Revision as of 13:10, 20 December 2024

Trigonometrie 1

Trigonometrie 2

Aufgaben Trigonometrie

Mit anderen Themenbereichen der Mathematik

Geometrie

Satze des Pythagoras

Geometrische Veranschaulichung

Potenzen

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Kehrwert

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Der Kehrwert (auch der reziproke Wert oder das Reziproke) einer von $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}">$ verschiedenen <a href="/wiki/Zahl" title="Zahl">Zahl</a> $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">$ ist in der <a href="/wiki/Arithmetik" title="Arithmetik">Arithmetik</a> diejenige Zahl, die mit $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">$ <a href="/wiki/Multiplikation" title="Multiplikation">multipliziert</a> die Zahl $Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">$ ergibt; er wird als $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{x}}}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3da30de216ba1a9649809913816f8b640eb26f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.776ex; height:3.343ex;" alt="{\displaystyle {\tfrac {1}{x}}}">$ oder $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-1}}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf91609f1a0b7847e108023b015cb6b0d567821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.662ex; height:2.676ex;" alt="{\displaystyle x^{-1}}">$ notiert.

Eigenschaften

Kernaussagen

<figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Hyperbola_one_over_x.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Hyperbola_one_over_x.svg/220px-Hyperbola_one_over_x.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Hyperbola_one_over_x.svg/330px-Hyperbola_one_over_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Hyperbola_one_over_x.svg/440px-Hyperbola_one_over_x.svg.png 2x" data-file-width="1600" data-file-height="1200" /></a><figcaption>Der Graph der Kehrwertfunktion ist eine <a href="/wiki/Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbel</a>.</figcaption></figure>

Je näher eine Zahl bei $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}">$ liegt, desto weiter ist ihr Kehrwert von $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}">$ entfernt. Die Zahl $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}">$ selbst hat keinen Kehrwert und ist auch kein Kehrwert. Die durch $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)={\tfrac {1}{x}}}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295255af2ebd89bd25ea6119f95ade0f789036a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.546ex; height:3.343ex;" alt="{\displaystyle y=f(x)={\tfrac {1}{x}}}">$ beschriebene Kehrwertfunktion (siehe Abbildung) hat dort eine <a href="/wiki/Polstelle" title="Polstelle">Polstelle</a>. Der Kehrwert einer positiven Zahl ist positiv, der Kehrwert einer negativen Zahl ist negativ. Dies findet seinen geometrischen Ausdruck darin, dass der Graph in zwei <a href="/wiki/Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbeläste</a> zerfällt, die im ersten bzw. dritten Quadranten liegen. Die Kehrwertfunktion ist eine <a href="/wiki/Involution_(Mathematik)" title="Involution (Mathematik)">Involution</a>, d. h., der Kehrwert des Kehrwerts von $Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">$ ist wieder $Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x.}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}">$ Ist eine Größe $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">$ <a href="/wiki/Umgekehrt_proportional" class="mw-redirect" title="Umgekehrt proportional">umgekehrt proportional</a> zu einer Größe $Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}">$ dann ist sie proportional zum Kehrwert von $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x.}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}">$

Den Kehrbruch eines <a href="/wiki/Bruchrechnung#Gemeine_Brüche" title="Bruchrechnung">Bruches</a>, also den Kehrwert eines <a href="/wiki/Quotient" title="Quotient">Quotienten</a> $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}">$ mit $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\neq 0,}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/491acb120732257985e2f7ab789fef7cdf54f767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.169ex; height:2.676ex;" alt="{\displaystyle a,b\neq 0,}">$ erhält man, indem man Zähler und Nenner vertauscht:

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\frac {a}{b}}}={\frac {b}{a}}}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/952a852fd53dd6a4539101d38db0e7d9d37d65f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:7.706ex; height:6.676ex;" alt="{\displaystyle {\frac {1}{\frac {a}{b}}}={\frac {b}{a}}}">

Daraus folgt die Rechenregel für das <a href="/wiki/Division_(Mathematik)" title="Division (Mathematik)">Dividieren</a> durch einen Bruch: Durch einen Bruch wird dividiert, indem man mit seinem Kehrwert multipliziert. Siehe auch <a href="/wiki/Bruchrechnung" title="Bruchrechnung">Bruchrechnung</a>.

Den Kehrwert $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{n}}}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee46f3d1f145f31319826905e4ce0750792d55b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.822ex; height:3.343ex;" alt="{\displaystyle {\tfrac {1}{n}}}">$ einer <a href="/wiki/Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürlichen Zahl</a> $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">$ nennt man einen <a href="/wiki/Stammbruch" title="Stammbruch">Stammbruch</a>.

