{"id":652,"date":"2016-11-01T09:30:27","date_gmt":"2016-11-01T09:30:27","guid":{"rendered":"http:\/\/ragipsahin.com\/?p=652"},"modified":"2016-10-22T09:59:53","modified_gmt":"2016-10-22T09:59:53","slug":"pisali-leonardo-fibonacci","status":"publish","type":"post","link":"https:\/\/www.ragipsahin.com.tr\/?p=652","title":{"rendered":"Pisal\u0131 Leonardo Fibonacci"},"content":{"rendered":"

\u00dcnl\u00fc matematik\u00e7i Fibonacci, google’de hi\u00e7 bu kadar \u00e7ok aranmam\u0131\u015ft\u0131r herhalde. 8.s\u0131n\u0131f matematik dersinde \u00f6zel say\u0131 dizileri konusunda ge\u00e7n Fibonacci say\u0131 dizisinin, proje ve performans \u00f6devi olarak \u00f6\u011frencilere verilmesi, T\u00fcrkiye’de Fibonacci’nin \u00fcn\u00fcne \u00fcn katm\u0131\u015f gibi g\u00f6z\u00fck\u00fcyor.<\/p>\n

Sizler i\u00e7in Fibonacci ve dizisi hakk\u0131nda 4yaz\u0131 derledik…<\/p>\n

1.yaz\u0131<\/span><\/h1>\n

Fibonacci<\/strong>,ya da daha do\u011frusu Leonardo da Pisa, \u0130.S. 1175 de \u0130talya\u2019n\u0131n Pisa kentinde do\u011fdu. Babas\u0131 Kuzey Afrika\u2019da g\u00fcmr\u00fck memurlu\u011fu da yapm\u0131\u015f bir t\u00fcccard\u0131. Cezayir, M\u0131s\u0131r ,Suriye ,Yunanistan ve Sicilya\u2019ya i\u015f yolculuklar\u0131 yapt\u0131. 1200 y\u0131l\u0131nda Pisa\u2019ya geri d\u00f6nd\u00fc ve yolculuklar\u0131 s\u0131ras\u0131nda edindi\u011fi bilgilerini kullanarak Avrupa\u2019ya onlu say\u0131 sistemini tan\u0131tt\u0131\u011f\u0131 \u201cLiber Abaci\u201d yi (Hesap Kitab\u0131) yazd\u0131. B\u00f6l\u00fcm 1\u2019in ilk k\u0131sm\u0131 \u015f\u00f6yle ba\u015flamaktayd\u0131 :\u00a0Bunlar Hintlilerin dokuz rakam\u0131d\u0131r : 9 8 7 6 5 4 3 2 1. Bunlar ve araplar\u0131n \u2018zephirum\u2019 dedikleri 0 i\u015fareti ile birlikte her say\u0131 yaz\u0131labilmektedir.<\/p>\n

