The Liber Abaci (Book of Calculation) was written by Leonardo Pisano in 1202 CE. The book was revised several times during Leonardo's lifetime, the latest was 1254. Leonardo Pisano is known today by his Latin name Fibonacci.

Leonardo was the son of a Pisa merchant. He often went with his father to Arab ports and other trading locations. On these trips Leonardo learned practical and abstract aspects of Arabic arithmetic. In addition, an older weights and measures system of units, in which Leonardo's father traded, was learned and documented. As a historical consequence, Fibonacci's medieval document presents both practical and theoretical aspects of Egyptian fraction arithmetic. The major features of medieval Egyptian fractions had been passed down through Greeks, Hellenes and Arabs, having been altered little over 3,200 years.

The Liber Abaci (LA) was a successful book during Leonardo's lifetime. It continued in use for 200 years after Leonardo's death. It presented in practical examples describing theoretical mathematical topics of interest to medieval Latin students. Today five copies of the book exist. The 800 year old book was reviewed by 20th century scholars in fragmentary and often confusing ways. Scholars had tended to only discuss modern LA math topics. Several of the Egyptian fraction topics had been omitted from serious reviews.

As a correction several LA oversights were included in a complete Latin translation, line by line, into English in 2002 by L.E. Sigler. Sigler's translation opened several medieval math threads, one being Egyptian fractions that were first recorded in 1,900 BCE Egypt.

One of the most frequently stressed modern LA themes stressed by scholars is its indirect role in creating our modern base 10 decimal system. Fibonacci directly reports elements of an 3,200 year old Egyptian fraction arithmetic that used no algorithm at all. Modern scholars may have stressed aspects of the LA's role in creating modern decimals based on the LA's 400 year old use of Arabic numerals. It took another 250 years to birth modern decimals, meaning that a 650 year gestation period took place.

Yet, to Fibonacci and adherents of traditional Egyptian fraction system, they may have fought 650 years against the rounding off any fraction in the form n/p and n/pq. The elements of Egyptian fraction arithmetic are important to review for several reasons. One is why did it take so long for decimals to emerge after Arabic numerals arrived on the world numeration stage? A second reason considers the older arithmetic context of Greek mathematics, such as a line magnitude being reported as an arithmetic multiple, topics that may provide additional reasons related to the 650 year gestation period.

Since 2002, scholars since have had easy access to the LA. Seven rational number conversion methods define the elements of Greek arithmetic. Sigler's footnotes begin to analyze the seven conversion methods. It should be stressed Sigler's footnotes are sparse, in part, caused by Sigler's untimely death. That is, the footnotes are only beginning points.

The beginning seven chapters, 126 pages, of the LA detail practical, and theoretical aspects of the 3,200 year old Egyptian fraction arithmetic. The initial 126 pages suggest a few of the hows, and whys of Egyptian fraction arithmetic. Egyptian fraction arithmetic was primarily written by Leonardo within one medieval notation, as broken down into seven rational number conversion methods. On its 500 English pages, Fibonacci reports two additional Egyptian fraction notations. All three notations expose other hows, and whys of Greeks Egyptian fraction arithmetic (arithmoi), and much more. Each of the three Egyptian fraction notations was used for a different purpose. In the interest of brevity, only the first notation will be discussed in terms of the oldest Egyptian fraction conversion methods, the focal of this narrative.

LEONARDO'S SEVEN EGYPTIAN FRACTION METHODS

Seven methods, or distinctions, generally converted rational numbers written into three Liber Abaci remainder arithmetic notations. Only the first of Leonardo's three notations will be discussed in this narrative. The second, and third notations seem to have been unique to medieval or Greek scribes. Therefore, the last two notations will not be considered, for present purposes, pertinent to the older Egyptian fraction definitions, the subject of this narrative.

Fibonacci's notations assisted his conversion of vulgar fractions to elegant and not-so-elegant Egyptian fractions series, practices that had been recorded in his travels. The intent of this narrative is to discuss relevant threads related to the first notation. The first notation describes seven rational number conversion methods, or distinctions (to use Sigler's term).

It will be shown that four of the seven methods defined within the first notation date to 2,000 BCE, as defined by:

1. Fibonacci's first method (distinction)

The first method contains three aspects—the simple, the second composite and the third reversed composite. Two remainder arithmetic notations are used in this method.

a. Simple means factoring ancient way of writing 1/2 of 9 in the oldest notation as 1/18 as (1/2*1/9), and then converting to an Egyptian fraction series, such as by 1/2 = 1/3 + 1/6, meaning 1/18 = 1/27 + 1/54.

b. The second composite uses a Greek or Arab notation that reports

c. The third reversed composite, continued to use a this Greek or Arab notation that allowed the denominators, 10 and 9, to be switched, stating that:

Note that

Sigler summarizes Fibonacci's first method (distinction) rule, by three aspects. The first cites Dunton and Grimm's use of k/kl = 1/l, an identity that captures very little of the Fibonacci' three part rule. The second, and third aspects include the use of a third reversed composite, aspects omitted by Dunton and Grimm.

2. Second method (distinction)

When greater numbers are not divisible by the lesser, a phrase offered by Fibonacci, was clarified by these examples:

a.

b.

c. A reverse composite is used to solve for

was used to solve example problems by applying tables of separations, as Fibonacci listed lists parts of 6, 8, 12, 20, 24, 60 and 100. This class of tables have been reported in the Coptic era, by David Fowler and others. In a broad sense, the RMP 2/nth table itself is such a table.

(Again, Sigler cites Dunton and Grimm per the statement

3. Third method (distinction)

a.

b.

c.

d.

meaning that a table of values created in distinction two can be used.

