{"id":15063,"date":"2016-08-01T16:44:48","date_gmt":"2016-08-01T13:44:48","guid":{"rendered":"http:\/\/109.232.216.219\/~bilimvegelecek\/?p=15063"},"modified":"2017-12-21T17:11:09","modified_gmt":"2017-12-21T14:11:09","slug":"matematigin-cesur-ve-guzeli","status":"publish","type":"post","link":"https:\/\/bilimvegelecek.com.tr\/index.php\/2016\/08\/01\/matematigin-cesur-ve-guzeli","title":{"rendered":"\u00a0Matemati\u011fin cesur ve g\u00fczeli"},"content":{"rendered":"

Zafer Ercan\u2019a<\/em><\/p>\n

On yedisinde verdi\u011fi karar\u0131n gerek\u00e7esi hayat\u0131n\u0131n sonras\u0131 i\u00e7in ipucu gibidir. \u00c7ok istemesine kar\u015f\u0131n fizik okumaktan vazge\u00e7ip matematik e\u011fitimi almay\u0131 tercih eder. Nedeniyse, Hiro\u015fima ve Nagasaki\u2019ye at\u0131lan atom bombalar\u0131ndan sonra \u00fcmitsizce, fizi\u011fin tehlikeli oldu\u011funu d\u00fc\u015f\u00fcnmesidir. Bu ad\u0131m\u0131 att\u0131\u011f\u0131nda, atom bombas\u0131 yap\u0131m\u0131nda do\u011frudan bir katk\u0131s\u0131 olmayan Albert Einstein\u2019\u0131n \u201cB\u00f6yle olaca\u011f\u0131n\u0131 bilseydim bir ayakkab\u0131 tamircisi olurdum.\u201d s\u00f6z\u00fcn\u00fc hen\u00fcz duymam\u0131\u015ft\u0131r.<\/p>\n

Her b\u00fcy\u00fck matematik\u00e7i gibi matemati\u011fi, kitaplarda yaz\u0131lanlardan ba\u011f\u0131ms\u0131z kendi ba\u015f\u0131na yarat\u0131rcas\u0131na ke\u015ffederek \u00f6\u011frenir, sadece birka\u00e7 kitapla \u00e7ok \u00e7al\u0131\u015f\u0131r.<\/p>\n

Doktorada hocas\u0131 Laurent Schwartz\u2019\u0131n, J. Dieudonn\u00e9\u2019la birlikte listeledikleri ve o g\u00fcne kadar \u00e7\u00f6z\u00fclememi\u015f olan on d\u00f6rt problem kendisine verildi\u011finde, hen\u00fcz yirmi \u00fc\u00e7 ya\u015f\u0131ndad\u0131r. \u0130ki aydan k\u0131sa bir s\u00fcrede problemlerin t\u00fcm\u00fcn\u00fc \u00e7\u00f6zer. O tarihte hocas\u0131 Schwartz matemati\u011fin Nobel\u2019i olarak g\u00f6r\u00fclen Fields Madalyas\u0131na sahip, zirvede olan \u00fcnl\u00fc bir matematik\u00e7idir.<\/p>\n

Hen\u00fcz kariyerinin ilk y\u0131llar\u0131ndad\u0131r, akademik hayata hapsolmu\u015f bir matematik\u00e7i olmayaca\u011f\u0131n\u0131n ilk i\u015faretini verir\u2026 Fransa\u2019n\u0131n Cezayir\u2019i i\u015fgalini \u015fiddetle protesto ederek, sessiz kalan Schwartz, Chevalley, Samuel, Cartier gibi meslekta\u015flar\u0131na tepki g\u00f6sterir. Bu konuyu Fransa\u2019y\u0131 terk edecek kadar ciddiye al\u0131r.<\/p>\n

Otuz sekizinde, bu kez Fields Madalyas\u0131 O\u2019na verilir, ama \u00f6d\u00fcl\u00fcn\u00fc almak i\u00e7in Moskova\u2019daki t\u00f6rene gitmeyip Sovyet militarizmini protesto eder.<\/p>\n

