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Number Representation and Arithmetic

GATE CSE 2016 Set 1 · Q7

NAT1 markEasy

Time

The 16-bit 2's complement representation of an integer is 1111 1111 1111 0101; its decimal representation is _________ .

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Official answer: -11 (±0)

Number Representation and Arithmetic

The given 16-bit binary number is:

Binary=1111 1111 1111 01012\text{Binary} = 1111\ 1111\ 1111\ 0101_2

Since the most significant bit (MSB) is 11, the number is negative in 2's complement representation.

To find its decimal value:

  1. Find the 1's complement of the number:

    0000 0000 0000 101020000\ 0000\ 0000\ 1010_2
  2. Add 11 to find the magnitude (2's complement of the negative number):

    0000 0000 0000 10102+1=0000 0000 0000 10112=11100000\ 0000\ 0000\ 1010_2 + 1 = 0000\ 0000\ 0000\ 1011_2 = 11_{10}
  3. Apply the negative sign:

    Decimal value=−11\text{Decimal value} = -11

Alternatively, calculate the weighted sum directly:

−215+∑i=014bi2i=−32768+32757=−11-2^{15} + \sum_{i=0}^{14} b_i 2^i = -32768 + 32757 = -11

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This Number Representation and Arithmetic previous year question is Q7 of the GATE CSE 2016 Set 1 paper. Find more GATE CSE PYQs on Number Representation and Arithmetic in Digital Logic, or browse all GATE CSE PYQs.

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Number Representation and Arithmetic in GATE CSE

Number Representation and Arithmetic has 33 GATE CSE PYQs across 20 papers from 2012 to 2026. 11 of them are 1-mark NAT questions like this one, and it is the only one in GATE CSE 2016 Set 1.

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The 16-bit 2's complement representation of an integer is 1111 1111 1111 0101; its decimal representation is …

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