Christian: Difference between revisions

From My Zacky Installer Website
Jump to navigation Jump to search
No edit summary
(Replaced content with "== Trigonometrie 1 == #[https://www.schule-bw.de/faecher-und-schularten/mathematisch-naturwissenschaftliche-faecher/mathematik/unterrichtsmaterialien/sekundarstufe1/geometrie/trig/trigors Landesbildungsserver BW - Einführung] #[https://www.geogebra.org/m/FJtrEDAr Visualisierung von Sin, Cos, Tan in Geogebra] #[https://www.geogebra.org/m/a4TpwVtm Visualisierung interaktiv] ##[https://www.youtube.com/watch?v=TPm55fbMQtM Merksatz für sin/cos von 0/30/45/60/90] == Tr...")
Tag: Replaced
Line 23: Line 23:
#[https://www.geogebra.org/m/D38a3Y7s Potenzen zusammenfassen]
#[https://www.geogebra.org/m/D38a3Y7s Potenzen zusammenfassen]


<!DOCTYPE html>
<html class="client-nojs" lang="de" dir="ltr">
<head>
<meta charset="UTF-8">
<title>Kehrwert – Wikipedia</title>
<script>(function(){var className="client-js";var cookie=document.cookie.match(/(?:^|; )dewikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t.",".\t,"],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","Januar","Februar","März","April","Mai","Juni","Juli","August","September","Oktober","November","Dezember"],"wgRequestId":"0f5e8c36-a60d-4870-ac80-6cedf3de63e6","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Kehrwert","wgTitle":"Kehrwert","wgCurRevisionId":245058555,"wgRevisionId":245058555,"wgArticleId":15271,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":[
"Division (Mathematik)"],"wgPageViewLanguage":"de","wgPageContentLanguage":"de","wgPageContentModel":"wikitext","wgRelevantPageName":"Kehrwert","wgRelevantArticleId":15271,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgFlaggedRevsParams":{"tags":{"accuracy":{"levels":1}}},"wgStableRevisionId":245058555,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"de","pageLanguageDir":"ltr","pageVariantFallbacks":"de"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":8000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":true,"wgVector2022LanguageInHeader":false,"wgULSisLanguageSelectorEmpty":false,
"wgWikibaseItemId":"Q216906","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.citeRef":"ready","ext.gadget.defaultPlainlinks":"ready","ext.gadget.dewikiCommonHide":"ready","ext.gadget.dewikiCommonLayout":"ready","ext.gadget.dewikiCommonStyle":"ready","ext.gadget.NavFrame":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.styles.legacy":"ready","ext.flaggedRevs.basic":"ready","mediawiki.codex.messagebox.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","codex-search-styles":"ready","ext.uls.interlanguage":
"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.legacy.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.flaggedRevs.advanced","ext.gadget.createNewSection","ext.gadget.WikiMiniAtlas","ext.gadget.OpenStreetMap","ext.gadget.CommonsDirekt","ext.gadget.donateLink","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.compactlinks","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script>
<script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"});
}];});});</script>
<link rel="stylesheet" href="/w/load.php?lang=de&amp;modules=codex-search-styles%7Cext.cite.styles%7Cext.flaggedRevs.basic%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cmediawiki.codex.messagebox.styles%7Cskins.vector.styles.legacy%7Cwikibase.client.init&amp;only=styles&amp;skin=vector">
<script async="" src="/w/load.php?lang=de&amp;modules=startup&amp;only=scripts&amp;raw=1&amp;skin=vector"></script>
<meta name="ResourceLoaderDynamicStyles" content="">
<link rel="stylesheet" href="/w/load.php?lang=de&amp;modules=ext.gadget.NavFrame%2CciteRef%2CdefaultPlainlinks%2CdewikiCommonHide%2CdewikiCommonLayout%2CdewikiCommonStyle&amp;only=styles&amp;skin=vector">
<link rel="stylesheet" href="/w/load.php?lang=de&amp;modules=site.styles&amp;only=styles&amp;skin=vector">
<meta name="generator" content="MediaWiki 1.44.0-wmf.8">
<meta name="referrer" content="origin">
<meta name="referrer" content="origin-when-cross-origin">
<meta name="robots" content="max-image-preview:standard">
<meta name="format-detection" content="telephone=no">
<meta name="viewport" content="width=1120">
<meta property="og:title" content="Kehrwert – Wikipedia">
<meta property="og:type" content="website">
<link rel="preconnect" href="//upload.wikimedia.org">
<link rel="alternate" media="only screen and (max-width: 640px)" href="//de.m.wikipedia.org/wiki/Kehrwert">
<link rel="alternate" type="application/x-wiki" title="Seite bearbeiten" href="/w/index.php?title=Kehrwert&amp;action=edit">
<link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png">
<link rel="icon" href="/static/favicon/wikipedia.ico">
<link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (de)">
<link rel="EditURI" type="application/rsd+xml" href="//de.wikipedia.org/w/api.php?action=rsd">
<link rel="canonical" href="https://de.wikipedia.org/wiki/Kehrwert">
<link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">
<link rel="alternate" type="application/atom+xml" title="Atom-Feed für „Wikipedia“" href="/w/index.php?title=Spezial:Letzte_%C3%84nderungen&amp;feed=atom">
<link rel="dns-prefetch" href="//meta.wikimedia.org" />
<link rel="dns-prefetch" href="login.wikimedia.org">
</head>
<body class="skin-vector-legacy mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Kehrwert rootpage-Kehrwert skin-vector action-view"><div id="mw-page-base" class="noprint"></div>
<div id="mw-head-base" class="noprint"></div>
<div id="content" class="mw-body" role="main">
<a id="top"></a>
<div id="siteNotice"><!-- CentralNotice --></div>
<div class="mw-indicators">
</div>
<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Kehrwert</span></h1>
<div id="bodyContent" class="vector-body">
<div id="siteSub" class="noprint">aus Wikipedia, der freien Enzyklopädie</div>
<div id="contentSub"><div id="mw-content-subtitle"></div></div>
<div id="contentSub2"></div>
<div id="jump-to-nav"></div>
<a class="mw-jump-link" href="#mw-head">Zur Navigation springen</a>
<a class="mw-jump-link" href="#searchInput">Zur Suche springen</a>
<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Der <b>Kehrwert</b> (auch der <b>reziproke Wert</b> oder das <b>Reziproke</b>) einer von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 0}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>0</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> verschiedenen <a href="/wiki/Zahl" title="Zahl">Zahl</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle x}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>x</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle x}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> ist in der <a href="/wiki/Arithmetik" title="Arithmetik">Arithmetik</a> diejenige Zahl, die mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle x}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>x</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle x}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> <a href="/wiki/Multiplikation" title="Multiplikation">multipliziert</a> die Zahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 1}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>1</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> ergibt; er wird als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\tfrac {1}{x}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mn>1</mn>
              <mi>x</mi>
            </mfrac>
          </mstyle>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{x}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3da30de216ba1a9649809913816f8b640eb26f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.776ex; height:3.343ex;" alt="{\displaystyle {\tfrac {1}{x}}}"></span> oder <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle x^{-1}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msup>
          <mi>x</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mo>&#x2212;<!-- − --></mo>
            <mn>1</mn>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle x^{-1}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf91609f1a0b7847e108023b015cb6b0d567821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.662ex; height:2.676ex;" alt="{\displaystyle x^{-1}}"></span> notiert.
</p>
<div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Eigenschaften"><span class="tocnumber">1</span> <span class="toctext">Eigenschaften</span></a>
<ul>
<li class="toclevel-2 tocsection-2"><a href="#Kernaussagen"><span class="tocnumber">1.1</span> <span class="toctext">Kernaussagen</span></a></li>
<li class="toclevel-2 tocsection-3"><a href="#Summe_aus_Zahl_und_Kehrwert"><span class="tocnumber">1.2</span> <span class="toctext">Summe aus Zahl und Kehrwert</span></a></li>
<li class="toclevel-2 tocsection-4"><a href="#Summe_zweier_Kehrwerte"><span class="tocnumber">1.3</span> <span class="toctext">Summe zweier Kehrwerte</span></a></li>
<li class="toclevel-2 tocsection-5"><a href="#Summe_aufeinanderfolgender_Kehrwerte"><span class="tocnumber">1.4</span> <span class="toctext">Summe aufeinanderfolgender Kehrwerte</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-6"><a href="#Beispiele"><span class="tocnumber">2</span> <span class="toctext">Beispiele</span></a></li>
<li class="toclevel-1 tocsection-7"><a href="#Verallgemeinerung"><span class="tocnumber">3</span> <span class="toctext">Verallgemeinerung</span></a></li>
<li class="toclevel-1 tocsection-8"><a href="#Verwandte_Themen"><span class="tocnumber">4</span> <span class="toctext">Verwandte Themen</span></a></li>
<li class="toclevel-1 tocsection-9"><a href="#Literatur"><span class="tocnumber">5</span> <span class="toctext">Literatur</span></a></li>
<li class="toclevel-1 tocsection-10"><a href="#Weblinks"><span class="tocnumber">6</span> <span class="toctext">Weblinks</span></a></li>
<li class="toclevel-1 tocsection-11"><a href="#Einzelnachweise"><span class="tocnumber">7</span> <span class="toctext">Einzelnachweise</span></a></li>
</ul>
</div>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kehrwert&amp;veaction=edit&amp;section=1" title="Abschnitt bearbeiten: Eigenschaften" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kehrwert&amp;action=edit&amp;section=1" title="Quellcode des Abschnitts bearbeiten: Eigenschaften"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div>
<div class="mw-heading mw-heading3"><h3 id="Kernaussagen">Kernaussagen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kehrwert&amp;veaction=edit&amp;section=2" title="Abschnitt bearbeiten: Kernaussagen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kehrwert&amp;action=edit&amp;section=2" title="Quellcode des Abschnitts bearbeiten: Kernaussagen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div>
<figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Hyperbola_one_over_x.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Hyperbola_one_over_x.svg/220px-Hyperbola_one_over_x.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Hyperbola_one_over_x.svg/330px-Hyperbola_one_over_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Hyperbola_one_over_x.svg/440px-Hyperbola_one_over_x.svg.png 2x" data-file-width="1600" data-file-height="1200" /></a><figcaption>Der Graph der Kehrwertfunktion ist eine <a href="/wiki/Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbel</a>.</figcaption></figure>
<p>Je näher eine Zahl bei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 0}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>0</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> liegt, desto weiter ist ihr Kehrwert von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 0}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>0</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> entfernt. Die Zahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 0}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>0</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> selbst hat keinen Kehrwert und ist auch kein Kehrwert. Die durch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle y=f(x)={\tfrac {1}{x}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>y</mi>
        <mo>=</mo>
        <mi>f</mi>
        <mo stretchy="false">(</mo>
        <mi>x</mi>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mn>1</mn>
              <mi>x</mi>
            </mfrac>
          </mstyle>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle y=f(x)={\tfrac {1}{x}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295255af2ebd89bd25ea6119f95ade0f789036a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.546ex; height:3.343ex;" alt="{\displaystyle y=f(x)={\tfrac {1}{x}}}"></span> beschriebene Kehrwertfunktion (siehe Abbildung) hat dort eine <a href="/wiki/Polstelle" title="Polstelle">Polstelle</a>. Der Kehrwert einer positiven Zahl ist positiv, der Kehrwert einer negativen Zahl ist negativ. Dies findet seinen geometrischen Ausdruck darin, dass der Graph in zwei <a href="/wiki/Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbeläste</a> zerfällt, die im ersten bzw. dritten Quadranten liegen. Die Kehrwertfunktion ist eine <a href="/wiki/Involution_(Mathematik)" title="Involution (Mathematik)">Involution</a>, d.&#160;h., der Kehrwert des Kehrwerts von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle x}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>x</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle x}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> ist wieder <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle x.}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>x</mi>