Alexander Grothendieck\u2026<\/p>\n

Yirminci Y\u00fczy\u0131l\u0131n ikinci yar\u0131s\u0131n\u0131n en b\u00fcy\u00fck matematik\u00e7isi olarak kabul edilir. Al\u0131\u015f\u0131lmad\u0131k yarat\u0131c\u0131l\u0131\u011f\u0131, kendini y\u00fczlerce sayfal\u0131k matematik literat\u00fcr\u00fc olarak ortaya koyar. Cebirsel geometri alan\u0131nda akademik stat\u00fckonun belirledi\u011fi \u00e7er\u00e7eve ve y\u00f6ntemleri reddederek devrim niteli\u011finde \u00e7al\u0131\u015fmalara imza atar.<\/p>\n

\u00d6te yandan \u201cmilitan bir aktivist\u201dtir. 1960’larda NATO, NASA ve benzeri savunma temelli kurulu\u015flar\u0131n destekledi\u011fi konferanslara kat\u0131lmay\u0131 reddeder. B\u00f6ylesi durumlarda, Grothendieck’in kat\u0131l\u0131m\u0131n\u0131 sa\u011flayabilmek i\u00e7in, konferans\u0131 d\u00fczenleyenlerin farkl\u0131 kurulu\u015flardan kaynak aray\u0131\u015f\u0131na gittikleri bilinmektedir.<\/p>\n

Otuz dokuzunda kendini vatans\u0131z bir d\u00fcnya vatanda\u015f\u0131 ilan eder. 1967\u2019nin Aral\u0131k ay\u0131nda Amerikan bombard\u0131man\u0131 alt\u0131ndaki Vietnam\u2019a gider. Bir orman s\u0131\u011f\u0131na\u011f\u0131nda Hanoi \u00dcniversitesi matematik b\u00f6l\u00fcm\u00fcnden 30-40 kadar dinleyiciye yakla\u015f\u0131k iki hafta boyunca homolojik cebir \u00fczerine dersler verir.<\/p>\n

En yarat\u0131c\u0131 oldu\u011fu d\u00f6nem Frans\u0131z \u0130leri Bilimsel Ara\u015ft\u0131rmalar\u0131 Enstit\u00fcs\u00fc\u2019nde (IH\u00c9S) ge\u00e7irdi\u011fi on iki y\u0131ld\u0131r. Enstit\u00fc, onun karizmas\u0131 ve liderli\u011fiyle yepyeni bir matematik okuluna d\u00f6n\u00fc\u015f\u00fcr, d\u00fcnya \u00e7ap\u0131nda bir cebirsel geometri merkezi haline gelir.<\/p>\n

IH\u00c9S\u2019in gelirlerinin y\u00fczde be\u015finin Frans\u0131z Savunma Bakanl\u0131\u011f\u0131 taraf\u0131ndan kar\u015f\u0131land\u0131\u011f\u0131n\u0131 \u00f6\u011frendi\u011finde enstit\u00fc kurucular\u0131yla ya\u015fad\u0131\u011f\u0131 \u015fiddetli tart\u0131\u015fmalar sonunda istifa eder; bir ay sonra, 1970 Haziran\u2019\u0131nda Paris \u00dcniversitesi\u2019nde \u00e7ok \u00e7arp\u0131c\u0131 bir konu\u015fma yapar. Y\u00fczlerce dinleyicinin \u00f6n\u00fcnde n\u00fckleer silahlar\u0131n yayg\u0131nl\u0131\u011f\u0131, silahlanma yar\u0131\u015f\u0131, teknolojik geli\u015fmenin insanl\u0131k i\u00e7in olu\u015fturdu\u011fu tehdit gibi kendisinin \u00e7ok \u00f6nem verdi\u011fi konular\u0131 ele al\u0131r. Daha da ileri giderek, teknolojik geli\u015fmelerin bir par\u00e7as\u0131 olmas\u0131ndan dolay\u0131 matematiksel ara\u015ft\u0131rmalardan uzak durulmas\u0131 gerekti\u011finden s\u00f6z eder. Sonradan bu konferans\u0131n i\u00e7eri\u011fi \u201cG\u00fcn\u00fcm\u00fcz D\u00fcnyas\u0131nda Bilim \u0130nsanlar\u0131n\u0131n Sorumlulu\u011fu: Militer Yap\u0131lanma ve Bilim \u0130nsan\u0131\u201d ba\u015fl\u0131kl\u0131 bir bro\u015f\u00fcr olarak yay\u0131mlan\u0131r.<\/p>\n

