128 = 1000 0000 reflect = 0111 1111 add one = 1000 0000 ??????

It looks like +128 and - 128 are represented by the same pattern.
This is not good.
A non-zero integer and its negative can't both be represented by the
same pattern.
So +128 can not be represented in eight bits.
The maximum positive integer that can be represented in
eight bits is 127_{10}.

What number is represented by 1000 0000?
Add the representation of 127_{10} to it:

1000 0000 = ?0111 1111= 127_{10}1111 1111 = -1_{10}

A good choice for ? is -128_{10}.
Therefore 1000 0000 represents -128_{10}.
Eight bits can be used to represent the numbers
-128_{10} ... 0 ... +127_{10}.

Range of N Bit 2's Complement |
---|

-( 2 |

For example, the range of integers that can be represented in eight bits using two's complement is:

-( 2^{(8-1)}) = -128 ... 0 ... 2^{(8-1)}- 1 = 127

Notice that one more negative integer can be represented than positive integers.

How many integers are there in the range -128 ... 0 ... +127 ?

How bit patterns can be formed with 8 bits?