{"id":62,"date":"2022-02-03T14:05:00","date_gmt":"2022-02-03T13:05:00","guid":{"rendered":""},"modified":"2023-08-04T22:45:23","modified_gmt":"2023-08-04T20:45:23","slug":"najkrajsia-matematicka-rovnica","status":"publish","type":"post","link":"https:\/\/www.jankafialka.sk\/?p=62","title":{"rendered":"Najkraj\u0161ia matematick\u00e1 rovnica"},"content":{"rendered":"\n

Dne\u0161n\u00fd pr\u00edspevok chcem venova\u0165 jednej rovnici, ktor\u00fa mnoh\u00ed matematici pova\u017euj\u00fa za t\u00fa najkraj\u0161iu. Paradoxne, v zozname dvan\u00e1stich najkraj\u0161\u00edch rovn\u00edc (nielen matematick\u00fdch) som ju nena\u0161la. Pravdou toti\u017e je, \u017ee jej, na rozdiel od napr\u00edklad Pytagorovej vety, be\u017en\u00fd \u010dlovek ve\u013emi nerozumie. Znie ale ve\u013emi jednoducho: e<\/i> umocnen\u00e9 na i\u03c0 +1=0. To je pod\u013ea m\u0148a t\u00e1 najkraj\u0161ia matematick\u00e1 rovnica.<\/p>\n\n\n\n

Najkraj\u0161ie matematick\u00e9 kon\u0161tanty<\/h2>\n\n\n\n

Pre\u010do je kr\u00e1sna? Jeden z d\u00f4vodov je, \u017ee obsahuje hne\u010f nieko\u013eko d\u00f4le\u017eit\u00fdch matematick\u00fdch kon\u0161t\u00e1nt: nulu, jednotku, e<\/i>, i a \u03c0. Prv\u00e9 dve pozn\u00e1 ka\u017ed\u00fd. Jednotka ako z\u00e1klad v\u0161etk\u00e9ho po\u010d\u00edtania. Nula ako koncept, ktor\u00e9ho vznik nebol v\u00f4bec jednoduch\u00fd. Ch\u00e1pete, urobi\u0165 nie\u010do z ni\u010doho. Znak pre nulu kedysi neexistoval. S pr\u00edchodom pozi\u010dnej \u010d\u00edselnej s\u00fastavy bola nula len pr\u00e1zdne miesto. Nesk\u00f4r sa zap\u00edsala ako mal\u00e9 koliesko (u Mayov pr\u00e1zdna last\u00fara) a zrazu z nej bolo nie\u010do.<\/p>\n\n\n\n

V\u0161eobecne zn\u00e1me je aj \u010d\u00edslo \u03c0. Nach\u00e1dza sa v mnoh\u00fdch vtipoch a ve\u013emi \u010dasto sa zamie\u0148a za hodnotu 3,14. Na to som tro\u0161ku alergick\u00e1. \u03c0 nie je 3,14! To je len pribli\u017en\u00e1 hodnota. \u03c0 m\u00e1 v skuto\u010dnosti nekone\u010dne ve\u013ea desatinn\u00fdch \u010d\u00edslic a neexistuje ani zlomok, ktor\u00fd by sa rovnal presne \u010d\u00edslu \u03c0. Tak\u017ee ani 22\/7 nie je \u03c0, ale odhad je to dobr\u00fd. To sa mus\u00ed Archimedovi uzna\u0165. On si bol vedom\u00fd, \u017ee je to pribli\u017en\u00e1 hodnota. Dokonca vedel aj, \u017ee je o nie\u010do v\u00e4\u010d\u0161ia ako samotn\u00e9 \u03c0. O \u010d\u00edsle \u03c0 by sa dala nap\u00edsa\u0165 cel\u00e1 kniha. A veru, \u017ee sa aj nejak\u00e9 nap\u00edsali. \ud83d\ude09<\/p>\n\n\n\n

