View source for Fibonacci

{{Short description|Italian mathematician (c. 1170 – c. 1240/50)}}
{{About||the number sequence|Fibonacci sequence|the Prison Break character|Otto Fibonacci}}
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{{Infobox person
| nickname           = {{ubl|Leonardo Fibonacci|Lionardo Fibonacci|Leonardo di Pisa|Leonardo Bigollo Pisano}}
| name               = Fibonacci
| image              = Leonardo Fibonacci.JPG
| caption            = Statue of Fibonacci (1863) by {{ill|Giovanni Paganucci|it}} in the [[Camposanto Monumentale]], Pisa{{efn|Fibonacci's actual appearance is not known.<ref name=statue>{{cite web |url=http://www.epsilones.com/paginas/artes/artes-027-historia-estatua-ingles.html |title=Fibonacci's Statue in Pisa |publisher=Epsilones.com |access-date=2010-08-02 |archive-date=2014-02-22 |archive-url=https://web.archive.org/web/20140222022051/http://www.epsilones.com/paginas/artes/artes-027-historia-estatua-ingles.html |url-status=live }}</ref>}}
| birth_name         = Leonardo Bonacci
| birth_date         = {{c.|1170}}
| birth_place        = [[Pisa]],<ref>{{cite book|last1=Smith|first1=David Eugene|last2=Karpinski|first2=Louis Charles|title=The Hindu–Arabic Numerals|year=1911|page=128|url=https://books.google.com/books?id=wEw6AAAAMAAJ&pg=PA128|location=Boston and London|publisher=Ginn and Company|access-date=2016-03-02|archive-date=2023-03-13|archive-url=https://web.archive.org/web/20230313121951/https://books.google.com/books?id=wEw6AAAAMAAJ&pg=PA128|url-status=live}}.</ref> [[Republic of Pisa]]
| death_date         = {{c.|lk=no}} {{death date and age|1250|||1170||}}
| death_place        = Pisa, Republic of Pisa
| occupation         = [[Mathematician]]
| known_for          = {{Plainlist|
* {{Lang|la|[[Liber Abaci]]}}
* Popularizing the [[Hindu–Arabic numeral system]] in Europe
* [[Congruum]]
* [[Fibonacci number]]s
* [[Greedy algorithm for Egyptian fractions|Fibonacci–Sylvester method]]
* [[Formulas_for_generating_Pythagorean_triples#Fibonacci's_method|Fibonacci method]]}}
| father             = Guglielmo Bonacci
}}

'''Leonardo Bonacci''' ({{c.|1170}} – {{c.|1240–50}}),<ref name=Knott>{{cite web |first=R. |last=MacTutor |url=http://www-history.mcs.st-and.ac.uk/Biographies/Fibonacci.html |title=Leonardo Pisano Fibonacci |publisher=www-history.mcs.st-and.ac.uk |access-date=2018-12-22 |archive-date=2019-10-28 |archive-url=https://web.archive.org/web/20191028210911/http://www-history.mcs.st-and.ac.uk/Biographies/Fibonacci.html |url-status=live }}</ref> commonly known as '''Fibonacci''',{{efn|{{IPAc-en|ˌ|f|ɪ|b|ə|ˈ|n|ɑː|tʃ|i}} {{respell|FIB|ə|NAH|chee}},<ref>{{Cite dictionary |url=http://www.lexico.com/definition/Fibonacci,+Leonardo |archive-url=https://web.archive.org/web/20210512082627/https://www.lexico.com/definition/fibonacci,_leonardo?s=t |url-status=dead |archive-date=2021-05-12 |title=Fibonacci, Leonardo |dictionary=[[Lexico]] UK English Dictionary |publisher=[[Oxford University Press]]}}</ref> {{IPAc-en|USalso|ˌ|f|iː|b|-}} {{respell|FEEB|-}},<ref>[https://www.collinsdictionary.com/dictionary/english/fibonacci-series "Fibonacci series"] {{Webarchive|url=https://web.archive.org/web/20190623162150/https://www.collinsdictionary.com/dictionary/english/fibonacci-series |date=2019-06-23 }} and {{cite web|url=https://www.collinsdictionary.com/dictionary/english/fibonacci-sequence|title=Fibonacci sequence|work=[[Collins English Dictionary]]|publisher=[[HarperCollins]]|access-date=23 June 2019|archive-date=12 June 2012|archive-url=https://web.archive.org/web/20120612123014/https://www.collinsdictionary.com/dictionary/english/fibonacci-sequence|url-status=live}}</ref><ref>{{Cite Merriam-Webster|Fibonacci number|access-date=23 June 2019}}</ref> {{IPA|it|fiboˈnattʃi|lang}}.}}{{efn|Also known as '''Leonardo of Pisa''' ({{langx|it|Leonardo di Pisa}}) or '''Leonardo Bigollo Pisano''' ('Leonardo the Traveller from Pisa').<ref name=livio/>}} was an [[Italians|Italian]] [[mathematician]] from the [[Republic of Pisa]], considered to be "the<!--The cited book uses a lower-case initial "t" in "the", and this is in quotation marks, so lower case should be used.--> most talented Western mathematician of the [[Middle Ages]]".<ref>[[Howard Eves|Eves, Howard]]. ''An Introduction to the History of Mathematics''. Brooks Cole, 1990: {{ISBN|0-03-029558-0}} (6th ed.), p. 261.</ref>

