Thursday, January 04, 2007

Liber Abaci and seven Egyptian fraction methods

The Liber Abaci (Book of Calculation) was written by Leonardo Pisano. Fibonacci was Leonardo's Latin name. Leonard was the son of a Pisa merchant, and often went with his father to Arab ports and other trading locations. On these trips Leonardo learned abstract and practical sides Hindu-Arabic numerals that were required to compute weights and measures systems of the Mediterranean world. As an important historical consequence, Fibonacci documentation of medieval trading units reported very old theoretical and practical Egyptian fraction arithmetic that allowed medieval schools and commerce units to be defined and administered.

The Liber Abaci (LA) was a very successful book. It was used in the medieval period for about 250 years to teach arithmetic and higher mathematics in European Latin schools. Today, five copies of the book exist. The 808 year old book has been partially translated and reviewed by 20th century scholars.  Scholar tend to take a single arithmetic LA subject, omitting other arithmetic subjects. The 500 page book therefore has not been fully translated into English and European languages related to ancient arithmetic implications.

One 20th century theme limited LA discussions to base 10 decimals foundations related to algorithms. The base 10 decimal theme, a subject that flowered in the late 1500's, minimally considered 3,200 year older Egyptian fraction arithmetic that did not rely on algorithms.

The 500 page LA book began by using the first 125 pages to expose aspects of the 3,200 year older Egyptian fraction arithmetic. Scholars had not fully translated, footnoted, and discussed Liber Abaci arithmetic. Silger's 2002 translation of the LA exposed several unifying aspects of the oldest form of Egyptian fraction arithmetic.

Sigler's important work began a process of  footnoting the LA's meta arithmetic topics, especially rational number notations and methods written in unit fractions. Sigler's footnotes are at times sparse partly based on his untimely death. Sigler's footnotes began to explain theoretical aspects of medieval arithmetic, as well as pointing out medieval theoretical foundations of factoring methods that connect to the Egyptian Middle Kingdom.

As a projection, the 500 page Sigler translation leads to meta math exposures of historical threads on which the medieval version of the fundamental theorem of arithmetic was understood by Arabs, as passed down from the Greeks, who in turn received from Egyptians.

The beginning seven chapters of the LA, 25% of the book, detail theoretical and practical aspects of the 3,200-year old system of Egyptian fractions. At various places Fibonacci used three different remainder arithmetic notations writing rational numbers into not-so-concise unit fraction series. Fibonacci's first arithmetic notation was dominant in the LA. The second two notations were  used for special factoring purposes.

The following paragraphs of this blog will stresses the first notation, and detail its intellectual scope.


This blog summarizes and analyzes seven methods, or distinctions, as written in three Liber Abaci remainder arithmetic notations. Fibonacci used the notations to assist in the conversion of vulgar fractions to elegant and not-so-elegant Egyptian fractions series, the practice that he had likely found in his travels. The intent in this blog is to discuss the oldest Egyptian fraction thread(s) written within the first notation. The first notation was parsed by Fibonacci into seven distinctions (Sigler's term).

The seven methods, or distinctions, were built by selecting a first partition, (I'll call it 1/m), a step that Ahmes also used (that Ahmes wrote its remainders in red). In Fibonacci' arithmetic operations subtraction steps were to reach final answers, understanding that

(n/pq - 1)/m = (mn - pq)/mnp

was solved by inspecting (mn-pq), an odd number,

with Fibonacci always taking

[(mn- pq) + 1]/2

to assist finding a second subtraction step, a method that was repeated until the problem was solved, as summarized by Silger's distinctions:

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 factoring by writing 1/2 of 9 in the oldest notation as 1/18 as 1/2 x1/9, and converting to an Egyptian fraction series (such as by 1/2 = 1/3 + 1/6, meant that 1/18 = 1/27 + 1/54, was an Egyptian fraction series that Fibonacci did not list).

b. The second composite uses a Greek or Arab notation that reports 1/18 = 1/2 0/9, which equals 5/10 0/9, as listed by Fibonacci. Aspects of this rule may date to the time of Ahmes, when (64/64), a hekat unity, allowed its partition into quotients and Egyptian fraction remainders. For example, Fibonacci wrote 1/2 as 5/10, and wrote other fractions to their equals, selecting least common denominators and other relationships to best complete his conversion of vulgar fractions to Egyptian fraction series written into the first notation. (It will be shown Ahmes achieved the same class of rational number answers by applying four closely related conversion methods.)

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:

1/18 = 5/10 0/9 = 5/9 0/10.

Note that 5/10 0/9 = 5/90 = 1/18 and 5/9 0/10 = 5/90 = 1/18.

