Chapter 2, Problem 11E

a)

Explanation of Solution

Converting “100001.111” 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.

$\begin{array}{l}\text{100001}{\text{.111 =1x2}}^5+0{\text{x2}}^{\text{4}}+0{\text{x2}}^3+0{\text{x2}}^2+0{\text{x2}}^1+{\text{1x2}}^0+{\text{1x2}}^{-1}+{\text{1x2}}^{-2}+{\text{1x2}}^{-3}\\ =32+0+0+0+0+1+0.5+0.25+0.125\\ =33.875\end{array}$

Thus, the decimal equivalent of 100001.111 is “$33.875_{10}$”.

b)

Explanation of Solution

Converting “111111.10011” 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.

$\begin{array}{l}\text{111111}{\text{.10011 =1x2}}^5{\text{+1x2}}^4{\text{+1x2}}^3+1{\text{x2}}^2+{\text{1x2}}^1+{\text{1x2}}^0+1{\text{x2}}^{-1}+0{\text{x2}}^{-2}+0{\text{x2}}^{-3}+{\text{1x2}}^{-4}+{\text{1x2}}^{-5}\\ =32+16+8+4+2+1+0.5+0+0+0.0625+0.03125\\ =63.59375\end{array}$

Thus, the decimal equivalent of 111111.10011 is “$63.59375_{10}$”.

c)

Explanation of Solution

Converting “1001100.1011” 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.

$\begin{array}{l}\text{1001100}{\text{.1011=1x2}}^6+0{\text{x2}}^5{\text{+0x2}}^{\text{4}}+1{\text{x2}}^3+1{\text{x2}}^2+0{\text{x2}}^1+0{\text{x2}}^0+{\text{1x2}}^{-1}+0{\text{x2}}^{-2}+1{\text{x2}}^{-3}+1{\text{x2}}^{-4}\\ =64+0+0+8+4+0+0+0.5+0+0.125+0.0625\\ =76.6875\end{array}$

Thus, the decimal equivalent of 1001100.1011 is “$76.6875_{10}$”.

d)

Explanation of Solution

Converting “10001001.0111” 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.

$\begin{array}{l}\text{10001001}{\text{.0111=1x2}}^7+0{\text{x2}}^6+0{\text{x2}}^5{\text{+0x2}}^{\text{4}}+1{\text{x2}}^3+0{\text{x2}}^2+0{\text{x2}}^1+1{\text{x2}}^0+0{\text{x2}}^{-1}+1{\text{x2}}^{-2}+1{\text{x2}}^{-3}+1{\text{x2}}^{-4}\\ =128+0+0+0+8+0+0+1+0+0.25+0.125+0.0625\\ =137.4375\end{array}$

Thus, the decimal equivalent of 10001001.0111 is “$137.4375_{10}$”.