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<!DOCTYPE html><html> <head> <title>Egyptian fraction</title>  <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <link rel="stylesheet" href="LaTeXML.css" type="text/css"> <link rel="stylesheet" href="ltx-article.css" type="text/css"> <link rel="stylesheet" href="https://cdn.rawgit.com/holtzermann17/3f71ceeb3b055e1ddc3b6c11fb1f074c/raw/2bb23e3b173ff96840797fc0c3bcb8c54085df8e/LaTeXML.css" type="text/css"> <link rel="stylesheet" href="https://cdn.rawgit.com/holtzermann17/4bda0365b30858ac2fb83623185fe3ec/raw/cedd84ed3e3ad597c5d293f443ecfe4803741c6b/ltx-article.css" type="text/css"> <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML" type="text/javascript"></script> </head> <body> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Egyptian fraction</h1> <div id="p1" class="ltx_para"> <br class="ltx_break"> <p class="ltx_p">This narrative reports <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">rational number</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/egyptianfraction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/liberabaci"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/rationalnumber"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> as a/b rather than in the standard LaTeX <a class="nnexus_concept" href="http://planetmath.org/fraction">fraction</a> form.</p> </div> <div id="p2" class="ltx_para"> <p class="ltx_p">Prior to 2050 BCE Old Kingdom Egyptian scribes rounded off rational numbers to six-terms binary representations for 1,000 years. The binary notation was stated in 1/2n units with a 1/64 unit thrown away. The <span class="ltx_text ltx_font_typewriter">http://en.<a class="nnexus_concept" href="http://planetmath.org/wikipedia">wikipedia</a>.org/wiki/Eye_of_Horus</span>Horus-Eye recorded rational numbers in the cursive pattern:</p> </div> <div id="p3" class="ltx_para"> <p class="ltx_p">1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + (1/64).</p> </div> <div id="p4" class="ltx_para"> <p class="ltx_p">Note that a <a class="nnexus_concept" href="http://planetmath.org/lamellarfield">potential</a> 7th term (1/64) was rounded-off and thrown way.</p> </div> <div id="p5" class="ltx_para"> <p class="ltx_p">After 2050 BCE an exact <a class="nnexus_concept" href="http://planetmath.org/numerationsystem">numeration system</a> discontinued the rounded-off Old Kingdom binary system. An exact hieratic weights and measures system reported rational numbers in 1/64 <a class="nnexus_concept" href="http://planetmath.org/division">quotient</a> and 1/320 <a class="nnexus_concept" href="http://planetmath.org/longdivision">remainder</a> units whenever possible.</p> </div> <div id="p6" class="ltx_para"> <p class="ltx_p">The new Middle Kingdom math system ”healed” rounded off binary series by several finite methods. Two weights and measures finite systems can be reported by:</p> </div> <div id="p7" class="ltx_para"> <p class="ltx_p">1. 1 hekat (a volume unit) used a unity (64/64)such that (32 + 16 + 8 + 4 + 2 + 1/64)hekat+ 5 ro</p> </div> <div id="p8" class="ltx_para"> <p class="ltx_p">and (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64)hekat + 5 ro</p> </div> <div id="p9" class="ltx_para"> <p class="ltx_p">meant (64/64)/n = Q/64 + (5R/n)ro</p> </div> <div id="p10" class="ltx_para"> <p class="ltx_p">Note that the hekat unity was generally divided by rational number n. To divide by 3 scribal long-hand would have written out</p> </div> <div id="p11" class="ltx_para"> <p class="ltx_p">(64/64)/3 = 21/64 hekat + 5/192 = (16 + 4 + 1)/64 hekat + 5/3 ro =</p> </div> <div id="p12" class="ltx_para"> <p class="ltx_p">(1/4 + 1/16 + 1/64)hekat + ( 1 + 2/3)ro</p> </div> <div id="p13" class="ltx_para"> <p class="ltx_p">2. (100-hekat)/70 = (6400/64)/70 = 91/64 hekat + 30/4480 = (64 + 16 + 8+ 2 + 1)/64 hekat + 150/70 ro =</p> </div> <div id="p14" class="ltx_para"> <p class="ltx_p">(1 + 1/4 + 1/8 + 1/32 + 1/64)hekat + (2 + 1/7)ro </p> </div> <div id="p15" class="ltx_para"> <p class="ltx_p">meant (6400/64)/n = Q/64 + (5R/n)ro was applied for almost any <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">hekat division</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/akhmimwoodentablet1"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/akhmimwoodentablet"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> problem.