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<!----------Name---------->
| name                  = Egyptian Mathematical Leather Roll (EMLR)
| location              = [[British Museum]] in London
<!----------Image---------->
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| width                =
| caption              =
<!----------General---------->
| Also known as        =
| Type                  =
| Date                  = ca 1650 BCE
| Place of origin      = [[Thebes, Egypt|Thebes]]
| Language(s)          = [[Hieratic]]
| Scribe(s)            =
| Author(s)            =
| Compiled by           =
| Illuminated by        =
| Patron                =
| Dedicated to         =
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| Size                  = Length: {{convert|10|in|cm}}<br>Width: {{convert|17|in|cm}}
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| Exemplar(s)          =
| Previously kept      =
| Discovered            =
| Other                =
| below                =
}}


The '''Egyptian Mathematical Leather Roll (EMLR)''' was a 10&nbsp;&times;&nbsp;17&nbsp;in (25&nbsp;&times;&nbsp;43&nbsp;cm) 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).
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The writing consists of [[Middle Kingdom of Egypt|Middle Kingdom]] [[hieratic]] characters written right to left. Scholars date the EMLR to the 17th century BCE.<ref name="Clagett">Clagett, Marshall.  Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society, 1999, pp. 17&ndash;18, 25, 37&ndash;38, 255&ndash;257</ref>
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==Mathematical content==
This leather roll is an aid for computing [[Egyptian fraction]]s. It contains 26 sums of unit fractions which equal another unit fraction. The sums appear in two columns, and are followed by two more columns which contain exactly the same sums.<ref name="Imhausen">Annette Imhausen, in The Mathematics of Egypt, Mesopotamia, China, India, and Islam:
A Sourcebook, Edited by Victor J. Katz, 2007, pp. 21&ndash;22</ref>
 