(again, Sigler cites Dunton and Grimm per the identity

4. Fourth method (distinction)

This method allows the use of Ahmes' 2/p method (present in the RMP 2/th table), where a large and highly composite denominator was selected to solve several examples. The vulgar fraction examples selected by Fibonacci were, 19/53, 5/11, 7/11, 6/19 and 7/29. This method was rediscovered in 1895 by F. Hultssch, and is now titled the <a href="http://planetmath.org/encyclopedia/HultschBruinsMethodEgyptianfractions2.html">Hultsch-Bruins method</a>.

a.

b.

c.

d.

e.

(Sigler did not comment on this distinction, though the method clearly represents vital facts used by Fibonacci.)

5. Fifth method (distinction)

a.

b. 11/26

c.

d.

(Again, Sigler cited nothing to summarize this distinction's importance.)

6. Sixth method (distinction)

a.

b.

7. Seventh method (distinction)

a.

b.

finds an elegant series by

c.

=

The factoring of 4/19 into (1/7)* (4/7) appears in Ahmes's RMP 2/95 example, where 2/95 = (1/5)* (2/19), with the solution to 2/19 being taken from the 2/nth table, or,

Silger offered a proposed algorithm by concluding that it "works for all examples." I, for one, do not see an algorithm at work here in all cases. Nor is it the simplest method that Leonard cited. I see an extension of the previous six arithmetic distinctions, connected in an easy to understand rules, with four rules that dating to Ahmes. . Silger discusses a possible error made by Fibonacci when he converted 4/49 by subtracting 1/13, by the statement:

Sigler suggested that 2/637 was the correct answer. However, it is clear that:

and Fibonacci's not so elegant Egyptian fractiion series. Using the fraction rule found in the seventh method a preferred elegant series can be found by:

REMOVING THE MEDIEVAL NOTATIONS, ALLOWING THE OLDEST METHODS TO SHINE

Translating the seven methods, or distinctions, into the older Egyptian fraction context, the first 126 pages summarize this information. The seven methods describe at least 10 medieval and older methods that Leonardo used to convert vulgar fractions to Egyptian fraction series.

Method one contains three methods, as noted above. The first method dates to Ahmes' scribal style of writing Egyptian fractions. Leonardo converted 1/18 by factoring (1/2) * (1/9), with 1/2 = 1/3 + 1/6 such that

The second two methods discuss medieval arithmetic and its short hand notation, all useful in finding elegant coversions. Leondard's medieval arithmetic notation, at times, wrote out elegant and not-so-elegant Egyptian fraction answers. These medieval arithmetic notations used by Leonardo have not been thoroughly discussed by math historians related to their use in finding elegant Egyptian fraction answers. Hopefully I will run across analyses of this class information in ways that connect to this discussion, the oldest form of Egyptian fraction arithmetic, and add it to this blog.

Method two wrote 5/6 as

Method three details 8/11 = 2/11 + 6/11, a tabular method that 400 AD Coptics used. The Coptic style was reported by David Fowler in Historia Mathematica in 1982 citing answers to problems from n/5 to n/31, and n/4 to n/32, or thereabouts. The RMP 2/nth table can also be seen as the 2/11 aspect of this method. Knowing any 2/n Egyptian fraction series, a second table entry can be found by doubling, such that

Methods four, five and six focus on the Hultsch-Bruins method, as used in all of Ahmes 2/p series. These three methods increasingly define complex definitions of the very old Hultsch-Bruins method, as first reported in the modern era by F. Hultsch in 1895.. The basic H-B method is discussed in method four. It was used by Ahmes to convert 2/p vulgar fractions to unit fraction series by first selecting a first partition with a highly composite denominator, and so forth, as explained elsewhere. Methods five and six show that the first partition need not have been a unit fraction, a style that Ahmes did not adopt. For example, Leonardo method six to convert 20/53 by subtracting 3/8 after raising it to a multiple of 6, 18/48, writing out an answer,

Method seven includes two methods. The first method is an extension of the Hultsch-Bruins method. Leonardo allowed a second subtraction, when the remainder's vulgar fraction could not be converted by method two, creating a not-so-elegant answer, possibly a form of recreational mathematics.

The second method discussed under method seven covers a factoring method first noted in the RMP 2/nth table, by Fowler and others, where 2/95 was factored as (1/5)* (2/19), with 2/19 taken from the 2/nth table. Adding method one (a) and method seven (b) generally factoring was available to Leonardo, Ahmes and everyone working with Egyptian fractions, any time during its 3,200 year recorded history.

In Leonardo's case Sigler footnotes stressed that an error had been made converting 4/49 by a quasi-greedy algorithm method. However, factoring

There is more to the story. For example, the RMP 2/pq method was not directly discussed by Leonardo. However, recalling that Ahmes used a (p+1)/(p+1) multiple as an aide in this work, it is easy to see that Leonardo used this relationship to solve Hultsch-Bruins-type n/pq problems in methods four, five and six.

SUMMARY

The first 125 pages of the 500 page Liber Abaci detail seven methods. The methods show that Fibonacci converted rational numbers to an Egyptian fraction series, as needed. Four of his methods may have originated in the Egyptian Middle Kingdom. Three of the methods were unique to Greeks, Arabs and medievals. The 1202 AD work of Leonardo continues to be parsed in surprising ways, at times, allowing previously unknown theoretical aspects of Egyptian fraction arithmetic to be exposed.

REFERENCES

1. Sigler, L,E„ " Fibonacci's Liber Abaci, Leonardo Pisano's Book of Calculations" Springer , New York, 2002, ISBN 0-387-40737-5.

2. Lüneburg, Heinz (1993). <i>Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers</i>. Mannheim: B. I. Wissenschaftsverlag.

3. Ore, Oystein (1948). Number Theory and its History. McGraw Hill.Dover version also available, 1988,