</p> </div> <div id="p16" class="ltx_para"> <p class="ltx_p">The hieratic word ro meant 1/320 of a hekat in a grain weights and measures system. Note that 5 ro meant 5/320 = 1/64.</p> </div> <div id="p17" class="ltx_para"> <p class="ltx_p">Generally, scribal shorthand recorded duplation aspects of mental calculations and fully recorded two-part hekat quotients and ro remainders.</p> </div> <div id="p18" class="ltx_para"> <p class="ltx_p">At other times 2/64 was scaled to 10/320 such that (8 + 2)/320 = 1/40 + 2 ro</p> </div> <div id="p19" class="ltx_para"> <p class="ltx_p">The 1900 BCE <a class="nnexus_concept" href="http://mathworld.wolfram.com/AkhmimWoodenTablet.html">Akhmim Wooden Tablet</a> (AWT) scaled two volume units in the <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">Egyptian fraction</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/rhindmathematicalpapyrus"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/egyptianmathematicalleatherroll"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/egyptianfraction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/hultschbruinsmethod"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/remainderarithmeticvsegyptianfractions"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/remainderarithmetic"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/unitfraction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> notation that allowed Hana Vymazalova in 2002 CE to open a long lost (64/64) hekat unity aspect of Egyptian fraction weights and measures.</p> </div> <div id="p20" class="ltx_para"> <p class="ltx_p">In 2006 scholars showed that the AWT scribe divided (64/64) by 3, 7, 10, 11 and 13 by writing exact 1/64 quotients and (5/5) scaled remainders to 1/320 unit fraction series. Each AWT answer was proven by multiplying by the <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">divisor</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Divisor.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/divisibility"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/divisortheory"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> returning (64/64) the initial hekat unity, the <a class="nnexus_concept" href="http://mathworld.wolfram.com/CentralPoint.html">central point</a> made by Vymazalova.</p> </div> <div id="p21" class="ltx_para"> <p class="ltx_p">A <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">complete</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/ordersinanumberfield"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/soundcomplete"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/completebinarytree"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/completegraph"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/kripkesemantics"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/maximallyconsistent"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> decoding of the AWT scribal methodology documented a solution to an Old Kingdom binary round-off problem. Ahmes used the AWT method over 40 times, <a class="nnexus_concept" href="http://planetmath.org/affinetransformation">scaling</a> a 1/10 hin unit to 1/320 ro unit 29 times, as well as solving several classes of problems.</p> </div> <div id="p22" class="ltx_para"> <p class="ltx_p">On a higher numeration level the finite hieratic system scaled rational numbers by least common <a class="nnexus_concept" href="http://mathworld.wolfram.com/Multiple.html">multiple</a> (LCM) m to optimized, but not optimal, unit fraction series. Middle Kingdom scribes scaled rational numbers to unit fraction series by the LCM that created a <span class="ltx_text ltx_font_typewriter">http://www-history.mcs.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html</span>numeration system in the AWT and other hieratic texts. The Egyptian fraction numeration system ciphered <a class="nnexus_concept" href="http://mathworld.wolfram.com/CountingNumber.html">counting numbers</a> 1:1 onto hieratic sound symbols replacing an Old Kingdom hieroglyphic many-to-one system numbers written in rounded-off binary numbers. A line was drawn over hieratic sound symbols to denote <a class="nnexus_concept" href="http://planetmath.org/hibehpapyrus">unit fractions</a>. <span class="ltx_text ltx_font_typewriter">http://www.jstor.org/pss/330598</span>Carl B. Boyer reported the significance of the ciphered numeration system. <span class="ltx_text ltx_font_typewriter">http://en.wikipedia.org/wiki/Egyptian_fractions</span>Egyptian fraction series represented rational numbers in ordered unit fraction series, writing the smallest to the largest unit fraction, from left to right.