You probably know that integers are represented in binary–in base 2. This is pretty straightforward for positive numbers, but it means you must choose an encoding for representing negatives. The encoding used by C++ (as well was by C and Java) is two’s complement.

In two’s complement, the first bit of a negative number is always 1. Otherwise, the number is 0 or postive. To find the bitstring representing a negative number, you take the bitstring representing the corresponding positive number and flip all the bits. Next, add 1 to the result.

In the following example, Ive used 4-bit numbers for simplicity:

  -5d = -(0101b) = (1010b + 1b) = 1011b

Notice that -1d is represented as all 1’s:

  -1d = -(0001b) = 1110b + 1 = 1111b

A nice property of this encoding is that you can subtract by negating and then adding the following:

  7d - 3d = 0111b          - 0011b  =>        0111b          + 1100b + 1  =>        0111b          + 1101b          = 0100b = 4d

Yet another nice property is that overflows and underflows wrap around and cancel one another out, like this:

  5d + 6d = 0101b          + 0110b          = 1011b          = -(0100b + 1)          = -0101b = -5d

If you subtract 6 from this result (by adding its negation), youll get 5 back. First, compute -6:

  -6d = -(0110b) = 1001b + 1 = 1010b

Then, add -6d to -5d to get the original value:

           1011b         + 1010b         = 0101