{| class="wikitable" border="1" cellpadding="5" align="center" | style="margin: 1em auto 1em auto; width: 75%"
|+'''The Egyptian Mathematical Leather Roll<ref name="Imhausen"/>'''
!|Column 1 !! |Column 2!!align="center" |Column 3 !! |Column 4
|-
| <math>\frac{1}{10} + \frac{1}{40} = \frac{1}{8}</math> || <math>\frac{1}{30} + \frac{1}{45} + \frac{1}{90}= \frac{1}{15}</math> || <math>\frac{1}{10} + \frac{1}{40} = \frac{1}{8}</math> || <math>\frac{1}{18} + \frac{1}{36} = \frac{1}{12}</math>
|-
| <math>\frac{1}{5} + \frac{1}{20} = \frac{1}{4}</math> || <math>\frac{1}{24} + \frac{1}{48} = \frac{1}{16}</math> || <math>\frac{1}{5} + \frac{1}{20} = \frac{1}{4}</math>  || <math>\frac{1}{21} + \frac{1}{42} = \frac{1}{14}</math>
|-
| <math>\frac{1}{4} + \frac{1}{12} = \frac{1}{3}</math> || <math>\frac{1}{18} + \frac{1}{36} = \frac{1}{12}</math> || <math>\frac{1}{4} + \frac{1}{12} = \frac{1}{3}</math>  || <math>\frac{1}{45} + \frac{1}{90} = \frac{1}{30}</math>
|-
| <math>\frac{1}{10} + \frac{1}{10} = \frac{1}{5}</math> || <math>\frac{1}{21} + \frac{1}{42} = \frac{1}{14}</math> || <math>\frac{1}{10} + \frac{1}{10} = \frac{1}{5}</math> || <math>\frac{1}{30} + \frac{1}{60} = \frac{1}{20}</math>
|-
| <math>\frac{1}{6} + \frac{1}{6} = \frac{1}{3}</math> || <math>\frac{1}{45} + \frac{1}{90} = \frac{1}{30}</math> || <math>\frac{1}{6} + \frac{1}{6} = \frac{1}{3}</math> || <math>\frac{1}{15} + \frac{1}{30} = \frac{1}{10}</math>
|-
| <math>\frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{1}{2}</math> || <math>\frac{1}{30} + \frac{1}{60} = \frac{1}{20}</math> || <math>\frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{1}{2}</math> || <math>\frac{1}{48} + \frac{1}{96} = \frac{1}{32}</math>
|-
| <math>\frac{1}{3} + \frac{1}{3} = \frac{2}{3}</math> || <math>\frac{1}{15} + \frac{1}{30} = \frac{1}{10}</math> || <math>\frac{1}{3} + \frac{1}{3} = \frac{2}{3}</math> || <math>\frac{1}{96} + \frac{1}{192} = \frac{1}{64}</math>
|-
| <math>\frac{1}{25} + \frac{1}{15} + \frac{1}{75} + \frac{1}{200} = \frac{1}{8}</math> || <math>\frac{1}{48} + \frac{1}{96} = \frac{1}{32}</math> || <math>\frac{1}{25} + \frac{1}{15} + \frac{1}{75} + \frac{1}{200} = \frac{1}{8}</math> ||
|-
| <math>\frac{1}{50} + \frac{1}{30} + \frac{1}{150} + \frac{1}{400} = \frac{1}{16}</math> || <math>\frac{1}{96} + \frac{1}{192} = \frac{1}{64}</math> || <math>\frac{1}{50} + \frac{1}{30} + \frac{1}{150} + \frac{1}{400} = \frac{1}{16}</math> ||
|-
| <math>\frac{1}{25} + \frac{1}{50} + \frac{1}{150}  = \frac{1}{6}</math> ||  || <math>\frac{1}{25} + \frac{1}{50} + \frac{1}{150}  = \frac{1}{6}</math> ||
|-
| <math>\frac{1}{9} + \frac{1}{18}  = \frac{1}{6}</math> ||  || <math>\frac{1}{9} + \frac{1}{18}  = \frac{1}{6}</math> ||
|-
| <math>\frac{1}{7} + \frac{1}{14} + \frac{1}{28}  = \frac{1}{4}</math> ||  || <math>\frac{1}{7} + \frac{1}{14} + \frac{1}{28}  = \frac{1}{4}</math> ||
|-
| <math>\frac{1}{12} + \frac{1}{24}  = \frac{1}{8}</math> ||  || <math>\frac{1}{12} + \frac{1}{24}  = \frac{1}{8}</math> ||
|-
| <math>\frac{1}{14} + \frac{1}{21} + \frac{1}{42}  = \frac{1}{7}</math> ||  || <math>\frac{1}{14} + \frac{1}{21} + \frac{1}{42}  = \frac{1}{7}</math> ||
|-
| <math>\frac{1}{18} + \frac{1}{27} + \frac{1}{54}  = \frac{1}{9}</math> ||  || <math>\frac{1}{18} + \frac{1}{27} + \frac{1}{54}  = \frac{1}{9}</math> ||
|-
| <math>\frac{1}{22} + \frac{1}{33} + \frac{1}{66}  = \frac{1}{11}</math> ||  || <math>\frac{1}{22} + \frac{1}{33} + \frac{1}{66}  = \frac{1}{11}</math> ||
|-
| <math>\frac{1}{28} + \frac{1}{49} + \frac{1}{196}  = \frac{1}{13}</math> ||  || <math>\frac{1}{28} + \frac{1}{49} + \frac{1}{196}  = \frac{1}{13}</math> ||
|-
|  ||  || <math>\frac{1}{30} + \frac{1}{45} + \frac{1}{90}  = \frac{1}{15}</math> ||
|-
|  ||  || <math>\frac{1}{24} + \frac{1}{48}  = \frac{1}{16}</math> ||
|}
 
Of the 26 rational numbers listed, ten are [[Eye of Horus]] numbers: 1/2, 1/4 (twice), 1/8 (thrice), 1/16 (twice), 1/32, 1/64 converted to Egyptian fractions. There are seven other even rational numbers converted to Egyptian fractions: 1/6 (listed twice–but wrong once), 1/10, 1/12, 1/14, 1/20 and 1/30. Finally, there were nine odd rational numbers converted to Egyptian fractions: 2/3, 1/3 (twice), 1/5, 1/7, 1/9, 1/11, 1/13 and 1/15, training patterns for scribal students to learn the [[RMP 2/n table]] method.
 
