Liber Abaci

Liber Abaci (1202) is an historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. Its title has two common translations, The Book of the Abacus or The Book of Calculation. In this work, Fibonacci introduced to Europe the Hindu-Arabic numerals, a major element of our decimal system, which he had learned by studying the older form of Egyptian fractions with Arabs while living in North Africa with his father, Guilielmo Bonaccio, who wished for him to become a merchant.

Liber Abaci was not the first Western book to describe Hindu-Arabic numerals, the first being by Pope Silvester II in 999, but by addressing tradesmen and academics, it discusseed the properties of a simpler system, a special case during his life-time, writing 3.1416 as 6/10 1/10 4/10 1/10 3. The medieval factoring system generally wrote composite numbers into its prime components without regards to the base number system, a method that had been reported in the older Greek and Egyptian numeration systems. In addition Fibonacci wrote out composite and prime vulgar fractions, n/pq and n/p, into the older Egyptian fraction system in other ways thereby providing several clues to the older methodolgies. The first section of the book introduces the Hindu-Arabic numeral system, and describes how to calculate with numbers of this type. The second section presents examples from commerce, such as conversions of currency and measurements, and calculations of profit and interest. The third section discusses a number of mathematical problems; for instance, it includes (ch. II.12) the Chinese remainder theorem, perfect numbers and Mersenne primes as well as formulas for arithmetic series and for square pyramidal numbers. Another example in this chapter, describing the growth of a population of rabbits, was the origin of the Fibonacci sequence for which the author is most famous today. The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots. The book also includes Euclidean geometric proofs, and a study of simultaneous linear equations. The seventh chapter details the medieval form of writing Egyptian fractions, rigorously writing out several methods that date to the oldest Egyptian and Greek scribes.

Fibonacci's notation for fractions

In reading Liber Abaci, it is helpful to understand Fibonacci's notation for rational numbers, a notation that is intermediate in form between the Egyptian and Greek Egyptian fractions that were commonly used until that time.

There are three key differences between Fibonacci's notation and the oldest Egyptian methods of writing Egyptian fractions. First, Fibonacci wrote his whole number and vulgar fractions into Egyptian fractions by only listing its prime factors. For example, 347/540 was written as 1/5 1/4 1/3 5/9, meaning 5/9 + 1/27 + 1/38 + 1/45. Egyptians would have written 5/9 as 1/2 + 1/18, and showed that 347/540 - 5/9 = 423/4860 = (180 + 135 + 108)/4860 = 1/27 + 1/36 + 1/45. The second and third differences relate to the manner in which the older radix information was recorded. One radix system was used for every day factoring of fractions, writing 1/805 as 1/5 0/7 0/23 = 1/(5x7x23), and 13/24 as 1/2 5/6 7/10 = 7/10 + 6/(6x10) + 1/(2x6x10). Thirdly, a second radix system was taken from Euclid, expressing 56/3 as 2/3 4/5 6/7 8/9 O = 8/9 + (6x8)/(7x9) + (4x6x8)/(5x7x9) + (2x4x6x8)/(3x5x7x9). Egyptians would have simply written 56/3 as a quotient 18 with a remainder 2/3.

In addition, there are also three key differences between Fibonacci's notation and modern fraction notation.

1. Where we generally write a fraction to the right of the whole number to which it is added, Fibonacci would write the same fraction to the left. That is, we write 7/3 as $\scriptstyle 2\,{\frac {1}{3}}$, while Fibonacci would write the same number as $\scriptstyle {\frac {1}{3}}\,2$. Fibonacci was following the oldest definition of division, where 2 was a quotient and 1/3 was a remainder, a structure that is not often used today.
2. Fibonacci used a composite fraction notation in which a sequence of numerators and denominators shared the same fraction bar; each such term represented an additional fraction of the given numerator divided by the product of all the denominators below and to the left of it. That is, $\scriptstyle {\frac {a\,\,b}{c\,\,d}}={\frac {a}{cd}}+{\frac {b}{d}},$ and $\scriptstyle {\frac {a\,\,b\,\,c}{d\,\,e\,\,f}}={\frac {a}{def}}+{\frac {b}{ef}}+{\frac {c}{f}}.$ This notation, a form of mixed radix notation, was very convenient for dealing with traditional systems of weights, measures, and currency. For instance, for units of length, a foot is 1/3 of a yard, and an inch is 1/12 of a foot, so a quantity of 5 yards, 2 feet, and $\scriptstyle 7{\frac {3}{4}}$ inches could be represented as a composite fraction: $\scriptstyle {\frac {3\ \,7\,\,2}{4\,\,12\,\,3}}\,5$ yards. However, typical notations for traditional measures, while similarly based on mixed radixes, do not write out the denominators explicitly; the explicit denominators in Fibonacci's notation allow him to use different radixes for different problems when convenient, a methodology that dates to 2,000 BCE. For instance Sigler points out an instance where Fibonacci uses composite fractions in which all denominators are 10, prefiguring modern decimal notation for fractions.
3. Fibonacci sometimes wrote several fractions next to each other, representing a sum of the given fractions. For instance, 1/3+1/4 = 7/12, so a notation like $\scriptstyle {\frac {1}{4}}\,{\frac {1}{3}}\,2$ would represent the number that would now more commonly be written $\scriptstyle 2\,{\frac {7}{12}}$, or simply the vulgar fraction 31/12. Notation of this form can be distinguished from sequences of numerators and denominators sharing a fraction bar by the visible break in the bar. If all numerators are 1 in a fraction written in this form, and all denominators are different from each other, the result is an Egyptian fraction representation of the number. This notation was also sometimes combined with the composite fraction notation: two composite fractions written next to each other would represent the sum of the fractions.

The complexity of this notation allows numbers to be written in many different ways, and Fibonacci described several methods for converting from one style of representation to another. In particular, chapter II.7 contains a list of methods for converting a vulgar fraction to an Egyptian fraction. If the number is not already a unit fraction, the first method in this list is to attempt to split the numerator of the fraction 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 $\scriptstyle {\frac {a}{ab-1}}={\frac {1}{b}}+{\frac {1}{b(ab-1)}}.$ If all these other methods fail, Fibonacci suggests a greedy algorithm in which one makes two subtractions of the largest possible unit fraction from the given fraction; he gives as examples the greedy expansions $\scriptstyle {\frac {4}{13}}={\frac {1}{468}}\,{\frac {1}{18}}\,{\frac {1}{4}}$ and $\scriptstyle {\frac {17}{29}}={\frac {1}{348}}\,{\frac {1}{12}}\,{\frac {1}{2}}.$ Following these methods, Fibonacci suggests instead expanding a fraction $a/b$ by searching for a number c having many divisors, with $b/2<c<b$, replacing $a/b$ by $ac/bc$, and expanding $ac$ as a sum of divisors of $bc$; a similar method was much more recently posited by Hultsch and Bruins as explaining the Egyptian fraction expansions appearing in the Egyptian papyri, a definition that shows the required first subtraction as including the modern definition of subtraction.

References

• Sigler, Laurence E. (trans.) (2002). Fibonacci's Liber Abaci. Springer-Verlag. ISBN 0-387-95419-8.

• Lüneburg, Heinz (1993). Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers. Mannheim: B. I. Wissenschaftsverlag. ISBN 3-411-15461-6.

Links

• This Book @ WikiSource