A\u015fa\u011f\u0131daki diyalog, \u00f6\u011frenciyle \u00f6\u011fretmeni aras\u0131nda ge\u00e7mektedir. Umar\u0131m matematiksel kan\u0131t\u0131n anlam ve \u00f6nemini anlat\u0131yordur.<\/p>\n
\u2013 Ge\u00e7enlerde Aritmeti\u011fin Temel Teoremi diye bir teoreme rastlad\u0131m: 1\u2019den b\u00fcy\u00fck her pozitif tamsay\u0131 sonlu say\u0131da asal say\u0131n\u0131n \u00e7arp\u0131m\u0131d\u0131r ve bu \u00e7arp\u0131m tek \u015fekilde yaz\u0131l\u0131r. \u00c7ok belirgin bir \u00f6nerme. Birileri bu denli a\u00e7\u0131k \u015feyleri kan\u0131tlamak i\u00e7in neden u\u011fra\u015f\u0131p duruyor? Kan\u0131tlamaya gerek var m\u0131?<\/p>\n
\u2013 Elbette gerek var, matemati\u011fi matematik yapan b\u00f6ylesi teoremler. \u00d6klid\u2019in \u00f6nermelerinin sonucu olan bu teoremle Euler, Legendre gibi bir\u00e7ok \u00fcnl\u00fc matematik\u00e7i ilgilenmi\u015f, ilk ve net kan\u0131t Gauss taraf\u0131ndan verilmi\u015f. Bence her matematiksever lise \u00f6\u011frencisi b\u00f6ylesi basit teoremler \u00fczerine kafa yormal\u0131, bir \u00f6nermenin do\u011frulu\u011funu nedenleriyle birlikte \u00f6\u011frenmeli.<\/p>\n
\u2013 Bir \u00f6nermenin do\u011frulu\u011funu nedenleriyle birlikte \u00f6\u011frenmek\u2026 Ben ve \u00e7o\u011fu \u00f6\u011frenci arkada\u015f\u0131m \u201cnedenleri\u201d merak etmiyoruz.<\/p>\n
\u2013 Hakl\u0131s\u0131n, maalesef \u00fclkemiz e\u011fitim sisteminde \u00f6\u011frenciler matemati\u011fi tepeden inme bir bi\u00e7imde belleyerek \u00f6\u011frenmek zorunda kal\u0131yor, bir s\u00fcre sonra ya matematikten kopuyor ya da h\u0131zla i\u015flem yapan, be\u015f se\u00e7enekten birini i\u015faretlemeyi \u00f6\u011frenen robotlara d\u00f6n\u00fc\u015f\u00fcyor.<\/p>\n
\u2013 \u0130lkokula ba\u015flad\u0131\u011f\u0131m y\u0131llarda merak etme duygumun ne denli g\u00fc\u00e7l\u00fc oldu\u011funu \u00e7ok iyi an\u0131ms\u0131yorum. Sonras\u0131nda \u00e7oktan se\u00e7meli s\u0131navlara haz\u0131rland\u0131m, bir soruyu bir dakikada \u00e7\u00f6zmeyi ba\u015far\u0131 olarak g\u00f6rd\u00fcm.<\/p>\n
\u2013 Kurallar nedenleri olmaks\u0131z\u0131n verildi\u011finde h\u0131zla unutulan, kolayca \u00f6z\u00fcmsenemeyen bilgi y\u0131\u011f\u0131n\u0131na d\u00f6n\u00fc\u015f\u00fcyor. Bir otomobili motorunun nas\u0131l \u00e7al\u0131\u015ft\u0131\u011f\u0131n\u0131 bilmeden kullanabilirsin, ama matematikte koyulan kurallar\u0131n nedenlerini bilmeden matematik yapamazs\u0131n.<\/p>\n
\u2013 Evet, bu s\u00f6zler \u00fczerine d\u00fc\u015f\u00fcnd\u00fc\u011f\u00fcmde verili bilgiyi sorgulama ihtiyac\u0131 hissetmedi\u011fimi anl\u0131yorum, \u201ckan\u0131ts\u0131z matematik\u201d yapmaya al\u0131\u015ft\u0131m. Ald\u0131\u011f\u0131m e\u011fitimin merak etme, sorgulama iste\u011fimi k\u00f6reltti\u011fini d\u00fc\u015f\u00fcn\u00fcyorum. Belki de bu y\u00fczden Aritmeti\u011fin Temel Teoremi\u2019ne \u015fa\u015f\u0131r\u0131yor ve kan\u0131tlamaya gerek var m\u0131 diye soruyorum.<\/p>\n
\u2013 Teoremlerin kan\u0131tlar\u0131n\u0131 yapmadan veya anlamadan ge\u00e7en bir \u00f6\u011frencinin matemati\u011fi sevmesi olanaks\u0131zd\u0131r. Kan\u0131tlamazsan sevemezsin!<\/p>\n
\u2013 Aritmeti\u011fin Temel Teoremi i\u00e7in ben \u015f\u00f6yle d\u00fc\u015f\u00fcnm\u00fc\u015ft\u00fcm: Hangi pozitif tamsay\u0131y\u0131 al\u0131rsak alal\u0131m sonlu say\u0131da asal\u0131n \u00e7arp\u0131m\u0131d\u0131r, \u00f6rne\u011fin 15 = 3\u00d75, 36 = 2\u00d72\u00d73\u00d73. O halde b\u00fct\u00fcn say\u0131lar sonlu asal\u0131n \u00e7arp\u0131m\u0131 olarak yaz\u0131labilir, kan\u0131tlanacak bir \u015fey yok.<\/p>\n
