{"id":26236,"date":"2022-06-14T19:07:19","date_gmt":"2022-06-14T17:07:19","guid":{"rendered":"https:\/\/mahifi.se\/?page_id=26236"},"modified":"2025-03-14T14:31:21","modified_gmt":"2025-03-14T12:31:21","slug":"vad-ar-egentligen-pi","status":"publish","type":"page","link":"https:\/\/mahifi.se\/?page_id=26236","title":{"rendered":"Vad \u00e4r egentligen pi?"},"content":{"rendered":"\t\t
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Vad \u00e4r egentligen \\pi <\/span>?<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\\pi <\/span> \u00e4r konstanten som representerar f\u00f6rh\u00e5llandet mellan omkretsen och diametern i en cirkel. \\pi <\/span>\u00a0approximeras till \\approx <\/span>\u00a0<\/span>3.14159265<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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St\u00e4llen d\u00e4r \\pi <\/span> dyker upp <\/b><\/p>

Leonhard Euler<\/b> formulerade en o\u00e4ndlig serie som visade sig ha ett r\u00e4tt ov\u00e4ntat resultat <\/b><\/p> \\frac{\\pi^2}{6}=\u00a0\\frac{1}{1^2}+\\frac{1}{2^2}+\\frac{1}{3^2}+\\frac{1}{4^2}… <\/span>

\u00a0<\/p>

Man kan ocks\u00e5 uttrycka vinklar i avdelar av \\pi <\/span> i enhet som heter radianer<\/b> (Radianer introduceras i kursen Matematik 4).\u00a0<\/p>

2\\pi = 360\\degree <\/span>.\u00a0<\/p>

Radianer blir v\u00e4ldigt praktiskt n\u00e4r man jobbar med trigonometri och komplexa tal. Om man vill omvandla grader till radianer kan man anv\u00e4nda f\u00f6ljande formel: radianer = grader \\cdot \\frac{\\pi}{180} <\/span>. Radianer anv\u00e4nds bland annat n\u00e4r man formulerar Eulers identitet e^{\\pi i}+1=0 <\/span>.<\/p>

Ett annat intressant resultat som resulterar i \\pi <\/span> \u00e4r\u00a0Gabriels horn<\/b>. Gabriels horn \u00e4r ett horn eller en trumpet som smalnar av och g\u00e5r mot o\u00e4ndligheten (se bilden nedan). Det finns paradoxer kopplat till Gabriels horn. Till exempel \u00e4r det om\u00f6jligt m\u00e5la hela trumpeten, arean p\u00e5 utsidan av hornet \u00e4r allts\u00e5 o\u00e4ndligt ( \\infty <\/span>). D\u00e4remot \u00e4r volymen p\u00e5 hornet \u00e4ndligt, man kan allts\u00e5 fylla hornet med en \u00e4ndlig m\u00e4ngd. Om vi antar att radien f\u00f6r den cirkul\u00e4ra \u00f6ppningen p\u00e5 hornet \u00e4r 1 l.e kommer volymen f\u00f6r Gabriels horn att vara\u00a0 \\pi <\/span>. B\u00e5da dessa slutsatser g\u00e5r att visa matematiskt.\u00a0<\/span><\/p>

Man kan visa att volymen \u00e4r\u00a0<\/span> \\pi <\/span> med hj\u00e4lp av rotationsvolymer. Den generella formeln f\u00f6r rotationsvolymer \u00e4r\u00a0<\/span>V = \\pi\\cdot\\int_a^b (f(x))^2 dx <\/span><\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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Om vi delar Gabriels horn i tv\u00e5 lika stora delar och tittar p\u00e5 hornet i tv\u00e5 dimensioner kan vi se att det ser ut som grafen till funktionen\u00a0<\/span> f(x)=\\frac{1}{x} <\/span>. Om vi sedan applicerar v\u00e5r formel f\u00f6r rotationsvolymer p\u00e5 funktionen\u00a0<\/span> f(x)=\\frac{1}{x} <\/span> f\u00e5r vi f\u00f6ljande uttryck:\u00a0<\/span>V = \\pi\\cdot\\int_a^b (\\frac{1}{x})^2 dx <\/span>. Vi beh\u00f6ver sedan definiera v\u00e5ra gr\u00e4nser f\u00f6r integralen\u00a0<\/span> a <\/span> och\u00a0<\/span> b <\/span>.\u00a0<\/span><\/p>

Vi antar att radien f\u00f6r den cirkul\u00e4ra \u00f6ppningen ska vara 1 l.e det betyder att vi beh\u00f6ver ha funktionsv\u00e4rdet 1 och ta reda p\u00e5 f\u00f6r vilket\u00a0<\/span> x <\/span> \u00e4r\u00a0<\/span> f(x)=1 <\/span> och det \u00e4r d\u00e5\u00a0<\/span> x=1 <\/span>. V\u00e5r nedre gr\u00e4ns blir s\u00e5ledes\u00a0<\/span> a=1 <\/span>. Den \u00f6vre gr\u00e4nsen d\u00e4remot ska vara o\u00e4ndligheten eftersom Gabriels horn \u00e4r o\u00e4ndligt l\u00e5ng. Vi f\u00e5r d\u00e5 integralen:\u00a0<\/span>V = \\pi\\cdot\\int_1^{\\infty} (\\frac{1}{x})^2 dx <\/span>. Om man sedan ber\u00e4knar den integralen f\u00e5r vi svaret V=\\pi <\/span><\/span>. Att visa att arean \u00e4r o\u00e4ndlig \u00e4r en sv\u00e5rare historia, det kommer jag inte visa i det h\u00e4r dokumentet. Om du vill se beviset f\u00f6r att arean p\u00e5 Gabriels horn \u00e4r o\u00e4ndlig och en f\u00f6rdjupad f\u00f6rst\u00e5else f\u00f6r matematiken bakom kan du kolla h\u00e4r<\/a>.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t

