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Subtraktion – Wikipedia
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type="application/atom+xml" title="Atom-Feed für „Wikipedia“" href="/w/index.php?title=Spezial:Letzte_%C3%84nderungen&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin-vector-legacy mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Subtraktion rootpage-Subtraktion skin-vector action-view"><div id="mw-page-base" class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice"><!-- CentralNotice --></div> <div class="mw-indicators"> </div> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Subtraktion</span></h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">aus Wikipedia, der freien Enzyklopädie</div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Zur Navigation springen</a> <a class="mw-jump-link" href="#searchInput">Zur Suche springen</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Subtraction01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Subtraction01.svg/220px-Subtraction01.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Subtraction01.svg/330px-Subtraction01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Subtraction01.svg/440px-Subtraction01.svg.png 2x" data-file-width="300" data-file-height="200" /></a><figcaption>Subtraktion 5 − 2 = 3 am Beispiel von Pfirsichen.</figcaption></figure> <p>Die <b>Subtraktion</b> (von <a href="/wiki/Lateinische_Sprache" class="mw-redirect" title="Lateinische Sprache">lat.</a> <i>subtrahere</i> „wegziehen“, „entfernen“), umgangssprachlich auch <b>Minus-Rechnen</b> genannt, ist eine der vier <a href="/wiki/Grundrechenarten" class="mw-redirect" title="Grundrechenarten">Grundrechenarten</a> der <a href="/wiki/Arithmetik" title="Arithmetik">Arithmetik</a>. Unter der Subtraktion versteht man das <b>Abziehen</b> einer <a href="/wiki/Zahl" title="Zahl">Zahl</a> von einer anderen. <a href="/wiki/Mathematik" title="Mathematik">Mathematisch</a> handelt es sich bei der Subtraktion um eine <a href="/wiki/Zweistellige_Verkn%C3%BCpfung" title="Zweistellige Verknüpfung">zweistellige Verknüpfung</a>. Die Subtraktion ist die <a href="/wiki/Umkehroperation" title="Umkehroperation">Umkehroperation</a> der <a href="/wiki/Addition" title="Addition">Addition</a>. Das <a href="/wiki/Rechenzeichen" class="mw-redirect" title="Rechenzeichen">Rechenzeichen</a> für die Subtraktion ist das <a href="/wiki/Minuszeichen" title="Minuszeichen">Minuszeichen</a> „−“. </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="#Sprachregelungen,_Grundeigenschaften_und_Notation"><span class="tocnumber">1</span> <span class="toctext">Sprachregelungen, Grundeigenschaften und Notation</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Mathematische_Definition"><span class="tocnumber">2</span> <span class="toctext">Mathematische Definition</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Basisverfahren"><span class="tocnumber">3</span> <span class="toctext">Basisverfahren</span></a> <ul> <li class="toclevel-2 tocsection-4"><a href="#Graphische_Methode"><span class="tocnumber">3.1</span> <span class="toctext">Graphische Methode</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Subtraktion-Subtraktion-Methode"><span class="tocnumber">3.2</span> <span class="toctext">Subtraktion-Subtraktion-Methode</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Subtraktion-Addition-Methode"><span class="tocnumber">3.3</span> <span class="toctext">Subtraktion-Addition-Methode</span></a></li> <li class="toclevel-2 tocsection-7"><a href="#Komplement-Methode"><span class="tocnumber">3.4</span> <span class="toctext">Komplement-Methode</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-8"><a href="#Schriftliche_Subtraktion"><span class="tocnumber">4</span> <span class="toctext">Schriftliche Subtraktion</span></a> <ul> <li class="toclevel-2 tocsection-9"><a href="#Vertikale_Subtraktion_mit_Überträgen"><span class="tocnumber">4.1</span> <span class="toctext">Vertikale Subtraktion mit Überträgen</span></a> <ul> <li class="toclevel-3 tocsection-10"><a href="#Ergänzungsverfahren"><span class="tocnumber">4.1.1</span> <span class="toctext">Ergänzungsverfahren</span></a></li> <li class="toclevel-3 tocsection-11"><a href="#Subtraktion_von_links_nach_rechts"><span class="tocnumber">4.1.2</span> <span class="toctext">Subtraktion von links nach rechts</span></a></li> <li class="toclevel-3 tocsection-12"><a href="#Entbündelungsverfahren"><span class="tocnumber">4.1.3</span> <span class="toctext">Entbündelungsverfahren</span></a></li> <li class="toclevel-3 tocsection-13"><a href="#Vorab-Entbündelung"><span class="tocnumber">4.1.4</span> <span class="toctext">Vorab-Entbündelung</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-14"><a href="#Vertikale_Subtraktion_ohne_Überträge"><span class="tocnumber">4.2</span> <span class="toctext">Vertikale Subtraktion ohne Überträge</span></a> <ul> <li class="toclevel-3 tocsection-15"><a href="#Teildifferenzen"><span class="tocnumber">4.2.1</span> <span class="toctext">Teildifferenzen</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-16"><a href="#Nicht-vertikale_Verfahren"><span class="tocnumber">4.3</span> <span class="toctext">Nicht-vertikale Verfahren</span></a> <ul> <li class="toclevel-3 tocsection-17"><a href="#Ausschreiten_der_Differenz"><span class="tocnumber">4.3.1</span> <span class="toctext">Ausschreiten der Differenz</span></a></li> <li class="toclevel-3 tocsection-18"><a href="#Zergliederung_des_Subtrahenden"><span class="tocnumber">4.3.2</span> <span class="toctext">Zergliederung des Subtrahenden</span></a></li> <li class="toclevel-3 tocsection-19"><a href="#Gleiche_Veränderung"><span class="tocnumber">4.3.3</span> <span class="toctext">Gleiche Veränderung</span></a></li> </ul> </li> </ul> </li> <li class="toclevel-1 tocsection-20"><a href="#Siehe_auch"><span class="tocnumber">5</span> <span class="toctext">Siehe auch</span></a></li> <li class="toclevel-1 tocsection-21"><a href="#Weblinks"><span class="tocnumber">6</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-22"><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="Sprachregelungen,_Grundeigenschaften_und_Notation"><span id="Sprachregelungen.2C_Grundeigenschaften_und_Notation"></span>Sprachregelungen, Grundeigenschaften und Notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=1" title="Abschnitt bearbeiten: Sprachregelungen, Grundeigenschaften und Notation" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Sprachregelungen, Grundeigenschaften und Notation"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Für die Elemente einer Subtraktion gibt es folgende Symbole und Sprechweisen: </p> <ul><li>Das Rechenzeichen für die Subtraktion ist das Minuszeichen „−“. Es wurde 1489 von <a href="/wiki/Johannes_Widmann_(Mathematiker)" title="Johannes Widmann (Mathematiker)">Johannes Widmann</a> eingeführt.</li> <li>Die Zahl, von der etwas abgezogen wird, heißt <i>Minuend</i> (lateinisch „das zu Verringernde“).</li> <li>Die Zahl, die abgezogen wird, heißt <i>Subtrahend</i> (lateinisch „das Abzuziehende“).</li> <li>Der Rechenausdruck (<a href="/wiki/Term" title="Term">Term</a>), der den Minuenden, das Minus-Zeichen und den Subtrahenden umfasst, heißt <i>Differenz</i>.</li> <li>Das Ergebnis einer Subtraktion ist der <i>Wert der Differenz</i> (auch <i>Differenzwert</i> oder auch kurz nur <i>Differenz</i>).</li> <li>Das Symbol für Differenzen als Terme ist der griechische Großbuchstabe <a href="/wiki/Delta" title="Delta">Delta</a> „Δ“, der auch als <a href="/wiki/Operator_(Mathematik)" title="Operator (Mathematik)">Operator</a> für die Differenzbildung benutzt wird (siehe <a href="/wiki/Differenz-Operator" title="Differenz-Operator">Differenz-Operator</a>). Häufig wird als <i>Differenz</i> – besonders im alltäglichen Sprachgebrauch – allerdings <i>nur</i> das <i>Ergebnis</i> dieser „Minusrechnung“, noch häufiger der <a href="/wiki/Betragsfunktion" title="Betragsfunktion">Betrag</a> dieses Ergebnisses bezeichnet. Beispiel: <i>Die Differenz zwischen 7 und 9 und die Differenz zwischen 5 und 3 beträgt 2</i>. Im Beispiel wird dies durch das Verb „beträgt“ betont.</li></ul> <p>Merkhilfe: <i>Minuend minus Subtrahend gleich Wert der Differenz</i> (<a href="/wiki/Eselsbr%C3%BCcke" class="mw-redirect" title="Eselsbrücke">Eselsbrücken</a>: <b>M</b>inuend kommt im Alphabet vor <b>S</b>ubtrahend; im Wort „<b>M</b>inu<b>s</b>“ wird durch Anfangs- und Endbuchstabe gezeigt, wie <b>M</b>inuend und <b>S</b>ubtrahend platziert sind) </p><p>Beispiele (mit Berücksichtigung des Vorzeichens): </p> <ul><li>4 minus 1 ist (gleich) 3 oder anders geschrieben: <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 4-1=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>−<!-- − --></mo> <mn>1</mn> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4-1=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28341eb1fc30c4f47b888a08f9b364fe915e22a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 4-1=3}"></span>.</li></ul> <p>Dabei ist 4 der Minuend, 1 stellt den Subtrahenden dar, der Rechenausdruck (Term) <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 4-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>−<!-- − --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b9974e7dee6170b79554e1cba720bee6f73014c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle 4-1}"></span> ist die Differenz und das Ergebnis 3 bildet den Wert der Differenz bzw. den Differenzwert. </p><p>Die Menge der <a href="/wiki/Nat%C3%BCrliche_Zahlen" class="mw-redirect" title="Natürliche Zahlen">natürlichen Zahlen</a> ist bezüglich der Subtraktion nicht <a href="/wiki/Abgeschlossenheit_(algebraische_Struktur)" title="Abgeschlossenheit (algebraische Struktur)">abgeschlossen</a>, das heißt mit der Subtraktion erzielt man eventuell ein Ergebnis, das außerhalb des Bereichs der <a href="/wiki/Nat%C3%BCrliche_Zahlen" class="mw-redirect" title="Natürliche Zahlen">natürlichen Zahlen</a> liegt. </p> <ul><li>Beispiel: <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 1-4=-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−<!-- − --></mo> <mn>4</mn> <mo>=</mo> <mo>−<!-- − --></mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-4=-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dba4e137b8b73edf6018bff940dea5d725728db3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.234ex; height:2.343ex;" alt="{\displaystyle 1-4=-3}"></span></li></ul> <p>Es gibt abkürzende Notationen 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}"> <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/1b80866c2bf2f1bc1f2e4c97e7937f5663150ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a-b}"></span>, beispielsweise <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|_{x=b}^{x=a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>a</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x|_{x=b}^{x=a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8807c411646e9d343dbef09050b0b7497bd7a13f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.297ex; height:3.176ex;" alt="{\displaystyle x|_{x=b}^{x=a}}"></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]_{b}^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x]_{b}^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3492a592e95eeb9e28a82ac9a49629371d3ce730" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.725ex; height:3.009ex;" alt="{\displaystyle [x]_{b}^{a}}"></span>, was vor allem bei Termen wie <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 f(a)-f(b)=f(x)|_{x=b}^{x=a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>a</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(a)-f(b)=f(x)|_{x=b}^{x=a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd34d5fb6986eb529cc7e1983350733ad791ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.727ex; height:3.176ex;" alt="{\displaystyle f(a)-f(b)=f(x)|_{x=b}^{x=a}}"></span> bzw. <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 F(a)-F(b)=[F(x)]_{b}^{a}=[F(x)]_{x=b}^{x=a}=F(x){\Big |}_{b}^{a}=F(x){\Big |}_{x=b}^{x=a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">[</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>a</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(a)-F(b)=[F(x)]_{b}^{a}=[F(x)]_{x=b}^{x=a}=F(x){\Big |}_{b}^{a}=F(x){\Big |}_{x=b}^{x=a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/990f7354ad5ce3caaeab710dc9f0792291fe55a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:56.807ex; height:4.509ex;" alt="{\displaystyle F(a)-F(b)=[F(x)]_{b}^{a}=[F(x)]_{x=b}^{x=a}=F(x){\Big |}_{b}^{a}=F(x){\Big |}_{x=b}^{x=a}}"></span> Anwendung findet. </p><p>Bei mehreren hintereinander auftretenden Subtraktionen wird der Ausdruck von links nach rechts ausgewertet; die Subtraktion ist daher per Konvention <a href="/wiki/Operatorassoziativit%C3%A4t" title="Operatorassoziativität">linksassoziativ</a>:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <ul><li><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-c=(a-b)-c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−<!