The British Museum examiners found no introduction or description to how or why the equivalent unit fraction series were computed.<ref>Gillings, Richard J. “The Egyptian Mathematical Leather Role–Line 8. How Did the Scribe Do it?” (Historia Mathematica 1981), 456–457.</ref> Equivalent unit fraction series are associated with fractions 1/3, 1/4, 1/8 and 1/16. There was a trivial error associated with the final 1/15 unit fraction series. The 1/15 series was listed as equal to 1/6. Another serious error was associated with 1/13, an issue that the 1927 examiners did not attempt to resolve.
 
==Modern analysis==
The original mathematical texts never explain where the procedures and formulas came from. This holds true for the EMLR as well. Scholars have attempted to deduce what techniques the ancient Egyptians may have used to construct both the unit fraction tables of the EMLR and the 2 /n tables known from the [[Rhind Mathematical Papyrus]] and the [[Lahun Mathematical Papyri]]. Both types of tables were used to aid in computations dealing with fractions, and for the conversion of measuring units.<ref name="Imhausen"/>
 
It has been noted that there are groups of unit fraction decompositions in the EMLR which are very similar. For instance lines 5 and 6 easily combine into the equation 1/3&nbsp;+&nbsp;1/6&nbsp;=&nbsp;1/2. It is easy to derive lines 11, 13, 24, 20, 21, 19, 23, 22, 25 and 26 by dividing this equation by 3, 4, 5, 6, 7, 8, 10, 15, 16 and 32 respectively.<ref>Gillings, Richard J., Mathematics in the Time of the Pharaohs, Dover Publications, 1982 reprint (1972) ISBN 0-486-24315-X</ref>
 
Some of the problems would lend themselves to a solution via an algorithm which involves multiplying both the numerator and the denominator by the same term and then further reducing the resulting equation:
: <math>\frac{1}{pq} = \frac{1}{N}\times\frac{N}{pq} </math>
This method leads to a solution for the fraction 1/8 as appears in the EMLR when using N=25 (using modern mathematical notation):
: <math>1/8 = 1/25 \times 25/8 = 1/5 \times 25/40 = 1/5 \times (3/5 + 1/40) </math>
: <math>= 1/5 \times (1/5 + 2/5 + 1/40) = 1/5 \times (1/5 + 1/3 + 1/15 + 1/40) = 1/25 + 1/15 + 1/75 + 1/200</math> <ref name="MG">Gardner, Milo. “The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term” History of the Mathematical Sciences”, Ivor Grattan-Guinness, B.C. Yadav (eds), New Delhi, Hindustan Book Agency, 2002:119&ndash;134.</ref>
 
==Modern conclusions==
 
The EMLR has been considered a student scribe test document since 1927, the year that the text was unrolled at the British Museum. The scribe practiced conversions of rational numbers 1/p and 1/pq to alternative unit fraction series. Reading available Middle Kingdom math records, [[RMP 2/n table]] being one, modern students of Egyptian arithmetic may see that trained scribes improved conversions of 2/n and n/p to concise unit fraction series by applying algorithmic and non-algorithmic methods.
 
==Chronology==
The following chronology shows several milestones that marked the recent progress toward reporting a clearer understanding of the EMLR's contents, related to the RMP 2/''n'' table.
 
