Processing Math: Done
Solution 1.1:7c
From Förberedande kurs i matematik 1
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{{NAVCONTENT_START}} | {{NAVCONTENT_START}} | ||
- | + | 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> | ||
- | + | 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. | |
::<math>\insteadof[right]{10000x}{x}{}=0\,,\,2\ 001\ 001\ 001\,\ldots</math> | ::<math>\insteadof[right]{10000x}{x}{}=0\,,\,2\ 001\ 001\ 001\,\ldots</math> | ||
{{NAVCONTENT_STEP}} | {{NAVCONTENT_STEP}} | ||
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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}} | ||
- | + | 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>( | + | ::<math>\phantom{10000x-10x}{} = 1999\,\mbox{.}\quad</math>(decimal parts cancel) |
{{NAVCONTENT_STEP}} | {{NAVCONTENT_STEP}} | ||
- | + | 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> | ||
{{NAVCONTENT_STOP}} | {{NAVCONTENT_STOP}} | ||
<!--<center> [[Image:1_1_7c-1(2).gif]] </center> | <!--<center> [[Image:1_1_7c-1(2).gif]] </center> | ||
<center> [[Image:1_1_7c-2(2).gif]] </center>--> | <center> [[Image:1_1_7c-2(2).gif]] </center>--> |
Revision as of 14:00, 14 September 2008
If we look more closely at this number, we see that the combination 001 is repeated from the second decimal place onwards,

and this reveals that the number is rational.
By multiplying a certain number of times by 10 we can move the decimal place step by step to the right.
x=0 2 001 001 001...
10x=2 001 001 001 1...
100x=20 01 001 001 1...
1000x=200 1 001 001 1...
10000x=2001 001 001 1...
In this list, we see that 10x and 10000x have the same decimal expansion, which means that
10000x−10x=2001 001 001 001...−2
001 001 001...
=1999. (decimal parts cancel)
As
9990x=1999 x=99901999.