twos complement (idea)

1. The 2's complement number system is a common number system used for expressing signed numbers in binary. In this system, the most significant bit acquires a negative value. Thus, if the most significant bit is a "1", the entire number is negative. All other digits represent positive numbers. For example, if we have the bitstring:

   1000 1111

   We can determine the (decimal) value as follows:

   1. The most significant bit is a 1, so it is negative. It is in the 7th place, so it has an absolute value of 2⁷. Thus, the most significant bit contributes -2⁷ = -128.
   2. The next three bits are zero, so they contribute nothing to the value.
   3. The last four bits represent the number 15. Since they are not in the most significant bit's place, 15 is positive.
   4. We add up all the numbers to get: -128+15 = -113.

   Note that we can only represent numbers from -128 to 127. Or, in general, for a bitstring of length x bits we can represent numbers from -2^x-1 up to 2^x-1-1.
   

2. Or "to take the 2's complement", meaning to take the opposite of a 2's complement number. Using our example above, we perform the following procedure:

   1. Take the 1's complement or invert each bit => We get: 0111 0000.
   2. Add 1 => We get 0111 0001

   Determining the decimal value as above, we see that the resulting number is indeed +113.

Other things to see: infinity, unsigned.