* 1895 &ndash; Hultsch suggested that all RMP 2/p series were coded by aliquot parts.<ref>Hultsch, F, Die Elemente der Aegyptischen Theihungsrechmun 8, Ubersich uber die Lehre von den Zerlegangen, (1895):167&ndash;71</ref>
* 1927 &ndash; Glanville concluded that EMLR arithmetic was purely additive.<ref>Glanville, S.R.K. "The Mathematical Leather Roll in the British Museum” Journal of Egyptian Archaeology 13, London (1927): 232–8</ref>
* 1929 &ndash; Vogel reported the EMLR to be more important (than the RMP), though it contains only 25 unit fraction series.<ref>Vogel, Kurt. “Erweitert die Lederolle unserer Kenntniss ägyptischer Mathematik Archiv fur Geschichte der Mathematik, V 2, Julius Schuster, Berlin (1929): 386&ndash;407</ref>
* 1950 &ndash; Bruins independently confirms Hultsch’s RMP 2/''p'' analysis (Bruins 1950)
* 1972 &ndash; Gillings found solutions to an easier RMP problem, the 2/''pq'' series (Gillings 1972: 95&ndash;96).
* 1982 &ndash; Knorr identifies  RMP  unit fractions 2/35, 2/91 and 2/95 as exceptions to the 2/''pq'' problem.<ref>[[Wilbur Knorr|Knorr, Wilbur R.]] “Techniques of Fractions in Ancient Egypt and Greece”. Historia Mathematica 9 Berlin, (1982): 133–171.</ref>
* 2002 &ndash; Gardner identifies five abstract EMLR patterns.<ref name="MG"/>
 
==See also==
Egyptian mathematical texts:
* [[Moscow Mathematical Papyrus]]
* [[Lahun Mathematical Papyri]]
* [[Berlin Papyrus 6619]]
* [[Akhmim Wooden Tablet]]
* [[Reisner Papyrus]]
* [[Rhind Mathematical Papyrus]]
 
Other:
*[[Liber Abaci]]
*[[:fr:Sylvia Couchoud|Sylvia Couchoud]] {{fr icon}}
 
==References==
<references/>
 
==Further reading==
*Brown, Kevin S. The Akhmin Papyrus 1995 &ndash; Egyptian Unit Fractions 1995
*Bruckheimer, Maxim and Y. Salomon. “Some Comments on R. J. Gillings’ Analysis of the 2/n Table in the Rhind Papyrus.” Historia Mathematica 4 Berlin (1977): 445–452.
*Bruins, Evert M.  “Platon et la table égyptienne 2/n”. Janus 46, Amsterdam, (1957): 253–263.
*Bruins, Evert M. “Egyptian Arithmetic.” Janus 68, Amsterdam, (1981): 33–52.
*Bruins, Evert M. “Reducible and Trivial Decompositions Concerning Egyptian Arithmetics”. Janus 68, Amsterdam, (1981): 281–297.
*Daressy, Georges. “Akhmim Wood Tablets”, Le Caire Imprimerie de l’Institut Francais d’Archeologie Orientale, 1901, 95–96.
*Gardner, Milo. "Mathematical Roll of Egypt", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, Nov. 2005.
*Gillings, Richard J. “The Egyptian Mathematical Leather Roll”. Australian Journal of Science 24 (1962): 339&ndash;344, Mathematics in the Time of the Pharaohs. Cambridge, Mass.: MIT Press, 1972. New York: Dover, reprint 1982.
*Gillings, Richard J. “The Recto of the Rhind Mathematical Papyrus: How Did the Ancient Egyptian Scribe Prepare It ?” Archive for History of Exact Sciences 12 (1974), 291–298.
*Gillings, Richard J. “The Recto of the RMP and the EMLR”, Historia Mathematica, Toronto 6 (1979), 442&ndash;447.
*Gillings, Richard J. “The Egyptian Mathematical Leather Role–Line 8. How Did the Scribe Do it?” (Historia Mathematica 1981), 456–457.
*[[Battiscombe Gunn|Gunn, Battiscombe George]]. Review of ”The Rhind Mathematical Papyrus” by T. E. Peet. The Journal of Egyptian Archaeology 12 London, (1926): 123–137.
*Imhausen, Annette. “Egyptian Mathematical Texts and their Contexts”, Science in Context, vol 16, Cambridge (UK), (2003): 367&ndash;389.
*Legon, John A.R. “A Kahun Mathematical Fragment”. Discussions in Egyptology, 24 Oxford, (1992).
*Lüneburg, H. “Zerlgung von Bruchen in Stammbruche” Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers, Wissenschaftsverlag, Mannheim, 1993. 81–85.
*Rees, C. S. “Egyptian Fractions”, Mathematical Chronicle 10, Auckland, (1981): 13–33.
*Roero, C. S. “Egyptian mathematics” Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences” I. Grattan-Guinness (ed), London, (1994): 30–45.
*Scott, A. and Hall, H.R., “Laboratory Notes: Egyptian Mathematical Leather Roll of the Seventeenth Century BC”, British Museum Quarterly, Vol 2, London, (1927): 56.
*Sylvester, J. J. “On a Point in the Theory of Vulgar Fractions”: American Journal Of Mathematics, 3 Baltimore (1880): 332–335, 388–389.
 
