Question

This 4 bit binary addition will result in overflow. 0110 + 0111

___ True   

___ False

Please justify your answer.

Answer 1

Answer

Normal binary addition: False (No overflow)
Two's complement addition:  True (overflow)

Explanation:
-------------
Adding 0110 and 0111 in binary
    0110
    0111
---------
 (0)1101
---------
So, sum of these numbers in binary is 1101

Verification
---------------
first = 0110
Converting 0110 to decimal
0110
=> 0x2^3+1x2^2+1x2^1+0x2^0
=> 0x8+1x4+1x2+0x1
=> 0+4+2+0
=> 6

second = 0111
Converting 0111 to decimal
0111
=> 0x2^3+1x2^2+1x2^1+1x2^0
=> 0x8+1x4+1x2+1x1
=> 0+4+2+1
=> 7

If this was a normal binary addition:
--------------------------------------
sum = 1101
Converting 1101 to decimal
1101
=> 1x2^3+1x2^2+0x2^1+1x2^0
=> 1x8+1x4+0x2+1x1
=> 8+4+0+1
=> 13
This is correct since we can verify that 6+7 = 13
So, there was no overflow.

If it was in two's complement:
-------------------------------
then calculation for result is
sum = 1101
since left most bit is 1, this number is negative number.
so, follow these steps below to convert this into a decimal value.
I. first flip all the bits. Flip all 0's to 1 and all 1's to 0.
   1101 is flipped to 0010
II. Add 1 to above result
0010 + 1 = 0011
III. Now convert this result to decimal value
Converting 11 to decimal
11
=> 1x2^1+1x2^0
=> 1x2+1x1
=> 2+1
=> 3
Answer: -3
6+7 must be 13
This is not correct since we can verify that 6+7 not equals -3
So, there was an overflow.