Liber Abaci

TheLiber AbaciorLiber Abbaci^[1](Latinfor "The Book of Calculation" ) was a 1202 Latin work onarithmeticby Leonardo of Pisa, posthumously known asFibonacci.It is primarily famous for helping popularizeArabic numeralsin Europe.

Premise

Liber Abaciwas among the first Western books to describe theHindu–Arabic numeral systemand to use symbols resembling modern "Arabic numerals".By addressing the applications of both commercial tradesmen and mathematicians, it promoted the superiority of the system, and the use of these glyphs.^[2]

Although the book's title is sometimes translated as "The Book of the Abacus",Sigler (2002)notes that it is an error to read this as referring to calculating devices called "abacus". Rather, the word "abacus" was used at the time to refer to calculation in any form; the spelling "abbacus" with two "b" s (which is how Leonardo spelled it in the original Latin manuscript) 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 anabacus,and asOre (1948)confirms, for centuries after its publication thealgorismists(followers of the style of calculation demonstrated inLiber Abaci) remained in conflict with the abacists (traditionalists who continued to use the abacus in conjunction with Roman numerals). The historian of mathematicsCarl Boyeremphasizes in hisHistory of Mathematicsthat although "Liber abaci...isnoton 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."^[3]

Summary of sections

The first section introduces the Hindu–Arabic numeral system, including methods for converting between different representation systems. This section also includes the first known description oftrial divisionfor testing whether a number iscompositeand, if so,factoringit.^[4]

The second section presents examples from commerce, such as conversions ofcurrencyand measurements, and calculations ofprofitandinterest.^{[citation needed]}

The third section discusses a number of mathematical problems; for instance, it includes (ch. II.12) theChinese remainder theorem,perfect numbersandMersenne primesas well as formulas forarithmetic seriesand forsquare 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 problem dates back long before Leonardo, its inclusion in his book is why theFibonacci sequenceis named after him today.^{[citation needed]}

The fourth section derives approximations, both numerical and geometrical, ofirrational numberssuch as square roots.^{[citation needed]}

The book also includes proofs inEuclidean geometry.Fibonacci's method of solving algebraic equations shows the influence of the early 10th-century Egyptian mathematicianAbū Kāmil Shujāʿ ibn Aslam.^[5]

Fibonacci's notation for fractions

In readingLiber Abaci,it is helpful to understand Fibonacci's notation for rational numbers, a notation that is intermediate in form between theEgyptian fractionscommonly used until that time and thevulgar fractionsstill in use today.^[6]

Fibonacci's notation differs from modern fraction notation in three key ways:^{[citation needed]}

Modern notation generally writes a fraction to the right of the whole number to which it is added, for instance $2\,{\tfrac {1}{3}}$ for 7/3. Fibonacci instead would write the same fraction to the left, i.e., ${\tfrac {1}{3}}\,2$ .^{[citation needed]}
Fibonacci used acomposite fractionnotation 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, ${\tfrac {b\,\,a}{d\,\,c}}={\tfrac {a}{c}}+{\tfrac {b}{cd}}$ ,and ${\tfrac {c\,\,b\,\,a}{f\,\,e\,\,d}}={\tfrac {a}{d}}+{\tfrac {b}{de}}+{\tfrac {c}{def}}$ .The notation was read from right to left. For example, 29/30 could be written as ${\tfrac {1\,\,2\,\,4}{2\,\,3\,\,5}}$ ,representing the value ${\tfrac {4}{5}}+{\tfrac {2}{3\times 5}}+{\tfrac {1}{2\times 3\times 5}}$ .This can be viewed as a form ofmixed radixnotation, and was very convenient for dealing with traditional systems of weights, measures, and currency. For instance, for units of length, afootis 1/3 of ayard,and aninchis 1/12 of a foot, so a quantity of 5 yards, 2 feet, and $7{\tfrac {3}{4}}$ inches could be represented as a composite fraction: ${\tfrac {3\ \,7\,\,2}{4\,\,12\,\,3}}\,5$ 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. Sigler also points out an instance where Fibonacci uses composite fractions in which all denominators are 10, prefiguring modern decimal notation for fractions.^{[citation needed]}
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 ${\tfrac {1}{4}}\,{\tfrac {1}{3}}\,2$ would represent the number that would now more commonly be written as the mixed number $2\,{\tfrac {7}{12}}$ ,or simply the improper fraction ${\tfrac {31}{12}}$ .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.^{[citation needed]}