==External links==
*[http://planetmath.org/encyclopedia/EgyptianMathematicalLeatherRoll2.html EMLR]
*[http://planetmath.org/encyclopedia/AhmesBirdFeedingRateMethod.html Theoretical (expected) economic control numbers]
*[http://planetmath.org/encyclopedia/RMP35To38PlusRMP66.html RMP 35&ndash;38 plus RMP 66]
 
[[Category:Egyptian fractions]]
[[Category:Ancient Egyptian literature]]
[[Category:Mathematics manuscripts]]
[[Category:Ancient Egyptian objects in the British Museum]]
[[Category:Egyptian mathematics]]

Latest revision as of 23:00, 24 December 2014

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If you have the ability to type quickly, you may consider transcription jobs. You need to have good headphones and the ability to keep up with what is being said. Often, the companies will have their own formatting guidelines. Familiarize yourself with these prior to beginning. Once you have, you can begin transcribing. ganhe dinheiro na internet What do you do? Do it for pay! A number of websites offer you points when you search through them and/or make purchases of items that you were planning to buy anyway. Take the points you earn and trade them for gift cards, gifts and sometimes even cash. Sites thatprovide this opportunity include Swagbucks and Mypoints.

Medical transcription can be a good way to make money online as a full time career. Formal training is required, and this can be costly. Additionally, it is necessary to have good computer and transcription equipment that works reliably. A great deal of work is available for people who are able and willing ganhar dinheiro to invest in training and good equipment. Pay is quite substantial. You can sell your wares on the Internet to make money. Certain sites will help you to do your selling. You can personalize the t-shirts for your client. You can advertise by putting fliers up around the neighborhood or by using Craigslist.

If you want to do something creative, consider ways to create viral videos. Brainstorm ideas that you think are rip riotously hilarious, then start putting them into action. Post them on YouTube, turn on ads and, if they work, watch the money start rolling in! Learn about marketing to see how you can get them even more popular. como conseguir dinheiro If you know how to prepare taxes, you can earn money online. Apply for the necessary government licenses and then begin offering your services online. Many of today's top producers work from home with only their laptop and an email address. For best results, continually advertise your services throughout the year.

that's a lie. Don't fall prey to these types of websites and always read the reviews before joining. Talking to experts in their field can help you to use ganhando dinheiro na internet the Internet successfully. Try to network with people who do what you want to do. Doing these things can help give you an advantage over your competition.

Manage your time wisely when you are trying to make money online. You may have a real-world job, which means you have to budget time wisely. Set a schedule that will allow you enough time to balance the two. Also make sure that you are being efficient in that time, so you don't run into stress ganhe dinheiro about working online. Becoming a ghost writer is a great way to earn money online. There are people who need content for their websites, but who lack good writing skills. You can write for them and earn money for what you write. Many people are earning a good income writing for others this way.

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