2的补码是表示负数的一种方式，大多数控制器和处理器都以2的补码形式存储负数。

补一词来源于完备性。在十进制世界中，数字0到9提供了一个数字或数字符号的补集(完整集)来表示所有的十进制数。在二进制世界中，数字0和1提供了一个数字的补数来表示所有二进制数。事实上，符号0和1必须用来表示所有东西(文本、图像等)以及正(0)和负(1)。 在我们的世界里，数字左边的空白被认为是零:

                  35=035=000000035.

In a computer storage location there is no blank space. All bits (binary digits) must be either 0 or 1. To efficiently use memory numbers may be stored as 8 bit, 16 bit, 32 bit, 64 bit, 128 bit representations. When a number that is stored as an 8 bit number is transferred to a 16 bit location the sign and magnitude (absolute value) must remain the same. Both 1's complement and 2's complement representations facilitate this. As a noun: Both 1's complement and 2's complement are binary representations of signed quantities where the most significant bit (the one on the left) is the sign bit. 0 is for positive and 1 is for negative. 2s complement does not mean negative. It means a signed quantity. As in decimal the magnitude is represented as the positive quantity. The structure uses sign extension to preserve the quantity when promoting to a register [] with more bits:

       [0101]=[00101]=[00000000000101]=5 (base 10)
       [1011]=[11011]=[11111111111011]=-5(base 10)

用作动词: 2的补语表示否定。这并不意味着消极。意思是如果负数变成正数;如果是正的就是负的。大小是绝对值:

        if a >= 0 then |a| = a
        if a < 0 then |a| = -a = 2scomplement of a

此功能允许使用先求负后加的有效二进制减法。 A -b = A + (-b)

1的补数的官方方法是每一位数用1减去它的值。

        1'scomp(0101) = 1010.

这与逐个翻转或反转每一位是一样的。结果是- 0，这是不受欢迎的，所以给te 1的补码加上1就解决了这个问题。 要求2s的补，先求1s的补，然后加1。

        Example 1                             Example 2
         0101  --original number              1101
         1's comp  1010                       0010
         add 1     0001                       0001
         2's comp  1011  --negated number     0011

在这些例子中，否定也适用于符号扩展数。

添加: 1110进位111110进位 0110与000110相同 1111年 111111年 Sum 0101 Sum 000101

减法:

    1110  Carry                      00000   Carry
     0110          is the same as     00110
    -0111                            +11001
  ----------                        ----------
sum  0101                       sum   11111

请注意，当使用2的补码时，数字左侧的空白区域对于正数用0填充，而对于负数用1填充。进位总是被加上，必须是1或0。

干杯

这是一种对负整数进行编码的聪明方法，该方法将数据类型中大约一半的位组合保留给负整数，并且将大多数负整数与其对应的正整数相加会导致进位溢出，使结果为二进制零。

因此，在2的补码中，如果1是0x0001，那么-1是0x1111，因为这将导致0x0000的组合和(溢出1)。

2's complement is essentially a way of coming up with the additive inverse of a binary number. Ask yourself this: Given a number in binary form (present at a fixed length memory location), what bit pattern, when added to the original number (at the fixed length memory location), would make the result all zeros ? (at the same fixed length memory location). If we could come up with this bit pattern then that bit pattern would be the -ve representation (additive inverse) of the original number; as by definition adding a number to its additive inverse always results in zero. Example: take 5 which is 101 present inside a single 8 bit byte. Now the task is to come up with a bit pattern which when added to the given bit pattern (00000101) would result in all zeros at the memory location which is used to hold this 5 i.e. all 8 bits of the byte should be zero. To do that, start from the right most bit of 101 and for each individual bit, again ask the same question: What bit should I add to the current bit to make the result zero ? continue doing that taking in account the usual carry over. After we are done with the 3 right most places (the digits that define the original number without regard to the leading zeros) the last carry goes in the bit pattern of the additive inverse. Furthermore, since we are holding in the original number in a single 8 bit byte, all other leading bits in the additive inverse should also be 1's so that (and this is important) when the computer adds "the number" (represented using the 8 bit pattern) and its additive inverse using "that" storage type (a byte) the result in that byte would be all zeros.

