two's complement

Question

plz help me i have spent hours trying to figure this out.

1. Solving 11₁₀ – 11₁₀ using 2’s complement will lead to a problem; by using 7-bit data representation. Explain what the problem is and suggest steps to overcome the problem.

Answer 1

Hi Lidia. A very interesting problem:

Solving 11₁₀ – 11₁₀ using 2’s complement will lead to a problem; by using 7-bit data representation. Explain what the problem is and suggest steps to overcome the problem.

Things to remember:

1. the subtraction has to be done by adding the 2's complement
2. the 2's complement is the 1's complement plus 1
3. if we have an overflow bit, we are allowed to throw it out

Step by step:

1. Convert 11₁₀ to binary 1011₂
2. find the 1's complement 0100₂
3. add 1 to get the 2's complement 0101₂
4. add 1011 and 0101 1011₂+0101₂=10000₂

In 4-bit representation, the 1 in 10000₂ gets thrown away, leaving 0000₂, or zero, the correct answer.

In 7-bit representation, the 1 in 10000₂ has to stay, leaving 10000₂, or 16₁₀, Yikes!😧

So we have identified the problem.

As to ways to avoid this, I can think of a few:

1. Don't use 7-bit data representation. If I had stayed with 4-bit I could have tossed out the overflow bit and gotten the correct answer. This is an obvious answer, but probably not the best. Can you guess why?
2. Use leading zeroes in the 7-bit representation. If we do that, the numbers become 0001011₂, 1110101₂ and 10000000₂. That leaves me an overflow bit of 1 which I can throw away, leaving 0000000₂ or zero.