Liber Abaci

The 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 (LA) 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, mostly citing factoring examples, Fibonacci summarized the arithmetic section in two pages citing seven rational number conversion methods. It is clear that Ahmes used four of the seven methods to create 2/n tables, and other rational number conversions to optimal Egyptian fraction series. Three of Fibonacci's methods define the Hultsch-Bruins method reported by F. Hultsch in 1895 AD. Fibonacci generalized a H-B conversion method in method four. That is, Fibonacci discussed, by example, four conversion methods that allowed Ahmes to generally convert n/p and n/pq vulgar fractions by selecting optimal multiples. Read the EMLR and the RMP 2/n Table methods for the older details. Methods five, six and seven are written in 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. The first four methods only raises the initial rational number to a multiple of itself.

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 rational numbers to elegant Egyptian fraction series as needed. Four of Fibonacci's methods likely originated in the Egyptian Middle Kingdom. All three of the notations used by Fibonacci were unique to Greeks, Arabs and medievals. Yet, the 1202 AD book continues to be parsed in surprising ways, allowing previously unknown 2,000 BCE and closely related medieval theoretical aspects, mostly factoring methods using multiples, to be exposed for serious review by modern scholars.

Bibliography

1

L.E. Sigler, Fibonacci's Liber Abaci, Leonardo Pisano's Book of Calculations, Springer, 2002.

2

Heinz Lüneburg, Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers, Mannheim: B. I. Wissenschaftsverlag , 1993.

3

Oystein Ore, Number Theory and its History, McGraw-Hill, 1948.