floating-point - 如何用二进制表示 float 。 IEEE

Question

类似于十进制二进制数也可以表示 float 。现在我读到它可以有那种花车

0.5:0.1 , 0.25:0.01 , 0.125:0.001 ...等等。但是，例如，0.1(十进制)如何用二进制表示？

此外，给定一个十进制 float ，如何将其转换为等效的十进制数，(假设它不是那么简单)。

编辑:所以我知道更好的问题是；如何将十进制 float 转换为二进制？现在我知道我们将小数部分相乘，直到它变为零。现在很有可能两个浮点可以具有相同的表示，对吧？

Answer 1

给定多少位？

0.1b:

0.00011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011...

如您所见，这是一个近似值。

Binary                                          Decimal
0.1     == 1/2^1         == 1/2              == 0.5
0.01    == 1/2^2         == 1/4              == 0.25
0.11    == 1/2^1 + 1/2^2 == 1/2 + 1/4 == 3/4 == 0.75

radix point 之后的每一位表示 1/2^(position_after_bit_string)。

postion:   |1|2|3|4|5|6|7|
         0.|0|0|0|0|0|0|1|

所以 0.0000001 = 1/2^7 = 0.0078125

伪代码:

decimal_value = 0 
for i, bit in enumerate(binary_string):
    if bit == 1
         decimal_value += 1/2**i

了解更多信息 Why can't decimal numbers be represented exactly in binary?