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 12710.
What number is represented by 1000 0000? Add the representation of 12710 to it:
1000 0000 = ? 0111 1111 = 12710 1111 1111 = -110
A good choice for ? is -12810. Therefore 1000 0000 represents -12810. Eight bits can be used to represent the numbers -12810 ... 0 ... +12710.
| Range of N Bit 2's Complement |
|---|
-( 2(N-1) ) ... 0 ... 2(N-1) - 1 |
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?