Several additional examples of the first method (distinction), and its three aspects, will be reviewed and published elsewhere (and will be linked to this site).

Sigler summarizes Fibonacci's first method (distinction) rule, and its three aspects, primarily through the assistance of Dunton and Grimm's "Fibonacci on Egyptian fraction" paper. Dunton and Grimm use the algebraic statement k/kl = 1/l, an algebraic identity, as a fair method (which it is not) to capture Fibonacci' three part rule, as cited above. Note that the third aspect of Fibonacci's method (distinction), titled "third reversed composite," is one of three Fibonacci aspect methods that Dunton and & Grimm oddly did not mention, a logical omission that Silger chose to adopt.

2. Second method (distinction)

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

a. 5/6 = (3 + 2)/6

= (1/2 1/3), a quotient name for a numerator

b. 7/8 = (4 + 2+ 1)/8 = (1/2 1/4 1/8)

c. A reverse composite is used to solve for

3/40 = (3/4 0/10) meaning that

3/4 = (3/10 0/4)

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 stresses Dunton and Grimm per the statement (k + 1)/klm = 1/lm + 1/km), an analysis that insufficiently captures Fibonacci's definitions, and examples.)

3. Third method (distinction)

a. 2/11 = (1/6 0/11), parts of 2/11

b. 3/11 = (1/4 0/11) = (1/11 1/4)

c. 6/11 = 1/22 1/2

d. 8/11 = 2/11 + 6/11

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

(Sigler consistently cited Dunton and Grimm per the identity k/(kl -1) = 1/l + 1/(kl -1), again omitting vital information offered by Fibonacci's examples).

4. Fourth method (distinction)

This method allows the use of Ahmes' 2/p method (presented 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. Hultsch, and is now titled the Hultsch-Bruins method.

a. 19/53 - 1/3 = (3 + 1))/(3 x 53) = 1/159 1/53, meant that

19/53 = 1/159 1/53 1/3, a statement found in Egyptian texts.

b. 5/11 - 1/3 = (3 + 1)/(3x 11) = 1/11 1/33, meant that

5/11 = 1/33 1/11 1/3, again a very old form of style and contents

c. 7/11 - 1/2 = (2 + 1)/(2 x 11) = 1/22 1/11 1/2

d. 6/19 - 1/4 = (4 + 1)/(4x 19) = 1/19 1/76, meant

6/19 = 1/76 1/19 1/4

e. 7/29 - 1/5 = (5 + 1)/(5x29) = 1/29 + 1/145, or

7/29 = 1/145 1/29 1/5 in Fibonacci's notation.

(Sigler did not comment on this distinction, though the method clearly represents vital facts understood and used by Fibonacci, information that will be expanded in the 7th distinction to solve 30/53 by solving 28/53 + 2/53 as Ahmes discussed 2,500 years earlier.

5. Fifth method (distinction)

a. 9/26 - 1/3 = (1/3 0/26 1/3) = 1/78 1/3

b. 11/26

c. 11/29 = (1/78 1/3 1/3) since

11/29 - 1/3 = (3 + 1)/(93 x 29) = 1/79 1/3

d. 11/62 = (0/62 1/31 1/7), since

11/62 - 1/7 = (14 + 1)/(7x 62), or

11/62 = (0/62 1/7 1/31 1/7), an alternate Fibonacci notation.

(Again, Sigler cited almost not historical or pertinent info to summarize this distinction's important set of examples, at least in the eyes of Fibonacci.)

6. Sixth method (distinction)

a. 17/27 - 3/27 = 14/27 - 1/2 = 1/54, meant

17/27 = 1/54 1/9 1/2, since 3/27 was found to reduce the vulgar fraction being converted.

b. 20/53 - 18/48 = (960 - 954)/(18x53), meant that

20/53 = 18/48 + 6/(18 x 53) = 18/48 1/8 0/53

"7. Seventh method (distinction)

a. 4/9 - 1/13 = 3/(13 x 49)

= (1/319 0/637 1/617 1/319 1/13), not elegant

b. 4/49 - 1/14 = 7/(14 x 49)

= (1/2 0/49 1/14), elegant

c. 4/49 = 1/7 x (4/7) = 1/7 x (4/7 - 1/2 = 1/14)

= (1/2 0/49 1/14), alternate elegant

A clear parallel is available in Ahmes' shorthand. In RMP 31 and RMP 36, two rational numbers 28/97 and 30/53 could not be converted in their original form. Ahmes applied the 2/n table to achieve a fast conversion to unit fraction series, a method copied by Fibonacci into a subtraction context. Ahmes' arithmetic considered the aliquot parts of 56, and 4 for 28/97 and 30 and 2 for 30/53 and writing:

