The format of the single-precision floating point representation of a real…
2021
The format of the single-precision floating point representation of a real number as per the IEEE 754
\(\begin{array}{|c|c|c|} \hline \text{sign} & \text{exponent} & \text{mantissa} \\ \hline \end{array}\)
Which one of the following choices is correct with respect to the smallest normalized positive number represented using the standard?
- A.
exponent =00000000 and mantissa =0000000000000000000000000
- B.
exponent =00000000 and mantissa =0000000000000000000000001
- C.
exponent =00000001 and mantissa =0000000000000000000000000
- D.
exponent =00000001 and mantissa =0000000000000000000000001
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Correct answer: C
Key insight: normalized IEEE 754 single-precision numbers have an exponent field not equal to all zeros. The value for normalized numbers is (−1)^sign × (1.fraction) × 2^(exponent − bias).
For single precision the exponent has 8 bits and the bias is 127.
The smallest normalized positive number uses exponent = 00000001 (e = 1) and mantissa = all zeros, giving value = 1.0 × 2^(1 − 127) = 2^-126 ≈ 1.17549435e-38.
Exponent = 00000000 indicates zero or subnormal numbers (denormals). The smallest positive subnormal has exponent = 00000000 and mantissa = ...0001 and equals 2^-149 ≈ 1.401298e-45, which is smaller than any normalized number but is not considered normalized.
Therefore, the correct representation for the smallest normalized positive single-precision number is exponent = 00000001 and mantissa = 0000000000000000000000000.