E\u0161te je tam Eulerovo \u010d\u00edslo e<\/i>, ktor\u00e9 je, podobne ako \u03c0, iracion\u00e1lne. (M\u00e1 nekone\u010dne ve\u013ea desatinn\u00fdch \u010d\u00edslic, ned\u00e1 sa zap\u00edsa\u0165 ako zlomok.) \u010c\u00edslo e je pribli\u017ene 2,718. O \u0148om by sa tie\u017e dalo nap\u00edsa\u0165 ve\u013emi ve\u013ea. Spomeniem len exponenci\u00e1lnu funkciu, preto\u017ee pr\u00e1ve t\u00e1 s\u00favis\u00ed s rovnicou, o ktorej je tento text. Exponenci\u00e1lna funkcia je funkcia e<\/i> na x<\/i>-t\u00fa. Zauj\u00edmav\u00e9 na nej je to, \u017ee je im\u00fanna vo\u010di derivovaniu a integrovaniu. Tak\u017ee jej deriv\u00e1cia je zasa e<\/i> na x<\/i>-t\u00fa. Samozrejme, ak derivujeme pod\u013ea x<\/i>. V tom spo\u010d\u00edta tie\u017e jeden zn\u00e1my vtip.<\/p>\n\n\n\n

Potom tam m\u00e1me i<\/i>. Toto i<\/i> sa naz\u00fdva imagin\u00e1rna jednotka a d\u00e1va n\u00e1m najavo, \u017ee cel\u00e1 rovnica bude v obore komplexn\u00fdch \u010d\u00edsel. Kto e\u0161te nepo\u010dul o komplexn\u00fdch \u010d\u00edslach, tak len podotknem, \u017ee pre i<\/i> plat\u00ed, \u017ee jej druh\u00e1 mocnina je -1. Alebo, inak povedan\u00e9 i<\/i> je odmocnina z m\u00ednus jednotky. \u00c1no, v re\u00e1lnych \u010d\u00edslach nemo\u017en\u00e9. Komplexn\u00e9 \u010d\u00edsla s\u00fa toti\u017e nadmno\u017einou t\u00fdch re\u00e1lnych. T.j. ka\u017ed\u00e9 re\u00e1lne \u010d\u00edslo je aj komplexn\u00e9, ale nie ka\u017ed\u00e9 komplexn\u00e9 \u010d\u00edslo je re\u00e1lne.<\/p>\n\n\n\n

Toto je len n\u00e1\u010drt toho, \u010do v\u0161etko sa matematikovi vynor\u00ed v mysli, ke\u010f sa na t\u00fato rovnicu d\u00edva. Kedysi som \u010d\u00edtala o experimente, v ktorom ukazovali matematikom v\u00fdber pekn\u00fdch rovn\u00edc a pritom pozorovali aktivitu ich mozgu na magnetickej rezonancii. Zistili, \u017ee je rovnak\u00e1 ako u umelcov sleduj\u00facich ve\u013ek\u00e9 umeleck\u00e9 diela. Ja to hovor\u00edm st\u00e1le: matematika je umenie<\/i>. Preto m\u00f4\u017eem opr\u00e1vnene naz\u00fdva\u0165 Eulerovu identitu jednou z najkraj\u0161\u00edch rovn\u00edc v\u00f4bec.<\/p>\n\n\n\n

D\u00f4kaz platnosti<\/h2>\n\n\n\n

Neodpust\u00edm si e\u0161te kr\u00e1tku pozn\u00e1mku o tom, pre\u010do vlastne t\u00e1to rovnos\u0165 plat\u00ed. Odvodi\u0165 ju nie je \u00fapln\u00e1 banalita. Mus\u00edme nie\u010do vedie\u0165 nielen o komplexn\u00fdch \u010d\u00edslach ale aj o Taylorov\u00fdch rozvojoch funkci\u00ed. Taylorov rozvoj je nekone\u010dn\u00fd s\u00fa\u010det, ktor\u00fdm mo\u017eno nahradi\u0165 p\u00f4vodn\u00fa funkciu. S\u010d\u00edtancov je v \u0148om s\u00edce nekone\u010dne ve\u013ea, ale s\u00fa jednoduch\u0161ie ako p\u00f4vodn\u00e1 funkcia. Taylorov rozvoj alebo Taylorov rad pre exponenci\u00e1lnu funkciu sem p\u00edsa\u0165 nejdem. (St\u00e1le p\u00ed\u0161em z mobilu.) \u013dahko si ho vyg\u00faglite. A pozrite si pritom aj Taylorov rad pre s\u00ednus a kos\u00ednus x<\/i>. \u010co m\u00e1 s t\u00fdm spolo\u010dn\u00e9 s\u00ednus a kos\u00ednus? Hm, ako to poda\u0165 jednoducho? Ka\u017ed\u00e9 komplexn\u00e9 \u010d\u00edslo sa d\u00e1 nap\u00edsa\u0165 v tzv. goniometrickom tvare: z = |z|<\/i>(cosα+i\u2022<\/i>sinα). Komplexn\u00e9 \u010d\u00edsla s\u00fa tak trochu dvojrozmern\u00e9. Aj sa tak zobrazuj\u00fa. Vodorovn\u00e1 os ozna\u010duje re\u00e1lnu \u010das\u0165, zvisl\u00e1 imagin\u00e1rnu. To z<\/i> v absol\u00fatnej hodnote je vzdialenos\u0165 od nuly a uhol α je uhol, ktor\u00fd zviera spojnica bodu z<\/i> s nulou s re\u00e1lnou osou.<\/p>\n\n\n\n

