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width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}"></span>, für die <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^{2}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e03acb68e3fcefd688ea86165a88d7219e432b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.645ex; height:2.676ex;" alt="{\displaystyle x^{2}=2}"></span> gilt. Diese Zahl ist eindeutig bestimmt, <a href="/wiki/Irrationale_Zahl" title="Irrationale Zahl">irrational</a> und wird durch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> dargestellt. Die ersten Stellen ihrer Dezimalbruchentwicklung sind: </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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> = 1,414213562…</dd></dl> <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="#Allgemeines"><span class="tocnumber">1</span> <span class="toctext">Allgemeines</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Irrationalität"><span class="tocnumber">1.1</span> <span class="toctext">Irrationalität</span></a></li> <li class="toclevel-2 tocsection-3"><a href="#Nachkommastellen"><span class="tocnumber">1.2</span> <span class="toctext">Nachkommastellen</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Kettenbruchentwicklung"><span class="tocnumber">1.3</span> <span class="toctext">Kettenbruchentwicklung</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Kettenwurzeleigenschaft"><span class="tocnumber">1.4</span> <span class="toctext">Kettenwurzeleigenschaft</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-6"><a href="#Geometrische_Konstruktion"><span class="tocnumber">2</span> <span class="toctext">Geometrische Konstruktion</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#Trigonometrie"><span class="tocnumber">3</span> <span class="toctext">Trigonometrie</span></a></li> <li class="toclevel-1 tocsection-8"><a href="#Geschichte"><span class="tocnumber">4</span> <span class="toctext">Geschichte</span></a></li> <li class="toclevel-1 tocsection-9"><a href="#Vorkommen_in_der_Natur"><span class="tocnumber">5</span> <span class="toctext">Vorkommen in der Natur</span></a></li> <li class="toclevel-1 tocsection-10"><a href="#Sonstiges"><span class="tocnumber">6</span> <span class="toctext">Sonstiges</span></a> <ul> <li class="toclevel-2 tocsection-11"><a href="#Merkhilfe_für_die_ersten_Nachkommastellen"><span class="tocnumber">6.1</span> <span class="toctext">Merkhilfe für die ersten Nachkommastellen</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-12"><a href="#Ganzzahligkeit_von_Ausdrücken"><span class="tocnumber">7</span> <span class="toctext">Ganzzahligkeit von Ausdrücken</span></a></li> <li class="toclevel-1 tocsection-13"><a href="#Weblinks"><span class="tocnumber">8</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-14"><a href="#Einzelnachweise"><span class="tocnumber">9</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Allgemeines">Allgemeines</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quadratwurzel_aus_2&veaction=edit&section=1" title="Abschnitt bearbeiten: Allgemeines" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Allgemeines"><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:Euclid_statue,_Oxford_University_Museum_of_Natural_History,_UK_-_20080315.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Euclid_statue%2C_Oxford_University_Museum_of_Natural_History%2C_UK_-_20080315.jpg/170px-Euclid_statue%2C_Oxford_University_Museum_of_Natural_History%2C_UK_-_20080315.jpg" decoding="async" width="170" height="263" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Euclid_statue%2C_Oxford_University_Museum_of_Natural_History%2C_UK_-_20080315.jpg/255px-Euclid_statue%2C_Oxford_University_Museum_of_Natural_History%2C_UK_-_20080315.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Euclid_statue%2C_Oxford_University_Museum_of_Natural_History%2C_UK_-_20080315.jpg/340px-Euclid_statue%2C_Oxford_University_Museum_of_Natural_History%2C_UK_-_20080315.jpg 2x" data-file-width="950" data-file-height="1472" /></a><figcaption>Euklid (fiktiv nach <a href="/wiki/Andr%C3%A9_Thevet" title="André Thevet">André Thevet</a>, 1584)</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Irrationalität"><span id="Irrationalit.C3.A4t"></span>Irrationalität</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quadratwurzel_aus_2&veaction=edit&section=2" title="Abschnitt bearbeiten: Irrationalität" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Irrationalität"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Quadratwurzel aus 2 ist wie die <a href="/wiki/Kreiszahl" title="Kreiszahl">Kreiszahl</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 \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> oder die <a href="/wiki/Eulersche_Zahl" title="Eulersche Zahl">eulersche Zahl</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {e} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">e</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {e} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc87a842dfc8cbab341df10d8d5133fd68ffc2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.032ex; height:1.676ex;" alt="{\displaystyle \mathbb {e} }"></span> irrational. Im Gegensatz zu den beiden ist sie jedoch nicht <a href="/wiki/Transzendente_Zahl" title="Transzendente Zahl">transzendent</a>, sondern <a href="/wiki/Algebraische_Zahl" title="Algebraische Zahl">algebraisch</a>. Bereits um 500 v. Chr. war dem Griechen <a