 1 1 1
 ----------
   1 0 1
 1 0 1 1 ---> additive inverse
  ---------
   0 0 0

就像我看到的大多数解释一样，上面的解释清楚地说明了如何使用2的补码，但并没有真正解释它们在数学上是什么。我会试着这么做，至少对整数来说是这样的，我会先介绍一些你们可能熟悉的背景知识。

回想一下它是如何用于十进制的:2345是2 × 103 + 3 × 102 + 4 × 101 + 5 × 100的一种写法。

同样地，二进制是一种只使用0和1来写数字的方法，遵循相同的思路，但把上面的10换成了2。然后在二进制中，1111是1 × 23 + 1 × 22 + 1 × 21 + 1 × 20的一种写法，如果你算出来，结果等于15(以10为底)。因为8+4+2+1 = 15。

这对于正数来说很好。它甚至适用于负数，如果你愿意在负数前面加一个负号，就像人类对待小数一样。在某种程度上，这甚至可以在计算机上完成，但我从20世纪70年代初就没见过这样的计算机了。我将把原因留到另一个讨论。

对于计算机来说，负数使用补表示法效率更高。这里有一些经常被忽视的东西。补表示法涉及到数字数字的某种反转，甚至是在正常正数之前隐含的零。这很尴尬，因为问题来了:所有这些?这可能是一个无限的数字要考虑。

幸运的是，计算机并不代表无穷。数字被限制在特定的长度(或者宽度，如果你喜欢)。所以让我们回到正二进制数，但有一个特定的大小。在这些例子中，我将使用8个数字(“位”)。所以我们的二进制数应该是00001111或者0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20

为了形成2的补负，我们首先将所有的(二进制)数字补成11110000，然后加上1，形成11110001，但我们如何理解这意味着-15?

The answer is that we change the meaning of the high-order bit (the leftmost one). This bit will be a 1 for all negative numbers. The change will be to change the sign of its contribution to the value of the number it appears in. So now our 11110001 is understood to represent  -1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 1 × 20Notice that "-" in front of that expression? It means that the sign bit carries the weight -27, that is -128 (base 10). All the other positions retain the same weight they had in unsigned binary numbers.

算出-15，就是-128 + 64 + 32 + 16 + 1用计算器试试。它是-15。

Of the three main ways that I've seen negative numbers represented in computers, 2's complement wins hands down for convenience in general use. It has an oddity, though. Since it's binary, there have to be an even number of possible bit combinations. Each positive number can be paired with its negative, but there's only one zero. Negating a zero gets you zero. So there's one more combination, the number with 1 in the sign bit and 0 everywhere else. The corresponding positive number would not fit in the number of bits being used.

关于这个数字更奇怪的是，如果你试图通过互补和加1来形成正数，你会得到相同的负数。0会这样做似乎很自然，但这是出乎意料的，完全不是我们习惯的行为，因为除了计算机，我们通常认为数字是无限供应的，而不是这种固定长度的算术。

这只是怪胎的冰山一角。表面之下还有更多的东西在等待着，但这就足够我们讨论了。如果你研究定点算术中的“溢出”，你可能会发现更多。如果你真的想深入了解它，你可能还会研究“模算术”。

我在Reddit上读到jng的一篇精彩的解释，用里程表做类比。

It is a useful convention. The same circuits and logic operations that add / subtract positive numbers in binary still work on both positive and negative numbers if using the convention, that's why it's so useful and omnipresent. Imagine the odometer of a car, it rolls around at (say) 99999. If you increment 00000 you get 00001. If you decrement 00000, you get 99999 (due to the roll-around). If you add one back to 99999 it goes back to 00000. So it's useful to decide that 99999 represents -1. Likewise, it is very useful to decide that 99998 represents -2, and so on. You have to stop somewhere, and also by convention, the top half of the numbers are deemed to be negative (50000-99999), and the bottom half positive just stand for themselves (00000-49999). As a result, the top digit being 5-9 means the represented number is negative, and it being 0-4 means the represented is positive - exactly the same as the top bit representing sign in a two's complement binary number. Understanding this was hard for me too. Once I got it and went back to re-read the books articles and explanations (there was no internet back then), it turned out a lot of those describing it didn't really understand it. I did write a book teaching assembly language after that (which did sell quite well for 10 years).