Question

Consider a 9-bit representation. Which of the following correctly gives the smallest number that can be represented in: (i) 1's complement, (ii) 2's complement?

• A. (i) -256, (ii) -255
• B. (i) -255, (ii) -255
• C. (i) -256, (ii) -256
• D. (i) -255, (ii) -256

Hint:

In 1's complement, the smallest number is \( -(2^{n-1} - 1) \), while in 2's complement, it is \( -2^{n-1} \).

Answer 1

The Correct Option is D

Step 1: Understand 1's complement and 2's complement representations.
- In 1's complement, the smallest number that can be represented in an \( n \)-bit system is \( -(2^{n-1} - 1) \).
- In 2's complement, the smallest number that can be represented is \( -2^{n-1} \).

Step 2: Apply to the 9-bit representation.
For a 9-bit representation:
- In 1's complement, the smallest number is \( -255 \) (since \( 2^{8} - 1 = 255 \)).
- In 2's complement, the smallest number is \( -256 \) (since \( -2^{8} = -256 \)).

Step 3: Conclusion.
Thus, the correct answer is (d) (i) -255, (ii) -256.