Chapter 2, Problem 8E

Explanation of Solution

To represent “7” using “5” bits:

• Using “5” bits, one can mean “$2^5$”, which means “32” unique integers.
• If those integers are numbers from “0” to “31, then one can mean the digits from “0” to “4” with bits.
• The “32” possible bit configurations is as follows,

16	8	4	2	1	Integer
0	0	0	0	0	0
0	0	0	0	1	1
0	0	0	1	0	2
0	0	0	1	1	3
0	0	1	0	0	4
0	0	1	0	1	5
0	0	1	1	0	6
0	0	1	1	1	7
0	1	0	0	0	8
0	1	0	0	1	9
0	1	0	1	0	10
0	1	0	1	1	11
0	1	1	0	0	12
0	1	1	0	1	13
0	1	1	1	0	14
0	1	1	1	1	15
1	0	0	0	0	16
1	0	0	0	1	17
1	0	0	1	0	18
1	0	0	1	1	19
1	0	1	0	0	20
1	0

a.

Explanation of Solution

Addition of 1’s complement representation of numbers:

• The 1’s complement representation of positive number “7” is simply expressed as binary number.
• The number “7” in 8-bit 1’s complement representation is “00000111”.
• The negative number “-7” is represented in 1’s complement.
• To represent “-7” in 1’s complement, express the corresponding positive number in binary and takes 1’s complement of that positive number.
• The binary representation of “7” is “00000111” and the 1’s complement of this number is “11111000”.
• Hence the 1’s complement of the number “-7” is “11111000”.
• Then the addition is performed as follows,

$\begin{array}{c}\text{1's complement of 7 }\Rightarrow \text{00000111}\\ \text{1's complement of -7 }\Rightarrow \end{array}$

b.

Explanation of Solution

Addition of signed magnitude representation of numbers:

• The signed binary numbers are commonly represented using signed magnitude notation.
• The MSB is used as the sign bit to denote whether the number is positive or negative.
• The number is positive, if the MSB is zero and the number is negative, if the MSB is one.
• The signed magnitude representation of positive number is simply expressed as binary number.
• Hence, the signed magnitude representation of the number “7” is “00111”.
• To represent the signed magnitude representation of negative number “-7”,
  • The binary representation of the number “7” is “00111” and to denote that the number is negative, the MSB of that number must be “1”.
  • Therefore, the representation of the number “-7” in signed magnitude is “10111”.
• Then the addition is performed as follows,

$\begin{array}{c}\text{Signed magnitude of 7 }\Rightarrow \text{00000111}\\ \end{array}$

c.

Explanation of Solution

Largest positive integer in n-bit 2’s complement code:

The largest positive integer in n-bit 2’s complement is “$2^{n-1}-1$”.

Explanation:

• One bit is used to indicate the sign of the value in 2’s complement code
• This bit is known as sign bit.
• The Left most Significant Bit (LSB) is used to denote the sign of the integer

d.

Explanation of Solution

Greatest magnitude negative integer in n-bit 2’s complement code:

The greatest magnitude negative integer in n-bit 2’s complement code is “$-2^{n-1}$”.

Explanation:

• One bit is used to indicate the sign of the value in 2’s complement code
• This bit is known as sign bit.
• The Left most Significant Bit (LSB) is used to denote the sign of the integer.
• The weight of LSB is the negative of corresponding equivalent power of “2”