【数学I】例題1.2.1：循環小数（One More）★

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% 例題I1.2.1：循環小数（One More）★
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(1) 次の分数を小数の形に直し，循環小数の表し方で書け．

(i) $\frac{1}{3}$

(ii) $\frac{3}{11}$

(iii) $\frac{13}{18}$

(2) 次の循環小数を分数の形で表せ．

(i) $0.\dot{2}$

(ii) $0.\dot{1}\dot{2}$

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% 例題I1.2.1の解答（One More）★
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(1)

(i) $\frac{1}{3}=0.33\ldots=0.\dot{3}$

(ii) $\frac{3}{11}=0.2727\ldots=0.\dot{2}\dot{7}$

(iii) $\frac{13}{18}=0.722\ldots=0.7\dot{2}$

(2)

(i) $x=0.\dot{2}$とおく．右のように計算して，

\[
9x=2
\]

よって，$x=\frac{2}{9}$

\[
\begin{array}{rrl}
&10x=&2.22\dots\\
-&x=&0.22\dots\\
\hline
&9x=&2
\end{array}
\]

(ii) $x=0.\dot{1}\dot{2}$とおく．右のように計算して，

\[
99x=12
\]

よって，$x=\frac{12}{99}=\frac{4}{33}$

\[
\begin{array}{rrl}
&100x=&12.121\dots\\
-&x=&0.121\dots\\
\hline
&99x=&12
\end{array}
\]

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% 問題I1.2.1：循環小数（One More）★
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(1) 次の分数を小数の形に直し，循環小数の表し方で書け．

(i) $\frac{2}{7}$

(ii) $\frac{5}{12}$

(iii) $\frac{7}{15}$

(2) 次の循環小数を分数の形で表せ．

(i) $0.\dot{4}$

(ii) $0.\dot{3}\dot{6}$

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% 問題I1.2.1の解答（One More）★
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(1)

(i) $\frac{2}{7}=0.285714285714\ldots=0.\dot{2}8571\dot{4}$

(ii) $\frac{5}{12}=0.41666\ldots=0.41\dot{6}$

(iii) $\frac{7}{15}=0.46666\ldots=0.4\dot{6}$

(2)

(i) $x=0.\dot{4}$とおく．右のように計算して，

\[
9x=4
\]

よって，$x=\frac{4}{9}$

\[
\begin{array}{rrl}
&10x=&4.44\dots\\
-&x=&0.44\dots\\
\hline
&9x=&4
\end{array}
\]

(ii) $x=0.\dot{3}\dot{6}$とおく．右のように計算して，

\[
99x=36
\]

よって，$x=\frac{36}{99}=\frac{4}{11}$

\[
\begin{array}{rrl}
&100x=&36.363\dots\\
-&x=&0.363\dots\\
\hline
&99x=&36
\end{array}
\]

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% 例題I1.2.1の側注：循環小数（One More）★
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% 側注1
筆算などで割り算を行う．

\[
\begin{array}{rrrrrr}
&&0.&3&3&\ldots\\\cline{2-5}
3&)&1&0&&\\
&&&9&&\\\cline{3-5}
&&&1&0&\\
&&&&9&\\\cline{4-5}
&&&&1&\ldots\\
\end{array}
\]

% 側注2
循環する部分が1桁であるので，両辺を$10^1(=10)$倍する．

% 側注3
循環する部分が2桁であるので，両辺を$10^2(=100)$倍する．なお，答えは既約分数にする．

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% 問題I1.2.1の側注：循環小数（One More）★
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% 側注1
循環する部分が1桁であるので，両辺を$10^1(=10)$倍する．

% 側注2
循環する部分が2桁であるので，両辺を$10^2(=100)$倍する．

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% 例題I1.2.1の考え方・補足など：循環小数（One More）★
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% 考え方
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(1) 割り算を行い，循環する部分の最初と最後の数字の上に$\bullet$をつけて表す．

(2) (i)$x=0.\dot{2}$とおくと，$10x=2.\dot{2}$となり，$10x$と$x$の差を考えることで循環する部分を消すことができる．

(ii) 循環する部分が2桁であるので，$x=0.\dot{1}\dot{2}$とおいて，$10^2x(=100x)$と$x$の差を考える．

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% One Point
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循環小数は，循環する桁数に着目して計算する．

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% 注意
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(2)の$\frac{2}{9},\frac{4}{33}$のように，分母と分子に1以外の公約数がなく，これ以上約分できない分数のことを既約分数という．