{"id":2548,"date":"2023-02-13T11:42:38","date_gmt":"2023-02-13T11:42:38","guid":{"rendered":"https:\/\/www.juniorjavadeveloper.pl\/?p=2548"},"modified":"2023-11-21T17:53:05","modified_gmt":"2023-11-21T17:53:05","slug":"korzen-kwadratu-a-postep-geometryczny-spotkanie-fibonacciego-z-feynmanem","status":"publish","type":"post","link":"https:\/\/www.juniorjavadeveloper.pl\/en\/2023\/02\/13\/korzen-kwadratu-a-postep-geometryczny-spotkanie-fibonacciego-z-feynmanem\/","title":{"rendered":"The root of the square and geometric progression - Fibonacci's meeting with Feynman"},"content":{"rendered":"<p class=\"wp-block-paragraph\">Artyku\u0142 ten b\u0119dzie inny ni\u017c poprzednie, nie b\u0119dzie o programowaniu w j\u0119zyku Java z perspektywy m\u0142odszego programisty, programistki. Nie b\u0119dzie on wyk\u0142adem z matematyki, ale opisem mojego podej\u015bcia do pr\u00f3by zrozumienia poj\u0119\u0107 i wzor\u00f3w matematycznych. Poj\u0119\u0107 i wzor\u00f3w, kt\u00f3re &#8220;dobrze znamy na pami\u0119\u0107&#8221; z lekcji matematyki, ale czy, aby na pewno dobrze je rozumiemy? Artyku\u0142 ten b\u0119dzie moj\u0105 pr\u00f3b\u0105 udokumentowania \u015bcie\u017cki, kt\u00f3r\u0105 przeszed\u0142em od korzenia kwadratu (pierwiastka kwadratowego), poprzez post\u0119p geometryczny (pol.: ci\u0105g, ang.: sequence), a\u017c do &#8220;spotkania&#8221; Fibonacciego z Feynmanem.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sk\u0105d w og\u00f3le pomys\u0142 na nauk\u0119 w inny spos\u00f3b ni\u017c ten z zaj\u0119\u0107 szkolnych, czy te\u017c ten ze studi\u00f3w? Po prostu uczenie si\u0119 na pami\u0119\u0107 poj\u0119\u0107 i wzor\u00f3w bez praktycznego zastosowania i dog\u0142\u0119bnego zrozumienia tematu nigdy nie mia\u0142o dla mnie sensu. Ucz\u0105c si\u0119 nowego poj\u0119cia, wzoru zawsze zadawa\u0142em sobie pytanie &#8220;Po co mi to? Do czego mog\u0119, to wykorzysta\u0107?&#8221;. Ciekawym podej\u015bciem jest <a rel=\"noreferrer noopener\" href=\"https:\/\/pl.wikipedia.org\/wiki\/Technika_Feynmana\" data-type=\"URL\" data-id=\"https:\/\/pl.wikipedia.org\/wiki\/Technika_Feynmana\" target=\"_blank\">Technika Feynmana<\/a>. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">W podr\u0119czniku do matematyki dla lice\u00f3w i technik\u00f3w z 2015 r., mo\u017cna znale\u017a\u0107 &#8220;praktyczne zastosowanie&#8221; ci\u0105gu geometrycznego na przyk\u0142adzie &#8220;Lokaty pieni\u0119\u017cne i kredyty bankowe&#8221;. <strong>Czy na prawd\u0119 trzeba by\u0142o ca\u0142y podrozdzia\u0142 po\u015bwi\u0119ci\u0107 temu jak bra\u0107 kredyty i oblicza\u0107 lokaty?<\/strong> Od razu przypominaj\u0105 mi si\u0119 moje lekcje &#8220;przedsi\u0119biorczo\u015bci&#8221;, gdzie wype\u0142niali\u015bmy PITy, czyli formularze deklaracji podatkowych. Zar\u00f3wno lokaty, kredyty oraz PITy opieraj\u0105 si\u0119 na czym\u015b wymy\u015blonym przez cz\u0142owiek i zmiennym w czasie. <strong>Potrzebowa\u0142em czego\u015b, co jest &#8220;od zawsze&#8221; w otaczaj\u0105cym nas \u015bwiecie i mo\u017cna spr\u00f3bowa\u0107, to ubra\u0107 w poj\u0119cia i wzory matematyczne.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Zdaj\u0119 sobie spraw\u0119, \u017ce na rynku wydawniczym jest ca\u0142a masa ksi\u0105\u017cek, kt\u00f3re lepiej lub gorzej pr\u00f3buj\u0105 wyt\u0142umaczy\u0107 skomplikowane poj\u0119cia i wzory matematyczne w spos\u00f3b przyst\u0119pny dla czytelnik\u00f3w o r\u00f3\u017cnym poziomie wiedzy. Na swojej drodze napotka\u0142em poni\u017csze ksi\u0105\u017cki, kt\u00f3re podejmuj\u0105 pr\u00f3b\u0119 wyja\u015bnienia poj\u0119\u0107 matematycznych w przyst\u0119pny spos\u00f3b:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a rel=\"noreferrer noopener\" href=\"http:\/\/helion.pl\/view\/20365c\/jatedo_ebook.htm\" data-type=\"URL\" data-id=\"http:\/\/helion.pl\/view\/20365c\/jatedo_ebook.htm\" target=\"_blank\">Jak tego dowie\u015b\u0107 &#8211; kr\u00f3tka opowie\u015b\u0107. Dowody matematyczne dla ka\u017cdego<\/a>,<\/li>\n\n\n\n<li><a href=\"http:\/\/helion.pl\/view\/20365c\/e_0nlv_ebook.htm\" data-type=\"URL\" data-id=\"http:\/\/helion.pl\/view\/20365c\/e_0nlv_ebook.htm\" target=\"_blank\" rel=\"noreferrer noopener\">Zanurzeni w liczbach. Jak matematyka kszta\u0142tuje nasze \u017cycie<\/a>,<\/li>\n\n\n\n<li><a href=\"http:\/\/helion.pl\/view\/20365c\/s_002g_ebook.htm\" data-type=\"URL\" data-id=\"http:\/\/helion.pl\/view\/20365c\/s_002g_ebook.htm\" target=\"_blank\" rel=\"noreferrer noopener\">Ciekawe zadania z matematyki. Zbi\u00f3r zada\u0144 dla zainteresowanego matematyk\u0105 licealisty<\/a>,<\/li>\n\n\n\n<li><a href=\"https:\/\/lubimyczytac.pl\/ksiazka\/193908\/17-rownan-ktore-zmienily-swiat\" data-type=\"URL\" data-id=\"https:\/\/lubimyczytac.pl\/ksiazka\/193908\/17-rownan-ktore-zmienily-swiat\" target=\"_blank\" rel=\"noreferrer noopener\">17 r\u00f3wna\u0144, kt\u00f3re zmieni\u0142y \u015bwiat<\/a>.