Lösung 3.2:6a - Versionsgeschichte

Moni um 16:35, 22. Aug. 2009

Moni — Sat, 22 Aug 2009 16:35:56 GMT

← Nächstältere Version		Version vom 16:35, 22. Aug. 2009
Zeile 3:		Zeile 3:
	[[Image:3_2_6_a.gif\|center]]		[[Image:3_2_6_a.gif\|center]]

-	Anstatt ~~dass wir die komplexe~~ Zahl ~~wie~~	+	Anstatt der komplexen Zahl in der Form
-	<math>x+iy</math> ~~schreiben~~, schreiben wir sie in Polarform, um unter anderen Multiplikation und Division zu ~~vereinfachen~~.	+	<math>x+iy</math>, schreiben wir sie in Polarform, um unter anderen Multiplikation und Division zu erleichtern.

	Die Polarform wird definiert durch:		Die Polarform wird definiert durch:
Zeile 10:		Zeile 10:
	{{Abgesetzte Formel\|\|<math>x+iy=r(\cos\alpha + i\sin\alpha)\,,</math>}}		{{Abgesetzte Formel\|\|<math>x+iy=r(\cos\alpha + i\sin\alpha)\,,</math>}}

-	wo <math>x</math> und <math>y</math> reelle Zahlen sind, und <math>r</math> und <math>\alpha</math> der Betrag und das Argument sind. Die rechte Seite der Formen benennt man die Polarform einer komplexen Zahl.	+	wobei <math>x</math> und <math>y</math> reelle Zahlen sind und <math>r</math> und <math>\alpha</math> der Betrag und das Argument sind. Die rechte Seite der Formen benennt man die Polarform einer komplexen Zahl.

-	Die Zahl 3 hat den Betrag 3, und das Argument <math>0</math>, nachdem sie auf der reellen Achse liegt. Daher ist	+	Die Zahl 3 hat den Betrag 3 und das Argument <math>0</math>, nachdem sie auf der reellen Achse liegt. Daher ist

	{{Abgesetzte Formel\|\|<math>3(\cos 0 + i\sin 0)\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>3(\cos 0 + i\sin 0)\,\textrm{.}</math>}}

Martin um 13:04, 13. Mai 2009

Martin — Wed, 13 May 2009 13:04:33 GMT

← Nächstältere Version		Version vom 13:04, 13. Mai 2009
Zeile 1:		Zeile 1:
-	~~Polar form is an alternative way to express complex numbers~~. ~~The idea is that it is sufficient to know a complex number's distance from the origin~~ (~~magnitude) and the angle which the line between the number and the origin makes with the positive real axis (argument~~), ~~in order to determine the number~~.	+	Die Polarform ist nur eine Art komplexe Zahlen zu schreiben. Nachdem ein Vektor eindeutig durch seinen Betrag und Argument (Winkel) definiert ist, gilt dasselbe auch für komplexe Zahlen.

	[[Image:3_2_6_a.gif\|center]]		[[Image:3_2_6_a.gif\|center]]

-	~~Instead of working with the number's rectangular form,~~	+	Anstatt dass wir die komplexe Zahl wie
-	<math>x+iy</math>, ~~we use its magnitude and argument to obtain a simpler and more natural description of e.g. multiplication and division~~.	+	<math>x+iy</math> schreiben, schreiben wir sie in Polarform, um unter anderen Multiplikation und Division zu vereinfachen.

-	~~The relation between a complex number in rectangular form and in polar form can be written as~~	+	Die Polarform wird definiert durch:

	{{Abgesetzte Formel\|\|<math>x+iy=r(\cos\alpha + i\sin\alpha)\,,</math>}}		{{Abgesetzte Formel\|\|<math>x+iy=r(\cos\alpha + i\sin\alpha)\,,</math>}}

-	~~where~~ <math>x</math> ~~and~~ <math>y</math> ~~are the number's real and imaginary parts respectively~~, ~~and where~~ <math>r</math> ~~and~~ <math>\alpha</math> ~~are the magnitude and the argument, respectively~~. ~~It is the right-hand side in this formula which is usually said to be the number's polar form~~.	+	wo <math>x</math> und <math>y</math> reelle Zahlen sind, und <math>r</math> und <math>\alpha</math> der Betrag und das Argument sind. Die rechte Seite der Formen benennt man die Polarform einer komplexen Zahl.

