The format of the single-precision floating-point representation of a real number as per the IEEE 754 standard is as follows:
\[ \begin{array}{|c|c|c|} \hline \text{sign} & \text{exponent} & \text{mantissa} \\ \hline \end{array}\] Which one of the following choices is correct with respect to the smallest normalized positive number represented using the standard?

Question

Assume that a 12-bit Hamming codeword consisting of 8-bit data and 4 check bits is $d_8 d_7 d_6 d_5 c_8 d_4 d_3 d_2 c_4 d_1 c_2 c_1$, where the data bits and the check bits are given in the following tables. Which one of the following choices gives the correct values of $x$ and $y$?

Answer 1

The question asks us to identify the correct configuration for the smallest normalized positive number represented using the IEEE 754 single-precision floating-point format. Let's explore this step-by-step:

IEEE 754 Single-Precision Format:
- This format is a 32-bit representation, which is divided into three parts: sign bit (1 bit), exponent (8 bits), and mantissa (23 bits).
- The general representation of the number is $ (-1)^{\text{sign}} \times 1.\text{mantissa} \times 2^{\text{exponent bias}} $.
Understanding the Bias and Exponent:
- In IEEE 754, the exponent is stored in a biased form. For single-precision, the bias is 127.
- An exponent of all zeros (00000000) indicates a denormalized number, and an exponent of all ones indicates special numbers such as infinity or NaN.
Normalized Numbers:
- For normalized numbers, the exponent must be between 00000001 and 11111110.
- The smallest normalized positive number corresponds to an exponent of 00000001 and a mantissa that is all zeros.
Explanation for the Correct Answer:
- Among the given options, the combination of exponent = 00000001 and mantissa = 00000000000000000000000 gives us the smallest normalized positive number. This corresponds to:
(-1)^0 \times 1.0 \times 2^{1 - 127} = 2^{-126}
Conclusion: Therefore, the correct choice is the "exponent = 00000001 and mantissa = 00000000000000000000000", which represents the smallest normalized positive number in single-precision IEEE 754 format.

Answer 2

The question asks us to identify the correct configuration for the smallest normalized positive number represented using the IEEE 754 single-precision floating-point format. Let's explore this step-by-step:

IEEE 754 Single-Precision Format:
- This format is a 32-bit representation, which is divided into three parts: sign bit (1 bit), exponent (8 bits), and mantissa (23 bits).
- The general representation of the number is $ (-1)^{\text{sign}} \times 1.\text{mantissa} \times 2^{\text{exponent bias}} $.
Understanding the Bias and Exponent:
- In IEEE 754, the exponent is stored in a biased form. For single-precision, the bias is 127.
- An exponent of all zeros (00000000) indicates a denormalized number, and an exponent of all ones indicates special numbers such as infinity or NaN.
Normalized Numbers:
- For normalized numbers, the exponent must be between 00000001 and 11111110.
- The smallest normalized positive number corresponds to an exponent of 00000001 and a mantissa that is all zeros.
Explanation for the Correct Answer:
- Among the given options, the combination of exponent = 00000001 and mantissa = 00000000000000000000000 gives us the smallest normalized positive number. This corresponds to:
(-1)^0 \times 1.0 \times 2^{1 - 127} = 2^{-126}
Conclusion: Therefore, the correct choice is the "exponent = 00000001 and mantissa = 00000000000000000000000", which represents the smallest normalized positive number in single-precision IEEE 754 format.

Show Hint

The Correct Option is C

Solution and Explanation

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