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=== Aufgaben Trigonometrie ===
=== Aufgaben Trigonometrie ===
#[https://www.matheaufgaben-loesen.de/ Mit anderen Themenbereichen der Mathematik]
#[https://www.matheaufgaben-loesen.de/ Mit anderen Themenbereichen der Mathematik]
#[https://mmf.univie.ac.at/fileadmin/user_upload/p_mathematikmachtfreunde/Materialien/AS-Trigonometrie.pdf Aufgaben 2]
#[https://henriks-mathewerkstatt.de/935.Trigonometrie.Aufgaben.pdf Transformationen der trig. Funktionen]
#[https://www.ccbuchner.de/_files_media/mediathek/downloads/4242.pdf Anwendungsaufgaben]


== Geometrie ==
== Geometrie ==
Line 23: Line 26:
#[https://www.geogebra.org/m/D38a3Y7s Potenzen zusammenfassen]
#[https://www.geogebra.org/m/D38a3Y7s Potenzen zusammenfassen]


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<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>
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<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>
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    <annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
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</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>
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    <annotation encoding="application/x-tex">{\displaystyle x}</annotation>
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</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>
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      <mstyle displaystyle="true" scriptlevel="0">
        <mi>x</mi>
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    <annotation encoding="application/x-tex">{\displaystyle x}</annotation>
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</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}">
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</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>
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        <mrow class="MJX-TeXAtom-ORD">
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{x}}}</annotation>
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</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>
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</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>
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    </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>
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    <annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
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</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}}}">
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    <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">
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              <mrow>
                <mn>3</mn>
                <mo>+</mo>
                <mn>4</mn>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="normal">i</mi>
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        <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>
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    <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>
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<span class="vector-menu-heading-label">In anderen Sprachen</span>
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<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>
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#[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]
Line 1,952: Line 36:
#[https://mathe.aufgabenfuchs.de/potenz/potenz.shtml Übungsblatt 2 - wieder einfache Übungen]
#[https://mathe.aufgabenfuchs.de/potenz/potenz.shtml Übungsblatt 2 - wieder einfache Übungen]
#[https://www.klassenarbeiten.de/realschule/klasse9/mathematik/potenzrechnung/4052.htm Leichte Übungen - 9. Klasse Gym]
#[https://www.klassenarbeiten.de/realschule/klasse9/mathematik/potenzrechnung/4052.htm Leichte Übungen - 9. Klasse Gym]
== Hinrichtung ==
#[https://www.youtube.com/watch?v=ykqzGEAZTb8 Hinrichtung 1]
#[https://www.youtube.com/watch?v=aJCSNIl2Pls Western]

Latest revision as of 23:45, 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
  2. Aufgaben 2
  3. Transformationen der trig. Funktionen
  4. Anwendungsaufgaben

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

Hinrichtung

  1. Hinrichtung 1
  2. Western
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