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Fibonacci

February 22, 2023

For the number sequence, see Fibonacci number. For the Prison Break character, see Otto Fibonacci.

Fibonacci (/ˌfɪbəˈnɑːtʃi/;[3] also US: /ˌfiːb-/,[4][5] Italian: [fiboˈnattʃi]; c. 1170 – c. 1240–50),[6] also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'[7]), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".[8]

Fibonacci
Statue of Fibonacci (1863) by Giovanni Paganucci in the Camposanto di Pisa[a]
Bornc. 1170
Pisa,[2] Republic of Pisa
Diedc. 1250(1250-00-00) (aged 79–80)
Pisa, Republic of Pisa
Other namesLeonardo Fibonacci, Leonardo Bonacci, Leonardo Pisano
OccupationMathematician
Known for
ParentGuglielmo "Bonacci" (father)

The name he is commonly called, Fibonacci, was made up in 1838 by the Franco-Italian historian Guillaume Libri[9][10] and is short for filius Bonacci ('son of Bonacci').[11][b] However, even earlier, in 1506, a notary of the Holy Roman Empire, Perizolo mentions Leonardo as "Lionardo Fibonacci".[12]

Fibonacci popularized the Indo–Arabic numeral system in the Western world primarily through his composition in 1202 of Liber Abaci (Book of Calculation).[13][14] He also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci.[15]

Biography

Fibonacci was born around 1170 to Guglielmo, an Italian merchant and customs official.[7] Guglielmo directed a trading post in Bugia (Béjaïa) in modern-day Algeria), the capital of the Hammadid empire.[16] Fibonacci travelled with him as a young boy, and it was in Bugia (Algeria) where he was educated that he learned about the Hindu–Arabic numeral system.[17][6]

Fibonacci travelled around the Mediterranean coast, meeting with many merchants and learning about their systems of doing arithmetic.[18] He soon realised the many advantages of the Hindu-Arabic system, which, unlike the Roman numerals used at the time, allowed easy calculation using a place-value system. In 1202, he completed the Liber Abaci (Book of Abacus or The Book of Calculation),[19] which popularized Hindu–Arabic numerals in Europe.[6]

Fibonacci was a guest of Emperor Frederick II, who enjoyed mathematics and science. A member of Frederick II's court, John of Palermo, posed several questions based on Arab mathematical works for Fibonacci to solve. In 1240, the Republic of Pisa honored Fibonacci (referred to as Leonardo Bigollo)[20] by granting him a salary in a decree that recognized him for the services that he had given to the city as an advisor on matters of accounting and instruction to citizens.[21][22]

Fibonacci is thought to have died between 1240[23] and 1250,[24] in Pisa.

Liber Abaci

Main article: Liber Abaci
 
A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled with Latin numbers and Roman numerals and the value in Hindu-Arabic numerals.

In the Liber Abaci (1202), Fibonacci introduced the so-called modus Indorum (method of the Indians), today known as the Hindu–Arabic numeral system,[25][26] with ten digits including a zero and positional notation. The book showed the practical use and value of this by applying the numerals to commercial bookkeeping, converting weights and measures, calculation of interest, money-changing, and other applications. The book was well-received throughout educated Europe and had a profound impact on European thought. Replacing Roman numerals, its ancient Egyptian multiplication method, and using an abacus for calculations, was an advance in making business calculations easier and faster, which assisted the growth of banking and accounting in Europe.[27][28]

The original 1202 manuscript is not known to exist.[29] In a 1228 copy of the manuscript, the first section introduces the numeral system and compares it with others, such as Roman numerals, and methods to convert numbers to it. The second section explains uses in business, for example converting different currencies, and calculating profit and interest, which were important to the growing banking industry. The book also discusses irrational numbers and prime numbers.[29][27][28]

Fibonacci sequence

Main article: Fibonacci number

Liber Abaci posed and solved a problem involving the growth of a population of rabbits based on idealized assumptions. The solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers. Although Fibonacci's Liber Abaci contains the earliest known description of the sequence outside of India, the sequence had been described by Indian mathematicians as early as the sixth century.[30][31][32][33]

In the Fibonacci sequence, each number is the sum of the previous two numbers. Fibonacci omitted the "0" and first "1" included today and began the sequence with 1, 2, 3, ... . He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to the next place, the value 377.[34][35] Fibonacci did not speak about the golden ratio as the limit of the ratio of consecutive numbers in this sequence.