</p> </div> <div id="p23" class="ltx_para"> <p class="ltx_p">One purpose of the Egyptian fraction system involved a finite weights and measures system applied to a commodity based monetary system established by Pharaoh. One theoretical aspect of the monetary system <a class="nnexus_concept" href="http://planetmath.org/superset">contained</a> Egyptian <span class="ltx_text ltx_font_typewriter">http://<a class="nnexus_concept" href="http://planetmath.org/planetmath">planetmath</a>.org/encyclopedia/EgyptianMultiplicationAndDivision.html</span><a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">multiplication</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/multiplication"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> and division <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">operations</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Operation.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/operation"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>. The scribal <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">arithmetic</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/arithmetic"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/arithmeticalhierarchy"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> was undervalued by historians for over 120 years, from 1879 to 1999, by only reporting the <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">additive</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/additivefunction1"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/additive"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> aspects of the raw transliterated data.</p> </div> <div id="p24" class="ltx_para"> <p class="ltx_p">Scribal unit fraction answers contained two sides, additive numerators, and LCM scaled reminders. Math historians, beginning with F. Hultsch in 1895, parsed aspects of the scribal <a class="nnexus_concept" href="http://planetmath.org/properdivisor">aliquot part</a> views of numerators and LCM denominators, though the unified aspects of the system remained vague for another 105 years. In 2002, the EMLR was validated containing aliquot parts of LCM denominators in none optimized manners. The LCM (m/m) was written as a unity becoming a multiplier that scribes used to scale rational numbers to optimized, but not optimal, unit fraction series in 2/n table tables and every day problem and answers. In 2006 the Akhmim Wooden Tablet was validated containing exact quotient and remainders within its <a class="nnexus_concept" href="http://mathworld.wolfram.com/Primary.html">primary</a> division division method, ancient fragments documented in the EMLR and RMP 2/n table. The EMLR and RMP 2/n table were fully decoded in <span class="ltx_text ltx_font_typewriter">http://www.academia.edu/617613/Egyptian_Fractions_Unit_Fractions_Hekats_and_Wages_-_an_Update</span>2011 as reported in Appendix I and II.</p> </div> <div id="p25" class="ltx_para"> <p class="ltx_p">During the Old Kingdom hieroglyphic writing scribes threw away 1/64 units within a 6-term binary notation named Horus-Eye. Around 1950 BCE the Akhmim Wooden Tablet (AWT) reported a hekat unity system solving the Old Kingdom round-off problem. Corrected volume units added back missing 1/64 units. Within the corrected notation multiplication of 400-hekat, 100-hekat, 4-hekat and 1-hekat substituted two classes of hekat units, one hekat = (64/64) (defined in the AWT and RMP 47) calculated binary quotients (Q/64) and scaled Egyptian fraction remainders (5R/n)ro; and one hekat = 320 ro (in RMP 35-38).