Lösung 1.1:7c - Versionsgeschichte

Moni: Ausdruck

2009-08-08T13:04:53Z

Ausdruck

← Nächstältere Version		Version vom 13:04, 8. Aug. 2009
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	und also ist sie rational.		und also ist sie rational.
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-	Indem wir die Zahl mit 10 multiplizieren, können wir das ~~komme~~ Schritt für Schritt nach rechts verschieben	+	Indem wir die Zahl mit 10 multiplizieren, können wir das Komma Schritt für Schritt nach rechts verschieben
	::<math>\insteadof[right]{10000x}{x}{}=0\,\textrm{.}\,2\ 001\ 001\ 001\,\ldots</math>		::<math>\insteadof[right]{10000x}{x}{}=0\,\textrm{.}\,2\ 001\ 001\ 001\,\ldots</math>
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	::<math>\insteadof[right]{10000x}{10000x}{}=2001\,\textrm{.}\,\underline{001}\ \underline{001}\ 1\ldots</math>		::<math>\insteadof[right]{10000x}{10000x}{}=2001\,\textrm{.}\,\underline{001}\ \underline{001}\ 1\ldots</math>
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-	Hier sieht man dass 10''x'' und 10000''x'' dieselbe Dezimalbruchentwicklung haben, ~~und~~ also ist	+	Hier sieht man, dass 10''x'' und 10000''x'' dieselbe Dezimalbruchentwicklung haben, also ist
	::<math>10000x-10x = 2001\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots - 2\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots</math>		::<math>10000x-10x = 2001\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots - 2\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	::<math>\phantom{10000x-10x}{} = 1999\,\mbox{.}\quad</math>(die ~~Dezimale kanzellieren~~)	+	::<math>\phantom{10000x-10x}{} = 1999\,\mbox{.}\quad</math>(die Nachkommastellen verschwinden)
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-	<math>10000x-10x = 9990x</math>, ~~und also~~ ist	+	<math>10000x-10x = 9990x</math>, daher ist
	::<math>9990x = 1999\quad\Leftrightarrow\quad x = \frac{1999}{9990}\,\mbox{.}</math>		::<math>9990x = 1999\quad\Leftrightarrow\quad x = \frac{1999}{9990}\,\mbox{.}</math>
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	<!--<center> [[Image:1_1_7c-1(2).gif]] </center>		<!--<center> [[Image:1_1_7c-1(2).gif]] </center>
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Tekbot: hat „Solution 1.1:7c“ nach „Lösung 1.1:7c“ verschoben: Robot: moved page

2008-10-22T13:24:24Z

hat „Solution 1.1:7c“ nach „Lösung 1.1:7c“ verschoben: Robot: moved page

← Nächstältere Version

Version vom 13:24, 22. Okt. 2008

Martin um 14:02, 18. Okt. 2008

2008-10-18T14:02:48Z

← Nächstältere Version		Version vom 14:02, 18. Okt. 2008
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-	~~If we look more closely at this number, we see that the combination~~ 001 ~~is repeated from the second decimal place onwards,~~	+	Man sieht dass die Zahl eine periodische Dezimalbruchentwicklung von 001 hat.
	<center><math>0\textrm{.}2\ \underline{001}\ \underline{001}\ \underline{001}\,\ldots</math></center>		<center><math>0\textrm{.}2\ \underline{001}\ \underline{001}\ \underline{001}\,\ldots</math></center>
-	~~and this reveals that the number is~~ rational.	+	und also ist sie rational.
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	~~By multiplying a certain number of times by~~ 10 ~~we can move the decimal place step by step to the right~~	+	Indem wir die Zahl mit 10 multiplizieren, können wir das komme Schritt für Schritt nach rechts verschieben
	::<math>\insteadof[right]{10000x}{x}{}=0\,\textrm{.}\,2\ 001\ 001\ 001\,\ldots</math>		::<math>\insteadof[right]{10000x}{x}{}=0\,\textrm{.}\,2\ 001\ 001\ 001\,\ldots</math>
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	::<math>\insteadof[right]{10000x}{10000x}{}=2001\,\textrm{.}\,\underline{001}\ \underline{001}\ 1\ldots</math>		::<math>\insteadof[right]{10000x}{10000x}{}=2001\,\textrm{.}\,\underline{001}\ \underline{001}\ 1\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	~~In this list, we see that~~ 10''x'' ~~and~~ 10000''x'' ~~have the same decimal expansion~~, ~~which means that~~	+	Hier sieht man dass 10''x'' und 10000''x'' dieselbe Dezimalbruchentwicklung haben, und also ist
	::<math>10000x-10x = 2001\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots - 2\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots</math>		::<math>10000x-10x = 2001\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots - 2\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	::<math>\phantom{10000x-10x}{} = 1999\,\mbox{.}\quad</math>(~~decimal parts cancel~~)	+	::<math>\phantom{10000x-10x}{} = 1999\,\mbox{.}\quad</math>(die Dezimale kanzellieren)
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	As <math>10000x-10x = 9990x</math> ~~then~~	+	<math>10000x-10x = 9990x</math>, und also ist
	::<math>9990x = 1999\quad\Leftrightarrow\quad x = \frac{1999}{9990}\,\mbox{.}</math>		::<math>9990x = 1999\quad\Leftrightarrow\quad x = \frac{1999}{9990}\,\mbox{.}</math>
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Tek: decimal comma --> decimal point