The name he is commonly called, ''Fibonacci'', is first found in a modern source in a 1838 text by the Franco-Italian mathematician [[Guglielmo Libri Carucci dalla Sommaja|Guglielmo Libri]]<ref>{{cite book |first= Keith|last= Devlin|author-link=Keith Devlin|title=Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World| publisher= Princeton University Press|year=2017|page=24}}</ref><ref name="Pask2015">{{cite book|authorlink= Colin Pask| first= Colin |last= Pask|title=Great Calculations: A Surprising Look Behind 50 Scientific Inquiries| url= https://books.google.com/books?id=fvjIBAAAQBAJ&pg=PT35|date=7 July 2015|publisher=Prometheus Books|isbn=978-1-63388-029-0|page=35|access-date=19 January 2020| archive-date=13 March 2023|archive-url=https://web.archive.org/web/20230313121952/https://books.google.com/books?id=fvjIBAAAQBAJ&pg=PT35|url-status= live}}</ref> and is short for {{lang|la|filius Bonacci}} ('son of Bonacci').<ref>{{cite book |first= Keith|last= Devlin| title= The Man of Numbers: Fibonacci's Arithmetic Revolution| publisher= A&C Black| year= 2012 | isbn= | page= 13}}</ref>{{efn|The etymology of ''Bonacci'' is 'good-natured', so the full name means 'son from a good-natured [family]'.<ref name=livio>{{cite book|last=Livio|first=Mario|author-link=Mario Livio|title=The Golden Ratio: The Story of Phi, the World's Most Astonishing Number|url=https://books.google.com/books?id=bUARfgWRH14C|orig-year=2002|edition=First trade paperback|year=2003|publisher=[[Random House|Broadway Books]]|location=New York City|isbn=0-7679-0816-3|pages=92–93|access-date=2018-12-19|archive-date= 2023-03-13|archive-url=https://web.archive.org/web/20230313121951/https://books.google.com/books?id=bUARfgWRH14C|url-status=live}}</ref>}} However, even as early as 1506, Perizolo, a notary of the [[Holy Roman Empire]], mentions him as "Lionardo Fibonacci".{{Sfn|Drozdyuk|Drozdyuk|2010|p=18}}
 
