Fizik \u201311 \/ b\u2013<\/p>\n
\u0130\u00e7erik: Ne yapt\u0131\u011f\u0131m\u0131z\u0131n fark\u0131nda olmak ya da ol_A_mamak!<\/p>\n
1 say\u0131s\u0131n\u0131 3\u2019e b\u00f6lelim.
\nAritmeti\u011fin b\u00f6lme i\u015flemine ili\u015fkin kurallar\u0131n\u0131 kullanal\u0131m. (*)
\n1\u2019de 3 yok.
\nARA SONU\u00c7: S\u0131f\u0131r sonra virg\u00fcl.
\n1\u2019i 10 ile \u00e7arpar\u0131m. 10\u2019da 3, 3 kez var. 10 b\u00f6l\u00fc 3 e\u015fittir 3, elde var 1.
\nARA SONU\u00c7: S\u0131f\u0131r sonra virg\u00fcl sonra 3.
\nEldeki 1\u2019i 10 ile \u00e7arpar 10 elde ederim.
\n10\u2019da 3, 3 kez var. 10 b\u00f6l\u00fc 3 e\u015fittir 3 elde var 1.
\nARA SONU\u00c7: S\u0131f\u0131r sonra virg\u00fcl sonra 3 sonra yine 3.
\nBu yolda s\u00fcrekli ilerleyerek 1\/3=0,333\u2026 elde ederim.<\/p>\n
\u015eimdi de sa\u011flamas\u0131n\u0131 yapal\u0131m.
\n0,333\u2026 = A olsun.
\n10×0,333\u2026 = 3,333\u2026 = 3+0,333\u2026=3+A = 10A olur.
\nDemek ki,
\n3 = 9A ve A = 1\/3.
\nBuraya dek hi\u00e7bir sorun yok.
\n0,333\u2026 + 0,333\u2026 = 0,666\u2026 oldu\u011funda da sorun yok.
\n0,333\u2026 + 0,333\u2026 + 0,333\u2026 = 0,999\u2026 oldu\u011funda da sorun yok.
\nAma, 0,333\u2026 + 0,333\u2026 + 0,333\u2026 = 1 midir? Buras\u0131 karanl\u0131k. \u00c7\u00fcnk\u00fc, 0,999\u2026 = 1 gibi bir sonu\u00e7 \u00e7\u0131kar ki, i\u015fte buras\u0131 Say\u0131 Kuram\u0131\u2019n\u0131n, dolay\u0131s\u0131 ile Say\u0131 Sistemi\u2019mizin, dolay\u0131s\u0131 ile Aritmeti\u011fin ve dolay\u0131s\u0131 ile Matemati\u011fin (\u00e7\u00f6kt\u00fc\u011f\u00fc demeyelim de) sorunlu b\u00f6lgelerindendir.<\/p>\n
\u00d6zetle; 0,999\u2026 = 1 yanl\u0131\u015ft\u0131r ama Matemati\u011fin kurallar\u0131 uygulanarak elde edilen alttaki analiz do\u011frudur. Yanl\u0131\u015f bir e\u015fitli\u011fi, yani bir e\u015fitsizli\u011fi do\u011fru bir e\u015fitlikmi\u015f gibi veren matemati\u011fe g\u00fcven olmaz!
\n0,999\u2026 = B olsun.
\n10×0,999\u2026 = 9,999\u2026 = 9 + 0,999\u2026 = 9 + B = 10B olur.
\nDemek ki,
\n9 = 9B ve B = 1\/3. (**)<\/p>\n
Eh, peki! Aritmetik\u2019te b\u00f6ylesi sorunlar var da Geometri pek mi sa\u011flam? De\u011fil!
\nEn kolay\u0131ndan, daha \u00f6nceki yaz\u0131lar\u0131m\u0131zda de\u011finmi\u015f idik, bir noktadan e\u015fit uzakl\u0131ktaki noktalar\u0131n olu\u015fturdu\u011fu \u00e7emberi tan\u0131mlamak kolay. Ama, evrende \u00e7ember bulmak m\u00fcmk\u00fcn de\u011fil. Yani evrende \u00e7ember yok.
\nEfendim? \u201c\u2014Pergelimin i\u011fne u\u00e7lu aya\u011f\u0131n\u0131 k\u00e2\u011f\u0131t gibi bir zemine bat\u0131r\u0131r, \u00f6b\u00fcr aya\u011f\u0131n\u0131 d\u00f6nd\u00fcrerek \u00e7ember elde ederim.\u201d diyen mi oldu? B\u00f6ylelikle elde edilen \u015fekil \u00e7ember tan\u0131m\u0131na uymaz ki. O \u015fekildeki noktalar, merkezden e\u015fit uzakl\u0131kta de\u011fil. Zaten pergelin sivri ucu da bir noktaya de\u011fil, \u015fu ya da bu geni\u015flikte bir y\u00fczeye batmaktad\u0131r.<\/p>\n
Yineleyelim; evrende \u00e7embere benzeyen pek \u00e7ok \u015fekil olabilir. Ama evrende \u00e7ember yoktur. \u00d6rne\u011fin Yer Ekvatoru, meridyen ve paralelleri, \u00e7embere benzer. Ama \u00e7ember de\u011fildirler.