        <mo>.</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle x.}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}"></span> Ist eine Größe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle y}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>y</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle y}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> <a href="/wiki/Umgekehrt_proportional" class="mw-redirect" title="Umgekehrt proportional">umgekehrt proportional</a> zu einer Größe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle x,}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>x</mi>
        <mo>,</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle x,}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"></span> dann ist sie proportional zum Kehrwert von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle x.}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>x</mi>
        <mo>.</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle x.}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}"></span>
</p><p>Den <b>Kehrbruch</b> eines <a href="/wiki/Bruchrechnung#Gemeine_Brüche" title="Bruchrechnung">Bruches</a>, also den Kehrwert eines <a href="/wiki/Quotient" title="Quotient">Quotienten</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\tfrac {a}{b}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mi>a</mi>
              <mi>b</mi>
            </mfrac>
          </mstyle>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}"></span> mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle a,b\neq 0,}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>a</mi>
        <mo>,</mo>
        <mi>b</mi>
        <mo>&#x2260;<!-- ≠ --></mo>
        <mn>0</mn>
        <mo>,</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle a,b\neq 0,}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/491acb120732257985e2f7ab789fef7cdf54f767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.169ex; height:2.676ex;" alt="{\displaystyle a,b\neq 0,}"></span> erhält man, indem man Zähler und Nenner vertauscht:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\frac {1}{\frac {a}{b}}}={\frac {b}{a}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mfrac>
              <mi>a</mi>
              <mi>b</mi>
            </mfrac>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mi>b</mi>
            <mi>a</mi>
          </mfrac>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\frac {a}{b}}}={\frac {b}{a}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/952a852fd53dd6a4539101d38db0e7d9d37d65f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:7.706ex; height:6.676ex;" alt="{\displaystyle {\frac {1}{\frac {a}{b}}}={\frac {b}{a}}}"></span></dd></dl>
<p>Daraus folgt die Rechenregel für das <a href="/wiki/Division_(Mathematik)" title="Division (Mathematik)">Dividieren</a> durch einen Bruch: <i>Durch einen Bruch wird dividiert, indem man mit seinem Kehrwert multipliziert.</i> Siehe auch <a href="/wiki/Bruchrechnung" title="Bruchrechnung">Bruchrechnung</a>.
</p><p>Den Kehrwert <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\tfrac {1}{n}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mn>1</mn>
              <mi>n</mi>
            </mfrac>
          </mstyle>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{n}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee46f3d1f145f31319826905e4ce0750792d55b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.822ex; height:3.343ex;" alt="{\displaystyle {\tfrac {1}{n}}}"></span> einer <a href="/wiki/Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürlichen Zahl</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle n}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>n</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle n}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> nennt man einen <a href="/wiki/Stammbruch" title="Stammbruch">Stammbruch</a>.
</p><p>Auch zu jeder von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 0}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>0</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> verschiedenen <a href="/wiki/Komplexe_Zahl" title="Komplexe Zahl">komplexen Zahl</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle z=a+b\mathrm {i} }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>z</mi>
        <mo>=</mo>
        <mi>a</mi>
        <mo>+</mo>
        <mi>b</mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="normal">i</mi>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle z=a+b\mathrm {i} }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d7f54052b27c21d6073ea59a31e499ea689970f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.901ex; height:2.343ex;" alt="{\displaystyle z=a+b\mathrm {i} }"></span> mit <i>reellen</i> Zahlen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle a,b}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>a</mi>
        <mo>,</mo>
        <mi>b</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span> gibt es einen Kehrwert <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\tfrac {1}{z}}.}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mn>1</mn>
              <mi>z</mi>
            </mfrac>
          </mstyle>
        </mrow>
        <mo>.</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{z}}.}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5770006851ba8ff951117476454da2731cd73c25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.305ex; height:3.343ex;" alt="{\displaystyle {\tfrac {1}{z}}.}"></span> Mit dem <a href="/wiki/Betragsfunktion#Komplexe_Betragsfunktion" title="Betragsfunktion">Absolutbetrag</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle |z|={\sqrt {a^{2}+b^{2}}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mo stretchy="false">|</mo>
        </mrow>
        <mi>z</mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mo stretchy="false">|</mo>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <msqrt>
            <msup>
              <mi>a</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>2</mn>
              </mrow>
            </msup>
            <mo>+</mo>
            <msup>
              <mi>b</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>2</mn>
              </mrow>
            </msup>
          </msqrt>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle |z|={\sqrt {a^{2}+b^{2}}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe94d0c3b0c3704e8771d0932fff6f983ef0082b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.98ex; height:3.509ex;" alt="{\displaystyle |z|={\sqrt {a^{2}+b^{2}}}}"></span> von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle z}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>z</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle z}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> und der zu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle z}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>z</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle z}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> <a href="/wiki/Komplexe_Konjugation" title="Komplexe Konjugation">konjugiert komplexen</a> Zahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\overline {z}}=a-b\mathrm {i} }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mover>
            <mi>z</mi>
            <mo accent="false">&#x00AF;<!-- ¯ --></mo>
          </mover>
        </mrow>
        <mo>=</mo>
        <mi>a</mi>
        <mo>&#x2212;<!-- − --></mo>
        <mi>b</mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="normal">i</mi>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\overline {z}}=a-b\mathrm {i} }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa7245b2db6d644ce58741004233134df972e3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.021ex; height:2.509ex;" alt="{\displaystyle {\overline {z}}=a-b\mathrm {i} }"></span> gilt:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\frac {1}{a+b\mathrm {i} }}={\frac {1}{z}}={\frac {\overline {z}}{z{\overline {z}}}}={\frac {\overline {z}}{|z|^{2}}}={\frac {a-b\mathrm {i} }{a^{2}+b^{2}}}={\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}\mathrm {i} }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mrow>
              <mi>a</mi>
              <mo>+</mo>
              <mi>b</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi mathvariant="normal">i</mi>
              </mrow>
            </mrow>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>z</mi>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mover>
              <mi>z</mi>
              <mo accent="false">&#x00AF;<!-- ¯ --></mo>
            </mover>
            <mrow>
              <mi>z</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mover>
                  <mi>z</mi>
                  <mo accent="false">&#x00AF;<!-- ¯ --></mo>
                </mover>
              </mrow>
            </mrow>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mover>
              <mi>z</mi>
              <mo accent="false">&#x00AF;<!-- ¯ --></mo>
            </mover>
            <mrow>
              <mrow class="MJX-TeXAtom-ORD">
                <mo stretchy="false">|</mo>
              </mrow>
              <mi>z</mi>
              <msup>
                <mrow class="MJX-TeXAtom-ORD">
                  <mo stretchy="false">|</mo>
                </mrow>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msup>
            </mrow>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <mi>a</mi>
              <mo>&#x2212;<!-- − --></mo>
              <mi>b</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi mathvariant="normal">i</mi>
              </mrow>
            </mrow>
            <mrow>
              <msup>
                <mi>a</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msup>
              <mo>+</mo>
              <msup>
                <mi>b</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msup>
            </mrow>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mi>a</mi>
            <mrow>
              <msup>
                <mi>a</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msup>
              <mo>+</mo>
              <msup>
                <mi>b</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msup>
            </mrow>
          </mfrac>
        </mrow>
        <mo>&#x2212;<!-- − --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mi>b</mi>
            <mrow>
              <msup>
                <mi>a</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msup>
              <mo>+</mo>
              <msup>
                <mi>b</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msup>
            </mrow>
          </mfrac>
        </mrow>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="normal">i</mi>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a+b\mathrm {i} }}={\frac {1}{z}}={\frac {\overline {z}}{z{\overline {z}}}}={\frac {\overline {z}}{|z|^{2}}}={\frac {a-b\mathrm {i} }{a^{2}+b^{2}}}={\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}\mathrm {i} }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e571b122897385c9f968daede3034bfb41ed961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.97ex; height:6.676ex;" alt="{\displaystyle {\frac {1}{a+b\mathrm {i} }}={\frac {1}{z}}={\frac {\overline {z}}{z{\overline {z}}}}={\frac {\overline {z}}{|z|^{2}}}={\frac {a-b\mathrm {i} }{a^{2}+b^{2}}}={\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}\mathrm {i} }"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Summe_aus_Zahl_und_Kehrwert">Summe aus Zahl und Kehrwert</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kehrwert&amp;veaction=edit&amp;section=3" title="Abschnitt bearbeiten: Summe aus Zahl und Kehrwert" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kehrwert&amp;action=edit&amp;section=3" title="Quellcode des Abschnitts bearbeiten: Summe aus Zahl und Kehrwert"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>Die Summe aus einer positiven <a href="/wiki/Reelle_Zahl" title="Reelle Zahl">reellen Zahl</a> und ihrem Kehrwert beträgt mindestens <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 2.}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>2.</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 2.}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2b3373a07e65d3312989163b5ebd400af86480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 2.}"></span><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">&#91;</span>1<span class="cite-bracket">&#93;</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">&#91;</span>2<span class="cite-bracket">&#93;</span></a></sup>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle x+{\frac {1}{x}}\geq 2}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>x</mi>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>x</mi>