Fibonacci popularized the [[Hindu–Arabic numeral system|Indo–Arabic numeral system]] in the Western world primarily through his composition in 1202 of {{Lang|la|[[Liber Abaci]]}} (''Book of Calculation'')<ref>{{cite web|url=http://www.halexandria.org/dward093.htm|title=Fibonacci Numbers| website= halexandria.org | publisher= |archive-date=2019-10-13 |archive-url= https://web.archive.org/web/20191013054050/http://halexandria.org/dward093.htm| url-status=live |access-date=2015-04-29}}</ref><ref>{{cite encyclopedia| url= https://www.britannica.com/eb/article-4153/Leonardo-Pisano |title= Leonardo Pisano: 'Contributions to number theory'| archiveurl= https://web.archive.org/web/20080617154015/https://www.britannica.com/eb/article-4153/Leonardo-Pisano |archivedate=2008-06-17| encyclopedia= [[Encyclopædia Britannica]]| year= 2006| page= 3| access-date= 18 September 2006}}</ref>  and also introduced Europe to the sequence of [[Fibonacci number]]s, which he used as an example in {{Lang|la|Liber Abaci}}.<ref>{{cite journal| last= Singh| first= Parmanand| title= Acharya Hemachandra and the (so called) Fibonacci Numbers| journal= [[Math (journal)|Math]]| publisher= Ed. Siwan| volume= 20| number= 1| pages= 28–30| year= 1986 |issn= 0047-6269}}</ref>

==Biography==
Fibonacci was born around 1170 to Guglielmo, an Italian merchant and customs official<ref name=livio/> who directed a trading post in [[Béjaïa|Bugia]], modern-day Béjaïa, [[Algeria]].<ref name= Germano>{{cite journal| first= G.| last= Germano| title= New editorial perspectives in Fibonacci's Liber abaci| journal= Reti medievali rivista| volume= 14| number= 2| url= http://www.rmoa.unina.it/2147/1/400-1462-3-PB.pdf | pages= 157–173| archiveurl= https://web.archive.org/web/20210709022418/http://www.rmoa.unina.it/2147/1/400-1462-3-PB.pdf |archivedate=2021-07-09| access-date=}}</ref> Fibonacci travelled with him as a young boy. He was educated in Bugia, where he learned about the [[Hindu–Arabic numeral system]].<ref name= "GlickLivesey2014">{{cite book| first1=Thomas F.| last1= Glick| first2= Steven| last2= Livesey| first3= Faith| last3= Wallis|title=Medieval Science, Technology, and Medicine: An Encyclopedia|url=https://books.google.com/books?id=77y2AgAAQBAJ&pg=PA172|year=2014|publisher=Routledge|isbn=978-1-135-45932-1|page=172|access-date=2018-12-07|archive-date=2023-03-13|archive-url=https://web.archive.org/web/20230313121951/https://books.google.com/books?id=77y2AgAAQBAJ&pg=PA172|url-status= live}}</ref><ref name="Knott" />

Fibonacci travelled around the [[Mediterranean]] coast, meeting with many merchants and learning about their systems of doing arithmetic.{{efn|In the ''Prologus'' of the ''Liber abacci'' he writes: 
{{quote|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."<ref name= Germano />}}}}
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 {{Lang|la|[[Liber Abaci]]}} (''Book of Abacus'' or ''The Book of Calculation''),{{efn|The English edition of the ''Liber abacci'' was published by L.E. Sigler, ''Leonardo Pisano's book of calculation'', New York, Springer-Verlag, 2003.}} which popularized Hindu–Arabic numerals in Europe.<ref name=Knott/>

