Using 2's compliment form, what is 42-18? Please show all the steps.

Answer 1

This question cannot be answered without knowing the word length.

I will assume that it is 8 bits.

If my assumption is wrong then hopefully you will still be able to extend the explanation

to the correct word length.

When using 2's compliment, subtraction is converted into addition.

There are two ways of thinking about it.

Method 1:

Given that the word length is 8, any pair of numbers that differ by exactly 256

have the same binary representation.

(It is 256 because 256 = 2⁸.)

So 42 - 18 has the same binary representation

as 42 - 18 + 256 = 42 + 238

Thus the binary representation of 42 - 18 is the same as

the binary representation of 42 + 238.

So we need to do binary addition of these two

00101010

11101110

Method 2:

This is a more mechanical approach which is actually implemented in hardware.

The binary representation of 18 is

00010010

The binary representation of -18 is obtained from the bitwise complement

11101101

and adding +1, resulting in

11101110

Notice that the representation of -18 is the same as the representation of 238.

And the reason is that they differ by 256.

Using either method we end up with the addition

00101010

11101110

--------------

00011000

which is the binary representation of 24.

Notice that the binary addition resulted in a carry out of the leading bit.

That does not represent an overflow -- in case of subtraction the carry is

the normal thing to happen.