          </mfrac>
        </mrow>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>2</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle x+{\frac {1}{x}}\geq 2}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5291484292966bff26e63e310e5a3fc6ba56f702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.597ex; height:5.176ex;" alt="{\displaystyle x+{\frac {1}{x}}\geq 2}"></span></dd></dl>
<p><i>Beweisvariante 1 (Figur 1):</i>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle \left(x+{\frac {1}{x}}\right)^{2}\geq 4\cdot x\cdot {\frac {1}{x}}\Leftrightarrow x+{\frac {1}{x}}\geq 2}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msup>
          <mrow>
            <mo>(</mo>
            <mrow>
              <mi>x</mi>
              <mo>+</mo>
              <mrow class="MJX-TeXAtom-ORD">
                <mfrac>
                  <mn>1</mn>
                  <mi>x</mi>
                </mfrac>
              </mrow>
            </mrow>
            <mo>)</mo>
          </mrow>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>4</mn>
        <mo>&#x22C5;<!-- ⋅ --></mo>
        <mi>x</mi>
        <mo>&#x22C5;<!-- ⋅ --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>x</mi>
          </mfrac>
        </mrow>
        <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo>
        <mi>x</mi>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>x</mi>
          </mfrac>
        </mrow>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>2</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \left(x+{\frac {1}{x}}\right)^{2}\geq 4\cdot x\cdot {\frac {1}{x}}\Leftrightarrow x+{\frac {1}{x}}\geq 2}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e07f845b4566d52630549d8b419941e8393ab70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.137ex; height:6.509ex;" alt="{\displaystyle \left(x+{\frac {1}{x}}\right)^{2}\geq 4\cdot x\cdot {\frac {1}{x}}\Leftrightarrow x+{\frac {1}{x}}\geq 2}"></span></dd></dl>
<p><i>Beweisvariante 2 (Figur 2):</i>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\frac {1}{x}}\geq 2-x\Leftrightarrow x+{\frac {1}{x}}\geq 2}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>x</mi>
          </mfrac>
        </mrow>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>2</mn>
        <mo>&#x2212;<!-- − --></mo>
        <mi>x</mi>
        <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo>
        <mi>x</mi>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>x</mi>
          </mfrac>
        </mrow>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>2</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x}}\geq 2-x\Leftrightarrow x+{\frac {1}{x}}\geq 2}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62b44aa71298390050daad2f39336a3e0514905e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.808ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{x}}\geq 2-x\Leftrightarrow x+{\frac {1}{x}}\geq 2}"></span></dd></dl>
<p><i>Beweisvariante 3 (Figur 3):</i>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle \left(x+{\frac {1}{x}}\right)^{2}=2^{2}+\left(x-{\frac {1}{x}}\right)^{2}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msup>
          <mrow>
            <mo>(</mo>
            <mrow>
              <mi>x</mi>
              <mo>+</mo>
              <mrow class="MJX-TeXAtom-ORD">
                <mfrac>
                  <mn>1</mn>
                  <mi>x</mi>
                </mfrac>
              </mrow>
            </mrow>
            <mo>)</mo>
          </mrow>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>=</mo>
        <msup>
          <mn>2</mn>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>+</mo>
        <msup>
          <mrow>
            <mo>(</mo>
            <mrow>
              <mi>x</mi>
              <mo>&#x2212;<!-- − --></mo>
              <mrow class="MJX-TeXAtom-ORD">
                <mfrac>
                  <mn>1</mn>
                  <mi>x</mi>
                </mfrac>
              </mrow>
            </mrow>
            <mo>)</mo>
          </mrow>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \left(x+{\frac {1}{x}}\right)^{2}=2^{2}+\left(x-{\frac {1}{x}}\right)^{2}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc7d3b39e90e80a51ba4b124ae9ef6e1336b98e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.778ex; height:6.509ex;" alt="{\displaystyle \left(x+{\frac {1}{x}}\right)^{2}=2^{2}+\left(x-{\frac {1}{x}}\right)^{2}}"></span> <i>(nach dem <a href="/wiki/Satz_des_Pythagoras" title="Satz des Pythagoras">Satz des Pythagoras</a>)</i></dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle \Leftrightarrow \left(x+{\frac {1}{x}}\right)^{2}\geq 2^{2}\Leftrightarrow x+{\frac {1}{x}}\geq 2}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo>
        <msup>
          <mrow>
            <mo>(</mo>
            <mrow>
              <mi>x</mi>
              <mo>+</mo>
              <mrow class="MJX-TeXAtom-ORD">
                <mfrac>
                  <mn>1</mn>
                  <mi>x</mi>
                </mfrac>
              </mrow>
            </mrow>
            <mo>)</mo>
          </mrow>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>&#x2265;<!-- ≥ --></mo>
        <msup>
          <mn>2</mn>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo>
        <mi>x</mi>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>x</mi>
          </mfrac>
        </mrow>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>2</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow \left(x+{\frac {1}{x}}\right)^{2}\geq 2^{2}\Leftrightarrow x+{\frac {1}{x}}\geq 2}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd31e4054fb61f750fabfe34d37f445ce23cad37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.306ex; height:6.509ex;" alt="{\displaystyle \Leftrightarrow \left(x+{\frac {1}{x}}\right)^{2}\geq 2^{2}\Leftrightarrow x+{\frac {1}{x}}\geq 2}"></span></dd></dl>
<p><i>Beweisvariante 4 (Figur 4):</i>
</p>
<dl><dd>Nach dem <a href="/wiki/Strahlensatz" title="Strahlensatz">Strahlensatz</a> sind die <a href="/wiki/Dreieck" title="Dreieck">Dreiecke</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle DEF}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>D</mi>
        <mi>E</mi>
        <mi>F</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle DEF}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e73d6f110c9dc2ee6ec8677a8e44f7e14ee3e37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.441ex; height:2.176ex;" alt="{\displaystyle DEF}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle DBC}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>D</mi>
        <mi>B</mi>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle DBC}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51aaac538474e68bf4652df3b42d258c164366e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.455ex; height:2.176ex;" alt="{\displaystyle DBC}"></span> <a href="/wiki/%C3%84hnlichkeit_(Geometrie)" title="Ähnlichkeit (Geometrie)">ähnlich</a>. Es gilt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\frac {x}{1}}={\frac {1}{\frac {1}{x}}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mi>x</mi>
            <mn>1</mn>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mfrac>
              <mn>1</mn>
              <mi>x</mi>
            </mfrac>
          </mfrac>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{1}}={\frac {1}{\frac {1}{x}}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/182401d6027f4887112049d46472d2b5954a331c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:7.877ex; height:6.509ex;" alt="{\displaystyle {\frac {x}{1}}={\frac {1}{\frac {1}{x}}}}"></span>. <a href="/wiki/Ohne_Beschr%C3%A4nkung_der_Allgemeinheit" title="Ohne Beschränkung der Allgemeinheit">Ohne Beschränkung der Allgemeinheit</a> wird hier <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle x\geq 1}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>x</mi>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>1</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle x\geq 1}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca3ced43f1713577888a8a7ade2d0aaf8354a4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 1}"></span> vorausgesetzt.</dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\frac {1}{2}}\cdot 1\cdot x+{\frac {1}{2}}\cdot 1\cdot {\frac {1}{x}}\geq 1\cdot 1\Leftrightarrow {\frac {x}{2}}+{\frac {1}{2x}}\geq 1\Leftrightarrow x^{2}+1\geq 2x\Leftrightarrow x+{\frac {1}{x}}\geq 2}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mn>2</mn>
          </mfrac>
        </mrow>
        <mo>&#x22C5;<!-- ⋅ --></mo>
        <mn>1</mn>
        <mo>&#x22C5;<!-- ⋅ --></mo>
        <mi>x</mi>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mn>2</mn>
          </mfrac>
        </mrow>
        <mo>&#x22C5;<!-- ⋅ --></mo>
        <mn>1</mn>
        <mo>&#x22C5;<!-- ⋅ --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>x</mi>
          </mfrac>
        </mrow>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>1</mn>
        <mo>&#x22C5;<!-- ⋅ --></mo>
        <mn>1</mn>
        <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mi>x</mi>
            <mn>2</mn>
          </mfrac>
        </mrow>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mrow>
              <mn>2</mn>
              <mi>x</mi>
            </mrow>
          </mfrac>
        </mrow>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>1</mn>
        <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo>
        <msup>
          <mi>x</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>+</mo>
        <mn>1</mn>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>2</mn>
        <mi>x</mi>
        <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo>
        <mi>x</mi>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>x</mi>
          </mfrac>
        </mrow>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>2</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\cdot 1\cdot x+{\frac {1}{2}}\cdot 1\cdot {\frac {1}{x}}\geq 1\cdot 1\Leftrightarrow {\frac {x}{2}}+{\frac {1}{2x}}\geq 1\Leftrightarrow x^{2}+1\geq 2x\Leftrightarrow x+{\frac {1}{x}}\geq 2}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14fabbfc730293a6e715f07f44a4ff52061cef82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:72.489ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}\cdot 1\cdot x+{\frac {1}{2}}\cdot 1\cdot {\frac {1}{x}}\geq 1\cdot 1\Leftrightarrow {\frac {x}{2}}+{\frac {1}{2x}}\geq 1\Leftrightarrow x^{2}+1\geq 2x\Leftrightarrow x+{\frac {1}{x}}\geq 2}"></span></dd></dl>
<div class="thumb tleft" style="margin-top: .5em; width:830px;"><div class="thumbinner"><div style="clear:both;font-weight:bold;text-align:center;">Grafische Veranschaulichung der Beweisvarianten</div><div style="float:left; padding:1px; width:202px;"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/Datei:Kehrwert_Summenungleichung_Beweis_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Kehrwert_Summenungleichung_Beweis_1.svg/200px-Kehrwert_Summenungleichung_Beweis_1.svg.png" decoding="async" width="200" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Kehrwert_Summenungleichung_Beweis_1.svg/300px-Kehrwert_Summenungleichung_Beweis_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Kehrwert_Summenungleichung_Beweis_1.svg/400px-Kehrwert_Summenungleichung_Beweis_1.svg.png 2x" data-file-width="308" data-file-height="291" /></a></span></div></div><div style="float:left; padding:1px; width:202px;"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/Datei:Kehrwert_Summenungleichung_Beweis_2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Kehrwert_Summenungleichung_Beweis_2.svg/200px-Kehrwert_Summenungleichung_Beweis_2.svg.png" decoding="async" width="200" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Kehrwert_Summenungleichung_Beweis_2.svg/300px-Kehrwert_Summenungleichung_Beweis_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Kehrwert_Summenungleichung_Beweis_2.svg/400px-Kehrwert_Summenungleichung_Beweis_2.svg.png 2x" data-file-width="686" data-file-height="708" /></a></span></div></div><div style="float:left; padding:1px; width:202px;"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/Datei:Kehrwert_Summenungleichung_Beweis_3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Kehrwert_Summenungleichung_Beweis_3.svg/200px-Kehrwert_Summenungleichung_Beweis_3.svg.png" decoding="async" width="200" height="86" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Kehrwert_Summenungleichung_Beweis_3.svg/300px-Kehrwert_Summenungleichung_Beweis_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Kehrwert_Summenungleichung_Beweis_3.svg/400px-Kehrwert_Summenungleichung_Beweis_3.svg.png 2x" data-file-width="636" data-file-height="272" /></a></span></div></div><div style="float:left; padding:1px; width:202px;"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/Datei:Kehrwert_Summenungleichung_Beweis_4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Kehrwert_Summenungleichung_Beweis_4.svg/200px-Kehrwert_Summenungleichung_Beweis_4.svg.png" decoding="async" width="200" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Kehrwert_Summenungleichung_Beweis_4.svg/300px-Kehrwert_Summenungleichung_Beweis_4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Kehrwert_Summenungleichung_Beweis_4.svg/400px-Kehrwert_Summenungleichung_Beweis_4.svg.png 2x" data-file-width="427" data-file-height="354" /></a></span></div></div><div style="clear:both;"></div>
