Solution 1.1:7b - Revision history

Tek: decimal comma --> decimal point

2008-09-19T07:13:55Z

decimal comma --> decimal point

←Older revision		Revision as of 07:13, 19 September 2008
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	A rational number always has a decimal expansion which, after a certain decimal place, repeats itself periodically.		A rational number always has a decimal expansion which, after a certain decimal place, repeats itself periodically.
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	In our case, the sequence is repeated indefinitely.	+	In our case, the sequence 1416 is repeated indefinitely
-	<center><math>3{,}\underline{1416}\ \underline{1416}\ \underline{1416}\,\ldots</math></center>	+	<center><math>3\textrm{.}\underline{1416}\ \underline{1416}\ \underline{1416}\,\ldots</math></center>
	In other words, the number is rational.		In other words, the number is rational.
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
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	If we write		If we write
-	::<math>\insteadof[right]{10000x}{x}{} = 3\,\~~color~~{~~red}{~~}\,\underline{1416}\ \underline{1416}\ \underline{1416}\,\ldots</math>	+	::<math>\insteadof[right]{10000x}{x}{} = 3\,\textrm{.}\,\underline{1416}\ \underline{1416}\ \underline{1416}\,\ldots</math>
	then		then
-	::<math>\insteadof[right]{10000x}{10x}{} = 31\,\~~color~~{~~red}{~~}\,4161\ 4161\ 4161\,\ldots</math>	+	::<math>\insteadof[right]{10000x}{10x}{} = 31\,\textrm{.}\,4161\ 4161\ 4161\,\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	::<math>\insteadof[right]{10000x}{100x}{} = 314\,\~~color~~{~~red}{~~}\,1614\ 1614\ 161\,\ldots</math>	+	::<math>\insteadof[right]{10000x}{100x}{} = 314\,\textrm{.}\,1614\ 1614\ 161\,\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	::<math>\insteadof[right]{10000x}{1000x}{} = 3141\,\~~color~~{~~red}{~~}\,6141\ 6141\ 61\,\ldots</math>	+	::<math>\insteadof[right]{10000x}{1000x}{} = 3141\,\textrm{.}\,6141\ 6141\ 61\,\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	::<math>\insteadof[right]{10000x}{10000x}{} = 31416\,\~~color~~{~~red}{~~}\,\underline{1416}\ \underline{1416}\ 1\,\ldots</math>	+	::<math>\insteadof[right]{10000x}{10000x}{} = 31416\,\textrm{.}\,\underline{1416}\ \underline{1416}\ 1\,\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	Note that, in 10000''x'' we have moved the decimal point sufficiently many places so that the decimal	+	Note that, in 10000''x'' we have moved the decimal point sufficiently many places so that the decimal expansion of 10000''x'' is in phase with the decimal expansion of ''x'', i.e. they have the same
-	expansion of	+
-	10000''x'' ~~has come~~ in phase with the decimal expansion of ''x'', i.e. they have the same	+
	decimal expansion.		decimal expansion.
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
	Therefore,		Therefore,
-	::<math>10000x-x = 31416\,{,}\,\underline{1416}\ \underline{1416}\,\ldots - 3\,{,}\,\underline{1416}\ \underline{1416}\,\ldots</math>	+	::<math>10000x-x = 31416\,\textrm{.}\,\underline{1416}\ \underline{1416}\,\ldots - 3\,\textrm{.}\,\underline{1416}\ \underline{1416}\,\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
	::<math>\phantom{10000x-x}{}= 31413\quad</math>(The decimal parts cancel out each other)		::<math>\phantom{10000x-x}{}= 31413\quad</math>(The decimal parts cancel out each other)

Ian at 13:54, 14 September 2008

2008-09-14T13:54:13Z

←Older revision		Revision as of 13:54, 14 September 2008
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-	~~Ett rationellt tal har alltid en decimalutveckling som från och med en viss~~ decimal ~~upprepar sig periodiskt~~.	+	A rational number always has a decimal expansion which, after a certain decimal place, repeats itself periodically.
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	~~I vårt fall så upprepas sekvensen 1416 i all oändlighet~~	+	In our case, the sequence is repeated indefinitely.
	<center><math>3{,}\underline{1416}\ \underline{1416}\ \underline{1416}\,\ldots</math></center>		<center><math>3{,}\underline{1416}\ \underline{1416}\ \underline{1416}\,\ldots</math></center>
-	~~Med andra ord är talet rationellt~~.	+	In other words, the number is rational.
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}

