Range of signed char

Question

Why the range of signed character is -128 to 127 but not -127 to 128 ?

Answer 1

I think an easy way to explain this for the common soul is :

A bit is a value 0 or 1, or 2 possibilities

A 2-bit holds two combinations or 0 and 1 for four possible values : 00, 01, 10, and 11.

A 3-bit holds three combinations for a total of eight possible values : 000 to 111.

Thus n-bits holds n combinations for a total of 2^n possible values. Therefore, an 8-bit value is 2^8 = 256 possible values.

For signed numbers, the most significant bit (the first one reading the value from left to right) is the sign bit; that leaves a possibility of 2^(n-1) possible values. For an 8-bit signed number, this is 2^7 = 128 possible values for each sign. But since the positive sign includes the zero (0 to 127 = 128 different values, and 128 + 128 = 2^8 = 256), the negative sign includes -1 to... -128 for 128 different values also. Where :

10000000 = -128
...
11111111 = -1
00000000 = 0
...
01111111 = 127