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Do powy\u017cszego wst\u0119pu dotycz\u0105cego samego sposobu uczenia si\u0119 dodam jeszcze kwesti\u0119 szybkiej dezaktualizacji, nieprawdziwo\u015bci informacji, kt\u00f3r\u0105 jeste\u015bmy obecnie zalewani we wsp\u00f3\u0142czesnym \u015bwiecie. Dlatego postanowi\u0142em, \u017ce moje \u017ar\u00f3d\u0142a wiedzy musz\u0105 by\u0107 &#8220;starsze ni\u017c 40 lat&#8221;, a wiedza w nich zawarta b\u0119dzie do\u015b\u0107 niezmienna w czasie.<\/p>\n\n\n\n<div class=\"wp-block-columns alignwide are-vertically-aligned-center is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\"><!DOCTYPE html PUBLIC \"-\/\/W3C\/\/DTD HTML 4.0 Transitional\/\/EN\" \"http:\/\/www.w3.org\/TR\/REC-html40\/loose.dtd\">\n<html><body><figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"318\" height=\"440\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Philosophy-and-Practice.jpg\" alt=\"\" class=\"wp-image-2628\" srcset=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Philosophy-and-Practice.jpg 318w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Philosophy-and-Practice-217x300.jpg 217w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Philosophy-and-Practice-289x400.jpg 289w\" sizes=\"auto, (max-width: 318px) 100vw, 318px\" \/><\/figure>\n<\/body><\/html>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"wp-block-paragraph\">Natrafi\u0142em na ksi\u0105\u017ck\u0119 <a rel=\"noreferrer noopener\" href=\"https:\/\/www.goodreads.com\/book\/show\/180660.Sacred_Geometry\" data-type=\"URL\" data-id=\"https:\/\/www.goodreads.com\/book\/show\/180660.Sacred_Geometry\" target=\"_blank\">Sacred Geometry: Philosophy and Practice &#8211; Robert Lawlor<\/a> (data pierwszego wydania 1982 rok, 41 lat temu), pomy\u015bla\u0142em, \u017ce, to co\u015b dla mnie, b\u0119dzie o jednym z dzia\u0142\u00f3w matematyki &#8211; geometria &#8211; do tego w kontek\u015bcie staro\u017cytnych cywilizacji.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#8220;The thinkers of ancient Egypt, Greece and India recognized that numbers governed much of what they saw in their world and hence provided an approach to its divine creator.&#8221;<\/p>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Opisuj\u0105c poni\u017cej poj\u0119cia matematyczne b\u0119d\u0119 r\u00f3wnolegle stosowa\u0142 polskie i angielskie okre\u015blenia termin\u00f3w matematycznych, gdy\u017c moim zdaniem niekt\u00f3re &#8220;t\u0142umaczenia&#8221; poj\u0119\u0107 gubi\u0105 pierwotne znaczenie w j\u0119zyku polskim.<\/strong> Oczywi\u015bcie, to moja subiektywna opinia, ale zawsze \u0142atwiej by\u0142o mi zrozumie\u0107 dane poj\u0119cie matematyczne, kiedy pozna\u0142em jego angloj\u0119zyczny odpowiednik dla j\u0119zyka polskiego.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Powy\u017csza ksi\u0105\u017cka &#8211; Sacred Geometry &#8211; pozwoli\u0142a mi zrozumie\u0107 poj\u0119cie pierwiastka kwadratowego bez u\u017cywania jakiegokolwiek wzoru matematycznego. Ca\u0142a wiedza dotycz\u0105ca pierwiastka kwadratowego oraz ci\u0105gu geometrycznego zosta\u0142a pokazana poprzez rysowanie po\u0142\u0105czonych &#8211; za pomoc\u0105 przek\u0105tnej &#8211; ze sob\u0105 kwadrat\u00f3w, z kt\u00f3rych pierwszy ma boki o d\u0142ugo\u015bci 1 cm.<\/p>\n\n\n<p>Obliczaj\u0105c na kalkulatorze <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-49a08670c0630c1ddc4a8404fe4a9300_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"29\" style=\"vertical-align: -4px;\"\/> otrzymujemy wynik 1,414213562. Tyle r\u00f3wnie\u017c wynosi przek\u0105tna kwadratu, kt\u00f3rego bok ma d\u0142ugo\u015b\u0107 1 cm.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Poni\u017cej rysunki prezentuj\u0105ce ci\u0105g geometryczny wykorzystuj\u0105cy pierwiastek kwadratowy (ang. square root). Rysunki wykonane zgodnie z instrukcjami z ksi\u0105\u017cki Sacred Geometry.<\/p>\n\n\n\n<div class=\"wp-block-columns alignwide are-vertically-aligned-center is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\"><!DOCTYPE html PUBLIC \"-\/\/W3C\/\/DTD HTML 4.0 Transitional\/\/EN\" \"http:\/\/www.w3.org\/TR\/REC-html40\/loose.dtd\">\n<html><body><figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"559\" height=\"944\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Zrzut-ekranu-z-2023-02-08-20-51-57.png\" alt=\"\" class=\"wp-image-2634\" srcset=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Zrzut-ekranu-z-2023-02-08-20-51-57.png 559w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Zrzut-ekranu-z-2023-02-08-20-51-57-178x300.png 178w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Zrzut-ekranu-z-2023-02-08-20-51-57-533x900.png 533w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Zrzut-ekranu-z-2023-02-08-20-51-57-474x800.png 