Chapter 1.6, Problem 2QE

a.

Explanation of Solution

Determine the two’s complement representation of the base ten representations 6:

• In the base ten representations 6, the number is a positive value.
• Convert the base ten representations to its equivalent two’s compliment representation using 8 bits pattern, divide the number 6 until the quotient is zero as shown below,

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

b.

Explanation of Solution

Determine the two’s complement representation of the base ten representations $-6$:

• In the base ten representations $-6$, the number is a negative value.
• Convert the base ten representations to its equivalent two’s compliment representation using 8 bits pattern, divide the number 6 until the quotient is zero as shown below,

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

• Write the binary representation from the above table in sequence of remainder from down to top, then padding the bit pattern to 8 bits

c.

Explanation of Solution

Determine the two’s complement representation of the base ten representations $-17$:

• In the base ten representations $-17$, the number is a negative value.
• Convert the base ten representations to its equivalent two’s compliment representation using 8 bit pattern, divide the number 17 until the quotient is zero as shown below,

$\begin{array}{ccc}2& 17& \\ 2& 8& 1\\ 2& 4& 0\\ 2& 2& 0\\ 2& 1& 0\\ & 0& 1\end{array}$

• Write the binary representation from the above table in sequence of remainder from down to top, then padding the bit pattern to 8 bits

d.

Explanation of Solution

Determine the two’s complement representation of the base ten representations 13:

• In the base ten representations 13, the number is a positive value.
• Convert the base ten representations to its equivalent two’s compliment representation using 8 bit pattern, divide the number 13 until the quotient is zero as shown below,

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

e.

Explanation of Solution

Determine the two’s complement representation of the base ten representations $-1$:

• In the base ten representations $-1$, the number is a negative value.
• Convert the base ten representations to its equivalent two’s compliment representation using 8 bits pattern, divide the number 1 until the quotient is zero as shown below,

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

• Write the binary representation from above table in sequence of remainder from down to top, then padding the bit pattern to 8 bits. The binary pattern of given number is,

$\begin{array}{c}1={\left(1\right)}_2\\ ={\left(00000001\right)}_2\end{array}$

• Convert the above binary representation to its equivalent two’s complement representation, complement the binary pattern and then add 1 to it

f.

Explanation of Solution

Determine the two’s complement representation of the base ten representations 0:

• In the base ten representations 0, the number is a positive value.
• Convert the base ten representations to its equivalent two’s compliment representation using 8 bits pattern, divide the number 0 until the quotient is zero as shown below,

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

• Write the binary representation from the above table in sequence of remainder from down to top, then padding the bit pattern to 8 bits