Legacy

In the 19th century, a statue of Fibonacci was set in Pisa. Today it is located in the western gallery of the Camposanto, historical cemetery on the Piazza dei Miracoli.[1][36]

There are many mathematical concepts named after Fibonacci because of a connection to the Fibonacci numbers. Examples include the Brahmagupta–Fibonacci identity, the Fibonacci search technique, and the Pisano period. Beyond mathematics, namesakes of Fibonacci include the asteroid 6765 Fibonacci and the art rock band The Fibonaccis.

Works

See also

Notes

  1. ^ Fibonacci's actual appearance is not known.[1]
  2. ^ The etymology of Bonacci is "good-natured", so the full name means "son from a good-natured [family]".[7]

References

  1. ^ a b "Fibonacci's Statue in Pisa". Epsilones.com. Retrieved 2010-08-02.
  2. ^ Smith, David Eugene; Karpinski, Louis Charles (1911), The Hindu–Arabic Numerals, Boston and London: Ginn and Company, p. 128.
  3. ^ . Lexico UK English Dictionary. Oxford University Press. Archived from the original on 2021-05-12.
  4. ^ "Fibonacci series" and "Fibonacci sequence". Collins English Dictionary. HarperCollins. Retrieved 23 June 2019.
  5. ^ "Fibonacci number". Merriam-Webster Dictionary. Retrieved 23 June 2019.
  6. ^ a b c MacTutor, R. "Leonardo Pisano Fibonacci". www-history.mcs.st-and.ac.uk. Retrieved 2018-12-22.
  7. ^ a b c Livio, Mario (2003) [2002]. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.). New York City: Broadway Books. pp. 92–93. ISBN 0-7679-0816-3.
  8. ^ Eves, Howard. An Introduction to the History of Mathematics. Brooks Cole, 1990: ISBN 0-03-029558-0 (6th ed.), p. 261.
  9. ^ Devlin, Keith (2017). Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World. Princeton University Press. p. 24.
  10. ^ Colin Pask (7 July 2015). Great Calculations: A Surprising Look Behind 50 Scientific Inquiries. Prometheus Books. p. 35. ISBN 978-1-63388-029-0.
  11. ^ Keith Devlin, The Man of Numbers: Fibonacci's Arithmetic Revolution, A&C Black, 2012 p. 13.
  12. ^ Drozdyuk, Andriy; Drozdyuk, Denys (2010). Fibonacci, his numbers and his rabbits. Toronto: Choven Pub. p. 18. ISBN 978-0-9866300-1-9. OCLC 813281753.
  13. ^ "Fibonacci Numbers". www.halexandria.org.
  14. ^ Leonardo Pisano: "Contributions to number theory". Encyclopædia Britannica Online, 2006. p. 3. Retrieved 18 September 2006.
  15. ^ Singh, Parmanand. "Acharya Hemachandra and the (so called) Fibonacci Numbers". Math. Ed. Siwan, 20(1):28–30, 1986. ISSN 0047-6269]
  16. ^ G. Germano, New editorial perspectives in Fibonacci's Liber abaci, «Reti medievali rivista» 14, 2, pp. 157–173.
  17. ^ Thomas F. Glick; Steven Livesey; Faith Wallis (2014). Medieval Science, Technology, and Medicine: An Encyclopedia. Routledge. p. 172. ISBN 978-1-135-45932-1.
  18. ^ In the Prologus of the Liber abacci he said: "Having been introduced there to this art with an amazing method of teaching by means of the nine figures of the Indians, I loved the knowledge of such an art to such an extent above all other arts and so much did I devote myself to it with my intellect, that I learned with very earnest application and through the technique of contradiction anything to be studied concerning it and its various methods used in Egypt, in Syria, in Greece, in Sicily, and in Provence, places I have later visited for the purpose of commerce" (translated by G. Germano, New editorial perspectives in Fibonacci's Liber abaci, «Reti medievali rivista» 14, 2, pp. 157–173.
