- Rhind Mathematical Papyrus 2/n table
The
Rhind Mathematical Papyrus contains, among other mathematical contents, atable ofEgyptian fraction s created from 2/"n". The text reports 51 rational numbers converted to short and conciseunit fraction series. The document was written in 1650 BCE byAhmes . Aspects of the document may have been copied from an unknown 1850 BCE text. Another ancient Egyptian papyrus containing a similar table of Egyptian fractions, theKahun Papyrus , written around 1850 BCE is about the age of one unknown source for the Rhind papyrus. The Kahun 2/"n" table differs in minor respects to the Rhind Papyrus' 2/"n" table.The 2/n table's 51 series were created from (red auxiliary) least common multiples. Scholars had confused the contents of the 2/n table by suggesting five methods may have been used. Ahmes was reported creating 2/"p" (where "p" is a
prime number ) by using two methods to convert 2/"p" (where "p" is aprime number ), and three methods to convert 2/"pq" with composite denominators. TheEgyptian Mathematical Leather Roll had not been clearly reported as a student scribe's introduction to an advanced 2/n table method.Today, it is clear that one multiple method was used to create EMLR and 2/n table series. The EMLR had used 8 multiples (2, 3, 4, 5, 6, 7, 10 and 25) converting 22 rational numbers to non-optimized unit fraction series. The RMP used 14 multiples (2, 3, 4, 6, 8, 12, 20, 24, 30, 36, 40, 56, 60 and 70) converting 51 rational numbers to nearly optimal unit fraction series.
The scholarly record on decoding the 2/n table table has been mixed. Prior to 2002 the Hultsch-Bruins method, named for F. Hultsch (1895) and E.M. Bruins (1945), was the best known method that parsed the denominator of the first partition into
aliquot parts. H-B allowed 2/p to stated as optimal unit fraction series. In 2002, a single method converted 2/101 by the multiple 6, creating 12/606 or (6 + 3 + 2 + 1)/606 = 1/101 + 1/202 + 1/303 + 1/606. TheEgyptian Mathematical Leather Roll used multiple of 6 to convert 1/4, 1/7, 1/9, 1/00 and 1/15. In addition,Eye of Horus 1/2, 1/4, 1/8, 1/16 numbers were converted in the EMLR by multiples reporting non-optimal Egyptian fraction series. As practice, 1/8 was converted by three multiples.In 2002, three scholarly 2/"pq" conversion methods reported a 'red auxiliary' least common multiple method. The multiple method had been used in the 250 year older
Egyptian Mathematical Leather Roll . The method converted 2/19 and 2/95 by a multiple of 12. Multiples 30 and 70 converted 2/35, and 2/91, respectively. Beyond the 2/"n" table, Egyptian fractions reported in Egyptian weights and measures, i.e. hin, dja, ro and hekat sub-units, followed the red auxiliary multiple method Egyptian fraction remainders. A longer time view of 2/n table methods includes four of seven conversion methods reported in the 1202 ADLiber Abaci , facts noticed in 2006.In summary, Ahmes, a KP scribe, and likely all other scribes converted 2/n entries to nearly optimal Egyptian fraction series by a single least common multiple method. The red auxiliary method created nearly optimal unit fraction series. As scribal students they were introduction to 2/n table multiples by working
Egyptian Mathematical Leather Roll non-optimal multiples.References
*Boyer, Carl B. (1968) "A History of Mathematics". New York: John Wiley.
*Brown, Kevin S. (1995) "The Akhmin Papyrus 1995 --- Egyptian Unit Fractions".
*Bruckheimer, Maxim, and Y. Salomon (1977) "Some Comments on R. J. Gillings' Analysis of the 2/n Table in the Rhind Papyrus," "Historia Mathematica" 4: 445-52.
*Bruins, Evert M. (1953) "Fontes matheseos: hoofdpunten van het prae-Griekse en Griekse wiskundig denken". Leiden: E. J. Brill.
*------- (1957) "Platon et la table égyptienne 2/n," "Janus" 46: 253-63.
*------- (1981) "Egyptian Arithmetic," "Janus" 68: 33-52.
*------- (1981) "Reducible and Trivial Decompositions Concerning Egyptian Arithmetics," "Janus" 68: 281-97.
*Burton, David M. (2003) "History of Mathematics: An Introduction". Boston Wm. C. Brown.
*Chace, Arnold Buffum, et al (1927) "The Rhind Mathematical Papyrus". Oberlin: Mathematical Association of America.
*Cooke, Roger (1997) "The History of Mathematics. A Brief Course". New York, John Wiley & Sons.
*Couchoud, Sylvia. “Mathématiques égyptiennes”. Recherches sur les connaissances mathématiques de l’Egypte pharaonique., Paris, Le Léopard d’Or, 1993.
*Daressy, Georges. “Akhmim Wood Tablets”, Le Caire Imprimerie de l’Institut Francais d’Archeologie Orientale, 1901, 95–96.
