Question

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?

• A. exponent = 00000000 and mantissa = 00000000000000000000000
• B. exponent = 00000000 and mantissa = 00000000000000000000001
• C. exponent = 00000001 and mantissa = 00000000000000000000000
• D. exponent = 00000001 and mantissa = 00000000000000000000001

Hint:

Exponent all zeros indicate denormalized numbers; normalized numbers always have a non-zero exponent.

Answer 1

The Correct Option is C

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:

1. 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 \( (-1)^{\text{sign}} \times 1.\text{mantissa} \times 2^{\text{exponent bias}} \).
2. 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.
3. 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.
4. 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}
5. 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.