  19. ^ The English edition of the Liber abacci was published by L.E. Sigler, Leonardo Pisano’s book of calculation, New York, Springer-Verlag, 2003
  20. ^ See the incipit of Flos: "Incipit flos Leonardi bigolli pisani..." (quoted in the MS Word document by David Singmaster, 18 March 2004 – emphasis added), in English: "Here starts 'the flower' by Leonardo the wanderer of Pisa..."
    The basic meanings of "bigollo" appear to be "bilingual" or "traveller". A. F. Horadam contends a connotation of "bigollo" is "absent-minded" (see first footnote of "Eight hundred years young"), which is also one of the connotations of the English word "wandering". The translation "the wanderer" in the quote above tries to combine the various connotations of the word "bigollo" in a single English word.
  21. ^ Keith Devlin (7 November 2002). "A man to count on". The Guardian. Retrieved 7 June 2016.
  22. ^ «Considerantes nostre civitatis et civium honorem atque profectum, qui eis, tam per doctrinam quam per sedula obsequia discreti et sapientis viri magistri Leonardi Bigolli, in abbacandis estimationibus et rationibus civitatis eiusque officialium et aliis quoties expedit, conferuntur; ut eidem Leonardo, merito dilectionis et gratie, atque scientie sue prerogativa, in recompensationem laboris sui quem substinet in audiendis et consolidandis estimationibus et rationibus supradictis, a Comuni et camerariis publicis, de Comuni et pro Comuni, mercede sive salario suo, annis singulis, libre xx denariorum et amisceria consueta dari debeant (ipseque pisano Comuni et eius officialibus in abbacatione de cetero more solito serviat), presenti constitutione firmamus». F. Bonaini, Memoria unica sincrona di Leonardo Fibonacci, novamente scoperta, «Giornale storico degli archivi toscani» 1, 4, 1857, pp. 239–246.
  23. ^ Koshy, Thomas (2011), Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, p. 3, ISBN 9781118031315.
  24. ^ Tanton, James Stuart (2005), Encyclopédia of Mathematics, Infobase Publishing, p. 192, ISBN 9780816051243.
  25. ^ a b Fibonacci's Liber Abaci, translated by Sigler, Laurence E., Springer-Verlag, 2002, ISBN 0-387-95419-8
  26. ^ Grimm 1973
  27. ^ a b "Fibonacci: The Man Behind The Math". NPR.org. Retrieved 2015-08-29.
  28. ^ a b Devlin, Keith. "The Man of Numbers: Fibonacci's Arithmetic Revolution [Excerpt]". Scientific American. Retrieved 2015-08-29.
  29. ^ a b Gordon, John Steele. "The Man Behind Modern Math". Retrieved 2015-08-28.
  30. ^ Singh, Pamanand (1985). "The so-called fibonacci numbers in ancient and medieval India". Historia Mathematica. 12 (3): 229–244. doi:10.1016/0315-0860(85)90021-7.
  31. ^ Goonatilake, Susantha (1998). Toward a Global Science. Indiana University Press. p. 126. ISBN 978-0-253-33388-9. Virahanka Fibonacci.
  32. ^ Knuth, Donald (2006). The Art of Computer Programming: Generating All Trees – History of Combinatorial Generation; Volume 4. Addison-Wesley. p. 50. ISBN 978-0-321-33570-8.
  33. ^ Hall, Rachel W. Math for poets and drummers 2012-02-12 at the Wayback Machine. Math Horizons 15 (2008) 10–11.
  34. ^ Sloane, N. J. A. (ed.). "Sequence A000045 (Fibonacci Numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  35. ^ Pisanus, Leonardus; Boncompagni, Baldassarre (1 January 1857). Scritti: Il Liber Abbaci. Tip. delle Scienze Fisiche e Matematiche. p. 231 – via Google Books.
  36. ^ Devlin, Keith (2010). "The Man of Numbers: In Search of Leonardo Fibonacci" (PDF). Mathematical Association of America. pp. 21–28.

