How is a max signed byte -1?

Question

I am really curious and confused:

how is 0xFFFF (0b11111111111111) or 0xFF (0b11111111) -1?

if 0b01111111 is 127 and the first bit indicates that it is a positive number then shouldn't 0b11111111 be -127?

Am I missing something???

Answer 1

Two's complement form is commonly used to represent signed integers. To swap the sign of a number this way, invert all the bits and add 1. It has the advantage that there is only one representation of zero.

01111111 = 127

To get -127 flip the bits:

10000000

And add 1:

10000001

To negate 1:

00000001 
11111110 // flip
11111111 // + 1

With 0:

00000000
11111111 // flip
00000000 // + 1 (the carry bit is discarded and it's still 0)

And just to show it works going the other way:

With -127:

10000001
01111110 // flip
01111111 // + 1 and you are back to +127.