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. 

    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:

   Start with 1001000101 and flip all bits to the left of the rightmost 1
 
to get  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.


      

    2. 
   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.


      

    3. 
   1001000101  -44310
  +1111011110   -3410
  ------------  ---
   1000100011  -47710   no overflow


      

    4. 
   1001000101  -44310
  +1101100000  -16010
  ------------  ---
   0110100101   42110  overflow


      

    5. 
   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.