Further reading

  • Devlin, Keith (2012). The Man of Numbers: Fibonacci's Arithmetic Revolution. Walker Books. ISBN 978-0802779083.
  • Goetzmann, William N. and Rouwenhorst, K.Geert (2005). The Origins of Value: The Financial Innovations That Created Modern Capital Markets. Oxford University Press Inc., US, ISBN 0-19-517571-9.
  • Goetzmann, William N., Fibonacci and the Financial Revolution (October 23, 2003), Yale School of Management International Center for Finance Working Paper No. 03–28
  • Grimm, R. E., "The Autobiography of Leonardo Pisano", Fibonacci Quarterly, Vol. 11, No. 1, February 1973, pp. 99–104.
  • Horadam, A. F. "Eight hundred years young," The Australian Mathematics Teacher 31 (1975) 123–134.
  • Gavin, J., Schärlig, A., extracts of Liber Abaci online and analyzed on BibNum [click 'à télécharger' for English analysis]

External links

 
Wikisource has the text of the 1911 Encyclopædia Britannica article "Leonardo of Pisa".

February 22, 2023
fibonacci, number, sequence, number, prison, break, character, otto, ɑː, also, italian, fiboˈnattʃi, 1170, 1240, also, known, leonardo, bonacci, leonardo, pisa, leonardo, bigollo, pisano, leonardo, traveller, from, pisa, italian, mathematician, from, republic,. For the number sequence see Fibonacci number For the Prison Break character see Otto Fibonacci Fibonacci ˌ f ɪ b e ˈ n ɑː tʃ i 3 also US ˌ f iː b 4 5 Italian fiboˈnattʃi c 1170 c 1240 50 6 also known as Leonardo Bonacci Leonardo of Pisa or Leonardo Bigollo Pisano Leonardo the Traveller from Pisa 7 was an Italian mathematician from the Republic of Pisa considered to be the most talented Western mathematician of the Middle Ages 8 FibonacciStatue of Fibonacci 1863 by Giovanni Paganucci in the Camposanto di Pisa a Bornc 1170 Pisa 2 Republic of PisaDiedc 1250 1250 00 00 aged 79 80 Pisa Republic of PisaOther namesLeonardo Fibonacci Leonardo Bonacci Leonardo PisanoOccupationMathematicianKnown forLiber Abaci Popularizing the Hindu Arabic numeral system in Europe Congruum Fibonacci numbers Fibonacci Sylvester method Fibonacci methodParentGuglielmo Bonacci father The name he is commonly called Fibonacci was made up in 1838 by the Franco Italian historian Guillaume Libri 9 10 and is short for filius Bonacci son of Bonacci 11 b However even earlier in 1506 a notary of the Holy Roman Empire Perizolo mentions Leonardo as Lionardo Fibonacci 12 Fibonacci popularized the Indo Arabic numeral system in the Western world primarily through his composition in 1202 of Liber Abaci Book of Calculation 13 14 He also introduced Europe to the sequence of Fibonacci numbers which he used as an example in Liber Abaci 15 Contents 1 Biography 2 Liber Abaci 3 Fibonacci sequence 4 Legacy 5 Works 6 See also 7 Notes 8 References 9 Further reading 10 External linksBiography EditFibonacci was born around 1170 to Guglielmo an Italian merchant and customs official 7 Guglielmo directed a trading post in Bugia Bejaia in modern day Algeria the capital of the Hammadid empire 16 Fibonacci travelled with him as a young boy and it was in Bugia Algeria where he was educated that he learned about the Hindu Arabic numeral system 17 6 Fibonacci travelled around the Mediterranean coast meeting with many merchants and learning about their systems of doing arithmetic 18 He soon realised the many advantages of the Hindu Arabic system which unlike the Roman numerals used at the time allowed easy calculation using a place value system In 1202 he completed the Liber Abaci Book of Abacus or The Book of Calculation 19 which popularized Hindu Arabic numerals in Europe 6 Fibonacci was a guest of Emperor Frederick II who enjoyed mathematics and science A member of Frederick II s court John of Palermo posed several questions based on Arab mathematical works for Fibonacci