474w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Zrzut-ekranu-z-2023-02-08-20-51-57-355x600.png 355w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Zrzut-ekranu-z-2023-02-08-20-51-57-237x400.png 237w\" sizes=\"auto, (max-width: 559px) 100vw, 559px\" \/><figcaption class=\"wp-element-caption\">Drawing from the book Sacred Geometry: Philosophy and Practice, page 26, figure 1.4 <\/figcaption><\/figure>\n<\/body><\/html>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\"><!DOCTYPE html PUBLIC \"-\/\/W3C\/\/DTD HTML 4.0 Transitional\/\/EN\" \"http:\/\/www.w3.org\/TR\/REC-html40\/loose.dtd\">\n<html><body><figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1998\" height=\"2560\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/1675883966775-scaled.jpg\" alt=\"\" class=\"wp-image-2635\" srcset=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/1675883966775-scaled.jpg 1998w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/1675883966775-234x300.jpg 234w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/1675883966775-799x1024.jpg 799w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/1675883966775-768x984.jpg 768w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/1675883966775-1199x1536.jpg 1199w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/1675883966775-1599x2048.jpg 1599w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/1675883966775-703x900.jpg 703w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/1675883966775-625x800.jpg 625w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/1675883966775-468x600.jpg 468w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/1675883966775-312x400.jpg 312w\" sizes=\"auto, (max-width: 1998px) 100vw, 1998px\" \/><figcaption class=\"wp-element-caption\">Moj odreczny rysunek zgodny z instrukcjami z Sacred Geometry: Philosophy and Practice, str. 25-26, rys. 1.1. &#8211; 1.4. <\/figcaption><\/figure>\n<\/body><\/html>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Podstaw\u0105 kolejnego kwadratu staje si\u0119 przek\u0105tna poprzedniego &#8211; w tym momencie znalaz\u0142em wyja\u015bnienie, pochodzenie poj\u0119cia &#8220;square root&#8221; &#8211; w wolnym t\u0142umaczeniu korze\u0144 kwadratu. Post\u0119puj\u0105c (ang. progression) kolejno i rysuj\u0105c nowe kwadraty, kt\u00f3rych podstaw\u0105 s\u0105 przek\u0105tne poprzednich kwadrat\u00f3w otrzymujemy ci\u0105g (ang. sequence) geometryczny (ang. geometric progression). <strong>W tym miejscu znajduje si\u0119 wyja\u015bnienie jednej z cz\u0119\u015bci &#8220;tajemniczego&#8221; tytu\u0142u mojego artyku\u0142u.<\/strong><\/p>\n\n\n\n<style id=\"bwg-style-0\">    #bwg_container1_0 #bwg_container2_0 .bwg-container-0.bwg-standard-thumbnails {      width: 1254px;              justify-content: center;        margin:0 auto !important;              background-color: rgba(255, 255, 255, 0.00);            padding-left: 4px;      padding-top: 4px;      max-width: 100%;            }        #bwg_container1_0 #bwg_container2_0 .bwg-container-0.bwg-standard-thumbnails .bwg-item {    justify-content: flex-start;      max-width: 250px;          }    #bwg_container1_0 #bwg_container2_0 .bwg-container-0.bwg-standard-thumbnails .bwg-item 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<\/script>    \n\n\n\n<p class=\"wp-block-paragraph\">Powy\u017csza galeria rysunk\u00f3w zawiera instrukcje z ksi\u0105zki Sacred Geometry: Philosophy and Practice, str. 25-26, rys. 1.1. &#8211; 1.4., kt\u00f3re pozwol\u0105 na samodzielne narysowanie korzenia kwadratu (ang. square root, pol. pierwiastek kwadratowy) w kontek\u015bcie ci\u0105gu geometrycznego (ang. geometric progression).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Poni\u017csze obrazy pokazuj\u0105 jak wygl\u0105da\u0142y narz\u0119dzia &#8211; &#8220;do geometrii&#8221; &#8211; u\u017cywane przez staro\u017cytnych oraz wsp\u00f3\u0142czesne odpowiedniki narz\u0119dzi, kt\u00f3rych sam u\u017cywam.<\/p>\n\n\n\n<div class=\"wp-block-columns alignwide are-vertically-aligned-center is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\"><!DOCTYPE html PUBLIC \"-\/\/W3C\/\/DTD HTML 4.0 Transitional\/\/EN\" \"http:\/\/www.w3.org\/TR\/REC-html40\/loose.dtd\">\n<html><body><figure class=\"wp-block-image size-full\"><a href=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Ancient.jpg\" target=\"_blank\" rel=\"noreferrer noopener\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"661\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Ancient.jpg\" alt=\"Sacred Geometry Tools - Ancient\" class=\"wp-image-2651\" srcset=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Ancient.jpg 1024w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Ancient-300x194.jpg 300w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Ancient-768x496.jpg 768w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Ancient-1000x646.jpg 1000w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Ancient-800x516.jpg 800w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Ancient-600x387.jpg 600w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Ancient-400x258.jpg 400w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><figcaption class=\"wp-element-caption\">Sacred Geometry Tools &#8211; starozytne narzedzia &#8220;do geometrii&#8221;.