Question

Consider the following statement about range of number in 9 bit 1's complement and 2's complement system. I. In 9 bits 1's complements the range is \(-255\) to \(+255\) and there exist two representations of zero. II. In 9 bits 2's complements the range is \(-256\) to \(+255\), both 1's complements and 2's complements can represent exactly \(512\) unique values. III. The maximum positive number representable is \(+255\) in both 1's complements and 2's complements 9 bit system. Options:

• A. I, III
• B. II, III
• C. I, II, III
• D. I, II

Hint:

Remember the standard ranges: 1's Complement: \[ -(2^{n-1}-1)\ \text{to}\ +(2^{n-1}-1) \] 2's Complement: \[ -2^{n-1}\ \text{to}\ +(2^{n-1}-1) \] Also, 1's complement has two zeros (\(+0\) and \(-0\)), whereas 2's complement has only one zero.

Answer 1

The Correct Option is C

Concept: Signed binary numbers can be represented using different schemes. Two of the most commonly used representations are:

• 1's Complement Representation

• 2's Complement Representation
For an \(n\)-bit system:

• In 1's complement: \[ \text{Range} = -(2^{n-1}-1)\ \text{to}\ +(2^{n-1}-1) \]

• In 2's complement: \[ \text{Range} = -2^{n-1}\ \text{to}\ +(2^{n-1}-1) \]
A key difference is that 1's complement has two representations of zero whereas 2's complement has only one representation of zero.

Step 1: Verify Statement I regarding 9-bit 1's complement representation.
For a 9-bit system, \[ n=9 \] Therefore, \[ 2^{n-1}=2^8=256 \] In 1's complement representation, \[ \text{Range} = -(2^{8}-1) \text{ to } +(2^{8}-1) \] \[ =-255 \text{ to } +255 \] Thus the first part of Statement I is correct. Now consider representation of zero. Positive zero: \[ 000000000 \] Negative zero: \[ 111111111 \] Hence two different representations of zero exist. Therefore Statement I is true.

Step 2: Verify Statement II regarding 9-bit 2's complement representation.
For 9-bit 2's complement: \[ \text{Range} = -2^{8} \text{ to } (2^{8}-1) \] \[ =-256 \text{ to } +255 \] Hence the first part of Statement II is correct. Now count the total values represented. A 9-bit binary system contains \[ 2^9=512 \] different bit patterns. Therefore both 1's complement and 2's complement systems contain \[ 512 \] possible binary patterns. Thus Statement II is also true.

Step 3: Verify Statement III regarding the maximum positive number.
In both representations, the largest positive value occurs when: \[ \text{Sign bit}=0 \] and all remaining bits are \(1\). Thus \[ 011111111 \] which equals \[ 255 \] Therefore the maximum positive integer representable is \[ +255 \] in both 9-bit 1's complement and 2's complement systems. Hence Statement III is true.

Step 4: Determine the correct option.
We have shown that: Statement I = True Statement II = True Statement III = True Therefore all three statements are correct. Hence the correct answer is \[ \boxed{(C)\ \text{I, II, III}} \]