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F\u00f6rs\u00f6ken att definiera \\pi <\/span><\/span><\/b><\/p>

Idag vet vi att \\pi <\/span> inte \u00e4r ett rationellt tal. Det betyder att \\pi <\/span> inte g\u00e5r att skrivas som en kvot \\frac{a}{b} <\/span><\/span><\/p>

Samtidigt har man genom historien f\u00f6rs\u00f6kt approximera \\pi <\/span>. I Babylonien som var en civilisation som hade sin h\u00f6gtid ungef\u00e4r 2000 f.kr approximera man \\pi <\/span> till \\frac{25}{8}= 3.125 <\/span><\/p>

F\u00f6rs\u00f6ken att definiera \\pi <\/span> fortsatte. Den k\u00e4nda matematikern Arkimedes kunde definiera en exakthet med 3 decimaler det vill s\u00e4ga \\approx <\/span> 3.141.\u00a0<\/span><\/p>

Ett stort genombrott i f\u00f6rs\u00f6ken att definiera <\/span> \\pi <\/span> var n\u00e4r den indiska matematikern Madhava (1340-1425)\u00a0b\u00f6rjade experimentera med o\u00e4ndliga serier. En o\u00e4ndlig serie inneb\u00e4r att man enligt ett visst m\u00f6nster adderar eller subtraherar tal som g\u00e5r mot o\u00e4ndligheten. Ett exempel p\u00e5 en o\u00e4ndlig serie \u00e4r <\/span> 1+\\frac{1}{2}+\\frac{1}{4} +\\frac{1}{8}+\\frac{1}{16}…<\/span> som visar sig vara lika med 2 om man drar det mot \u00e4ndligheten. Med Madhavas\u00a0o\u00e4ndliga serie <\/span>1-\\frac{1}{3}+\\frac{1}{5}-\\frac{1}{7}… <\/span> kunde han visa att det var lika med <\/span> \\frac{\\pi}{4} <\/span> om man drog det mot o\u00e4ndligheten. Idag ger man ofta \u00e4ran f\u00f6r den serien till Gottfried Leibniz (1646-1716) men Madhava var f\u00f6rst med att formulera den.<\/span><\/p>

Ett annat genombrott var n\u00e4r Johan\u00a0Heinrich Lambert 1761 visade att <\/span> \\pi <\/span> var en irrationellt tal. Efter det slutade\u00a0man f\u00f6rs\u00f6ka definiera <\/span> \\pi <\/span> som en kvot. Men man har lyckats definiera det utifr\u00e5n flera andra s\u00e4tt nedan finns n\u00e5gra exempel.<\/span><\/p> \\frac{2}{\\pi} = \\frac{\\sqrt{2}}{2}\\cdot\\frac{\\sqrt{2+\\sqrt{2}}}{2}… <\/span>

\u00a0<\/p> \\frac{\\pi}{2}=\\frac{2}{1}\\cdot\\frac{2}{3}\\cdot\\frac{4}{3}\\cdot\\frac{4}{5}\\cdot\\frac{6}{5}\\cdot\\frac{6}{7}… <\/span>

\u00a0<\/p>

\\pi =\\int_ {-\\infty}^{\\infty} \\frac{sinx}{x} dx <\/span><\/span><\/p>

\u00a0<\/div>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"

Vad \u00e4r egentligen ? \u00e4r konstanten som representerar f\u00f6rh\u00e5llandet mellan omkretsen och diametern i en cirkel. \u00a0approximeras till \u00a03.14159265 St\u00e4llen d\u00e4r dyker upp Leonhard Euler formulerade en o\u00e4ndlig serie som visade sig ha ett r\u00e4tt ov\u00e4ntat resultat \u00a0 Man kan ocks\u00e5 uttrycka vinklar i avdelar av i enhet som heter radianer (Radianer introduceras i kursen […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_uag_custom_page_level_css":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"site-sidebar-layout":"no-sidebar","site-content-layout":"page-builder","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"disabled","ast-breadcrumbs-content":"","ast-featured-img":"disabled","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"class_list":["post-26236","page","type-page","status-publish","hentry"],"aioseo_notices":[],"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false},"uagb_author_info":{"display_name":"joji9555@hotmail.com","author_link":"https:\/\/mahifi.se\/?author=1"},"uagb_comment_info":0,"uagb_excerpt":"Vad \u00e4r egentligen ? \u00e4r konstanten som representerar f\u00f6rh\u00e5llandet mellan omkretsen och diametern i en cirkel. \u00a0approximeras till \u00a03.14159265 St\u00e4llen d\u00e4r dyker upp Leonhard Euler formulerade en o\u00e4ndlig serie som visade sig ha ett r\u00e4tt ov\u00e4ntat resultat \u00a0 Man kan ocks\u00e5 uttrycka vinklar i avdelar av i enhet som heter radianer (Radianer introduceras i kursen…","_links":{"self":[{"href":"https:\/\/mahifi.se\/index.php?rest_route=\/wp\/v2\/pages\/26236","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mahifi.se\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mahifi.se\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mahifi.se\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mahifi.se\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=26236"}],"version-history":[{"count":358,"href":"https:\/\/mahifi.se\/index.php?rest_route=\/wp\/v2\/pages\/26236\/revisions"}],"predecessor-version":[{"id":34262,"href":"https:\/\/mahifi.se\/index.php?rest_route=\/wp\/v2\/pages\/26236\/revisions\/34262"}],"wp:attachment":[{"href":"https:\/\/mahifi.se\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=26236"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}