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Encyclopedia > 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 with Arabs while living in North Africa with his father, Guilielmo Bonaccio, who wished for him to become a merchant. // Events August 1 - Arthur of Brittany captured in Mirebeau, north of Poitiers Beginning of the Fourth Crusade. ... Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and simplest branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... Leonardo of Pisa (1170s or 1180s – 1250), also known as Leonardo Pisano, or Leonardo Bonacci ,Leonardo Fibonacci, or simply Fibonacci, was an Italian mathematician, considered by some the most talented mathematician of the Middle Ages. ... Hindu-Arabic numerals also known as Arabic Numerals, Hindu numerals, European numerals, and Western numerals are the most common set of symbols used to represent numbers around the world. ... The Arabs (Arabic: عرب) are a heterogeneous ethnic group who are predominantly speakers of the Arabic language, mainly found throughout the Middle East and North Africa. ...  Northern Africa (UN subregion)  geographic, including above North Africa or Northern Africa is the northernmost region of the African continent. ...


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 began to convince the public of the superiority of the new system. The first section 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. Gerbert of Aurillac, later known as pope Silvester II, (or Sylvester II), (ca. ... Profit, from Latin meaning to make progress, is defined in two different ways. ... Interest is the rent paid to borrow money. ... link titleThe Chinese remainder theorem (CRT) is the name for several related results in abstract algebra and number theory. ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... In mathematics, a Mersenne number is a number that is one less than a power of two. ... In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. ... A pyramidal number, or square pyramidal number, is a figurate number that represents a pyramid with a base and four sides. ... In mathematics, the Fibonacci numbers form a sequence defined recursively by: In words: you start with 0 and 1, and then produce the next Fibonacci number by adding the two previous Fibonacci numbers. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 − x3 = 1 2x1 − 2x2 + 4x3 = −2 −x1 + ½x2 − x3 = 0. ...


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 fractions commonly used until that time and the vulgar fractions still in use today. There are three key differences between Fibonacci's notation and modern fraction notation. An Egyptian fraction is the sum of distinct unit fractions, such as . ... In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ...

  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 scriptstyle2,frac13, while Fibonacci would write the same number as scriptstylefrac13,2, defining 2 as a quotient and 1/3 as a remainder.
  2. Fibonacci used his remainder arithmetic notation that wrote out addiitonal sequences of quotients and remainders, such as when subtracting two fractions a/b - c/d; with each quotient represented by the smaller fraction, c/d, and each remainder represented by the cross multiplication of the numerator of the larger fraction and the denominator of the smaller fraction, ad, divided by the cross multiplication of the denominator of the larger number and the number of the smaller fraction, bc. That is, a/b - c/d = ad/bc, was written as ad/bc + c/d = a/b. This notation has been reported as a mixed radix notation, using several expanded arithmetic forms, one being a/b, where a > b, as noted by 7/3 = 2 1/3, our modern definition of division. The remainder arithmetic structure was very convenient for dealing with non-decimal 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 frac34 inches could be represented as a composite fraction: scriptstylefrac{3 ,7,,2}{4,,12,,3},5 yards.
  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 scriptstylefrac14,frac13,2 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 scriptstylefrac{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 subtracts 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. Mixed radix numeral systems are more general than the usual ones in that the numerical base may vary from position to position. ... A foot (plural: feet; symbol or abbreviation: ft or, sometimes, ′ – a prime) is a unit of length, in a number of different systems, including English units, Imperial units, and United States customary units. ... A yard (abbreviation: yd) is the name of a unit of length in a number of different systems, including English units, Imperial units, and United States customary units. ... Mid-19th century tool for converting between different standards of the inch An inch is an Imperial and U.S. customary unit of length. ... An Egyptian fraction is the sum of distinct unit fractions, such as . ... A practical number or panarithmic number is a positive integer n such that all preceding positive integers are a sum of distinct divisors of n. ... In mathematics, an Egyptian fraction is a representation of a natural number as a sum of unit fractions, as e. ...


References

  • Sigler, Laurence E. (trans.) (2002). Fibonacci's Liber Abaci. Springer-Verlag. ISBN 0-387-95419-8.
  • L├╝neburg, Heinz (1993). Leonardi Pisani Liber Abbaci oder Lesevergn├╝gen eines Mathematikers. Mannheim: B. I. Wissenschaftsverlag. ISBN 3-411-15461-6.

Links

  • This Book @ WikiSource

  Results from FactBites:
 
The 800th birthday of the book that brought numbers to the west (2831 words)
Liber abaci, the book that gave numbers to the western world, is exactly 800 years old this year.
Liber quadratorum ("The book of squares"), published in 1225, is a book on advanced algebra and number theory, and is Fibonacci's most mathematically impressive work, revealing his substantial mathematical abilities.
The title Liber abaci is sometimes mistranslated as "book of the abacus", but it is more accurately rendered as "book of calculation", since, not only did it say nothing about using an abacus, it described methods that eliminated the need for such a device.
  More results at FactBites »

 
 

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