Chapter 2, Problem 12E

a)

Explanation of Solution

Converting “110001.10101” 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.

$\begin{array}{l}\text{110001}{\text{.10101 =1x2}}^5+{\text{1x2}}^{\text{4}}+0{\text{x2}}^3+0{\text{x2}}^2+0{\text{x2}}^1+{\text{1x2}}^0+{\text{1x2}}^{-1}+0{\text{x2}}^{-2}+{\text{1x2}}^{-3}+0{\text{x2}}^{-4}{\text{+1x2}}^{-5}\\ =32+16+0+0+0+1+0.5+0+0.125+0+0.03125\\ =49.65625\end{array}$

Thus, the decimal equivalent of 10001.10101 is “$49.65625_{10}$”.

b)

Explanation of Solution

Converting “111001.001011” 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.

$\begin{array}{l}\text{111001}{\text{.001011 =1x2}}^5{\text{+1x2}}^4{\text{+1x2}}^3+0{\text{x2}}^2+0{\text{x2}}^1+{\text{1x2}}^0+0{\text{x2}}^{-1}+0{\text{x2}}^{-2}+1{\text{x2}}^{-3}+0{\text{x2}}^{-4}+{\text{1x2}}^{-5}+{\text{1x2}}^{-6}\\ =32+16+8+0+0+1+0+0+0.125+0+0.03125\text{}+0.015625\\ =57.171875\end{array}$

Thus, the decimal equivalent of 111001.001011 is “$57.171875_{10}$”.

c)

Explanation of Solution

Converting “1001001.10101” 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.

$\begin{array}{l}\text{1001001}{\text{.10101=1x2}}^6+0{\text{x2}}^5{\text{+0x2}}^{\text{4}}+1{\text{x2}}^3+0{\text{x2}}^2+0{\text{x2}}^1+1{\text{x2}}^0+{\text{1x2}}^{-1}+0{\text{x2}}^{-2}+1{\text{x2}}^{-3}+0{\text{x2}}^{-4}+1{\text{x2}}^{-5}\\ =64+0+0+8+0+0+1+0.5+0+0.125+0+0.03125\\ =73.65625\end{array}$

Thus, the decimal equivalent of 1001001.10101 is “$73.65625_{10}$”.

d)

Explanation of Solution

Converting “11101001.110001” 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.

$\begin{array}{l}\text{11101001}{\text{.110001=1x2}}^7+1{\text{x2}}^6+1{\text{x2}}^5{\text{+0x2}}^{\text{4}}+1{\text{x2}}^3+0{\text{x2}}^2+0{\text{x2}}^1+1{\text{x2}}^0+1{\text{x2}}^{-1}+1{\text{x2}}^{-2}+0{\text{x2}}^{-3}+0{\text{x2}}^{-4}+0{\text{x2}}^{-5}+1{\text{x2}}^{-6}\\ =128+64+32+0+8+0+0+1+0.5+0.25+0+0+0+0.015625\\ =233.765625\end{array}$

Thus, the decimal equivalent of 11101001.110001 is “$233.765625_{10}$”.