Question

Consider the two numbers A=3.0 and B=−3.5. We would like to use the IEEE 754 standard representation. In this problem, we assume an 8-bit representation with 1 bit for sign, 3 for exponent in excess-3 notation (means the bias is 3), and 4 bits for the mantissa. What is the minimum positive number that can be represented if the significand is normalized?

Answer 1

The smallest positive normalized number representable in the specified 8-bit IEEE 754 format with 1 sign bit, 3 exponent bits (excess-3 notation), and 4 mantissa bits is 0.125.

Step-by-step explanation:

Understanding IEEE 754 Representation

The question pertains to the IEEE 754 standard for floating-point arithmetic, specifically an 8-bit representation. With 1 sign bit, 3 exponent bits (excess-3 notation), and 4 mantissa bits used in a normalized representation, the minimum positive number is determined.

In a normalized form, the mantissa does not have leading zeros and the first digit is always 1. Since we are finding the minimum positive number, assume the exponent is the smallest possible and the sign bit is 0 for a positive number. The exponent field is in excess-3 notation, so an exponent value of 000 represents -3 (0 in excess-3 notation).

The minimum positive number here is with the exponent field 000 (-3 after removing the bias) and the smallest non-zero normalized mantissa, which is 1000 for our 4-bit mantissa. Therefore, the representation would be 0 000 1000 (sign bit, exponent, mantissa).

Minimum Positive Normalized Number

The smallest number we can represent in this format is 1.0 × 2-3 (since the mantissa is normalized and starts with 1), which in decimal is 0.125. This is the minimum positive normalized number representable using the given 8-bit IEEE 754 configuration.