−1 at 8, 16 and 32 bits — two's complement
exam standardall ones at every width
Answer
11111111 / 1111111111111111 / 11111111111111111111111111111111
Why this example is worth doing
Minus one is all ones regardless of width, which is the most useful single fact about the representation and the reason ~0 is the idiomatic all-bits-set constant. The page shows it at three widths to make the width-independence visible, and connects it to the bitwise NOT example on the NOT gate page, where the same pattern arrives from the logic side.
Try your own input in the Two’s complement. Encode and decode signed binary at any width, with overflow and sign extension.
How the answer is reached
-1 in 8-bit two's complement
Encode -1 in 8-bit two's complement. Representable range: -128 to 127.
|-1| in binary00000001
invert every bit11111110
add 111111111— invert-and-add-one is the negation rule
8-bit pattern1111 11110xFF
check11111111 reads 255 unsigned and -1 signed
-1 in 16-bit two's complement
Encode -1 in 16-bit two's complement. Representable range: -32768 to 32767.
|-1| in binary0000000000000001
invert every bit1111111111111110
add 11111111111111111— invert-and-add-one is the negation rule
16-bit pattern1111 1111 1111 11110xFFFF
check1111111111111111 reads 65535 unsigned and -1 signed
-1 in 32-bit two's complement
Encode -1 in 32-bit two's complement. Representable range: -2147483648 to 2147483647.
|-1| in binary00000000000000000000000000000001
invert every bit11111111111111111111111111111110
add 111111111111111111111111111111111— invert-and-add-one is the negation rule
32-bit pattern1111 1111 1111 1111 1111 1111 1111 11110xFFFFFFFF
check11111111111111111111111111111111 reads 4294967295 unsigned and -1 signed