-- − --></mo> <mi>b</mi> <mo>−<!-- − --></mo> <mi>c</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−<!-- − --></mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b-c=(a-b)-c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/668f94d650191c50497d12c3e5296ce350b78788" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.738ex; height:2.843ex;" alt="{\displaystyle a-b-c=(a-b)-c}"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Mathematische_Definition">Mathematische Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=2" title="Abschnitt bearbeiten: Mathematische Definition" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Mathematische Definition"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Subtraktion ist die <a href="/wiki/Umkehroperation" title="Umkehroperation">Umkehroperation</a> der <a href="/wiki/Addition" title="Addition">Addition</a>. In <a href="/wiki/Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppen</a> lässt sich zu jedem gegebenen <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> genau ein <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> finden, so dass 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 b+x=a\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>+</mo> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b+x=a\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e4ba89e9cdd9acbbfb1f84fa6e04baf633853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.334ex; width:9.442ex; height:2.343ex;" alt="{\displaystyle b+x=a\!}"></span></dd></dl> <p>Die Bestimmung 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> heißt <i>Subtraktion</i>. <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> lässt sich bestimmen, indem man <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> 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 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> <i>subtrahiert</i> („abzieht“): </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=a-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mo>−<!-- − --></mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a-b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07016de924468f14e5584579884655286b306b77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.496ex; height:2.343ex;" alt="{\displaystyle x=a-b}"></span></dd></dl> <p><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> heißt der <i>Minuend</i>, <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> der <i>Subtrahend</i>. Das Ergebnis einer Subtraktion, 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}"> <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>, heißt <i>Wert der Differenz</i>. Eine Subtraktion wird mit dem Minuszeichen notiert: </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 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/1b80866c2bf2f1bc1f2e4c97e7937f5663150ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a-b}"></span></dd></dl> <p>Die Subtraktion <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/1b80866c2bf2f1bc1f2e4c97e7937f5663150ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a-b}"></span> kann auch als Addition der <a href="/wiki/Gegenzahl" class="mw-redirect" title="Gegenzahl">Gegenzahl</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 -b=(-1)\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−<!-- − --></mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -b=(-1)\cdot b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46512d9b196cba1074e03bdce9a8701f7e95eef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.361ex; height:2.843ex;" alt="{\displaystyle -b=(-1)\cdot b}"></span> des Subtrahenden zum Minuenden <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> definiert werden: </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 a-b=a+(-1)\cdot b=a+(-b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−<!-- − --></mo> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b=a+(-1)\cdot b=a+(-b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a0890dd8c6c1e3bfcd2147385f59ce4f265657" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.477ex; height:2.843ex;" alt="{\displaystyle a-b=a+(-1)\cdot b=a+(-b)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Basisverfahren">Basisverfahren</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=3" title="Abschnitt bearbeiten: Basisverfahren" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Basisverfahren"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Graphische_Methode">Graphische Methode</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=4" title="Abschnitt bearbeiten: Graphische Methode" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Graphische Methode"><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:Vektormethode.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Vektormethode.svg/220px-Vektormethode.svg.png" decoding="async" width="220" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Vektormethode.svg/330px-Vektormethode.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Vektormethode.svg/440px-Vektormethode.svg.png 2x" data-file-width="320" data-file-height="116" /></a><figcaption>Graphische Methode mit Vektoren</figcaption></figure> <p>Bei der graphischen Methode werden die Zahlenwerte als Balken, Linien, Punkte oder andere abstrakte Objekte dargestellt. Eine weitere Möglichkeit ist die Darstellung mit <a href="/wiki/Vektor" title="Vektor">Vektoren</a>, wobei die Richtung des Subtrahend-Vektors umgekehrt und die Vektoren anschließend aufaddiert werden. </p> <dl><dt>Beispiel</dt></dl> <table> <tbody><tr> <td></td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td align="right">(13) </td></tr> <tr> <td>-</td> <td></td> <td></td> <td></td> <td></td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td align="right">(9) </td></tr> <tr> <td>=</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td style="background-color:black;">•</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td align="right">(4) </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Subtraktion-Subtraktion-Methode">Subtraktion-Subtraktion-Methode</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=5" title="Abschnitt bearbeiten: Subtraktion-Subtraktion-Methode" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Subtraktion-Subtraktion-Methode"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bei der <i>Subtraktion-Subtraktion-Methode</i> wird so lange ein Teilbetrag des Subtrahenden von Subtrahend und Minuend abgezogen, bis der Subtrahend 0 ist. Dabei wird meist eine Zehnerstelle als Zwischenschritt gewählt. </p> <dl><dt>Beispiel</dt> <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 13-9=(13-3)-(9-3)=10-6=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>13</mn> <mo>−<!-- − --></mo> <mn>9</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>13</mn> <mo>−<!-- − --></mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mo stretchy="false">(</mo> <mn>9</mn> <mo>−<!-- − --></mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>10</mn> <mo>−<!-- − --></mo> <mn>6</mn> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13-9=(13-3)-(9-3)=10-6=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7fdc4505a740939c17695ed3e396d4c58cc0a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.065ex; height:2.843ex;" alt="{\displaystyle 13-9=(13-3)-(9-3)=10-6=4}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Subtraktion-Addition-Methode">Subtraktion-Addition-Methode</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=6" title="Abschnitt bearbeiten: Subtraktion-Addition-Methode" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Subtraktion-Addition-Methode"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bei der <i>Subtraktion-Addition-Methode</i> werden Subtrahend und Minuend in Teilkomponenten zerlegt, von diesen subtrahiert, und anschließend die Teilbeträge wieder addiert. </p> <dl><dt>Beispiel</dt> <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 13-9=(10+3)-9=(10-9)+3=1+3=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>13</mn> <mo>−<!-- − --></mo> <mn>9</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mn>9</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo>−<!-- − --></mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13-9=(10+3)-9=(10-9)+3=1+3=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d75d86806b92878fc47a7fa72ce57a075b4e363" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.329ex; height:2.843ex;" alt="{\displaystyle 13-9=(10+3)-9=(10-9)+3=1+3=4}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Komplement-Methode">Komplement-Methode</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=7" title="Abschnitt bearbeiten: Komplement-Methode" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Komplement-Methode"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bei der <i>Komplement-Methode</i> wird vom Subtrahenden das zugehörige <a href="/wiki/Komplement_(Mengenlehre)" title="Komplement (Mengenlehre)">Komplement</a> berechnet. Anschließend werden der Minuend und das Komplement addiert. Das Verfahren wird insbesondere in der <a href="/wiki/Technische_Informatik" title="Technische Informatik">technischen Informatik</a>, etwa beim mechanischen Feld-Tarrant-<a href="/wiki/Comptometer" title="Comptometer">Comptometer</a>, dem mechanischen <a href="/w/index.php?title=Hoffritz-Addierer&action=edit&redlink=1" class="new" title="Hoffritz-Addierer (Seite nicht vorhanden)">Hoffritz-Addierer</a>, sowie elektronischen <a href="/wiki/Addierwerk" title="Addierwerk">Addierwerken</a> in modernen Computersystemen, angewendet.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dt>Beispiel</dt></dl> <p>Ausgangsformel (linke Seite im <a href="/wiki/Dezimalsystem" title="Dezimalsystem">Dezimalsystem</a>, rechte Seite im <a href="/wiki/Dualsystem" title="Dualsystem">Dualsystem</a>): </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 13_{10}-9_{10}=1101_{2}-1001_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>=</mo> <msub> <mn>1101</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mn>1001</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13_{10}-9_{10}=1101_{2}-1001_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58adf4e11433f0525f914bc6930b64505480c7da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.427ex; height:2.509ex;" alt="{\displaystyle 13_{10}-9_{10}=1101_{2}-1001_{2}}"></span></dd></dl> <p>Dies entspricht: </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 13_{10}+\left(-9_{10}\right)=1101_{2}+\left(-1001_{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−<!-- − --></mo> <msub> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mn>1101</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−<!-- − --></mo> <msub> <mn>1001</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13_{10}+\left(-9_{10}\right)=1101_{2}+\left(-1001_{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d153913e172d7f457209253fb0d70a2b7280e02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.662ex; height:2.843ex;" alt="{\displaystyle 13_{10}+\left(-9_{10}\right)=1101_{2}+\left(-1001_{2}\right)}"></span></dd></dl> <p>Berechnung des Komplements im Dezimalsystem (<a href="/w/index.php?title=Zehnerkomplement&action=edit&redlink=1" class="new" title="Zehnerkomplement (Seite nicht vorhanden)">Zehnerkomplement</a>) und im Dualsystem (<a href="/wiki/Zweierkomplement" title="Zweierkomplement">Zweierkomplement</a>): </p> <table class="wikitable"> <caption>Berechnung des Komplements </caption> <tbody><tr> <th rowspan="2">Operation</th> <th colspan="2">Ergebniswert </th></tr> <tr> <th>Zehnerkomplement</th> <th>Zweierkomplement </th></tr> <tr> <td>Ausgangswert</td> <td align="right"><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 \ldots \ 000\ 009,0_{10}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…<!-- … --></mo> <mtext> </mtext> <mn>000</mn> <mtext> </mtext> <mn>009</mn> <mo>,</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots \ 000\ 009,0_{10}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c67bfae3bff5651e884a77bbae950d6df4b4e568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.319ex; height:2.509ex;" alt="{\displaystyle \ldots \ 000\ 009,0_{10}}"></span></td> <td align="right"><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 \ldots \ 0000\ 1001,0_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…<!-- … --></mo> <mtext> </mtext> <mn>0000</mn> <mtext> </mtext> <mn>1001</mn> <mo>,</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots \ 0000\ 1001,0_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2794d88d46824baa98c31b2c681eba001c64931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.822ex; height:2.509ex;" alt="{\displaystyle \ldots \ 0000\ 1001,0_{2}}"></span> </td></tr> <tr> <td>Invertierung</td> <td align="right"><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 \ldots \ 999\ 990,{\dot {9}}_{10}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…<!