</p> </div> <div id="p26" class="ltx_para"> <p class="ltx_p">Concerning hekat arithmetic details, the <span class="ltx_text ltx_font_typewriter">http://akhmimwoodentablet.blogspot.com</span>AWT discuss the hekat unity. The unity was partitioned by rational numbers. Multiplication and division answers recorded binary quotients and scaled (5R/n)ro Egyptian fraction remainders, hekat units as 1/10 (hinu), 1/64 (dja), 1/320 (ro), and other units within modified quotients and remainders. The secondary hekat system converted m/n to <a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">integer</a> quotients and non-scaled Egyptian fraction remainders written as m/n ”name”. For example, Ahmes created 1/10 units writing m = 10 and n= 3 by using the <a class="nnexus_concept" href="http://planetmath.org/expression">expression</a> 10/3 hin, meaning (3 + 1/3)hin.</p> </div> <div id="p27" class="ltx_para"> <p class="ltx_p">The AWT’s binary quotient and Egyptian fraction <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">partitions</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Partition.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/partition1"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/integerpartition"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/partition"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> were proven by multiplying each quotient and remainder answer by the initial divisor. The exact proof calculation returned each quotient and scaled remainder answer to an initial (64/64) unity value. The AWT proof may be the first proof recorded in Western math history. Ahmes applied the proof methid in RMP 35-38, 320 ro problems.</p> </div> <div id="p28" class="ltx_para"> <p class="ltx_p">Length and area units (cubits, khet, setat and mh) were partitioned into 1/8 setat quotients and mh (1 cubit by 100 cubit strips) remainders. <span class="ltx_text ltx_font_typewriter">http://planetmath.org/encyclopedia/CubitsEgyptianGeometryAreasCalculatedIn.html</span>RMP 53, 54, and 55 discuss Ahmes’ <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">geometry</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/moscowmathematicalpapyrus"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/mscclassificationofobjectsarticlessearch"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> and associated arithmetic.</p> </div> <div id="p29" class="ltx_para"> <p class="ltx_p">The <a class="nnexus_concept" href="http://planetmath.org/reisnerpapyrus">Reisner Papyrus</a>(RP), circa 1800 BCE, defines a labor efficiency division by 10 quotient and unscaled remainder rate. The RMP’s first six problems used the RP division by 10 method. The RMP is one of several texts that confirm the scribal use of a second form of quotient and <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">remainder arithmetic</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/remainderarithmeticvsegyptianfractions"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/remainderarithmetic"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>. Modern historians, going beyond 1920’s additive transliteration limitations, by considering meta Egyptian fractions, agree with the RP method, and other forms of meta (unified) Egyptian fraction mathematics.</p> </div> <div id="p30" class="ltx_para"> <p class="ltx_p">A second purpose of the Egyptian fraction notation exactly solved one and two <a class="nnexus_concept" href="http://planetmath.org/variable">variable</a> first degree algebra problems. Rational number answers were written into integer quotients and Egyptian fraction remainders. The Rhind Mathematical Papyrus (<span class="ltx_text ltx_font_typewriter">http://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus</span>RMP) and the <span class="ltx_text ltx_font_typewriter">http://en.wikipedia.org/wiki/Berlin_Papyrus</span>Berlin Papyrus cite several problems and solutions, each with hard-to-read intermediate steps reaching Egyptian fraction answers.</p> </div> <div id="p31" class="ltx_para"> <p class="ltx_p">A third purpose of the Egyptian fraction notation created a commodity and metal based monetary system. The system is outlined by Mahmoud Ezzamel <span class="ltx_text ltx_font_typewriter">http://www.blackwell-synergy.com/doi/abs/10.1111/1467-6281.00107</span>using modern accounting methods. Yet, Ezzamel fairly parses <span class="ltx_text ltx_font_typewriter">http://www.reshafim.org.il/ad/egypt/texts/heqanakht.htm</span>the four Heqanakht Papers by discussing two absentee landlords’ <a class="nnexus_concept" href="http://planetmath.org/formalgrammar">production</a> and management considerations of profit written in ancient Egyptian fractions.