Fizik \u201311 \/ a\u2013<\/p>\n
\u0130\u00e7erik: Ne yapt\u0131\u011f\u0131m\u0131z\u0131n fark\u0131nda olmak ya da ol_A_mamak!<\/p>\n
\u201cParadoks mu absurd mu? \u2013 b \u2013\u201c b\u00f6l\u00fcm\u00fcnde g\u00f6rd\u00fck (*); s\u0131f\u0131rdan sonraki virg\u00fclden sonraki basamaklar\u0131n hepsinde 9 olan say\u0131 0,999\u2026 i\u00e7in 1\u2019e e\u015fit diyen ve bu dedi\u011fini kan\u0131tlad\u0131\u011f\u0131 iddias\u0131nda olan hayli \u00e7ok say\u0131da makale mevcut. (**) Ayn\u0131 sa\u00e7mal\u0131k \u00e7er\u00e7evesinde 0,999\u2026 say\u0131s\u0131n\u0131n 1\u2019e (e\u015fit olmasa da) en yak\u0131n say\u0131 oldu\u011funu iddia eden makale say\u0131s\u0131 da hayli \u00e7ok. Bu makalelerin hayli \u00e7o\u011funun gayet sayg\u0131n say\u0131lan dergilerde yay\u0131mlanm\u0131\u015fl\u0131\u011f\u0131 da hayli ilgi \u00e7ekici.<\/p>\n
Nedense, \u015fu husus g\u00f6z ard\u0131 edilmekte; A herhangi bir ger\u00e7ek say\u0131 (\u2018real number\u2019) olsun. E\u011fer, 0,999\u2026 = 1 olsayd\u0131, A+0,999\u2026 = A+1 olurdu. Yani, \u00f6rne\u011fin, 1,999\u2026 = 2, 2,999\u2026 = 3 vb. olurdu. Bu da her say\u0131n\u0131n birden \u00e7ok de\u011feri oldu\u011fu anlam\u0131na gelirdi ki, b\u00f6yle bir durumda Say\u0131 Kuram\u0131 (\u2018Number Theory\u2019) ve dolay\u0131s\u0131 ile Say\u0131 Sistemi var olamazd\u0131.
\nA\u00e7\u0131kt\u0131r ki, 0,999\u2026 say\u0131s\u0131n\u0131n 1\u2019e e\u015fitli\u011fi, matematikten \u00f6te inan\u00e7 (hatta bo\u015f inan\u00e7) krall\u0131\u011f\u0131n\u0131n egemenli\u011findedir. Bu krall\u0131\u011f\u0131n sultas\u0131ndaki \u00f6\u011fretmenlerin \u00f6\u011frencileri i\u00e7in \u00fcz\u00fclmemek m\u00fcmk\u00fcn m\u00fc?<\/p>\n
Oysa, bu sat\u0131rlar\u0131n yazan\u0131 bundan alt\u0131 y\u0131l kadar \u00f6nce ni\u00e7in hemen alttaki i\u015flemler dizisinin do\u011fru oldu\u011funu ama 0,999\u2026 say\u0131s\u0131n\u0131n 1\u2019e e\u015fit olmad\u0131\u011f\u0131n\u0131 bulmu\u015ftu. Dahas\u0131, bu tuhafl\u0131\u011f\u0131n say\u0131 sisteminin hangi \u00f6zelli\u011finden kaynakland\u0131\u011f\u0131n\u0131 da bulmu\u015ftu. Peki yay\u0131mlad\u0131 m\u0131? Yoo, hay\u0131r. Te\u015febb\u00fcs bile etmedi, \u201c\u2014Bo\u015fver, biri bulur nas\u0131l olsa.\u201d gerek\u00e7esiyle umursamad\u0131. Gel gelelim ba\u015fka bulan da, daha do\u011frusu herhangi bir yay\u0131mlayan ki\u015fi de \u00e7\u0131kmad\u0131. (Bu ama\u00e7l\u0131 bir literat\u00fcr taramas\u0131na gerek yok; \u00f6yle biri \u00e7\u0131ksayd\u0131, o andan itibaren \u20180,999\u2026 = 1\u2019 konulu makale yay\u0131nlanmaz olurdu. )<\/p>\n
0,999\u2026 = B olsun;
\n10×0.999\u2026 = 10B olur. Sonu\u00e7 da
\n9.999\u2026 = 9 + 0.999\u2026 = 9 + B = 10B veya
\n9 = 9B, yani
\nB = 1 olarak saptan\u0131r m\u0131?<\/p>\n
\u201cHadi bari \u015fu makaleyi yaz\u0131p bir Matematik dergisine yollayal\u0131m.\u201d denir mi acep yak\u0131nda? Denmese ne olur ki? \u00c7ok mu acaip olur? 0,999\u2026 say\u0131s\u0131n\u0131n 1\u2019e e\u015fit oldu\u011funa inanan Matematik\u00e7i say\u0131s\u0131n\u0131n b\u00fcy\u00fckl\u00fc\u011f\u00fcndeki acaiplik yan\u0131nda mini minnac\u0131k kal\u0131r.
\nAritmeti\u011fin temeli olan Say\u0131 Sistemi\u2019nde durum yukar\u0131daki kadar vahim de Geometri de farkl\u0131 m\u0131 acaba?<\/p>\n
G\u00fcn\u00fcm\u00fczden 2300 y\u0131l kadar \u00f6nce \u0130skenderiye\u2019de ya\u015fam\u0131\u015f \u00d6klid\u2019in \u201cElementler\u201d (***) adl\u0131 kitab\u0131nda nokta \u2018hi\u00e7bir par\u00e7as\u0131 olmayan \u015fey\u2019 olarak tan\u0131mlanm\u0131\u015ft\u0131r. Par\u00e7as\u0131 olmad\u0131\u011f\u0131na g\u00f6re, nokta par\u00e7alanamaz demektir. Bu kadar\u0131 a\u00e7\u0131k da, \u2018\u015fey\u2019 ne demek? \u0130\u015fte buras\u0131 tan\u0131ms\u0131z. Belirtmeye gerek yok, pek \u00e7ok \u015fey var \u2018\u015fey\u2019 olan. Yani \u2018\u015fey\u2019 k\u00fcmesi evrensel bir k\u00fcmedir.