-- … --></mo> <mtext> </mtext> <mn>999</mn> <mtext> </mtext> <mn>990</mn> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>9</mn> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots \ 999\ 990,{\dot {9}}_{10}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c19174a3bf3499f6d41aae68cef6f21ee5318f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.319ex; height:3.009ex;" alt="{\displaystyle \ldots \ 999\ 990,{\dot {9}}_{10}}"></span></td> <td align="right"><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 \ldots \ 1111\ 0110,{\dot {1}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…<!-- … --></mo> <mtext> </mtext> <mn>1111</mn> <mtext> </mtext> <mn>0110</mn> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots \ 1111\ 0110,{\dot {1}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73095af60a316f12fc50fcc5d764b72249be6e28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.822ex; height:3.009ex;" alt="{\displaystyle \ldots \ 1111\ 0110,{\dot {1}}_{2}}"></span> </td></tr> <tr> <td>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 0,{\dot {9}}_{10}=0,{\dot {1}}_{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>9</mn> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,{\dot {9}}_{10}=0,{\dot {1}}_{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab6bfda930bd5e150999556a8a962bcc5124a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.008ex; height:3.009ex;" alt="{\displaystyle 0,{\dot {9}}_{10}=0,{\dot {1}}_{2}=1}"></span></td> <td align="right"><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 \ldots \ 999\ 991,0_{10}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…<!-- … --></mo> <mtext> </mtext> <mn>999</mn> <mtext> </mtext> <mn>991</mn> <mo>,</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots \ 999\ 991,0_{10}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c85c2598b22073e79bbd8d8570618812a5571a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.319ex; height:2.509ex;" alt="{\displaystyle \ldots \ 999\ 991,0_{10}}"></span></td> <td align="right"><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 \ldots \ 1111\ 0111,0_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…<!-- … --></mo> <mtext> </mtext> <mn>1111</mn> <mtext> </mtext> <mn>0111</mn> <mo>,</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots \ 1111\ 0111,0_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a479a6f19503e7cea9a64bb146c550b9fad7cad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.822ex; height:2.509ex;" alt="{\displaystyle \ldots \ 1111\ 0111,0_{2}}"></span> </td></tr></tbody></table> <p>Addition: </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 {\begin{array}{rrrcrrr}&13_{10}&&&1101_{2}\\+&\ldots \ 999\ 991_{10}&&+&\ldots \ 1111\ 0111_{2}\\=&\ldots \ 000\ 004_{10}&&=&\ldots \ 0000\ 0100_{2}\\=&4_{10}&&=&100_{2}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right center right right right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd /> <mtd> <msub> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mtd> <mtd /> <mtd /> <mtd> <msub> <mn>1101</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> </mtd> <mtd> <mo>…<!-- … --></mo> <mtext> </mtext> <mn>999</mn> <mtext> </mtext> <msub> <mn>991</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mtd> <mtd /> <mtd> <mo>+</mo> </mtd> <mtd> <mo>…<!-- … --></mo> <mtext> </mtext> <mn>1111</mn> <mtext> </mtext> <msub> <mn>0111</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mo>…<!-- … --></mo> <mtext> </mtext> <mn>000</mn> <mtext> </mtext> <msub> <mn>004</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mtd> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo>…<!-- … --></mo> <mtext> </mtext> <mn>0000</mn> <mtext> </mtext> <msub> <mn>0100</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mtd> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mn>100</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rrrcrrr}&13_{10}&&&1101_{2}\\+&\ldots \ 999\ 991_{10}&&+&\ldots \ 1111\ 0111_{2}\\=&\ldots \ 000\ 004_{10}&&=&\ldots \ 0000\ 0100_{2}\\=&4_{10}&&=&100_{2}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab5723d39b0c78eb01e23960784cf08b2e0ca430" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:41.406ex; height:12.509ex;" alt="{\displaystyle {\begin{array}{rrrcrrr}&13_{10}&&&1101_{2}\\+&\ldots \ 999\ 991_{10}&&+&\ldots \ 1111\ 0111_{2}\\=&\ldots \ 000\ 004_{10}&&=&\ldots \ 0000\ 0100_{2}\\=&4_{10}&&=&100_{2}\end{array}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Schriftliche_Subtraktion">Schriftliche Subtraktion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=8" title="Abschnitt bearbeiten: Schriftliche Subtraktion" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Schriftliche Subtraktion"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die schriftliche Subtraktion ist neben der <a href="/wiki/Addition#Schriftliche_Addition" title="Addition">schriftlichen Addition</a> eine der grundlegenden <a href="/wiki/Kulturtechnik" title="Kulturtechnik">Kulturtechniken</a>, die bereits in den ersten Schuljahren der Grundschule erlernt wird. Die Beherrschung der schriftlichen Subtraktion ist Voraussetzung für das Erlernen der <a href="/wiki/Schriftliche_Division" title="Schriftliche Division">schriftlichen Division</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Vertikale_Subtraktion_mit_Überträgen"><span id="Vertikale_Subtraktion_mit_.C3.9Cbertr.C3.A4gen"></span>Vertikale Subtraktion mit Überträgen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=9" title="Abschnitt bearbeiten: Vertikale Subtraktion mit Überträgen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Vertikale Subtraktion mit Überträgen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In den Grundschulen werden heute meist Verfahren gelehrt, bei denen die einander entsprechenden <a href="/wiki/Stellenwertsystem" title="Stellenwertsystem">Stellen</a> der Minuenden und Subtrahenden <i>übereinander</i> stehen. Die Stellen werden nacheinander abgearbeitet, meist von rechts nach links. </p><p>Für das schriftliche Subtrahieren muss der Minuend (Zahl oben) größer oder gleich dem Subtrahenden (Zahl(en) unten) sein. Negative Ergebnisse sind somit direkt nicht möglich. </p><p>Wenn der Minuend doch kleiner ist als der Subtrahend, dann können die Vorzeichen zum Rechnen vertauscht werden. Der Subtrahend wird so zum Minuend (oben geschrieben) und der Minuend zum Subtrahend (unten geschrieben). Es kann dann mit den unten beschriebenen Verfahren gerechnet werden. Das Ergebnis muss aber zum Schluss mit einem Minus versehen werden, denn es ist immer negativ (keine natürliche Zahl). Damit wird der zuvor zum Berechnen durchgeführte Vorzeichenwechsel wieder rückgängig gemacht. </p><p>Wenn die einzelnen Stellen der Subtrahenden größer sind als die gleichen Stellen der Minuenden, müssen <a href="/wiki/%C3%9Cbertrag" title="Übertrag">Überträge</a> gehandhabt werden. Das heißt, der Minuend wird, um die Subtraktion zu ermöglichen, um 10 erhöht; um dies auszugleichen, muss in der links benachbarten Spalte entweder der Minuend erniedrigt (Entbündelungsverfahren; Vorabberechnung der Überträge) oder der Subtrahend erhöht werden (Ergänzungsverfahren; Subtraktion von rechts nach links). Im deutschsprachigen Raum hat sich mit dem Ergänzungsverfahren die letztgenannte Vorgehensweise durchgesetzt. Im Jahr 2000 trat in einigen <a href="/wiki/Land_(Deutschland)" title="Land (Deutschland)">Bundesländern</a> ein neuer Lehrplan in Kraft, der nun statt des Ergänzens das Entbündeln als Standard vorschreibt. </p> <div class="mw-heading mw-heading4"><h4 id="Ergänzungsverfahren"><span id="Erg.C3.A4nzungsverfahren"></span>Ergänzungsverfahren</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=10" title="Abschnitt bearbeiten: Ergänzungsverfahren" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Ergänzungsverfahren"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Beim Ergänzungsverfahren, das auch Auffülltechnik oder (in den USA) <i>Austrian method</i> („Österreichische Methode“) genannt wird, wird keine Subtraktion vorgenommen, sondern der Subtrahend umgekehrt bis zum Minuenden <i>erhöht</i>. Falls dies nicht möglich ist, wird der Minuend um 10 erhöht. Die 10 wird nicht „geborgt“, sondern als 1 zum Subtrahenden der nächsten Teilberechnung addiert. Im deutschsprachigen Raum wird dieses Verfahren an den Grundschulen als Standardmethode gelehrt. Einer der Vorteile des Verfahrens besteht darin, dass es den Umgang mit Aufgaben vorbereitet, bei denen von einem Minuenden mehrere Subtrahenden abgezogen werden sollen. </p> <dl><dt>Beispiel</dt></dl> <table class="wikitable" style="text-align:center"> <tbody><tr> <th></th> <th>Beschreibung </th></tr> <tr> <td><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 {\begin{array}{r}75{\color {red}3}\\-{\underset {}{4}}9{\color {red}1}\\\hline \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="bottom"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>75</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>3</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>4</mn> <mrow /> </munder> </mrow> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>1</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </menclose> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}75{\color {red}3}\\-{\underset {}{4}}9{\color {red}1}\\\hline \end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d97d6b7c45dc515ab898b11fc9b22acb3230bc9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:6.549ex; height:7.843ex;" alt="{\displaystyle {\begin{array}{r}75{\color {red}3}\\-{\underset {}{4}}9{\color {red}1}\\\hline \end{array}}}"></span> </td> <td>1 + … = 3 </td></tr> <tr> <td><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 {\begin{array}{r}7{\color {red}5}3\\-{\underset {}{4}}{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>5</mn> </mstyle> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>4</mn> <mrow /> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>9</mn> </mstyle> </mrow> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>2</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}7{\color {red}5}3\\-{\underset {}{4}}{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53d2cdafa69410197f133833976b1eb2d2c90f09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:7.905ex; height:11.509ex;" alt="{\displaystyle {\begin{array}{r}7{\color {red}5}3\\-{\underset {}{4}}{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}"></span> </td> <td>Das Ergebnis wird unter den Strich geschrieben. </td></tr> <tr> <td><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 {\begin{array}{r}7{\color {red}5}3\\-{\underset {}{4}}{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>5</mn> </mstyle> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>4</mn> <mrow /> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>9</mn> </mstyle> </mrow> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>2</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}7{\color {red}5}3\\-{\underset {}{4}}{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53d2cdafa69410197f133833976b1eb2d2c90f09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:7.905ex; height:11.509ex;" alt="{\displaystyle {\begin{array}{r}7{\color {red}5}3\\-{\underset {}{4}}{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}"></span> </td> <td>9 + … = 5<br />Die angepeilte Summe (5) ist zu klein! </td></tr> <tr> <td><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 {\begin{array}{r}7{\color {red}5}3\\-{\underset {\color {red}1}{4}}{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>5</mn> </mstyle> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>4</mn> <mstyle mathcolor="red"> <mn>1</mn> </mstyle> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>9</mn> </mstyle> </mrow> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>2</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}7{\color {red}5}3\\-{\underset {\color {red}1}{4}}{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad1cdf253d3b825633008fd5f7c644137149a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:7.905ex; height:11.509ex;" alt="{\displaystyle {\begin{array}{r}7{\color {red}5}3\\-{\underset {\color {red}1}{4}}{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}"></span> </td> <td>Sie wird darum um 10 erhöht. Die 1 wird unter den nächsten Subtrahenden geschrieben. </td></tr> <tr> <td><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 {\begin{array}{r}7{\color {red}5}3\\-{\underset {\color {red}1}{4}}{\color {red}9}1\\\hline {\color {Gray}62}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>5</mn> </mstyle> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>4</mn> <mstyle