Chapter 1, Problem 31CRP

a.

Explanation of Solution

Convert 13 in two’s complement representation in which each value is represented by 7 bits:

The process to convert base ten to two’s representation.

• Check the number is positive or negative.
• Convert the number from base ten to binary if the number is positive.
• Divide the number by 2 until the quotient is zero.
• If the number is negative, then first convert the number from base ten to binary and then complement the binary and add 1.
• Convert number $13$ to binary by following the above process.

$\begin{array}{ccc}2& 13& \\ 2& 6& 1\\ 2& 3& 0\\ 2& 1& 1\\ 2& 0& 1\end{array}$

• The binary rep

b.

Explanation of Solution

Convert $-13$ in two’s complement representation in which each value is represented by 7 bits:

The process to convert base ten to two’s representation.

• Check the number if it is positive or negative.
• Convert the number from base ten to binary if the number is positive.
• Divide the number by 2 until the quotient is zero.
• When the number is negative, then first convert the number from base ten to binary and then complement the binary and add 1.
• Convert number $-13$ to binary by following the above process.

$\begin{array}{ccc}2& 13& \\ 2& 6& 1\\ 2& 3& 0\\ 2& 1& 1\\ 2& 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom ${\left(-13\right)}_{10}={\left(1101\right)}_2$
• To make the above binary representation to 7 bits, put 0 before the binary number ${\left(-13\right)}_2={\left(0001101\right)}_2$
• Convert the above binary number into two’s complement representation

c.

Explanation of Solution

Convert $-1$ in two’s complement representation in which each value is represented by 7 bits:

The process to convert base ten to two’s representation.

• Check the number if it is positive or negative.
• Convert the number from base ten to binary if the number is positive.
• Divide the number by 2 until the quotient is zero.
• When the number is negative, then first convert the number from base ten to binary and then complement the binary and add 1.
• Convert number $-1$ to binary by following the above process.

$\begin{array}{ccc}2& 1& \\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom ${\left(1\right)}_{10}={\left(1\right)}_2$
• To make the above binary representation to 7 bits, put 0 before the binary number ${\left(1\right)}_2={\left(0000001\right)}_2$
• Convert the above binary number into two’s complement representation.
•  Complement the binary number then add 1 to the complement binary number

d.

Explanation of Solution

Convert $0$ in two’s complement representation in which each value is represented by 7 bits:

The process to convert base ten to two’s representation.

• Check the number is positive or negative.
• Convert the number from base ten to binary if the number is positive.
• Divide the number by 2 until the quotient is zero.
• When the number is negative, then first convert the number from base ten to binary and then complement the binary and add 1.
• Convert number $0$ to binary by following the above process.

$\begin{array}{ccc}2& 0& \\ & 0& 0\end{array}$

• The binary representation is the sequence of the remainder from down to bottom ${}_{}$

e.

Explanation of Solution

Convert $16$  in two’s complement representation in which each value is represented by 7 bits:

The process to convert base ten to two’s representation.

• Check the number is positive or negative.
• Convert the number from base ten to binary if the number is positive.
• Divide the number by 2 until the quotient is zero.
• When the number is negative, then first convert the number from base ten to binary and then complement the binary and add 1.
• Convert number $16$ to binary by following the above process.

$\begin{array}{ccc}2& 16& \\ 2& 8& 0\\ 2& 4& 0\\ 2& 2& 0\\ 2& 1& 0\\ & 0\end{array}$