<div class="thumbcaption" style="float:left; padding:1px; width:202px !important;"><i>Figur 1</i></div><div class="thumbcaption" style="float:left; padding:1px; width:202px !important;"><i>Figur 2</i></div><div class="thumbcaption" style="float:left; padding:1px; width:202px !important;"><i>Figur 3</i></div><div class="thumbcaption" style="float:left; padding:1px; width:202px !important;"><i>Figur 4</i></div>
<div style="clear:both;"></div>
</div></div>
<div style="clear:both;"></div>
<div class="mw-heading mw-heading3"><h3 id="Summe_zweier_Kehrwerte">Summe zweier Kehrwerte</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kehrwert&amp;veaction=edit&amp;section=4" title="Abschnitt bearbeiten: Summe zweier Kehrwerte" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kehrwert&amp;action=edit&amp;section=4" title="Quellcode des Abschnitts bearbeiten: Summe zweier Kehrwerte"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div>
<figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Kehrwertsumme_Planfigur.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Kehrwertsumme_Planfigur.svg/220px-Kehrwertsumme_Planfigur.svg.png" decoding="async" width="220" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Kehrwertsumme_Planfigur.svg/330px-Kehrwertsumme_Planfigur.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Kehrwertsumme_Planfigur.svg/440px-Kehrwertsumme_Planfigur.svg.png 2x" data-file-width="297" data-file-height="295" /></a><figcaption><i>Figur 5</i></figcaption></figure>
<p>Die Summe der Kehrwerte zweier positiver reeller Zahlen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle a}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>a</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle a}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle b}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>b</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle b}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> mit der Summe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 1}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>1</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> beträgt mindestens <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 4}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>4</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 4}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}"></span>:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\frac {1}{a}}+{\frac {1}{b}}\geq 4}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>a</mi>
          </mfrac>
        </mrow>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>b</mi>
          </mfrac>
        </mrow>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>4</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a}}+{\frac {1}{b}}\geq 4}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3248531e2a57ff3479d1eac67299a17b088b686c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.166ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{a}}+{\frac {1}{b}}\geq 4}"></span> für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle a+b=1}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>a</mi>
        <mo>+</mo>
        <mi>b</mi>
        <mo>=</mo>
        <mn>1</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle a+b=1}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a8061cad08a2f1206af42fb3e0389fcf4353e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.329ex; height:2.343ex;" alt="{\displaystyle a+b=1}"></span>.</dd></dl>
<p><i>Beweis:</i>
</p><p>Gemäß <i>Figur 5</i> gilt:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 4ab\leq 1\Leftrightarrow {\frac {1}{ab}}\geq 4}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>4</mn>
        <mi>a</mi>
        <mi>b</mi>
        <mo>&#x2264;<!-- ≤ --></mo>
        <mn>1</mn>
        <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mrow>
              <mi>a</mi>
              <mi>b</mi>
            </mrow>
          </mfrac>
        </mrow>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>4</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 4ab\leq 1\Leftrightarrow {\frac {1}{ab}}\geq 4}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/149079010eed654fc2f606f1a0f92ec6c346de20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.589ex; height:5.343ex;" alt="{\displaystyle 4ab\leq 1\Leftrightarrow {\frac {1}{ab}}\geq 4}"></span></dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\frac {1}{a}}+{\frac {1}{b}}={\frac {a+b}{ab}}={\frac {1}{ab}}\geq 4}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>a</mi>
          </mfrac>
        </mrow>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>b</mi>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <mi>a</mi>
              <mo>+</mo>
              <mi>b</mi>
            </mrow>
            <mrow>
              <mi>a</mi>
              <mi>b</mi>
            </mrow>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mrow>
              <mi>a</mi>
              <mi>b</mi>
            </mrow>
          </mfrac>
        </mrow>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>4</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a}}+{\frac {1}{b}}={\frac {a+b}{ab}}={\frac {1}{ab}}\geq 4}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c515c6b999f47ecbe6b512157fb97c0b4a4291b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.33ex; height:5.509ex;" alt="{\displaystyle {\frac {1}{a}}+{\frac {1}{b}}={\frac {a+b}{ab}}={\frac {1}{ab}}\geq 4}"></span>,</dd></dl>
<p><a href="/wiki/Quod_erat_demonstrandum" title="Quod erat demonstrandum">was zu beweisen war</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">&#91;</span>3<span class="cite-bracket">&#93;</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Summe_aufeinanderfolgender_Kehrwerte">Summe aufeinanderfolgender Kehrwerte</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kehrwert&amp;veaction=edit&amp;section=5" title="Abschnitt bearbeiten: Summe aufeinanderfolgender Kehrwerte" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kehrwert&amp;action=edit&amp;section=5" title="Quellcode des Abschnitts bearbeiten: Summe aufeinanderfolgender Kehrwerte"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>Für jede <a href="/wiki/Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürliche Zahl</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle n&gt;1}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>n</mi>
        <mo>&gt;</mo>
        <mn>1</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle n&gt;1}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n&gt;1}"></span> gilt
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\frac {1}{n}}+{\frac {1}{n+1}}+{\frac {1}{n+2}}+...+{\frac {1}{n^{2}}}&gt;1}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>n</mi>
          </mfrac>
        </mrow>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mrow>
              <mi>n</mi>
              <mo>+</mo>
              <mn>1</mn>
            </mrow>
          </mfrac>
        </mrow>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mrow>
              <mi>n</mi>
              <mo>+</mo>
              <mn>2</mn>
            </mrow>
          </mfrac>
        </mrow>
        <mo>+</mo>
        <mo>.</mo>
        <mo>.</mo>
        <mo>.</mo>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <msup>
              <mi>n</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>2</mn>
              </mrow>
            </msup>
          </mfrac>
        </mrow>
        <mo>&gt;</mo>
        <mn>1</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}+{\frac {1}{n+1}}+{\frac {1}{n+2}}+...+{\frac {1}{n^{2}}}&gt;1}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da40156d1060ae455bc5c45838b258cad7ea1a98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:34.643ex; height:5.509ex;" alt="{\displaystyle {\frac {1}{n}}+{\frac {1}{n+1}}+{\frac {1}{n+2}}+...+{\frac {1}{n^{2}}}&gt;1}"></span>.</dd></dl>
<p>Den Beweis liefert die Abschätzung
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\frac {1}{n}}+{\frac {1}{n+1}}+{\frac {1}{n+2}}+...+{\frac {1}{n^{2}}}&gt;{\frac {1}{n}}+\left({\frac {1}{n^{2}}}+{\frac {1}{n^{2}}}+...+{\frac {1}{n^{2}}}\right)={\frac {1}{n}}+{\frac {1}{n^{2}}}\left(n^{2}-n\right)={\frac {1}{n}}+1-{\frac {1}{n}}=1}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>n</mi>
          </mfrac>
        </mrow>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mrow>
              <mi>n</mi>
              <mo>+</mo>
              <mn>1</mn>
            </mrow>
          </mfrac>
        </mrow>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mrow>
              <mi>n</mi>
              <mo>+</mo>
              <mn>2</mn>
            </mrow>
          </mfrac>
        </mrow>
        <mo>+</mo>
        <mo>.</mo>
        <mo>.</mo>
        <mo>.</mo>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <msup>
              <mi>n</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>2</mn>
              </mrow>
            </msup>
          </mfrac>
        </mrow>
        <mo>&gt;</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>n</mi>
          </mfrac>
        </mrow>
        <mo>+</mo>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mrow class="MJX-TeXAtom-ORD">
              <mfrac>
                <mn>1</mn>
                <msup>
                  <mi>n</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>2</mn>
                  </mrow>
                </msup>
              </mfrac>
            </mrow>
            <mo>+</mo>
            <mrow class="MJX-TeXAtom-ORD">
              <mfrac>
                <mn>1</mn>
                <msup>
                  <mi>n</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>2</mn>
                  </mrow>
                </msup>
              </mfrac>
            </mrow>
            <mo>+</mo>
            <mo>.</mo>
            <mo>.</mo>
            <mo>.</mo>
            <mo>+</mo>
            <mrow class="MJX-TeXAtom-ORD">
              <mfrac>
                <mn>1</mn>
                <msup>
                  <mi>n</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>2</mn>
                  </mrow>
                </msup>
              </mfrac>
            </mrow>
          </mrow>
          <mo>)</mo>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>n</mi>
          </mfrac>
        </mrow>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <msup>
              <mi>n</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>2</mn>
              </mrow>
            </msup>
          </mfrac>
        </mrow>
        <mrow>
          <mo>(</mo>
          <mrow>
            <msup>
              <mi>n</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>2</mn>
              </mrow>
            </msup>
            <mo>&#x2212;<!-- − --></mo>
            <mi>n</mi>
          </mrow>
          <mo>)</mo>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>n</mi>
          </mfrac>
        </mrow>
        <mo>+</mo>
        <mn>1</mn>
        <mo>&#x2212;<!-- − --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mi>n</mi>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mn>1</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}+{\frac {1}{n+1}}+{\frac {1}{n+2}}+...+{\frac {1}{n^{2}}}&gt;{\frac {1}{n}}+\left({\frac {1}{n^{2}}}+{\frac {1}{n^{2}}}+...+{\frac {1}{n^{2}}}\right)={\frac {1}{n}}+{\frac {1}{n^{2}}}\left(n^{2}-n\right)={\frac {1}{n}}+1-{\frac {1}{n}}=1}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38db5e46fb393fb9cc42f28547ec6f7e91241a7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:100.707ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{n}}+{\frac {1}{n+1}}+{\frac {1}{n+2}}+...+{\frac {1}{n^{2}}}&gt;{\frac {1}{n}}+\left({\frac {1}{n^{2}}}+{\frac {1}{n^{2}}}+...+{\frac {1}{n^{2}}}\right)={\frac {1}{n}}+{\frac {1}{n^{2}}}\left(n^{2}-n\right)={\frac {1}{n}}+1-{\frac {1}{n}}=1}"></span>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">&#91;</span>4<span class="cite-bracket">&#93;</span></a></sup></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kehrwert&amp;veaction=edit&amp;section=6" title="Abschnitt bearbeiten: Beispiele" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kehrwert&amp;action=edit&amp;section=6" title="Quellcode des Abschnitts bearbeiten: Beispiele"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div>
