How signed integers are represented using 16 bits

Source: http://www.xkcd.com/571/

This probably requires a little explanation with the nuances of how integers are stored in computers. A 16-bit signed integer simulates binary digits with 16 digits, where the first digit represents either positive () or negative (). Because this first digit represents positive or negative, the counting system is a little different than regular counting in binary.

For starters,

represents

represents

represents

represents

For the next number, , there’s a catch. The first means that this should represent a negative number. However, there’s no need for this to stand for , since we already have a representation for . So, to prevent representing the same number twice, we’ll say that this number represents , and we’ll follow this rule for all representations starting with . So

represents

represents

represents

represents

Because of this nuance, the following C computer program will result in the unexpected answer of (symbolized by the sheep going backwards in the comic strip).

main()

{

      short x = 32767;

      printf(“%d \n”, x + 1);

}

For more details, see http://en.wikipedia.org/wiki/Integer_%28computer_science%29