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Official PYQ · 2021Set 1 · Q24Number Representations (Fixed and Floating Point)NAT1 mark

GATE CS 2021 Set 1 Q24 — Number Representations (Fixed and Floating Point)

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Consider the following representation of a number in IEEE 754\text{IEEE 754} single-precision floating point format with a bias of 127127.S:1E:  10000001F:  11110000000000000000000S: 1\quad\quad E:\; 10000001\quad\quad F:\;11110000000000000000000 Here S,  ES, \;E and FF denote the sign, exponent, and fraction components of the floating point representation.

The decimal value corresponding to the above representation (rounded to 22 decimal places) is ____________.
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Answer: -7.75 : -7.75

The value of a number represented in the IEEE 754 single-precision floating-point format is given by:

Value=(−1)S×(1.F)×2E−bias\text{Value} = (-1)^S \times (1.F) \times 2^{E - \text{bias}}

Given components from the question:

  • Sign bit (SS): 11 (indicates a negative number)
  • Biased Exponent (EE): 10000001210000001_2 E=27+20=128+1=12910E = 2^7 + 2^0 = 128 + 1 = 129_{10}
  • Fraction (FF): 11110000000000000000000211110000000000000000000_2 F=0.11112F = 0.1111_2

Now, we calculate the actual exponent (ee):

e=E−bias=129−127=2e = E - \text{bias} = 129 - 127 = 2

Substitute these values into the formula:

Value=(−1)1×(1.11112)×22\text{Value} = (-1)^1 \times (1.1111_2) \times 2^2

Multiplying by 222^2 shifts the binary point two positions to the right:

Value=−(111.112)\text{Value} = - (111.11_2)

Convert the binary value 111.112111.11_2 to decimal:

  • Integer part: 1112=4+2+1=7111_2 = 4 + 2 + 1 = 7
  • Fractional part: 0.112=12+14=0.5+0.25=0.750.11_2 = \frac{1}{2} + \frac{1}{4} = 0.5 + 0.25 = 0.75
Value=−7.75\text{Value} = -7.75

Thus, the decimal value corresponding to the representation is -7.75.

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