href="/wiki/Hippasos_von_Metapont" title="Hippasos von Metapont">Hippasos von Metapont</a> die Irrationalität bekannt. Den wohl bekanntesten <a href="/wiki/Beweis_der_Irrationalit%C3%A4t_der_Wurzel_aus_2_bei_Euklid" title="Beweis der Irrationalität der Wurzel aus 2 bei Euklid">Beweis der Irrationalität der Quadratwurzel aus 2</a> veröffentlichte um 300 v. Chr. der Grieche <a href="/wiki/Euklid" title="Euklid">Euklid</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Nachkommastellen">Nachkommastellen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quadratwurzel_aus_2&veaction=edit&section=3" title="Abschnitt bearbeiten: Nachkommastellen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Nachkommastellen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Da Wurzel 2 irrational ist, hat die Zahl in jedem <a href="/wiki/Stellenwertsystem" title="Stellenwertsystem">Stellenwertsystem</a> unendlich viele nichtperiodische Nachkommastellen und lässt sich deshalb auch im <a href="/wiki/Dezimalsystem" title="Dezimalsystem">Dezimalsystem</a> nur näherungsweise darstellen. Die ersten 50 dezimalen Nachkommastellen lauten: </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 {\sqrt {2}}=1{,}41421\,35623\,73095\,04880\,16887\,24209\,69807\,85696\,71875\,37694\,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mn>1,414</mn> <mn>21</mn> <mspace width="thinmathspace" /> <mn>35623</mn> <mspace width="thinmathspace" /> <mn>73095</mn> <mspace width="thinmathspace" /> <mn>04880</mn> <mspace width="thinmathspace" /> <mn>16887</mn> <mspace width="thinmathspace" /> <mn>24209</mn> <mspace width="thinmathspace" /> <mn>69807</mn> <mspace width="thinmathspace" /> <mn>85696</mn> <mspace width="thinmathspace" /> <mn>71875</mn> <mspace width="thinmathspace" /> <mn>37694</mn> <mspace width="thinmathspace" /> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}=1{,}41421\,35623\,73095\,04880\,16887\,24209\,69807\,85696\,71875\,37694\,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a04d7efc9955d93b7dc1c80fb446f8d5d8431de7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:73.11ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}=1{,}41421\,35623\,73095\,04880\,16887\,24209\,69807\,85696\,71875\,37694\,\ldots }"></span> (Folge <a href="//oeis.org/A002193" class="extiw" title="oeis:A002193">A002193</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Kettenbruchentwicklung">Kettenbruchentwicklung</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quadratwurzel_aus_2&veaction=edit&section=4" title="Abschnitt bearbeiten: Kettenbruchentwicklung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Kettenbruchentwicklung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eine andere Möglichkeit, reelle Zahlen darzustellen, ist die <a href="/wiki/Kettenbruch" title="Kettenbruch">Kettenbruchentwicklung</a>. Die Kettenbruchdarstellung von Wurzel 2 ist – im Gegensatz zur Kreiszahl <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 \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> – <a href="/wiki/Kettenbruch#Periodische_Kettenbrüche" title="Kettenbruch">periodisch</a>, denn Wurzel 2 ist eine <a href="/wiki/Quadratische_Irrationalzahl" class="mw-redirect" title="Quadratische Irrationalzahl">quadratische Irrationalzahl</a>. Für die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-te Wurzel aus 2 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 n\geq 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73136e4a27fe39c123d16a7808e76d3162ce42bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 3}"></span> trifft dies jedoch nicht zu. </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 {\sqrt {2}}=[1;\,2,\,2,\,2,\,2,\,2,\,\dotsc ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>;</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}=[1;\,2,\,2,\,2,\,2,\,2,\,\dotsc ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac01476f2563cb539eec051d8cec616783fba328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.715ex; height:3.176ex;" alt="{\displaystyle {\sqrt {2}}=[1;\,2,\,2,\,2,\,2,\,2,\,\dotsc ]}"></span> (Folge <a href="//oeis.org/A040000" class="extiw" title="oeis:A040000">A040000</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <p>Diese periodische Entwicklung ergibt sich aus folgenden einfachen Tatsachen (mit der Gaußschen <a href="/wiki/Abrundungsfunktion" class="mw-redirect" title="Abrundungsfunktion">Abrundungsfunktion</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto \lfloor x\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto \lfloor x\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3b60078378682c77f591f9e387cbea7151dbe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.338ex; height:2.843ex;" alt="{\displaystyle x\mapsto \lfloor x\rfloor }"></span>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor {\sqrt {2}}\rfloor =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo fence="false" stretchy="false">⌋</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor {\sqrt {2}}\rfloor =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/632e14b9f07d4cfaeb91c6a0c62c2fbd67389985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.424ex; height:3.176ex;" alt="{\displaystyle \lfloor {\sqrt {2}}\rfloor =1}"></span> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{{\sqrt {2}}-1}}={\sqrt {2}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{{\sqrt {2}}-1}}={\sqrt {2}}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b37c21861bc66fbba8a484da2b9b43ade8be28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:18.137ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{{\sqrt {2}}-1}}={\sqrt {2}}+1}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor {\sqrt {2}}+1\rfloor =2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌋</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor {\sqrt {2}}+1\rfloor =2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4ea93a47edf5e096b411e182bae672c7a33f1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.427ex; height:3.176ex;" alt="{\displaystyle \lfloor {\sqrt {2}}+1\rfloor =2}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}+1-2={\sqrt {2}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+1-2={\sqrt {2}}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1325f5ae9dc16ed72d5979fcb68b63ecfde27ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.304ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+1-2={\sqrt {2}}-1}"></span></dd></dl></dd></dl> <p>Die ersten Näherungsbrüche der Kettenbruchentwicklung 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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> sind: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{1}},\,{\frac {3}{2}},\,{\frac {7}{5}},\,{\frac {17}{12}},\,{\frac {41}{29}},\,{\frac {99}{70}},\,\dotsc }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>5</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>17</mn> <mn>12</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>41</mn> <mn>29</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>99</mn> <mn>70</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1}},\,{\frac {3}{2}},\,{\frac {7}{5}},\,{\frac {17}{12}},\,{\frac {41}{29}},\,{\frac {99}{70}},\,\dotsc }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99b43b2d142dd9447c39e4587ac3e1e5ddae3afb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.728ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{1}},\,{\frac {3}{2}},\,{\frac {7}{5}},\,{\frac {17}{12}},\,{\frac {41}{29}},\,{\frac {99}{70}},\,\dotsc }"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Kettenwurzeleigenschaft">Kettenwurzeleigenschaft</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quadratwurzel_aus_2&veaction=edit&section=5" title="Abschnitt bearbeiten: Kettenwurzeleigenschaft" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Kettenwurzeleigenschaft"><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:Kettenwurzel.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Kettenwurzel.png/440px-Kettenwurzel.png" decoding="async" width="440" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Kettenwurzel.png/660px-Kettenwurzel.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Kettenwurzel.png/880px-Kettenwurzel.png 2x" data-file-width="1817" data-file-height="887" /></a><figcaption>Grafische Veranschaulichung der Kettenwurzeldarstellung von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span></figcaption></figure> <p>Die Zahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> lässt sich folgendermaßen als unendlich fortgesetzte Kettenwurzel darstellen:<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> </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 {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+\dotsb }}}}}}}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mo>⋯</mo> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+\dotsb }}}}}}}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64739ab02fddaa4a1e28331e485477be51ba46e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.903ex; height:7.509ex;" alt="{\displaystyle {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+\dotsb }}}}}}}}=2}"></span></dd></dl> <p>Die Figur verdeutlicht die Konvergenz gegen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> anhand der Funktionswerte der Wurzelfunktion <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4850536e7a37db22aacbc552b03f195a3eceaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.782ex; height:3.009ex;" alt="{\displaystyle f(x)={\sqrt {x}}}"></span> unter Einbeziehung der Hilfsgeraden <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7865dcf9685cbeb2aad8346052d48fb4544f781e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.586ex; height:2.509ex;" alt="{\displaystyle y=x-2}"></span>. </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Geometrische_Konstruktion">Geometrische Konstruktion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quadratwurzel_aus_2&veaction=edit&section=6" title="Abschnitt bearbeiten: Geometrische Konstruktion" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Geometrische Konstruktion"><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:Construction_of_square_root_of_2_on_the_line_number.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Construction_of_square_root_of_2_on_the_line_number.svg/220px-Construction_of_square_root_of_2_on_the_line_number.svg.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Construction_of_square_root_of_2_on_the_line_number.svg/330px-Construction_of_square_root_of_2_on_the_line_number.