Chapter 3, Problem 40E

a.

Program Plan Intro

Given information:

• The value of A is given as 11111110.
• The value of B is given as 00000010.

Two’s complement:

• Change the entire bit “0” to bit“1” and all the bit “1” to bit “0”.
• Add the Value “1” to the above converted binary string.
• It results into two’s complement of a given binary number.

Explanation of Solution

Calculating value for an expression “A+B” using two’s complement:

To evaluate the result of the given expression “A+B”, substitute the value of A as “11111110” and B as “00000010” in the expression.

A+B=>

In this expression two’s complement is not needed since A and B is non-negative values.

$\begin{array}{r}\text{11111110}\\ \text{+}\\ \underset{\_}{\text{00000010}}\\ \text{00000000}\end{array}$

Therefore, the calculated value of given expression “A+B” is “00000000”.

Explanation of Solution

b.

Calculating value for an expression “A-B” using two’s complement:

To evaluate the result of the given expression “A-B”, substitute the value of A as “11111110” and B as “00000010” in the given expression.

A-B=>

In this expression two’s complement is needed for B since it contains negative value.

To calculate two’s complement of value B=>

$\begin{array}{c}\text{00000010}\\ \Downarrow \\ 11111101\\ +\\ \underset{\_}{1}\\ 11111110\end{array}$

Two’s complement of value B value is “11111110”.

 Adding the values of A and converted B is,

$\begin{array}{r}\text{11111110}\\ \text{+}\\ \underset{\_}{\text{11111110}}\\ \text{11111100}\end{array}$

Overflow bit has occurred in the above calculation. This extra bit is ignored to assign the result in its allocated bits.

Therefore, the calculated value of given expression “A-B” is “11111100”.

Explanation of Solution

c.

Calculating value for an expression “B-A” using two’s complement:

To evaluate the result of the given expression “B-A”, substitute the value of A as “11111110” and B as “00000010” in the given expression.

B-A=>

In this expression two’s complement is needed for A since it contains negative value.

To calculate two’s complement of value A=>

$\begin{array}{c}\text{11111110}\\ \Downarrow \\ 00000001\\ +\\ \underset{\_}{1}\\ 00000010\end{array}$

Two’s complement of value B value is “00000010”.

Adding the values of converted A and B is,

$\begin{array}{r}\text{00000010}\\ \text{+}\\ \underset{\_}{\text{00000010}}\\ \text{00000100}\end{array}$

Therefore, the calculated value of given expression “B-A” is “00000100”.

Explanation of Solution

d.

Calculating value for an expression “-B” using two’s complement:

To evaluate the result of the given expression “-B”, substitute the value of B as “00000010” in the given expression.

-B=>

In this expression two’s complement is needed for B since it contains negative value.

To calculate two’s complement of value B=>

$\begin{array}{c}\text{00000010}\\ \Downarrow \\ 11111101\\ +\\ \underset{\_}{1}\\ 11111110\end{array}$

Two’s complement of value B value is “11111110”.

Therefore, the calculated value of given expression “-B” is “11111110”.

Explanation of Solution

e.

Calculating value for an expression “-(-A)” using two’s complement:

The given expression can be written as,

-(-A)=+A

Two’s complement is not needed for the value “A” since it contains positive value.

Therefore, the calculated value of given expression “-(-A)” is same as the given value “11111110”.