LiberAbaci 2007-12-10 11:33:21 2009-10-12 16:12:22 Definition Egyptian fractions % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The \PMlinkexternal{Liber Abaci}{http://en.wikipedia.org/wiki/Liber_Abaci} (Book of Calculation) was written by Leonardo Pisano (Fibonacci) in 1202 CE. The book was revised several times during Leonardo's life time. The book was used for over 250 years as Europe's arithmetic book in its Latin schools. Today five copies of the Latin language book exist. L.E. Silger translated the 500 page Liber Abaci (\PMlinkexternal{LA}{http://liberabaci.blogspot.com}) to English in 2002 AD in time for the book's 800th anniversary. One of Sigler's footnotes mentioned a Fibonacci error, converting 4/49 to an Egyptian fractions series. Actually, Fibonacci properly factored 4/49 in a manner that an unexpected exact elegant series was calculated. Hence Fibonacci had not erred. That is, Sigler misunderstood several number theory aspects related to the factoring methods presented in the LA's arithmetic section. There is more to the story. In the first 125 pages, citing factoring examples, Fibonacci summarized the arithmetic section in two pages citing seven rational number conversion methods. Ahmes used five of the seven methods to create 2/n tables, and other rational number conversions to optimal Egyptian fraction series following modifications of Ahmes methods. Three of Fibonacci's methods defined Hultsch-Bruins type methods reported by F. Hultsch in 1895 AD. Fibonacci generalized a H-B conversion method in method four. That is, Fibonacci modified, by example, five conversion methods that had allowed Ahmes 2,850 years earlier to convert n/p and n/pq (vulgar fractions) by selecting optimal multiples in a Medieval subtraction context. Read the \PMlinkexternal{EMLR}{http://emlr.blogspot.com}, the \PMlinkexternal{RMP 2/n Table}{http://en.wikipedia.org/wiki/RMP_2/n_table} and \PMlinkexternal{Ahmes Papyrus}{http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html} for the older details. Methods five,and six used a style that Ahmes did not originate. For example, Leonardo's method six converted 20/53 by subtracting 3/8 after raising 3/8 to a multiple of 6, 18/48, writing 18/48 1/8 0/53, using a Greek, Arab and medieval notation. Yet, in method 4, 20/53 was written as 1/53 + 19/53, with 19/53 written as a multiple of 3 nearer to Ahmes' thinking. The first four methods raised the initial rational number to a multiple following a Arab/Medieval subtraction context. The seventh distinction modified Ahmes' 2/n table by converting n/p = 1/p + (n -1)/p (i.e. 20/53) and (n/p - m1/m2) (i.e. 30/53 - 7/11 = (330 - 318)/603 = (11 + 1)/603, and so forth in Fibonacci's notation. Considering Egyptian fractions as its parent, Arabs and Fibonacci translated Ahmes Papyrus n/pq*(m/m) scaled multiplication context that included optimizing red number numerators into an easier subtraction context.The medieval subtraction context converted n/pq by a LCM, scaling n/pq by a subtraction step: n/pq - 1/m = (mn - pq)/mpq, with the divisors of mpq used to find a (mn-pq), usually 1. For difficult rational number conversion problems that did not equal one the 7th method (distinction) applied a second LCM, a step that Sigler reported in a footnote always worked. SUMMARY At the end of the first 125 pages of the 500 page Liber Abaci seven rational number conversion methods were detailed. The seven sets of examples show that Fibonacci converted any rational number to elegant Egyptian fraction series. Major aspects of five of the methods originated in the Egyptian Middle Kingdom. Only two of Fibonacci's notations were unique to Greeks, Arabs and medieval scribes. The Liber Abaci continues to be parsed in surprising ways allowing 1650 BCE and medieval arithmetic aspects to be directly compared by each era's unifying unit fraction threads. \begin{thebibliography}{3} \bibitem{1}L.E. Sigler, \emph{Fibonacci's Liber Abaci, Leonardo Pisano's Book of Calculations}, Springer, 2002. \bibitem{2} Heinz Leueneburg, \emph{Leonardi Pisani Liber Abbaci oder Lesevergngen eines Mathematikers}, Mannheim: B. I. Wissenschaftsverlag , 1993. \bibitem{3} Oystein Ore, \emph{Number Theory and its History}, McGraw-Hill, 1948. \end{thebibliography}