mathcolor="red"> <mn>1</mn> </mstyle> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>9</mn> </mstyle> </mrow> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>62</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}7{\color {red}5}3\\-{\underset {\color {red}1}{4}}{\color {red}9}1\\\hline {\color {Gray}62}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe96a3e667895945c28a980e94f69b233b9d65c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:7.905ex; height:11.509ex;" alt="{\displaystyle {\begin{array}{r}7{\color {red}5}3\\-{\underset {\color {red}1}{4}}{\color {red}9}1\\\hline {\color {Gray}62}\end{array}}}"></span> </td> <td>9 + … = 15<br />Die Berechnung kann jetzt durchgeführt werden, das Ergebnis wird unter den Strich geschrieben. </td></tr> <tr> <td><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 {\begin{array}{r}{\color {red}7}53\\-{\color {red}{\underset {1}{4}}}91\\\hline {\color {Gray}62}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>7</mn> </mstyle> </mrow> <mn>53</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>4</mn> <mn>1</mn> </munder> </mrow> </mstyle> </mrow> <mn>91</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>62</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}{\color {red}7}53\\-{\color {red}{\underset {1}{4}}}91\\\hline {\color {Gray}62}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93637dd7bbf5e24178cf7af6d12eb8e103be648b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:7.905ex; height:11.509ex;" alt="{\displaystyle {\begin{array}{r}{\color {red}7}53\\-{\color {red}{\underset {1}{4}}}91\\\hline {\color {Gray}62}\end{array}}}"></span> </td> <td>(4 + 1) + … = 7 </td></tr> <tr> <td><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 {\begin{array}{r}{\color {red}7}53\\-{\color {red}{\underset {1}{4}}}91\\\hline {\color {Gray}262}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>7</mn> </mstyle> </mrow> <mn>53</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>4</mn> <mn>1</mn> </munder> </mrow> </mstyle> </mrow> <mn>91</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>262</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}{\color {red}7}53\\-{\color {red}{\underset {1}{4}}}91\\\hline {\color {Gray}262}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04cbdab1b5ecd449dc9286734dc542c379f340a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:7.905ex; height:11.509ex;" alt="{\displaystyle {\begin{array}{r}{\color {red}7}53\\-{\color {red}{\underset {1}{4}}}91\\\hline {\color {Gray}262}\end{array}}}"></span> </td> <td>Das Ergebnis wird unter den Strich geschrieben. </td></tr> <tr> <td><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 {\begin{array}{r}753\\-{\underset {1}{4}}91\\\hline {\color {Gray}262}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mn>753</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>4</mn> <mn>1</mn> </munder> </mrow> <mn>91</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>262</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}753\\-{\underset {1}{4}}91\\\hline {\color {Gray}262}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/086004ae157e0411535d19cda9b9a800b271f98f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:7.905ex; height:11.509ex;" alt="{\displaystyle {\begin{array}{r}753\\-{\underset {1}{4}}91\\\hline {\color {Gray}262}\end{array}}}"></span> </td> <td>Das Gesamtergebnis. </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Subtraktion_von_links_nach_rechts">Subtraktion von links nach rechts</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=11" title="Abschnitt bearbeiten: Subtraktion von links nach rechts" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Subtraktion von links nach rechts"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Subtraktion kann auch von links nach rechts durchgeführt werden. Bei diesem ungewöhnlichen Verfahren, das eine Variante des Ergänzungsverfahrens ist, werden die Überträge abgearbeitet, bevor die Differenz genau ausgerechnet wird. Da die Überträge weder notiert noch gemerkt werden müssen, ist die Methode nicht nur vergleichsweise resistent gegen Flüchtigkeitsfehler, sondern auch sehr schnell und sogar fürs Kopfrechnen geeignet. </p> <dl><dt>Beispiel</dt></dl> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Datei:LeftToRight_Subtraction_Step_1.svg" class="mw-file-description" title="7 − 4 = 3 Dieser Wert wird nur gemerkt, nicht notiert."><img alt="7 − 4 = 3 Dieser Wert wird nur gemerkt, nicht notiert." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/LeftToRight_Subtraction_Step_1.svg/88px-LeftToRight_Subtraction_Step_1.svg.png" decoding="async" width="88" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/LeftToRight_Subtraction_Step_1.svg/133px-LeftToRight_Subtraction_Step_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/LeftToRight_Subtraction_Step_1.svg/177px-LeftToRight_Subtraction_Step_1.svg.png 2x" data-file-width="273" data-file-height="370" /></a></span></div> <div class="gallerytext">7 − 4 = 3<br />Dieser Wert wird nur gemerkt, nicht notiert.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Datei:LeftToRight_Subtraction_Step_2.JPG" class="mw-file-description" title="Da in der folgenden Spalte der Minuend kleiner ist als der Subtrahend, wird der soeben errechnete Wert um 1 erniedrigt."><img alt="Da in der folgenden Spalte der Minuend kleiner ist als der Subtrahend, wird der soeben errechnete Wert um 1 erniedrigt." src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/LeftToRight_Subtraction_Step_2.JPG/88px-LeftToRight_Subtraction_Step_2.JPG" decoding="async" width="88" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/LeftToRight_Subtraction_Step_2.JPG/131px-LeftToRight_Subtraction_Step_2.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/LeftToRight_Subtraction_Step_2.JPG/175px-LeftToRight_Subtraction_Step_2.JPG 2x" data-file-width="272" data-file-height="372" /></a></span></div> <div class="gallerytext">Da in der folgenden Spalte der Minuend kleiner ist als der Subtrahend, wird der soeben errechnete Wert um 1 erniedrigt.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Datei:LeftToRight_Subtraction_Step_3.JPG" class="mw-file-description" title="15 − 9 = 6"><img alt="15 − 9 = 6" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/LeftToRight_Subtraction_Step_3.JPG/88px-LeftToRight_Subtraction_Step_3.JPG" decoding="async" width="88" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/LeftToRight_Subtraction_Step_3.JPG/132px-LeftToRight_Subtraction_Step_3.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/LeftToRight_Subtraction_Step_3.JPG/176px-LeftToRight_Subtraction_Step_3.JPG 2x" data-file-width="272" data-file-height="371" /></a></span></div> <div class="gallerytext">15 − 9 = 6</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Datei:LeftToRight_Subtraction_Step_4.JPG" class="mw-file-description" title="Da in der folgenden Spalte der Minuend nicht kleiner ist als der Subtrahend, bleibt es bei diesem Wert."><img alt="Da in der folgenden Spalte der Minuend nicht kleiner ist als der Subtrahend, bleibt es bei diesem Wert." src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/LeftToRight_Subtraction_Step_4.JPG/88px-LeftToRight_Subtraction_Step_4.JPG" decoding="async" width="88" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/LeftToRight_Subtraction_Step_4.JPG/131px-LeftToRight_Subtraction_Step_4.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/LeftToRight_Subtraction_Step_4.JPG/175px-LeftToRight_Subtraction_Step_4.JPG 2x" data-file-width="271" data-file-height="371" /></a></span></div> <div class="gallerytext">Da in der folgenden Spalte der Minuend nicht kleiner ist als der Subtrahend, bleibt es bei diesem Wert.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Datei:LeftToRight_Subtraction_Step_5.JPG" class="mw-file-description" title="3 − 1 = 2"><img alt="3 − 1 = 2" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/LeftToRight_Subtraction_Step_5.JPG/88px-LeftToRight_Subtraction_Step_5.JPG" decoding="async" width="88" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/LeftToRight_Subtraction_Step_5.JPG/131px-LeftToRight_Subtraction_Step_5.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/LeftToRight_Subtraction_Step_5.JPG/175px-LeftToRight_Subtraction_Step_5.JPG 2x" data-file-width="271" data-file-height="371" /></a></span></div> <div class="gallerytext">3 − 1 = 2</div> </li> </ul> <p>Findet sich eine Spalte oder eine Sequenz von mehreren Spalten, in denen zwei gleiche Ziffern stehen, und rechts daneben eine Spalte mit einem Minuend, der kleiner als der Subtrahend ist, so muss die bei diesem Verfahren routinemäßige „Vorausschau“ nicht nur die zwei gleichen Ziffern, sondern auch die darauf folgenden Spalten umfassen. Jede Spalte mit den gleichen Ziffern erhält dann eine Neun statt einer Null als Ergebnis. </p><p>Die Vorausschau über mehreren Spalten in den oben geschilderten Fällen ist eine Schwachstelle dieser Methode. </p> <div class="mw-heading mw-heading4"><h4 id="Entbündelungsverfahren"><span id="Entb.C3.BCndelungsverfahren"></span>Entbündelungsverfahren</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=12" title="Abschnitt bearbeiten: Entbündelungsverfahren" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Entbündelungsverfahren"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Abziehen mit „Entbündeln“ bedeutet, dass der zu kleine Minuend bei seinem linken Nachbarn eine „Anleihe“ macht. Der Minuend wird um 10 erhöht und der linke Nachbar um 1 erniedrigt. Das Verfahren wird an den Grundschulen z. B. der <a href="/wiki/Vereinigte_Staaten" title="Vereinigte Staaten">Vereinigten Staaten</a> als Standardmethode gelehrt. Der reine Rechenaufwand ist ähnlich wie beim Ergänzungsverfahren; wenn von einer Null „geliehen“ werden muss, muss diese jedoch bei ihrem eigenen linken Nachbarn eine „Anleihe“ machen – eine Technik, die zusätzlich erlernt werden muss (beim Ergänzungsverfahren wird sie nicht gebraucht). Außerdem muss beim Entbündeln mehr geschrieben werden. </p> <dl><dt>Beispiel</dt></dl> <table class="wikitable"> <tbody><tr> <th></th> <th>Beschreibung </th></tr> <tr> <td><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 {\begin{array}{r}75{\color {red}3}\\-49{\color {red}1}\\\hline \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="bottom"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>75</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>3</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mn>49</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>1</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </menclose> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}75{\color {red}3}\\-49{\color {red}1}\\\hline \end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/def8f45af6dabe3e3eda9bc6a8609369d8b01323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:6.549ex; height:6.843ex;" alt="{\displaystyle {\begin{array}{r}75{\color {red}3}\\-49{\color {red}1}\\\hline \end{array}}}"></span> </td> <td>3 − 1 = … </td></tr> <tr> <td><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 {\begin{array}{r}75{\color {red}3}\\-49{\color {red}1}\\\hline {\color {Gray}2}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mn>75</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>3</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mn>49</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>1</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>2</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}75{\color {red}3}\\-49{\color {red}1}\\\hline {\color {Gray}2}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99b89b42773fce7bfef12d168137252e6ed3751c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:7.905ex; height:10.176ex;" alt="{\displaystyle {\begin{array}{r}75{\color {red}3}\\-49{\color {red}1}\\\hline {\color {Gray}2}\end{array}}}"></span> </td> <td>Das Ergebnis wird unter den Strich geschrieben. </td></tr> <tr> <td><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 {\begin{array}{r}7{\color {red}5}3\\-4{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>5</mn> </mstyle> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>9</mn> </mstyle> </mrow> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>2</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}7{\color {red}5}3\\-4{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a5f3c758b72d18247f103f82a4e797cf82761a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:7.905ex; height:10.176ex;" alt="{\displaystyle {\begin{array}{r}7{\color {red}5}3\\-4{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}"></span> </td> <td>5 − 9 = …<br />Der Minuend (5) ist zu klein! </td></tr> <tr> <td><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 {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {red}15}{\cancel {5}}}3\\-4{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>7</mn> </menclose> <mstyle