K\u00f6k Bulma<\/span><\/h1>\n

Fibonacci olduk\u00e7a dikkate de\u011fer bir hesaplama yetene\u011fine sahipti. A\u015fa\u011f\u0131da verilen k\u00fcbik e\u015fitli\u011fin pozitif \u00e7\u00f6z\u00fcm\u00fcn\u00fc bulabilmi\u015fti :\u00a0\u00a0As\u0131l dikkate de\u011fer olan ise t\u00fcm \u00e7al\u0131\u015fmalar\u0131n\u0131 60 taban\u0131n\u0131 kullanan Babil Sistemi ile yapmas\u0131 idi. \u00c7\u00f6z\u00fcm\u00fc 1,22,7,42,33,4,40 olarak vermi\u015fti. Bu ise\u00a0\u00a0\u00a0ifadesine e\u015fitti. Bu sonucu nas\u0131l elde etti\u011fi bilinmemektedir, fakat b\u00f6yle kesin bir sonucu ba\u015fka matematik\u00e7iler bulmadan 300 y\u0131l \u00f6nce bulmu\u015ftu.\u00a0Fibonacci Dizisi<\/strong>\u00a0Fibonacci g\u00fcn\u00fcm\u00fczde daha \u00e7ok \u2018Liber Abaci\u2019 adl\u0131 eserinde tan\u0131tt\u0131\u011f\u0131 ve sonradan onun ad\u0131yla an\u0131lmaya ba\u015flanan Fibonacci Say\u0131lar\u0131 ile tan\u0131nmaktad\u0131r. Dizi 1 ve 1 ile ba\u015flar. Ard\u0131ndan \u015fu basit kural uygulan\u0131r :\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u201cBir sonraki say\u0131y\u0131 bulmak i\u00e7in son iki say\u0131y\u0131 topla.\u201d\u00a01, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,…\u00a0Bu dizinin neden ortaya \u00e7\u0131kt\u0131\u011f\u0131 merak edilebilir. Fibonacci\u2019nin zaman\u0131nda matematik yar\u0131\u015fmalar\u0131 olduk\u00e7a yayg\u0131nd\u0131. Fibonacci 1225 y\u0131l\u0131nda Kral 2. Frederick\u2019 in d\u00fczenledi\u011fi bir turnuvaya kat\u0131lm\u0131\u015ft\u0131.\u00a0\u0130\u015fte bu tarz bir yar\u0131\u015fmada a\u015fa\u011f\u0131daki problem ortaya \u00e7\u0131kt\u0131 :\u00a0Tek bir \u00e7ift tav\u015fan ile ba\u015flayarak her ay \u00fcretken \u00e7ift, yeni bir tav\u015fan \u00e7ifti olu\u015fturursa ve yeni tav\u015fanlar bir ay sonra \u00fcretken oluyorsa, \u2018n\u2019 ay sonra toplam ka\u00e7 tav\u015fan olur?\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (Sorunun cevab\u0131 yaz\u0131n\u0131n sonundad\u0131r.)\u00a0Alt\u0131n B\u00f6l\u00fcm<\/strong>\u00a0Fibonacci serisi ile yak\u0131ndan ilgili \u00f6zel bir de\u011fer de \u2018alt\u0131n b\u00f6l\u00fcm\u2019 d\u00fcr. Bu de\u011fer serideki ard\u0131\u015f\u0131k terimlerin birbirine oran\u0131 ile elde edilir :\u00a0\u00a0\u00a0Fibonacci serisi ile yak\u0131ndan ilgili \u00f6zel bir de\u011fer de \u2018alt\u0131n b\u00f6l\u00fcm\u2019 d\u00fcr. Bu de\u011fer serideki ard\u0131\u015f\u0131k terimlerin birbirine oran\u0131 ile elde edilir :\u00a0\u00a0\u015eekilde g\u00f6r\u00fcld\u00fc\u011f\u00fc gibi bu oran bir de\u011fere yak\u0131nsamaktad\u0131r. Asl\u0131nda bu de\u011fer ikinci dereceden bir e\u015fitli\u011fin (a\u015fa\u011f\u0131da g\u00f6steriliyor) pozitif k\u00f6k\u00fcd\u00fcr ve alt\u0131n b\u00f6l\u00fcm, alt\u0131n oran ya da bazen alt\u0131n anlam olarak adland\u0131r\u0131lmaktad\u0131r.\u00a0\u00a0\u00a0Alt\u0131n b\u00f6l\u00fcm Yunan harfi \u2018phi\u2019 ile sembolize edilir. Asl\u0131nda, Plato zaman\u0131n\u0131n (\u0130.\u00d6. 400) matematik\u00e7ileri onu anlam ta\u015f\u0131yan bir de\u011fer olarak kabul ediyorlard\u0131 ve yunanl\u0131 mimarlar 1\/phi oran\u0131n\u0131 tasar\u0131mlar\u0131n\u0131n ayr\u0131lmaz bir par\u00e7as\u0131 olarak kullan\u0131yorlard\u0131. Bu mimari yap\u0131lardan en \u00fcnl\u00fcs\u00fc Atina\u2019daki Parthenon\u2019dur.\u00a0\u00a0Parthenon, Atina\u00a0Tav\u015fan Probleminin \u00c7\u00f6z\u00fcm\u00fc :<\/span><\/strong>\u00a0Farz edelim ki \u2018n\u2019 ay sonra Xn \u00e7ift tav\u015fan olsun. \u2018n+1\u2019.aydaki tav\u015fan \u00e7ifti say\u0131s\u0131 (problemde tav\u015fanlar\u0131n \u00f6lmedi\u011fi kabul edilmektedir) yeni do\u011fan tav\u015fan \u00e7iftleri ve Xn nin toplam\u0131 kadard\u0131r. Fakat yeni \u00e7iftler, en az bir ayl\u0131k \u00e7iftler taraf\u0131ndan do\u011frulaca\u011f\u0131ndan toplam tav\u015fan \u00e7ifti say\u0131s\u0131 xn<\/sub><\/em>+1<\/sub> = xn<\/sub><\/em> + xn<\/sub><\/em>-1<\/sub>olarak hesaplan\u0131r. Bu ifade ise asl\u0131nda Fibonacci say\u0131lar\u0131n\u0131n \u00fcretilmesinde kullan\u0131lan kurald\u0131r.<\/p>\n

