A) Sign-magnitude representation of signed numbers: We can represent $0$ using $4$-bit.
For magnitude(MSB bit), we can classify
$0\implies(+)\: \text{Positive}$
$1\implies(-)\: \text{Negative}$
- In the signed magnitude, the most significant bit is used to represent the sign. The rest of the bits are used to represent the magnitude of the number.
$(0)_{10}=(0000)_{2} = (+0)_{10}$
$(0)_{10}=(1000)_{2} = (-0)_{10}$
- Note that this leads to having two representations for the number zero.
$\implies$Range of Sign-magnitude representation($n$-bit signed numbers) is $[-(2^{n-1}-1),\:\:2^{n-1}-1]$.
B) $1's$ complement representation of signed numbers: We can represent $0$ using $4$-bit.
For magnitude(MSB bit), we can classify
$0\implies(+)\: \text{Positive}$
$1\implies(-)\: \text{Negative}$
- In the $1's$ complement, the most significant bit is used to represent the sign. The rest of the bits are used to represent the magnitude of the number.
$(0)_{10}=(0000)_{2}=(+0)_{10}$
$1's $ complement of $(0)_{10}$ is $(1111)_{2}=(-0)_{10}$.
- Note that this leads to having two representations for the number zero.
$\implies$Range of Sign-magnitude representation($n$-bit signed numbers) is $[-(2^{n-1}-1),\:\:2^{n-1}-1]$.
C) $2's$ complement representation of signed numbers: We can represent $0$ using $4$-bit.
In the $2's$ complement, the most significant bit is used to represent the sign.
$(0)_{10}=(0000)_{2}$
$2's$ complement of $(0)_{10} = (0000)_{2}$
So, $0$(Zero) has unique representation.
$\implies$Range of Sign-magnitude representation($n$-bit signed numbers) is $[-2^{n-1},\:\:2^{n-1}-1]$.
D) $9's$ complement representation of signed numbers: We can represent $0$ using $4$-bit.
$(0)_{10}=(0000)_{10}$
$9's$ complement of $(0)_{10} = (9999)_{10}$
Note that this leads to having two representations for the number zero.
So,the correct answer is $(C)$.