<\/figcaption><\/figure>\n<\/body><\/html>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\"><!DOCTYPE html PUBLIC \"-\/\/W3C\/\/DTD HTML 4.0 Transitional\/\/EN\" \"http:\/\/www.w3.org\/TR\/REC-html40\/loose.dtd\">\n<html><body><figure class=\"wp-block-image size-full\"><a href=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Contemporary.jpg\" target=\"_blank\" rel=\"noreferrer noopener\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"766\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Contemporary.jpg\" alt=\"Sacred Geometry Tools - Contemporary\" class=\"wp-image-2653\" srcset=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Contemporary.jpg 1024w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Contemporary-300x224.jpg 300w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Contemporary-768x575.jpg 768w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Contemporary-1000x748.jpg 1000w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Contemporary-800x598.jpg 800w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Contemporary-600x449.jpg 600w, https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Sacred-Geometry_Tools-Contemporary-400x299.jpg 400w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><figcaption class=\"wp-element-caption\">My modern equivalents of geometry tools.<\/figcaption><\/figure>\n<\/body><\/html>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Dodatkowo w trakcie pisania tego artyku\u0142u uczy\u0142em si\u0119 w praktyce jak zapisywa\u0107 wzory matematyczne z wykorzystaniem <a rel=\"noreferrer noopener\" href=\"https:\/\/pl.wikipedia.org\/wiki\/LaTeX\" data-type=\"URL\" data-id=\"https:\/\/pl.wikipedia.org\/wiki\/LaTeX\" target=\"_blank\">LaTeX<\/a>. &#8220;LaTeX jest szeroko stosowany w \u015brodowisku akademickim do komunikacji i publikacji dokument\u00f3w naukowych z wielu dziedzin, w tym matematyki, informatyki [&#8230;]. Odgrywa r\u00f3wnie\u017c znacz\u0105c\u0105 rol\u0119 w przygotowywaniu i publikowaniu ksi\u0105\u017cek i artyku\u0142\u00f3w [&#8230;].&#8221;. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Tak wygl\u0105da wz\u00f3r matematyczny przed przetworzeniem przez LaTeX &#8211; <code>a_{n}=a \\cdot r^{n-1}<\/code> &#8211; tak wygl\u0105da po przetworzeniu <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-5916f5495c0ea30fa5099014278abf50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#110;&#125;&#61;&#97;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#114;&#94;&#123;&#110;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"122\" style=\"vertical-align: -3px;\"\/>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pokrzepiony i zainspirowany &#8220;odkryciem znaczenia&#8221; pierwiastka kwadratowego zabra\u0142em si\u0119 za zg\u0142\u0119bianie praktycznycznego zastosowania ci\u0105gu geometrycznego (ang. geometric progression). Ci\u0105g nazywamy geometrycznym, je\u015bli stosunek mi\u0119dzy kolejnymi wyrazami jest sta\u0142y. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pami\u0119ta\u0142em dwa wzory dla ci\u0105gu geometrycznego:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote has-normal-font-size is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"wp-block-paragraph\">Wz\u00f3r na <code>n<\/code>-ty wyraz ci\u0105gu geometrycznego &#8211; posta\u0107 jawna (ang. closed formula):<\/p>\n<cite><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-5916f5495c0ea30fa5099014278abf50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#110;&#125;&#61;&#97;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#114;&#94;&#123;&#110;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"122\" style=\"vertical-align: -3px;\"\/><\/cite><\/blockquote>\n\n\n\n<blockquote class=\"wp-block-quote has-normal-font-size is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"wp-block-paragraph\">Wz\u00f3r na <code>n<\/code>-ty wyraz ci\u0105gu geometrycznego &#8211; posta\u0107 rekurencyjna (ang. recursive formula):<\/p>\n<cite><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-daeee5fd695ed9d5ea8f3717fb775ad3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#110;&#125;&#61;&#97;&#95;&#123;&#110;&#45;&#49;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"124\" style=\"vertical-align: -3px;\"\/><\/cite><\/blockquote>\n\n\n\n<p class=\"wp-block-paragraph\">Przyk\u0142ad ci\u0105gu geometrycznego: <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-a78e1eddb5a22d43cc686badd3d9cf6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#44;&#32;&#54;&#44;&#32;&#49;&#50;&#44;&#32;&#50;&#52;&#44;&#32;&#92;&#100;&#111;&#116;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"130\" style=\"vertical-align: -5px;\"\/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sprawdz\u0119 czy mam do czynienia z ci\u0105giem geometrycznym dziel\u0105c ka\u017cdy wyraz przez jego poprzedni wyraz: <code>6\/3 = 2, 12\/6 = 2, 24\/12 = 2<\/code>. Zak\u0142adaj\u0105c, \u017ce ten stosunek jest sta\u0142y, znalaz\u0142em <code>r = 2<\/code> &#8211; ci\u0105g nazywamy geometrycznym, je\u015bli stosunek mi\u0119dzy kolejnymi wyrazami jest sta\u0142y.