Why is signed char 0xff = -1?

Question

I don't understand it very well.

Answer 1

Signed integers are represented in two's complement. It's easier to think about this in terms of bits vs. the hexadecimal equivalent, so let's consider the decimal integers +127 and -127 represented by 8 bits:

|----------------------High-ordered Bit

|.......|--------------Low-ordered Bit

v.......v

0111 1111 = +127 (0x7F)

1000 0001 = -127 (0x81)

The sign is stored in the high-ordered bit for integers--a 0 in the high-ordered bit indicates a positive number whereas a 1 would indicate the value is a negative number. That's why a signed integer stored in 8 bits has a decimal range of -128 to +127. +127 has all bits turned on except the high-ordered bit. If we added 1 to this value, the high-ordered bit would turn on indicating a -128 signed integer. If we add one bit to get the two's complement, we would end up with -127 (even though the unsigned decimal conversion would be 129)

1000 0000 = -128 (0x80) -> Unsigned: 128

1000 0001 = -127 (0x81) -> Unsigned: 129

1000 0010 = -126 (0x82) -> Unsigned: 130

etc.

How do we find the two's complement of a particular number? One way is to get the one's complement and then add 1 to it...getting the two's complement using the integer +2 as an example:

.0000 0010 = +2 (0x02)

.|||| ||||

.1111 1101 <- One's complement of +2

+0000 0001 <- add one

----------

.1111 1110 = -2 (0xFE)

Note that the addition of one to the ones column carries over to the twos column to the left of it.

Now, let's look a the bits for the decimal integers +1 and -1:

|----------------------High-ordered Bit

|.......|--------------Low-ordered Bit

v.......v

0000 0001 = +1

1111 1111 = -1

And let's see how working through the one's complement "plus one" to arrive at the two's complement:

.0000 0001 = +1

.|||| ||||

.1111 1110 <- One's complement

+0000 0001 <- add one

----------

.1111 1111 = -1 (or 0xFF)

That is a short explanation on why 0xFF (or 1111 1111) is -1 as a signed integer.