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Construction_of_square_root_of_2_on_the_line_number.svg/440px-Construction_of_square_root_of_2_on_the_line_number.svg.png 2x" data-file-width="500" data-file-height="250" /></a><figcaption>Konstruktion von Wurzel 2 auf der Zahlengeraden</figcaption></figure> <p>Da irrationale Zahlen eine unendlich lange Dezimaldarstellung haben, ist es unmöglich, eine solche Zahl mit dem Lineal genau abzumessen. Es ist aber möglich, die Zahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> <a href="/wiki/Konstruktion_mit_Zirkel_und_Lineal" title="Konstruktion mit Zirkel und Lineal">mit Zirkel und Lineal zu konstruieren</a>: Die Diagonale eines Quadrates ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>-mal so lang wie seine Seitenlänge. Es reicht auch ein rechtwinkliges, gleichschenkliges Dreieck, bei dem die Katheten jeweils 1 Einheit lang sind. Die Länge der Hypotenuse beträgt dann <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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> Einheiten. Um dies zu beweisen, reicht der <a href="/wiki/Satz_des_Pythagoras" title="Satz des Pythagoras">Satz des Pythagoras</a>: Für die Länge <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> der Diagonale gilt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=1^{2}+1^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=1^{2}+1^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeeb5a2ab15fa78cf445135c05919888ba98ca68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.756ex; height:2.843ex;" alt="{\displaystyle x^{2}=1^{2}+1^{2}}"></span>. </p><p>Das genannte Dreieck ist auch der Beginn der <a href="/wiki/Wurzelschnecke" title="Wurzelschnecke">Wurzelschnecke</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Trigonometrie">Trigonometrie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quadratwurzel_aus_2&veaction=edit&section=7" title="Abschnitt bearbeiten: Trigonometrie" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Trigonometrie"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ähnlich 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 {\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b78ccdb7e18e02d4fc567c66aac99bf524acb5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {5}}}"></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 {\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}}"></span> kommt die Quadratwurzel aus 2 bei exakten trigonometrischen Werten spezieller Winkel vor, insbesondere bei den <a href="/wiki/Sinus" class="mw-redirect" title="Sinus">Sinus</a>- und <a href="/wiki/Cosinus" class="mw-redirect" title="Cosinus">Cosinus</a>-Werten. Einfache Beispiele sind: </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{aligned}\sin {\frac {\pi }{4}}=\sin 45^{\circ }&={\tfrac {1}{2}}{\sqrt {2}},\\[5pt]\cos {\frac {\pi }{4}}=\cos 45^{\circ }&={\tfrac {1}{2}}{\sqrt {2}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin {\frac {\pi }{4}}=\sin 45^{\circ }&={\tfrac {1}{2}}{\sqrt {2}},\\[5pt]\cos {\frac {\pi }{4}}=\cos 45^{\circ }&={\tfrac {1}{2}}{\sqrt {2}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3ff5e3bfddcea8b705b8d58d89a42e5954606a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.545ex; margin-bottom: -0.293ex; width:24.895ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\sin {\frac {\pi }{4}}=\sin 45^{\circ }&={\tfrac {1}{2}}{\sqrt {2}},\\[5pt]\cos {\frac {\pi }{4}}=\cos 45^{\circ }&={\tfrac {1}{2}}{\sqrt {2}}.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Geschichte">Geschichte</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quadratwurzel_aus_2&veaction=edit&section=8" title="Abschnitt bearbeiten: Geschichte" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Geschichte"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bereits die alten Hochkulturen haben sich Gedanken über die Wurzel aus 2 gemacht. Die alten Inder schätzen <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 {\sqrt {2}}\approx {\tfrac {577}{408}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>577</mn> <mn>408</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\approx {\tfrac {577}{408}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b5f1ebc8c6e0a04473ab572248bebdc21f28d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.499ex; height:3.843ex;" alt="{\displaystyle {\sqrt {2}}\approx {\tfrac {577}{408}}}"></span> = 1,<b>41421</b>5686… Diese Näherung stimmt auf fünf Nachkommastellen mit dem tatsächlichen Wert 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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> überein, die Abweichung beträgt nur +0,0001502 Prozent. Von ihrer Irrationalität wussten sie wahrscheinlich nichts. Die Babylonier wie auch die Sumerer schätzten um 1950 v. Chr. die Wurzel aus 2 umgerechnet noch auf 1,41. Aus der Zeit um 1800 v. Chr. ist von den Babyloniern eine weitere Näherung überliefert. Sie benutzten in ihrer <a href="/wiki/Keilschrift" title="Keilschrift">Keilschrift</a> ein Stellenwertsystem zur Basis 60 und berechneten die Näherung mit<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> </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 1\cdot 60^{0}+24\cdot 60^{-1}+51\cdot 60^{-2}+10\cdot 60^{-3}={\frac {30547}{21600}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mn>24</mn> <mo>⋅</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>51</mn> <mo>⋅</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>10</mn> <mo>⋅</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>30547</mn> <mn>21600</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot 60^{0}+24\cdot 60^{-1}+51\cdot 60^{-2}+10\cdot 60^{-3}={\frac {30547}{21600}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91dc40049afe73b06682b458d9cf4508ee9d1090" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.474ex; height:5.176ex;" alt="{\displaystyle 1\cdot 60^{0}+24\cdot 60^{-1}+51\cdot 60^{-2}+10\cdot 60^{-3}={\frac {30547}{21600}}}"></span> = 1,<b>41421</b>2962…</dd></dl> <p>Diese