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 thegreedy algorithm for Egyptian fractions,also known as the Fibonacci–Sylvester expansion.^{[citation needed]}

Modus Indorum

In theLiber Abaci,Fibonacci says the following introducing the affirmativeModus Indorum(the method of the Indians), today known asHindu–Arabic numeral systemor base-10 positional notation. It also introduced digits that greatly resembled the modernArabic numerals.^{[citation needed]}

As my father was a public official away from our homeland in theBugiacustomshouse established for the Pisan merchants who frequently gathered there, he had me in my youth brought to him, looking to find for me a useful and comfortable future; there he wanted me to be in the study of mathematics and to be taught for some days. There from a marvelous instruction in the art of the nine Indian figures, the introduction and knowledge of the art pleased me so much above all else, and I learnt from them, whoever was learned in it, from nearby Egypt, Syria, Greece, Sicily and Provence, and their various methods, to which locations of business I travelled considerably afterwards for much study, and I learnt from the assembled disputations. But this, on the whole, the algorithm and even the Pythagorean arcs, I still reckoned almost an error compared to the Indian method. Therefore strictly embracing the Indian method, and attentive to the study of it, from mine own sense adding some, and some more still from the subtle Euclidean geometric art, applying the sum that I was able to perceive to this book, I worked to put it together in xv distinct chapters, showing certain proof for almost everything that I put in, so that further, this method perfected above the rest, this science is instructed to the eager, and to the Italian people above all others, who up to now are found without a minimum. If, by chance, something less or more proper or necessary I omitted, your indulgence for me is entreated, as there is no one who is without fault, and in all things is altogether circumspect.^[7]

The nine Indian figures are:
9 8 7 6 5 4 3 2 1
With these nine figures, and with the sign 0 which the Arabs call zephir any number whatsoever is written...^[8]

In other words, in his book he advocated the use of the digits 0–9, and ofplace value.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", takingmany more centuriesto 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.^{[citation needed]}

Textual history

The first appearance of the manuscript was in 1202. No copies of this version are known. A revised version ofLiber Abaci,dedicated toMichael Scot,appeared in 1227 CE.^[9]^[10]There are at least nineteen manuscripts extant containing parts of this text.^[11]There are three complete versions of this manuscript from the thirteenth and fourteenth centuries.^[12]There are a further nine incomplete copies known between the thirteenth and fifteenth centuries, and there may be more not yet identified.^[12]^[11]

There were no known printed version ofLiber Abaciuntil Boncompagni's Italian translation of 1857.^[11]The first complete English translation was Sigler's text of 2002.^[11]

