Fibonacci: Malsamoj inter versioj
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Linio 11:
Fibonacci veturis etende ĉirkaŭ la marbordo de la [[Mediteraneo]], laŭ kiu li lernis kun multaj komercantoj kaj lernis pri ties sistemoj fari aritmetikon. Li tuj konstatis la multajn avantaĝojn de la hind-araba sistemo. En 1202 li kompletigis la ''[[Liber Abaci]]'' (''Libro de Abako'' aŭ ''Libro de Kalkulado'') kiu popularigis hind-arabajn numeralojn en Eŭropo.<ref name=Knott/>
Fibonacci gastiĝis ĉe [[Frederiko la 2-a (Sankta Romia Imperio)|imperiestro Frederiko la 2-a]], kiu ĝuis el matematiko kaj scienco. En 1240 la [[Pizo (Italio)|Respubliko Pisa]] honorigis Fibonacci (referencita kiel Leonardo Bigollo)<ref>Vidu la komencon de ''Flos'': "Incipit flos Leonardi '''bigolli''' pisani..." (citita en la dokumento de [[MS Word]] nome [http://www.g4g4.com/MyCD5/SOURCES/SOURCE1.DOC ''Sources in Recreational Mathematics: An Annotated Bibliography''] de David Singmaster, 18a de Marto 2004 – emfazo aldonita), en angla: "Here starts 'the flower' by Leonardo the wanderer of Pisa..."<br>
La dato de la morto de Fibonacci ne estas konata, sed oni ĉirkaŭkalkulis inter 1240<ref>{{citation|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}}.</ref> kaj 1250,<ref>{{citation|title=Encyclopédia of Mathematics|first=James Stuart|last=Tanton|publisher=Infobase Publishing|year=2005|isbn=9780816051243|page=192|url=https://books.google.com/books?id=MfKKMSuthacC&pg=PA192}}.</ref> plej verŝajne en Pisa.
==''Liber Abaci'' (1202)==
{{redaktata}}
[[Image:Liber abbaci magliab f124r.jpg|thumb|A page of Fibonacci's ''[[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 in Roman numerals and the value in Hindu-Arabic numerals.]]
{{Ĉefartikolo|Liber Abaci}}
In the ''Liber Abaci'' (1202), Fibonacci introduced the so-called ''modus Indorum'' (method of the Indians), today known as Hindu-Arabic numerals.<ref name="Sigler2002">{{citation | title = Fibonacci's Liber Abaci | last = Sigler | first = Laurence E. (trans.) | publisher = Springer-Verlag | year = 2002 | isbn = 0-387-95419-8}}</ref><ref>Grimm 1973</ref> The book advocated numeration with the digits 0–9 and [[place value]]. The book showed the practical use and value of the new Arabic [[numeral system]] 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. No copies of the 1202 edition are known to exist.<ref name=":1" />
The 1228 edition, first section, introduces the Arabic numeral system and compares the system with other systems, such as Roman numerals, and methods to convert the other numeral systems into Arabic numerals. Replacing the Roman numeral system, its [[ancient Egyptian multiplication]] method, and using an [[abacus]] for calculations, with an Arabic numeral system, was an advance in making business calculations easier and faster, which led to 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 = http://www.npr.org/2011/07/16/137845241/fibonaccis-numbers-the-man-behind-the-math|website = NPR.org|accessdate = 2015-08-29}}</ref><ref name=":0">{{Cite web|title = The Man of Numbers: Fibonacci's Arithmetic Revolution [Excerpt]|url = http://www.scientificamerican.com/article/the-man-of-numbers-fibona/|accessdate = 2015-08-29|first = Keith|last = Devlin}}</ref>
The second section explains the uses of Arabic numerals 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=":1">{{Cite web|title = The Man Behind Modern Math|url = http://www.barrons.com/articles/the-man-behind-modern-math-1440227497|accessdate = 2015-08-28|first = John Steele|last = Gordon}}</ref><ref name="Fibonacci: The Man Behind The Math"/><ref name=":0" />