Auch zu jeder von $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}">$ verschiedenen <a href="/wiki/Komplexe_Zahl" title="Komplexe Zahl">komplexen Zahl</a> $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+b\mathrm {i} }</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d7f54052b27c21d6073ea59a31e499ea689970f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.901ex; height:2.343ex;" alt="{\displaystyle z=a+b\mathrm {i} }">$ mit reellen Zahlen $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}">$ gibt es einen Kehrwert $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{z}}.}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5770006851ba8ff951117476454da2731cd73c25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.305ex; height:3.343ex;" alt="{\displaystyle {\tfrac {1}{z}}.}">$ Mit dem <a href="/wiki/Betragsfunktion#Komplexe_Betragsfunktion" title="Betragsfunktion">Absolutbetrag</a> $Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|={\sqrt {a^{2}+b^{2}}}}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe94d0c3b0c3704e8771d0932fff6f983ef0082b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.98ex; height:3.509ex;" alt="{\displaystyle |z|={\sqrt {a^{2}+b^{2}}}}">$ von $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">$ und der zu $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">$ <a href="/wiki/Komplexe_Konjugation" title="Komplexe Konjugation">konjugiert komplexen</a> Zahl $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {z}}=a-b\mathrm {i} }</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa7245b2db6d644ce58741004233134df972e3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.021ex; height:2.509ex;" alt="{\displaystyle {\overline {z}}=a-b\mathrm {i} }">$ gilt:

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mi>z</mi> <mo accent="false">&#x00AF;<!-- ¯ --></mo> </mover> <mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">&#x00AF;<!-- ¯ --></mo> </mover> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mi>z</mi> <mo accent="false">&#x00AF;<!-- ¯ --></mo> </mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>&#x2212;<!-- − --></mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>&#x2212;<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a+b\mathrm {i} }}={\frac {1}{z}}={\frac {\overline {z}}{z{\overline {z}}}}={\frac {\overline {z}}{|z|^{2}}}={\frac {a-b\mathrm {i} }{a^{2}+b^{2}}}={\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}\mathrm {i} }</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e571b122897385c9f968daede3034bfb41ed961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.97ex; height:6.676ex;" alt="{\displaystyle {\frac {1}{a+b\mathrm {i} }}={\frac {1}{z}}={\frac {\overline {z}}{z{\overline {z}}}}={\frac {\overline {z}}{|z|^{2}}}={\frac {a-b\mathrm {i} }{a^{2}+b^{2}}}={\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}\mathrm {i} }">

Summe aus Zahl und Kehrwert

Die Summe aus einer positiven <a href="/wiki/Reelle_Zahl" title="Reelle Zahl">reellen Zahl</a> und ihrem Kehrwert beträgt mindestens $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2.}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2b3373a07e65d3312989163b5ebd400af86480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 2.}">$ ^{<a href="#cite_note-1">[1]</a>}^{<a href="#cite_note-2">[2]</a>}

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>&#x2265;<!-- ≥ --></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+{\frac {1}{x}}\geq 2}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5291484292966bff26e63e310e5a3fc6ba56f702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.597ex; height:5.176ex;" alt="{\displaystyle x+{\frac {1}{x}}\geq 2}">

Beweisvariante 1 (Figur 1):

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>&#x2265;<!-- ≥ --></mo> <mn>4</mn> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>x</mi> <mo>&#x22C5;<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>&#x2265;<!-- ≥ --></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x+{\frac {1}{x}}\right)^{2}\geq 4\cdot x\cdot {\frac {1}{x}}\Leftrightarrow x+{\frac {1}{x}}\geq 2}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e07f845b4566d52630549d8b419941e8393ab70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.137ex; height:6.509ex;" alt="{\displaystyle \left(x+{\frac {1}{x}}\right)^{2}\geq 4\cdot x\cdot {\frac {1}{x}}\Leftrightarrow x+{\frac {1}{x}}\geq 2}">

Beweisvariante 2 (Figur 2):

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>&#x2265;<!-- ≥ --></mo> <mn>2</mn> <mo>&#x2212;<!-- − --></mo> <mi>x</mi> <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>&#x2265;<!-- ≥ --></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x}}\geq 2-x\Leftrightarrow x+{\frac {1}{x}}\geq 2}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62b44aa71298390050daad2f39336a3e0514905e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.808ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{x}}\geq 2-x\Leftrightarrow x+{\frac {1}{x}}\geq 2}">

Beweisvariante 3 (Figur 3):