2008-09-19T07:18:00Z

decimal comma --> decimal point

← Nächstältere Version		Version vom 07:18, 19. Sep. 2008
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	If we look more closely at this number, we see that the combination 001 is repeated from the second decimal place onwards,		If we look more closely at this number, we see that the combination 001 is repeated from the second decimal place onwards,
-	<center><math>0{,}2\ \underline{001}\ \underline{001}\ \underline{001}\,\ldots</math></center>	+	<center><math>0\textrm{.}2\ \underline{001}\ \underline{001}\ \underline{001}\,\ldots</math></center>
	and this reveals that the number is rational.		and this reveals that the number is rational.
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	By multiplying a certain number of times by 10 we can move the decimal place step by step to the right.	+	By multiplying a certain number of times by 10 we can move the decimal place step by step to the right
-	::<math>\insteadof[right]{10000x}{x}{}=0\,,\,2\ 001\ 001\ 001\,\ldots</math>	+	::<math>\insteadof[right]{10000x}{x}{}=0\,\textrm{.}\,2\ 001\ 001\ 001\,\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	::<math>\insteadof[right]{10000x}{10x}{}=2\,,\,\underline{001}\ \underline{001}\ \underline{001}\ 1\ldots</math>	+	::<math>\insteadof[right]{10000x}{10x}{}=2\,\textrm{.}\,\underline{001}\ \underline{001}\ \underline{001}\ 1\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	::<math>\insteadof[right]{10000x}{100x}{}=20\,,\,01\ 001\ 001\ 1\ldots</math>	+	::<math>\insteadof[right]{10000x}{100x}{}=20\,\textrm{.}\,01\ 001\ 001\ 1\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	::<math>\insteadof[right]{10000x}{1000x}{}=200\,,\,1\ 001\ 001\ 1\ldots</math>	+	::<math>\insteadof[right]{10000x}{1000x}{}=200\,\textrm{.}\,1\ 001\ 001\ 1\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	::<math>\insteadof[right]{10000x}{10000x}{}=2001\,,\,\underline{001}\ \underline{001}\ 1\ldots</math>	+	::<math>\insteadof[right]{10000x}{10000x}{}=2001\,\textrm{.}\,\underline{001}\ \underline{001}\ 1\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
	In this list, we see that 10''x'' and 10000''x'' have the same decimal expansion, which means that		In this list, we see that 10''x'' and 10000''x'' have the same decimal expansion, which means that
-	::<math>10000x-10x = 2001{,}\underline{001}\ \underline{001}\ \underline{001}\,\ldots - 2{,}\underline{001}\ \underline{001}\ \underline{001}\,\ldots</math>	+	::<math>10000x-10x = 2001\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots - 2\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots</math>
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	::<math>\phantom{10000x-10x}{} = 1999\,\mbox{.}\quad</math>(decimal parts cancel)		::<math>\phantom{10000x-10x}{} = 1999\,\mbox{.}\quad</math>(decimal parts cancel)

Ian um 14:00, 14. Sep. 2008

2008-09-14T14:00:17Z

← Nächstältere Version		Version vom 14:00, 14. Sep. 2008
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-	~~Tittar vi närmare på talet så ser vi att sifferkombinationen~~ 001 ~~upprepas från och med den andra decimalen~~	+	If we look more closely at this number, we see that the combination 001 is repeated from the second decimal place onwards,
	<center><math>0{,}2\ \underline{001}\ \underline{001}\ \underline{001}\,\ldots</math></center>		<center><math>0{,}2\ \underline{001}\ \underline{001}\ \underline{001}\,\ldots</math></center>
-	~~och det avslöjar att talet är rationellt~~.	+	and this reveals that the number is rational.
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	~~Genom att sedan multiplicera talet med~~ 10 ~~ett antal gånger kan vi skifta decimalkommat stegvis åt höger~~	+	By multiplying a certain number of times by 10 we can move the decimal place step by step to the right.
	::<math>\insteadof[right]{10000x}{x}{}=0\,,\,2\ 001\ 001\ 001\,\ldots</math>		::<math>\insteadof[right]{10000x}{x}{}=0\,,\,2\ 001\ 001\ 001\,\ldots</math>
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	::<math>\insteadof[right]{10000x}{10000x}{}=2001\,,\,\underline{001}\ \underline{001}\ 1\ldots</math>		::<math>\insteadof[right]{10000x}{10000x}{}=2001\,,\,\underline{001}\ \underline{001}\ 1\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	~~I denna lista ser vi att talen~~ 10''x'' ~~och~~ 10000''x'' ~~har samma decimalutveckling~~, ~~och det betyder att~~	+	In this list, we see that 10''x'' and 10000''x'' have the same decimal expansion, which means that
	::<math>10000x-10x = 2001{,}\underline{001}\ \underline{001}\ \underline{001}\,\ldots - 2{,}\underline{001}\ \underline{001}\ \underline{001}\,\ldots</math>		::<math>10000x-10x = 2001{,}\underline{001}\ \underline{001}\ \underline{001}\,\ldots - 2{,}\underline{001}\ \underline{001}\ \underline{001}\,\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	::<math>\phantom{10000x-10x}{} = 1999\,\mbox{.}\quad</math>(~~decimalerna tar ut varandra~~)	+	::<math>\phantom{10000x-10x}{} = 1999\,\mbox{.}\quad</math>(decimal parts cancel)
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	~~Eftersom~~ <math>10000x-10x = 9990x</math> ~~så är~~	+	As <math>10000x-10x = 9990x</math> then
	::<math>9990x = 1999\quad\Leftrightarrow\quad x = \frac{1999}{9990}\,\mbox{.}</math>		::<math>9990x = 1999\quad\Leftrightarrow\quad x = \frac{1999}{9990}\,\mbox{.}</math>
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Tekbot: Lösning 1.1:7c moved to Solution 1.1:7c: Robot: moved page