Näherung stimmt auf fünf Nachkommastellen mit dem tatsächlichen Wert 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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> überein, die Abweichung beträgt nur −0,0000424 Prozent. </p><p>Im späten 6. oder frühen 5. Jahrhundert v. Chr. entdeckte <a href="/wiki/Hippasos_von_Metapont" title="Hippasos von Metapont">Hippasos von Metapont</a>, ein <a href="/wiki/Pythagoreer" title="Pythagoreer">Pythagoreer</a>, entweder an einem Quadrat oder an einem regelmäßigen Fünfeck, dass das Verhältnis von Seitenlänge zu Diagonale nicht mit ganzen Zahlen darzustellen ist. Damit bewies er die Existenz inkommensurabler Größen. Eine antike Legende, wonach die Veröffentlichung dieser Erkenntnis von den Pythagoreern als Geheimnisverrat betrachtet wurde, ist nach heutigem Forschungsstand unglaubwürdig. </p> <div class="mw-heading mw-heading2"><h2 id="Vorkommen_in_der_Natur">Vorkommen in der Natur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quadratwurzel_aus_2&veaction=edit&section=9" title="Abschnitt bearbeiten: Vorkommen in der Natur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Vorkommen in der Natur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Im Gehirn gibt es Gitterzellen, die 2005 von einer Gruppe um <a href="/wiki/May-Britt_Moser" title="May-Britt Moser">May-Britt</a> und <a href="/wiki/Edvard_Moser" title="Edvard Moser">Edvard Moser</a> entdeckt wurden: „Die Gitterzellen wurden in dem Kortexbereich gefunden, der sich direkt neben dem Hippocampus befindet […]. An einem Ende dieses kortikalen Bereichs ist die Maschenweite klein und am anderen Ende sehr groß. Die Maschenweite nimmt jedoch nicht zufällig zu, sondern von einem Bereich zum nächsten jeweils um den Faktor Quadratwurzel aus zwei.“<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> </p> <div class="mw-heading mw-heading2"><h2 id="Sonstiges">Sonstiges</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quadratwurzel_aus_2&veaction=edit&section=10" title="Abschnitt bearbeiten: Sonstiges" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Sonstiges"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Der Rekord liegt seit dem 19. Juni 2016 bei 10 Billionen Nachkommastellen, erzielt von Ron Watkins (Stand: 4. Februar 2019).<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></li> <li>Das Verhältnis der beiden Seitenlängen eines Blattes im <a href="/wiki/Papierformat" title="Papierformat">DIN-A-Format</a> beträgt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f539855d8bdbadfd437df59db26fb28500894993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.027ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {2}}}}"></span> mit Rundung auf ganze Millimeter (und hat entgegen mancher Annahme nichts mit dem <a href="/wiki/Goldener_Schnitt" title="Goldener Schnitt">Goldenen Schnitt</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1+{\sqrt {5}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1+{\sqrt {5}}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1007a206eed6b61a892ae0f0c2ce3587e0d5c5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.128ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1+{\sqrt {5}}}{2}}}"></span> zu tun). Dadurch ist sichergestellt, dass bei Halbierung des Blattes entlang der längeren Seite wieder ein Blatt im DIN-A-Format (mit um eins erhöhter Nummerierung) entsteht.</li> <li>Die Wurzel aus 2 ist das Frequenzverhältnis zweier Töne in der Musik bei <a href="/wiki/Gleichstufige_Stimmung" title="Gleichstufige Stimmung">gleichschwebender Stimmung</a>, die einen <a href="/wiki/Tritonus" title="Tritonus">Tritonus</a>, also eine halbe <a href="/wiki/Oktave" title="Oktave">Oktave</a> bilden.</li> <li>In der Elektrotechnik enthält die Beziehung zwischen <a href="/wiki/Scheitelwert" title="Scheitelwert">Scheitelwert</a> und Effektivwert von sinusförmiger Wechselspannung ebenfalls die Konstante <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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Merkhilfe_für_die_ersten_Nachkommastellen"><span id="Merkhilfe_f.C3.BCr_die_ersten_Nachkommastellen"></span>Merkhilfe für die ersten Nachkommastellen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quadratwurzel_aus_2&veaction=edit&section=11" title="Abschnitt bearbeiten: Merkhilfe für die ersten Nachkommastellen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Merkhilfe für die ersten Nachkommastellen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 {\sqrt {2}}\approx 1{,}4\,14\,21\,35\,623\,7\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>≈</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>4</mn> <mspace width="thinmathspace" /> <mn>14</mn> <mspace width="thinmathspace" /> <mn>21</mn> <mspace width="thinmathspace" /> <mn>35</mn> <mspace width="thinmathspace" /> <mn>623</mn> <mspace width="thinmathspace" /> <mn>7</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\approx 1{,}4\,14\,21\,35\,623\,7\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5492ab7c9bfd563fa3bef2c3661817a2064e0cc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.116ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}\approx 1{,}4\,14\,21\,35\,623\,7\,}"></span> gilt:<br /> Die ersten vier Zweierblöcke 1,4 | 14 | 21 | 35 der dezimalen Stellen sind, aufgefasst als zweistellige Zahlen, alle durch 7 teilbar.