Egyptian Fraction - Medieval Mathematics

Medieval Mathematics

For more information on this subject, see Liber Abaci and Greedy algorithm for Egyptian fractions.

Egyptian fraction notation continued to be used in Greek times and into the Middle Ages (Struik 1967), despite complaints as early as Ptolemy's Almagest about the clumsiness of the notation compared to alternatives such as the Babylonian base-60 notation. An important text of medieval mathematics, the Liber Abaci (1202) of Leonardo of Pisa (more commonly known as Fibonacci), provides some insight into the uses of Egyptian fractions in the Middle Ages, and introduces topics that continue to be important in modern mathematical study of these series.

The primary subject of the Liber Abaci is calculations involving decimal and vulgar fraction notation, which eventually replaced Egyptian fractions. Fibonacci himself used a complex notation for fractions involving a combination of a mixed radix notation with sums of fractions. Many of the calculations throughout Fibonacci's book involve numbers represented as Egyptian fractions, and one section of this book (Sigler 2002, chapter II.7) provides a list of methods for conversion of vulgar fractions to Egyptian fractions. If the number is not already a unit fraction, the first method in this list is to attempt to split the numerator into a sum of divisors of the denominator; this is possible whenever the denominator is a practical number, and Liber Abaci includes tables of expansions of this type for the practical numbers 6, 8, 12, 20, 24, 60, and 100.

The next several methods involve algebraic identities such as For instance, Fibonacci represents the fraction by splitting the numerator into a sum of two numbers, each of which divides one plus the denominator: Fibonacci applies the algebraic identity above to each these two parts, producing the expansion Fibonacci describes similar methods for denominators that are two or three less than a number with many factors.

In the rare case that these other methods all fail, Fibonacci suggests a greedy algorithm for computing Egyptian fractions, in which one repeatedly chooses the unit fraction with the smallest denominator that is no larger than the remaining fraction to be expanded: that is, in more modern notation, we replace a fraction x/y by the expansion

where represents the ceiling function.

Fibonacci suggests switching to another method after the first such expansion, but he also gives examples in which this greedy expansion was iterated until a complete Egyptian fraction expansion was constructed: and

As later mathematicians showed, each greedy expansion reduces the numerator of the remaining fraction to be expanded, so this method always terminates with a finite expansion. However, compared to ancient Egyptian expansions or to more modern methods, this method may produce expansions that are quite long, with large denominators, and Fibonacci himself noted the awkwardness of the expansions produced by this method. For instance, the greedy method expands

while other methods lead to the much better expansion

Sylvester's sequence 2, 3, 7, 43, 1807, ... can be viewed as generated by an infinite greedy expansion of this type for the number one, where at each step we choose the denominator instead of, and sometimes Fibonacci's greedy algorithm is attributed to Sylvester.

After his description of the greedy algorithm, Fibonacci suggests yet another method, expanding a fraction by searching for a number c having many divisors, with, replacing by, and expanding as a sum of divisors of, similar to the method proposed by Hultsch and Bruins to explain some of the expansions in the Rhind papyrus.