为什么8位字节中的〜1等于-2？

Question

I would expect that a, due to a NOT 00000001 would turn into 11111110, otherwise known as 127, or -126 if counting the far left bit as the sign, if sign&magnitude was used.

Even in the instance of 2s compliment, I would expect the answer to result in -127

Why is it that the result is -2?

Answer 1

In two's complement:

-x = ~x + 1

By subtracting one from both sides we can see that:

~x = -x - 1

And so in your example, if we set x = 1 we get:

~1 = -1 - 1 = -2

Answer 2

Consider how the numbers wrap around.

If we start with 00000010 (2) and take away one then it is:

  00000010
- 00000001
 ---------
  00000001

Which is 1. We "borrow 1" from the column to the left just as we do with decimal subtraction, except that because it's binary 10 - 1 is 1 rather than 9 .

Take 1 away again and we of course get zero:

  00000001
- 00000001
 ---------
  00000000

Now, take 1 away from that, and we're borrowing 1 from the column to the left every time, and that borrowing wraps us around, so 0 - 1 = -1 is:

  00000000
- 00000001
-----------
  11111111

So -1 is all-ones.

This is even easier to see in the other direction, in that 11111111 plus one must be 00000000 as it keeps carrying one until it is lost to the left, so if x is 11111111 then it must be the case that x + 1 == 0 , so it must be -1.

Take away another one and we have:

  11111111
- 00000001
  --------
  11111110

So -2 is 1111110 , and of course ~1 means flipping every bit of 00000001 , which is also 11111110 . So ~1 must be -2 .

Another factor to note here though is that arithmetic and complements in C# always converts up to int for anything smaller. For a byte the value 11111110 is 254 , but because ~ casts up to int first you get -2 rather than 254.

byte b = 1;
var i = ~b; // i is an int, and is -2
b = unchecked((byte)~b); // Forced back into a byte, now is 254

Answer 3

To convert a negative 2-compliment number to its decimal representation we have to:

• start scanning the bitstring from right to left, until the first '1' is encountered
• start inverting every bit to the left of that first '1'

Thus, in 11111110 we see the sign bit is 1 (negative number), and above method yields the number 000000010 , which is a decimal 2. In total, we thus get -2 .