Fibonacci was a guest of [[Frederick II, Holy Roman Emperor|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){{efn| In ''Flos'': "Incipit flos Leonardi '''bigolli''' pisani..." [Here starts 'the flower' by Leonardo the wanderer of Pisa...]<ref>{{cite book| format= [[MS Word]] document| url= http://www.g4g4.com/MyCD5/SOURCES/SOURCE1.DOC| archiveurl= https://web.archive.org/web/20040722081047/http://www.g4g4.com/MyCD5/SOURCES/SOURCE1.DOC |title= Sources in Recreational Mathematics: An Annotated Bibliography| first= David| last= Singmaster| date= 18 March 2004| archivedate= 2004-07-22| access-date= }}</ref> The basic meanings of ''bigollo'' appear to be "bilingual" or "traveller". A. F. Horadam contends a connotation of ''bigollo'' is "absent-minded",<ref>{{cite journal| series= First footnote | url= http://faculty.evansville.edu/ck6/bstud/fibo.html |title= Eight hundred years young| first= A. F. |last= Horadam| journal= [[The Australian Mathematics Teacher]]| volume= 31 |year= 1975| pages= 123–134| via= faculty.evansville.edu| archiveurl= https://web.archive.org/web/20081219100546/http://faculty.evansville.edu/ck6/bstud/fibo.html |archivedate=2008-12-19 | access-date= 2025-07-15}}</ref> 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.}} 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.<ref>{{cite news | url= https://www.theguardian.com/education/2002/nov/07/research.science | title=A man to count on | first= Keith| last= Devlin | date=7 November 2002 | newspaper= The Guardian | access-date=7 June 2016 | archive-date=17 September 2016 | archive-url= https://web.archive.org/web/20160917004540/https://www.theguardian.com/education/2002/nov/07/research.science | url-status=live }}</ref><ref>{{cite journal| quote= 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.| first= F.| last= Bonaini| title= Memoria unica sincrona di Leonardo Fibonacci, novamente scoperta| journal= Giornale storico degli archivi toscani| volume= 1| number= 4| year= 1857| pages= 239–246}}</ref>

Fibonacci is thought to have died between 1240<ref>{{cite book |title=Fibonacci and Lucas Numbers with Applications|first=Thomas|last=Koshy|publisher=John Wiley & Sons|year=2011|isbn=9781118031315|url=https://books.google.com/books?id=1iDKKceqD2sC&pg=PA3|page=3|access-date=2015-12-12|archive-date=2023-03-13| archive-url=https://web.archive.org/web/20230313121951/https://books.google.com/books?id=1iDKKceqD2sC&pg=PA3|url-status=live}}.</ref> and 1250,<ref>{{cite book |title= Encyclopédia of Mathematics|first=James Stuart|last=Tanton|publisher=Infobase Publishing |year= 2005| isbn= 9780816051243| page=192}}</ref> in Pisa.

==''Liber Abaci''==
{{Main|Liber Abaci}}
[[Image:Liber abbaci magliab f124r.jpg|thumb|A page of Fibonacci's {{Lang|la|[[Liber Abaci]]}} from the [[National Central Library (Florence)|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 {{Lang|la|Liber Abaci}} (1202), Fibonacci introduced the so-called ''modus Indorum'' (method of the [[India]]ns), today known as the [[Hindu–Arabic numeral system]],<ref name="Sigler2002">{{cite book | title = Fibonacci's Liber Abaci | translator-last = Sigler | translator-first = Laurence E.  | publisher = Springer-Verlag | year = 2002 | isbn = 0-387-95419-8}}</ref><ref name= "Grimm 1973">{{cite journal| last= Grimm| first= R. E.
| url= http://www.fq.math.ca/Scanned/11-1/grimm.pdf| title= The Autobiography of Leonardo Pisano| journal= [[Fibonacci Quarterly]]| volume= 11| number= 1| date= February 1973| via= fq.math.ca| pages= 99–104
| doi= 10.1080/00150517.1973.12430873
}}</ref> 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 influence 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.<ref name="Fibonacci: The Man Behind The Math">{{Cite web|title = Fibonacci: The Man Behind The Math |url = https://www.npr.org/2011/07/16/137845241/fibonaccis-numbers-the-man-behind-the-math| website = [[Weekend Edition]] | publisher= NPR |date= 2011-07-16 | access-date = 2015-08-29|archive-date = 2011-07-16|archive-url = https://web.archive.org/web/20110716211651/https://www.npr.org/2011/07/16/137845241/fibonaccis-numbers-the-man-behind-the-math|url-status = live}}</ref><ref name="Man of Numbers excerpt">{{Cite web|title = The Man of Numbers: Fibonacci's Arithmetic Revolution [Excerpt]|url = http://www.scientificamerican.com/article/the-man-of-numbers-fibona/|first = Keith|last = Devlin |website = [[Scientific American]]| archive-date = 2014-06-18|archive-url = https://web.archive.org/web/20140618102122/http://www.scientificamerican.com/article/the-man-of-numbers-fibona/ |url-status = live| access-date = 2015-08-29}}</ref>