<ul><li>Der Kehrwert von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 1}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>1</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> ist wiederum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 1}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>1</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>.</li>
<li>Der Kehrwert von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 0{,}001}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>0,001</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 0{,}001}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37b02aa6542167e2202fec98516bf3237cd35b86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.297ex; height:2.509ex;" alt="{\displaystyle 0{,}001}"></span> ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 1000}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>1000</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 1000}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9060e16491f890b9fbcce0194c8d454cbee309ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.65ex; height:2.176ex;" alt="{\displaystyle 1000}"></span>.</li>
<li>Der Kehrwert von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 2}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>2</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 2}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\tfrac {1}{2}}=0{,}5}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mn>1</mn>
              <mn>2</mn>
            </mfrac>
          </mstyle>
        </mrow>
        <mo>=</mo>
        <mn>0</mn>
        <mrow class="MJX-TeXAtom-ORD">
          <mo>,</mo>
        </mrow>
        <mn>5</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}=0{,}5}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0e7fd12728cb5e48baf2932b97faf654f0afa42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.728ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}=0{,}5}"></span>.</li>
<li>Der Kehrwert des Bruches <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\tfrac {2}{5}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mn>2</mn>
              <mn>5</mn>
            </mfrac>
          </mstyle>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{5}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb22be2c480d6bb96c97cc2b6a1a796f8396489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{5}}}"></span> ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\tfrac {5}{2}}=2{\tfrac {1}{2}}=2{,}5}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mn>5</mn>
              <mn>2</mn>
            </mfrac>
          </mstyle>
        </mrow>
        <mo>=</mo>
        <mn>2</mn>
        <mrow class="MJX-TeXAtom-ORD">
          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mn>1</mn>
              <mn>2</mn>
            </mfrac>
          </mstyle>
        </mrow>
        <mo>=</mo>
        <mn>2</mn>
        <mrow class="MJX-TeXAtom-ORD">
          <mo>,</mo>
        </mrow>
        <mn>5</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\tfrac {5}{2}}=2{\tfrac {1}{2}}=2{,}5}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6377800ff02edf1c0cf48ab2e6fb5568f2b6b640" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.647ex; height:3.509ex;" alt="{\displaystyle {\tfrac {5}{2}}=2{\tfrac {1}{2}}=2{,}5}"></span>.</li>
<li>Der Kehrwert der komplexen Zahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 3+4\mathrm {i} }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>3</mn>
        <mo>+</mo>
        <mn>4</mn>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="normal">i</mi>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 3+4\mathrm {i} }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3ab335ff1f5595bf3cf91ef4241f78a48593ce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.812ex; height:2.343ex;" alt="{\displaystyle 3+4\mathrm {i} }"></span> ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\tfrac {1}{3+4\mathrm {i} }}={\tfrac {3}{25}}-{\tfrac {4}{25}}\mathrm {i} }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mn>1</mn>
              <mrow>
                <mn>3</mn>
                <mo>+</mo>
                <mn>4</mn>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="normal">i</mi>
                </mrow>
              </mrow>
            </mfrac>
          </mstyle>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mn>3</mn>
              <mn>25</mn>
            </mfrac>
          </mstyle>
        </mrow>
        <mo>&#x2212;<!-- − --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mn>4</mn>
              <mn>25</mn>
            </mfrac>
          </mstyle>
        </mrow>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="normal">i</mi>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{3+4\mathrm {i} }}={\tfrac {3}{25}}-{\tfrac {4}{25}}\mathrm {i} }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5abfc5e0e00b1a2871bd13d96da7cf097730a53b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.762ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{3+4\mathrm {i} }}={\tfrac {3}{25}}-{\tfrac {4}{25}}\mathrm {i} }"></span>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Verallgemeinerung">Verallgemeinerung</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kehrwert&amp;veaction=edit&amp;section=7" title="Abschnitt bearbeiten: Verallgemeinerung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kehrwert&amp;action=edit&amp;section=7" title="Quellcode des Abschnitts bearbeiten: Verallgemeinerung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>Eine Verallgemeinerung des Kehrwerts ist das <a href="/wiki/Inverses_Element" title="Inverses Element">multiplikativ Inverse</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle x^{-1}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msup>
          <mi>x</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mo>&#x2212;<!-- − --></mo>
            <mn>1</mn>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle x^{-1}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf91609f1a0b7847e108023b015cb6b0d567821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.662ex; height:2.676ex;" alt="{\displaystyle x^{-1}}"></span> zu einer <a href="/wiki/Einheit_(Mathematik)" title="Einheit (Mathematik)">Einheit</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle x}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>x</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle x}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> eines <a href="/wiki/Ring_(Algebra)" title="Ring (Algebra)">unitären Ringes</a>. Es ist ebenfalls durch die Eigenschaft <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle x^{-1}\cdot \ x=x\cdot \ x^{-1}=1}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msup>
          <mi>x</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mo>&#x2212;<!-- − --></mo>
            <mn>1</mn>
          </mrow>
        </msup>
        <mo>&#x22C5;<!-- ⋅ --></mo>
        <mtext>&#xA0;</mtext>
        <mi>x</mi>
        <mo>=</mo>
        <mi>x</mi>
        <mo>&#x22C5;<!-- ⋅ --></mo>
        <mtext>&#xA0;</mtext>
        <msup>
          <mi>x</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mo>&#x2212;<!-- − --></mo>
            <mn>1</mn>
          </mrow>
        </msup>
        <mo>=</mo>
        <mn>1</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle x^{-1}\cdot \ x=x\cdot \ x^{-1}=1}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa9f878a343f6121e1c85011d9146ce0a29921b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.863ex; height:2.676ex;" alt="{\displaystyle x^{-1}\cdot \ x=x\cdot \ x^{-1}=1}"></span> definiert, wobei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 1}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>1</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> das Einselement des Ringes bezeichnet.
</p><p>Wenn es sich z.&#160;B. um einen Ring von Matrizen handelt, so ist das Einselement nicht die Zahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 1,}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mn>1</mn>
        <mo>,</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle 1,}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc5fd8163a83100c5330622e9e317fa4e872403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 1,}"></span> sondern die <a href="/wiki/Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a>. Matrizen, zu denen keine <a href="/wiki/Inverse_Matrix" title="Inverse Matrix">inverse Matrix</a> existiert, heißen <a href="/wiki/Singul%C3%A4re_Matrix" class="mw-redirect" title="Singuläre Matrix">singulär</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Verwandte_Themen">Verwandte Themen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kehrwert&amp;veaction=edit&amp;section=8" title="Abschnitt bearbeiten: Verwandte Themen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kehrwert&amp;action=edit&amp;section=8" title="Quellcode des Abschnitts bearbeiten: Verwandte Themen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div>
<ul><li>Ist eine Größe <a href="/wiki/Proportionalit%C3%A4t" title="Proportionalität">proportional</a> zum Kehrwert einer anderen, liegt <a href="/wiki/Reziproke_Proportionalit%C3%A4t" title="Reziproke Proportionalität">reziproke Proportionalität</a> vor.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kehrwert&amp;veaction=edit&amp;section=9" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kehrwert&amp;action=edit&amp;section=9" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>Hintergrundwissen für Lehramtsstudenten zur Arithmetik:
</p>
<ul><li>Friedhelm Padberg&#58; <cite style="font-style:italic">Didaktik der Arithmetik. Für Lehrerausbildung und Lehrerfortbildung. <i>3. erweiterte völlig überarbeitete Auflage, Nachdruck</i></cite>. Spektrum Akademischer Verlag, München 2009, <a href="/wiki/Spezial:ISBN-Suche/9783827409935" class="internal mw-magiclink-isbn">ISBN 978-3-8274-0993-5</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Kehrwert&amp;rft.au=Friedhelm+Padberg&amp;rft.btitle=Didaktik+der+Arithmetik.+F%C3%BCr+Lehrerausbildung+und+Lehrerfortbildung.+3.+erweiterte+v%C3%B6llig+%C3%BCberarbeitete+Auflage%2C+Nachdruck&amp;rft.date=2009&amp;rft.genre=book&amp;rft.isbn=9783827409935&amp;rft.place=M%C3%BCnchen&amp;rft.pub=Spektrum+Akademischer+Verlag" style="display:none">&#160;</span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kehrwert&amp;veaction=edit&amp;section=10" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kehrwert&amp;action=edit&amp;section=10" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div>
<div class="sisterproject" style="margin:0.1em 0 0 0;"><span class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wiktionary"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/16px-Wiktfavicon_en.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/24px-Wiktfavicon_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/32px-Wiktfavicon_en.svg.png 2x" data-file-width="16" data-file-height="16" /></span></span></span><b><a href="https://de.wiktionary.org/wiki/Kehrwert" class="extiw" title="wikt:Kehrwert">Wiktionary: Kehrwert</a></b>&#160;– Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kehrwert&amp;veaction=edit&amp;section=11" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kehrwert&amp;action=edit&amp;section=11" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Roger B. Nelsen: <i>Beweise ohne Worte</i>, Deutschsprachige Ausgabe herausgegeben von Nicola Oswald, <a href="/wiki/Springer_Spektrum" title="Springer Spektrum">Springer Spektrum</a>, Springer-Verlag <a href="/wiki/Berlin" title="Berlin">Berlin</a> <a href="/wiki/Heidelberg" title="Heidelberg">Heidelberg</a> 2016, <a href="/wiki/Spezial:ISBN-Suche/9783662503300" class="internal mw-magiclink-isbn">ISBN 978-3-662-50330-0</a>, Seite 145</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Roger B. Nelsen: <i>Proof without Words: The Sum of a Positive Number and Its Reciprocal Is at Least Two (four proofs)</i> Mathematics Magazine, vol. 67, no. 5 (Dec. 1994), S. 374</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Claudi Alsina, Roger B. Nelsen: <i>Perlen der Mathematik - 20 geometrische Figuren als Ausgangspunkte für mathematische Erkundungsreisen</i>, <a href="/wiki/Springer_Spektrum" title="Springer Spektrum">Springer Spektrum</a>, Springer-Verlag GmbH <a href="/wiki/Berlin" title="Berlin">Berlin</a> 2015, <a href="/wiki/Spezial:ISBN-Suche/9783662454602" class="internal mw-magiclink-isbn">ISBN 978-3-662-45460-2</a>, Seiten 237 und 301</span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><a href="/wiki/Ross_Honsberger" title="Ross Honsberger">Ross Honsberger</a>: <i>Gitter - Reste - Würfel</i> <a href="/wiki/Vieweg_Verlag" title="Vieweg Verlag">Friedrich Vieweg &amp; Sohn Verlagsgesellschaft mbH</a>, <a href="/wiki/Braunschweig" title="Braunschweig">Braunschweig</a> 1984, <a href="/wiki/Spezial:ISBN-Suche/9783528084769" class="internal mw-magiclink-isbn">ISBN 978-3-528-08476-9</a>, S. 155</span>
</li>
