Chapter 2, Problem 45E

a)

Explanation of Solution

Finding the largest positive number and the smallest positive number:

To store the number in normalized form, the decimal point should be moved either left or right so that only single digit will be left before the decimal point.

Largest positive number:

Largest positive number can be formed by placing all ones in the significand position and placing “3” in the exponent position.

$0.1111\times 2^3={111.1}_2$

Converting binary fraction (111.1) to decimal:

Step 1: First separate the given binary fractions into two parts, left binary part and right binary fractions.

Step 2: To convert the binary fraction into decimal number, multiply the left part of the binary number with the positive integers for base 2.

Step 3: Multiply the right part of the binary number with the negative integers for base 2.

Step 4: Add all the values to get the final result.

$\begin{array}{l}\text{111}\text{.1 }{\text{=1x2}}^2+{\text{1x2}}^1+{\text{1x2}}^0+{\text{1x2}}^{-1}\\ =4+2+1+0.5\\ =7+0.5\\ =7.5\end{array}$

Thus, the largest positive number formed by 3-bit exponent and 4-bit significant is ${7.5}_{10}$

Smallest positive number:

Smallest positive number can be formed by placing one at the LSB bit of the significand position and placing 4 at the exponent position.

${0.1}_2\times 2^{-4}=0_{}$

b)

Explanation of Solution

Bias used in the exponent when all exponents are non-negative:

Assuming that all the exponents are non-negative the bias used in the exponent will be 4...