IH\u00c9S\u2019ten ayr\u0131ld\u0131ktan sonra, iyi \u00fcniversite ya da enstit\u00fclerde iki nedenden dolay\u0131, \u00e7al\u0131\u015fma olana\u011f\u0131 bulamaz. Yetkililer ya onun ko\u015fulllar\u0131na uygun ortam\u0131 sa\u011flayamazlar; ya da \u201cate\u015fli itibar\u0131\u201d kar\u015f\u0131s\u0131nda \u00e7ekingen kal\u0131rlar.<\/p>\n

1970\u2019lerin ba\u015f\u0131nda Survivre et Vivre isimli sava\u015f ve emperyalizm kar\u015f\u0131t\u0131 \u00e7evreci bir hareket ba\u015flat\u0131r, fakat bu hareket kitleselle\u015fmeyi ba\u015faramaz.<\/p>\n

Kendi belirledi\u011fi ilkeler do\u011frultusunda sol ve anar\u015fist bir \u00e7izgide m\u00fccadele verir. Haks\u0131zl\u0131klara, bilimin kirletilmesine ba\u015fkald\u0131r\u0131r. G\u00fc\u00e7 ve iktidara kar\u015f\u0131 hep savunmas\u0131zlar\u0131n yan\u0131ndad\u0131r.<\/p>\n

Konforlu yerlerde mutsuz; yoksullar\u0131n aras\u0131nda kendini iyi hissetti\u011fini s\u00f6yler. Evi sokaktakilere, d\u0131\u015flanm\u0131\u015flara daima a\u00e7\u0131kt\u0131r. Matematik\u00e7i arkada\u015f\u0131 Barry Mazur, Grothendieck’in otob\u00fcs dura\u011f\u0131nda kar\u015f\u0131la\u015ft\u0131\u011f\u0131 ve kalacak yere ihtiyac\u0131 olan bir aileyi evine davet ederek i\u015f kurmalar\u0131na yard\u0131mc\u0131 oldu\u011funu yazm\u0131\u015ft\u0131r. Mazur, O\u2019nun bir al\u0131\u015fveri\u015f merkeziyle kar\u015f\u0131la\u015ft\u0131\u011f\u0131nda hep \u201cka\u00e7al\u0131m buradan\u201d dedi\u011fini de belirtir.<\/p>\n

1973\u2019te akademik kariyerini ani bir kararla sona erdirerek k\u00fc\u00e7\u00fck bir kasabada inzivaya \u00e7ekilir. Neden b\u00f6ylesi bir karara vard\u0131\u011f\u0131, sonras\u0131nda yarat\u0131c\u0131 ve durmak bilmeyen ruhunun neyle me\u015fgul oldu\u011fu gibi sorular hi\u00e7bir zaman tam olarak yan\u0131tlanamam\u0131\u015ft\u0131r.<\/p>\n

1980\u2019ler boyunca matematiksel olan ve olmayan binlerce sayfa eser b\u0131rak\u0131r. Bunlar\u0131n i\u00e7inde, kar\u015f\u0131tlar\u0131na y\u00f6nelik kaleme ald\u0131\u011f\u0131 felsefi s\u00f6vg\u00fcler, matematik d\u00fcnyas\u0131 hakk\u0131ndaki ho\u015fnutsuzluklar\u0131ndan s\u00f6z etti\u011fi yaz\u0131lar\u0131, otobiyografik eseri olan R\u00e9coltes et Semailles<\/em> gibi \u00e7ok farkl\u0131 alanlara ait metinler vard\u0131r. \u00d6te yandan \u201canabelian geometri\u201d olarak bilinen yeni bir alan\u0131n temel \u00e7al\u0131\u015fmalar\u0131n\u0131 yapt\u0131\u011f\u0131 La Clef des Songes<\/em> eseri de bu d\u00f6nemin \u00fcr\u00fcn\u00fcd\u00fcr. Bir\u00e7ok alanda hi\u00e7 yay\u0131n yapmamas\u0131na kar\u015f\u0131n fikirleri ba\u015fka matematik\u00e7ilere ilham kayna\u011f\u0131 olur ve b\u00f6ylece yeni teoriler do\u011far.<\/p>\n