\u2013 Evet ama, sezgilerimizle ve deneysel olarak ula\u015ft\u0131\u011f\u0131m\u0131z bir sonu\u00e7 matematik i\u00e7in yeterli de\u011fildir. Matematik bizden bu \u00f6nermenin neden do\u011fru oldu\u011funu t\u00fcm say\u0131lar i\u00e7in g\u00f6stermemizi ister.<\/p>\n
\u2013 Peki, \u015fimdi matemati\u011fin sesine kulak verelim ve Aritmeti\u011fin Temel Teoremi\u2019ni birlikte kan\u0131tlayal\u0131m. Ge\u00e7en g\u00fcn elime ge\u00e7en o matematik kitab\u0131nda teoremin kan\u0131t\u0131n\u0131 \u00e7eli\u015fme y\u00f6ntemiyle yapm\u0131\u015f, yani teoremin do\u011fru olmad\u0131\u011f\u0131n\u0131 varsay\u0131p bir \u00e7eli\u015fki elde ederek teoremi kan\u0131tlam\u0131\u015f. Varsayal\u0131m ki sonlu say\u0131da asal\u0131n \u00e7arp\u0131m\u0131 olmayan pozitif tamsay\u0131lar var. Bu say\u0131lar\u0131n olu\u015fturdu\u011fu k\u00fcmeye A ve bu k\u00fcmenin en k\u00fc\u00e7\u00fck eleman\u0131na n diyelim. \u2013 Bir dakika\u2026 Bu k\u00fcmenin en k\u00fc\u00e7\u00fck eleman\u0131 oldu\u011funu nereden biliyoruz? Belki b\u00f6yle bir say\u0131 yok.<\/p>\n
\u2013 Yok art\u0131k, bundan da m\u0131 ku\u015fku duyaca\u011f\u0131z? Elbette ku\u015fkulanaca\u011f\u0131z, \u00e7\u00fcnk\u00fc bu ad\u0131m\u0131 atlayarak yaratt\u0131\u011f\u0131m\u0131z bo\u015fluk yapaca\u011f\u0131m\u0131z kan\u0131t\u0131n \u00e7\u00f6kmesine neden olacak. Hi\u00e7bir a\u00e7\u0131k, hi\u00e7bir belirsizlik olmamal\u0131. K\u00fcmeler kuram\u0131nda \u0130yi S\u0131ralama \u0130lkesi olarak bilinen bir \u00f6nerme var: Bo\u015fk\u00fcme olmayan her do\u011fal say\u0131 k\u00fcmesinin en k\u00fc\u00e7\u00fck eleman\u0131 vard\u0131r. Bu \u00f6nerme k\u00fcmeler kuram\u0131n\u0131n baz\u0131 modellerinde bir teorem, yani kan\u0131tlanabiliyor, baz\u0131lar\u0131ndaysa aksiyom olarak yer al\u0131yor. Biz bu ilkeyi do\u011fal say\u0131lar k\u00fcmesine uygulad\u0131\u011f\u0131m\u0131zdan teorem olarak ele almal\u0131y\u0131z, yani burada bizim i\u00e7in yard\u0131mc\u0131 teorem olmal\u0131. Bu teoremin kan\u0131t\u0131 konumuz d\u0131\u015f\u0131nda, sonraya b\u0131rakal\u0131m. Hadi \u015fimdi kald\u0131\u011f\u0131m\u0131z yerden devam edelim.<\/p>\n
\u2013 Hijyen! Matematiksel kan\u0131t\u0131n bu denli \u00f6zen ve titizlik i\u00e7inde yap\u0131laca\u011f\u0131n\u0131 hi\u00e7 d\u00fc\u015f\u00fcnmemi\u015ftim. Peki, devam ediyorum. Art\u0131k s\u00f6z\u00fcn\u00fc etti\u011fim o k\u00fcmenin en k\u00fc\u00e7\u00fck eleman\u0131n oldu\u011funu biliyoruz ve bu elemana n <\/em>demi\u015ftik. n <\/em>say\u0131s\u0131 bir asal say\u0131ya b\u00f6l\u00fcn\u00fcr. Bu kez ya\u015f tahtaya basm\u0131yorum, \u00e7\u00fcnk\u00fc 1 d\u0131\u015f\u0131nda her tamsay\u0131 bir asala b\u00f6l\u00fcn\u00fcr \u00f6nermesinin de bir teorem oldu\u011funu ve kan\u0131t\u0131n\u0131n yine \u00e7eli\u015fme y\u00f6ntemiyle yap\u0131ld\u0131\u011f\u0131n\u0131 biliyorum.<\/p>\n \u2013 Bravo! Ne g\u00fczel, matematiksel kan\u0131t\u0131n gereklerini yerine getirerek ilerliyorsun.