<\/h1>\n

2.yaz\u0131<\/span><\/h1>\n
\"\"<\/div>\n
\n

8. S\u0131n\u0131f \u00f6\u011frencileri, Matematik dersi performans g\u00f6revi i\u00e7in orta \u00e7a\u011f\u0131n en yetenekli matematik\u00e7isi olarak kabul edilen \u0130talyan matematik\u00e7i Leonardo Fibonacci\u2019nin hayat\u0131n\u0131, matemati\u011fe katk\u0131lar\u0131n\u0131, Fibonacci dizisi ile birlikte, do\u011fa ve matematik aras\u0131ndaki ili\u015fkiyi incelediler.<\/p>\n

\u00d6zellikle do\u011fada rastlanan Fibonacci say\u0131lar\u0131, bitki yapraklar\u0131, bitki tohumlar\u0131, \u00e7i\u00e7ek yapraklar\u0131 ve kozalaklarda s\u0131k\u00e7a kar\u015f\u0131m\u0131za \u00e7\u0131kmaktad\u0131r. Bu say\u0131lara Pascal veya Binom \u00fc\u00e7geninde, Mimar Sinan\u2019\u0131n eserlerinde, Da Vinci\u2019nin resimlerinde de rastlanmaktad\u0131r. Da Vinci\u2019nin yap\u0131t\u0131nda, Mona Lisa\u2019n\u0131n ba\u015f\u0131 etraf\u0131na bir d\u00f6rtgen \u00e7izildi\u011finde, sa\u011flanan d\u00f6rtgen alt\u0131n orana uymakta olup resmin boyutlar\u0131 da alt\u0131n oran\u0131 vermektedir. T\u00fct\u00fcn bitkisi yapraklar\u0131n\u0131n dizili\u015findeki Fibonacci dizisi ise, bitkinin g\u00fcne\u015ften ve havadaki karbondioksitten optimum d\u00fczeyde faydalanmas\u0131n\u0131 sa\u011flayarak, y\u00fcksek d\u00fczeyde fotosentez yapmas\u0131na olanak verir. Bu \u00f6zellik e\u011frelti otunda da g\u00f6zlemlenmektedir. Ay\u00e7i\u00e7e\u011finin \u00fcst\u00fcndeki spiral \u015feklinde dizilmi\u015f tohumlar\u0131 saat y\u00f6n\u00fcnde ve tersi y\u00f6nde sayd\u0131\u011f\u0131m\u0131zda ard\u0131\u015f\u0131k iki Fibonacci say\u0131s\u0131na ula\u015f\u0131r\u0131z. Papatya \u00e7i\u00e7e\u011finde de ayn\u0131 Fibonacci dizisi g\u00f6zlenmektedir. Kubbe ve kule tasar\u0131mlar\u0131 i\u00e7eren ve genellikle eski \u00e7a\u011flara ait mimari eserlerde de Fibonacci dizisi g\u00f6zlemlenir. Mimar Sinan\u2019\u0131n yapm\u0131\u015f oldu\u011fu Selimiye ve S\u00fcleymaniye camilerinin, kubbe ve minarelerinde alt\u0131n oran g\u00f6zlenmektedir.<\/p>\n