1988\u2019de \u0130sve\u00e7 Akademisi taraf\u0131ndan Crafoord \u00f6d\u00fcl\u00fc ile m\u00fck\u00e2fatland\u0131r\u0131l\u0131r. Bu \u00f6d\u00fcl, Alfred Nobel’in matematik alan\u0131nda bir \u00f6d\u00fcl belirlememi\u015f olmas\u0131n\u0131 telafi etmek i\u00e7in d\u00fc\u015f\u00fcn\u00fclm\u00fc\u015ft\u00fcr ve 160.000 Dolar de\u011ferindedir. \u00d6d\u00fcl\u00fc reddederek matematik d\u00fcnyas\u0131n\u0131 \u015fa\u015fk\u0131nl\u0131\u011fa u\u011frat\u0131r. Le Monde’ a g\u00f6nderdi\u011fi mektupta, \u00f6d\u00fcl\u00fc reddetme gerek\u00e7esini a\u00e7\u0131klarken, bilimsel ve politik d\u00fcnyan\u0131n kirlili\u011finden, yozla\u015fm\u0131\u015fl\u0131\u011f\u0131ndan ve d\u00fcr\u00fcst olmay\u0131\u015f\u0131ndan s\u00f6z eder.<\/p>\n

Altm\u0131\u015fl\u0131 ya\u015flar\u0131 boyunca ciddi derecede psikolojik sorunlar ya\u015far. 1991’de, \u00e7al\u0131\u015fmalar\u0131n\u0131n yer ald\u0131\u011f\u0131 binlerce sayfa metni bir arkada\u015f\u0131n\u0131n evinin bah\u00e7esinde yakt\u0131ktan sonra, tamamen ortal\u0131ktan kaybolur. Arkada\u015flar\u0131, ailesi ve meslekta\u015flar\u0131yla olan ba\u011flar\u0131n\u0131 kopar\u0131r, t\u00fcm sosyal ili\u015fkilerini sonland\u0131r\u0131r.<\/p>\n

Sonraki y\u0131llarda hakk\u0131nda bir\u00e7ok s\u00f6ylenti ortaya \u00e7\u0131kar. Baz\u0131lar\u0131 Budist oldu\u011funu, baz\u0131lar\u0131 ke\u00e7ilere \u00e7obanl\u0131k yapt\u0131\u011f\u0131n\u0131 ve radikal ekolojik teorilerle u\u011fra\u015ft\u0131\u011f\u0131n\u0131 s\u00f6ylerler. Ba\u015fka bir s\u00f6ylenti ise \u201c\u00f6zg\u00fcr iradenin fiziksel yap\u0131s\u0131\u201d gibi bir\u00e7ok felsefi konuyu i\u00e7eren elli ciltlik bir eser \u00fczerine \u00e7al\u0131\u015ft\u0131\u011f\u0131d\u0131r.<\/p>\n

2010\u2019da \u00f6\u011frencilerinden birine yazd\u0131\u011f\u0131 mektupta t\u00fcm \u00e7al\u0131\u015fmalar\u0131n\u0131n k\u00fct\u00fcphanelerden kald\u0131r\u0131lmas\u0131n\u0131 ve yeni bas\u0131mlar\u0131n\u0131n yap\u0131lmamas\u0131n\u0131 ister.<\/p>\n

2014\u2019te, seksen alt\u0131 ya\u015f\u0131ndayken g\u00fcneybat\u0131 Fransa\u2019daki bir hastanede hayata g\u00f6zlerini yumar.<\/p>\n

Atom bombas\u0131 y\u00fcz\u00fcnden fizik e\u011fitimi almaktan vazge\u00e7ecek kadar derin bir duyarl\u0131l\u0131kla ba\u015flayan, \u201ck\u00f6t\u00fcye ve k\u00f6t\u00fcl\u00fcklere\u201d meydan okuyarak devam eden bir yolculuk\u2026 Bir b\u00fcy\u00fck matematik\u00e7i, itiraz ve ba\u015fkald\u0131r\u0131\u2026 Cesur ve g\u00fczel\u2026<\/p>\n

Kaynaklar<\/strong><\/p>\n

1) W. Scharlau, Notices<\/em>, V.55, N.8.<\/p>\n

2) B. Mazur, Thinking about Grothendieck<\/em>. www.math.harvard.edu<\/p>\n

3)http:\/\/www.telegraph.co.uk\/news\/obituaries\/Alexander-Grothendieck-obituary.html<\/p>\n