-	~~Nästa~~ problem ~~är att skriva om talet som ett bråktal och då utnyttjar vi att multiplikation med~~ 10 ~~flyttar decimalkommat ett steg åt höger~~.	+	The next problem is to rewrite the number as a fraction, for which we use the fact that multiplication by 10 moves the decimal point one place to the right.
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	~~Om vi skriver~~	+	If we write
	::<math>\insteadof[right]{10000x}{x}{} = 3\,\color{red}{}\,\underline{1416}\ \underline{1416}\ \underline{1416}\,\ldots</math>		::<math>\insteadof[right]{10000x}{x}{} = 3\,\color{red}{}\,\underline{1416}\ \underline{1416}\ \underline{1416}\,\ldots</math>
-	~~så är därför~~	+	then
	::<math>\insteadof[right]{10000x}{10x}{} = 31\,\color{red}{}\,4161\ 4161\ 4161\,\ldots</math>		::<math>\insteadof[right]{10000x}{10x}{} = 31\,\color{red}{}\,4161\ 4161\ 4161\,\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
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	::<math>\insteadof[right]{10000x}{10000x}{} = 31416\,\color{red}{}\,\underline{1416}\ \underline{1416}\ 1\,\ldots</math>		::<math>\insteadof[right]{10000x}{10000x}{} = 31416\,\color{red}{}\,\underline{1416}\ \underline{1416}\ 1\,\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	~~Notera att i~~ 10000''x'' ~~har vi flyttat decimalkommat tillräckligt många steg så att decimalutvecklingen av~~ 10000''x'' ~~har kommit i fas med decimalutvecklingen av~~ ''x'', ~~dvs~~. ~~de har samma decimalutveckling~~.	+	Note that, in 10000''x'' we have moved the decimal point sufficiently many places so that the decimal
		+	expansion of
		+	10000''x'' has come in phase with the decimal expansion of ''x'', i.e. they have the same
		+	decimal expansion.
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	~~Därför är~~	+	Therefore,
	::<math>10000x-x = 31416\,{,}\,\underline{1416}\ \underline{1416}\,\ldots - 3\,{,}\,\underline{1416}\ \underline{1416}\,\ldots</math>		::<math>10000x-x = 31416\,{,}\,\underline{1416}\ \underline{1416}\,\ldots - 3\,{,}\,\underline{1416}\ \underline{1416}\,\ldots</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	::<math>\phantom{10000x-x}{}= 31413\quad</math>(~~decimalerna tar ut varandra~~)	+	::<math>\phantom{10000x-x}{}= 31413\quad</math>(The decimal parts cancel out each other)
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	~~och eftersom~~ <math>10000x-x = 9999x</math> ~~så har vi alltså sambandet~~	+	and as <math>10000x-x = 9999x</math> we get that
	::<math>9999x = 31413\,\mbox{.}</math>		::<math>9999x = 31413\,\mbox{.}</math>
	{{NAVCONTENT_STEP}}		{{NAVCONTENT_STEP}}
-	~~Löser vi ut~~ ''x'' ~~ur detta samband får vi~~ ''x'' ~~som en kvot mellan två heltal~~	+	Solving for ''x'' in this relationship we find ''x'' as a quotient between two integers
	::<math>x = \frac{31413}{9999}\quad\biggl({}= \frac{10471}{3333}\biggr)\,\mbox{.}</math>		::<math>x = \frac{31413}{9999}\quad\biggl({}= \frac{10471}{3333}\biggr)\,\mbox{.}</math>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
	<!--<center> [[Image:1_1_7b-1(2).gif]] </center>		<!--<center> [[Image:1_1_7b-1(2).gif]] </center>
	<center> [[Image:1_1_7b-2(2).gif]] </center>-->		<center> [[Image:1_1_7b-2(2).gif]] </center>-->