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">W podr\u0119cznikach do matematyki i na polskoj\u0119zycznej <a href=\"https:\/\/pl.wikipedia.org\/wiki\/Ci%C4%85g_geometryczny\" data-type=\"URL\" data-id=\"https:\/\/pl.wikipedia.org\/wiki\/Ci%C4%85g_geometryczny\" target=\"_blank\" rel=\"noreferrer noopener\">Wikipedii<\/a> zamiast <code>r<\/code> u\u017cywane jest <code>q<\/code>, ale dla mnie \u0142atwiej by\u0142o zrozumie\u0107 wersj\u0119 r\u00f3wnania z <code>r<\/code>, bo jest, to pierwsza litera od angielskiego wyrazu <code>ratio<\/code> &#8211; pol. stosunek, wsp\u00f3\u0142czynnik.<\/p>\n\n\n\n<div class=\"wp-block-columns alignwide are-vertically-aligned-top is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-vertically-aligned-top is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"wp-block-paragraph\">Wz\u00f3r na <code>n<\/code>-ty wyraz \u2013 posta\u0107 jawna.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-5916f5495c0ea30fa5099014278abf50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#110;&#125;&#61;&#97;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#114;&#94;&#123;&#110;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"122\" style=\"vertical-align: -3px;\"\/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Podstawiam <code>r = 2<\/code>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-c3945a58e91ec3213e7231f9fd1d675c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#110;&#125;&#61;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#94;&#123;&#110;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"122\" style=\"vertical-align: -3px;\"\/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Obliczam dla kolejnych wyraz\u00f3w ci\u0105gu.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-63a4442b3fd3df78b16ed54ad8751f79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#49;&#125;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -3px;\"\/><br><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-01382fe9b69dd951c1fa8749062b0b50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#50;&#125;&#61;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#94;&#123;&#50;&#45;&#49;&#125;&#61;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#94;&#123;&#49;&#125;&#32;&#61;&#32;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#32;&#61;&#32;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"306\" style=\"vertical-align: -3px;\"\/><br><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-cbd6f2bbaff31a1c6935febb53ff60ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#51;&#125;&#61;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#94;&#123;&#51;&#45;&#49;&#125;&#61;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#94;&#123;&#50;&#125;&#32;&#61;&#32;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#52;&#32;&#61;&#32;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"316\" style=\"vertical-align: -3px;\"\/><br><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-bbba192594667b06d2fd53b41ffebb4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#52;&#125;&#61;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#94;&#123;&#52;&#45;&#49;&#125;&#61;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#94;&#123;&#51;&#125;&#32;&#61;&#32;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#56;&#32;&#61;&#32;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"317\" style=\"vertical-align: -3px;\"\/><br><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-ebe88494998f3b41b49ae8bc79833fcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#53;&#125;&#61;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#94;&#123;&#53;&#45;&#49;&#125;&#61;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#94;&#123;&#52;&#125;&#32;&#61;&#32;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#49;&#54;&#32;&#61;&#32;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"328\" style=\"vertical-align: -3px;\"\/><\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-top is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"wp-block-paragraph\">Wz\u00f3r na <code>n<\/code>-ty wyraz \u2013 posta\u0107 rekurencyjna.