2008-09-09T08:14:03Z

Lösning 1.1:7c moved to Solution 1.1:7c: Robot: moved page

← Nächstältere Version

Version vom 08:14, 9. Sep. 2008

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

2008-08-21T06:22:49Z

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

← Nächstältere Version		Version vom 06:22, 21. Aug. 2008
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	::<math>9990x = 1999\quad\Leftrightarrow\quad x = \frac{1999}{9990}\,\mbox{.}</math>		::<math>9990x = 1999\quad\Leftrightarrow\quad x = \frac{1999}{9990}\,\mbox{.}</math>
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-	<!--<center> [[~~Bild~~:1_1_7c-1(2).gif]] </center>	+	<!--<center> [[Image:1_1_7c-1(2).gif]] </center>
-	<center> [[~~Bild~~:1_1_7c-2(2).gif]] </center>-->	+	<center> [[Image:1_1_7c-2(2).gif]] </center>-->

Tek um 08:16, 29. Apr. 2008

2008-04-29T08:16:53Z

← Nächstältere Version		Version vom 08:16, 29. Apr. 2008
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-	<center> ~~[[Bild:1_1_7c-1(~~2).~~gif~~]] </~~center~~>	+	Tittar vi närmare på talet så ser vi att sifferkombinationen 001 upprepas från och med den andra decimalen
-	<~~center~~> [[~~Bild~~:~~1_1_7c~~-2(2).~~gif]]~~ </~~center~~>	+	<center><math>0{,}2\ \underline{001}\ \underline{001}\ \underline{001}\,\ldots</math></center>
		+	och det avslöjar att talet är rationellt.
		+	{{NAVCONTENT_STEP}}
		+	Genom att sedan multiplicera talet med 10 ett antal gånger kan vi skifta decimalkommat stegvis åt höger
		+	::<math>\insteadof[right]{10000x}{x}{}=0\,,\,2\ 001\ 001\ 001\,\ldots</math>
		+	{{NAVCONTENT_STEP}}
		+	::<math>\insteadof[right]{10000x}{10x}{}=2\,,\,\underline{001}\ \underline{001}\ \underline{001}\ 1\ldots</math>
		+	{{NAVCONTENT_STEP}}
		+	::<math>\insteadof[right]{10000x}{100x}{}=20\,,\,01\ 001\ 001\ 1\ldots</math>
		+	{{NAVCONTENT_STEP}}
		+	::<math>\insteadof[right]{10000x}{1000x}{}=200\,,\,1\ 001\ 001\ 1\ldots</math>
		+	{{NAVCONTENT_STEP}}
		+	::<math>\insteadof[right]{10000x}{10000x}{}=2001\,,\,\underline{001}\ \underline{001}\ 1\ldots</math>
		+	{{NAVCONTENT_STEP}}
		+	I denna lista ser vi att talen 10''x'' och 10000''x'' har samma decimalutveckling, och det betyder att
		+	::<math>10000x-10x = 2001{,}\underline{001}\ \underline{001}\ \underline{001}\,\ldots - 2{,}\underline{001}\ \underline{001}\ \underline{001}\,\ldots</math>
		+	{{NAVCONTENT_STEP}}
		+	::<math>\phantom{10000x-10x}{} = 1999\,\mbox{.}\quad</math>(decimalerna tar ut varandra)
		+	{{NAVCONTENT_STEP}}
		+	Eftersom <math>10000x-10x = 9990x</math> så är
		+	::<math>9990x = 1999\quad\Leftrightarrow\quad x = \frac{1999}{9990}\,\mbox{.}</math>
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		+	<!--<center> [[Bild:1_1_7c-1(2).gif]] </center>
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2008-03-20T11:27:59Z

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