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Official PYQ · 2021Set 2 · Q4Number Representations (Fixed and Floating Point)MCQ1 mark

GATE CS 2021 Set 2 Q4 — Number Representations (Fixed and Floating Point)

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The format of the single-precision floating point representation of a real number as per the IEEE 754\text{IEEE 754} standard is as follows:

signexponentmantissa\begin{array}{|c|c|c|} \hline \text{sign} & \text{exponent} & \text{mantissa} \\ \hline \end{array}

Which one of the following choices is correct with respect to the smallest normalized positive number represented using the standard?

Choose one option
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Answer: C

In the IEEE 754 single-precision floating-point standard:

  • The exponent field has 88 bits.
  • The mantissa (fraction) field has 2323 bits.

For normalized numbers:

  • The exponent field EE cannot be all 00s or all 11s.
    • E=000000002E = 00000000_2 is reserved for subnormal (denormalized) numbers and zero.
    • E=111111112E = 11111111_2 is reserved for infinity and NaN (Not a Number).
  • Therefore, the minimum possible exponent value for a normalized number is E=000000012E = 00000001_2 (which represents an unbiased exponent of 1−127=−1261 - 127 = -126).

To find the smallest positive normalized number, we must also minimize the mantissa value. The minimum value for the fractional part of the mantissa is all 00s.

Thus, the smallest positive normalized number has:

  • Exponent: 0000000100000001
  • Mantissa: 0000000000000000000000000000000000000000000000 (all zeros)

This corresponds to the value 1.0×2−1261.0 \times 2^{-126}.

Therefore, the correct choice is C.

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