The original 1202 manuscript is not known to exist.<ref name="JSG" /> 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]].<ref name="JSG">{{Cite web|title = The Man Behind Modern Math|url = http://www.barrons.com/articles/the-man-behind-modern-math-1440227497|access-date = 2015-08-28|first = John Steele|last = Gordon|author-link = John Steele Gordon|archive-date = 2015-08-23|archive-url = https://web.archive.org/web/20150823132758/http://www.barrons.com/articles/the-man-behind-modern-math-1440227497|url-status = live}}</ref><ref name="Fibonacci: The Man Behind The Math"/><ref name="Man of Numbers excerpt" />

==Fibonacci sequence==
{{Main|Fibonacci sequence}}
{{Lang|la|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 number]]s. Although Fibonacci's {{Lang|la|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.<ref>{{cite journal|first=Pamanand|last=Singh|title=The so-called fibonacci numbers in ancient and medieval India|journal=Historia Mathematica|volume=12|issue=3|year=1985|pages=229–244|doi=10.1016/0315-0860(85)90021-7|doi-access=free}}</ref><ref>{{cite book |title = Toward a Global Science | first = Susantha | last = Goonatilake |publisher = Indiana University Press |year = 1998 |page = [https://archive.org/details/towardglobalscie0000goon/page/126 126] |isbn = 978-0-253-33388-9 |url = https://archive.org/details/towardglobalscie0000goon |url-access = registration |quote = Virahanka Fibonacci. }}</ref><ref>{{cite book |title=The Art of Computer Programming: Generating All Trees – History of Combinatorial Generation; Volume 4 |first=Donald |last=Knuth |publisher=Addison-Wesley |year=2006 |isbn=978-0-321-33570-8 |page=50 |url=https://books.google.com/books?id=56LNfE2QGtYC&q=rhythms&pg=PA50 |access-date=2020-11-11 |archive-date=2023-03-13 |archive-url=https://web.archive.org/web/20230313121953/https://books.google.com/books?id=56LNfE2QGtYC&q=rhythms&pg=PA50 |url-status=live }}</ref><ref>Hall, Rachel W. [http://www.sju.edu/~rhall/mathforpoets.pdf Math for poets and drummers] {{Webarchive|url=https://web.archive.org/web/20120212145748/http://www.sju.edu/~rhall/mathforpoets.pdf |date=2012-02-12 }}. ''Math Horizons'' '''15''' (2008) 10–11.</ref>

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.<ref>{{Cite OEIS|1=A000045|2=Fibonacci Numbers}}</ref><ref>{{cite book |url=https://books.google.com/books?id=gvRFAAAAcAAJ&pg=PA231 |title=Scritti: Il Liber Abbaci |first1=Leonardus |last1=Pisanus |first2=Baldassarre |last2=Boncompagni |date=1 January 1857 |page=231 |publisher=Tip. delle Scienze Fisiche e Matematiche |via=Google Books |access-date=20 December 2018 |archive-date=13 March 2023 |archive-url=https://web.archive.org/web/20230313121953/https://books.google.com/books?id=gvRFAAAAcAAJ&pg=PA231 |url-status=live }}</ref> Fibonacci did not speak about the [[golden ratio]] as the limit of the ratio of consecutive numbers in this sequence.

