Take the following byte:

00000011

This byte represents the numeral 3. If we want to turn this number into -3 we have to perform the Two's Complement operation.

1. Write the absolute value of a number in binary.
2. Flip all the bits (0 becomes 1, and 1 becomes 0).
3. Add 1 to the result.

So it follows that the numeral -3 is represented in a byte by:

11111101

However, in certain cases we want to cast this byte (int8_t) to a larger quantity, say for example a (int16_t).

For that to happen, we need to perform a sign extension.

Signed extension

The procedure is as follow:

1. Grab the value of the sign bit (highmost bit).
2. Extend the binary number by filling the high bits with the sign bit.

For the numeral 3 we have:

00000000 00000011

And for the numeral -3 we have:

11111111 11111101

The numeral -3 has been extended, and it is still the Two's Complement of 3.

In C

C will automatically sign extend the numbers when casting an integer with more storage space.

I am not entirely sure right now if this is implementation dependent or if it is a defined behaviour expected by the language.

#include <stdint.h>

int main() {
  int8_t num = -3;
  // (gdb) p /t num
  // > 11111101

  int16_t num16 = num;
  // (gdb) p /t num16
  // > 11111111 11111101

  // (gdb) p /t num32
  // > 11111111 11111111 11111111 11111101
  int num32 = num;

  // (gdb) p /t num64
  // > 11111111 11111111 11111111 11111111 11111111 1111111 11111111 111111101
  long long num64 = num;
}