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Viewing Version
42
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'Liber Abaci'
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| Title of object: |
Liber Abaci |
| Canonical Name: |
LiberAbaci |
| Type: |
Definition |
| Created on: |
2007-12-10 11:33:21 |
| Modified on: |
2009-08-10 13:20:41 |
| Classification: |
msc:01A35 |
| Defines: |
Egyptian fractions |
| Synonyms: |
Liber Abaci=rational numbers |
Preamble:
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Content:
The \PMlinkexternal{Liber Abaci}{http://en.wikipedia.org/wiki/Liber_Abaci}
(Book of Calculation) was written by Leonardo Pisano in 1202 CE. The book was revised several times during Leonardo's
life time. It was used for over 250 years as Europe's arithmetic book in its Latin schools.
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, mostly 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 out an answer, 18/48 1/8 0/53, using a Greek, Arab or medieval notation in the 4th distinction. Yet, in method 4, 20/53 was written as 1/53 + 19/53, with 19/53 written as a multiple of 3, near Ahmes' thinking. The first four methods raised the initial rational number to a multiple of itself following a 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 = 2/p + (n -2)/p (i.e. 30/53).
SUMMARY
At the end of he first 125 pages of the 500 page Liber Abaci Fibonacci detailed seven rational number conversion methods. The methods show that Fibonacci easily converted any rational number to elegant Egyptian fraction series. Five of Fibonacci's 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 theoretical arithmetic aspects to be compared and analyzed as unifying threads by serious modern reviews.
\begin{thebibliography}{3}
\bibitem{1}L.E. Sigler, \emph{Fibonacci's Liber Abaci, Leonardo Pisano's Book of Calculations}, Springer, 2002.
\bibitem{2} Heinz Lüneburg, \emph{Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers}, Mannheim: B. I. Wissenschaftsverlag , 1993.
\bibitem{3} Oystein Ore, \emph{Number Theory and its History}, McGraw-Hill, 1948.
\end{thebibliography}
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