binary - Adding and subtracting two's complement -

using six-bit one's , two's complement representation trying solve following problem:

12 - 7  

now, take 12 in binary , 7 in binary first.

12 = 001100 - 6 bit  7 =  000111 - 6 bit 

then, flip bit two's complement , add one?

12 = 110011 ones complement       +    1     -------      001101  7  = 111000 ones complement      +     1    ---------       111001 

then, add 2 complement together

 001101 +111001 ------- 1000110 = overflow? discard last digit?  if 5 

now, if have number like

-15 + 2 

i add sign magnitude on msb if it's zero?

like:

-15 = 001111 6 bit 

would add 1 @ end here before flip bits?

  = 101111 

using two's complement represent negative values has benefit subtraction , addition same. in case, can think of 12 - 7 12 + (-7). hence need find two's complement representation of -7 , add +12:

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

then discard carry (indicates overflow), , have result: 000101 equals 5 expected.

for example of -15 + 2, follow same procedure two's complement representation of -15:

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

now addition usual:

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

to see res indeed equals -13, can see negative (msb set). magnitude, convert positive (invert bits, add 1):

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

hence magnitude 13 expected.