Solution 1.1:7c

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m (decimal comma --> decimal point)
 
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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,
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<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.
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{{NAVCONTENT_STEP}}
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By multiplying a certain number of times by 10 we can move the decimal place step by step to the right.
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By multiplying a certain number of times by 10 we can move the decimal place step by step to the right
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::<math>\insteadof[right]{10000x}{x}{}=0\,,\,2\ 001\ 001\ 001\,\ldots</math>
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::<math>\insteadof[right]{10000x}{x}{}=0\,\textrm{.}\,2\ 001\ 001\ 001\,\ldots</math>
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::<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>
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::<math>\insteadof[right]{10000x}{100x}{}=20\,,\,01\ 001\ 001\ 1\ldots</math>
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::<math>\insteadof[right]{10000x}{100x}{}=20\,\textrm{.}\,01\ 001\ 001\ 1\ldots</math>
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::<math>\insteadof[right]{10000x}{1000x}{}=200\,,\,1\ 001\ 001\ 1\ldots</math>
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::<math>\insteadof[right]{10000x}{1000x}{}=200\,\textrm{.}\,1\ 001\ 001\ 1\ldots</math>
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::<math>\insteadof[right]{10000x}{10000x}{}=2001\,,\,\underline{001}\ \underline{001}\ 1\ldots</math>
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::<math>\insteadof[right]{10000x}{10000x}{}=2001\,\textrm{.}\,\underline{001}\ \underline{001}\ 1\ldots</math>
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{{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
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::<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)

Current revision

If we look more closely at this number, we see that the combination 001 is repeated from the second decimal place onwards,

\displaystyle 0\textrm{.}2\ \underline{001}\ \underline{001}\ \underline{001}\,\ldots

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

\displaystyle \insteadof[right]{10000x}{x}{}=0\,\textrm{.}\,2\ 001\ 001\ 001\,\ldots
\displaystyle \insteadof[right]{10000x}{10x}{}=2\,\textrm{.}\,\underline{001}\ \underline{001}\ \underline{001}\ 1\ldots
\displaystyle \insteadof[right]{10000x}{100x}{}=20\,\textrm{.}\,01\ 001\ 001\ 1\ldots
\displaystyle \insteadof[right]{10000x}{1000x}{}=200\,\textrm{.}\,1\ 001\ 001\ 1\ldots
\displaystyle \insteadof[right]{10000x}{10000x}{}=2001\,\textrm{.}\,\underline{001}\ \underline{001}\ 1\ldots

In this list, we see that 10x and 10000x have the same decimal expansion, which means that

\displaystyle 10000x-10x = 2001\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots - 2\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots
\displaystyle \phantom{10000x-10x}{} = 1999\,\mbox{.}\quad(decimal parts cancel)

As \displaystyle 10000x-10x = 9990x then

\displaystyle 9990x = 1999\quad\Leftrightarrow\quad x = \frac{1999}{9990}\,\mbox{.}
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