Answers to Two's Complement Exercises

1. The largest number that can be represented in 10-bit two's complement is 2⁹-1 = 511; the smallest number is -2⁹ = -512.

2. 1.    0101011010   34610
        +1001000101  -44310
        ------------  ---
         1110011111   -9710    no overflow
      
      Explanation:
      -----------
      0101011010: This is positive (leftmost bit is 0), so just interpret this
      as a base 2 number => 34610.
      
      1001000101: This is negative (leftmost bit is 1), so take the two's complement
      and then negate the base 2 interpretation of the result:
            1001000101
      
            0110111010 flip all bits
            +        1
            ----------
            0110111011 = 44310, so original number is -443.
      
      Add bits to get 1110011111, then interpret it like the second number above,
      since it is negative.  This gives -97, the correct result; no overflow
      occurred.
      
      
      
         0101011010   34610
        +0011000110   19810
        ------------  ---
         1000100000  -48010    overflow
      
      Explanation:
      -----------
      0101011010: This is positive, so interpret it as a base 2 number => 34610.
      
      0011000110: This is positive,  so interpret it as a base 2 number => 198.10.
      
      Add bits to get 1000100000.  This is negative, so take two's complement,
      interpret as a base 2 number and negate => -480.  The sum of two positive
      numbers cannot produce a negative number, so overflow must have occurred.
      Furthermore, the correct sum 544 is too large to fit in 10-bit twos complement
      so the sum overflows.
      
      
      
         1001000101  -44310
        +1111011110   -3410
        ------------  ---
         1000100011  -47710   no overflow
      
      
      
         1001000101  -44310
        +1101100000  -16010
        ------------  ---
         0110100101   42110  overflow
      
      
      
         0101011010   34610
        +1111011110   -3410
        ------------  ---
         0100111000   31210   no overflow
      

3. False. A 1 carried out of the leftmost position doesn't mean anything; overflow may or may not have occurred. Note that it will always happen when you add two negatives, and may happen when you add a negative and a positive.

4. No. The result will always be closer to 0 than either number, and so cannot produce overflow.