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>&#x2212;<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x+{\frac {1}{x}}\right)^{2}=2^{2}+\left(x-{\frac {1}{x}}\right)^{2}}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc7d3b39e90e80a51ba4b124ae9ef6e1336b98e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.778ex; height:6.509ex;" alt="{\displaystyle \left(x+{\frac {1}{x}}\right)^{2}=2^{2}+\left(x-{\frac {1}{x}}\right)^{2}}">

(nach dem <a href="/wiki/Satz_des_Pythagoras" title="Satz des Pythagoras">Satz des Pythagoras</a>)

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>&#x2265;<!-- ≥ --></mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>&#x2265;<!-- ≥ --></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow \left(x+{\frac {1}{x}}\right)^{2}\geq 2^{2}\Leftrightarrow x+{\frac {1}{x}}\geq 2}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd31e4054fb61f750fabfe34d37f445ce23cad37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.306ex; height:6.509ex;" alt="{\displaystyle \Leftrightarrow \left(x+{\frac {1}{x}}\right)^{2}\geq 2^{2}\Leftrightarrow x+{\frac {1}{x}}\geq 2}">

Beweisvariante 4 (Figur 4):

Nach dem <a href="/wiki/Strahlensatz" title="Strahlensatz">Strahlensatz</a> sind die <a href="/wiki/Dreieck" title="Dreieck">Dreiecke</a>

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>E</mi> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DEF}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e73d6f110c9dc2ee6ec8677a8e44f7e14ee3e37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.441ex; height:2.176ex;" alt="{\displaystyle DEF}">

und

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DBC}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51aaac538474e68bf4652df3b42d258c164366e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.455ex; height:2.176ex;" alt="{\displaystyle DBC}">

<a href="/wiki/%C3%84hnlichkeit_(Geometrie)" title="Ähnlichkeit (Geometrie)">ähnlich</a>. Es gilt

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>1</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{1}}={\frac {1}{\frac {1}{x}}}}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/182401d6027f4887112049d46472d2b5954a331c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:7.877ex; height:6.509ex;" alt="{\displaystyle {\frac {x}{1}}={\frac {1}{\frac {1}{x}}}}">

. <a href="/wiki/Ohne_Beschr%C3%A4nkung_der_Allgemeinheit" title="Ohne Beschränkung der Allgemeinheit">Ohne Beschränkung der Allgemeinheit</a> wird hier

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>&#x2265;<!-- ≥ --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq 1}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca3ced43f1713577888a8a7ade2d0aaf8354a4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 1}">

vorausgesetzt.

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>&#x22C5;<!-- ⋅ --></mo> <mn>1</mn> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>&#x22C5;<!-- ⋅ --></mo> <mn>1</mn> <mo>&#x22C5;<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>&#x2265;<!-- ≥ --></mo> <mn>1</mn> <mo>&#x22C5;<!-- ⋅ --></mo> <mn>1</mn> <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>&#x2265;<!-- ≥ --></mo> <mn>1</mn> <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>&#x2265;<!-- ≥ --></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>&#x2265;<!-- ≥ --></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\cdot 1\cdot x+{\frac {1}{2}}\cdot 1\cdot {\frac {1}{x}}\geq 1\cdot 1\Leftrightarrow {\frac {x}{2}}+{\frac {1}{2x}}\geq 1\Leftrightarrow x^{2}+1\geq 2x\Leftrightarrow x+{\frac {1}{x}}\geq 2}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14fabbfc730293a6e715f07f44a4ff52061cef82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:72.489ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}\cdot 1\cdot x+{\frac {1}{2}}\cdot 1\cdot {\frac {1}{x}}\geq 1\cdot 1\Leftrightarrow {\frac {x}{2}}+{\frac {1}{2x}}\geq 1\Leftrightarrow x^{2}+1\geq 2x\Leftrightarrow x+{\frac {1}{x}}\geq 2}">

Grafische Veranschaulichung der Beweisvarianten

Figur 1

Figur 2

Figur 3

Figur 4

Summe zweier Kehrwerte

<figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Kehrwertsumme_Planfigur.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Kehrwertsumme_Planfigur.svg/220px-Kehrwertsumme_Planfigur.svg.png" decoding="async" width="220" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Kehrwertsumme_Planfigur.svg/330px-Kehrwertsumme_Planfigur.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Kehrwertsumme_Planfigur.svg/440px-Kehrwertsumme_Planfigur.svg.png 2x" data-file-width="297" data-file-height="295" /></a><figcaption>Figur 5</figcaption></figure>

Die Summe der Kehrwerte zweier positiver reeller Zahlen $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}">$ und $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">$ mit der Summe $Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">$ beträgt mindestens $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}">$ :

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> </mrow> <mo>&#x2265;<!-- ≥ --></mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a}}+{\frac {1}{b}}\geq 4}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3248531e2a57ff3479d1eac67299a17b088b686c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.166ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{a}}+{\frac {1}{b}}\geq 4}">

für

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=1}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a8061cad08a2f1206af42fb3e0389fcf4353e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.329ex; height:2.343ex;" alt="{\displaystyle a+b=1}">

.