*Eves, Howard (1961) "An Introduction to the History of Mathematics". New York, Holt, Rinehard & Winston.
*Fowler, David H. (1999) "The mathematics of Plato's Academy: a new reconstruction". Oxford Univ. Press.
*Gardiner, Alan H. (1957) "Egyptian Grammar being an Introduction to the Study of Hieroglyphs". Oxford University Press.
*Gardner, Milo (2002) "The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term" in "History of the Mathematical Sciences",
Ivor Grattan-Guinness , B.C. Yadav (eds), New Delhi, Hindustan Book Agency:119-34.*-------- "Mathematical Roll of Egypt" in "Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures". Springer, Nov. 2005.
*Gillings, Richard J. (1962) "The Egyptian Mathematical Leather Roll," "Australian Journal of Science" 24: 339-44. Reprinted in his (1972) "Mathematics in the Time of the Pharaohs". MIT Press. Reprinted by Dover Publications, 1982.
*-------- (1974) "The Recto of the Rhind Mathematical Papyrus: How Did the Ancient Egyptian Scribe Prepare It?" "Archive for History of Exact Sciences" 12: 291-98.
*-------- (1979) "The Recto of the RMP and the EMLR," "Historia Mathematica", Toronto 6 (1979), 442-447.
*-------- (1981) "The Egyptian Mathematical Leather Role–Line 8. How Did the Scribe Do it?" "Historia Mathematica": 456-57.
*Glanville, S.R.K. "The Mathematical Leather Roll in the British Museum” "Journal of Egyptian Archaeology" 13, London (1927): 232–8
*Griffith, Francis Llewelyn. "The Petrie Papyri. Hieratic Papyri from Kahun and Gurob (Principally of the Middle Kingdom), Vols. 1, 2". Bernard Quaritch, London, 1898.
*Gunn, Battiscombe George. Review of T"he Rhind Mathematical Papyrus" by T. E. Peet. "The Journal of Egyptian Archaeology" 12 London, (1926): 123–137.
*Hultsch, F, "Die Elemente der Aegyptischen Theihungsrechmun 8, Ubersich uber die Lehre von den Zerlegangen", (1895):167-71.
*Imhausen, Annette. “Egyptian Mathematical Texts and their Contexts”, "Science in Context" 16, Cambridge (UK), (2003): 367-389.
*Joseph, George Gheverghese. "The Crest of the Peacock/the non-European Roots of Mathematics", Princeton, Princeton University Press, 2000
*Klee, Victor, and Wagon, Stan. "Old and New Unsolved Problems in Plane Geometry and Number Theory", Mathematical Association of America, 1991.
*Knorr, Wilbur R. “Techniques of Fractions in Ancient Egypt and Greece”. "Historia Mathematica" 9 Berlin, (1982): 133–171.
*Legon, John A.R. “A Kahun Mathematical Fragment”. "Discussions in Egyptology", 24 Oxford, (1992).
*Lüneburg, H. (1993) "Zerlgung von Bruchen in Stammbruche" Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers, Wissenschaftsverlag, Mannheim: 81=85.
*Neugebauer, Otto (1957) "The Exact Sciences in Antiquity". Brown University.
*Ore, Oystein (1948) "Number Theory and its History". New York, McGraw-Hill.
*Rees, C. S. (1981) "Egyptian Fractions," "Mathematical Chronicle" 10: 13-33.
*Robins, Gay. and Charles Shute (1987) "The Rhind Mathematical Papyrus: an Ancient Egyptian Text". London, British Museum Press.
*Roero, C. S. (1994) "Egyptian mathematics" in
Ivor Grattan-Guinness , ed, "Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences". London: 30-45.*Sarton, George (1927) "Introduction to the History of Science, Vol I". New York: Williams & Son
*Scott, A. and Hall, H.R. (1927) "Laboratory Notes: Egyptian Mathematical Leather Roll of the Seventeenth Century," "British Museum Quarterly" 2: 56.
*Sylvester, J. J. (1880) "On a Point in the Theory of Vulgar Fractions," "American Journal Of Mathematics" 3: 332-35, 388-89.
*Vogel, Kurt (1929) "Erweitert die Lederolle unserer Kenntniss ägyptischer Mathematik," "Archiv fur Geschichte der Mathematik" 2: 386-407
*van der Waerden, Bartel Leendert (1963) "Science Awakening".
Links
*http://rmprectotable.blogspot.com/
*http://planetmath.org/encyclopedia/EgyptianMath3.html
*http://www.mathorigins.com/image%20grid/awta.htm
*http://weekly.ahram.org.eg/2007/844/heritage.htm
*http://planetmath.org/encyclopedia/EgyptianMathematicalLeatherRoll2.html
*http://planetmath.org/encyclopedia/RationalNumbers.html
Wikimedia Foundation. 2010.