Ge\u00e7en say\u0131daki problemlerin \u00e7\u00f6z\u00fcm\u00fc<\/strong><\/p>\n

\u00c7apk\u0131n yazar. <\/strong>A\u015fklar\u0131n mektuplarla ilan edildi\u011fi zamanlarda \u00e7apk\u0131nl\u0131\u011f\u0131yla \u00fcnl\u00fc bir yazar be\u015f sevgilisine g\u00f6nderece\u011fi mektuplar\u0131 g\u00f6zlerini kapatarak be\u015f ayr\u0131 zarfa geli\u015fi g\u00fczel koyup, postalam\u0131\u015f. Mektuplar\u0131n be\u015finin de yanl\u0131\u015f adreslere gitmesi olas\u0131l\u0131\u011f\u0131 ka\u00e7t\u0131r?<\/p>\n

\u00c7\u00f6z\u00fcm. <\/strong>Mektuplar\u0131 M1<\/sub>, M2<\/sub>, M3<\/sub>, M4<\/sub>, M5<\/sub><\/em> , sevgilileri de S1<\/sub>, S2<\/sub>, S3<\/sub>, S4<\/sub>, S5<\/sub><\/em> \u00a0ile g\u00f6sterelim. Mektuplar\u0131n rastgele g\u00f6nderilmesi halinde 5!=120 \u00a0farkl\u0131 durum vard\u0131r.<\/p>\n

Mektuplar\u0131n be\u015finin de farkl\u0131 adreslere g\u00f6nderildi\u011fi durumlar\u0131n say\u0131s\u0131 S(5)\u00a0olsun. \u00a0M1<\/sub><\/em>\u2019in S2<\/sub><\/em>\u2019ye gitti\u011fini varsayal\u0131m. Bu durumda M2<\/sub><\/em>\u2019nin \u00a0S1<\/sub><\/em>\u2019e gitti\u011fi ve gitmedi\u011fi iki se\u00e7ene\u011fi inceleyelim. M2<\/sub><\/em>\u2019nin \u00a0S1<\/sub><\/em>\u2019e gitti\u011fi se\u00e7enekte mektuplar\u0131n farkl\u0131 adreslere gitti\u011fi durumlar\u0131n say\u0131s\u0131 S(3), di\u011fer se\u00e7enekte ise istenen durumlar\u0131n say\u0131s\u0131 S(4)olur. B\u00f6ylece \u00a0M<\/em> 1<\/sub>\u2019in S<\/em>1<\/sub>\u2019e de\u011fil \u00a0S2<\/sub><\/em>\u2019ye gitti\u011finde di\u011fer mektuplar\u0131n t\u00fcm\u00fcn\u00fc farkl\u0131 adreslere g\u00f6nderen e\u015flemelerin say\u0131s\u0131
\nS(3)+S(4) olur. M1<\/sub><\/em>\u2019in gidece\u011fi \u00a0adres oldu\u011funa g\u00f6re
\nS(4)=[S(3)+S(4)] olur. Benzer bir ak\u0131l y\u00fcr\u00fctmeyle
\nS(4)=3[S(2)+S(3)], S(2)=1, S(3)=2 oldu\u011fu kolayl\u0131kla bulunabilir. Buradan S(4)=9 ve S(5)=44.<\/p>\n

\u0130stenen olas\u0131l\u0131k, 44\/120=11\/30\u00a0olur. (Genel \u00e7\u00f6z\u00fcm i\u00e7in A. T\u00f6r\u00fcn, Matemati\u011fin (M)izah\u0131<\/em>\u2019na bak\u0131n\u0131z.)<\/p>\n

Yolcu otob\u00fcs\u00fc. <\/strong>40 ki\u015filik bir yolcu otob\u00fcs\u00fcne s\u0131rada bekleyen 40 yolcunun teker teker binmesi ve her yolcunun elindeki bilette yaz\u0131l\u0131 numaral\u0131 koltu\u011fa oturmas\u0131 gerekiyor. Ama ilk yolcu elindeki bilete bakmaks\u0131z\u0131n tamamen rastlant\u0131sal bir koltu\u011fa oturuyor. Sonraki yolcularsa biletlerinde yazan koltu\u011fa oturuyorlar, tabii o yer bo\u015fsa. E\u011fer doluysa onlar da bo\u015f koltuklar aras\u0131nda rastgele bir koltu\u011fa oturuyor. Otob\u00fcse son binen yolcunun kendi yerine oturma olas\u0131l\u0131\u011f\u0131 ka\u00e7t\u0131r?<\/p>\n