==Fibonacci-sekvenco==
{{Ĉefartikolo|Fibonaĉi-nombro}}
{| style="float:right;"
|Tiel la unuaj fibonaĉi-nombroj estas:
|-
|
{| class="wikitable" style="float:right;"
! n !! F(n)
|-
| 1 || 1
|-
| 2 || 1
|-
| 3 || 2
|-
| 4 || 3
|-
| 5 || 5
|-
| 6 || 8
|-
| 7 || 13
|-
| 8 || 21
|-
| 9 || 34
|-
| 10 || 55
|-
| 11 || 89
|-
| 12 || 144
|-
| 13 || 233
|-
| 14 || 377
|-
| 15 || 610
|-
| 16 || 987
|-
| 17 || 1597
|-
| 18 || 2584
|-
| 19 || 4181
|-
| 20 || 6765
|-
| 21 || 10946
|-
| 22 || 17711
|-
| 23 || 28657
|-
| 24 || 46368
|-
| 25 || 75025
|-
| 26 || 121393
|-
| 27 || 196418
|-
| 28 || 317811
|-
| 29 || 514229
|-
| 30 || 832040
|-
| 31 || 1346269
|-
| 32 || 2178309
|-
| 33 || 3524578
|-
| 34 || 5702887
|-
| 35 || 9227465
|-
| 36 || 14930352
|-
| 37 || 24157817
|-
| 38 || 39088169
|-
| 39 || 63245986
|-
| 40 || 102334155
|-
| 41 || 165580141
|-
| 42 || 267914296
|-
| 43 || 433494437
|-
| 44 || 701408733
|-
| 45 || 1134903170
|-
| 46 || 1836311903
|-
| 47 || 2971215073
|-
| 48 || 4807526976
|-
| 49 || 7778742049
|-
| 50 || 12586269025
|-
| 51 || 20365011074
|-
| 52 || 32951280099
|-
| 53 || 53316291173
|-
| 54 || 86267571272
|-
| 55 || 139583862445
|-
| 56 || 225851433717
|-
| 57 || 365435296162
|-
| 58 || 591286729879
|-
| 59 || 956722026041
|-
| 60 || 1548008755920
|-
| 61 || 2504730781961
|-
| 62 || 4052739537881
|-
| 63 || 6557470319842
|-
| 64 || 10610209857723
|-
| 65 || 17167680177565
|-
| 66 || 27777890035288
|-
| 67 || 44945570212853
|-
| 68 || 72723460248141
|-
| 69 || 117669030460994
|-
| 70 || 190392490709135
|-
| 71 || 308061521170129
|-
| 72 || 498454011879264
|-
| 73 || 806515533049393
|-
| 74 || 1304969544928657
|-
| 75 || 2111485077978050
|-
| 76 || 3416454622906707
|-
| 77 || 5527939700884757
|-
| 78 || 8944394323791464
|-
| 79 || 14472334024676221
|-
| 80 || 23416728348467685
|-
| 81 || 37889062373143906
|-
| 82 || 61305790721611591
|-
| 83 || 99194853094755497
|-
| 84 || 160500643816367088
|-
| 85 || 259695496911122585
|-
| 86 || 420196140727489673
|-
| 87 || 679891637638612258
|-
| 88 || 1100087778366101931
|-
| 89 || 1779979416004714189
|-
| 90 || 2880067194370816120
|}
|}
La '''fibonaĉi-nombroj''', tiel nomataj pro la itala [[matematikisto]] [[Fibonacci]], konsistigas [[progresio]]n kies [[termo]]j difinitas per:
:<math>F(n + 2) = F(n) + F(n + 1)</math>,
kaj la ekkondiĉoj:
: ''F''(0) = 0
: ''F''(1) = 1
alinome:
:<math>
F_n := F(n):=
\begin{cases}
0 & \mbox{dla } n = 0; \\
1 & \mbox{dla } n = 1; \\
F(n-1)+F(n-2) & \mbox{dla } n > 1. \\
\end{cases}
</math>
== Propraĵoj ==
:<math>F_n = \sum_{k=1}^n{n-k \choose k-1}</math>
:<math>\sum_{k=1}^n F_k = F_{n+2}-1</math>
:<math>\sum_{i=0}^n iF_i = nF_{n+2} - F_{n+3} + 2</math>
{|class=wikitable
| [[Dosiero:FibonacciBlocks.svg|180px]] <br> Kaheligo de [[ortangulo]] per [[kvadrato (geometrio)|kvadratoj]] kies longoj de lateroj estas fibonaĉi-nombroj
|}
:<math>\sum_{k=1}^n F_k^2 = F_{n+1}F_n</math> (kaheligo de [[ortangulo]] montrita sur bildo)
:<math>\sum_{k=1}^n F_k^3 = (F_{3n+2}+ (-1)^{n+1}6 F_{n-1}+5 )/10</math>
:<math>F_{2n} = F_{n+1}^2 - F_{n-1}^2</math>
:<math>F_{2n-1} = {F_n}^2 + {F_{n-1}}^2</math>
:<math>F_{n+1}F_{n-1} - F_n^2 = (-1)^n</math>
:<math>F_{n+1}F_{m} + F_n F_{m-1} = F_{m+n}\, </math>
* [[Teoremo de Carmichael]] pri [[prima faktoro|primaj faktoroj]].
==Notoj==
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