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This is the user page for Milo Gardner, whose login name on Wikipedia is User:Milogardner
Military language and math based code breaking methods were applied to Russian, Arabic and math topics in the late 1950s. College math and economics courses were taken from 1959 to 1964 included medieval mathematics, history of mathematics (Eves), Theory of Equations (Borofsky), history of number theory (Ore) http://books.google.com/books?id=Sl_6BPp7S0AC&printsec=frontcover&dq=subject:%22Mathematics+-+History%22&hl=en&ei=niYvTYjiD426sQPlleWmCQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CDgQ6AEwBA#v=onepage&q&f=false, European/US history of economic thought linked the Hanseatic League with capitalism economic history threads that birthed Adam Smith's Wealth of Nations and the USA economy (and political system). A 1970 MBA sharpened modern business threads that connect to ancient monetary and economic systems in subtle ways.
In 1988 independent studies re-dated zero's arrival to the West (Egypt and Babylon) by 2,000 BCE. Six months were spent reading math books in local university libraries. It is clear that theoretical zero was used in 2,000 BCE Egypt (nfr) and diffused the idea to classical Greece (two dots), India, and the medieval Arab world. Pope Sylvester II adopted Arab math that included non-positional fragments of zero in 999 AD and implemented the finite arithmetic after the Crusades by Fibonacci's Liber Abaci. By 1585 AD a theoretical zero was formalized in Europe by positional numbers within Simon Stevin's algorithmic base 10 decimal system points cited by Eves.
Meeting Noel Braymer in 1989, a retired SRI International electrical engineer, the RMP 2/n table and Egyptian math became interests that searched for scribal methods that directly calculated scribal mathematics. Military cryptanalysis empowered several studies of hieratic (Middle Kingdom Egyptian) math patterns. A dozen studies over 20 years have cleared up Middle Kingdom math text transliteration errors. Scribal arithmetic patterns have been re-transliterated and translated into modern arithmetic statements in new ways. Gathering an approval from Worcester Polytechnic Institute's history of science department, followed by reviews by David Pingree and two Historia Mathematica experts, three number theory submissions freshly decoded 50 Rhind Mathematical Papyrus 2/n table patterns by modern number theory recorded in least common multiples, one of the new methods. A third rejection recommended that Egyptian number theory be limited to internet discussions. For 15 years internet discussion groups that included David Fowler (mathematician) discussed ancient Egyptian number theory. Internet discussions identified ancient number theory and history of zero threads as theoretical statements.
In 2004 an Egyptian Mathematical Leather Roll (EMLR) journal paper parsed 26 patterns by an incomplete modern 'splitting method'. In 2005 anAkhmim Wooden Tablet (AWT) journal paper discussedab initio aspects of Georges Daressy's [] 1906 1/11 and 1/13 oversights. The (64/64) proof side of five AWT patterns was published by Hana Vymazalova in 2002. Vymazalova validated ab initio (64/64) units were scaled into hekat units. In 2011 Ahmes's least common multiple m completely scaled rational numbers related to commodity transactions recorded in hekat and pesu units. The 2011 paper updated the 2004 Egyptian Mathematical Leather Roll and 2006 AWT papers and added scribal aspects of the RMP 2/n table related to LCM construction methods.
Several MK weights and measures and unit fraction patterns have been translated into basic number theory statements. As background, Egypt's 2050 BCE ciphered numerals were replaced by 800 AD when Hindu-Islamic numerals (as used in the first 124 pages of the 500 pages of Fibonacci's 1202 AD Liber Abaci) were reported by Arab mathematics. The Liber Abaci scaled rational numbers n/p by a subtraction method, (n/p - 1/m) = (mn -p)/mp. The medieval method replaced Ahmes (n/p)(m/m) = mn/mp multiplication method that wrote out concise unit fraction series, a context in which weights and measures for over 3,500 years were written in the Ancient Near East and the medieval world.
2006, 2009, 2010, and 2011 studies were completed in association with Bruce Friedman. The 2006 AWT study reported scribal division as identical to RMP division. Ancient division was inverse to multiplication in the modern sense, invert the divisor n to 1/n and multiply. In RMP 83 Ahmes the RMP scribe used quotient and remainder statements(http://planetmath.org/encyclopedia/AhmesBirdFeedingRateMethod.html in RMP 83. Three bird feeding portions:(1) 2 geese, and a crane ate (1/8 + 1/32) hekat + 3 1/3 ro, (2) a set-duck ate (1/32 + 1/64) hekat + 1 ro, and (3) a set-goose, dove and quail each ate (1/64) hekat + 3 ro units of grain in one day. A ro was 1/320 of a hekat. Ahmes' question asked: how much grain did the seven birds eat in one day? (answer 5/8 of a hekat]. In 2010 the Berlin Papyrus gave up Schack-Schackenberg's 1900 view of RMP 69 and the pesu, as well as freshly reporting RMP 36 and RMP 37 red auxiliary numbers' scribal secrets connected to the Egyptian Mathematical Leather Roll. RMP 36 reported 5/53 as (60/636 - 1/12) = 7/212 = (4 + 2 + 1)/636 = 1/519 + 1/318 + 1/636 and 3/53 = (60/1060 - 1/20) = 7/1060 = (4 + 2 + 1)/1060 = 1/265 + 1/530 + 1/1060. Tanja Pemmerening's 2002 and 2005 Ebers Papyrus papers scaled a dja to 1/64 of a hekat, and a 2009 advocacy of interdisciplinary approaches are greatly appreciated.