Beweis:

Gemäß Figur 5 gilt:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>a</mi> <mi>b</mi> <mo>&#x2264;<!-- ≤ --></mo> <mn>1</mn> <mo stretchy="false">&#x21D4;<!-- ⇔ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>&#x2265;<!-- ≥ --></mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4ab\leq 1\Leftrightarrow {\frac {1}{ab}}\geq 4}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/149079010eed654fc2f606f1a0f92ec6c346de20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.589ex; height:5.343ex;" alt="{\displaystyle 4ab\leq 1\Leftrightarrow {\frac {1}{ab}}\geq 4}">

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mrow> <mi>a</mi> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>&#x2265;<!-- ≥ --></mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a}}+{\frac {1}{b}}={\frac {a+b}{ab}}={\frac {1}{ab}}\geq 4}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c515c6b999f47ecbe6b512157fb97c0b4a4291b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.33ex; height:5.509ex;" alt="{\displaystyle {\frac {1}{a}}+{\frac {1}{b}}={\frac {a+b}{ab}}={\frac {1}{ab}}\geq 4}">

,

<a href="/wiki/Quod_erat_demonstrandum" title="Quod erat demonstrandum">was zu beweisen war</a>.^{<a href="#cite_note-3">[3]</a>}

Summe aufeinanderfolgender Kehrwerte

Für jede <a href="/wiki/Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürliche Zahl</a> $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}">$ gilt

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>&gt;</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}+{\frac {1}{n+1}}+{\frac {1}{n+2}}+...+{\frac {1}{n^{2}}}&gt;1}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da40156d1060ae455bc5c45838b258cad7ea1a98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:34.643ex; height:5.509ex;" alt="{\displaystyle {\frac {1}{n}}+{\frac {1}{n+1}}+{\frac {1}{n+2}}+...+{\frac {1}{n^{2}}}>1}">

.

Den Beweis liefert die Abschätzung

Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>&gt;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>&#x2212;<!-- − --></mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> <mo>&#x2212;<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}+{\frac {1}{n+1}}+{\frac {1}{n+2}}+...+{\frac {1}{n^{2}}}&gt;{\frac {1}{n}}+\left({\frac {1}{n^{2}}}+{\frac {1}{n^{2}}}+...+{\frac {1}{n^{2}}}\right)={\frac {1}{n}}+{\frac {1}{n^{2}}}\left(n^{2}-n\right)={\frac {1}{n}}+1-{\frac {1}{n}}=1}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38db5e46fb393fb9cc42f28547ec6f7e91241a7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:100.707ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{n}}+{\frac {1}{n+1}}+{\frac {1}{n+2}}+...+{\frac {1}{n^{2}}}>{\frac {1}{n}}+\left({\frac {1}{n^{2}}}+{\frac {1}{n^{2}}}+...+{\frac {1}{n^{2}}}\right)={\frac {1}{n}}+{\frac {1}{n^{2}}}\left(n^{2}-n\right)={\frac {1}{n}}+1-{\frac {1}{n}}=1}">

.^{<a href="#cite_note-4">[4]</a>}

Beispiele

Der Kehrwert von $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">$ ist wiederum $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">$ .
Der Kehrwert von $Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0,001</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0{,}001}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37b02aa6542167e2202fec98516bf3237cd35b86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.297ex; height:2.509ex;" alt="{\displaystyle 0{,}001}">$ ist $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1000}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9060e16491f890b9fbcce0194c8d454cbee309ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.65ex; height:2.176ex;" alt="{\displaystyle 1000}">$ .
Der Kehrwert von $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}">$ ist $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}=0{,}5}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0e7fd12728cb5e48baf2932b97faf654f0afa42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.728ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}=0{,}5}">$ .
Der Kehrwert des Bruches $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{5}}}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb22be2c480d6bb96c97cc2b6a1a796f8396489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{5}}}">$ ist $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {5}{2}}=2{\tfrac {1}{2}}=2{,}5}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6377800ff02edf1c0cf48ab2e6fb5568f2b6b640" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.647ex; height:3.509ex;" alt="{\displaystyle {\tfrac {5}{2}}=2{\tfrac {1}{2}}=2{,}5}">$ .
Der Kehrwert der komplexen Zahl $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3+4\mathrm {i} }</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3ab335ff1f5595bf3cf91ef4241f78a48593ce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.812ex; height:2.343ex;" alt="{\displaystyle 3+4\mathrm {i} }">$ ist $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>25</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>25</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{3+4\mathrm {i} }}={\tfrac {3}{25}}-{\tfrac {4}{25}}\mathrm {i} }</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5abfc5e0e00b1a2871bd13d96da7cf097730a53b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.762ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{3+4\mathrm {i} }}={\tfrac {3}{25}}-{\tfrac {4}{25}}\mathrm {i} }">$ .