to solve In 1240 the Republic of Pisa honored Fibonacci referred to as Leonardo Bigollo 20 by granting him a salary in a decree that recognized him for the services that he had given to the city as an advisor on matters of accounting and instruction to citizens 21 22 Fibonacci is thought to have died between 1240 23 and 1250 24 in Pisa Liber Abaci EditMain article Liber Abaci A page of Fibonacci s Liber Abaci from the Biblioteca Nazionale di Firenze showing in box on right the Fibonacci sequence with the position in the sequence labeled with Latin numbers and Roman numerals and the value in Hindu Arabic numerals In the Liber Abaci 1202 Fibonacci introduced the so called modus Indorum method of the Indians today known as the Hindu Arabic numeral system 25 26 with ten digits including a zero and positional notation The book showed the practical use and value of this by applying the numerals to commercial bookkeeping converting weights and measures calculation of interest money changing and other applications The book was well received throughout educated Europe and had a profound impact on European thought Replacing Roman numerals its ancient Egyptian multiplication method and using an abacus for calculations was an advance in making business calculations easier and faster which assisted the growth of banking and accounting in Europe 27 28 The original 1202 manuscript is not known to exist 29 In a 1228 copy of the manuscript the first section introduces the numeral system and compares it with others such as Roman numerals and methods to convert numbers to it The second section explains uses in business for example converting different currencies and calculating profit and interest which were important to the growing banking industry The book also discusses irrational numbers and prime numbers 29 27 28 Fibonacci sequence EditMain article Fibonacci number Liber Abaci posed and solved a problem involving the growth of a population of rabbits based on idealized assumptions The solution generation by generation was a sequence of numbers later known as Fibonacci numbers Although Fibonacci s Liber Abaci contains the earliest known description of the sequence outside of India the sequence had been described by Indian mathematicians as early as the sixth century 30 31 32 33 In the Fibonacci sequence each number is the sum of the previous two numbers Fibonacci omitted the 0 and first 1 included today and began the sequence with 1 2 3 He carried the calculation up to the thirteenth place the value 233 though another manuscript carries it to the next place the value 377 34 35 Fibonacci did not speak about the golden ratio as the limit of the ratio of consecutive numbers in this sequence Legacy EditIn the 19th century a statue of Fibonacci was set in Pisa Today it is located in the western gallery of the Camposanto historical cemetery on the Piazza dei Miracoli 1 36 There are many mathematical concepts named after Fibonacci because of a connection to the Fibonacci numbers Examples include the Brahmagupta Fibonacci identity the Fibonacci search technique and the Pisano period Beyond mathematics namesakes of Fibonacci include the asteroid 6765 Fibonacci and the art rock band The Fibonaccis Works EditLiber Abaci 1202 a book on calculations English translation by Laurence Sigler 2002 25 Practica Geometriae 1220 a compendium of techniques in surveying the measurement and partition of areas and volumes and other topics in practical geometry English translation by Barnabas Hughes Springer 2008 Flos 1225 solutions to problems posed by Johannes of Palermo Liber quadratorum The Book of Squares on Diophantine equations dedicated to Emperor Frederick II See in particular congruum and the Brahmagupta Fibonacci identity Di minor guisa on commercial