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

Step 1: Set up the bit budget.
We have $9$ bits. One bit is the sign, leaving $8$ magnitude bits, so $2^{8}=256$ is the key number to remember throughout.
Step 2: Range of 1's complement (Statement I).
In 1's complement the largest magnitude is $2^{8}-1 = 255$ on both sides, so the range is $-255$ to $+255$. The first half of Statement I is correct.
Step 3: The two zeros in 1's complement (Statement I).
$000000000$ is $+0$ and flipping every bit gives $111111111 = -0$. Two distinct codes mean the same value, so 1's complement really does have two zeros. Statement I is fully true.
Step 4: Range of 2's complement (Statement II).
2's complement is asymmetric: the most negative value is $-2^{8} = -256$ and the most positive is $+2^{8}-1 = +255$, giving $-256$ to $+255$.
Step 5: Count of patterns (Statement II).
Nine bits give $2^{9}=512$ distinct patterns regardless of the scheme, so both systems use exactly $512$ codes. Statement II is true.
Step 6: Maximum positive value (Statement III).
The biggest positive number in either scheme is $011111111 = 255$, since the sign bit must stay $0$. Statement III is true.
Step 7: Combine.
All three statements check out, so the answer is the option naming I, II and III.
\[ \boxed{\text{I, II, III}} \]