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-daeee5fd695ed9d5ea8f3717fb775ad3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#110;&#125;&#61;&#97;&#95;&#123;&#110;&#45;&#49;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"124\" style=\"vertical-align: -3px;\"\/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Podstawiam <code>r = 2<\/code>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-37e5d7d7c6a6a436b13426f600f2f8e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#110;&#125;&#61;&#97;&#95;&#123;&#110;&#45;&#49;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -3px;\"\/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Obliczam dla kolejnych wyraz\u00f3w ci\u0105gu.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-63a4442b3fd3df78b16ed54ad8751f79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#49;&#125;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -3px;\"\/><br><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-fd7cc751a82ebf8c6d006e042efc77ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#50;&#125;&#61;&#97;&#95;&#123;&#50;&#45;&#49;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#32;&#61;&#32;&#97;&#95;&#123;&#49;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#32;&#61;&#32;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#32;&#61;&#32;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"307\" style=\"vertical-align: -3px;\"\/><br><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-cc87ddaff13176561cd1462cc3b3203b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#51;&#125;&#61;&#97;&#95;&#123;&#51;&#45;&#49;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#32;&#61;&#32;&#97;&#95;&#123;&#50;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#32;&#61;&#32;&#54;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#32;&#61;&#32;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"317\" style=\"vertical-align: -3px;\"\/><br><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-6f1d0905ed6b4c6db9e6aaedf9fc3542_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#52;&#125;&#61;&#97;&#95;&#123;&#52;&#45;&#49;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#32;&#61;&#32;&#97;&#95;&#123;&#51;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#32;&#61;&#32;&#49;&#50;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#32;&#61;&#32;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -3px;\"\/><br><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/ql-cache\/quicklatex.com-feb2259a672c8e595964671106897da5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#53;&#125;&#61;&#97;&#95;&#123;&#53;&#45;&#49;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#32;&#61;&#32;&#97;&#95;&#123;&#52;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#32;&#61;&#32;&#50;&#52;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#32;&#61;&#32;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -3px;\"\/><\/p>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Poni\u017cej zapis wzoru na <code>n<\/code>-ty wyraz ci\u0105gu geometrycznego w postaci jawnej &#8211; kod j\u0119zyka Java. U\u017cywam p\u0119tli <code>for<\/code> do obliczania kolejnych wyraz\u00f3w  w ci\u0105gu geometrycznym.<\/p>\n\n\n\n<div class=\"wp-block-columns alignwide is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"java\" data-enlighter-theme=\"\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">static int geometricProgressionLoop(int a_n, int r, int terms) {\n    int result = 0;\n    for (int n = 0; n &lt; terms; n++) {\n        result = (int) (a_n * Math.pow(r, n));\n    }\n    return result;\n}<\/pre>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Poni\u017cej zapis wzoru na <code>n<\/code>-ty wyraz ci\u0105gu geometrycznego w postaci rekurencyjnej &#8211; kod j\u0119zyka Java. Metoda <code>geometricProgressionRecursive()<\/code> wywo\u0142uje sam\u0105 siebie do obliczania kolejnych wyraz\u00f3w w ci\u0105gu geometrycznym.<\/p>\n\n\n\n<div class=\"wp-block-columns alignwide is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"java\" data-enlighter-theme=\"\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">static int geometricProgressionRecursive(int a_n, int r, int terms) {\n    if (a_n == terms) return 1;\n    return a_n * r * geometricProgressionRecursive(a_n + 1, r, terms);\n}<\/pre>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Poni\u017cej przyk\u0142ady praktycznego zastosowania ci\u0105gu geometrycznego:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Calculation of interest earned,<\/li>\n\n\n\n<li>Obliczenie kwoty na naszym koncie oszcz\u0119dno\u015bciowym,<\/li>\n\n\n\n<li>Obliczanie wielko\u015bci wyk\u0142adniczego wzrostu populacji, na przyk\u0142ad bakterii na szalce Petriego,<\/li>\n\n\n\n<li>Sko\u0144czony ci\u0105g geometryczny jest przyk\u0142adem odbijaj\u0105cej si\u0119 pi\u0142ki. Wysoko\u015b\u0107 pi\u0142ki zmniejsza si\u0119 o po\u0142ow\u0119 przy ka\u017cdym odbiciu.