<br /> Die vier darauf folgenden Ziffern lassen sich in zwei Blöcke 623 | 7 aufteilen, die ebenfalls durch 7 teilbar sind. </p> <div class="mw-heading mw-heading2"><h2 id="Ganzzahligkeit_von_Ausdrücken"><span id="Ganzzahligkeit_von_Ausdr.C3.BCcken"></span>Ganzzahligkeit von Ausdrücken</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quadratwurzel_aus_2&veaction=edit&section=12" title="Abschnitt bearbeiten: Ganzzahligkeit von Ausdrücken" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Ganzzahligkeit von Ausdrücken"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Für alle ganzen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}"></span> ist nach dem <a href="/wiki/Binomischer_Lehrsatz" title="Binomischer Lehrsatz">binomischen Lehrsatz</a> das allgemeine Glied der <a href="/wiki/Pell-Folge" title="Pell-Folge">Pell-Folge</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 P(n)={\frac {\left(1+{\sqrt {2}}\right)^{n}-\left(1-{\sqrt {2}}\right)^{n}}{2\cdot {\sqrt {2}}}}=\sum _{k=0}^{\infty }{\binom {n}{2k+1}}\cdot 2^{k}=\sum _{0\leq k<{\frac {n+1}{2}}}{\binom {n}{2k+1}}\cdot 2^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>≤</mo> <mi>k</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)={\frac {\left(1+{\sqrt {2}}\right)^{n}-\left(1-{\sqrt {2}}\right)^{n}}{2\cdot {\sqrt {2}}}}=\sum _{k=0}^{\infty }{\binom {n}{2k+1}}\cdot 2^{k}=\sum _{0\leq k<{\frac {n+1}{2}}}{\binom {n}{2k+1}}\cdot 2^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce97e7089e595a41379c7994aa49ddfb75ed46a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:77.983ex; height:9.176ex;" alt="{\displaystyle P(n)={\frac {\left(1+{\sqrt {2}}\right)^{n}-\left(1-{\sqrt {2}}\right)^{n}}{2\cdot {\sqrt {2}}}}=\sum _{k=0}^{\infty }{\binom {n}{2k+1}}\cdot 2^{k}=\sum _{0\leq k<{\frac {n+1}{2}}}{\binom {n}{2k+1}}\cdot 2^{k}}"></span></dd></dl> <p>eine natürliche Zahl (für ganzzahliges <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 k>{\tfrac {n-1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>{\tfrac {n-1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fef56524eb3a8ecc37fd2d4a4cb5d279a9663058" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.233ex; height:3.676ex;" alt="{\displaystyle k>{\tfrac {n-1}{2}}}"></span> gilt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {n}{2k+1}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {n}{2k+1}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e92af18dd024c2483b270a509c2f5a812b0da34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.17ex; height:3.343ex;" alt="{\displaystyle {\tbinom {n}{2k+1}}=0}"></span>). <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 P(n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2846b7cbc67a6e521b30d90ba22d6400eb10c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.952ex; height:2.843ex;" alt="{\displaystyle P(n+1)}"></span> ist der Nenner des <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-ten <a href="#Kettenbruchentwicklung">Näherungsbruches der Kettenbruchentwicklung</a> 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 {\sqrt {2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4b9a6710637a4b6fac61388a46180a132d66c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.745ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}.}"></span> </p> <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=Quadratwurzel_aus_2&veaction=edit&section=13" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=13" 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; 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Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PythagorassConstant.html"><i>Pythagoras’s Constant</i>.</a> In: <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li> <li>Folge <a href="//oeis.org/A028254" class="extiw" title="oeis:A028254">A028254</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a> (<a href="/wiki/Engel-Entwicklung" title="Engel-Entwicklung">Engel-Entwicklung</a> (englisch <span lang="en"><i><a href="https://en.wikipedia.org/wiki/Engel_expansion" class="extiw" title="en:Engel expansion">Engel expansion</a></i></span>) von √2)</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=Quadratwurzel_aus_2&veaction=edit&section=14" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quadratwurzel_aus_2&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Roger B. Nelsen: <i>Beweise ohne Worte.</i> Deutschsprachige Ausgabe, herausgegeben von Nicola Oswald, <a href="/wiki/Springer_Spektrum" title="Springer Spektrum">Springer Spektrum</a>, Springer-Verlag, Berlin/Heidelberg 2016, <a href="/wiki/Spezial:ISBN-Suche/9783662503300" class="internal mw-magiclink-isbn">ISBN 978-3-662-50330-0</a>, Seite 179.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><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/20070509055700/http://www.do.nw.schule.de/mbr/2020/geschichte.htm"><i>Kleiner Geschichtsabriss zur Computer-, Technik-, Kommunikations- und Mediengeschichte.</i></a> (<a href="/wiki/Web-Archivierung#Begrifflichkeiten" title="Web-Archivierung"><span class="webarchiv-memento">Memento</span></a> vom 9. Mai 2007 im <i><a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a></i>). Beitrag zum Schülerprojekt <i>Meine Welt 2020. Reportagen aus der Zukunft.</i> 31. März 2000, abgerufen am 4. Februar 2024.