<\/p>\n \u2013 n <\/em>say\u0131s\u0131n\u0131 b\u00f6len asala p <\/em>diyelim. B\u00f6ylece bir m <\/em>do\u011fal say\u0131s\u0131 i\u00e7in<\/p>\n n = pm <\/em><\/p>\n olur. Bu e\u015fitlikte m <\/em>s\u0131f\u0131ra e\u015fit olamaz, \u00e7\u00fcnk\u00fc bu durumda n <\/em>de s\u0131f\u0131r olur. E\u011fer m <\/em>= 1 ise n <\/em>= p <\/em>olur ki, bu durumdan say\u0131s\u0131 tek bir asal\u0131n yani p<\/em>\u2019nin \u00e7arp\u0131m\u0131 olarak yaz\u0131l\u0131r.<\/p>\n E\u011fer m <\/em>\u2265 2 ise n <\/em>= pm <\/em>e\u015fitli\u011finden dolay\u0131 m <\/em>< n <\/em>olur. Bu durumda n<\/em>, A<\/em>\u2019n\u0131n (sonlu say\u0131da asal\u0131n \u00e7arp\u0131m\u0131 olmayan pozitif tamsay\u0131lar\u0131n t\u00fcm\u00fcn\u00fc eleman kabul eden k\u00fcmenin) en k\u00fc\u00e7\u00fck eleman\u0131 oldu\u011fundan m<\/em>, A<\/em>\u2019n\u0131n eleman\u0131 olamaz. O halde m <\/em>sonlu say\u0131da asal\u0131n \u00e7arp\u0131m\u0131 olarak yaz\u0131l\u0131r. \u00d6te yandan n=pm oldu\u011fundan n\u2019nin de asallar\u0131n \u00e7arp\u0131m\u0131 olarak yaz\u0131labilece\u011fi sonucuna ula\u015f\u0131r\u0131z ki, bu da bir \u00e7eli\u015fkidir. Demek ki A bo\u015f k\u00fcmeymi\u015f. B\u00f6ylece teorem kan\u0131tlanm\u0131\u015f olur.<\/p>\n \u2013 G\u00fczel, tebrikler. Tabii burada teoremin \u201cbu \u00e7arp\u0131m tek \u015fekilde yaz\u0131l\u0131r\u201d b\u00f6l\u00fcm\u00fcn\u00fc hen\u00fcz kan\u0131tlamad\u0131k.<\/p>\n \u2013 Evet ayr\u0131ca, att\u0131\u011f\u0131m\u0131z ad\u0131mlar aras\u0131nda kan\u0131t\u0131n\u0131 vermeden ge\u00e7ti\u011fimiz iki yard\u0131mc\u0131 teorem daha var. Ama matematiksel kan\u0131t\u0131n titiz mant\u0131k y\u00fcr\u00fctme anlay\u0131\u015f\u0131n\u0131n ne oldu\u011funu ve gereklili\u011fini \u00e7ok iyi anlam\u0131\u015f bulunuyorum. Te\u015fekk\u00fcrler. Ya\u015fas\u0131n matematiksel kan\u0131t!<\/p>\n","protected":false},"excerpt":{"rendered":" A\u015fa\u011f\u0131daki diyalog, \u00f6\u011frenciyle \u00f6\u011fretmeni aras\u0131nda ge\u00e7mektedir. Umar\u0131m matematiksel kan\u0131t\u0131n anlam ve \u00f6nemini anlat\u0131yordur. \u2013 Ge\u00e7enlerde Aritmeti\u011fin Temel Teoremi diye bir teoreme rastlad\u0131m: 1\u2019den b\u00fcy\u00fck her pozitif tamsay\u0131 sonlu say\u0131da asal say\u0131n\u0131n \u00e7arp\u0131m\u0131d\u0131r ve bu \u00e7arp\u0131m tek \u015fekilde yaz\u0131l\u0131r. \u00c7ok belirgin bir \u00f6nerme. Birileri bu denli a\u00e7\u0131k \u015feyleri kan\u0131tlamak i\u00e7in neden u\u011fra\u015f\u0131p duruyor? Kan\u0131tlamaya gerek var […]<\/p>\n","protected":false},"author":375,"featured_media":16567,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[180,25,514],"tags":[1967,208,1572],"class_list":["post-16566","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-143-sayi","category-matematik","category-matematik-sohbetleri","tag-kanit","tag-matematik","tag-problemler"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/posts\/16566","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/users\/375"}],"replies":[{"embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/comments?post=16566"}],"version-history":[{"count":0,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/posts\/16566\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/media\/16567"}],"wp:attachment":[{"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/media?parent=16566"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/categories?post=16566"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/tags?post=16566"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}