Verallgemeinerung

Eine Verallgemeinerung des Kehrwerts ist das <a href="/wiki/Inverses_Element" title="Inverses Element">multiplikativ Inverse</a> $Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-1}}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf91609f1a0b7847e108023b015cb6b0d567821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.662ex; height:2.676ex;" alt="{\displaystyle x^{-1}}">$ zu einer <a href="/wiki/Einheit_(Mathematik)" title="Einheit (Mathematik)">Einheit</a> $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">$ eines <a href="/wiki/Ring_(Algebra)" title="Ring (Algebra)">unitären Ringes</a>. Es ist ebenfalls durch die Eigenschaft $Failed to parse (syntax error): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mtext> </mtext> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>⋅</mo> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-1}\cdot \ x=x\cdot \ x^{-1}=1}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa9f878a343f6121e1c85011d9146ce0a29921b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.863ex; height:2.676ex;" alt="{\displaystyle x^{-1}\cdot \ x=x\cdot \ x^{-1}=1}">$ definiert, wobei $Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">$ das Einselement des Ringes bezeichnet.

Wenn es sich z. B. um einen Ring von Matrizen handelt, so ist das Einselement nicht die Zahl $Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,}</annotation> </semantics> } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc5fd8163a83100c5330622e9e317fa4e872403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 1,}">$ sondern die <a href="/wiki/Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a>. Matrizen, zu denen keine <a href="/wiki/Inverse_Matrix" title="Inverse Matrix">inverse Matrix</a> existiert, heißen <a href="/wiki/Singul%C3%A4re_Matrix" class="mw-redirect" title="Singuläre Matrix">singulär</a>.

Literatur

Hintergrundwissen für Lehramtsstudenten zur Arithmetik:

Friedhelm Padberg: Didaktik der Arithmetik. Für Lehrerausbildung und Lehrerfortbildung. 3. erweiterte völlig überarbeitete Auflage, Nachdruck. Spektrum Akademischer Verlag, München 2009, <a href="/wiki/Spezial:ISBN-Suche/9783827409935" class="internal mw-magiclink-isbn">ISBN 978-3-8274-0993-5</a>.

Weblinks

<img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/16px-Wiktfavicon_en.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/24px-Wiktfavicon_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/32px-Wiktfavicon_en.svg.png 2x" data-file-width="16" data-file-height="16" /><a href="https://de.wiktionary.org/wiki/Kehrwert" class="extiw" title="wikt:Kehrwert">Wiktionary: Kehrwert</a> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen

Einzelnachweise

<a href="#cite_ref-1">↑</a> Roger B. Nelsen: Beweise ohne Worte, Deutschsprachige Ausgabe herausgegeben von Nicola Oswald, <a href="/wiki/Springer_Spektrum" title="Springer Spektrum">Springer Spektrum</a>, Springer-Verlag <a href="/wiki/Berlin" title="Berlin">Berlin</a> <a href="/wiki/Heidelberg" title="Heidelberg">Heidelberg</a> 2016, <a href="/wiki/Spezial:ISBN-Suche/9783662503300" class="internal mw-magiclink-isbn">ISBN 978-3-662-50330-0</a>, Seite 145
<a href="#cite_ref-2">↑</a> Roger B. Nelsen: Proof without Words: The Sum of a Positive Number and Its Reciprocal Is at Least Two (four proofs) Mathematics Magazine, vol. 67, no. 5 (Dec. 1994), S. 374
<a href="#cite_ref-3">↑</a> Claudi Alsina, Roger B. Nelsen: Perlen der Mathematik - 20 geometrische Figuren als Ausgangspunkte für mathematische Erkundungsreisen, <a href="/wiki/Springer_Spektrum" title="Springer Spektrum">Springer Spektrum</a>, Springer-Verlag GmbH <a href="/wiki/Berlin" title="Berlin">Berlin</a> 2015, <a href="/wiki/Spezial:ISBN-Suche/9783662454602" class="internal mw-magiclink-isbn">ISBN 978-3-662-45460-2</a>, Seiten 237 und 301
<a href="#cite_ref-4">↑</a> <a href="/wiki/Ross_Honsberger" title="Ross Honsberger">Ross Honsberger</a>: Gitter - Reste - Würfel <a href="/wiki/Vieweg_Verlag" title="Vieweg Verlag">Friedrich Vieweg & Sohn Verlagsgesellschaft mbH</a>, <a href="/wiki/Braunschweig" title="Braunschweig">Braunschweig</a> 1984, <a href="/wiki/Spezial:ISBN-Suche/9783528084769" class="internal mw-magiclink-isbn">ISBN 978-3-528-08476-9</a>, S. 155

