Chapter 1, Problem 33CRP

a.

Explanation of Solution

Convert $5+1$ into two’s complement notation, converting any subtraction problem to an equivalent addition problem and perform the addition and convert the answer to base ten notations:

The condition of overflow is:

• Check the number is positive or negative.
• When the number is positive, convert the number from base ten to binary.
• When the number is negative, convert the number from base ten to binary, complement the binary number and add 1.
• Convert number 1 into two’s complement notation.

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

• 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 5 bits, put 0 before the binary number ${\left(1\right)}_2={\left(00001\right)}_2$
• The two’s complement notation of 1 is $00001$.
• Convert number 5 into two’s complement notation.

$\begin{array}{ccc}2& 5& \\ 2& 2& 1\\ 2& 1& 0\\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom

b.

Explanation of Solution

Convert $5-1$ into two’s complement notation, converting any subtraction problem to an equivalent addition problem and perform the addition and convert the answer to base ten notations:

The condition of overflow is:

• Check the number is positive or negative.
• When the number is positive, convert the number from base ten to binary.
• When the number is negative, convert the number from base ten to binary, complement the binary number and add 1.
• Convert number 1 into two’s complement notation.

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

• 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 5 bits, put 0 before the binary number ${\left(1\right)}_2={\left(00001\right)}_2$
• Take the complement of the above binary number.

$\begin{array}{ccccc}0& 0& 0& 0& 1\\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ 1& 1& 1& 1& 0\end{array}$

• Add 1 to the above complement binary number.

$\begin{array}{cccccc}& 1& 1& 1& 1& 0\\ +& & & & & 1\\ & 1& 1& 1& 1& 1\end{array}$

• The two’s complement notation of ${\left(-1\right)}_2$ is $11111$
• Convert number 5 into two’s complement notation

c.

Explanation of Solution

Convert $12-5$ into two’s complement notation, converting any subtraction problem to an equivalent addition problem and perform the addition and convert the answer to base ten notations:

The condition of overflow is:

• Check the number is positive or negative.
• When the number is positive, convert the number from base ten to binary.
• When the number is negative, convert the number from base ten to binary, complement the binary number and add 1.
• Convert number 12 into two’s complement notation.

$\begin{array}{ccc}2& 12& \\ 2& 6& 0\\ 2& 3& 0\\ 2& 1& 1\\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom ${\left(12\right)}_{10}={\left(1100\right)}_2$ .
• To make the above binary representation to 5 bits, put 0 before the binary number ${\left(12\right)}_{10}={\left(01100\right)}_2$ .
• The two’s complement notation of ${\left(12\right)}_{10}$ is $01100$.
• Convert number $-5$ into two’s complement notation.

$\begin{array}{ccc}2& 5& \\ 2& 2& 1\\ 2& 1& 0\\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom ${\left(5\right)}_{10}={\left(101\right)}_2$
• To make the above binary representation to 5 bits, put 0 before the binary number ${\left(5\right)}_{10}={\left(00101\right)}_2$
• Take the complement of the above binary number

d.

Explanation of Solution

Convert $8-7$ into two’s complement notation, converting any subtraction problem to an equivalent addition problem and perform the addition and convert the answer to base ten notations:

The condition of overflow is:

• Check the number is positive or negative.
• When the number is positive, convert the number from base ten to binary.
• When the number is negative, convert the number from base ten to binary, complement the binary number and add 1.
• Convert number 8 into two’s complement notation.

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

• The binary representation is the sequence of the remainder from down to bottom ${\left(8\right)}_{10}={\left(1000\right)}_2$
• To make the above binary representation to 5 bits, put 0 before the binary number ${\left(8\right)}_{10}={\left(01000\right)}_2$
• The two’s complement notation of ${\left(8\right)}_{10}$ is $01000$.
• Convert number $7$ into two’s complement notation.

$\begin{array}{ccc}2& 7& \\ 2& 3& 1\\ 2& 1& 1\\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom ${\left(7\right)}_{10}={\left(111\right)}_2$
• To make the above binary representation to 5 bits, put 0 before the binary number ${\left(7\right)}_{10}={\left(00111\right)}_2$
• Take the complement of the above binary number

e.

Explanation of Solution

Convert $12+5$ into two’s complement notation, converting any subtraction problem to an equivalent addition problem and perform the addition and convert the answer to base ten notations:

The condition of overflow is:

• Check the number is positive or negative.
• When the number is positive, convert the number from base ten to binary.
• When the number is negative, convert the number from base ten to binary, complement the binary number and add 1.
• Convert the number 12 into two’s complement notation.

$\begin{array}{ccc}2& 12& \\ 2& 6& 0\\ 2& 3& 0\\ 2& 1& 1\\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom ${\left(12\right)}_{10}={\left(1100\right)}_2$
• To make the above binary representation to 5 bits, put 0 before the binary number ${\left(12\right)}_{10}={\left(01100\right)}_2$
• Take the complement of the above binary number.

$\begin{array}{ccccc}0& 1& 1& 0& 0\\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ 1& 0& 0& 1& 1\end{array}$

• Add 1 to the above complement binary number.

$\begin{array}{cccccc}& 1& 0& 0& 1& 1\\ +& & & & & 1\\ & 1& 0& 0& 0& 0\end{array}$

• The two’s complement notation of ${\left(-11\right)}_{10}$ is $10101$

f.

Explanation of Solution

Convert $5-11$ into two’s complement notation, converting any subtraction problem to an equivalent addition problem and perform the addition and convert the answer to base ten notations:

The condition of overflow is:

• Check the number is positive or negative.
• When the number is positive, convert the number from base ten to binary.
• When the number is negative, convert the number from base ten to binary, complement the binary number and add 1.
• Convert the number $-11$  into two’s complement notation.

$\begin{array}{ccc}2& 11& \\ 2& 5& 1\\ 2& 2& 1\\ 2& 1& 0\\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom ${\left(11\right)}_{10}={\left(1011\right)}_2$
• To make the above binary representation to 5 bits, put 0 before the binary number ${\left(11\right)}_{10}={\left(01011\right)}_2$
• Take the complement of the above binary number.

$\begin{array}{ccccc}0& 1& 0& 1& 1\\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ 1& 0& 1& 0& 0\end{array}$

• Add 1 to the above complement binary number.

$\begin{array}{cccccc}& 1& 0& 1& 0& 0\\ +& & & & & 1\\ & 1& 0& 1& 0& 1\end{array}$

• The two’s complement notation of ${\left(-11\right)}_{10}$ is $10101$