Adding and subtracting two's complement

Question

Using six-bit one\'s and two\'s complement representation I am trying to solve the following problem:

12 - 7 

Now, i take 12 in binary and

Answer 1

Using two's complement to represent negative values has the benefit that subtraction and addition are the same. In your case, you can think of 12 - 7 as 12 + (-7). Hence you only need to find the two's complement representation of -7 and add it to +12:

12  001100
-7  111001   -- to get this, invert all bits of 7 (000111) and add 1
----------
 5 1000101

Then discard the carry (indicates overflow), and you have your result: 000101 which equals to 5 as expected.

For your example of -15 + 2, simply follow the same procedure to get the two's complement representation of -15:

15  001111
    110000   -- inverted bits
    110001   -- add 1

Now do the addition as usual:

-15  110001
  2  000010
-----------
res  110011

To see that res indeed equals -13, you can see that it is negative (MSB set). For the magnitude, convert to positive (invert bits, add 1):

res  110011
     001100  -- inverted bits
     001101  -- add 1

Hence the magnitude is 13 as expected.