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<a href="https://fr.wikipedia.org/wiki/Inverse" title="Inverse – Französisch" lang="fr" hreflang="fr" data-title="Inverse" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target">Français</a>
<a href="https://frr.wikipedia.org/wiki/Kiarw%C3%A4%C3%A4rs" title="Kiarwäärs – Nordfriesisch" lang="frr" hreflang="frr" data-title="Kiarwäärs" data-language-autonym="Nordfriisk" data-language-local-name="Nordfriesisch" class="interlanguage-link-target">Nordfriisk</a>
<a href="https://gl.wikipedia.org/wiki/Inverso_multiplicativo" title="Inverso multiplicativo – Galicisch" lang="gl" hreflang="gl" data-title="Inverso multiplicativo" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target">Galego</a>
<a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%94%D7%95%D7%A4%D7%9B%D7%99" title="מספר הופכי – Hebräisch" lang="he" hreflang="he" data-title="מספר הופכי" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target">עברית</a>
<a href="https://hu.wikipedia.org/wiki/Reciprok" title="Reciprok – Ungarisch" lang="hu" hreflang="hu" data-title="Reciprok" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target">Magyar</a>
<a href="https://id.wikipedia.org/wiki/Invers_perkalian" title="Invers perkalian – Indonesisch" lang="id" hreflang="id" data-title="Invers perkalian" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target">Bahasa Indonesia</a>
<a href="https://is.wikipedia.org/wiki/Umhverfa" title="Umhverfa – Isländisch" lang="is" hreflang="is" data-title="Umhverfa" data-language-autonym="Íslenska" data-language-local-name="Isländisch" class="interlanguage-link-target">Íslenska</a>
<a href="https://it.wikipedia.org/wiki/Reciproco" title="Reciproco – Italienisch" lang="it" hreflang="it" data-title="Reciproco" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target">Italiano</a>
<a href="https://ja.wikipedia.org/wiki/%E9%80%86%E6%95%B0" title="逆数 – Japanisch" lang="ja" hreflang="ja" data-title="逆数" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target">日本語</a>
<a href="https://ko.wikipedia.org/wiki/%EA%B3%B1%EC%85%88_%EC%97%AD%EC%9B%90" title="곱셈 역원 – Koreanisch" lang="ko" hreflang="ko" data-title="곱셈 역원" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target">한국어</a>
<a href="https://lmo.wikipedia.org/wiki/Invers" title="Invers – Lombardisch" lang="lmo" hreflang="lmo" data-title="Invers" data-language-autonym="Lombard" data-language-local-name="Lombardisch" class="interlanguage-link-target">Lombard</a>
<a href="https://lt.wikipedia.org/wiki/Atvirk%C5%A1tinis_skai%C4%8Dius" title="Atvirkštinis skaičius – Litauisch" lang="lt" hreflang="lt" data-title="Atvirkštinis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target">Lietuvių</a>
<a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B5%D1%86%D0%B8%D0%BF%D1%80%D0%BE%D1%87%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82" title="Реципрочна вредност – Mazedonisch" lang="mk" hreflang="mk" data-title="Реципрочна вредност" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" class="interlanguage-link-target">Македонски</a>
<a href="https://ms.wikipedia.org/wiki/Salingan" title="Salingan – Malaiisch" lang="ms" hreflang="ms" data-title="Salingan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malaiisch" class="interlanguage-link-target">Bahasa Melayu</a>
<a href="https://nds.wikipedia.org/wiki/Kehrweert" title="Kehrweert – Niederdeutsch" lang="nds" hreflang="nds" data-title="Kehrweert" data-language-autonym="Plattdüütsch" data-language-local-name="Niederdeutsch" class="interlanguage-link-target">Plattdüütsch</a>
<a href="https://nl.wikipedia.org/wiki/Omgekeerde" title="Omgekeerde – Niederländisch" lang="nl" hreflang="nl" data-title="Omgekeerde" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target">Nederlands</a>
<a href="https://nn.wikipedia.org/wiki/Resiprok" title="Resiprok – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Resiprok" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target">Norsk nynorsk</a>
<a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%81%E0%A8%A3%E0%A8%BE%E0%A8%A4%E0%A8%AE%E0%A8%95_%E0%A8%89%E0%A8%B2%E0%A8%9F" title="ਗੁਣਾਤਮਕ ਉਲਟ – Punjabi" lang="pa" hreflang="pa" data-title="ਗੁਣਾਤਮਕ ਉਲਟ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a>
<a href="https://pl.wikipedia.org/wiki/Liczba_odwrotna" title="Liczba odwrotna – Polnisch" lang="pl" hreflang="pl" data-title="Liczba odwrotna" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target">Polski</a>