\u00c7\u00f6z\u00fcm. <\/strong>Son yolcunun otob\u00fcse binmeden \u00f6nceki durumu d\u00fc\u015f\u00fcnelim. 2\u2019den 39\u2019a kadar numaral\u0131 38 koltu\u011fun t\u00fcm\u00fc dolu olmak zorunda, \u00e7\u00fcnk\u00fc bu yerlere ya sahibi ya da ba\u015fkas\u0131 taraf\u0131ndan oturulmu\u015f. O halde ya 1 ya da 40 numaral\u0131 koltuklar bo\u015f. \u0130stenen olas\u0131l\u0131k 1\/2.<\/p>\n

Sevgililerin bulu\u015fmas\u0131. <\/strong>\u0130ki sevgili telefon g\u00f6r\u00fc\u015fmesinde bir pastanede bulu\u015fmay\u0131 kararla\u015ft\u0131r\u0131yorlar. Sevgililerin her biri o pastaneye en az 10, en \u00e7ok 30 dakikada gidebiliyor. G\u00f6r\u00fc\u015fmenin hemen ard\u0131ndan yola \u00e7\u0131kan sevgililerin en \u00e7ok 20 dakika sonra bulu\u015fma olas\u0131l\u0131\u011f\u0131 ka\u00e7t\u0131r?<\/p>\n

\u00c7\u00f6z\u00fcm. <\/strong>Say\u0131 do\u011frusunda 0\u2019\u0131n sa\u011f\u0131nda 10, 20 ve say\u0131lar\u0131n\u0131 g\u00f6sterelim. Bulu\u015fma, 10 ile 30\u2019uncu dakikalar aras\u0131nda olaca\u011f\u0131ndan sevgililerden birinin bu aral\u0131\u011f\u0131n i\u00e7inde bulunma olas\u0131l\u0131\u011f\u0131 10\/20, ayn\u0131 \u015fekilde di\u011ferinin de bulunma olas\u0131l\u0131\u011f\u0131 10\/20. B\u00f6ylece istenen olas\u0131l\u0131k
\n10\/20×10\/20=1\/4 olur.<\/p>\n

Y\u00fcz\u00fcc\u00fcler. <\/strong>Denizin ortas\u0131nda, k\u0131y\u0131ya uzakl\u0131\u011f\u0131 2 kilometreden fazla olan bir duba var. Dubaya uzakl\u0131\u011f\u0131 1 kilometre olan herhangi bir noktaya A <\/em>y\u00fcz\u00fcc\u00fcs\u00fc, A<\/em>\u2019dan 1 kilometre uzakta herhangi bir noktaya da B<\/em> y\u00fcz\u00fcc\u00fcs\u00fc b\u0131rak\u0131l\u0131yor. H\u0131zlar\u0131 e\u015fit olan bu y\u00fcz\u00fcc\u00fcler en k\u0131sa yoldan dubaya do\u011fru y\u00fczerlerse, B<\/em>\u2019nin dubaya A<\/em> ile ayn\u0131 anda ya da daha \u00f6nce ula\u015fma olas\u0131l\u0131\u011f\u0131 ka\u00e7t\u0131r?<\/p>\n

\u00c7\u00f6z\u00fcm. <\/strong>Verilenlere uygun 1 birim yar\u0131\u00e7apl\u0131 iki e\u015f \u00e7ember \u00e7izilirse kesi\u015fim b\u00f6lgesindeki yaylar\u0131n \u00f6l\u00e7\u00fclerinin 120\u2019\u015fer derece oldu\u011fu g\u00f6r\u00fcl\u00fcr. \u0130stenen olas\u0131l\u0131k 120\/360 = 1\/3.<\/p>\n