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Poni\u017cej najbardziej znany przyk\u0142ad u\u017cycia rekurencji, czyli <a rel=\"noreferrer noopener\" href=\"https:\/\/pl.wikipedia.org\/wiki\/Ci%C4%85g_Fibonacciego\" data-type=\"URL\" data-id=\"https:\/\/pl.wikipedia.org\/wiki\/Ci%C4%85g_Fibonacciego\" target=\"_blank\">ci\u0105g Fibonacciego<\/a>. Kod \u017ar\u00f3d\u0142owy w j\u0119zyku Java oraz Python. <\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"java\" data-enlighter-theme=\"\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">int fibonacci(int n) {\n    if (n &lt; 2) return n;\n    return fibonacci(n - 1) + fibonacci(n - 2);\n}<\/pre>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">def fibonacci(n):\n    if n &lt; 2:\n        return n\n    return fibonacci(n - 1) + fibonacci(n - 2)<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Osoby pocz\u0105tkuj\u0105ce mog\u0105 skorzysta\u0107 z narz\u0119dzi online, kt\u00f3re przedstawiaj\u0105 w formie graficznej &#8211; wizualizuj\u0105 &#8211; dzia\u0142ania algorytm\u00f3w:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.recursionvisualizer.com\/?function_definition=def%20virfib%28n%29%3A%0A%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20return%200%0A%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20return%201%0A%20%20else%3A%0A%20%20%20%20return%20virfib%28n%20-%201%29%20%2B%20virfib%28n%20-%202%29%0A%20%20%20%20&amp;function_call=virfib%283%29\" data-type=\"URL\" data-id=\"https:\/\/www.recursionvisualizer.com\/?function_definition=def%20virfib%28n%29%3A%0A%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20return%200%0A%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20return%201%0A%20%20else%3A%0A%20%20%20%20return%20virfib%28n%20-%201%29%20%2B%20virfib%28n%20-%202%29%0A%20%20%20%20&amp;function_call=virfib%283%29\" target=\"_blank\" rel=\"noreferrer noopener\">Visualize a recursive function<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/algorithm-visualizer.org\/dynamic-programming\/fibonacci-sequence\">https:\/\/algorithm-visualizer.org\/dynamic-programming\/fibonacci-sequence<\/a><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-dots\"\/>\n\n\n\n<p class=\"has-text-align-center has-large-font-size wp-block-paragraph\"><strong>Tym samym uzyska\u0142em obiecane w tytule artyku\u0142u spotkanie Fibonacciego z Feynmanem.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Podejmowa\u0142em ju\u017c r\u00f3\u017cne pr\u00f3by powtarzania i utrwalania wiedzy zdobytej w szkole, ale zawsze brakowa\u0142o mi, tego praktycznego i namacalnego zastosowania poj\u0119\u0107 i wzor\u00f3w matematycznych. Na dalszych etapach nauki zapewne spotkam si\u0119 z abstrakcyjnymi poj\u0119ciami i wzorami, kt\u00f3re nie maj\u0105 odzwierciedlenia w \u015bwiecie rzeczywistym, wtedy zastanowi\u0119 si\u0119 nad tym czy matematyka, aby nie jest tworem ludzkiego umys\u0142u i jest oderwana od rzeczywisto\u015bci. <a href=\"https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman\" data-type=\"URL\" data-id=\"https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman\" target=\"_blank\" rel=\"noreferrer noopener\">Feynman<\/a> twierdzi\u0142, \u017ce matematyka nie jest nauk\u0105 &#8211; <a href=\"https:\/\/www.goodreads.com\/quotes\/1459603-mathematics-is-not-a-science-from-our-point-of-view\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/www.goodreads.com\/quotes\/1459603-mathematics-is-not-a-science-from-our-point-of-view<\/a>:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"wp-block-paragraph\">Richard P. Feynman, The Feynman Lectures on Physics Vol 1<\/p>\n<cite>\u201cMathematics is not a science from our point of view, in the sense that it is not a natural science. The test of its validity is not experiment.\u201d.<\/cite><\/blockquote>\n\n\n\n<p class=\"wp-block-paragraph\">Patrz\u0105c na przyk\u0142ad z korzeniem kwadratu (ang. square root) znalezionego w ksi\u0105\u017cce &#8220;Sacred Geometry&#8230;&#8221; wraca moja nadzieja, \u017ce matematyka mo\u017ce by\u0107 pr\u00f3b\u0105 znalezienia sposobu na zapisanie wzorami otaczaj\u0105cego nas \u015bwiata. Tymczasem ja szukam innych \u017ar\u00f3de\u0142 &#8211; ni\u017c matematyka &#8211; opisuj\u0105cych otaczaj\u0105cy nas \u015bwiat.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Podsumowuj\u0105c, artyku\u0142 ten pozwoli\u0142 mi zastosowa\u0107 w praktyce: technik\u0119 Feynmana, zapis wzor\u00f3w matematycznych z wykorzystaniem LaTeX, kod j\u0119zyka programowania w odniesieniu do ci\u0105gu geometrycznego (ang. geometric progression). Wiedz\u0119, kt\u00f3r\u0105 tu przedstawi\u0142em nie przyswoi\u0142em w &#8220;jeden dzie\u0144&#8221;, jest ona wynikiem &#8220;kompilacji&#8221; wiedzy z r\u00f3\u017cnych \u017ar\u00f3de\u0142 z celowym pomini\u0119ciem &#8211; &#8220;dobrze przeszkolonego pracownik informacji turystycznej&#8221; &#8211; sztucznej inteligencji. Jestem ciekawy czy, taka forma artyku\u0142u si\u0119 spodoba?