Pomocou t\u00fdch troch Taylorov\u00fdch radov zist\u00edme, \u017ee e<\/i> umocnen\u00e9 na i<\/i>\u03c0 je to ist\u00e9 ako cos \u03c0 + i\u2022<\/i>sin \u03c0. To plat\u00ed pre ak\u00fdko\u013evek uhol, nielen \u03c0. No, a ke\u010f\u017ee \u03c0 je uhol 180\u00b0, jeho kos\u00ednus je -1 a s\u00ednus nula, hne\u010f vid\u00edme platnos\u0165 Eulerovej identity.<\/p>\n\n\n\n

Kedysi mi e\u0161te nebolo zrejm\u00e9, pre\u010do je 180\u00b0 pr\u00e1ve \u03c0, tak to sem e\u0161te dod\u00e1m. Uhly sa meraj\u00fa v stup\u0148och z historick\u00fdch d\u00f4vodov. Vedecky spr\u00e1vne je ale mera\u0165 uhly v radi\u00e1noch. Jeden radi\u00e1n je uhol, ktor\u00fd zodpoved\u00e1 obl\u00fa\u010diku d\u013a\u017eky jeden na jednotkovej kru\u017enici. (To je kru\u017enica s polomerom 1.) Radi\u00e1nu sa hovor\u00ed aj obl\u00fakov\u00e1 miera, preto\u017ee uhly s\u00fa reprezentovan\u00e9 d\u013a\u017ekami obl\u00fa\u010dikov na jednotkovej kru\u017enici. A ke\u010f\u017ee cel\u00e1 kru\u017enica s polomerom 1 m\u00e1 d\u013a\u017eku 2\u03c0, tak je jasn\u00e9, \u017ee jej polovica bude \u03c0 a to je z\u00e1rove\u0148 uhol 180\u00b0.<\/p>\n\n\n\n

\"\"
Nie\u010do pekn\u00e9<\/figcaption><\/figure>\n\n\n\n

\u010co doda\u0165? Myslela som, \u017ee po mojom pr\u00edspevku o logaritmick\u00fdch tabu\u013ek\u00e1ch nap\u00ed\u0161em nie\u010do jednoduch\u0161ie. No, nevydalo. \ud83d\ude04<\/p>\n","protected":false},"excerpt":{"rendered":"

Dne\u0161n\u00fd pr\u00edspevok chcem venova\u0165 jednej rovnici, ktor\u00fa mnoh\u00ed matematici pova\u017euj\u00fa za t\u00fa najkraj\u0161iu. Paradoxne, v zozname dvan\u00e1stich najkraj\u0161\u00edch rovn\u00edc (nielen matematick\u00fdch) som ju nena\u0161la. Pravdou toti\u017e je, \u017ee jej, na rozdiel od napr\u00edklad Pytagorovej vety, be\u017en\u00fd \u010dlovek ve\u013emi nerozumie. Znie ale ve\u013emi jednoducho: e umocnen\u00e9 na i\u03c0 +1=0. To je pod\u013ea m\u0148a t\u00e1 najkraj\u0161ia matematick\u00e1 […]<\/p>\n","protected":false},"author":2,"featured_media":1952,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[11,12,13,14,15,16],"class_list":["post-62","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-nezaradene","tag-e","tag-eulerova-identita","tag-komplexne-cisla","tag-krasne-rovnice","tag-oblukova-miera","tag-16"],"yoast_head":"\nNajkraj\u0161ia matematick\u00e1 rovnica - jankafialka<\/title>\n<meta name=\"description\" content=\"Najkraj\u0161ia matematick\u00e1 rovnica obsahuje hne\u010f nieko\u013eko zauj\u00edmav\u00fdch \u010dlenov. 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