==Legacy==
[[File:Fibonacci5.jpg|thumb|Modern engraving of Fibonacci, not based on authentic sources.<ref>{{Cite book |last=Smith |first=David Eugene |author-link=David Eugene Smith |url=http://archive.org/details/historyofmathema0001davi_h8a3 |title=History of Mathematics |date= |publisher=[[Ginn and Company]] |year=1951 |volume=I |page=214 |orig-year=1923 |via=[[Internet Archive]]}}</ref> There are no known authentic descriptions or portraits of Fibonacci.{{Sfn|Drozdyuk|Drozdyuk|2010|p=2}}]]
In the 19th century, a statue of Fibonacci was set in Pisa. Today it is located in the western gallery of the [[Camposanto Monumentale|Camposanto]], historical cemetery on the [[Piazza dei Miracoli]].<ref name=statue/><ref>{{cite web| url= https://www.maa.org/external_archive/devlin/Fibonacci.pdf|title=The Man of Numbers: In Search of Leonardo Fibonacci |last= Devlin| first=Keith |year=2010 |website= maa.org| publisher= [[Mathematical Association of America]]| pages=21–28 |archive-url= https://web.archive.org/web/20150907132015/https://www.maa.org/external_archive/devlin/Fibonacci.pdf|url-status=live |archive-date=2015-09-07| access-date=2018-12-21}}</ref>

There are many mathematical [[List of things named after Fibonacci|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==
* {{Lang|la|[[Liber Abaci]]}} (1202), a book on calculations (English translation by Laurence Sigler, 2002)<ref name="Sigler2002" />
* ''Practica Geometriae'' (1220), a compendium of techniques in [[surveying]], the [[measurement]] and partition of [[area]]s and [[volume]]s, 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 equation]]s, dedicated to [[Frederick II, Holy Roman Emperor|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==
* [[Fibonacci numbers in popular culture]]
* [[Republic of Pisa]]
* [[Adelard of Bath]]

==Notes==
{{Notelist}}

==References==
{{Reflist|30em}}

=== Bibliography ===
{{Refbegin}}
* {{Cite book|url=https://nrc-publications.canada.ca/eng/view/fulltext/?id=fbfdddf4-d0e9-45fe-bef6-fe6d3cbecfa0|title=Fibonacci, his numbers and his rabbits| last1= Drozdyuk| first1= Andriy |last2= Drozdyuk| first2= Denys|date=2010|publisher=Choven Pub|isbn=978-0-9866300-1-9|location=Toronto|oclc=813281753|access-date=2020-01-26|archive-date=2020-02-17|archive-url=https://web.archive.org/web/20200217035426/https://nrc-publications.canada.ca/eng/view/fulltext/?id=fbfdddf4-d0e9-45fe-bef6-fe6d3cbecfa0|url-status=live}}
{{Refend}}

==Further reading==
* 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., ''[https://ssrn.com/abstract=461740 Fibonacci and the Financial Revolution]'' (October 23, 2003), [[Yale School of Management]] International Center for Finance Working Paper No. 03–28
* Gavin, J., Schärlig, A., extracts of {{Lang|la|Liber Abaci}} online and analyzed on [http://www.bibnum.education.fr/mathematiques/algebre/leonard-de-pise-un-sujet-anodin-devenu-reference-et-une-contribution ''BibNum''] <small>[click 'à télécharger' for English analysis]</small>

==External links==
{{EB1911 poster|Leonardo of Pisa}}
* {{cite encyclopedia| title= Fibonacci, Leonardo, or Leonardo of Pisa| encyclopedia= Complete Dictionary of Scientific Biography| year= 2008| via= Encyclopedia.com| access-date= April 20, 2015| url= http://www.encyclopedia.com/doc/1G2-2830901418.html}}
*[https://web.archive.org/web/20070715032716/http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=630&bodyId=1002 Fibonacci] at Convergence
*{{MacTutor|id=Fibonacci|title=Leonardo Pisano Fibonacci}}
*[http://lhldigital.lindahall.org/cdm/search/collection/math/searchterm/fibonacci/field/creato/mode/all/conn/and/order/nosort Fibonacci (2 vol., 1857 & 1862) ''Il liber abaci'' and ''Practica Geometriae''] – digital facsimile from the [[Linda Hall Library]]
*[http://www.hs-augsburg.de/~harsch/Chronologia/Lspost13/Fibonacci/fib_intr.html Fibonacci, Liber abbaci] [[Bibliotheca Augustana]]
{{Fibonacci}}
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