</ol></div><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&amp;type=1x1&amp;usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript>
<div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Kehrwert&amp;oldid=245058555">https://de.wikipedia.org/w/index.php?title=Kehrwert&amp;oldid=245058555</a>“</div></div>
<div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorie</a>: <ul><li><a href="/wiki/Kategorie:Division_(Mathematik)" title="Kategorie:Division (Mathematik)">Division (Mathematik)</a></li></ul></div></div>
</div>
</div>
<div id="mw-navigation">
<h2>Navigationsmenü</h2>
<div id="mw-head">
<nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label"  >
<h3
id="p-personal-label"
class="vector-menu-heading "
>
<span class="vector-menu-heading-label">Meine Werkzeuge</span>
</h3>
<div class="vector-menu-content">
<ul class="vector-menu-content-list">
<li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&amp;returnto=Kehrwert" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. Es ist jedoch nicht zwingend erforderlich."><span>Benutzerkonto erstellen</span></a></li><li id="pt-login" class="mw-list-item"><a href="/w/index.php?title=Spezial:Anmelden&amp;returnto=Kehrwert" title="Anmelden ist zwar keine Pflicht, wird aber gerne gesehen. [o]" accesskey="o"><span>Anmelden</span></a></li>
</ul>
</div>
</nav>
<div id="left-navigation">
<nav id="p-namespaces" class="mw-portlet mw-portlet-namespaces vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-namespaces-label"  >
<h3
id="p-namespaces-label"
class="vector-menu-heading "
>
<span class="vector-menu-heading-label">Namensräume</span>
</h3>
<div class="vector-menu-content">
<ul class="vector-menu-content-list">
<li id="ca-nstab-main" class="selected mw-list-item"><a href="/wiki/Kehrwert" title="Seiteninhalt anzeigen [c]" accesskey="c"><span>Artikel</span></a></li><li id="ca-talk" class="mw-list-item"><a href="/wiki/Diskussion:Kehrwert" rel="discussion" title="Diskussion zum Seiteninhalt [t]" accesskey="t"><span>Diskussion</span></a></li>
</ul>
</div>
</nav>
<nav id="p-variants" class="mw-portlet mw-portlet-variants emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-variants-label"  >
<input type="checkbox"
id="p-variants-checkbox"
role="button"
aria-haspopup="true"
data-event-name="ui.dropdown-p-variants"
class="vector-menu-checkbox"
aria-labelledby="p-variants-label"
>
<label
id="p-variants-label"
class="vector-menu-heading "
>
<span class="vector-menu-heading-label">Deutsch</span>
</label>
<div class="vector-menu-content">
<ul class="vector-menu-content-list">
</ul>
</div>
</nav>
</div>
<div id="right-navigation">
<nav id="p-views" class="mw-portlet mw-portlet-views vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-views-label"  >
<h3
id="p-views-label"
class="vector-menu-heading "
>
<span class="vector-menu-heading-label">Ansichten</span>
</h3>
<div class="vector-menu-content">
<ul class="vector-menu-content-list">
<li id="ca-view" class="selected mw-list-item"><a href="/wiki/Kehrwert"><span>Lesen</span></a></li><li id="ca-ve-edit" class="mw-list-item"><a href="/w/index.php?title=Kehrwert&amp;veaction=edit" title="Diese Seite mit dem VisualEditor bearbeiten [v]" accesskey="v"><span>Bearbeiten</span></a></li><li id="ca-edit" class="collapsible mw-list-item"><a href="/w/index.php?title=Kehrwert&amp;action=edit" title="Den Quelltext dieser Seite bearbeiten [e]" accesskey="e"><span>Quelltext bearbeiten</span></a></li><li id="ca-history" class="mw-list-item"><a href="/w/index.php?title=Kehrwert&amp;action=history" title="Frühere Versionen dieser Seite [h]" accesskey="h"><span>Versionsgeschichte</span></a></li>
</ul>
</div>
</nav>
<nav id="p-cactions" class="mw-portlet mw-portlet-cactions emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-cactions-label"  title="Weitere Optionen" >
<input type="checkbox"
id="p-cactions-checkbox"
role="button"
aria-haspopup="true"
data-event-name="ui.dropdown-p-cactions"
class="vector-menu-checkbox"
aria-labelledby="p-cactions-label"
>
<label
id="p-cactions-label"
class="vector-menu-heading "
>
<span class="vector-menu-heading-label">Weitere</span>
</label>
<div class="vector-menu-content">
<ul class="vector-menu-content-list">
</ul>
</div>
</nav>
<div id="p-search" role="search" class="vector-search-box-vue  vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box">
<h3 >Suche</h3>
<form action="/w/index.php" id="searchform" class="vector-search-box-form">
<div id="simpleSearch"
class="vector-search-box-inner"
data-search-loc="header-navigation">
<input class="vector-search-box-input"
type="search" name="search" placeholder="Wikipedia durchsuchen" aria-label="Wikipedia durchsuchen" autocapitalize="sentences" title="Durchsuche die Wikipedia [f]" accesskey="f" id="searchInput"
>
<input type="hidden" name="title" value="Spezial:Suche">
<input id="mw-searchButton"
class="searchButton mw-fallbackSearchButton" type="submit" name="fulltext" title="Suche nach Seiten, die diesen Text enthalten" value="Suchen">
<input id="searchButton"
class="searchButton" type="submit" name="go" title="Gehe direkt zu der Seite mit genau diesem Namen, falls sie vorhanden ist." value="Artikel">
</div>
</form>
</div>
</div>
</div>
<div id="mw-panel" class="vector-legacy-sidebar">
<div id="p-logo" role="banner">
<a class="mw-wiki-logo" href="/wiki/Wikipedia:Hauptseite"
title="Hauptseite"></a>
</div>
<nav id="p-navigation" class="mw-portlet mw-portlet-navigation vector-menu-portal portal vector-menu" aria-labelledby="p-navigation-label"  >
<h3
id="p-navigation-label"
class="vector-menu-heading "
>
<span class="vector-menu-heading-label">Navigation</span>
</h3>
<div class="vector-menu-content">
<ul class="vector-menu-content-list">
<li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Wikipedia:Hauptseite" title="Hauptseite besuchen [z]" accesskey="z"><span>Hauptseite</span></a></li><li id="n-topics" class="mw-list-item"><a href="/wiki/Portal:Wikipedia_nach_Themen"><span>Themenportale</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Spezial:Zuf%C3%A4llige_Seite" title="Zufällige Seite aufrufen [x]" accesskey="x"><span>Zufälliger Artikel</span></a></li>
</ul>
</div>
</nav>
<nav id="p-Mitmachen" class="mw-portlet mw-portlet-Mitmachen vector-menu-portal portal vector-menu" aria-labelledby="p-Mitmachen-label"  >
<h3
id="p-Mitmachen-label"
class="vector-menu-heading "
>
<span class="vector-menu-heading-label">Mitmachen</span>
</h3>
<div class="vector-menu-content">
<ul class="vector-menu-content-list">
<li id="n-Artikel-verbessern" class="mw-list-item"><a href="/wiki/Wikipedia:Beteiligen"><span>Artikel verbessern</span></a></li><li id="n-Neuerartikel" class="mw-list-item"><a href="/wiki/Hilfe:Neuen_Artikel_anlegen"><span>Neuen Artikel anlegen</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Autorenportal" title="Info-Zentrum über Beteiligungsmöglichkeiten"><span>Autorenportal</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Hilfe:%C3%9Cbersicht" title="Übersicht über Hilfeseiten"><span>Hilfe</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Spezial:Letzte_%C3%84nderungen" title="Liste der letzten Änderungen in Wikipedia [r]" accesskey="r"><span>Letzte Änderungen</span></a></li><li id="n-contact" class="mw-list-item"><a href="/wiki/Wikipedia:Kontakt" title="Kontaktmöglichkeiten"><span>Kontakt</span></a></li><li id="n-sitesupport" class="mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&amp;wmf_medium=sidebar&amp;wmf_campaign=de.wikipedia.org&amp;uselang=de" title="Unterstütze uns"><span>Spenden</span></a></li>
</ul>
</div>
</nav>
<nav id="p-tb" class="mw-portlet mw-portlet-tb vector-menu-portal portal vector-menu" aria-labelledby="p-tb-label"  >
<h3
id="p-tb-label"
class="vector-menu-heading "
>
<span class="vector-menu-heading-label">Werkzeuge</span>
</h3>
<div class="vector-menu-content">
<ul class="vector-menu-content-list">
<li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Spezial:Linkliste/Kehrwert" title="Liste aller Seiten, die hierher verlinken [j]" accesskey="j"><span>Links auf diese Seite</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Spezial:%C3%84nderungen_an_verlinkten_Seiten/Kehrwert" rel="nofollow" title="Letzte Änderungen an Seiten, die von hier verlinkt sind [k]" accesskey="k"><span>Änderungen an verlinkten Seiten</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Spezial:Spezialseiten" title="Liste aller Spezialseiten [q]" accesskey="q"><span>Spezialseiten</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Kehrwert&amp;oldid=245058555" title="Dauerhafter Link zu dieser Seitenversion"><span>Permanenter Link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Kehrwert&amp;action=info" title="Weitere Informationen über diese Seite"><span>Seiten­­informationen</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Spezial:Zitierhilfe&amp;page=Kehrwert&amp;id=245058555&amp;wpFormIdentifier=titleform" title="Hinweise, wie diese Seite zitiert werden kann"><span>Artikel zitieren</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Spezial:URL-K%C3%BCrzung&amp;url=https%3A%2F%2Fde.wikipedia.org%2Fwiki%2FKehrwert"><span>Kurzlink</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Spezial:QrKodu&amp;url=https%3A%2F%2Fde.wikipedia.org%2Fwiki%2FKehrwert"><span>QR-Code herunterladen</span></a></li>
</ul>
</div>
</nav>
<nav id="p-coll-print_export" class="mw-portlet mw-portlet-coll-print_export vector-menu-portal portal vector-menu" aria-labelledby="p-coll-print_export-label"  >
<h3
id="p-coll-print_export-label"
class="vector-menu-heading "
>
<span class="vector-menu-heading-label">Drucken/​exportieren</span>
</h3>
<div class="vector-menu-content">
<ul class="vector-menu-content-list">
<li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Spezial:DownloadAsPdf&amp;page=Kehrwert&amp;action=show-download-screen"><span>Als PDF herunterladen</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Kehrwert&amp;printable=yes" title="Druckansicht dieser Seite [p]" accesskey="p"><span>Druckversion</span></a></li>
</ul>
</div>
</nav>
<nav id="p-wikibase-otherprojects" class="mw-portlet mw-portlet-wikibase-otherprojects vector-menu-portal portal vector-menu" aria-labelledby="p-wikibase-otherprojects-label"  >
<h3
id="p-wikibase-otherprojects-label"
class="vector-menu-heading "
>
<span class="vector-menu-heading-label">In anderen Projekten</span>
</h3>
<div class="vector-menu-content">
<ul class="vector-menu-content-list">
<li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q216906" title="Link zum verbundenen Objekt im Datenrepositorium [g]" accesskey="g"><span>Wikidata-Datenobjekt</span></a></li>
</ul>
</div>
</nav>
<nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label"  >
<h3
id="p-lang-label"
class="vector-menu-heading "
>
<span class="vector-menu-heading-label">In anderen Sprachen</span>
</h3>
<div class="vector-menu-content">
<ul class="vector-menu-content-list">
<li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Omgekeerde" title="Omgekeerde – Afrikaans" lang="af" hreflang="af" data-title="Omgekeerde" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%82%D9%84%D9%88%D8%A8_%D8%B9%D8%AF%D8%AF" title="مقلوب عدد – Arabisch" lang="ar" hreflang="ar" data-title="مقلوب عدد" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Inversu_multiplicativu" title="Inversu multiplicativu – Asturisch" lang="ast" hreflang="ast" data-title="Inversu multiplicativu" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B5%D1%86%D0%B8%D0%BF%D1%80%D0%BE%D1%87%D0%BD%D0%B0_%D1%81%D1%82%D0%BE%D0%B9%D0%BD%D0%BE%D1%81%D1%82" title="Реципрочна стойност – Bulgarisch" lang="bg" hreflang="bg" data-title="Реципрочна стойност" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Invers_multiplicatiu" title="Invers multiplicatiu – Katalanisch" lang="ca" hreflang="ca" data-title="Invers multiplicatiu" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/P%C5%99evr%C3%A1cen%C3%A1_hodnota" title="Převrácená hodnota – Tschechisch" lang="cs" hreflang="cs" data-title="Převrácená hodnota" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D1%83%D1%82%C4%83%D0%BD%D0%BB%D0%B0_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Кутăнла хисеп – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Кутăнла хисеп" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cilydd" title="Cilydd – Walisisch" lang="cy" hreflang="cy" data-title="Cilydd" data-language-autonym="Cymraeg" data-language-local-name="Walisisch" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Reciprok" title="Reciprok – Dänisch" lang="da" hreflang="da" data-title="Reciprok" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%BF%CE%BB%CE%BB%CE%B1%CF%80%CE%BB%CE%B1%CF%83%CE%B9%CE%B1%CF%83%CF%84%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CE%BD%CF%84%CE%AF%CF%83%CF%84%CF%81%CE%BF%CF%86%CE%BF%CF%82" title="Πολλαπλασιαστικός αντίστροφος – Griechisch" lang="el" hreflang="el" data-title="Πολλαπλασιαστικός αντίστροφος" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Multiplicative_inverse" title="Multiplicative inverse – Englisch" lang="en" hreflang="en" data-title="Multiplicative