Tekbot: Lösning 1.1:7b moved to Solution 1.1:7b: Robot: moved page

2008-09-09T08:13:43Z

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

←Older revision

Revision as of 08:13, 9 September 2008

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

2008-08-21T06:22:29Z

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

←Older revision		Revision as of 06:22, 21 August 2008
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	::<math>x = \frac{31413}{9999}\quad\biggl({}= \frac{10471}{3333}\biggr)\,\mbox{.}</math>		::<math>x = \frac{31413}{9999}\quad\biggl({}= \frac{10471}{3333}\biggr)\,\mbox{.}</math>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
-	<!--<center> [[~~Bild~~:1_1_7b-1(2).gif]] </center>	+	<!--<center> [[Image:1_1_7b-1(2).gif]] </center>
-	<center> [[~~Bild~~:1_1_7b-2(2).gif]] </center>-->	+	<center> [[Image:1_1_7b-2(2).gif]] </center>-->

Tek at 08:14, 29 April 2008

2008-04-29T08:14:02Z

←Older revision		Revision as of 08:14, 29 April 2008
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	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:~~1_1_7b-1(2).gif~~]] </~~center~~>	+	Ett rationellt tal har alltid en decimalutveckling som från och med en viss decimal upprepar sig periodiskt.
-	<~~center~~> [~~[Bild~~:~~1_1_7b~~-2(2).~~gif]]~~ </~~center~~>	+	{{NAVCONTENT_STEP}}
		+	I vårt fall så upprepas sekvensen 1416 i all oändlighet
		+	<center><math>3{,}\underline{1416}\ \underline{1416}\ \underline{1416}\,\ldots</math></center>
		+	Med andra ord är talet rationellt.
		+	{{NAVCONTENT_STEP}}
		+
		+	Nästa problem är att skriva om talet som ett bråktal och då utnyttjar vi att multiplikation med 10 flyttar decimalkommat ett steg åt höger.
		+	{{NAVCONTENT_STEP}}
		+	Om vi skriver
		+	::<math>\insteadof[right]{10000x}{x}{} = 3\,\color{red}{}\,\underline{1416}\ \underline{1416}\ \underline{1416}\,\ldots</math>
		+	så är därför
		+	::<math>\insteadof[right]{10000x}{10x}{} = 31\,\color{red}{}\,4161\ 4161\ 4161\,\ldots</math>
		+	{{NAVCONTENT_STEP}}
		+	::<math>\insteadof[right]{10000x}{100x}{} = 314\,\color{red}{}\,1614\ 1614\ 161\,\ldots</math>
		+	{{NAVCONTENT_STEP}}
		+	::<math>\insteadof[right]{10000x}{1000x}{} = 3141\,\color{red}{}\,6141\ 6141\ 61\,\ldots</math>
		+	{{NAVCONTENT_STEP}}
		+	::<math>\insteadof[right]{10000x}{10000x}{} = 31416\,\color{red}{}\,\underline{1416}\ \underline{1416}\ 1\,\ldots</math>
		+	{{NAVCONTENT_STEP}}
		+	Notera att i 10000''x'' har vi flyttat decimalkommat tillräckligt många steg så att decimalutvecklingen av 10000''x'' har kommit i fas med decimalutvecklingen av ''x'', dvs. de har samma decimalutveckling.
		+	{{NAVCONTENT_STEP}}
		+	Därför är
		+	::<math>10000x-x = 31416\,{,}\,\underline{1416}\ \underline{1416}\,\ldots - 3\,{,}\,\underline{1416}\ \underline{1416}\,\ldots</math>
		+	{{NAVCONTENT_STEP}}
		+	::<math>\phantom{10000x-x}{}= 31413\quad</math>(decimalerna tar ut varandra)
		+	{{NAVCONTENT_STEP}}
		+	och eftersom <math>10000x-x = 9999x</math> så har vi alltså sambandet
		+	::<math>9999x = 31413\,\mbox{.}</math>
		+	{{NAVCONTENT_STEP}}
		+	Löser vi ut ''x'' ur detta samband får vi ''x'' som en kvot mellan två heltal
		+	::<math>x = \frac{31413}{9999}\quad\biggl({}= \frac{10471}{3333}\biggr)\,\mbox{.}</math>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
		+	<!--<center> [[Bild:1_1_7b-1(2).gif]] </center>
		+	<center> [[Bild:1_1_7b-2(2).gif]] </center>-->

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2008-03-20T11:27:07Z

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