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GO Classes DPP | GATE CS | Digital Logic | Two’s Complement

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4 4 votes

Which of the following $2$’s complement bit strings represents the smallest decimal value? Each bit string must be interpreted using its own given bit-width.

  1. $1001011$ as a $7$-bit number
     
  2. $11001010$ as an $8$-bit number
     
  3. $111000101$ as a $9$-bit number
     
  4. $101111$ as a $6$-bit number

2 Answers

1 1 vote

In an $n$-bit $2$’s complement system, if the leftmost bit is $1$, the number is negative.

For a negative $n$-bit number, signed value $=$ unsigned value $-2^n$.


A: $1001011$ is a $7$-bit number.

Unsigned value $=64+8+2+1=75$.

Signed value $=75-2^7=75-128=-53$.


B: $11001010$ is an $8$-bit number.

Unsigned value $=128+64+8+2=202$.

Signed value $=202-2^8=202-256=-54$.


C: $111000101$ is a $9$-bit number.

Unsigned value $=256+128+64+4+1=453$.

Signed value $=453-2^9=453-512=-59$.


D: $101111$ is a $6$-bit number.

Unsigned value $=32+8+4+2+1=47$.

Signed value $=47-2^6=47-64=-17$.


Now compare the values:

A $=-53$, B $=-54$, C $=-59$, D $=-17$.

The smallest value is $-59$.


Final Answer: C.

0 0 votes

its very simple do calculation of 2's complement the eaiest method is - in 2's complement the first bit is -ve and rest are positive .

for option d = 101111 => -32 + 8 +4+ 2+1 = -17 is smallest here

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