Hedef tahtas\u0131.<\/strong> ABC<\/em> \u00fc\u00e7gensel hedef tahtas\u0131nda AB=4, AC=6 birimdir. At\u0131lan okun tahtaya vurdu\u011fu biliniyorsa [AB]\u2019ye g\u00f6re [AB]\u2019ye daha yak\u0131n bir noktaya vurma olas\u0131l\u0131\u011f\u0131 ka\u00e7t\u0131r?<\/p>\n

\u00c7\u00f6z\u00fcm. <\/strong>A a\u00e7\u0131s\u0131n\u0131n a\u00e7\u0131ortay\u0131 [BC]\u2019yi N noktas\u0131nda kessin. Ok, ANC \u00fc\u00e7gensel b\u00f6lgesine isabet ederse [AB]\u2019ye g\u00f6re [AC]\u2019ye daha yak\u0131n bir noktaya vurmu\u015f olur. Ve ABC \u00fc\u00e7genlerinin alanlar\u0131n\u0131n \u00f6l\u00e7\u00fclerinin oran\u0131ndan istenen olas\u0131l\u0131k 3\/5 olarak bulunur.<\/p>\n

Sinema salonu. <\/strong>Koltuklar\u0131n numaral\u0131 oldu\u011fu bir sinema salonunda 8 s\u0131ra ve her s\u0131rada 10 koltuk bulunmaktad\u0131r. Birbirlerinden habersiz bilet alan iki arkada\u015f\u0131n yan yana oturma olas\u0131l\u0131\u011f\u0131 ka\u00e7t\u0131r?<\/p>\n

\u00c7\u00f6z\u00fcm. <\/strong>\u0130ki arkada\u015f\u0131n kenardaki koltuklara ya da kenarlara oturmad\u0131klar\u0131n\u0131 d\u00fc\u015f\u00fcnerek hesaplanan iki ayr\u0131 olas\u0131l\u0131\u011f\u0131n toplam\u0131 sorunun yan\u0131t\u0131d\u0131r: 16\/80×1\/79+64\/80×2\/79 = 9\/395<\/p>\n

Anahtarl\u0131k. <\/strong>Bir kap\u0131y\u0131, i\u00e7inde 6 anahtar bulunan bir anahtarl\u0131ktan sadece 2 tanesi a\u00e7maktad\u0131r. Bu kap\u0131y\u0131 a\u00e7may\u0131 deneyen birinin en \u00e7ok \u00fc\u00e7\u00fcnc\u00fc denemede kap\u0131y\u0131 a\u00e7ma olas\u0131l\u0131\u011f\u0131 ka\u00e7t\u0131r?<\/p>\n

\u00c7\u00f6z\u00fcm. <\/strong>Kap\u0131n\u0131n ilk denemede a\u00e7\u0131lma olas\u0131l\u0131\u011f\u0131 2\/6. \u0130kinci denemeye ge\u00e7ildiyse bir anahtar<\/p>\n

elenir ve kap\u0131n\u0131n a\u00e7\u0131lma olas\u0131l\u0131\u011f\u0131 4\/6×2\/5=4\/15 olur. Benzer yolla kap\u0131n\u0131n \u00fc\u00e7\u00fcnc\u00fc denemede a\u00e7\u0131lma olas\u0131l\u0131\u011f\u0131 1\/5 olacakt\u0131r. Bu \u00fc\u00e7 olas\u0131l\u0131\u011f\u0131n toplam\u0131 sorunun yan\u0131t\u0131d\u0131r: 4\/5.<\/p>\n

\u00dc\u00e7gen.<\/strong> Birim uzunluktaki (1 metre olabilir) bir ipin iki yerden kesilmesiyle elde edilen par\u00e7alar\u0131n \u00fc\u00e7gen olma olas\u0131l\u0131\u011f\u0131 ka\u00e7t\u0131r?<\/p>\n

