Two's Complement Notation

Two's complement number representation is used for signed numbers on most modern computers. This notation allows a computer to add and subtract numbers using the same operations (thus we do not need to implement adders and subtractors). We can characterize two's complement notation as:

• A fixed number of bits are used to represent numbers
• The most significant bit is called the sign bit
• This same notation is used to represent both positive and negative numbers

Positive numbers represented normally

Example: Using a 4 bit representation 5 in 2's complement

Example: Using an 8 bit representation 5 in 2's complement

Example: Using an 8 bit representation 24 base 16

Negative numbers

Are represented using a 2's complement form. To obtain the 2's complement of a number:

1. Complement the bits
2. Add one to the result

Examples: (4 bits)

Represent -6 in 2's complement

+6         0110
complement 1001
add 1      0001
-6 =       1010

Represent -3 in 2's complement

+3         0011
complement 1100
add 1      0001
-3 = 	   1101
        

Example: (5 bits)

Represent -13 in 2's complement

+13        01101
complement 10010
add 1      00001
-3 =	   10011
        

Addition and subtraction in 2's complement notation

Is performed by doing the simple binary addition of the two numbers. Subtraction is accomplished by first performing the 2's complement operation on the number being subtracted then adding the two numbers.

Examples: 5 bits

   8       01000    
  +4       00100
  --       -----
  12       01100

  -8       11000
+ -4       11100
  --       -----
 -12       10100

   8       01000
+ -4       11100
  --       -----
   4       00100

Example: 8 bits

   25   0001 1001               19
+ -29   1110 0011               E3       
  ---   ---------
   -4   1111 1100               FC

Overflow

Since we are working with numbers contained in a fixed number of bits, we must be able to detect overflow following an operation.

1. No overflow occurs when the value of the bit carried into the most significant bit is the same as the value carried out of the most significant bit.
2. Overflow occurs when the value of the bit carried into the most significant bit is not the same as the bit carried out of the most significant bit.

Examples: 4 bits

            6                   0110
          + 1                   0001
         -----                 ------
            7                   0111



            7                   0111
          + 1                   0001
        -----                  ------
            8                   1000

but, in the second case is 8 the correct answer?