Questions on Integer Representations

1. What is the 8-bit, unsigned representation for the decimal value:
   
   A. 0
   00000000
   
   B. 12
   00001100
   
   C. 64
   01000000
   
   D. 1000
   The decimal value 1000 does not fit into 8 bits (unsigned). Therefore we say that overflow occurs.
   
   E. -1
   Unsigned representaion does not exist for negative numbers, so this part has no answer.
2. What decimal value is represented by the 8-bit unsigned representation:
   
   A. 10000000
   128
   
   B. 00001111
   15
   
   C. 01111111
   127
3. What is the 8-bit, two's complement representation for the decimal value:
   
   A. 0
   00000000
   
   B. 12
   00001100
   
   C. 64
   01000000
   
   D. 1000
   Just as for unsigned representation, the decimal value 1000 does not fit into 8 bits (2's complement). Therefore we say that overflow occurs.
   
   E. -1
   11111111

   Hint: memorize that the all ones bit pattern represents the decimal value -1 in any number of bits (2's complement).
   
   F. -18
   11101110

       How this representation may be calculated:
   
       Start with the 8-bit representation for +18:   00010010
   
       Find the additive inverse (also called "taking the two's complement"). 
   
       00010010
   
       11101101  (the bitwise complement)
     +        1
     ----------
       11101110  (the additive inverse)
   

4. What is the two's complement representation for the largest magnitude, positive value that can be represented in 8 bits? What decimal value is being represented?
   01111111

   This represents the decimal value 127.

5. What is the two's complement representation for the largest magnitude, negative value that can be represented in 8 bits? What decimal value is being represented?
   10000000

   This represents the decimal value -128.

6. What is the 2's complement representation for the decimal value -2 in 6 bits? In 7 bits? In 8 bits? What operation is done on this representation to increase the number of bits, without changing the value being represented?
   111110, 1111110, and 11111110.

   The operation is called sign extension.

7. What range of decimal integers can be represented by an n-bit two's complement representation?
   For n bits, the range of values is -(2^n-1) to +(2^n-1 - 1).