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If signed numbers are used in binary arithmetic, then which one of the following notations would have unique representation for zero?

  1. Sign-magnitude
  2. $1$’s complement
  3. $2$’s complement
  4. $9$’s complement
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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 $(C)$.

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