<a href="https://pt.wikipedia.org/wiki/Inverso_multiplicativo" title="Inverso multiplicativo – Portugiesisch" lang="pt" hreflang="pt" data-title="Inverso multiplicativo" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target">Português</a>
<a href="https://qu.wikipedia.org/wiki/T%27ikrasqa_yupay" title="T'ikrasqa yupay – Quechua" lang="qu" hreflang="qu" data-title="T'ikrasqa yupay" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target">Runa Simi</a>
<a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Обратное число – Russisch" lang="ru" hreflang="ru" data-title="Обратное число" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target">Русский</a>
<a href="https://simple.wikipedia.org/wiki/Reciprocal" title="Reciprocal – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Reciprocal" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target">Simple English</a>
<a href="https://sk.wikipedia.org/wiki/Prevr%C3%A1ten%C3%A1_hodnota" title="Prevrátená hodnota – Slowakisch" lang="sk" hreflang="sk" data-title="Prevrátená hodnota" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" class="interlanguage-link-target">Slovenčina</a>
<a href="https://sl.wikipedia.org/wiki/Recipro%C4%8Dna_vrednost" title="Recipročna vrednost – Slowenisch" lang="sl" hreflang="sl" data-title="Recipročna vrednost" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target">Slovenščina</a>
<a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B5%D1%86%D0%B8%D0%BF%D1%80%D0%BE%D1%87%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82" title="Реципрочна вредност – Serbisch" lang="sr" hreflang="sr" data-title="Реципрочна вредност" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target">Српски / srpski</a>
<a href="https://sv.wikipedia.org/wiki/Reciprok_(matematik)" title="Reciprok (matematik) – Schwedisch" lang="sv" hreflang="sv" data-title="Reciprok (matematik)" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target">Svenska</a>
<a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%86%E0%AE%B0%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%B2%E0%AF%8D_%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AF%8D%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%81" title="பெருக்கல் நேர்மாறு – Tamil" lang="ta" hreflang="ta" data-title="பெருக்கல் நேர்மாறு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a>
<a href="https://th.wikipedia.org/wiki/%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B8%9C%E0%B8%81%E0%B8%9C%E0%B8%B1%E0%B8%99%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%84%E0%B8%B9%E0%B8%93" title="ตัวผกผันการคูณ – Thailändisch" lang="th" hreflang="th" data-title="ตัวผกผันการคูณ" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target">ไทย</a>
<a href="https://tl.wikipedia.org/wiki/Kabaligtarang_pamparami" title="Kabaligtarang pamparami – Tagalog" lang="tl" hreflang="tl" data-title="Kabaligtarang pamparami" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a>
<a href="https://uk.wikipedia.org/wiki/%D0%9E%D0%B1%D0%B5%D1%80%D0%BD%D0%B5%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Обернене число – Ukrainisch" lang="uk" hreflang="uk" data-title="Обернене число" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target">Українська</a>
<a href="https://ur.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA%DB%8C_%D9%85%D9%82%D9%84%D9%88%D8%A8" title="ریاضیاتی مقلوب – Urdu" lang="ur" hreflang="ur" data-title="ریاضیاتی مقلوب" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a>
<a href="https://vi.wikipedia.org/wiki/Ngh%E1%BB%8Bch_%C4%91%E1%BA%A3o_ph%C3%A9p_nh%C3%A2n" title="Nghịch đảo phép nhân – Vietnamesisch" lang="vi" hreflang="vi" data-title="Nghịch đảo phép nhân" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target">Tiếng Việt</a>
<a href="https://wuu.wikipedia.org/wiki/%E5%80%92%E6%95%B0" title="倒数 – Wu" lang="wuu" hreflang="wuu" data-title="倒数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a>
<a href="https://zh.wikipedia.org/wiki/%E5%80%92%E6%95%B0" title="倒数 – Chinesisch" lang="zh" hreflang="zh" data-title="倒数" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target">中文</a>
<a href="https://zh-classical.wikipedia.org/wiki/%E5%80%92%E6%95%B8" title="倒數 – Klassisches Chinesisch" lang="lzh" hreflang="lzh" data-title="倒數" data-language-autonym="文言" data-language-local-name="Klassisches Chinesisch" class="interlanguage-link-target">文言</a>
<a href="https://zh-yue.wikipedia.org/wiki/%E5%80%92%E6%95%B8_(%E6%95%B8)" title="倒數 (數) – Kantonesisch" lang="yue" hreflang="yue" data-title="倒數 (數)" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target">粵語</a>