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Kaja Nordengen: <cite class="lang" lang="nb" dir="auto" style="font-style:italic">Hjernen er sternen. Ditt eneste uerstattelige organ</cite>. Kagge Forlag AS, 2016, <a href="/wiki/Spezial:ISBN-Suche/9788248920182" class="internal mw-magiclink-isbn">ISBN 978-82-489-2018-2</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>81</span> (norwegisch (Bokmål)): <span class="lang" lang="nb" dir="auto">“Gittercellene ble funnet barkområdet som ligger rett ved hippocampus […]. I den ene enden av dette barkområdet er maskestørrelsen liten og i den andre er den kjempe stor. Økningen i maskestørrelse er imidlertid ikke overlatt tilfeldighetene, men øker med kvadratroten av to, fra ett område til det neste.”</span><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:Quadratwurzel+aus+2&rft.au=Kaja+Nordengen&rft.btitle=Hjernen+er+sternen.+Ditt+eneste+uerstattelige+organ&rft.date=2016&rft.genre=book&rft.isbn=9788248920182&rft.pages=81&rft.pub=Kagge+Forlag+AS" style="display:none"> </span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://numberworld.org/digits/Sqrt(2)/"><i>Square Root of 2.</i></a> In: <i>numberworld.org.</i> 9. Januar 2017, abgerufen am 4. Februar 2024.</span> </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" 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=Quadratwurzel_aus_2&oldid=247256195">https://de.wikipedia.org/w/index.php?title=Quadratwurzel_aus_2&oldid=247256195</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">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Besondere_Zahl" title="Kategorie:Besondere Zahl">Besondere Zahl</a></li><li><a href="/wiki/Kategorie:Wurzel_(Mathematik)" title="Kategorie:Wurzel (Mathematik)">Wurzel (Mathematik)</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Quadratwurzel+aus+2" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q389813" title="Link zum verbundenen Objekt im Datenrepositorium [g]" accesskey="g"><span>Wikidata-Datenobjekt</span></a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">In anderen Sprachen</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%84%D8%AC%D8%B0%D8%B1_%D8%A7%D9%84%D8%AA%D8%B1%D8%A8%D9%8A%D8%B9%D9%8A_%D9%84_2" title="الجذر التربيعي ل 2 – Arabisch" lang="ar" hreflang="ar" data-title="الجذر التربيعي ل 2" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A7%A8-%E0%A6%8F%E0%A6%B0_%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%97%E0%A6%AE%E0%A7%82%E0%A6%B2" title="২-এর বর্গমূল – Bengalisch" lang="bn" hreflang="bn" data-title="২-এর বর্গমূল" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kvadratni_korijen_iz_2" title="Kvadratni korijen iz 2 – Bosnisch" lang="bs" hreflang="bs" data-title="Kvadratni korijen iz 2" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Arrel_quadrada_de_2" title="Arrel quadrada de 2 – Katalanisch" lang="ca" hreflang="ca" data-title="Arrel quadrada de 2" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%DB%95%DA%AF%DB%8C_%D8%AF%D9%88%D9%88%DB%95%D9%85%DB%8C_%D9%A2" title="ڕەگی دووەمی ٢ – Zentralkurdisch" lang="ckb" hreflang="ckb" data-title="ڕەگی دووەمی ٢" data-language-autonym="کوردی" data-language-local-name="Zentralkurdisch" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%BA%D0%BA%C4%95%D0%BD_%D1%82%C4%83%D0%B2%D0%B0%D1%82%D0%BA%D0%B0%D0%BB%D0%BB%D0%B0_%D1%82%D1%8B%D0%BC%D0%B0%D1%80%C4%95" title="Иккĕн тăваткалла тымарĕ – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Иккĕн тăваткалла тымарĕ" 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/Ail_isradd_2" title="Ail isradd 2 – Walisisch" lang="cy" hreflang="cy" data-title="Ail isradd 2" data-language-autonym="Cymraeg" data-language-local-name="Walisisch" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CE%B5%CF%84%CF%81%CE%B1%CE%B3%CF%89%CE%BD%CE%B9%CE%BA%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1_%CF%84%CE%BF%CF%85_2" title="Τετραγωνική ρίζα του 2 – Griechisch" lang="el" hreflang="el" data-title="Τετραγωνική ρίζα του 2" 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/Square_root_of_2" title="Square root of 2 – Englisch" lang="en" hreflang="en" data-title="Square root of 2" 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/Kvadrata_radiko_de_2" title="Kvadrata radiko de 2 – Esperanto" lang="eo" hreflang="eo" data-title="Kvadrata radiko de 2" 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/Ra%C3%ADz_cuadrada_de_dos" title="Raíz cuadrada de dos – Spanisch" lang="es" hreflang="es" data-title="Raíz cuadrada de dos" 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/Ruutjuur_kahest" title="Ruutjuur kahest – Estnisch" lang="et" hreflang="et" data-title="Ruutjuur kahest" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%DB%8C%D8%B4%D9%87_%D8%AF%D9%88%D9%85_%DB%B2" title="ریشه دوم ۲ – Persisch" lang="fa" hreflang="fa" data-title="ریشه دوم ۲" 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/Neli%C3%B6juuri_2" title="Neliöjuuri 2 – Finnisch" lang="fi" hreflang="fi" data-title="Neliöjuuri 2" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="lesenswerter Artikel"><a href="https://fr.wikipedia.org/wiki/Racine_carr%C3%A9e_de_deux" title="Racine carrée de deux – Französisch" lang="fr" hreflang="fr" data-title="Racine carrée de deux" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%A9%D7%95%D7%A8%D7%A9_%D7%94%D7%A8%D7%99%D7%91%D7%95%D7%A2%D7%99_%D7%A9%D7%9C_2" title="השורש הריבועי של 2 – Hebräisch" lang="he" hreflang="he" data-title="השורש הריבועי של 2" 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%A5%A8_%E0%A4%95%E0%A4%BE_%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97%E0%A4%AE%E0%A5%82%E0%A4%B2" title="२ का वर्गमूल – Hindi" lang="hi" hreflang="hi" data-title="२ का वर्गमूल" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/N%C3%A9gyzetgy%C3%B6k_2" title="Négyzetgyök 2 – Ungarisch" lang="hu" hreflang="hu" data-title="Négyzetgyök 2" 