-	~~The number <math>\text{~~3~~}</math> has magnitude <math>\text{~~3~~}</math> (distance to the origin)~~, ~~and argument~~ <math>0</math> ~~because the number lies on the positive real axis~~. ~~The number's polar form is therefore~~	+	Die Zahl 3 hat den Betrag 3, und das Argument <math>0</math>, nachdem sie auf der reellen Achse liegt. Daher ist

	{{Abgesetzte Formel\|\|<math>3(\cos 0 + i\sin 0)\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>3(\cos 0 + i\sin 0)\,\textrm{.}</math>}}

Tekbot: Solution 3.2:6a moved to Lösung 3.2:6a: Robot: moved page

Tekbot — Wed, 11 Mar 2009 10:40:35 GMT

Solution 3.2:6a moved to Lösung 3.2:6a: Robot: moved page

← Nächstältere Version

Version vom 10:40, 11. Mär. 2009

Tekbot: Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

Tekbot — Tue, 10 Mar 2009 13:09:44 GMT

Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

← Nächstältere Version		Version vom 13:09, 10. Mär. 2009
Zeile 8:		Zeile 8:
	The relation between a complex number in rectangular form and in polar form can be written as		The relation between a complex number in rectangular form and in polar form can be written as

-	{{~~Displayed math~~\|\|<math>x+iy=r(\cos\alpha + i\sin\alpha)\,,</math>}}	+	{{Abgesetzte Formel\|\|<math>x+iy=r(\cos\alpha + i\sin\alpha)\,,</math>}}

	where <math>x</math> and <math>y</math> are the number's real and imaginary parts respectively, and where <math>r</math> and <math>\alpha</math> are the magnitude and the argument, respectively. It is the right-hand side in this formula which is usually said to be the number's polar form.		where <math>x</math> and <math>y</math> are the number's real and imaginary parts respectively, and where <math>r</math> and <math>\alpha</math> are the magnitude and the argument, respectively. It is the right-hand side in this formula which is usually said to be the number's polar form.
Zeile 14:		Zeile 14:
	The number <math>\text{3}</math> has magnitude <math>\text{3}</math> (distance to the origin), and argument <math>0</math> because the number lies on the positive real axis. The number's polar form is therefore		The number <math>\text{3}</math> has magnitude <math>\text{3}</math> (distance to the origin), and argument <math>0</math> because the number lies on the positive real axis. The number's polar form is therefore

-	{{~~Displayed math~~\|\|<math>3(\cos 0 + i\sin 0)\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>3(\cos 0 + i\sin 0)\,\textrm{.}</math>}}

Tek um 12:47, 29. Okt. 2008

Tek — Wed, 29 Oct 2008 12:47:23 GMT

← Nächstältere Version		Version vom 12:47, 29. Okt. 2008
Zeile 1:		Zeile 1:
-	Polar form is an alternative way to express complex numbers. The idea is that it is sufficient to know a complex number's distance from the origin (magnitude) and the angle which the line between the number and the origin makes with the positive real axis (argument), in order to determine the number.	+	Polar form is an alternative way to express complex numbers. The idea is that it is sufficient to know a complex number's distance from the origin (magnitude) and the angle which the line between the number and the origin makes with the positive real axis (argument), in order to determine the number.

	[[Image:3_2_6_a.gif\|center]]		[[Image:3_2_6_a.gif\|center]]
-

	Instead of working with the number's rectangular form,		Instead of working with the number's rectangular form,
Zeile 9:		Zeile 8:
	The relation between a complex number in rectangular form and in polar form can be written as		The relation between a complex number in rectangular form and in polar form can be written as

		+	{{Displayed math\|\|<math>x+iy=r(\cos\alpha + i\sin\alpha)\,,</math>}}

-	<math>x~~+iy=~~r~~\left( \cos~~ \alpha ~~+i\sin \alpha \right)~~</math>	+	where <math>x</math> and <math>y</math> are the number's real and imaginary parts respectively, and where <math>r</math> and <math>\alpha</math> are the magnitude and the argument, respectively. It is the right-hand side in this formula which is usually said to be the number's polar form.

