[Solved] Convert fractions from decimal to binary

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“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.

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"

solved Convert fractions from decimal to binary