将分数从十进制转换为二进制 [英] Convert fractions from decimal to binary

问题描述

我想将数字8.7转换为二进制.

I want to convert the number 8.7 to binary.

我知道命令

(format nil "~b" (rationalize 8.7)) ===>
1010111/1010

或

(format nil "~b" (/ 87 10))====> 1010111/1010

我们观察是否对商1010111/1010做商得到1000.1011001100110011001100110011.

We observe if we do the quotient binary 1010111/1010 we obtain 1000.1011001100110011001100110011.

有可能在Lisp中获得

Is possible to obtain in Lisp

(8.7)_2 ~ 1000.1011001100110011001100110011?

如果是，怎么办?

推荐答案

"2.718 ..."等于2 * 10^1 + 7 * 10^-1 + 1 * 10^-2 + 8 * 10^-3 ...这意味着您可以通过连接string(n / base^i)的反向过程来生成字符串，其中 i 是字符串的索引，而 n 是仍需要转换为 base 的值.本质上，这是一种贪婪的变更决策算法.

"2.718..." is equal to 2 * 10^1 + 7 * 10^-1 + 1 * 10^-2 + 8 * 10^-3... This means that you can generate the string by the reverse process of concatenating string(n / base^i) where i is the index into the string and n is the value that still needs to be converted to base. It's essentially a greedy change-making algorithm.

以下内容大致可行，但不能保证它会产生确切的IEEE 754分数.它应该与实现的浮动一样准确

The following roughly works, with no guarantee that it produces the exact IEEE 754 fraction. It should be as accurate as your implementation's floats

(defun fractional-binary (fractional &optional (place 1) (result "."))
  (if (zerop fractional)
      result
      (let ((fraction (expt 2 (- place))))
        (if (<= fraction fractional)
            (fractional-binary (- fractional fraction)
                               (1+ place)
                               (concatenate 'string result "1"))
            (fractional-binary fractional
                               (1+ place)
                               (concatenate 'string result "0"))))))

CL-USER> (fractional-binary .7)
".101100110011001100110011"

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