<\/h1>\n

3.yaz\u0131<\/span><\/h1>\n

\u0130talyan matematik\u00e7i Fibonacci yazd\u0131\u011f\u0131 matematik kitaplar\u0131ndan birinde tav\u015fan \u00e7iftli\u011fi olan bir arkada\u015f\u0131yla ilgili oldu\u011funu iddia etti\u011fi bir problem sorar. Bu probleme g\u00f6re arkada\u015f\u0131n\u0131n \u00e7iftli\u011findeki tav\u015fanlar do\u011fduklar\u0131 ilk iki ay yavru yapmazlar. \u00dc\u00e7\u00fcnc\u00fc aydan itibaren her \u00e7ift her ay bir \u00e7ift yavru yapar. Buna g\u00f6re Fibonacci’nin arkada\u015f\u0131 bir \u00e7ift tav\u015fanla ba\u015flarsa ka\u00e7 ay sonra ka\u00e7 \u00e7ift tav\u015fan\u0131 olur?<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u0130lk ay yeni do\u011fmu\u015f bir \u00e7ift tav\u015fan\u0131m\u0131z olsun. Matematik problemlerinde bu yavrular\u0131n anas\u0131z babas\u0131z nas\u0131l b\u00fcy\u00fct\u00fclecekleri konusuna pek girilmez. \u0130kinci ayda bu tav\u015fanlar hen\u00fcz yavrulamad\u0131klar\u0131 i\u00e7in hala bir \u00e7ift tav\u015fan\u0131m\u0131z var. \u00dc\u00e7\u00fcnc\u00fc ay bunlar bir \u00e7ift yavru verecek ve iki \u00e7ift tav\u015fan\u0131m\u0131z olacak. Yeni do\u011fan \u00e7ift d\u00f6rd\u00fcnc\u00fc ay do\u011furmayacak, oysa ana babalar\u0131 yeniden bir \u00e7ift yavru yapacak ve toplam \u00fc\u00e7 \u00e7ift tav\u015fan\u0131m\u0131z olacak. Bu \u015fekilde devam edersek pek bir yere varamayaca\u011f\u0131z galiba. D\u00fc\u015f\u00fcnsenize 100.aya kadar hesab\u0131 b\u00f6yle g\u00f6t\u00fcrmemiz m\u00fcmk\u00fcn m\u00fc? \u00d6rne\u011fin 100.ayda ka\u00e7 tav\u015fan\u0131m\u0131z olaca\u011f\u0131n\u0131 do\u011frudan hesaplamaya \u00e7al\u0131\u015fal\u0131m. 99.ayda ka\u00e7 tav\u015fan\u0131m\u0131z varsa onlar\u0131n hepsi 100. ayda da olacak. Bunlar\u0131n bir k\u0131sm\u0131 yavrulayacak. Yavrulayacak olanlar\u0131n en az iki ayl\u0131k olmas\u0131 gerekti\u011fine g\u00f6re 100. ayda yavrulayacak olanlar 98.ayda sahip oldu\u011fumuz tav\u015fanlar\u0131n hepsi olacak. Demek ki 100. aydaki tav-\u015fan say\u0131s\u0131n\u0131 bulmak i\u00e7in 98.aydaki tav\u015fan say\u0131s\u0131yla 99.aydaki tav\u015fan say\u0131s\u0131n\u0131 toplamak gerekiyor.<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Bu hesaba baz\u0131 itirazlar y\u00fckselebilir. Biz sadece 100. aydaki say\u0131y\u0131 merak ediyorduk. \u015eimdi onu bulmak i\u00e7in hem 98. hem de 99. aylardaki say\u0131y\u0131 bulmam\u0131z gerekecek. Bu hesab\u0131 100. ayda de\u011fil de \u00fc\u00e7\u00fcnc\u00fc aydan itibaren yapal\u0131m. Birinci ve ikinci aylarda birer \u00e7ift tav\u015fan\u0131m\u0131z vard\u0131. Demek ki \u00fc\u00e7\u00fcnc\u00fc ay\u00a0iki \u00e7ift tav\u015fan\u0131m\u0131z olacak. \u0130kinci aydaki bir \u00e7ift ile \u00fc\u00e7\u00fcnc\u00fc aydaki iki \u00e7ifti toplarsak d\u00f6rd\u00fcnc\u00fc ay \u00fc\u00e7 \u00e7ifti bulaca\u011f\u0131z.<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Buna g\u00f6re Fibonacci dizisi \u015f\u00f6yle tan\u0131mlan\u0131r:<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 F1<\/sub> = 1<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 F2<\/sub> = 1<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Fn<\/sub> = Fn-1<\/sub> + Fn-2<\/sub> ,\u00a0 n>2<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Buna g\u00f6re Fibonacci say\u0131lar\u0131n\u0131n ilk birka\u00e7 tanesi \u015f\u00f6yle s\u0131ralan\u0131r:<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946…<\/strong><\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Bu arada unutmadan 100.ayda ka\u00e7 \u00e7ift tav\u015fan\u0131 olacak sorusunun cevab\u0131 da \u015f\u00f6yle:<\/p>\n