mathcolor="#949698"> <mn>6</mn> </mstyle> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>5</mn> </menclose> <mstyle mathcolor="red"> <mn>15</mn> </mstyle> </mover> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>9</mn> </mstyle> </mrow> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>2</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {red}15}{\cancel {5}}}3\\-4{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d27bd0dedb60a3d05a622163db4d74d77850a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:8.652ex; height:12.843ex;" alt="{\displaystyle {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {red}15}{\cancel {5}}}3\\-4{\color {red}9}1\\\hline {\color {Gray}2}\end{array}}}"></span> </td> <td>Er wird darum um 10 erhöht. Diese 10 wird von der links daneben stehenden Ziffer (7) „geliehen“; diese wird um 1 erniedrigt. </td></tr> <tr> <td><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 {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {red}15}{\cancel {5}}}3\\-4{\color {red}9}1\\\hline {\color {Gray}62}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>7</mn> </menclose> <mstyle mathcolor="#949698"> <mn>6</mn> </mstyle> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>5</mn> </menclose> <mstyle mathcolor="red"> <mn>15</mn> </mstyle> </mover> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>9</mn> </mstyle> </mrow> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>62</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {red}15}{\cancel {5}}}3\\-4{\color {red}9}1\\\hline {\color {Gray}62}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c79412d3806064e8665ff3e59551c8382f2b32c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:8.652ex; height:12.843ex;" alt="{\displaystyle {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {red}15}{\cancel {5}}}3\\-4{\color {red}9}1\\\hline {\color {Gray}62}\end{array}}}"></span> </td> <td>15 − 9 = …<br />Die Subtraktion kann jetzt durchgeführt werden. Das Ergebnis wird unter den Strich geschrieben. </td></tr> <tr> <td><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 {\begin{array}{r}{\overset {\color {Red}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}3\\-{\color {red}4}91\\\hline {\color {Gray}62}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>7</mn> </menclose> <mstyle mathcolor="#ff5555"> <mn>6</mn> </mstyle> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>5</mn> </menclose> <mstyle mathcolor="#949698"> <mn>15</mn> </mstyle> </mover> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>4</mn> </mstyle> </mrow> <mn>91</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>62</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}{\overset {\color {Red}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}3\\-{\color {red}4}91\\\hline {\color {Gray}62}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc7787b5b7ed296069b861486b1729383c5bbd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:8.652ex; height:12.843ex;" alt="{\displaystyle {\begin{array}{r}{\overset {\color {Red}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}3\\-{\color {red}4}91\\\hline {\color {Gray}62}\end{array}}}"></span> </td> <td>6 − 4 = … </td></tr> <tr> <td><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 {\begin{array}{r}{\overset {\color {Red}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}3\\-{\color {red}4}91\\\hline {\color {Gray}262}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>7</mn> </menclose> <mstyle mathcolor="#ED1B23"> <mn>6</mn> </mstyle> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>5</mn> </menclose> <mstyle mathcolor="#949698"> <mn>15</mn> </mstyle> </mover> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>4</mn> </mstyle> </mrow> <mn>91</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>262</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}{\overset {\color {Red}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}3\\-{\color {red}4}91\\\hline {\color {Gray}262}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/743a0c7b5bf19d412a653f41c87d81f806617b4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:8.652ex; height:12.843ex;" alt="{\displaystyle {\begin{array}{r}{\overset {\color {Red}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}3\\-{\color {red}4}91\\\hline {\color {Gray}262}\end{array}}}"></span> </td> <td>Das Ergebnis wird unter den Strich geschrieben. </td></tr> <tr> <td><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 {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}3\\-491\\\hline {\color {Gray}262}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>7</mn> </menclose> <mstyle mathcolor="#949698"> <mn>6</mn> </mstyle> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>5</mn> </menclose> <mstyle mathcolor="#949698"> <mn>15</mn> </mstyle> </mover> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mn>491</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>262</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}3\\-491\\\hline {\color {Gray}262}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d63320bb3ec54d73f543b71ab1e735a22566e80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:8.652ex; height:12.843ex;" alt="{\displaystyle {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}3\\-491\\\hline {\color {Gray}262}\end{array}}}"></span> </td> <td>Das Gesamtergebnis. </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Vorab-Entbündelung"><span id="Vorab-Entb.C3.BCndelung"></span>Vorab-Entbündelung</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=13" title="Abschnitt bearbeiten: Vorab-Entbündelung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Vorab-Entbündelung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eine Variante des Entbündelungsverfahrens besteht darin, dass alle Stellen in einem ersten Arbeitsgang vollständig entbündelt werden, sodass für den zweiten Arbeitsgang, bei dem nur noch subtrahiert wird, hinreichend große Minuenden zur Verfügung stehen.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dt>Beispiel</dt></dl> <table class="wikitable"> <tbody><tr> <th></th> <th>Beschreibung </th></tr> <tr> <td><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 {\begin{array}{r}75{\color {red}3}\\-49{\color {red}1}\\\hline \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="bottom"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>75</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>3</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mn>49</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>1</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </menclose> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}75{\color {red}3}\\-49{\color {red}1}\\\hline \end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/def8f45af6dabe3e3eda9bc6a8609369d8b01323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:6.549ex; height:6.843ex;" alt="{\displaystyle {\begin{array}{r}75{\color {red}3}\\-49{\color {red}1}\\\hline \end{array}}}"></span> </td> <td>3 − 1 = möglich.<br />Kein „leihen“ von der links daneben stehenden Ziffer notwendig. </td></tr> <tr> <td><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="{\textstyle {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {\color {red}5}}}3\\-4{\color {red}9}1\\\hline \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="bottom"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>7</mn> </menclose> <mstyle mathcolor="#949698"> <mn>6</mn> </mstyle> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mstyle mathcolor="red"> <mn>5</mn> </mstyle> </menclose> <mstyle mathcolor="#949698"> <mn>15</mn> </mstyle> </mover> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>9</mn> </mstyle> </mrow> <mn>1</mn> </mtd> </mtr> </mtable> </menclose> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {\color {red}5}}}3\\-4{\color {red}9}1\\\hline \end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb32144be43f5be23cd9efd0eec988f3b3960398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:7.295ex; height:9.176ex;" alt="{\textstyle {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {\color {red}5}}}3\\-4{\color {red}9}1\\\hline \end{array}}}"></span> </td> <td>5 − 9 = nicht möglich.<br />Die 5 wird um 10 erhöht. Da die 10 bei der links benachbarten 7 „geliehen“ ist, muss diese um 1 erniedrigt werden. </td></tr> <tr> <td><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 {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}{\color {red}3}\\-49{\color {red}1}\\\hline {\color {Gray}2}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>7</mn> </menclose> <mstyle mathcolor="#949698"> <mn>6</mn> </mstyle> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>5</mn> </menclose> <mstyle mathcolor="#949698"> <mn>15</mn> </mstyle> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>3</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mn>49</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>1</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>2</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}{\color {red}3}\\-49{\color {red}1}\\\hline {\color {Gray}2}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e663c20f66afb0efb0d1e498e7ba79c99321c9bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:8.652ex; height:12.843ex;" alt="{\displaystyle {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}{\color {red}3}\\-49{\color {red}1}\\\hline {\color {Gray}2}\end{array}}}"></span> </td> <td>Abarbeitung der Stellen:<br />3 − 1 = 2 </td></tr> <tr> <td><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 {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {red}15}{\cancel {5}}}3\\-4{\color {red}9}1\\\hline {\color {Gray}62}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>7</mn> </menclose> <mstyle mathcolor="#949698"> <mn>6</mn> </mstyle> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>5</mn> </menclose> <mstyle mathcolor="red"> <mn>15</mn> </mstyle> </mover> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>9</mn> </mstyle> </mrow> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>62</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {red}15}{\cancel {5}}}3\\-4{\color {red}9}1\\\hline {\color {Gray}62}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c79412d3806064e8665ff3e59551c8382f2b32c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:8.652ex; height:12.843ex;" alt="{\displaystyle {\begin{array}{r}{\overset {\color {Gray}6}{\cancel {7}}}{\overset {\color {red}15}{\cancel {5}}}3\\-4{\color {red}9}1\\\hline {\color {Gray}62}\end{array}}}"></span> </td> <td>15 − 9 = 6 </td></tr> <tr> <td><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 {\begin{array}{r}{\overset {\color {red}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}3\\-{\color {red}4}91\\\hline {\color {Gray}262}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>7</mn> </menclose> <mstyle mathcolor="red"> <mn>6</mn> </mstyle> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <menclose notation="updiagonalstrike"> <mn>5</mn> </menclose> <mstyle mathcolor="#949698"> <mn>15</mn> </mstyle> </mover> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>4</mn> </mstyle> </mrow> <mn>91</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>262</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{r}{\overset {\color {red}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}3\\-{\color {red}4}91\\\hline {\color {Gray}262}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe30b0bcdf02f4ca4e7b39003ca51a14a09d06c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:8.652ex; height:12.843ex;" alt="{\displaystyle {\begin{array}{r}{\overset {\color {red}6}{\cancel {7}}}{\overset {\color {Gray}15}{\cancel {5}}}3\\-{\color {red}4}91\\\hline {\color {Gray}262}\end{array}}}"></span> </td> <td>6 − 4 = 2 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Vertikale_Subtraktion_ohne_Überträge"><span id="Vertikale_Subtraktion_ohne_.C3.9Cbertr.C3.A4ge"></span>Vertikale Subtraktion ohne Überträge</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=14" title="Abschnitt bearbeiten: Vertikale Subtraktion ohne Überträge" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Vertikale Subtraktion ohne Überträge"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Teildifferenzen">Teildifferenzen</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=15" title="Abschnitt bearbeiten: Teildifferenzen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Teildifferenzen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die <i>Partial Differences</i>-Methode unterscheidet sich von anderen vertikalen Subtraktionsmethoden dadurch, dass keine Überträge verwendet werden. An deren Stelle treten Teildifferenzen, die – je nachdem, ob in einer Spalte der Minuend oder der Subtrahend größer ist – ein Plus- oder ein Minuszeichen erhalten. Die Summe der Teildifferenzen ergibt die Gesamtdifferenz.