\u00c7\u00f6z\u00fcm. <\/strong>\u0130p \u00fczerinde x \u2264 y ko\u015fulunu sa\u011flayan rastgele iki nokta alal\u0131m ve ipi bu noktalardan keselim. (x, y) noktalar\u0131 [0,1]x[0,1] \u00a0karesinin i\u00e7inde ve k\u00f6\u015fegeninin (soldan sa\u011fa \u00e7izilen) \u00fcst taraf\u0131nda bulunur. Par\u00e7alar\u0131n uzunluklar\u0131 x, y-x ve 1-y olur. \u00dc\u00e7gen e\u015fitsizli\u011fi yaz\u0131l\u0131rsa x\u22641\/2, y\u22651\/2 ve y\u2264x+1\/2 e\u015fitsizlikleri elde edilir. Bu e\u015fitsizliklerin sa\u011fland\u0131\u011f\u0131 b\u00f6lgenin alan\u0131n \u00f6l\u00e7\u00fcs\u00fcn\u00fc karenin k\u00f6\u015fegeninin \u00fcst\u00fcndeki alan\u0131n \u00f6l\u00e7\u00fcs\u00fcne b\u00f6lersek istenen olas\u0131l\u0131k \u00bc olur.<\/p>\n

Say\u0131 tutma. <\/strong>\u0130ki ki\u015fi 1\u2019den 80\u2019e kadar (1 ve 80 dahil) do\u011fal say\u0131lar aras\u0131ndan birer say\u0131 tutuyorlar. Birinin tutu\u011fu say\u0131n\u0131n di\u011ferinin tuttu\u011fu say\u0131n\u0131n 2 kat\u0131 olma olas\u0131l\u0131\u011f\u0131 ka\u00e7t\u0131r?<\/p>\n

\u00c7\u00f6z\u00fcm. <\/strong>M\u00fcmk\u00fcn hallerin say\u0131s\u0131 80×80, uygun hallerin say\u0131s\u0131 \u0130stenen olas\u0131l\u0131k 1\/80.<\/p>\n

Zar oyunu. <\/strong>A<\/em> ve B<\/em> gibi iki oyuncu zar at\u0131yorlar ve ilk 6 atan oyunu kazan\u0131yor. Zar\u0131 \u00f6nce A<\/em> at\u0131yor, sonra B<\/em>, daha sonra A<\/em> ve sonra tekrar B.<\/em> Oyun bu \u015fekilde ilk 6 gelene kadar devam ediyor. A<\/em> ve B<\/em>\u2019nin oyunu kazanma olas\u0131l\u0131klar\u0131 ka\u00e7t\u0131r?<\/p>\n

\u00c7\u00f6z\u00fcm. <\/strong>A<\/em>\u2019n\u0131n oyunu kazanma olas\u0131l\u0131\u011f\u0131 6p <\/em>iken B<\/em>\u2019ninki 5p<\/em>\u2019dir. \u00d6rnek uzay\u0131n olas\u0131l\u0131\u011f\u0131 1 oldu\u011fundan 6p<\/em> + 5p<\/em> = 1 ve p<\/em> = 1\/11 olur. \u0130stenen olas\u0131l\u0131klar 6\/11 ve 5\/11.<\/p>\n

Not:\u00a0<\/strong>Yukar\u0131daki sorular\u0131 yan\u0131tlayan, \u00f6d\u00fcl olarak Matematik\u00e7i Portreleri<\/em> isimli kitab\u0131 almaya hak kazanan Mustafa Berke Yelten ve Sevgi Ufaker\u2019i tebrik ediyoruz.<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"

Zafer Ercan\u2019a On yedisinde verdi\u011fi karar\u0131n gerek\u00e7esi hayat\u0131n\u0131n sonras\u0131 i\u00e7in ipucu gibidir. \u00c7ok istemesine kar\u015f\u0131n fizik okumaktan vazge\u00e7ip matematik e\u011fitimi almay\u0131 tercih eder. Nedeniyse, Hiro\u015fima ve Nagasaki\u2019ye at\u0131lan atom bombalar\u0131ndan sonra \u00fcmitsizce, fizi\u011fin tehlikeli oldu\u011funu d\u00fc\u015f\u00fcnmesidir. Bu ad\u0131m\u0131 att\u0131\u011f\u0131nda, atom bombas\u0131 yap\u0131m\u0131nda do\u011frudan bir katk\u0131s\u0131 olmayan Albert Einstein\u2019\u0131n \u201cB\u00f6yle olaca\u011f\u0131n\u0131 bilseydim bir ayakkab\u0131 tamircisi olurdum.\u201d […]<\/p>\n","protected":false},"author":375,"featured_media":15065,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[187,25,514],"tags":[1737,1736,208],"class_list":["post-15063","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-150-sayi","category-matematik","category-matematik-sohbetleri","tag-aktivizm","tag-alexander-grothendieck","tag-matematik"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t