F100<\/sub> = 354 224 848 179 261 915 075<\/p>\n

<\/h1>\n

4.yaz\u0131<\/span><\/h1>\n

Pisal\u0131 Leonardo Fibonacci R\u00f6nesans \u00f6ncesi Avrupa’n\u0131n en \u00f6nde gelen Matematik\u00e7isidir. Fibonacci i\u00e7in, “Matematik’i Araplar’dan al\u0131p, Avrupa’ya aktaran ki\u015fi” denilebilir.<\/p>\n

Fibonacci’nin ya\u015fam\u0131 hakk\u0131nda matematik yaz\u0131lar\u0131 d\u0131\u015f\u0131nda pek az \u015fey biliniyor. \u0130lk ve en iyi bilinen kitab\u0131 Liber Abaci<\/span><\/strong>‘nin yaz\u0131ld\u0131\u011f\u0131 1202 tarihine bak\u0131l\u0131rsa, 1170 dolay\u0131nda do\u011fmu\u015f olabilece\u011fi san\u0131l\u0131yor. Bu y\u00f6nde pek kan\u0131t olmamakla birlikte \u0130talya’n\u0131n Pisa kentinde do\u011fmu\u015f olmas\u0131 olas\u0131l\u0131\u011f\u0131 var. Fibonacci hen\u00fcz \u00e7ocuk ya\u015ftayken, Pisa’l\u0131 bir t\u00fcccar olan babas\u0131 Guglielmo, Pisal\u0131 t\u00fcccarlar\u0131n ya\u015fad\u0131\u011f\u0131 Bugia adl\u0131 Kuzey Afrika liman\u0131na Kons\u00fcl olarak atan\u0131r. (Bu liman, \u015fimdiki Bejaya’d\u0131r ve Cezayir’dedir.) Babas\u0131 burada o\u011fluna hesap \u00f6\u011fretmesi i\u00e7in bir Arap hoca tutar. Fibonacci daha sonra Liber Abaci’de hocas\u0131ndan “Dokuz Hint Rakam\u0131n\u0131n Sanat\u0131n\u0131” \u00f6\u011frenirken duydu\u011fu mutlulu\u011fu anlatacakt\u0131r.<\/p>\n

Fibonacci’nin Liber Abaci adl\u0131 kitab\u0131n\u0131n yay\u0131nland\u0131\u011f\u0131 y\u0131llarda, Hindu-Arap say\u0131lar\u0131, Avrupa’da Harzemli Muhammed Bin Musa’n\u0131n eserlerinin \u00e7evirilerini okuyabilmi\u015f bir ka\u00e7 “ayd\u0131n” d\u0131\u015f\u0131nda bilinmiyordu. Fibonacci, kitab\u0131nda bu rakamlar\u0131 anlatmaya \u015f\u00f6yle ba\u015flar: “Dokuz Hint Rakam\u0131 9 8 7 6 5 4 3 2 1 dir. Bu dokuz rakama “0” i\u015faretinin de eklenmesiyle, her hangi bir say\u0131 yaz\u0131labilir.”<\/p>\n

