問52
頻出度★★☆
2進数 1.101 を 10 進数で表現したものはどれか。
平成22年度 春期 ITパスポート試験 問52
テクノロジ系基礎理論 › 離散数学
This question's explanation and your answers so far.
解説
問 52
Incorrectア
1.2 is wrong. The values that can be expressed by a 3-digit binary fraction are limited to integer multiples of 0.125 (= 2^-3), so only values in steps of 0.125, such as 1.000, 1.125, 1.250, and so on, can be formed. Since 1.2 does not fall on one of these steps, it is a value that cannot be represented at all by a finite binary fraction like 1.101. Conversely, converting 1.2 into binary gives 1.0011001100..., which repeats and never terminates in a finite number of digits.
Incorrectイ
1.5 is the decimal value of the binary number 1.1 (= 1 + 0.5); it is the answer you get when you fail to count the 1 (= 0.125) in the third fractional place. Because 1.101 has a 1 in the third fractional place, 0.125 must be added, so 1.5 falls short by exactly 0.125.
Incorrectウ
1.505 is the value that comes out when you mistakenly divide the weights of the fractional part by 10 each time (0.5, 0.05, 0.005) instead of by 2 each time (0.5, 0.25, 0.125). That calculation gives 1 + 0.5 + 0 + 0.005 = 1.505, but the base of the place values in binary is 2, not 10, so this weighting is incorrect.
- エ
正解 The correct answer is エ, 1.625. In a binary fraction, the digits carry the weights (place values) 2^0, 2^-1, 2^-2, 2^-3, and so on, starting from the digit to the left of the binary point. In other words, each digit to the right of the point is worth half the one before it: 0.5, 0.25, 0.125. Breaking 1.101 down digit by digit: the 1 in the integer part is 1×2^0 = 1, the 1 in the first fractional place is 1×2^-1 = 0.5, the 0 in the second fractional place is 0×2^-2 = 0, and the 1 in the third fractional place is 1×2^-3 = 0.125. Adding these together gives 1 + 0.5 + 0 + 0.125 = 1.625. Checking with fractions gives the same result: 1 + 1/2 + 1/8 = 8/8 + 4/8 + 1/8 = 13/8 = 1.625.
ここがポイント
- Place values in binary fractions: the weights of the digits to the right of the binary point are 2^-1 = 0.5, 2^-2 = 0.25, 2^-3 = 0.125, halving with each digit. To convert to decimal, simply add up the weights of the digits whose value is 1.
- How radix conversion works: in base n, each digit weight is a power of the radix n, and the fractional part goes n^-1, n^-2, and so on. Keeping a feel for the fact that a binary fraction with up to three fractional digits can only express values in steps of 0.125 lets you narrow down the choices quickly.
用語