Help understanding Binary Representation for Integers?

I'm trying to understand these illustrations but there are parts which I don't understand:

• "But the computer had to count backwards for the negative numbers"

  • Why does adding a 1 to the front of a binary mean the computer has to count backwards?

• "Flip the bits and add开发者_高级运维 1!"

  • What does that mean add 1?

woops: http://csillustrated.berkeley.edu/PDFs/integer-representations.pdf

This may be easiest to show by example. Here are the numbers from -4 to 4 represented in binary:

4   0000 0100
3   0000 0011
2   0000 0010
1   0000 0001
0   0000 0000
-1  1111 1111
-2  1111 1110
-3  1111 1101
-4  1111 1100

So say we want to go from 1 to -1. We first flip all the bits of 1

1 0000 0001
  flip bits
-----------
  1111 1110

Then we add a 1:

  1111 1110
  +       1
-----------
  1111 1111

We now have -1.

I'm not seeing the illustrations, but you're probably talking about Two's complement representation. (http://en.wikipedia.org/wiki/Two's_complement)

1. Why does adding a 1 to the front mean the computer has to count backwards?

   Due to how carrying works, FFFFFFFFF + 1 == 0 and 0 - 1 == FFFFFFFF. All the bits get flipped, including the first bit. If you simply define the negative numbers as those starting with a 1 bit (80000000 - FFFFFFFF) then you get a nice uniform behavior for addition with a natural overflow.

2. Flip the bits and add 1: in 2's complement, this negates a number

   ~x+1 == -x; // always true

What you're talking about is called signed integers.