【数学A】例題4.2.12：記数法（One More）★

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% 例題A4.2.12：記数法（One More）★
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(1) $10110_{(2)},543_{(6)}$をそれぞれ10進法で表せ．

(2) 10進法で表された数58を，2進法，3進法，6進法でそれぞれ表せ．

(3) $3.24_{(5)}$を10進法で表せ．

(4) 10進法で表された小数$0.375$を2進法で表せ．

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% 例題A4.2.12の解答（One More）★
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(1)

\begin{align*}
10110_{(2)}&=1\times 2^4+0\times 2^3+1\times 2^2+1\times 2^1+0\times 2^0\\
&=16+0+4+2+0=22,
\end{align*}

\[
543_{(6)}=5\times 6^2+4\times 6^1+3\times 6^0=180+24+3=207
\]

(2)

\begin{align*}
58&=1\times 2^5+1\times 2^4+1\times 2^3+0\times 2^2+1\times 2^1+0\times 2^0=111010_{(2)},\\
58&=2\times 3^3+0\times 3^2+1\times 3^1+1\times 3^0=2011_{(3)},\\
58&=1\times 6^2+3\times 6^1+4\times 6^0=134_{(6)}
\end{align*}

(3) $3.24_{(5)}=3+2\times\frac{1}{5}+4\times\frac{1}{5^2}=3+0.4+0.16=3.56$

(4) $0.375=0\times\frac{1}{2}+1\times\frac{1}{2^2}+1\times\frac{1}{2^3}=0.011_{(2)}$

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% 例題A4.2.12の別解
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$0.375=0.abc\ldots_{(2)}$とおくと，

\[
0.375=a\times\frac{1}{2}+b\times\frac{1}{4}+c\times\frac{1}{8}+\cdots
\]

両辺に2を掛けると，$0.75=a+b\times\frac{1}{2}+c\times\frac{1}{4}+\cdots$より，$a=0$

これを代入して，両辺に2を掛けると，$1.5=b+c\times\frac{1}{2}+d\times\frac{1}{4}+\cdots$より，$b=1$

これを代入して，両辺に2を掛けると，$1=c+d\times\frac{1}{2}+e\times\frac{1}{4}+\cdots$より，$c=1$

よって，$0.375=0.011_{(2)}$

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% 問題A4.2.12：記数法（One More）★
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(1) $11001_{(2)},354_{(6)}$をそれぞれ10進法で表せ．

(2) 10進法で表された数42を，2進法，3進法，6進法でそれぞれ表せ．

(3) $2.13_{(5)}$を10進法で表せ．

(4) 10進法で表された小数$0.625$を2進法で表せ．

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% 問題A4.2.12の解答（One More）★
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(1)

\[
11001_{(2)}=1\times 2^4+1\times 2^3+0\times 2^2+0\times 2^1+1\times 2^0=25,
\]

\[
354_{(6)}=3\times 6^2+5\times 6^1+4\times 6^0=142
\]

(2)

\begin{align*}
42&=1\times 2^5+0\times 2^4+1\times 2^3+0\times 2^2+1\times 2^1+0\times 2^0=101010_{(2)},\\
42&=1\times 3^3+1\times 3^2+2\times 3^1+0\times 3^0=1120_{(3)},\\
42&=1\times 6^2+1\times 6^1+0\times 6^0=110_{(6)}
\end{align*}

(3) $2.13_{(5)}=2+1\times\frac{1}{5}+3\times\frac{1}{5^2}=2+0.2+0.12=2.32$

(4) $0.625=1\times\frac{1}{2}+0\times\frac{1}{2^2}+1\times\frac{1}{2^3}=0.101_{(2)}$

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% 問題A4.2.12の別解
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$0.625=0.abc\ldots_{(2)}$とおくと，

\[
0.625=a\times\frac{1}{2}+b\times\frac{1}{4}+c\times\frac{1}{8}+\cdots
\]

両辺に2を掛けると，$1.25=a+b\times\frac{1}{2}+c\times\frac{1}{4}+\cdots$より，$a=1$

これを代入して，両辺に2を掛けると，$0.5=b+c\times\frac{1}{2}+d\times\frac{1}{4}+\cdots$より，$b=0$

これを代入して，両辺に2を掛けると，$1=c+d\times\frac{1}{2}+e\times\frac{1}{4}+\cdots$より，$c=1$

よって，$0.625=0.101_{(2)}$

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% 例題A4.2.12の側注：記数法（One More）★
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% 側注1
58を$2^5$で割り，その余りを$2^4$で割り，$\cdots$という操作を繰り返す．

% 側注2
2を掛ける操作を繰り返し，整数部分を取り出してその数字を各位の数にする．

\[
\begin{array}{r}0.375\\\times 2\\\hline 0.750\\\times 2\\\hline 1.50\\\times 2\\\hline 1.0\end{array}
\]

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% 問題A4.2.12の側注：記数法（One More）★
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% 側注1
42を$2^5$で割り，その余りを$2^4$で割り，$\cdots$という操作を繰り返す．

% 側注2
2を掛ける操作を繰り返し，整数部分を取り出してその数字を各位の数にする．

\[
\begin{array}{r}0.625\\\times 2\\\hline 1.25\\\times 2\\\hline 0.50\\\times 2\\\hline 1.0\end{array}
\]

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% 例題A4.2.12の考え方・補足など：記数法（One More）★
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% 考え方
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$n$進法で$a_ka_{k-1}\ldots a_2a_1a_0$と表されるとき，10進法では$a_kn^k+a_{k-1}n^{k-1}+\cdots+a_1n+a_0$と表されることを利用する．

(2) 解答では，例えば58を2進法で表すには，58を$2^5$で割り，その余りを$2^4$で割り，$\cdots$という操作を繰り返すことによって求めている．なお，10進法で表された自然数を$n$進法で表すには，右のように筆算を用いて，商が割る数より小さくなるまで$n$で割る操作を繰り返し，最後の商と余りを下から順に並べることでも求めることができる．

(4) 別解のように，$0.375$に2を掛けて，整数部分を取り出し，残った小数部分に2を掛けて，整数部分を取り出し，$\cdots$という操作を繰り返して，取り出した整数部分を順に並べることでも求めることができる．