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dt>Beispiel</dt></dl> <table class="wikitable"> <tbody><tr> <th></th> <th>Beschreibung </th></tr> <tr> <td><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 {\begin{array}{rr}&{\color {red}7}53\\-&{\color {red}4}91\\\hline {\color {Gray}+}&{\color {Gray}300}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right" rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>7</mn> </mstyle> </mrow> <mn>53</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>4</mn> </mstyle> </mrow> <mn>91</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mo>+</mo> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>300</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rr}&{\color {red}7}53\\-&{\color {red}4}91\\\hline {\color {Gray}+}&{\color {Gray}300}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2bf64501a80191bc6d1a44c5b402e796c4d87c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:10.228ex; height:10.176ex;" alt="{\displaystyle {\begin{array}{rr}&{\color {red}7}53\\-&{\color {red}4}91\\\hline {\color {Gray}+}&{\color {Gray}300}\end{array}}}"></span> </td> <td>Die kleinere Zahl wird von der größeren abgezogen:<br />700 − 400 = 300<br />Weil der Minuend größer ist als der Subtrahend, erhält die Differenz ein Pluszeichen. </td></tr> <tr> <td><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 {\begin{array}{rr}&7{\color {red}5}3\\-&4{\color {red}9}1\\\hline {\color {Gray}+}&{\color {Gray}300}\\{\color {Gray}-}&{\color {Gray}40}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right" rowspacing="4pt" columnspacing="1em" rowlines="none solid none"> <mtr> <mtd /> <mtd> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>5</mn> </mstyle> </mrow> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> </mtd> <mtd> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>9</mn> </mstyle> </mrow> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mo>+</mo> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>300</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mo>−<!-- − --></mo> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>40</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rr}&7{\color {red}5}3\\-&4{\color {red}9}1\\\hline {\color {Gray}+}&{\color {Gray}300}\\{\color {Gray}-}&{\color {Gray}40}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b28db2632f5009de052c4fad2548977fa6799c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:10.228ex; height:13.509ex;" alt="{\displaystyle {\begin{array}{rr}&7{\color {red}5}3\\-&4{\color {red}9}1\\\hline {\color {Gray}+}&{\color {Gray}300}\\{\color {Gray}-}&{\color {Gray}40}\end{array}}}"></span> </td> <td>Die kleinere Zahl wird von der größeren abgezogen:<br />90 − 50 = 40<br />Weil der Subtrahend größer ist als der Minuend, erhält die Differenz ein Minuszeichen. </td></tr> <tr> <td><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 {\begin{array}{rr}&75{\color {red}3}\\-&49{\color {red}1}\\\hline {\color {Gray}+}&{\color {Gray}300}\\{\color {Gray}-}&{\color {Gray}40}\\{\color {Gray}+}&{\color {Gray}2}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right" rowspacing="4pt" columnspacing="1em" rowlines="none solid none"> <mtr> <mtd /> <mtd> <mn>75</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>3</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> </mtd> <mtd> <mn>49</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>1</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mo>+</mo> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>300</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mo>−<!-- − --></mo> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>40</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mo>+</mo> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>2</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rr}&75{\color {red}3}\\-&49{\color {red}1}\\\hline {\color {Gray}+}&{\color {Gray}300}\\{\color {Gray}-}&{\color {Gray}40}\\{\color {Gray}+}&{\color {Gray}2}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69663cb70c109f1cc2b97aae959712659acc8b4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:10.228ex; height:16.843ex;" alt="{\displaystyle {\begin{array}{rr}&75{\color {red}3}\\-&49{\color {red}1}\\\hline {\color {Gray}+}&{\color {Gray}300}\\{\color {Gray}-}&{\color {Gray}40}\\{\color {Gray}+}&{\color {Gray}2}\end{array}}}"></span> </td> <td>Die kleinere Zahl wird von der größeren abgezogen:<br />3 − 1 = 2<br />Weil der Minuend größer ist als der Subtrahend, erhält die Differenz ein Pluszeichen. </td></tr> <tr> <td><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 {\begin{array}{rr}&753\\-&491\\\hline {\color {red}+}&{\color {red}300}\\{\color {red}-}&{\color {red}40}\\{\color {red}+}&{\color {red}2}\\\hline &{\color {Gray}262}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right" rowspacing="4pt" columnspacing="1em" rowlines="none solid none none solid"> <mtr> <mtd /> <mtd> <mn>753</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> </mtd> <mtd> <mn>491</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mo>+</mo> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>300</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mo>−<!-- − --></mo> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>40</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mo>+</mo> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>2</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mn>262</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rr}&753\\-&491\\\hline {\color {red}+}&{\color {red}300}\\{\color {red}-}&{\color {red}40}\\{\color {red}+}&{\color {red}2}\\\hline &{\color {Gray}262}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1003c89ea0bb442e953dc7f86b999436216f4907" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:10.228ex; height:20.176ex;" alt="{\displaystyle {\begin{array}{rr}&753\\-&491\\\hline {\color {red}+}&{\color {red}300}\\{\color {red}-}&{\color {red}40}\\{\color {red}+}&{\color {red}2}\\\hline &{\color {Gray}262}\end{array}}}"></span> </td> <td>+ 300 − 40 + 2 = 262 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Nicht-vertikale_Verfahren">Nicht-vertikale Verfahren</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=16" title="Abschnitt bearbeiten: Nicht-vertikale Verfahren" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=16" title="Quellcode des Abschnitts bearbeiten: Nicht-vertikale Verfahren"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Ausschreiten_der_Differenz">Ausschreiten der Differenz</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=17" title="Abschnitt bearbeiten: Ausschreiten der Differenz" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=17" title="Quellcode des Abschnitts bearbeiten: Ausschreiten der Differenz"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Berechnung einer Differenz muss nicht Stelle für Stelle erfolgen. Meist umständlich, aber möglich ist es auch, den zwischen einem Subtrahenden und einem Minuenden liegenden Zahlenraum auszuschreiten.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dt>Beispiel</dt></dl> <p>1234 − 567 = kann über folgende Schritte errechnet werden: </p> <ul><li>567 + <b>3</b> = 570</li> <li>570 + <b>30</b> = 600</li> <li>600 + <b>400</b> = 1000</li> <li>1000 + <b>234</b> = 1234</li></ul> <p>Um die Differenz zu ermitteln, werden die Werte der Einzelschritte addiert: 3 + 30 + 400 + 234 = 667. </p> <div class="mw-heading mw-heading4"><h4 id="Zergliederung_des_Subtrahenden">Zergliederung des Subtrahenden</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=18" title="Abschnitt bearbeiten: Zergliederung des Subtrahenden" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=18" title="Quellcode des Abschnitts bearbeiten: Zergliederung des Subtrahenden"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eine weitere Vorgehensweise, die sich gleichermaßen für die schriftliche Subtraktion wie für das Kopfrechnen eignet, ist die Zergliederung des Subtrahenden, der in Einzelschritten vom Minuenden abgezogen wird.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <dl><dt>Beispiel</dt></dl> <p>„1234 − 567 =“ kann über folgende Schritte errechnet werden: </p> <ul><li>1234 − <b>500</b> = 734</li> <li>734 − <b>60</b> = 674</li> <li>674 − <b>7</b> = 667</li></ul> <div class="mw-heading mw-heading4"><h4 id="Gleiche_Veränderung"><span id="Gleiche_Ver.C3.A4nderung"></span>Gleiche Veränderung</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=19" title="Abschnitt bearbeiten: Gleiche Veränderung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=19" title="Quellcode des Abschnitts bearbeiten: Gleiche Veränderung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Grundlage der <i>Same change</i>-Subtraktion ist die Beobachtung, dass eine Subtraktion einfach durchzuführen ist, wenn am Ende des Subtrahenden eine oder mehrere Nullen stehen. Der Subtrahend wird bei diesem Verfahren darum auf den nächstliegenden Zehner erhöht oder erniedrigt; da der Minuend um dieselbe Differenz erhöht oder erniedrigt wird, nimmt die Manipulation auf die Differenz keinen Einfluss. Wenn die Aufgabe danach immer noch zu schwer ist, kann die Operation wiederholt werden.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <dl><dt>Beispiel</dt></dl> <p>„1234 − 567 =“ kann über folgende Schritte errechnet werden: </p> <ul><li>1234 − 567 = 1237 − 570 = 1267 − 600 = 667</li></ul> <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subtraktion&veaction=edit&section=20" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=20" title="Quellcode des Abschnitts bearbeiten: Siehe auch"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Konstruktion_mit_Zirkel_und_Lineal#Algebraische_Operationen" title="Konstruktion mit Zirkel und Lineal">Subtraktion zweier Zahlen mit Zirkel und Lineal</a></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=Subtraktion&veaction=edit&section=21" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=21" 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;"><div class="noresize 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="Commons"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div><b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Subtraction?uselang=de"><span lang="en">Commons</span>: Subtraction</a></span></b> – Sammlung von Bildern, Videos und Audiodateien</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/Subtraktion" class="extiw" title="wikt:Subtraktion">Wiktionary: Subtraktion</a></b> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120130170553/http://www.lehrplan-bayern.de/pdf/Mathe_11.pdf">Beispiele für das Abziehen mit Entbündeln und Erweiterungstechnik</a> (PDF-Datei; 39 kB)</li></ul> <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=Subtraktion&veaction=edit&section=22" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subtraktion&action=edit&section=22" 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">Education Place: <a rel="nofollow" class="external text" href="http://eduplace.com/math/mathsteps/4/a/index.html">The Order of Operations</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="/wiki/Khan_Academy" title="Khan Academy">Khan Academy</a>: <a rel="nofollow" class="external text" href="https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-order-of-operations/v/introduction-to-order-of-operations">The Order of Operations</a> (<a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=ClYdw4d4OmA&t=5m40s">Video, ab 05:40</a>)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a href="/w/index.php?title=Virginia_Department_of_Education&action=edit&redlink=1" class="new" title="Virginia Department of Education (Seite nicht vorhanden)">Virginia Department of Education</a>: <style data-mw-deduplicate="TemplateStyles:r246413598">.mw-parser-output .webarchiv-memento{color:var(--color-base,#202122)!important}</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20220716062834/http://www.doe.virginia.gov/instruction/mathematics/middle/algebra_readiness/curriculum_companion/order-operations.pdf#page=3">Using Order of Operations and Exploring Properties</a> (<a href="/wiki/Web-Archivierung#Begrifflichkeiten" title="Web-Archivierung"><span class="webarchiv-memento">Memento</span></a> vom 16. Juli 2022 im <i><a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a></i>), Absatz 9</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/Technische_Universit%C3%A4t_Chemnitz" title="Technische Universität Chemnitz">Technische Universität Chemnitz</a>: <a rel="nofollow" class="external text" href="https://www.tu-chemnitz.de/urz/archiv/kursunterlagen/C/kap2/vorrang.htm">Vorrangregeln und Assoziativität</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Donald E. Knuth: <cite style="font-style:italic">The Art of Computer Programming, Volume 2: Seminumerical Algorithms</cite>. 