Binary 2's Complement

Question

I'm facing a problem. when we want to subtract a number from another using 2's complement we can do that. I don't know how to subtract fractional number using 2's complement.

5 is in binary form 101 and 2 is 10. if we want to subtract 2 from 5 we need to find out 2's complement of 2 2's complement of 2-> 11111110

so if we now add with binary of 5 we can get the subtraction result. If I want to get the result of 5.5-2.125. what would be the procedure.

Answer 1

Fixed point numbers can be used and it is still common to find them in embedded code or hardware.

Their use is identical to integers, but you need to specify where your "point" is. For instance, assume that you want 3 bits after after the point and that your data is 8 bits, bits 7..3 are the integer part (left of "point") and bits 2..0 the fractional part. The interpretation of integer part is as usual the binary decomposition of this integer: bits 3 correspond to 2⁰, bits 4 to 2¹, etc.

For the fractional part, the decomposition is in negative powers or two. bits 2 correspond to 2^-1, bits 1 to 2^-2 and bit 0 to 2^-3.

So for you problem, 5.5=4+1+1/2=2²+2⁰+2^-1 and its code is 00101(.)100. Similarly 2.125=2+1/8 and its code is 00010(.)001 (note (.) is just an help to understand the coding).

Indeed they are just integers, but you must take into account that all your numbers are multiplied by 2^-3. This will have no impact for addition, but results of multiplication and division must be adjusted. Taking into account the position of point and managing over and underflows is the difficulty of arithmetic with fixed point, but it allows to do fractional computations even if your hardware does not provide floating point support (for instance with low end microcontrollers or FPGA systems).

Two complement is similar to integers and its computation is identical. If code of 2.125 is 00010(.)001, than -2.125==11101(.)111. Operations are as usual.

+5     00101(.)100
-2.125 11101(.)111
       00011(.)011

and 00011(.)011=2+1+1/4+1/8=3,375

For the record, two complement first use was for fixed point fractional numbers and two complement name comes from that. If a fractional number if represented by, say 0(.)1100000 (0.75), its negative counter part will be 1(.)0100000 (-0.75 or 1.25 if interpreted as unsigned) and we always have x+(unsigned)-x=2. For this coding, the negative value of a fractional number x is the number y that must be added to x to get a 2, hence the name that y is 2's complement of x.