inverse" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Inverso" title="Inverso – Esperanto" lang="eo" hreflang="eo" data-title="Inverso" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Inverso_multiplicativo" title="Inverso multiplicativo – Spanisch" lang="es" hreflang="es" data-title="Inverso multiplicativo" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/P%C3%B6%C3%B6rdv%C3%A4%C3%A4rtus" title="Pöördväärtus – Estnisch" lang="et" hreflang="et" data-title="Pöördväärtus" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Alderantzizko_zenbaki" title="Alderantzizko zenbaki – Baskisch" lang="eu" hreflang="eu" data-title="Alderantzizko zenbaki" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%88%D8%A7%D8%B1%D9%88%D9%86_%D8%B6%D8%B1%D8%A8%DB%8C" title="وارون ضربی – Persisch" lang="fa" hreflang="fa" data-title="وارون ضربی" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/K%C3%A4%C3%A4nteisluku" title="Käänteisluku – Finnisch" lang="fi" hreflang="fi" data-title="Käänteisluku" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Inverse" title="Inverse – Französisch" lang="fr" hreflang="fr" data-title="Inverse" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kiarw%C3%A4%C3%A4rs" title="Kiarwäärs – Nordfriesisch" lang="frr" hreflang="frr" data-title="Kiarwäärs" data-language-autonym="Nordfriisk" data-language-local-name="Nordfriesisch" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Inverso_multiplicativo" title="Inverso multiplicativo – Galicisch" lang="gl" hreflang="gl" data-title="Inverso multiplicativo" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%94%D7%95%D7%A4%D7%9B%D7%99" title="מספר הופכי – Hebräisch" lang="he" hreflang="he" data-title="מספר הופכי" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Reciprok" title="Reciprok – Ungarisch" lang="hu" hreflang="hu" data-title="Reciprok" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Invers_perkalian" title="Invers perkalian – Indonesisch" lang="id" hreflang="id" data-title="Invers perkalian" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Umhverfa" title="Umhverfa – Isländisch" lang="is" hreflang="is" data-title="Umhverfa" data-language-autonym="Íslenska" data-language-local-name="Isländisch" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Reciproco" title="Reciproco – Italienisch" lang="it" hreflang="it" data-title="Reciproco" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%80%86%E6%95%B0" title="逆数 – Japanisch" lang="ja" hreflang="ja" data-title="逆数" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%B1%EC%85%88_%EC%97%AD%EC%9B%90" title="곱셈 역원 – Koreanisch" lang="ko" hreflang="ko" data-title="곱셈 역원" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Invers" title="Invers – Lombardisch" lang="lmo" hreflang="lmo" data-title="Invers" data-language-autonym="Lombard" data-language-local-name="Lombardisch" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Atvirk%C5%A1tinis_skai%C4%8Dius" title="Atvirkštinis skaičius – Litauisch" lang="lt" hreflang="lt" data-title="Atvirkštinis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B5%D1%86%D0%B8%D0%BF%D1%80%D0%BE%D1%87%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82" title="Реципрочна вредност – Mazedonisch" lang="mk" hreflang="mk" data-title="Реципрочна вредност" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Salingan" title="Salingan – Malaiisch" lang="ms" hreflang="ms" data-title="Salingan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malaiisch" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Kehrweert" title="Kehrweert – Niederdeutsch" lang="nds" hreflang="nds" data-title="Kehrweert" data-language-autonym="Plattdüütsch" data-language-local-name="Niederdeutsch" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Omgekeerde" title="Omgekeerde – Niederländisch" lang="nl" hreflang="nl" data-title="Omgekeerde" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Resiprok" title="Resiprok – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Resiprok" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%81%E0%A8%A3%E0%A8%BE%E0%A8%A4%E0%A8%AE%E0%A8%95_%E0%A8%89%E0%A8%B2%E0%A8%9F" title="ਗੁਣਾਤਮਕ ਉਲਟ – Punjabi" lang="pa" hreflang="pa" data-title="ਗੁਣਾਤਮਕ ਉਲਟ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczba_odwrotna" title="Liczba odwrotna – Polnisch" lang="pl" hreflang="pl" data-title="Liczba odwrotna" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Inverso_multiplicativo" title="Inverso multiplicativo – Portugiesisch" lang="pt" hreflang="pt" data-title="Inverso multiplicativo" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/T%27ikrasqa_yupay" title="T&#039;ikrasqa yupay – Quechua" lang="qu" hreflang="qu" data-title="T&#039;ikrasqa yupay" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Обратное число – Russisch" lang="ru" hreflang="ru" data-title="Обратное число" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Reciprocal" title="Reciprocal – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Reciprocal" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Prevr%C3%A1ten%C3%A1_hodnota" title="Prevrátená hodnota – Slowakisch" lang="sk" hreflang="sk" data-title="Prevrátená hodnota" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Recipro%C4%8Dna_vrednost" title="Recipročna vrednost – Slowenisch" lang="sl" hreflang="sl" data-title="Recipročna vrednost" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B5%D1%86%D0%B8%D0%BF%D1%80%D0%BE%D1%87%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82" title="Реципрочна вредност – Serbisch" lang="sr" hreflang="sr" data-title="Реципрочна вредност" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Reciprok_(matematik)" title="Reciprok (matematik) – Schwedisch" lang="sv" hreflang="sv" data-title="Reciprok (matematik)" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%86%E0%AE%B0%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%B2%E0%AF%8D_%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AF%8D%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%81" title="பெருக்கல் நேர்மாறு – Tamil" lang="ta" hreflang="ta" data-title="பெருக்கல் நேர்மாறு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B8%9C%E0%B8%81%E0%B8%9C%E0%B8%B1%E0%B8%99%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%84%E0%B8%B9%E0%B8%93" title="ตัวผกผันการคูณ – Thailändisch" lang="th" hreflang="th" data-title="ตัวผกผันการคูณ" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Kabaligtarang_pamparami" title="Kabaligtarang pamparami – Tagalog" lang="tl" hreflang="tl" data-title="Kabaligtarang pamparami" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D0%B1%D0%B5%D1%80%D0%BD%D0%B5%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Обернене число – Ukrainisch" lang="uk" hreflang="uk" data-title="Обернене число" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA%DB%8C_%D9%85%D9%82%D9%84%D9%88%D8%A8" title="ریاضیاتی مقلوب – Urdu" lang="ur" hreflang="ur" data-title="ریاضیاتی مقلوب" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ngh%E1%BB%8Bch_%C4%91%E1%BA%A3o_ph%C3%A9p_nh%C3%A2n" title="Nghịch đảo phép nhân – Vietnamesisch" lang="vi" hreflang="vi" data-title="Nghịch đảo phép nhân" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%80%92%E6%95%B0" title="倒数 – Wu" lang="wuu" hreflang="wuu" data-title="倒数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%80%92%E6%95%B0" title="倒数 – Chinesisch" lang="zh" hreflang="zh" data-title="倒数" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%80%92%E6%95%B8" title="倒數 – Klassisches Chinesisch" lang="lzh" hreflang="lzh" data-title="倒數" data-language-autonym="文言" data-language-local-name="Klassisches Chinesisch" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%80%92%E6%95%B8_(%E6%95%B8)" title="倒數 (數) – Kantonesisch" lang="yue" hreflang="yue" data-title="倒數 (數)" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target"><span>粵語</span></a></li>
</ul>
<div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q216906#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen bearbeiten" class="wbc-editpage">Links bearbeiten</a></span></div>
</div>
</nav>
</div>
</div>
<footer id="footer" class="mw-footer" >
<ul id="footer-info">
<li id="footer-info-lastmod"> Diese Seite wurde zuletzt am 17. Mai 2024 um 15:20 Uhr bearbeitet.</li>
<li id="footer-info-copyright"><div id="footer-info-copyright-stats" class="noprint"><a rel="nofollow" class="external text" href="https://pageviews.wmcloud.org/?pages=Kehrwert&amp;project=de.wikipedia.org">Abrufstatistik</a>&#160;· <a rel="nofollow" class="external text" href="https://xtools.wmcloud.org/authorship/de.wikipedia.org/Kehrwert?uselang=de">Autoren</a> </div><div id="footer-info-copyright-separator"><br /></div><div id="footer-info-copyright-info">
<p>Der Text ist unter der Lizenz <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">„Creative-Commons Namensnennung – Weitergabe unter gleichen Bedingungen“</a> verfügbar; Informationen zu den Urhebern und zum Lizenzstatus eingebundener Mediendateien (etwa Bilder oder Videos) können im Regelfall durch Anklicken dieser abgerufen werden. Möglicherweise unterliegen die Inhalte jeweils zusätzlichen Bedingungen. Durch die Nutzung dieser Website erklären Sie sich mit den <span class="plainlinks"><a class="external text" href="https://foundation.wikimedia.org/wiki/Policy:Terms_of_Use/de">Nutzungsbedingungen</a> und der <a class="external text" href="https://foundation.wikimedia.org/wiki/Policy:Privacy_policy/de">Datenschutzrichtlinie</a></span> einverstanden.<br />
</p>
Wikipedia® ist eine eingetragene Marke der Wikimedia Foundation Inc.</div></li>
</ul>
<ul id="footer-places">
<li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy/de">Datenschutz</a></li>
<li id="footer-places-about"><a href="/wiki/Wikipedia:%C3%9Cber_Wikipedia">Über Wikipedia</a></li>
<li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:Impressum">Impressum</a></li>
<li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Verhaltenskodex</a></li>
<li id="footer-places-developers"><a href="https://developer.wikimedia.org">Entwickler</a></li>
<li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/de.wikipedia.org">Statistiken</a></li>
<li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Stellungnahme zu Cookies</a></li>
<li id="footer-places-mobileview"><a href="//de.m.wikipedia.org/w/index.php?title=Kehrwert&amp;mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile Ansicht</a></li>
</ul>
<ul id="footer-icons" class="noprint">
<li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li>
<li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li>
</ul>
</footer>
<script>(RLQ=window.RLQ||[]).push(function(){mw.log.warn("This page is using the deprecated ResourceLoader module \"codex-search-styles\".\n[1.43] Use a CodexModule with codexComponents to set your specific components used: https://www.mediawiki.org/wiki/Codex#Using_a_limited_subset_of_components");mw.config.set({"wgHostname":"mw-web.eqiad.main-57c8968774-v9qg6","wgBackendResponseTime":146,"wgPageParseReport":{"limitreport":{"cputime":"0.189","walltime":"0.326","ppvisitednodes":{"value":913,"limit":1000000},"postexpandincludesize":{"value":5597,"limit":2097152},"templateargumentsize":{"value":1110,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":4565,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00%  115.686      1 -total"," 51.38%  59.445      1 Vorlage:Literatur"," 34.26%  39.637      1 Vorlage:Mehrere_Bilder"," 21.84%  25.262      8 Vorlage:Str_replace"," 21.43%  24.797      4 Vorlage:Mehrere_Bilder/Align"," 10.17%  11.763      1 Vorlage:Wiktionary","  6.93%    8.019      1 Vorlage:Booland","  1.98%    2.293      1 Vorlage:Absatz"]},"scribunto":{"limitreport-timeusage":{"value":"0.043","limit":"10.000"},"limitreport-memusage":{"value":1986840,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-cc877b49b-l4ht7","timestamp":"20241127132206","ttl":2592000,"transientcontent":false}}});});</script>
<script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Kehrwert","url":"https:\/\/de.wikipedia.org\/wiki\/Kehrwert","sameAs":"http:\/\/www.wikidata.org\/entity\/Q216906","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q216906","author":{"@type":"Organization","name":"Autoren der Wikimedia-Projekte"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-12-20T22:00:51Z","headline":"Begriff aus der Arithmetik"}</script>
</body>
</html>
#[https://www.ingoostwald.de/v2_14/downloads/unterricht_fr/82211.pdf Potenzgesetze]
#[https://www.ingoostwald.de/v2_14/downloads/unterricht_fr/82211.pdf Potenzgesetze]
#[https://www.schule-bw.de/faecher-und-schularten/mathematisch-naturwissenschaftliche-faecher/mathematik/unterrichtsmaterialien/sekundarstufe1/zahl/ter/term9/potenzen/potenzgesetze.pdf/@@download/file/potenzgesetze.pdf Potenzgesetze in einfacher Darstellung - automatischer Download]
#[https://www.schule-bw.de/faecher-und-schularten/mathematisch-naturwissenschaftliche-faecher/mathematik/unterrichtsmaterialien/sekundarstufe1/zahl/ter/term9/potenzen/potenzgesetze.pdf/@@download/file/potenzgesetze.pdf Potenzgesetze in einfacher Darstellung - automatischer Download]