<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-wp-embed is-provider-poradnik-dla-przysz-ych-junior-java-developer-039-w wp-block-embed-poradnik-dla-przysz-ych-junior-java-developer-039-w\"><div class=\"wp-block-embed__wrapper\">\n<blockquote class=\"wp-embedded-content\" data-secret=\"Os1ZYUVgm5\"><a href=\"https:\/\/www.juniorjavadeveloper.pl\/en\/kontakt\/\">Contact<\/a><\/blockquote><iframe loading=\"lazy\" class=\"wp-embedded-content\" sandbox=\"allow-scripts\" security=\"restricted\" style=\"position: absolute; visibility: hidden;\" title=\"\u201eKontakt\u201d \u2014 Poradnik dla przysz\u0142ych Junior Java Developer&amp;apos;\u00f3w\" src=\"https:\/\/www.juniorjavadeveloper.pl\/kontakt\/embed\/#?secret=bBaHcSRtbw#?secret=Os1ZYUVgm5\" data-secret=\"Os1ZYUVgm5\" width=\"600\" height=\"338\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\"><\/iframe>\n<\/div><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-wp-embed is-provider-poradnik-dla-przysz-ych-junior-java-developer-039-w wp-block-embed-poradnik-dla-przysz-ych-junior-java-developer-039-w\"><div class=\"wp-block-embed__wrapper\">\n<blockquote class=\"wp-embedded-content\" data-secret=\"6zwK89ZuOk\"><a href=\"https:\/\/www.juniorjavadeveloper.pl\/en\/mentoring-rezerwuj-termin\/\">Mentoring - book an appointment<\/a><\/blockquote><iframe loading=\"lazy\" class=\"wp-embedded-content\" sandbox=\"allow-scripts\" security=\"restricted\" style=\"position: absolute; visibility: hidden;\" title=\"\u201eMentoring \u2013 rezerwuj termin\u201d \u2014 Poradnik dla przysz\u0142ych Junior Java Developer&amp;apos;\u00f3w\" src=\"https:\/\/www.juniorjavadeveloper.pl\/mentoring-rezerwuj-termin\/embed\/#?secret=3YRe3DulNm#?secret=6zwK89ZuOk\" data-secret=\"6zwK89ZuOk\" width=\"600\" height=\"338\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\"><\/iframe>\n<\/div><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Zdj\u0119cie autorstwa <a href=\"https:\/\/www.pexels.com\/photo\/top-view-photo-of-curved-road-surrounded-by-trees-1834403\/\" data-type=\"URL\" data-id=\"https:\/\/www.pexels.com\/photo\/top-view-photo-of-curved-road-surrounded-by-trees-1834403\/\" target=\"_blank\" rel=\"noreferrer noopener\">Yaroslav Shuraev<\/a> z <a rel=\"noreferrer noopener\" href=\"https:\/\/www.pexels.com\/pl-pl\/\" target=\"_blank\">Pexels<\/a>.<\/p>","protected":false},"excerpt":{"rendered":"<p>Artyku\u0142 ten b\u0119dzie inny ni\u017c poprzednie, nie b\u0119dzie o programowaniu w j\u0119zyku Java z perspektywy m\u0142odszego programisty, programistki. Nie b\u0119dzie on wyk\u0142adem z matematyki, ale opisem mojego podej\u015bcia do pr\u00f3by zrozumienia poj\u0119\u0107 i wzor\u00f3w matematycznych. Poj\u0119\u0107 i wzor\u00f3w, kt\u00f3re &#8220;dobrze znamy na pami\u0119\u0107&#8221; z lekcji matematyki, ale czy, aby na pewno dobrze je rozumiemy? Artyku\u0142 [&hellip;]<\/p>","protected":false},"author":1,"featured_media":2668,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_ppp_document_settings_meta":"","footnotes":"","jetpack_publicize_message":"","jetpack_is_tweetstorm":false,"jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false}}},"categories":[6],"tags":[],"class_list":["post-2548","post","type-post","status-publish","format-standard","has-post-thumbnail","category-felieton"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"https:\/\/www.juniorjavadeveloper.pl\/wp-content\/uploads\/2023\/02\/Korzen-kwadratu_Fibonacci-Feynman_Mentor-Junior-Java-Developer.jpg","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.juniorjavadeveloper.pl\/en\/wp-json\/wp\/v2\/posts\/2548","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.juniorjavadeveloper.pl\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.juniorjavadeveloper.pl\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.juniorjavadeveloper.pl\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.juniorjavadeveloper.pl\/en\/wp-json\/wp\/v2\/comments?post=2548"}],"version-history":[{"count":104,"href":"https:\/\/www.juniorjavadeveloper.pl\/en\/wp-json\/wp\/v2\/posts\/2548\/revisions"}],"predecessor-version":[{"id":5663,"href":"https:\/\/www.juniorjavadeveloper.pl\/en\/wp-json\/wp\/v2\/posts\/2548\/revisions\/5663"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.juniorjavadeveloper.pl\/en\/wp-json\/wp\/v2\/media\/2668"}],"wp:attachment":[{"href":"https:\/\/www.juniorjavadeveloper.pl\/en\/wp-json\/wp\/v2\/media?parent=2548"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.juniorjavadeveloper.pl\/en\/wp-json\/wp\/v2\/categories?post=2548"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.juniorjavadeveloper.pl\/en\/wp-json\/wp\/v2\/tags?post=2548"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}