\nBak\u0131n\u0131z \u201cMathematics and the metaphysicians\u201d adl\u0131 kitab\u0131nda Bertrand Russell ne demi\u015f?
\n\u201cHipotezimiz herhangi bir \u015fey hakk\u0131ndaysa ve belirli bir veya daha fazla \u015feyle ilgili de\u011filse, \u00e7\u0131kar\u0131mlar\u0131m\u0131z matemati\u011fi olu\u015fturur. Dolay\u0131s\u0131yla matematik, ne hakk\u0131nda konu\u015ftu\u011fumuzu asla bilmedi\u011fimiz bir konu veya s\u00f6yledi\u011fimizin do\u011fru olup olmad\u0131\u011f\u0131 olarak tan\u0131mlanabilir.\u201d (****)<\/p>\n
\u00d6klid\u2019in nokta ve \u00e7izgi tan\u0131mlar\u0131 i\u00e7in \u015fu ifadeler de mevcut: \u2018Nokta, b\u00fcy\u00fckl\u00fc\u011f\u00fc olmayand\u0131r.\u2019 ve \u2018\u00c7izgi, eni olmayan uzunluktur.\u2019 (#) Dikkate \u015fayand\u0131r, burada (Kitap 1, sayfa 1, Tan\u0131mlar 1) \u2018Bir \u00e7izginin u\u00e7lar\u0131 noktalard\u0131r.\u2019 (Tan\u0131m 3) denmi\u015f olsa da \u00e7izginin noktalardan olu\u015ftu\u011funa dair herhangi bir ifade mevcut de\u011fildir. Ama her \u00e7izgi rastgele par\u00e7alara b\u00f6l\u00fcnebilir oldu\u011fu ve her par\u00e7an\u0131n u\u00e7lar\u0131 da noktalar olaca\u011f\u0131ndan, \u00e7izginin noktalardan olu\u015ftu\u011fu gibi bir mant\u0131ksal \u00e7\u0131karsamaya ula\u015f\u0131labilir.<\/p>\n
Ayn\u0131 kaynakta, 15. ve 16. Tan\u0131mlar \u00e7ember ve \u00e7ember merkezine ili\u015fkindir. Meali \u015f\u00f6yledir: Bir noktadan e\u015fit uzakl\u0131ktaki noktalar \u00e7ember olu\u015fturur. S\u00f6z\u00fc edilen ilk noktaya da \u00e7emberin merkezi denir.<\/p>\n
Bu konudaki bir sonraki yaz\u0131m\u0131zda (##), buradan ba\u015flay\u0131p \u00d6klid\u2019i ilk and\u0131\u011f\u0131m\u0131z sat\u0131ra dek geri gidece\u011fiz.<\/p>\n
(*) https:\/\/blog.metu.edu.tr\/caglart\/2024\/10\/16\/paradoks-mu-absurd-mu-b\/
\n(**) https:\/\/www.jstor.org\/action\/doBasicSearch?Query=0.999&so=rel
\n(***)https:\/\/web.archive.org\/web\/20170701145939\/http:\/\/aleph0.clarku.edu\/~djoyce\/java\/elements\/elements.html
\n(***) Bu al\u0131nt\u0131n\u0131n kayna\u011f\u0131 i\u00e7in Bkz., https:\/\/tr.wikipedia.org\/wiki\/%C3%96klid_geometrisi
\n(#) \u00d6klid\u2019in Elemanlar\u0131, T\u00fcrk\u00e7esi ve notlar: Ali Sinan Sert\u00f6z
\nhttp:\/\/mat.msgsu.edu.tr\/~dpierce\/Dersler\/113\/2018\/oklid-kitap-i-sinan-sertoz-2018-5-8.pdf
\n(##) SAYI, \u015eEK\u0130L, \u00d6L\u00c7\u00dcM \u2013b\u2013
\nFizik \u201311 \/ b\u2013<\/p>\n","protected":false},"excerpt":{"rendered":"
Fizik \u201311 \/ a\u2013 \u0130\u00e7erik: Ne yapt\u0131\u011f\u0131m\u0131z\u0131n fark\u0131nda olmak ya da ol_A_mamak! \u201cParadoks mu absurd mu? \u2013 b \u2013\u201c b\u00f6l\u00fcm\u00fcnde g\u00f6rd\u00fck (*); s\u0131f\u0131rdan sonraki virg\u00fclden sonraki basamaklar\u0131n hepsinde 9 olan say\u0131 0,999\u2026 i\u00e7in 1\u2019e e\u015fit diyen ve bu dedi\u011fini kan\u0131tlad\u0131\u011f\u0131 iddias\u0131nda olan hayli \u00e7ok say\u0131da makale mevcut. (**) Ayn\u0131 sa\u00e7mal\u0131k \u00e7er\u00e7evesinde 0,999\u2026 say\u0131s\u0131n\u0131n 1\u2019e (e\u015fit […]<\/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-374","post","type-post","status-publish","format-standard","hentry","category-genel"],"_links":{"self":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/posts\/374","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=374"}],"version-history":[{"count":0,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/posts\/374\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/media?parent=374"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/categories?post=374"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/tags?post=374"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}