arithmetic lost Commentary on Book X of Euclid s Elements lost See also EditFibonacci numbers in popular culture Republic of Pisa Adelard of BathNotes Edit Fibonacci s actual appearance is not known 1 The etymology of Bonacci is good natured so the full name means son from a good natured family 7 References Edit a b Fibonacci s Statue in Pisa Epsilones com Retrieved 2010 08 02 Smith David Eugene Karpinski Louis Charles 1911 The Hindu Arabic Numerals Boston and London Ginn and Company p 128 Fibonacci Leonardo Lexico UK English Dictionary Oxford University Press Archived from the original on 2021 05 12 Fibonacci series and Fibonacci sequence Collins English Dictionary HarperCollins Retrieved 23 June 2019 Fibonacci number Merriam Webster Dictionary Retrieved 23 June 2019 a b c MacTutor R Leonardo Pisano Fibonacci www history mcs st and ac uk Retrieved 2018 12 22 a b c Livio Mario 2003 2002 The Golden Ratio The Story of Phi the World s Most Astonishing Number First trade paperback ed New York City Broadway Books pp 92 93 ISBN 0 7679 0816 3 Eves Howard An Introduction to the History of Mathematics Brooks Cole 1990 ISBN 0 03 029558 0 6th ed p 261 Devlin Keith 2017 Finding Fibonacci The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World Princeton University Press p 24 Colin Pask 7 July 2015 Great Calculations A Surprising Look Behind 50 Scientific Inquiries Prometheus Books p 35 ISBN 978 1 63388 029 0 Keith Devlin The Man of Numbers Fibonacci s Arithmetic Revolution A amp C Black 2012 p 13 Drozdyuk Andriy Drozdyuk Denys 2010 Fibonacci his numbers and his rabbits Toronto Choven Pub p 18 ISBN 978 0 9866300 1 9 OCLC 813281753 Fibonacci Numbers www halexandria org Leonardo Pisano Contributions to number theory Encyclopaedia Britannica Online 2006 p 3 Retrieved 18 September 2006 Singh Parmanand Acharya Hemachandra and the so called Fibonacci Numbers Math Ed Siwan 20 1 28 30 1986 ISSN 0047 6269 G Germano New editorial perspectives in Fibonacci s Liber abaci Reti medievali rivista 14 2 pp 157 173 Thomas F Glick Steven Livesey Faith Wallis 2014 Medieval Science Technology and Medicine An Encyclopedia Routledge p 172 ISBN 978 1 135 45932 1 In the Prologus of the Liber abacci he said Having been introduced there to this art with an amazing method of teaching by means of the nine figures of the Indians I loved the knowledge of such an art to such an extent above all other arts and so much did I devote myself to it with my intellect that I learned with very earnest application and through the technique of contradiction anything to be studied concerning it and its various methods used in Egypt in Syria in Greece in Sicily and in Provence places I have later visited for the purpose of commerce translated by G Germano New editorial perspectives in Fibonacci s Liber abaci Reti medievali rivista 14 2 pp 157 173 The English edition of the Liber abacci was published by L E Sigler Leonardo Pisano s book of calculation New York Springer Verlag 2003 See the incipit of Flos Incipit flos Leonardi bigolli pisani quoted in the MS Word document Sources in Recreational Mathematics An Annotated Bibliography by David Singmaster 18 March 2004 emphasis added in English Here starts the flower by Leonardo the wanderer of Pisa The basic meanings of bigollo appear to be bilingual or traveller A F Horadam contends a connotation of bigollo is absent minded see first footnote of Eight hundred years young which is also one of the connotations of the English word wandering The translation the wanderer in the quote above tries to combine the various connotations of the word bigollo in a single English word Keith Devlin 7 November 2002 A man to count on The Guardian Retrieved 7 June 2016 Considerantes nostre civitatis et civium honorem atque profectum qui