</p> </div> <div id="p32" class="ltx_para"> <p class="ltx_p">A fourth purpose of the Egyptian fraction notation that generally converted rational numbers to optimized, but not optimal, unit fraction series. The <span class="ltx_text ltx_font_typewriter">http://rmprectotable.blogspot.com</span>RMP 2/n table was written by Ahmes in 1650 BCE. Translators assumed that Ahmes had intuitively used ’red auxiliary’ multiples to write optimized, but not optimal, unit fraction series. When the RMP was first published in 1879, historians began the task of breaking the 2/n table code, a project that was not completed until 2005. The earliest nearly successful code breaking effort was published by F. Hultsch in 1895. With E.M. Bruins confirming Hultsch’s aliquot part approach in 1944. Today F. Hultsch and E.M. Bruins are honored by the <a class="nnexus_concept" href="http://planetmath.org/hultschbruinsmethod">Hultsch-Bruins method</a>.</p> </div> <div id="p33" class="ltx_para"> <p class="ltx_p">Returning to the 1650 BCE and the Rhind Mathematical Papyrus it began with a 2/n table. A 200 year older text, the <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">Egyptian Mathematical Leather Roll</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/EgyptianMathematicalLeatherRoll.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/egyptianmathematicalleatherroll"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> (EMLR) was a student’s <a class="nnexus_concept" href="http://planetmath.org/introduction">introduction</a> to the 2/n table subject. A student converted 26 rational numbers to non-optimal Egyptian fraction series. The EMLR converted rational numbers such as 1/8 by using multiples 3/3, 5/5 and 25/25. Two out-of order series are also discussed on <span class="ltx_text ltx_font_typewriter">http://emlr.blogspot.com</span>EMLR.</p> </div> <div id="p34" class="ltx_para"> <p class="ltx_p">Several Egyptian unit fraction notations were continuously used from 2050 BCE to 1637 AD. For about 3,700 years Egyptian fraction methods unitized rational numbers in nearby Mediterranean cultures. Greeks used the hieratic numeration systems by <a class="nnexus_concept" href="http://planetmath.org/mapping">mapping</a> the counting numbers onto Ionian and Dorian <a class="nnexus_concept" href="http://planetmath.org/alphabet">alphabets</a>. Unit fractions were denoted by a <a class="nnexus_concept" href="http://planetmath.org/greekalphabet">Greek letter</a> followed by ’, or beta’ = 1/2. Greeks, and Hellenes fully used the Egyptian fraction method of converting rational numbers to Egyptian fraction series for several purposes altering Ahmes’ methods in minor ways.</p> </div> <div id="p35" class="ltx_para"> <p class="ltx_p">Archimedes followed a one-fourth geometric series tradition established by Eudoxus that found the area of parabola first by an infinite series:</p> </div> <div id="p36" class="ltx_para"> <p class="ltx_p">4A/3 = A + A/4 + A/16 + A/64 + …,</p> </div> <div id="p37" class="ltx_para"> <p class="ltx_p">and by second finite Egyptian fraction proof:</p> </div> <div id="p38" class="ltx_para"> <p class="ltx_p">4A/3 = A + A/4 + A/12</p> </div> <div id="p39" class="ltx_para"> <p class="ltx_p">Heiberg in 1906 reported Archimedes’ two-level <a class="nnexus_concept" href="http://mathworld.wolfram.com/Calculus.html">calculus</a> method decoded from an 1100 AD Byzantine vellum document. </p> </div> <div id="p40" class="ltx_para"> <p class="ltx_p">Around 800 AD a major change took place in the Mediterranean region. Modern base 10 numerals diffused from India and Arab trade began to enter Europe. By 999 AD, a Catholic Pope adopted Arab mathematics with its base 10 numerals and Egyptian fraction arithmetic. By the time of Leonardo de Pisa, <a class="nnexus_concept" href="http://planetmath.org/leonardodapisa">Fibonacci</a>, and the 1202 AD Liber Abaci, European weights an measures were also written in <a class="nnexus_concept" href="http://planetmath.org/arabicnumerals">Arabic numerals</a>.