3. Auflage. Addison-Wesley, New York 1997, <a href="/wiki/Spezial:ISBN-Suche/9780201896848" class="internal mw-magiclink-isbn">ISBN 978-0-201-89684-8</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Subtraktion&rft.au=Donald+E.+Knuth&rft.btitle=The+Art+of+Computer+Programming%2C+Volume+2%3A+Seminumerical+Algorithms&rft.date=1997&rft.edition=3&rft.genre=book&rft.isbn=9780201896848&rft.place=New+York&rft.pub=Addison-Wesley" style="display:none"> </span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.math.nyu.edu/~braams/links/em-arith.html">The Many Ways of Arithmetic in UCSMP Everyday Mathematics</a> Subtraction: Trade First</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r246413598"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140623021239/http://ouronlineschools.org/Schools/NC/Demoschool/4thGrade/Math/PartialDifferences.htm">Partial-Differences Subtraction</a> (<a href="/wiki/Web-Archivierung#Begrifflichkeiten" title="Web-Archivierung"><span class="webarchiv-memento">Memento</span></a> vom 23. Juni 2014 im <i><a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a></i>); <a rel="nofollow" class="external text" href="http://www.math.nyu.edu/~braams/links/em-arith.html">The Many Ways of Arithmetic in UCSMP Everyday Mathematics</a> Subtraction: Partial Differences</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.math.nyu.edu/~braams/links/em-arith.html">The Many Ways of Arithmetic in UCSMP Everyday Mathematics</a> Subtraction: Counting Up</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.math.nyu.edu/~braams/links/em-arith.html">The Many Ways of Arithmetic in UCSMP Everyday Mathematics</a> Subtraction: Left to Right Subtraction</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.math.nyu.edu/~braams/links/em-arith.html">The Many Ways of Arithmetic in UCSMP Everyday Mathematics</a> Subtraction: Same Change Rule</span> </li> </ol> <div class="hintergrundfarbe1 rahmenfarbe1 navigation-not-searchable normdaten-typ-s" style="border-style: solid; border-width: 1px; clear: left; margin-bottom:1em; margin-top:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;" id="normdaten"> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div> Normdaten (Sachbegriff): <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <span class="plainlinks-print"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4359078-0">4359078-0</a></span> <span class="noprint">(<a rel="nofollow" class="external text" href="https://lobid.org/gnd/4359078-0">lobid</a>, <a rel="nofollow" class="external text" href="https://swb.bsz-bw.de/DB=2.104/SET=1/TTL=1/CMD?retrace=0&trm_old=&ACT=SRCHA&IKT=2999&SRT=RLV&TRM=4359078-0">OGND</a><span class="metadata">, <a rel="nofollow" class="external text" href="https://prometheus.lmu.de/gnd/4359078-0">AKS</a></span>)</span> <span class="metadata"></span></div> </div></div></div><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Subtraktion&oldid=250220601">https://de.wikipedia.org/w/index.php?title=Subtraktion&oldid=250220601</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorie</a>: <ul><li><a href="/wiki/Kategorie:Subtraktion" title="Kategorie:Subtraktion">Subtraktion</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Subtraktion" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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title="Subtraktion – Schweizerdeutsch" lang="gsw" hreflang="gsw" data-title="Subtraktion" data-language-autonym="Alemannisch" data-language-local-name="Schweizerdeutsch" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Resta" title="Resta – Aragonesisch" lang="an" hreflang="an" data-title="Resta" data-language-autonym="Aragonés" data-language-local-name="Aragonesisch" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D8%B1%D8%AD" title="طرح – Arabisch" lang="ar" hreflang="ar" data-title="طرح" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%B7%D8%B1%D8%AD" title="طرح – Ägyptisches Arabisch" lang="arz" hreflang="arz" data-title="طرح" data-language-autonym="مصرى" data-language-local-name="Ägyptisches Arabisch" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Resta" title="Resta – Asturisch" lang="ast" hreflang="ast" data-title="Resta" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Jakhuqawi" title="Jakhuqawi – Aymara" lang="ay" hreflang="ay" data-title="Jakhuqawi" data-language-autonym="Aymar aru" data-language-local-name="Aymara" class="interlanguage-link-target"><span>Aymar aru</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%87%C4%B1xma" title="Çıxma – Aserbaidschanisch" lang="az" hreflang="az" data-title="Çıxma" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaidschanisch" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DA%86%DB%8C%D8%AE%D9%85%D8%A7" title="چیخما – Südaserbaidschanisch" lang="azb" hreflang="azb" data-title="چیخما" data-language-autonym="تۆرکجه" data-language-local-name="Südaserbaidschanisch" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%90%D0%BB%D1%8B%D1%83" title="Алыу – Baschkirisch" lang="ba" hreflang="ba" data-title="Алыу" data-language-autonym="Башҡортса" data-language-local-name="Baschkirisch" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Pag-ina" title="Pag-ina – Zentralbikolano" lang="bcl" hreflang="bcl" data-title="Pag-ina" data-language-autonym="Bikol Central" data-language-local-name="Zentralbikolano" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B4%D0%BD%D1%96%D0%BC%D0%B0%D0%BD%D0%BD%D0%B5" title="Адніманне – Belarussisch" lang="be" hreflang="be" data-title="Адніманне" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%B4%D1%8B%D0%BC%D0%B0%D0%BD%D1%8C%D0%BD%D0%B5" title="Адыманьне – Weißrussisch (Taraschkewiza)" lang="be-tarask" hreflang="be-tarask" data-title="Адыманьне" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Weißrussisch (Taraschkewiza)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%B7%D0%B2%D0%B0%D0%B6%D0%B4%D0%B0%D0%BD%D0%B5" title="Изваждане – Bulgarisch" lang="bg" hreflang="bg" data-title="Изваждане" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A7%8B%E0%A6%97" title="বিয়োগ – Bengalisch" lang="bn" hreflang="bn" data-title="বিয়োগ" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A0%E0%BD%95%E0%BE%B2%E0%BD%B2%E0%BC%8B%E0%BD%A2%E0%BE%A9%E0%BD%B2%E0%BD%A6%E0%BC%8D" title="འཕྲི་རྩིས། – Tibetisch" lang="bo" hreflang="bo" data-title="འཕྲི་རྩིས།" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetisch" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Lamadur" title="Lamadur – Bretonisch" lang="br" hreflang="br" data-title="Lamadur" data-language-autonym="Brezhoneg" data-language-local-name="Bretonisch" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Oduzimanje" title="Oduzimanje – Bosnisch" lang="bs" hreflang="bs" data-title="Oduzimanje" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%A5%D0%B0%D2%BB%D0%B0%D0%BB%D1%82%D0%B0" title="Хаһалта – Russisches Burjatisch" lang="bxr" hreflang="bxr" data-title="Хаһалта" data-language-autonym="Буряад" data-language-local-name="Russisches Burjatisch" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Resta" title="Resta – Katalanisch" lang="ca" hreflang="ca" data-title="Resta" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/G%C4%93ng-hu%C3%A1k" title="Gēng-huák – Min Dong" lang="cdo" hreflang="cdo" data-title="Gēng-huák" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Min Dong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-chr mw-list-item"><a href="https://chr.wikipedia.org/wiki/%E1%8F%97%E1%8E%AA%E1%8F%A3%E1%8E%B4%E1%8F%8D%E1%8F%97" title="ᏗᎪᏣᎴᏍᏗ – Cherokee" lang="chr" hreflang="chr" data-title="ᏗᎪᏣᎴᏍᏗ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="Cherokee" class="interlanguage-link-target"><span>ᏣᎳᎩ</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%84%DB%8E%D8%AF%DB%95%D8%B1%DA%A9%D8%B1%D8%AF%D9%86" title="لێدەرکردن – Zentralkurdisch" lang="ckb" hreflang="ckb" data-title="لێدەرکردن" data-language-autonym="کوردی" data-language-local-name="Zentralkurdisch" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Od%C4%8D%C3%ADt%C3%A1n%C3%AD" title="Odčítání – Tschechisch" lang="cs" hreflang="cs" data-title="Odčítání" 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%C4%83%D0%BB%D0%B0%D1%80%D0%B0%D1%81%D1%81%D0%B8" title="Кăларасси – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Кăларасси" 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/Tynnu" title="Tynnu – Walisisch" lang="cy" hreflang="cy" data-title="Tynnu" 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/Subtraktion" title="Subtraktion – Dänisch" lang="da" hreflang="da" data-title="Subtraktion" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-dz mw-list-item"><a href="https://dz.wikipedia.org/wiki/%E0%BD%9F%E0%BD%91%E0%BC%8B%E0%BD%A0%E0%BD%82%E0%BE%B2%E0%BD%BC%E0%BC%8B%E0%BD%A6%E0%BE%90%E0%BE%B1%E0%BD%B2%E0%BD%93%E0%BC%8B%E0%BD%A0%E0%BD%82%E0%BE%B2%E0%BD%B4%E0%BD%A3%E0%BC%8B%E0%BD%82%E0%BE%B1%E0%BD%B2%E0%BC%8B%E0%BD%91%E0%BD%BC%E0%BD%93%E0%BC%8B%E0%BD%A3%E0%BD%B4%E0%BC%8B" title="ཟད་འགྲོ་སྐྱིན་འགྲུལ་གྱི་དོན་ལུ་ – Dzongkha" lang="dz" hreflang="dz" data-title="ཟད་འགྲོ་སྐྱིན་འགྲུལ་གྱི་དོན་ལུ་" data-language-autonym="ཇོང་ཁ" data-language-local-name="Dzongkha" class="interlanguage-link-target"><span>ཇོང་ཁ</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%86%CE%B1%CE%AF%CF%81%CE%B5%CF%83%CE%B7" title="Αφαίρεση – Griechisch" lang="el" hreflang="el" data-title="Αφαίρεση" 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/Subtraction" title="Subtraction – Englisch" lang="en" hreflang="en" data-title="Subtraction" 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/Subtraho" title="Subtraho – Esperanto" lang="eo" hreflang="eo" data-title="Subtraho" 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/Resta" title="Resta – Spanisch" lang="es" hreflang="es" data-title="Resta" 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/Lahutamine" title="Lahutamine – Estnisch" lang="et" hreflang="et" data-title="Lahutamine" 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/Kenketa" title="Kenketa – Baskisch" lang="eu" hreflang="eu" data-title="Kenketa" 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/%D8%AA%D9%81%D8%B1%DB%8C%D9%82" title="تفریق – Persisch" lang="fa" hreflang="fa" data-title="تفریق" 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/V%C3%A4hennyslasku" title="Vähennyslasku – Finnisch" lang="fi" hreflang="fi" data-title="Vähennyslasku" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Subtraction" title="Subtraction – Fidschi" lang="fj" hreflang="fj" data-title="Subtraction" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fidschi" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Soustraction" title="Soustraction – Französisch" lang="fr" hreflang="fr" data-title="Soustraction" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%B8%9B%E6%B3%95" title="減法 – Gan" lang="gan" hreflang="gan" data-title="減法" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Soustraksyon" title="Soustraksyon – Französisch-Guayana Kreolisch" lang="gcr" hreflang="gcr" data-title="Soustraksyon" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Französisch-Guayana Kreolisch" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Toirt_air_falbh" title="Toirt air falbh – Gälisch (Schottland)" lang="gd" hreflang="gd" data-title="Toirt air falbh" data-language-autonym="Gàidhlig" data-language-local-name="Gälisch (Schottland)" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Subtracci%C3%B3n" title="Subtracción – Galicisch" lang="gl" hreflang="gl" data-title="Subtracción" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/K%C3%A1m-fap" title="Kám-fap – Hakka" lang="hak" hreflang="hak" data-title="Kám-fap" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="Hakka" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%99%D7%A1%D7%95%D7%A8" title="חיסור – Hebräisch" lang="he" hreflang="he" data-title="חיסור" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%98%E0%A4%9F%E0%A4%BE%E0%A4%A8%E0%A4%BE" title="घटाना – Hindi" lang="hi" hreflang="hi" data-title="घटाना" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Ghataana" title="Ghataana – Fidschi-Hindi" lang="hif" hreflang="hif" data-title="Ghataana" data-language-autonym="Fiji Hindi" data-language-local-name="Fidschi-Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Oduzimanje" title="Oduzimanje – Kroatisch" lang="hr" hreflang="hr" data-title="Oduzimanje" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kivon%C3%A1s" title="Kivonás – Ungarisch" lang="hu" hreflang="hu" data-title="Kivonás" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%B6%D5%B8%D6%82%D5%B4" title="Հանում – Armenisch" lang="hy" hreflang="hy" data-title="Հանում" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pengurangan" title="Pengurangan – Indonesisch" lang="id" hreflang="id" data-title="Pengurangan" 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/Fr%C3%A1dr%C3%A1ttur" title="Frádráttur – Isländisch" lang="is" hreflang="is" data-title="Frádráttur" 