Chapter 3, Problem 1RQ

a.

Explanation of Solution

Conversion of binary numbers to decimal numbers:

• Given, binary number is 10.
• First, multiply the given binary number 10 with multiples of 2, with respect to their positions and then, add all the values to get the final answer.
• The following calculation describes the decimal notation of the binary number 10.

$\begin{array}{l}10=2^1*1+2^0*0\\ =2*1+1*0\\ =2+0\\ =2\end{array}$

• Therefore, the decimal notation of binary number ${\left(10\right)}_2$ is “${\left(2\right)}_{10}$”.

b.

Explanation of Solution

Conversion of binary numbers to decimal numbers:

• Given, binary number is 100.
• First, multiply the given binary number 100 with multiples of 2, with respect to their positions and then, add all the values to get the final answer.
• The following calculation describes the decimal notation of the binary number 100.

$\begin{array}{l}100=2^2*1+2^1*0+2^0*0\\ =4*1+2*0+1*0\\ =4+0+0\\ =4\end{array}$

• Therefore, the decimal notation of binary number ${\left(100\right)}_2$ is “${\left(4\right)}_{10}$”.

c.

Explanation of Solution

Conversion of binary numbers to decimal numbers:

• Given, binary number is 111.
• First, multiply the given binary number 111 with multiples of 2, with respect to their positions and then, add all the values to get the final answer.
• The following calculation describes the decimal notation of the binary number 111.

$\begin{array}{l}111=2^2*1+2^1*1+2^0*1\\ =4*1+2*1+1*1\\ =4+2+1\\ =7\end{array}$

• Therefore, the decimal notation of binary number ${\left(111\right)}_2$ is “${\left(7\right)}_{10}$”.

d.

Explanation of Solution

Conversion of binary numbers to decimal numbers:

• Given, binary number is 1011.
• First, multiply the given binary number 1011 with multiples of 2, with respect to their positions and then, add all the values to get the final answer.
• The following calculation describes the decimal notation of the binary number 1011.

$\begin{array}{l}1011=2^3*1+2^2*0+2^1*1+2^0*1\\ =8*1+4*0+2*1+1*1\\ =8+0+2+1\\ =11\end{array}$

• Therefore, the decimal notation of binary number ${\left(1011\right)}_2$ is “${\left(11\right)}_{10}$”.

e.

Explanation of Solution

Conversion of binary numbers to decimal numbers:

• Given, binary number is 1100.
• First, multiply the given binary number 1100 with multiples of 2, with respect to their positions and then, add all the values to get the final answer.
• The following calculation describes the decimal notation of the binary number 1100.

$\begin{array}{l}1100=2^3*1+2^2*1+2^1*0+2^0*0\\ =8*1+4*1+2*0+1*0\\ =8+4+0+0\\ =12\end{array}$

• Therefore, the decimal notation of binary number ${\left(1100\right)}_2$  is “${\left(12\right)}_{10}$”.

f.

Explanation of Solution

Conversion of binary numbers to decimal numbers:

• Given, binary number is 10010.
• First, multiply the given binary number 10010 with multiples of 2, with respect to their positions and then, add all the values to get the final answer.
• The following calculation describes the decimal notation of the binary number 10010.

$\begin{array}{l}10010=2^4*1+2^3*0+2^2*0+2^1*1+2^0*0\\ =16*1+8*0+4*0+2*1+1*0\\ =16+0+0+2+0\\ =18\end{array}$

• Therefore, the decimal notation of binary number ${\left(10010\right)}_2$ is “${\left(18\right)}_{10}$”.

g.

Explanation of Solution

Conversion of binary numbers to decimal numbers:

• Given, binary number is 10101.
• First, multiply the given binary number 10101 with multiples of 2, with respect to their positions and then, add all the values to get the final answer.
• The following calculation describes the decimal notation of the binary number 10101.

$\begin{array}{l}10101=2^4*1+2^3*0+2^2*1+2^1*0+2^0*1\\ =16*1+8*0+4*1+2*0+1*1\\ =16+0+4+0+1\\ =21\end{array}$

• Therefore, the decimal notation of binary number ${\left(10101\right)}_2$ is “${\left(21\right)}_{10}$”.

h.

Explanation of Solution

Conversion of binary numbers to decimal numbers:

• Given, binary number is 11111.
• First, multiply the given binary number 11111 with multiples of 2, with respect to their positions and then, add all the values to get the final answer.
• The following calculation describes the decimal notation of the binary number 11111.

$\begin{array}{l}11111=2^4*1+2^3*1+2^2*1+2^1*1+2^0*1\\ =16*1+8*1+4*1+2*1+1*1\\ =16+8+4+2+1\\ =31\end{array}$

• Therefore, the decimal notation of binary number ${\left(11111\right)}_2$ is “${\left(31\right)}_{10}$”.

h.

Explanation of Solution

Conversion of binary numbers to decimal numbers:

• Given, binary number is 11001101.
• First, multiply the given binary number 11001101 with multiples of 2, with respect to their positions and then, add all the values to get the final answer.
• The following calculation describes the decimal notation of the binary number 11001101.

$\begin{array}{l}11001101=2^7*1+2^6*1+2^5*0+2^4*0+2^3*1+2^2*1+2^1*0+2^0*1\\ =128*1+64*1+32*0+16*0+8*1+4*1+2*0+1*1\\ =128+64+0+0+8+4+0+1\\ =205\end{array}$

• Therefore, the decimal notation of binary number ${\left(11001101\right)}_2$ is “${\left(205\right)}_{10}$”.

j.

Explanation of Solution

Conversion of binary numbers to decimal numbers:

• Given, binary number is 11100011.
• First, multiply the given binary number 11100011 with multiples of 2, with respect to their positions and then, add all the values to get the final answer.
• The following calculation describes the decimal notation of the binary number 11100011.

$\begin{array}{l}11100011=2^7*1+2^6*1+2^5*1+2^4*0+2^3*0+2^2*0+2^1*1+2^0*1\\ =128*1+64*1+32*1+16*0+8*0+4*0+2*1+1*1\\ =128+64+32+0+0+0+2+1\\ =227\end{array}$

• Therefore, the decimal notation of binary number ${\left(11100011\right)}_2$ is “${\left(227\right)}_{10}$”.