moscou 14.jpg|thumb|left]]
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| The 14th problem of the Moscow Mathematical calculates the volume of a [[frustum]].
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| Problem 14 states that a pyramid has been truncated in such a way that the top area is a square of length 2 units, the bottom a square of length 4 units, and the height 6 units, as shown. The volume is found to be 56 cubic units, which is correct.<ref name="Clagett"/>
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| The text of the example runs like this: "If you are told: a truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top: You are to square the 4; result 16. You are to double 4; result 8. You are to square this 2; result 4. You are to add the 16 and the 8 and the 4; result 28. You are to take 1/3 of 6; result 2. You are to take 28 twice; result 56. See, it is of 56. You will find [it] right" <ref>as given in Gunn & Peet, ''Journal of Egyptian Archaeology,'' 1929, 15: 176. See also, Van der Waerden, 1961, Plate 5</ref>
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| The solution to the problem indicates that the Egyptians knew the correct formula for obtaining the [[volume]] of a [[frustum|truncated pyramid]]:
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| :<math>V = \frac{1}{3} h(a^2 + a b +b^2).</math>
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| It remains unknown how the Egyptians arrived at the formula for the volume of a [[frustum]].{{Citation needed|date=May 2012}}
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| ==Other papyri==
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| Other mathematical texts from Ancient Egypt include:
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| *[[Berlin Papyrus 6619]]
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| *[[Egyptian Mathematical Leather Roll]]
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| *[[Lahun Mathematical Papyri]]
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| *[[Rhind Mathematical Papyrus]]
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| General papyri:
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| *[[Papyrus Harris I]]
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| *[[Rollin Papyrus]]
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| For the 2/n tables see:
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| *[[RMP 2/n table]]
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| ==References==
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| <references/>
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| ===Full Text of the Moscow Mathematical Papyrus===
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| *Struve, Vasilij Vasil'evič, and [[Boris Turaev]]. 1930. ''Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau''. Quellen und Studien zur Geschichte der Mathematik; Abteilung A: Quellen 1. Berlin: J. Springer
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| ===Other references===
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| *Allen, Don. April 2001. [http://www.math.tamu.edu/~don.allen/history/egypt/node4.html ''The Moscow Papyrus''] and [http://www.math.tamu.edu/~don.allen/history/egypt/node5.html ''Summary of Egyptian Mathematics''].
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| *Imhausen, A., Ägyptische Algorithmen. Eine Untersuchung zu den mittelägyptischen mathematischen Aufgabentexten, Wiesbaden 2003.
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| *Mathpages.com. [http://www.mathpages.com/home/kmath189/kmath189.htm ''The Prismoidal Formula''].
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| *O'Connor and Robertson, 2000. [http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Egyptian_papyri.html ''Mathematics in Egyptian Papyri''].
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| *Truman State University, Math and Computer Science Division. '''Mathematics and the Liberal Arts:''' [http://math.truman.edu/~thammond/history/AncientEgypt.html ''Ancient Egypt''] and [http://math.truman.edu/~thammond/history/MoscowPapyrus.html ''The Moscow Mathematical Papyrus''].
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| *Williams, Scott W. [http://www.math.buffalo.edu/mad/index.html ''Mathematicians of the African Diaspora''], containing a page on [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html ''Egyptian Mathematics Papyri''].
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| *Zahrt, Kim R. W. [http://www.iusb.edu/~journal/static/volumes/2000/zahrt.html ''Thoughts on Ancient Egyptian Mathematics''].
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| [[Category:Egyptian mathematics]]
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| [[Category:Egyptian fractions]]
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| [[Category:Ancient Egyptian literature]]
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| [[Category:Egyptian papyri]]
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