Liber Abaci
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Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
for "The Book of Calculation") was a 1202 Latin work on
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
by Leonardo of Pisa, posthumously known as
Fibonacci Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci ...
. It is primarily famous for introducing both base-10 positional notation and the symbols known as
Arabic numerals The ten Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) are the most commonly used symbols for writing numbers. The term often also implies a positional notation number with a decimal base, in particular when contrasted with Roman numera ...
in Europe.


Premise

was among the first Western books to describe the
Hindu–Arabic numeral system The Hindu–Arabic numeral system (also known as the Indo-Arabic numeral system, Hindu numeral system, and Arabic numeral system) is a positional notation, positional Decimal, base-ten numeral system for representing integers; its extension t ...
and to use symbols resembling modern "
Arabic numerals The ten Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) are the most commonly used symbols for writing numbers. The term often also implies a positional notation number with a decimal base, in particular when contrasted with Roman numera ...
". By addressing the applications of both commercial tradesmen and mathematicians, it promoted the superiority of the system and the use of these glyphs. Although the book's title is sometimes translated as "The Book of the Abacus", notes that it is an error to read this as referring to the
abacus An abacus ( abaci or abacuses), also called a counting frame, is a hand-operated calculating tool which was used from ancient times in the ancient Near East, Europe, China, and Russia, until the adoption of the Hindu–Arabic numeral system. A ...
as a calculating device. Rather, the word "abacus" was used at the time to refer to calculation in any form; the spelling "abbacus" with two "b"s was, and still is in Italy, used to refer to calculation using Hindu-Arabic numerals, which can avoid confusion. The book describes methods of doing calculations without aid of an abacus, and as confirms, for centuries after its publication the
algorism Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist. This positional notation system ...
ists (followers of the style of calculation demonstrated in ) remained in conflict with the abacists (traditionalists who continued to use the abacus in conjunction with Roman numerals). The historian of mathematics Carl Boyer emphasizes in his ''History of Mathematics'' that although "''Liber abaci''...is ''not'' on the abacus" '' per se'', nevertheless "...it is a very thorough treatise on algebraic methods and problems in which the use of the Hindu-Arabic numerals is strongly advocated."


Summary of sections

The first section introduces the Hindu–Arabic numeral system, including its arithmetic and methods for converting between different representation systems. This section also includes the first known description of trial division for testing whether a number is composite and, if so, factoring it. The second section presents examples from commerce, such as conversions of
currency A currency is a standardization of money in any form, in use or circulation as a medium of exchange, for example banknotes and coins. A more general definition is that a currency is a ''system of money'' in common use within a specific envi ...
and measurements, and calculations of
profit Profit may refer to: Business and law * Profit (accounting), the difference between the purchase price and the costs of bringing to market * Profit (economics), normal profit and economic profit * Profit (real property), a nonpossessory inter ...
and
interest In finance and economics, interest is payment from a debtor or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is distinct f ...
. The third section discusses a number of mathematical problems; for instance, it includes the
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
,
perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfec ...
s and
Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...
s as well as formulas for
arithmetic series An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...
and for square pyramidal numbers. Another example in this chapter involves the growth of a population of rabbits, where the solution requires generating a numerical sequence. Although the resulting
Fibonacci sequence In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...
dates back long before Leonardo, its inclusion in his book is why the sequence is named after him today. The fourth section derives approximations, both numerical and geometrical, of
irrational number In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
s such as square roots. The book also includes proofs in
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
. Fibonacci's method of solving algebraic equations shows the influence of the early 10th-century Egyptian mathematician Abū Kāmil Shujāʿ ibn Aslam.


Fibonacci's notation for fractions

In reading , 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. Fibonacci's notation differs from modern fraction notation in three key ways: # Modern notation generally writes a fraction to the right of the whole number to which it is added, for instance 2\,\tfrac13 for 7/3. Fibonacci instead would write the same fraction to the left, i.e., \tfrac13\,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 right of it. That is, \tfrac = \tfrac + \tfrac, and \tfrac = \tfrac + \tfrac + \tfrac. The notation was read from right to left. For example, 29/30 could be written as \tfrac, representing the value \tfrac45+\tfrac2+\tfrac1. This can be viewed as a form of mixed radix notation and was very convenient for dealing with traditional systems of weights, measures, and currency. For instance, for units of length, a
foot The foot (: feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a limb which bears weight and allows locomotion. In many animals with feet, the foot is an organ at the terminal part of the leg made up o ...
is 1/3 of a
yard The yard (symbol: yd) is an English units, English unit of length in both the British imperial units, imperial and US United States customary units, customary systems of measurement equalling 3 foot (unit), feet or 36 inches. Sinc ...
, and an
inch The inch (symbol: in or prime (symbol), ) is a Units of measurement, unit of length in the imperial units, British Imperial and the United States customary units, United States customary System of measurement, systems of measurement. It is eq ...
is 1/12 of a foot, so a quantity of 5 yards, 2 feet, and 7 \tfrac34 inches could be represented as a composite fraction: \tfrac\,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. Sigler also points out an instance where Fibonacci uses composite fractions in which all denominators are 10, prefiguring modern decimal notation for fractions. # 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 \tfrac14\,\tfrac13\,2 would represent the number that would now more commonly be written as the mixed number 2\,\tfrac, or simply the improper fraction \tfrac. 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 an improper fraction to an Egyptian fraction, including the
greedy algorithm for Egyptian fractions In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a representation of an irreducible fraction as a s ...
, also known as the Fibonacci–Sylvester expansion.


''Modus Indorum''

In the , Fibonacci says the following introducing the affirmative ''Modus Indorum'' (the method of the Indians), today known as
Hindu–Arabic numeral system The Hindu–Arabic numeral system (also known as the Indo-Arabic numeral system, Hindu numeral system, and Arabic numeral system) is a positional notation, positional Decimal, base-ten numeral system for representing integers; its extension t ...
or base-10 positional notation. It also introduced digits that greatly resembled the modern
Arabic numerals The ten Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) are the most commonly used symbols for writing numbers. The term often also implies a positional notation number with a decimal base, in particular when contrasted with Roman numera ...
. In other words, in his book he advocated the use of the digits 0–9, and of
place value Place may refer to: Geography * Place (United States Census Bureau), defined as any concentration of population ** Census-designated place, a populated area lacking its own municipal government * "Place", a type of street or road name ** O ...
. Until this time Europe used Roman numerals, making modern mathematics almost impossible. The book thus made an important contribution to the spread of decimal numerals. The spread of the Hindu-Arabic system, however, as Ore writes, was "long-drawn-out", taking many more centuries to spread widely, and did not become complete until the later part of the 16th century, accelerating dramatically only in the 1500s with the advent of printing.


Textual history

The first appearance of the manuscript was in 1202. No copies of this version are known. A revised version of dedicated to Michael Scot, appeared in 1227 CE. There are at least nineteen manuscripts extant containing parts of this text. There are three complete versions of this manuscript from the thirteenth and fourteenth centuries. There are a further nine incomplete copies known between the thirteenth and fifteenth centuries, and there may be more not yet identified. There were no known printed versions of until Boncompagni's Italian translation of 1857. The first complete English translation was Sigler's text of 2002.


See also

* '' The Book of Squares''


References


External links

* . {{Fibonacci 1202 books 13th century in science 13th-century books in Latin Mathematics books