In MMP 10, RMP 41, RMP 42 and RMP 43 reported the area of a circle formula that solved for Diameter (D) by replacing radius (R) with D/2 and pi with 256/81 such that: A = [(8/9)D]^2 cubit^2. In RMP 42 Ahmes multiplied height (H) to compute Volume (V) a cubit^3 unit. By increasing the cubit^3 value by 3/2 a khar unit appeared. In RMP 43 and the Kahun Papyrus scribes scaled A^1/2 = (8/9)D by 3/2 and obtained V = (2/3)H[(4/3)D]^2 khar. In RMP 44 1500 khar/20 = (75) 400-hekat. In RMP 47 400-hekat times 1/10, and 1/20 = (10) 4-hekat and (5)4-hekat, respectively, and 100-hekat times 1/40, 1/50, 1/80, and 1/100 reported binary (Q/64)hekat quotients and (5R/n) ro remainders were understood by Clagett (1999). Clagett did not report an understanding of 100-hekat multiplications, facts reported by 1/30, 1/60, 1/70 and 1/90 discussed by: http://mathforum.org/kb/message.jspa?messageID=7259234&tstart=0 . A Dec. 6, 2010 New York Times article reported puzzle implications of four Egyptian texts per: http://www.nytimes.com/2010/12/07/science/07first.html?ref=science written in an economic context citing wage payments.
The 2011 update of the AWT, EMLR and RMP 2/n table corrected least common multiples (http://planetmath.org/encyclopedia/Arithmetics.html). Middle Kingdom scribes scaled 50 members of the RMP 2/n table and 26 lines of EMLR text in the same manner. RMP 36 http://planetmath.org/encyclopedia/RMP36AndThe2nTable.html was parsed in 2008 and 2011 that added red auxiliary numbers in an AWT hekat context to the scribal math tool kit. The update replaced inappropriate modern EMLR 'splitting methods' with a fully written out RMP 36 longhand method. The ab initio aspects of scribal longhand included initial, intermediate and final unit fraction series. The fragmented shorthand texts have been translated into readable modern arithmetic sentences by replaced missing scribal steps. For example 26 EMLR lines, 51 RMP 2/n table entries, and about 200 Egyptian math problems, 87 in the RMP, 25 in the Kahun Papyrus, 25 in the Moscow Mathematical Papyrus, two in the Berlin Papyrus, two in the Reisner Papyrus, and the remainder in Ebers Papyrus-type prescriptions recorded shorthand LCMs and red auxiliary numbers that applied the scribal longhand method. The longhand scribal method solved a wide range of issues.
About 1,500 later the 300 BCE Greek Hibeh Papyrus (http://planetmath.org/encyclopedia/HibehPapyrus.html) mentioned 27 Greek festival days within an Egyptian civil calendar, and other topics. The length of each festival day and night was encoded to Egyptian unit fraction series that applied LCM 1, 2 and 4. A 500 AD Coptic text (http://www.mathorigins.com/image%20grid/ACHMIM_GR_001.htm) reported the Egyptian-Greek-Coptic LCM arithmetic in the Akhmim Papyrus (http://mathforum.org/kb/message.jspa?messageID=7485114&tstart=0), a ciphered number system that scaled rational numbers n/17 and n/19 by LCM 2, 3, 4, 5, 6, 10, 12, 30, 60 and 120, as needed. The 2,800 year old numeration system ended in 800 AD with the rise of Arabic numerals.
A related approach was applied by Fibonacci's rational number conversion method, the simplest of three Arabic notations. One of the Arabic notations was named after Euclid. Ahmes (the RMP scribe)scaled rational numbers n/p by LCM m to mn/mp selectrf the best divisors of denominator mp that summed to numerator mn calculated 2-term. 3-term, 4-term, or 5-term series. Fibonacci's n/p conversion method also used a related LCM m method that scaled (n/p - 1/m) = (mn -p)/m with numerators (mn-p) set to unity that reported 2-term series. When (mn-p) could not be found a second LCM m was subtracted from the remainder that calculated a 3-term exact unit fraction series (i.e 4/13 = 1/4 + 1/18 + 1/468), unit fraction system that ended in 1585 AD with Simon Stevin's use of zero as a place-holder in an algorithmic base 10 decimal system.
In summary, Middle Kingdom Egyptian and Fibonacci arithmetic systems were finite. Middle Kingdom arithmetic defined economic and theoretical arithmetic methods that empowered scribal wage payments (made in hekats of grain, and other commodity units) that were extended to the medieval era (with minor modifications). Middle Kingdom Egyptian arithmetic was confirmed in the 21st century AD by decoding the EMLR, RMP (http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html) and other texts. Egyptian scribes used prime numbers, rational numbers, least common multiples, quotients, remainders, arithmetical progressions, and later Archimedes square root http://planetmath.org/squarerootof3567and29 that reduced square root approximation errors to 4- 6 places.
BIO: http://milorgardner.blogspot.com/2008/08/milo-gardner-personal-info.html links 24 ancient Egyptian, three classical Greek, two medieval and Mesoamerican topics (http://planetmath.org/mayanseasonalalmanac) are also published in Planetmath's math encyclopedia. A 12/6/2010 New York Times Science section article reported four Egyptian math paper puzzles per: http://www.nytimes.com/2010/12/07/science/07first.html?ref=science. In 2011 an interview captures personal snapshots of my life: http://www.eimacs.com/blog/2011/11/milo-gardner-cryptanalyst-code-breaker/ .