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Aufgaben zu Potenzgesetzen

[1] <a href="#cite_ref-1">↑</a> Roger B. Nelsen: Beweise ohne Worte, Deutschsprachige Ausgabe herausgegeben von Nicola Oswald, <a href="/wiki/Springer_Spektrum" title="Springer Spektrum">Springer Spektrum</a>, Springer-Verlag <a href="/wiki/Berlin" title="Berlin">Berlin</a> <a href="/wiki/Heidelberg" title="Heidelberg">Heidelberg</a> 2016, <a href="/wiki/Spezial:ISBN-Suche/9783662503300" class="internal mw-magiclink-isbn">ISBN 978-3-662-50330-0</a>, Seite 145

[2] <a href="#cite_ref-2">↑</a> Roger B. Nelsen: Proof without Words: The Sum of a Positive Number and Its Reciprocal Is at Least Two (four proofs) Mathematics Magazine, vol. 67, no. 5 (Dec. 1994), S. 374

[3] <a href="#cite_ref-3">↑</a> Claudi Alsina, Roger B. Nelsen: Perlen der Mathematik - 20 geometrische Figuren als Ausgangspunkte für mathematische Erkundungsreisen, <a href="/wiki/Springer_Spektrum" title="Springer Spektrum">Springer Spektrum</a>, Springer-Verlag GmbH <a href="/wiki/Berlin" title="Berlin">Berlin</a> 2015, <a href="/wiki/Spezial:ISBN-Suche/9783662454602" class="internal mw-magiclink-isbn">ISBN 978-3-662-45460-2</a>, Seiten 237 und 301

[4] <a href="#cite_ref-4">↑</a> <a href="/wiki/Ross_Honsberger" title="Ross Honsberger">Ross Honsberger</a>: Gitter - Reste - Würfel <a href="/wiki/Vieweg_Verlag" title="Vieweg Verlag">Friedrich Vieweg & Sohn Verlagsgesellschaft mbH</a>, <a href="/wiki/Braunschweig" title="Braunschweig">Braunschweig</a> 1984, <a href="/wiki/Spezial:ISBN-Suche/9783528084769" class="internal mw-magiclink-isbn">ISBN 978-3-528-08476-9</a>, S. 155

Christian: Difference between revisions

Revision as of 13:10, 20 December 2024

Contents

Trigonometrie 1

Trigonometrie 2

Aufgaben Trigonometrie

Geometrie

Satze des Pythagoras

Potenzen

Kehrwert

Inhaltsverzeichnis

Eigenschaften

Kernaussagen

Summe aus Zahl und Kehrwert

Summe zweier Kehrwerte

Summe aufeinanderfolgender Kehrwerte

Beispiele

Verallgemeinerung

Verwandte Themen

Literatur

Weblinks

Einzelnachweise

Navigationsmenü

Meine Werkzeuge

Namensräume

Ansichten

Navigation

Mitmachen

Werkzeuge

Drucken/exportieren

In anderen Projekten

In anderen Sprachen

Aufgaben zu Potenzgesetzen

Navigation menu

Christian: Difference between revisions

Revision as of 13:10, 20 December 2024

Trigonometrie 1

Trigonometrie 2

Aufgaben Trigonometrie

Geometrie

Satze des Pythagoras

Potenzen

Kehrwert

Eigenschaften

Kernaussagen

Summe aus Zahl und Kehrwert

Summe zweier Kehrwerte

Summe aufeinanderfolgender Kehrwerte

Beispiele

Verallgemeinerung

Verwandte Themen

Literatur

Weblinks

Einzelnachweise

Navigationsmenü

Meine Werkzeuge

Namensräume

Ansichten

Suche

Navigation

Mitmachen

Werkzeuge

Drucken/​exportieren

In anderen Projekten

In anderen Sprachen

Aufgaben zu Potenzgesetzen

Navigation menu

Search

Drucken/exportieren