</p> </div> <div id="p41" class="ltx_para"> <p class="ltx_p">In 1585 AD, the beginning of our modern base 10 decimals, modern base 10 decimals literally erased 3,600 years of Egyptian fraction arithmetic history. By 1900 AD European and Arab scholars were unable to read medieval Egyptian fractions texts as well as the older Egyptian mathematical texts.</p> </div> <div id="p42" class="ltx_para"> <p class="ltx_p">Ahmes’ 84 problems have been slowly read by scholars after the RMP was published in 1877. The 2/n table was key to scholar research. But what method or methods did Ahmes use to create his 2/n table? Ahmes seemingly left few clues to assist scholars in decoding the 2/n table’s construction method(s). Post-1877 scholars relied on intuition and personalized mathematical senses to report suspected details, often confusing the subject. Scholarly debates have correctly focused on the 2/n table and its construction methods. Yet, confirmed 2/n table threads continue to be controversial. Egyptologists, and math historians, i.e. Neugebauer, Exact Sciences in Antiquity, had inappropriately proposed that the RMP 2/n table and related Egyptian fraction methods represented forms of intellectual decline.</p> </div> <div id="p43" class="ltx_para"> <p class="ltx_p">However, one simple RMP 2/n table construction method, multiples, was reported in 2002 that changed the debate. An advanced idea had created finite unit fraction statements. By 2006 it was shown that Egyptian fraction statements had solved an Old Kingdom infinite series round-off problem. Scholars continue to parse several theoretical fragments of a group of ancient texts by considering related arithmetic patterns. As a <a class="nnexus_concept" href="http://planetmath.org/logicalimplication">consequence</a> 130 years of confusion related to Egyptian fraction arithmetic is lifting an intellectual fog.</p> </div> <div id="p44" class="ltx_para"> <p class="ltx_p">For example, Ahmes, the RMP scribe, has gained the majority of scholarly attention since the RMP was published in 1877. Sylvester in 1891 incorrectly suggested that the greedy algorithm was present in the medieval Liber Abaci and by <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">implication</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Implication.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/implication"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> the RMP. Hultsch in 1895 began to parse Ahmes’ 2/p conversion patterns by using aliquot parts. It took Bruins in 1944 to confirm Hultsch’s earlier work, now known as the H-B method.</p> </div> <div id="p45" class="ltx_para"> <p class="ltx_p">Ahmes, therefore, converted 2/p rational numbers into optimal or elegant Egyptian fraction series using a form of the H-B method. Ahmes’ shorthand indicated that 2/43, and other 2/n table members, were converted to an Egyptian fraction series by selecting optimal multiples, in the 2/43 case the multiple 42. Ahmes use of the multiple 42 allowed 1/42 to become the first partition. The remaining Egyptian fraction were found by considering the divisors (aliquot parts) of 42 or (21, 14, 7, 6, 3, 2, 1). Ahmes’ fragmented shorthand indicates:</p> </div> <div id="p46" class="ltx_para"> <p class="ltx_p">2/43*42/42 = (42 + 21 + 14 + 6)/(42*43),</p> </div> <div id="p47" class="ltx_para"> <p class="ltx_p">such that:</p> </div> <div id="p48" class="ltx_para"> <p class="ltx_p">2/43 = 1/42 + 1/86 + 1/129 + 1/301</p> </div> <div id="p49" class="ltx_para"> <p class="ltx_p">in clear hieratic script.</p> </div> <div id="p50" class="ltx_para"> <p class="ltx_p">Today, 20th century additive scholarly adherents oddly do not accept the H-B method, or related methods used by Egyptians, Greeks, Hellenes, Arabs, medievals and others for over 3,000 years. It is clear that H-B type methods used modern <a class="nnexus_concept" href="http://planetmath.org/addition">addition</a>, <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">subtraction</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/subtraction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/additionandsubtractionformulasforsineandcosine"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>, multiplication and division operations and correctly parse 3,700 years of rational numbers conversions to Egyptian fraction series. <span class="ltx_text ltx_font_typewriter">http://en.wikipedia.org/wiki/Occam's_Razor</span>Occam’s Razor , the simplest method is likely the historical method, is slowly changing scholarly minds.