\nDemek ki, \u00e7ember ger\u00e7ek evrenin (***) unsuru, par\u00e7as\u0131 vb. de\u011fildir.
\nH\u0131mm. Peki, \u00e7ember ger\u00e7ek evrende var de\u011filse; dik \u00fc\u00e7gen var m\u0131d\u0131r, e\u015fkenar \u00fc\u00e7gen var m\u0131d\u0131r? Dik koni de yoktur, kare de yoktur, vd.
\nH\u0131mmmm. Peki, \u00f6yleyse Geometri ni\u00e7in var? Yoksa, \u00d6klid\u2019in (hi\u00e7 de\u011filse baz\u0131) postulatlar\u0131 m\u0131 yanl\u0131\u015f? Bkz., (****)<\/p>\n
(*) ODT\u00dc 1. s\u0131n\u0131f \u00f6\u011frencisi (\u2018freshman\u2019) iken \u0130ngilizce dersinin kredisi 9 idi ve okuma kitab\u0131m\u0131zda Isaac Asimov\u2019dan \u00e7e\u015fitli \u00f6yk\u00fcler vard\u0131. Bunlardan birinde, bir teknisyenin delikanl\u0131 ya\u015flar\u0131ndaki o\u011flu, depoda buldu\u011fu bir hesaplay\u0131c\u0131 (\u2018calculator\u2019) sayesinde, hesaplay\u0131c\u0131n\u0131n birka\u00e7 d\u00fc\u011fmesine bas\u0131larak elde edilen aritmetikteki d\u00f6rt i\u015flemin elde edili\u015findeki basamaklar\u0131 \u00f6\u011frenir. Bunu babas\u0131na, sonra da alan\u0131n uzmanlar\u0131na kar\u015f\u0131 yapt\u0131\u011f\u0131 g\u00f6steri ile tan\u0131tlar. Elle yap\u0131lan i\u015flemler, (bilgisayar atas\u0131 say\u0131lan) hesaplay\u0131c\u0131 ile yap\u0131lanlarla ay\u0131 sonu\u00e7lar\u0131 verince herkes hayretlere d\u00fc\u015fer.
\n(**) Bu \u00e7eli\u015fkili sonucun nereden kaynakland\u0131\u011f\u0131n\u0131 (a\u00e7\u0131klayan makaleyi yaz\u0131p bir dergiye g\u00f6ndermek yerine, zira yetersiz edit\u00f6r ve hakemlere rastlama olas\u0131l\u0131\u011f\u0131 hayli y\u00fcksek) bu sitede a\u00e7\u0131klamay\u0131 ye\u011flerim.
\n(***) M\u00fczik evreni, \u015fiir evreni, moda evreni gibi deyimlerden ayr\u0131\u015ft\u0131rmak amac\u0131yla.
\n(***) SAYI, \u015eEK\u0130L, \u00d6L\u00c7\u00dcM \u2013c\u2013
\nFizik \u201311 \/ c\u2013<\/p>\n","protected":false},"excerpt":{"rendered":"
Fizik \u201311 \/ b\u2013 \u0130\u00e7erik: Ne yapt\u0131\u011f\u0131m\u0131z\u0131n fark\u0131nda olmak ya da ol_A_mamak! 1 say\u0131s\u0131n\u0131 3\u2019e b\u00f6lelim. Aritmeti\u011fin b\u00f6lme i\u015flemine ili\u015fkin kurallar\u0131n\u0131 kullanal\u0131m. (*) 1\u2019de 3 yok. ARA SONU\u00c7: S\u0131f\u0131r sonra virg\u00fcl. 1\u2019i 10 ile \u00e7arpar\u0131m. 10\u2019da 3, 3 kez var. 10 b\u00f6l\u00fc 3 e\u015fittir 3, elde var 1. ARA SONU\u00c7: S\u0131f\u0131r sonra virg\u00fcl sonra 3. […]<\/p>\n","protected":false},"author":1425,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[1],"tags":[],"class_list":["post-377","post","type-post","status-publish","format-standard","hentry","category-genel"],"_links":{"self":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/posts\/377","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/users\/1425"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/comments?post=377"}],"version-history":[{"count":0,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/posts\/377\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/media?parent=377"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/categories?post=377"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/tags?post=377"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}