28/97 = 2/97 + 26/97 = 2/97*(56/56) + 26/97*(4/4) = (97 + 8 + 7 ) + (97 + 4 + 2 + 1)/388

30/53 = 2/53 + 28/53 = 2/53 *(30/30) + 28/53*(2/2) = (53 + 5 + 2)/1590 + (53 + 2 + 1)/106

Fibonacci also consider the aliquot parts (divisors) of the same denominators. Sigler did not discuss 30/53, a vulgar fraction that can not be solved by Fibonacci's 29/53 + 1/53 rule. There is no first partition that solves 29/53. Try 1/2, means 29/53 - 1/2 =(53 + 2 + 2 + 1)/106 are needed, thereby breaking a repetition rule.

Fibonacci's rule 7 obviously was extended to include 2/53, meaning 2/53 was converted by:

2/53 - 1/30 = (53 +5 + 2)/1590 = 1/30 + 1/318 + 1/795

exactly following Ahmes 2/n table method.

Summary: Sigler offered no proof that an algorithm was used by Fibonacci. The best way to explain Fibonacci's seventh method is the to clarify a context of the two 28/97 and 30/97 examples. Fibonacci fairly concluded that the seventh conversion method works for all examples, a fact that Ahmes has been attested by applying a 2/n table rule in RMP 31 and RMP 36..

As a footnote, Sigler unfairly discussed a possible error made by Fibonacci when Fibonacci converted 4/49 by subtracting 1/13, by the statement:

4/49 - 1/13 = 3/637

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

4/49 - 1/13 = (52 - 49)/(13 x 49) = 3/637

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

4/49 = 1/7 x 4/7 = 1/7 x (1/2 + 1/14)

= 1/14 + 1/98.

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. Additional discussions on this suhbect can be found at::

Method one contains three methods, as noted above. The first method may date to Ahmes and his Egyptian scribal style of writing parts of a fraction, in long hand, as the EMLR and RMP used multiples and factoring to convert several vulgar fractions. Leonardo converted 1/18 by factoring 1/2 x 1/9, with 1/2 = 1/3 + 1/6 such that
1/18 = 1/27 + 1/64. The EMLR used this method four times, and the RMP used it to convert 2/101.

The second two methods discuss medieval arithmetic and its short hand notation, all useful in finding elegant conversions. Leonardo'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 (3 + 2)/6 = 1/2 + 1/3 as the EMLR wrote out all of its 1/p and 1/pq answers, and Ahmes used over and over again. The EMLR increased 1/p and 1/pq to multiples of 2,3, 4, 5, 7, and 25, such that conversions were completed as this LA section details in n/p and n/pq problems, and completed the finding of Egyptian fraction series by parsing the numerator in this manner.

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 2/p + (n- 2)/p = n/p.

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, 18/48 1/8 0/53. This answer is confirmed by subtracting each fraction, given a little practice, exactly as method four and five were solved.

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 x 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 4/49 to 1/7 x 4/7 with 4/7 = 1/2 + 1/14 such that an elegant unit fraction answer 4/49 = 1/14 + 1/98 was preferred by Leonardo. Again, the not-so-elegant 'quasi greedy algorithm' version of the 4/49 problem may have been recreational in scope.

There is more to the story. For example the RMP 2/pq method seemingly was not discussed by Leonardo. However, recalling that Ahmes used a (p + 1) multples was an aide in this work, it is easy to see that Leonardo also used the (p + 1) relationship to solve his Hultsch-Bruins and n/pq problems in methods four, five and six. Additional research will be conducted to determine if Leonardo also wrote his 2/pq elegant answers in a manner that Ahmes wrote his 2/pq answers.

Thee seven LA rational number conversion methods suggests that Leonardo de Pisa could also parse aspects of older Egyptian fraction arithmetic. The 1202 AD meta math of Leonardo parses additional aspects of older Egyptian mathematical texts, providing important meta hints to better decode the EMLR,  the RMP and Kahun Papyrus 2/nth tables, and other texts in improved ways.

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

2. L├╝neburg, Heinz (1993). Leonardi Pisani Liber Abbaci oder Lesevergn├╝gen eines Mathematikers. Mannheim: B. I. Wissenschaftsverlag.

3. Ore, Oystein (1948). Number Theory and its History. McGraw Hill.
Dover version also available, 1988, ISBN 978-0486656205.


1 Liber Abaci (Wikipedia)

2. Ahmes Papyrus

3. EMLR (Wikipedia)

4. EMLR (Planetmath)

5. RMP 2/n Table (Wikipedia)

6. Breaking the RMP 2/n Table Code (blog)

7 Egyptian fractions (Planetmath)

author: Milo Gardner