Liber Abaci, 13.yy. Avrupas\u0131nda b\u00fcy\u00fck ilgi g\u00f6r\u00fcr, \u00e7ok say\u0131da kopya edilir ve kilisenin yasaklamas\u0131na kar\u015f\u0131n Arap say\u0131lar\u0131 \u0130talyan t\u00fcccarlar aras\u0131nda yay\u0131l\u0131r. Kitap Kutsal Roma \u0130mparatoru II. Frderick’in dikkatini \u00e7eker. Frederick bilime d\u00fc\u015fk\u00fcn bir imparatordur. Bilim adamlar\u0131n\u0131 korur. Bu nedenle kendisine Stupor Mudi (D\u00fcnya Harikas\u0131) <\/strong>denilmektedir. 1220 y\u0131l\u0131nda Fibonacci huzura \u00e7a\u011fr\u0131l\u0131r. Frderick’in bilim adamlar\u0131ndan biri taraf\u0131ndan s\u0131nava \u00e7ekilir. Sonunda Fibonacci g\u00f6ze girer. Y\u0131llarca hem imparatorla, hem de imparatorun dostlar\u0131yla yaz\u0131\u015f\u0131r. 1225 y\u0131l\u0131nda yazd\u0131\u011f\u0131 Liber Quadratornum<\/span><\/strong>‘u (Kare Say\u0131lar\u0131n Kitab\u0131)<\/strong> imparatora ithaf eder. “Diyofantus Denklemleri<\/span>“ne ayr\u0131lan bu kitap Fibonacci’nin ba\u015f yap\u0131t\u0131d\u0131r. Her ne kadar Liber Abaci’ye \u00e7ok daha dar bir \u00e7evrenin ilgisini \u00e7ekerse de kitap say\u0131lar kuram\u0131na b\u00fcy\u00fck katk\u0131 getirir.<\/p>\n

1228’de Fibonacci, Liber Abaci’yi yeniden g\u00f6zden ge\u00e7irir ve kitab\u0131n bu ikinci yaz\u0131l\u0131m\u0131n\u0131 imparatorun ba\u015f bilimcisi Michael Socott’a ithaf eder. Bu tarihten 1240 y\u0131l\u0131na kadar Fibonacci hakk\u0131nda hi\u00e7 bir \u015fey bilinmiyor. 1240’ta Pisa kenti kendisine kente yapt\u0131\u011f\u0131 hizmetlerden dolay\u0131 “20 Pisa Liras\u0131” y\u0131ll\u0131k ba\u011flar. Bundan sonra Matematik\u00e7imiz ne kadar ya\u015fad\u0131, o da bilinmiyor.<\/p>\n

Leonardo Fibonacci, Arap Matematik’ini kullan\u0131\u015fl\u0131 Hindu-Arap say\u0131lar\u0131n\u0131 Bat\u0131’ya tan\u0131tmakla \u00e7ok b\u00fcy\u00fck bir katk\u0131da bulundu. Ancak ilgin\u00e7tir, \u00e7a\u011f\u0131m\u0131z matematik\u00e7ileri Fibonacci’nin ad\u0131n\u0131. daha \u00e7ok, Liber Abaci’de yer alan bir problemde ortaya \u00e7\u0131kan bir say\u0131 dizisi <\/span><\/em><\/strong>nedeniyle bilirler. Dolay\u0131s\u0131yla Fibonacci’yi anlatan bir yaz\u0131da “Fibonacci Say\u0131lar\u0131<\/span>“ndan ya da “Fibonacci Dizisi<\/span>“nden s\u00f6z etmemek olmaz.Bu nedenle biz de bu b\u00f6l\u00fcm\u00fcn geri kalan kesimini bu diziye ay\u0131raca\u011f\u0131z…<\/p>\n

\u00a0PEK\u0130 YA NED\u0130R BU FIBONACCI D\u0130Z\u0130S\u0130?<\/strong><\/span><\/p>\n

Liber Abaci’de yer alan problemin metni a\u015fa\u011f\u0131 yukar\u0131 \u015f\u00f6yle;<\/p>\n

“Adam\u0131n biri, d\u00f6rt bir yan\u0131 duvarla \u00e7evrili yere bir \u00e7ift tav\u015fan koymu\u015f. Her \u00e7ift tav\u015fan\u0131n bir ay i\u00e7inde yeni bir \u00e7ift tav\u015fan peydahlad\u0131\u011f\u0131, her yeni \u00e7iftin de erginle\u015fmesi i\u00e7in bir ay gerekti\u011fi ve tav\u015fanlar\u0131n \u00f6lmedi\u011fi var say\u0131l\u0131rsa, 100 ay sonunda d\u00f6rt duvar\u0131n aras\u0131nda ka\u00e7 \u00e7ift tav\u015fan olur?”<\/strong><\/em><\/p>\n