</p> </div> <div id="p51" class="ltx_para"> <p class="ltx_p">As another long term verification of Ahmes’ arithmetic Fibonacci’s 1202 AD <span class="ltx_text ltx_font_typewriter">http://liberabaci.blogspot.com</span>Liber Abaci reports a very old style of writing multiples within the first four of seven rational number conversion methods.</p> </div> <div id="p52" class="ltx_para"> <p class="ltx_p">Summary: Today, interdisciplinary studies groups are oddly created by ’invitation only’. As meta aspects of ancient mathematics are randomly studied, without invitation, based on reading entire bodies of mathematical texts, great academic progress in understanding ancient mathematics will take place. Currently, common abstract mathematics themes are deposited in fragmented story lines, often transliterating a small set of the ancient records, not noticing closely related mathematical methods. Moreover, university math history and philology departments often do not professionally share common meta themes (found by parsing numerical data) even when their offices are in the same building.</p> </div> <div id="p53" class="ltx_para"> <p class="ltx_p">Broadly considering Old Kingdom edicts several Pharaohs likely had requested exact numerical systems to control and allocate vital inventories of grain and its <a class="nnexus_concept" href="http://planetmath.org/product">products</a> including beer and bread are often mentioned. Egyptian scribes exactly converted rational numbers to optimal or elegant unit fraction series. The Middle Kingdom innovation exactly scaled weights and measure units to two Egyptian fraction systems, one defined in the Reisner Papyrus and the second in the Akhmim Wooden Tablet.</p> </div> <div id="p54" class="ltx_para"> <p class="ltx_p">Since 2002 AD scholarly debates are increasingly correcting Middle Kingdom shorthand omissions of Egyptian fraction mathematics by adding back the missing data in scribal longhand notations. The 2050 BCE point of departure date formalized Egyptian fraction innovations within abstract arithmetic and practical weights and measures notations. Greeks and Arab scribes modified the theoretical arithmetic and weights and measures notations. Medieval scribes by Fibonacci recorded Greek and Arab versions of the oldest Egyptian fraction mathematics. For example, Sigler’s 2002 <a class="nnexus_concept" href="http://mathworld.wolfram.com/Translation.html">translation</a> of the 1202AD Liber Abaci includes four 1650 BCE conversion methods. Fibonacci’s era used 3,000 year old Egyptian fraction conversion methods that modified an ancient multiplication <a class="nnexus_concept" href="http://planetmath.org/conceptlattice">context</a> to a medieval subtraction context. Today, 20th century Egyptian fraction debates, that fragments analyzed mathematical theory and practical statements are being unified. The oldest Egyptian fraction mathematics are being <a class="nnexus_concept" href="http://planetmath.org/connectedgraph">connected</a> to the medieval era in interesting ways. The largest set of ancient texts that describe economic and weights and measures texts are linked to the 1950 BCE hieratic texts and to the 1202 AD Liber Abaci. Over 3150 years of common uses of rational number and unit fraction monetary methods.</p> </div> <section id="bib" class="ltx_bibliography"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li id="bib.bib1" class="ltx_bibitem"> <span class="ltx_bibtag ltx_role_refnum">1</span> <span class="ltx_bibblock"> Mahmoud Ezzamel, <em class="ltx_emph ltx_font_italic">Accounting for Private Estates and the Household in the 20th Century BC Middle Kingdom</em>, Abacus Vol 38 pp 235-263, 2002 </span> </li> <li id="bib.bib2" class="ltx_bibitem"> <span class="ltx_bibtag ltx_role_refnum">2</span> <span class="ltx_bibblock"> Milo Gardner, <em class="ltx_emph ltx_font_italic">The Egyptian Mathematical Leather Roll Attested Short Term and Long Term, History of Mathematical Sciences</em>, Hindustan Book Company, 2002. </span> </li> <li id="bib.bib3" class="ltx_bibitem"> <span class="ltx_bibtag ltx_role_refnum">3</span> <span