-	~~where~~	+	The number <math>\text{3}</math> has magnitude <math>\text{3}</math> (distance to the origin), and argument <math>0</math> because the number lies on the positive real axis. The number's polar form is therefore
-	<math>x</math>	+
-	~~and~~	+
-	<math>y</math>	+
-	~~are~~ the ~~number's real and imaginary parts respectively~~, and ~~where~~	+
-	<math>r</math>	+
-	~~and~~	+
-	~~<math>\alpha </math>~~	+
-	~~are~~ the ~~magnitude and~~ the ~~argument, respectively~~. ~~It is the right-hand side in this formula which is usually said to be the~~ number's polar form.	+

-	~~The number~~	+	{{Displayed math\|\|<math>3(\cos 0 + i\sin 0)\,\textrm{.}</math>}}
-	~~<math>\text~~{~~3}</~~math>	+
-	~~and magnitude~~ <math>~~\text{~~3~~}</math>~~ (~~distance to the origin~~), ~~and argument~~	+
-	~~<math>0~~</math>	+
-	~~because the number lies on the positive real axis. The number's polar form is therefore~~	+

Ian um 10:11, 23. Okt. 2008

Ian — Thu, 23 Oct 2008 10:11:54 GMT

← Nächstältere Version		Version vom 10:11, 23. Okt. 2008
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-	~~{{NAVCONTENT_START}}~~	+	Polar form is an alternative way to express complex numbers. The idea is that it is sufficient to know a complex number's distance from the origin (magnitude) and the angle which the line between the number and the origin makes with the positive real axis (argument), in order to determine the number.
-	~~<center> [[Image:3_2_6a-1~~(2)~~.gif]] </center>~~	+
-	~~{{NAVCONTENT_STOP}}~~	+
-	~~{{NAVCONTENT_START}}~~	+
-	~~<center> [[Image:3_2_6a-2~~(2).~~gif]] </center>~~	+
-	~~{{NAVCONTENT_STOP}}~~	+

	[[Image:3_2_6_a.gif\|center]]		[[Image:3_2_6_a.gif\|center]]
		+
		+
		+	Instead of working with the number's rectangular form,
		+	<math>x+iy</math>, we use its magnitude and argument to obtain a simpler and more natural description of e.g. multiplication and division.
		+
		+	The relation between a complex number in rectangular form and in polar form can be written as
		+
		+
		+	<math>x+iy=r\left( \cos \alpha +i\sin \alpha \right)</math>
		+
		+	where
		+	<math>x</math>
		+	and
		+	<math>y</math>
		+	are the number's real and imaginary parts respectively, and where
		+	<math>r</math>
		+	and
		+	<math>\alpha </math>
		+	are the magnitude and the argument, respectively. It is the right-hand side in this formula which is usually said to be the number's polar form.
		+
		+	The number
		+	<math>\text{3}</math>
		+	and magnitude <math>\text{3}</math> (distance to the origin), and argument
		+	<math>0</math>
		+	because the number lies on the positive real axis. The number's polar form is therefore

Tekbot: Lösning 3.2:6a moved to Solution 3.2:6a: Robot: moved page

Tekbot — Tue, 16 Sep 2008 15:01:01 GMT

Lösning 3.2:6a moved to Solution 3.2:6a: Robot: moved page

← Nächstältere Version

Version vom 15:01, 16. Sep. 2008

Tekbot: Robot: Automated text replacement (-[[Bild: +[[Image:)

Tekbot — Thu, 21 Aug 2008 08:10:14 GMT

Robot: Automated text replacement (-[[Bild: +[[Image:)

← Nächstältere Version		Version vom 08:10, 21. Aug. 2008
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-	<center> [[~~Bild~~:3_2_6a-1(2).gif]] </center>	+	<center> [[Image:3_2_6a-1(2).gif]] </center>
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-	[[~~Bild~~:3_2_6_a.gif\|center]]	+	[[Image:3_2_6_a.gif\|center]]

Madde um 15:36, 19. Aug. 2008

Madde — Tue, 19 Aug 2008 15:36:23 GMT

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		+	[[Bild:3_2_6_a.gif\|center]]

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Oskar — Mon, 07 Apr 2008 12:16:10 GMT

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