References

Citations

^"Fibonacci's Liber Abaci (Book of Calculation)".The University of Utah.13 December 2009.Retrieved2018-11-27.
^Devlin, Keith (2012).The Man of Numbers: Fibonacci's Arithmetic Revolution.Walker Books.ISBN 978-0802779083.
^Boyer, Carl (1968).A History of Mathematics(PDF).New York, London, Sydney: John Wiley & Sons. p. 280.
^Mollin, Richard A. (2002). "A brief history of factoring and primality testing B. C. (before computers)".Mathematics Magazine.75(1): 18–29.doi:10.2307/3219180.JSTOR 3219180.MR 2107288.See also Sigler, pp. 65–66.
^O'Connor, John J.; Robertson, Edmund F. (1999)."Abu Kamil Shuja ibn Aslam".MacTutor History of Mathematics archive.
^Moyon, Marc; Spiesser, Maryvonne (3 June 2015). "L'arithmétique des fractions dans l'œuvre de Fibonacci: fondements & usages".Archive for History of Exact Sciences.69(4): 391–427.doi:10.1007/s00407-015-0155-y.
^Devlin, Keith (2019).Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World.Princeton, N.J.: Princeton University Press. pp. 92–93 (quoted on).ISBN 9780691192307.OCLC 975288613.Retrieved10 July2024.
^Sigler 2002;seeGrimm 1973for another translation
^Scott, T. C.; Marketos, P.,"Michael Scot",in O'Connor, John J.;Robertson, Edmund F.(eds.),MacTutor History of Mathematics Archive,University of St Andrews
^Scott, T. C.; Marketos, P. (March 2014),On the Origin of the Fibonacci Sequence(PDF),MacTutor History of Mathematics archive,University of St Andrews
^^a ^b ^c ^dGermano, Giuseppe (2013). "New Editorial Perspectives on Fibonacci's Liber Abaci".Reti Medievali Rivista.doi:10.6092/1593-2214/400(inactive 1 November 2024).{{cite journal}}:CS1 maint: DOI inactive as of November 2024 (link)
^^a ^bDictionary of Scientific Biography(PDF).

General and cited references

Grimm, R. E. (1973),"The Autobiography of Leonardo Pisano"(PDF),The Fibonacci Quarterly,11(1): 99–104.
Ore, Øystein(1948),Number Theory and Its History,McGraw Hill.Dover version also available, 1988,ISBN 978-0-486-65620-5.
Sigler, L. E. (trans.) (2002),Fibonacci's Liber Abaci,Springer-Verlag,ISBN 0-387-95419-8.

External links

Pisano, Leonardo (1202),Incipit liber Abbaci compositus to Lionardo filio Bonaccii Pisano in year Mccij [Manuscript],Museo Galileo.

[1] "Fibonacci's Liber Abaci (Book of Calculation)".The University of Utah.13 December 2009.Retrieved2018-11-27.

[2] Devlin, Keith (2012).The Man of Numbers: Fibonacci's Arithmetic Revolution.Walker Books.ISBN 978-0802779083.

[3] Boyer, Carl (1968).A History of Mathematics(PDF).New York, London, Sydney: John Wiley & Sons. p. 280.

[4] Mollin, Richard A. (2002). "A brief history of factoring and primality testing B. C. (before computers)".Mathematics Magazine.75(1): 18–29.doi:10.2307/3219180.JSTOR 3219180.MR 2107288.See also Sigler, pp. 65–66.

[5] O'Connor, John J.; Robertson, Edmund F. (1999)."Abu Kamil Shuja ibn Aslam".MacTutor History of Mathematics archive.

[6] Moyon, Marc; Spiesser, Maryvonne (3 June 2015). "L'arithmétique des fractions dans l'œuvre de Fibonacci: fondements & usages".Archive for History of Exact Sciences.69(4): 391–427.doi:10.1007/s00407-015-0155-y.

[7] Devlin, Keith (2019).Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World.Princeton, N.J.: Princeton University Press. pp. 92–93 (quoted on).ISBN 9780691192307.OCLC 975288613.Retrieved10 July2024.

[8] Sigler 2002;seeGrimm 1973for another translation

[9] Scott, T. C.; Marketos, P.,"Michael Scot",in O'Connor, John J.;Robertson, Edmund F.(eds.),MacTutor History of Mathematics Archive,University of St Andrews

[10] Scott, T. C.; Marketos, P. (March 2014),On the Origin of the Fibonacci Sequence(PDF),MacTutor History of Mathematics archive,University of St Andrews

[:0-11] Germano, Giuseppe (2013). "New Editorial Perspectives on Fibonacci's Liber Abaci".Reti Medievali Rivista.doi:10.6092/1593-2214/400(inactive 1 November 2024).{{cite journal}}:CS1 maint: DOI inactive as of November 2024 (link)

[:1-12] Dictionary of Scientific Biography(PDF).

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