eis tam per doctrinam quam per sedula obsequia discreti et sapientis viri magistri Leonardi Bigolli in abbacandis estimationibus et rationibus civitatis eiusque officialium et aliis quoties expedit conferuntur ut eidem Leonardo merito dilectionis et gratie atque scientie sue prerogativa in recompensationem laboris sui quem substinet in audiendis et consolidandis estimationibus et rationibus supradictis a Comuni et camerariis publicis de Comuni et pro Comuni mercede sive salario suo annis singulis libre xx denariorum et amisceria consueta dari debeant ipseque pisano Comuni et eius officialibus in abbacatione de cetero more solito serviat presenti constitutione firmamus F Bonaini Memoria unica sincrona di Leonardo Fibonacci novamente scoperta Giornale storico degli archivi toscani 1 4 1857 pp 239 246 Koshy Thomas 2011 Fibonacci and Lucas Numbers with Applications John Wiley amp Sons p 3 ISBN 9781118031315 Tanton James Stuart 2005 Encyclopedia of Mathematics Infobase Publishing p 192 ISBN 9780816051243 a b Fibonacci s Liber Abaci translated by Sigler Laurence E Springer Verlag 2002 ISBN 0 387 95419 8 Grimm 1973 a b Fibonacci The Man Behind The Math NPR org Retrieved 2015 08 29 a b Devlin Keith The Man of Numbers Fibonacci s Arithmetic Revolution Excerpt Scientific American Retrieved 2015 08 29 a b Gordon John Steele The Man Behind Modern Math Retrieved 2015 08 28 Singh Pamanand 1985 The so called fibonacci numbers in ancient and medieval India Historia Mathematica 12 3 229 244 doi 10 1016 0315 0860 85 90021 7 Goonatilake Susantha 1998 Toward a Global Science Indiana University Press p 126 ISBN 978 0 253 33388 9 Virahanka Fibonacci Knuth Donald 2006 The Art of Computer Programming Generating All Trees History of Combinatorial Generation Volume 4 Addison Wesley p 50 ISBN 978 0 321 33570 8 Hall Rachel W Math for poets and drummers Archived 2012 02 12 at the Wayback Machine Math Horizons 15 2008 10 11 Sloane N J A ed Sequence A000045 Fibonacci Numbers The On Line Encyclopedia of Integer Sequences OEIS Foundation Pisanus Leonardus Boncompagni Baldassarre 1 January 1857 Scritti Il Liber Abbaci Tip delle Scienze Fisiche e Matematiche p 231 via Google Books Devlin Keith 2010 The Man of Numbers In Search of Leonardo Fibonacci PDF Mathematical Association of America pp 21 28 Further reading EditDevlin Keith 2012 The Man of Numbers Fibonacci s Arithmetic Revolution Walker Books ISBN 978 0802779083 Goetzmann William N and Rouwenhorst K Geert 2005 The Origins of Value The Financial Innovations That Created Modern Capital Markets Oxford University Press Inc US ISBN 0 19 517571 9 Goetzmann William N Fibonacci and the Financial Revolution October 23 2003 Yale School of Management International Center for Finance Working Paper No 03 28 Grimm R E The Autobiography of Leonardo Pisano Fibonacci Quarterly Vol 11 No 1 February 1973 pp 99 104 Horadam A F Eight hundred years young The Australian Mathematics Teacher 31 1975 123 134 Gavin J Scharlig A extracts of Liber Abaci online and analyzed on BibNum click a telecharger for English analysis External links Edit Wikisource has the text of the 1911 Encyclopaedia Britannica article Leonardo of Pisa Fibonacci Leonardo or Leonardo of Pisa Complete Dictionary of Scientific Biography 2008 Encyclopedia com April 20 2015 1 Fibonacci at Convergence O Connor John J Robertson Edmund F Leonardo Pisano Fibonacci MacTutor History of Mathematics archive University of St Andrews Fibonacci 2 vol 1857 amp 1862 Il liber abaci and Practica Geometriae digital facsimile from the Linda Hall Library Fibonacci Liber abbaci Bibliotheca Augustana Retrieved from https en wikipedia org w index php title Fibonacci amp oldid 1140445963, wikipedia, wiki, book, books, library,

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