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/Sottrazione" title="Sottrazione – Italienisch" lang="it" hreflang="it" data-title="Sottrazione" 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/%E6%B8%9B%E6%B3%95" title="減法 – Japanisch" lang="ja" hreflang="ja" data-title="減法" 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-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Sobchakshan" title="Sobchakshan – Jamaikanisch-Kreolisch" lang="jam" hreflang="jam" data-title="Sobchakshan" data-language-autonym="Patois" data-language-local-name="Jamaikanisch-Kreolisch" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Pangurangan" title="Pangurangan – Javanisch" lang="jv" hreflang="jv" data-title="Pangurangan" data-language-autonym="Jawa" data-language-local-name="Javanisch" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%90%E1%83%9B%E1%83%9D%E1%83%99%E1%83%9A%E1%83%94%E1%83%91%E1%83%90" title="გამოკლება – Georgisch" lang="ka" hreflang="ka" data-title="გამოკლება" data-language-autonym="ქართული" data-language-local-name="Georgisch" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D0%B7%D0%B0%D0%B9%D1%82%D1%83_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Азайту (математика) – Kasachisch" lang="kk" hreflang="kk" data-title="Азайту (математика)" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B3%8D%E0%B2%AF%E0%B2%B5%E0%B2%95%E0%B2%B2%E0%B2%A8" title="ವ್ಯವಕಲನ – Kannada" lang="kn" hreflang="kn" data-title="ವ್ಯವಕಲನ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%BA%84%EC%85%88" title="뺄셈 – Koreanisch" lang="ko" hreflang="ko" data-title="뺄셈" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%B5%D0%BC%D0%B8%D1%82%D2%AF%D2%AF" title="Кемитүү – Kirgisisch" lang="ky" hreflang="ky" data-title="Кемитүү" data-language-autonym="Кыргызча" data-language-local-name="Kirgisisch" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Subtractio" title="Subtractio – Latein" lang="la" hreflang="la" data-title="Subtractio" data-language-autonym="Latina" data-language-local-name="Latein" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%81%E0%BA%B2%E0%BA%99%E0%BA%AB%E0%BA%B1%E0%BA%81%E0%BA%A5%E0%BA%BB%E0%BA%9A_(%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B2%E0%BA%94)" title="ການຫັກລົບ (ຄະນິດສາດ) – Laotisch" lang="lo" hreflang="lo" data-title="ການຫັກລົບ (ຄະນິດສາດ)" data-language-autonym="ລາວ" data-language-local-name="Laotisch" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Atimtis" title="Atimtis – Litauisch" lang="lt" hreflang="lt" data-title="Atimtis" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/At%C5%86em%C5%A1ana" title="Atņemšana – Lettisch" lang="lv" hreflang="lv" data-title="Atņemšana" data-language-autonym="Latviešu" data-language-local-name="Lettisch" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9E%D0%B4%D0%B7%D0%B5%D0%BC%D0%B0%D1%9A%D0%B5" title="Одземање – Mazedonisch" lang="mk" hreflang="mk" data-title="Одземање" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B5%8D%E0%B4%AF%E0%B4%B5%E0%B4%95%E0%B4%B2%E0%B4%A8%E0%B4%82" title="വ്യവകലനം – Malayalam" lang="ml" hreflang="ml" data-title="വ്യവകലനം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%9C%E0%A4%BE%E0%A4%AC%E0%A4%BE%E0%A4%95%E0%A5%80" title="वजाबाकी – Marathi" lang="mr" hreflang="mr" data-title="वजाबाकी" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Penolakan" title="Penolakan – Malaiisch" lang="ms" hreflang="ms" data-title="Penolakan" 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-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Tnaqqis" title="Tnaqqis – Maltesisch" lang="mt" hreflang="mt" data-title="Tnaqqis" data-language-autonym="Malti" data-language-local-name="Maltesisch" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%98%E0%A4%9F%E0%A4%BE%E0%A4%89" title="घटाउ – Nepalesisch" lang="ne" hreflang="ne" data-title="घटाउ" data-language-autonym="नेपाली" data-language-local-name="Nepalesisch" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Aftrekken_(wiskunde)" title="Aftrekken (wiskunde) – Niederländisch" lang="nl" hreflang="nl" data-title="Aftrekken (wiskunde)" 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/Subtraksjon" title="Subtraksjon – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Subtraksjon" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Subtraksjon" title="Subtraksjon – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Subtraksjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Subtraktione" title="Subtraktione – Novial" lang="nov" hreflang="nov" data-title="Subtraktione" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Sostraccion" title="Sostraccion – Okzitanisch" lang="oc" hreflang="oc" data-title="Sostraccion" data-language-autonym="Occitan" data-language-local-name="Okzitanisch" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%98%E0%A8%9F%E0%A8%BE%E0%A8%85" title="ਘਟਾਅ – Punjabi" lang="pa" hreflang="pa" data-title="ਘਟਾਅ" 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/Odejmowanie" title="Odejmowanie – Polnisch" lang="pl" hreflang="pl" data-title="Odejmowanie" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Sotrassion" title="Sotrassion – Piemontesisch" lang="pms" hreflang="pms" data-title="Sotrassion" data-language-autonym="Piemontèis" data-language-local-name="Piemontesisch" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AA%D9%81%D8%B1%D9%8A%D9%82" title="تفريق – Paschtu" lang="ps" hreflang="ps" data-title="تفريق" data-language-autonym="پښتو" data-language-local-name="Paschtu" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Subtra%C3%A7%C3%A3o" title="Subtração – Portugiesisch" lang="pt" hreflang="pt" data-title="Subtração" 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/Qichuy" title="Qichuy – Quechua" lang="qu" hreflang="qu" data-title="Qichuy" 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-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sc%C4%83dere" title="Scădere – Rumänisch" lang="ro" hreflang="ro" data-title="Scădere" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D1%8B%D1%87%D0%B8%D1%82%D0%B0%D0%BD%D0%B8%D0%B5" title="Вычитание – Russisch" lang="ru" hreflang="ru" data-title="Вычитание" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9A%D3%A9%D2%95%D2%AF%D1%80%D1%8D%D1%82%D0%B8%D0%B8" title="Көҕүрэтии – Jakutisch" lang="sah" hreflang="sah" data-title="Көҕүрэтии" data-language-autonym="Саха тыла" data-language-local-name="Jakutisch" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%B5%E1%B1%B7%E1%B1%AE%E1%B1%9C%E1%B1%AE%E1%B1%AB" title="ᱵᱷᱮᱜᱮᱫ – Santali" lang="sat" hreflang="sat" data-title="ᱵᱷᱮᱜᱮᱫ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="Santali" class="interlanguage-link-target"><span>ᱥᱟᱱᱛᱟᱲᱤ</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Suttrazzioni" title="Suttrazzioni – Sizilianisch" lang="scn" hreflang="scn" data-title="Suttrazzioni" data-language-autonym="Sicilianu" data-language-local-name="Sizilianisch" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Oduzimanje" title="Oduzimanje – Serbokroatisch" lang="sh" hreflang="sh" data-title="Oduzimanje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Subtraction" title="Subtraction – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Subtraction" 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/Od%C4%8D%C3%ADtanie" title="Odčítanie – Slowakisch" lang="sk" hreflang="sk" data-title="Odčítanie" 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/Od%C5%A1tevanje" title="Odštevanje – Slowenisch" lang="sl" hreflang="sl" data-title="Odštevanje" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Kubvisa" title="Kubvisa – Shona" lang="sn" hreflang="sn" data-title="Kubvisa" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Faraq" title="Faraq – Somali" lang="so" hreflang="so" data-title="Faraq" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Zbritja_(matematik%C3%AB)" title="Zbritja (matematikë) – Albanisch" lang="sq" hreflang="sq" data-title="Zbritja (matematikë)" data-language-autonym="Shqip" data-language-local-name="Albanisch" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D0%B4%D1%83%D0%B7%D0%B8%D0%BC%D0%B0%D1%9A%D0%B5" title="Одузимање – Serbisch" lang="sr" hreflang="sr" data-title="Одузимање" 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/Subtraktion" title="Subtraktion – Schwedisch" lang="sv" hreflang="sv" data-title="Subtraktion" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Utoaji" title="Utoaji – Suaheli" lang="sw" hreflang="sw" data-title="Utoaji" data-language-autonym="Kiswahili" data-language-local-name="Suaheli" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%B4%E0%AE%BF%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%B2%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="கழித்தல் (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="கழித்தல் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A4%E0%B1%80%E0%B0%B8%E0%B0%BF%E0%B0%B5%E0%B1%87%E0%B0%A4" title="తీసివేత – Telugu" lang="te" hreflang="te" data-title="తీసివేత" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A2%D0%B0%D1%80%D2%B3_(%D0%B0%D0%BC%D0%B0%D0%BB%D0%B8_%D1%80%D0%B8%D1%91%D0%B7%D3%A3)" title="Тарҳ (амали риёзӣ) – Tadschikisch" lang="tg" hreflang="tg" data-title="Тарҳ (амали риёзӣ)" data-language-autonym="Тоҷикӣ" data-language-local-name="Tadschikisch" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%A5%E0%B8%9A" title="การลบ – Thailändisch" lang="th" hreflang="th" data-title="การลบ" 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/Pagbabawas" title="Pagbabawas – Tagalog" lang="tl" hreflang="tl" data-title="Pagbabawas" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%87%C4%B1karma" title="Çıkarma – Türkisch" lang="tr" hreflang="tr" data-title="Çıkarma" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D1%96%D0%B4%D0%BD%D1%96%D0%BC%D0%B0%D0%BD%D0%BD%D1%8F" title="Віднімання – Ukrainisch" lang="uk" hreflang="uk" data-title="Віднімання" 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%AA%D9%81%D8%B1%DB%8C%D9%82_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="تفریق (ریاضی) – Urdu" lang="ur" hreflang="ur" data-title="تفریق (ریاضی)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ayirish" title="Ayirish – Usbekisch" lang="uz" hreflang="uz" data-title="Ayirish" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Usbekisch" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Sotra" title="Sotra – Venetisch" lang="vec" hreflang="vec" data-title="Sotra" data-language-autonym="Vèneto" data-language-local-name="Venetisch" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_tr%E1%BB%AB" title="Phép trừ – Vietnamesisch" lang="vi" hreflang="vi" data-title="Phép trừ" 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-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Pag-iban-iban" title="Pag-iban-iban – Waray" lang="war" hreflang="war" data-title="Pag-iban-iban" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%87%8F%E6%B3%95" title="减法 – Wu" lang="wuu" hreflang="wuu" data-title="减法" 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-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%A5%D0%B0%D1%81%D0%BB%D0%B0%D2%BB%D0%B0%D0%BD" title="Хаслаһан – Kalmückisch" lang="xal" hreflang="xal" data-title="Хаслаһан" data-language-autonym="Хальмг" data-language-local-name="Kalmückisch" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Ukuthabatha" title="Ukuthabatha – Xhosa" lang="xh" hreflang="xh" data-title="Ukuthabatha" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%90%D7%A8%D7%90%D7%A4%D7%A0%D7%A2%D7%9D" title="אראפנעם – Jiddisch" lang="yi" hreflang="yi" data-title="אראפנעם" data-language-autonym="ייִדיש" data-language-local-name="Jiddisch" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/%C3%8Cy%E1%BB%8Dk%C3%BAr%C3%B2" title="Ìyọkúrò – Yoruba" lang="yo" hreflang="yo" data-title="Ìyọkúrò" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-za mw-list-item"><a href="https://za.wikipedia.org/wiki/Gemjfap" title="Gemjfap – Zhuang" lang="za" hreflang="za" data-title="Gemjfap" data-language-autonym="Vahcuengh" data-language-local-name="Zhuang" class="interlanguage-link-target"><span>Vahcuengh</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%B8%9B%E6%B3%95" title="減法 – Chinesisch" lang="zh" hreflang="zh" data-title="減法" 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-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ki%C3%A1m-hoat" title="Kiám-hoat – Min Nan" lang="nan" hreflang="nan" data-title="Kiám-hoat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Min Nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%B8%9B" title="減 – Kantonesisch" lang="yue" hreflang="yue" data-title="減" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q40754#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen bearbeiten" class="wbc-editpage">Links bearbeiten</a></span></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Diese Seite wurde zuletzt am 10. 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