Knuth dostumuza g\u00f6re, Fibonacci bu problemi kitab\u0131na biyoloji biliminde bir uygulama olsun diye ya da n\u00fcfus patlamas\u0131 sorununa bir \u00e7\u00f6z\u00fcm getirsin diye koymam\u0131\u015f (Ben de ayn\u0131 kan\u0131day\u0131m…). Toplama al\u0131\u015ft\u0131rmas\u0131 olarak d\u00fc\u015f\u00fcnm\u00fc\u015f bunu, besbelli. Her neyse biraz d\u00fc\u015f\u00fcn\u00fcnce tav\u015fan \u00e7iftlerinin aylara g\u00f6re \u015f\u00f6yle \u00e7o\u011falaca\u011f\u0131 ortaya \u00e7\u0131k\u0131yor:<\/p>\n

1,1,2,3,5,8,13,21,34,55,89,…<\/strong><\/p>\n

Yani her ay sonundaki tav\u015fan \u00e7ifti say\u0131s\u0131 o aydan hemen \u00f6nceki iki aydaki say\u0131lar\u0131n toplam\u0131na e\u015fit.<\/p>\n

Neyse her halde sorumuzun cevab\u0131n\u0131 merak ediyorsunuz… Al\u0131n size cevap… Bak\u0131n bakal\u0131m, ka\u00e7 tav\u015fan olu\u015furmu\u015f 100 ayda???<\/p>\n

CEVAP —>>> 354.224.848.179.261.915.075 <\/strong>TANE TAV\u015eAN OLU\u015eUR….<\/p>\n

\u00a0FIBONACCI D\u0130Z\u0130S\u0130 (B\u0130RAZ DAHA CEB\u0130RSEL)<\/strong><\/span><\/p>\n

*** <\/strong>Fibonacci Dizisi’nin \u00f6zelli\u011fi \u015fu; Fibonacci Dizisindeki bir terim kendinden \u00f6nceki iki terimin toplam\u0131na e\u015fittir.<\/p>\n

FIBONACCI D\u0130Z\u0130S\u0130’ni yazal\u0131m…<\/p>\n

…………….1,1,2,3,5,8,13,21,34,55,89,144………….<\/strong><\/p>\n

G\u00f6r\u00fcld\u00fc\u011f\u00fc gibi bir terim kendinden \u00f6nceki iki terimin toplam\u0131na e\u015fittir. Mesela;<\/p>\n

1+1=2\u00a0 2+3=5\u00a0 3+5=8 \u00a0 5+8=13 8+13=21\u00a0 13+21=34 ……… 89+144=233 gibi.<\/p>\n

\u0130sterseniz bir de bu Fibonacci Dizisinin Form\u00fcl\u00fcn\u00fc Yazal\u0131m:::<\/p>\n

FIBONACCI D\u0130Z\u0130S\u0130N\u0130N G\u00d6R\u00dcLD\u00dc\u011e\u00dc VE KULLANILDI\u011eI YERLER:<\/span><\/strong><\/p>\n

1) Ay\u00e7i\u00e7e\u011fi: <\/strong>Ay\u00e7i\u00e7e\u011fi’nin merkezinden d\u0131\u015far\u0131ya do\u011fru sa\u011fdan sola ve soldan sa\u011fa do\u011fru taneler say\u0131ld\u0131\u011f\u0131nda \u00e7\u0131kan say\u0131lar Fibonacci Dizisinin ard\u0131\u015f\u0131k terimleridir.<\/p>\n

2) Papatya \u00c7i\u00e7e\u011fi: <\/strong>Papatya \u00c7i\u00e7e\u011finde de ay\u00e7i\u00e7e\u011finde oldu\u011fu gibi bir Fibonacci Dizisi mevcuttur.<\/p>\n

3) Fibonacci Dizisinin Fark Dizisi:<\/strong> Fibonacci Dizisindeki ard\u0131\u015f\u0131k terimlerin fark\u0131yla olu\u015fan dizi de Fibonacci Dizisidir.<\/p>\n

