Chapter 2, Problem 10E

a)

Explanation of Solution

Converting “10111.1101” to decimal equivalent:

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.

$\begin{array}{l}\text{10111}{\text{.1101 =1x2}}^{\text{4}}+0{\text{x2}}^3+{\text{1x2}}^2+{\text{1x2}}^1+{\text{1x2}}^0+{\text{1x2}}^{-1}+{\text{1x2}}^{-2}+{\text{1x2}}^{-3}+{\text{1x2}}^{-4}\\ =16+0+4+2+1+0.5+0.25+0+0.0625\\ =23.8125\end{array}$

Thus, the decimal equivalent of 10111.1101 is “$23.8125_{10}$”.

b)

Explanation of Solution

Converting “100011.10011” to decimal equivalent:

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.

$\begin{array}{l}\text{100011}{\text{.10011 =1x2}}^5{\text{+0x2}}^4{\text{+0x2}}^3+0{\text{x2}}^2+{\text{1x2}}^1+{\text{1x2}}^0+1{\text{x2}}^{-1}+0{\text{x2}}^{-2}+0{\text{x2}}^{-3}+{\text{1x2}}^{-4}+{\text{1x2}}^{-5}\\ =32+0+0+0+2+1+0.5+0+0+0.0625+0.03125\\ =35.59375\end{array}$

Thus, the decimal equivalent of 100011.10011 is “$23.8125_{10}$”.

c)

Explanation of Solution

Converting “1010011.10001” to decimal equivalent:

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.

$\begin{array}{l}\text{1010011}{\text{.10001=1x2}}^6+0{\text{x2}}^5{\text{+1x2}}^{\text{4}}+0{\text{x2}}^3+0{\text{x2}}^2+{\text{1x2}}^1+{\text{1x2}}^0+{\text{1x2}}^{-1}+0{\text{x2}}^{-2}+0{\text{x2}}^{-3}+0{\text{x2}}^{-4}+{\text{1x2}}^{-5}\\ =64+0+16+0+0+2+1+0.5+0+0+0+0+0.03125\\ =83.53125\end{array}$

Thus, the decimal equivalent of 1010011.10001 is “${83.53125}_{10}$”.

d)

Explanation of Solution

Converting “11000010.111” to decimal equivalent:

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.

$\begin{array}{l}\text{11000010}{\text{.111=1x2}}^7+1{\text{x2}}^6+0{\text{x2}}^5{\text{+0x2}}^{\text{4}}+0{\text{x2}}^3+0{\text{x2}}^2+1{\text{x2}}^1+0{\text{x2}}^0+{\text{1x2}}^{-1}+1{\text{x2}}^{-2}+1{\text{x2}}^{-3}\\ =128+64+0+0+0+0+2+0.5+0.25+0.125\\ =194.875\end{array}$

Thus, the decimal equivalent of 11000010.111 is “${194.875}_{10}$”.