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%94%D5%A1%D5%BC%D5%A1%D5%AF%D5%B8%D6%82%D5%BD%D5%AB_%D5%A1%D6%80%D5%B4%D5%A1%D5%BF_%D5%A5%D6%80%D5%AF%D5%B8%D6%82%D5%BD%D5%AB%D6%81" title="Քառակուսի արմատ երկուսից – Armenisch" lang="hy" hreflang="hy" data-title="Քառակուսի արմատ երկուսից" 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/Akar_kuadrat_dari_2" title="Akar kuadrat dari 2 – Indonesisch" lang="id" hreflang="id" data-title="Akar kuadrat dari 2" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Radice_quadrata_di_2" title="Radice quadrata di 2 – Italienisch" lang="it" hreflang="it" data-title="Radice quadrata di 2" 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/2%E3%81%AE%E5%B9%B3%E6%96%B9%E6%A0%B9" title="2の平方根 – Japanisch" lang="ja" hreflang="ja" data-title="2の平方根" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%9C%EA%B3%B1%EA%B7%BC_2" title="제곱근 2 – Koreanisch" lang="ko" hreflang="ko" data-title="제곱근 2" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kvadratin%C4%97_%C5%A1aknis_i%C5%A1_2" title="Kvadratinė šaknis iš 2 – Litauisch" lang="lt" hreflang="lt" data-title="Kvadratinė šaknis iš 2" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Punca_kuasa_dua_untuk_nombor_2" title="Punca kuasa dua untuk nombor 2 – Malaiisch" lang="ms" hreflang="ms" data-title="Punca kuasa dua untuk nombor 2" 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-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wortel_2" title="Wortel 2 – Niederländisch" lang="nl" hreflang="nl" data-title="Wortel 2" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kvadratroten_av_2" title="Kvadratroten av 2 – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Kvadratroten av 2" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pierwiastek_kwadratowy_z_2" title="Pierwiastek kwadratowy z 2 – Polnisch" lang="pl" hreflang="pl" data-title="Pierwiastek kwadratowy z 2" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Raiz_quadrada_de_dois" title="Raiz quadrada de dois – Portugiesisch" lang="pt" hreflang="pt" data-title="Raiz quadrada de dois" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-roa-tara mw-list-item"><a href="https://roa-tara.wikipedia.org/wiki/Radice_quadrate_de_2_(100_mila_cifre)" title="Radice quadrate de 2 (100 mila cifre) – Tarandíne" lang="nap-x-tara" hreflang="nap-x-tara" data-title="Radice quadrate de 2 (100 mila cifre)" data-language-autonym="Tarandíne" data-language-local-name="Tarandíne" class="interlanguage-link-target"><span>Tarandíne</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D1%8B%D0%B9_%D0%BA%D0%BE%D1%80%D0%B5%D0%BD%D1%8C_%D0%B8%D0%B7_2" title="Квадратный корень из 2 – Russisch" lang="ru" hreflang="ru" data-title="Квадратный корень из 2" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/2_%E0%B7%84%E0%B7%92_%E0%B7%80%E0%B6%BB%E0%B7%8A%E0%B6%9C_%E0%B6%B8%E0%B7%96%E0%B6%BD%E0%B6%BA" title="2 හි වර්ග මූලය – Singhalesisch" lang="si" hreflang="si" data-title="2 හි වර්ග මූලය" data-language-autonym="සිංහල" data-language-local-name="Singhalesisch" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Square_root_of_2" title="Square root of 2 – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Square root of 2" 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kvadratni_koren_%C5%A1tevila_2" title="Kvadratni koren števila 2 – Slowenisch" lang="sl" hreflang="sl" data-title="Kvadratni koren števila 2" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%B8_%D0%BA%D0%BE%D1%80%D0%B5%D0%BD_%D0%B8%D0%B7_2" title="Квадратни корен из 2 – Serbisch" lang="sr" hreflang="sr" data-title="Квадратни корен из 2" 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/Kvadratroten_ur_2" title="Kvadratroten ur 2 – Schwedisch" lang="sv" hreflang="sv" data-title="Kvadratroten ur 2" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B2%E0%B8%81%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B8%AA%E0%B8%AD%E0%B8%87%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%AA%E0%B8%AD%E0%B8%87" title="รากที่สองของสอง – Thailändisch" lang="th" hreflang="th" data-title="รากที่สองของสอง" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Karek%C3%B6k_2" title="Karekök 2 – Türkisch" lang="tr" hreflang="tr" data-title="Karekök 2" 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%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D1%80%D1%96%D0%BD%D1%8C_%D0%B7_%D0%B4%D0%B2%D0%BE%D1%85" title="Квадратний корінь з двох – Ukrainisch" lang="uk" hreflang="uk" data-title="Квадратний корінь з двох" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C4%83n_b%E1%BA%ADc_hai_c%E1%BB%A7a_2" title="Căn bậc hai của 2 – Vietnamesisch" lang="vi" hreflang="vi" data-title="Căn bậc hai của 2" 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-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/2%E7%9A%84%E7%AE%97%E8%A1%93%E5%B9%B3%E6%96%B9%E6%A0%B9" title="2的算術平方根 – Chinesisch" lang="zh" hreflang="zh" data-title="2的算術平方根" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%95%A2%E9%81%94%E5%93%A5%E6%8B%89%E6%96%AF%E5%B8%B8%E6%95%B8" title="畢達哥拉斯常數 – Klassisches Chinesisch" lang="lzh" hreflang="lzh" data-title="畢達哥拉斯常數" data-language-autonym="文言" data-language-local-name="Klassisches Chinesisch" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%96%8B%E6%96%B92" title="開方2 – Kantonesisch" lang="yue" hreflang="yue" data-title="開方2" 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/Q389813#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 31. 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