In a 32-bit floating number with normalized mantissa and excess-64 exponent base 16, the number $16^{-65}$ denotes

Question

In a 32-bit floating number with normalized mantissa and excess-64 exponent base 16, the number $16^{-65}$ denotes

1. Floating point overflow.

2. Negative floating point overflow.

3. All 0's in the exponent and mantissa fields.

4. The minimum representable positive number .

I think that minimum representable number should be $1 \times 16^{-63}$ because the minimum mantissa should be 1 and and the possible exponent range in bias form is from 1 to 127 (where 1 corresponds to most negative exponent i.e. -63, and 127 corresponds to most positive exponent i.e. 63)

So according to me, the answer is: A positive floating point underflow. Please correct me if i am wrong. The IEEE-754 representation is confusing me.

Someone also told me something along the lines of " the mantissa part is always taken as 0.M if the base is something other than 2". However I don't have any reference for this statement.

Answer 1

The format you are using is not IEEE-754 but an IBM FP format. The mantissa is fractional, so the binary representation 0001_0000_0000_0000_0000_0000 means 0.1 (base 16). With a zero value in the exponent field, the exponent would be -64. So you can see that $16^{-65}$ would be the smallest normalized positive value ($1 * 16^{-1} * 16^{-64}$)