4) \u00d6mer Hayyam veya Pascal veya Binom \u00dc\u00e7geni: <\/strong>\u00d6mer Hayyam \u00fc\u00e7genindeki t\u00fcm katsay\u0131lar veya terimler yaz\u0131l\u0131p \u00e7apraz toplamlar\u0131 al\u0131nd\u0131\u011f\u0131nda Fibonacci Dizisi ortaya \u00e7\u0131kar.<\/p>\n

5) Tav\u015fan: <\/strong>Zaten sorumuz tav\u015fanla alakal\u0131…<\/p>\n

6) \u00c7am Kozala\u011f\u0131: <\/strong>\u00c7am kozala\u011f\u0131ndaki taneler kozala\u011f\u0131n alt\u0131ndaki sabit bir noktadan kozala\u011f\u0131n tepesindeki ba\u015fka bir sabit noktaya do\u011fru spiraller (e\u011friler) olu\u015fturarak \u00e7\u0131karlar. \u0130\u015fte bu taneler soldan sa\u011fa ve sa\u011fdan sola say\u0131ld\u0131\u011f\u0131nda \u00e7\u0131kan say\u0131lar, Fibonacci Dizisi’nin ard\u0131\u015f\u0131k terimleridir.<\/p>\n

7) T\u00fct\u00fcn Bitkisi: <\/strong>T\u00fct\u00fcn Bitkisinin yapraklar\u0131n\u0131n dizili\u015finde bir Fibonacci Dizisi s\u00f6z konusudur; yani yapraklar\u0131n diziliminde bu dizi mevcuttur. Bundan dolay\u0131 t\u00fct\u00fcn bitkisi G\u00fcne\u015f’ten en iyi \u015fekilde g\u00fcne\u015f \u0131\u015f\u0131\u011f\u0131 ve havadan en iyi \u015fekilde Karbondioksit alarak Fotosentez’i m\u00fckemmel bir \u015fekilde ger\u00e7ekle\u015ftirir.<\/p>\n

8) E\u011frelti Otu: <\/strong>T\u00fct\u00fcn Bitkisindeki ayn\u0131 \u00f6zellik E\u011frelti Otu’nda da vard\u0131r.<\/p>\n

9) <\/strong>M\u0130MAR S\u0130NAN<\/span><\/strong><\/em>:<\/strong> Mimar Sinan’\u0131n da bir \u00e7ok eserinde Fibonacci Dizisi g\u00f6r\u00fclmektedir. Mesela S\u00fcleymaniye ve Selimiye Camileri’nin minarelerinde bu dizi mevcuttur.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

\u00dcnl\u00fc matematik\u00e7i Fibonacci, google’de hi\u00e7 bu kadar \u00e7ok aranmam\u0131\u015ft\u0131r herhalde. 8.s\u0131n\u0131f matematik dersinde \u00f6zel say\u0131 dizileri konusunda ge\u00e7n Fibonacci say\u0131 dizisinin, proje ve performans \u00f6devi olarak \u00f6\u011frencilere verilmesi, T\u00fcrkiye’de Fibonacci’nin \u00fcn\u00fcne \u00fcn katm\u0131\u015f gibi g\u00f6z\u00fck\u00fcyor. Sizler i\u00e7in Fibonacci ve dizisi hakk\u0131nda 4yaz\u0131 derledik… 1.yaz\u0131 Fibonacci,ya da daha do\u011frusu Leonardo da Pisa, \u0130.S. 1175 de \u0130talya\u2019n\u0131n …<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[1],"tags":[160,161,162],"class_list":["post-652","post","type-post","status-publish","format-standard","hentry","category-genel","tag-fibonacci","tag-pisali-leonardo-fibonacci","tag-pisali-leonardo-fibonacci-kimdir"],"aioseo_notices":[],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p2YBEC-aw","jetpack_sharing_enabled":true,"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.ragipsahin.com.tr\/index.php?rest_route=\/wp\/v2\/posts\/652","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ragipsahin.com.tr\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ragipsahin.com.tr\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ragipsahin.com.tr\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ragipsahin.com.tr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=652"}],"version-history":[{"count":0,"href":"https:\/\/www.ragipsahin.com.tr\/index.php?rest_route=\/wp\/v2\/posts\/652\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.ragipsahin.com.tr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=652"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ragipsahin.com.tr\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=652"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ragipsahin.com.tr\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=652"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}