class="ltx_bibblock"> Milo Gardner, <em class="ltx_emph ltx_font_italic">An Ancient Egyptian Problem and its Innovative Solution, Ganita Bharati</em>, MD Publications Pvt Ltd, 2006. </span> </li> <li id="bib.bib4" class="ltx_bibitem"> <span class="ltx_bibtag ltx_role_refnum">4</span> <span class="ltx_bibblock">Richard Gillings, <em class="ltx_emph ltx_font_italic">Mathematics in the Time of the Pharaohs</em>, Dover Books, 1992. </span> </li> <li id="bib.bib5" class="ltx_bibitem"> <span class="ltx_bibtag ltx_role_refnum">5</span> <span class="ltx_bibblock"> T.E. Peet, <em class="ltx_emph ltx_font_italic">Arithmetic in the Middle Kingdom</em>, Journal Egyptian Archeology, 1923. </span> </li> <li id="bib.bib6" class="ltx_bibitem"> <span class="ltx_bibtag ltx_role_refnum">6</span> <span class="ltx_bibblock"> Tanja Pommerening, <em class="ltx_emph ltx_font_italic">”Altagyptische Holmasse Metrologish neu Interpretiert” and relevant phramaceutical and medical knowledge, an abstract, Phillips-Universtat, Marburg, 8-11-2004, taken from ”Die Altagyptschen Hohlmass</em>, Buske-Verlag, 2005. </span> </li> <li id="bib.bib7" class="ltx_bibitem"> <span class="ltx_bibtag ltx_role_refnum">7</span> <span class="ltx_bibblock"> L.E. Sigler, <em class="ltx_emph ltx_font_italic">Fibonacci’s Liber Abaci: Leonardo Pisano’s Book of Calculation</em>, Springer, 2002. </span> </li> <li id="bib.bib8" class="ltx_bibitem"> <span class="ltx_bibtag ltx_role_refnum">8</span> <span class="ltx_bibblock"> Hana Vymazalova, <em class="ltx_emph ltx_font_italic">The Wooden Tablets from Cairo:The Use of the Grain Unit HK3T in Ancient Egypt, Archiv Orientalai</em>, Charles U Prague, 2002. </span> </li> </ul> </section> <div id="p55" class="ltx_para ltx_align_right"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle"> <tbody class="ltx_tbody"> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">Title</th> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">Egyptian fraction</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l"><a class="nnexus_concept" href="http://planetmath.org/canonical">Canonical</a> name</th> <td class="ltx_td ltx_align_left ltx_border_r">EgyptianFraction</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Date of creation</th> <td class="ltx_td ltx_align_left ltx_border_r">2013-03-22 17:38:57</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Last modified on</th> <td class="ltx_td ltx_align_left ltx_border_r">2013-03-22 17:38:57</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Owner</th> <td class="ltx_td ltx_align_left ltx_border_r">milogardner (13112)</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Last modified by</th> <td class="ltx_td ltx_align_left ltx_border_r">milogardner (13112)</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Numerical id</th> <td class="ltx_td ltx_align_left ltx_border_r">181</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Author</th> <td class="ltx_td ltx_align_left ltx_border_r">milogardner (13112)</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Entry type</th> <td class="ltx_td ltx_align_left ltx_border_r"><a class="nnexus_concept" href="http://planetmath.org/definition">Definition</a></td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">Classification</a></th> <td class="ltx_td ltx_align_left ltx_border_r">msc 01A35</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Classification</th> <td class="ltx_td ltx_align_left ltx_border_r">msc 01A30</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Classification</th> <td class="ltx_td ltx_align_left ltx_border_r">msc 01A20</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Classification</th> <td class="ltx_td ltx_align_left ltx_border_r">msc 01A16</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Synonym</th> <td class="ltx_td ltx_align_left ltx_border_r">unit fraction series</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">Defines</th> <td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">rational numbers</td> </tr> </tbody> </table> </div> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Thu Feb 8 19:08:22 2018 by <a href="http://dlmf.nist.gov/LaTeXML/">LaTeXML <img src="data:image/png;base64,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" alt="[LOGO]"></a> </div></footer> </div> </body> </html>