Revision as of 18:48, 22 December 2024

Trigonometrie 1

  1. Landesbildungsserver BW - Einführung
  2. Visualisierung von Sin, Cos, Tan in Geogebra
  3. Visualisierung interaktiv
    1. Merksatz für sin/cos von 0/30/45/60/90

Trigonometrie 2

  1. Trigonometrische Funktionen als Potenzreihen
  2. Geogebra - Approximation trig. Funktionen durch Polynome (interaktiv)

Aufgaben Trigonometrie

  1. Mit anderen Themenbereichen der Mathematik

Geometrie

Satze des Pythagoras

  1. Geometrische Veranschaulichung


Potenzen

  1. Potenzen mit neg. Exponenten umformen
  2. Potenzwerte berechnen
  3. Termwerte berechnen (0,9 * 10^8)
  4. Potenzen zusammenfassen
  1. Potenzgesetze
  2. Potenzgesetze in einfacher Darstellung - automatischer Download
  3. Speziell zur Umwandlung von Logarithmen und log10
  4. Logarithmengesetze mit Beispielen und anklickbaren Lösungen

https://drive.google.com/drive/folders/1wNKPr9Cww3qrHnykiepCS6gl5CkquYkM

Aufgaben zu Potenzgesetzen

  1. Übungsblatt 1
  2. Übungsblatt 2 - wieder einfache Übungen
  3. Leichte Übungen - 9. Klasse Gym
Retrieved from "http://caski.atwebpages.com/index.php?title=Christian&oldid=815"