Chapter 2, Problem 27E

Program Plan Intro

Types of representation for signed number:

• Signed magnitude representation
• One’s complement
• Two’s complement
• Excess-M representation

Signed magnitude representation:

In signed magnitude representation, an extra bit is used to represent the sign of the number. This extra bit is called as sign bit.

• The Most Significant Bit (MSB) in the binary number is referred as sign bit.
• In the signed magnitude representation, if the MSB is 0, then it is treated as positive sign and if the MSB is 1, then it is treated as negative sign.

Example:

Consider the decimal number 36 that can be represented as 00100100 and the decimal number -36 can be represented as 10100100.

One’s complement:

The binary numbers can be represented using one’s complement. Here, the value of every binary digit is complemented that is if the value is 1, it becomes 0 and if the value is 0 it becomes 1.

Example:

Consider the binary number 1101; the one’s complement of the given number is 0010.

Two’s complement:

Two’s complement is another way of representing the binary numbers. To implement the two’s complement to the number, the given binary number should be one’s complemented and then add 1 to the result obtained after one’s complement.

Example:

Consider the binary number 1011. First implement the one’s complement to the given binary number. The number becomes 0100. Then add 1 to the resultant obtained after the one’s complement. The result becomes 0101.

Excess-M Representation:

To get the excess-M representation of a given number,

• Add the given number with the M.
• Convert the obtained value to the binary representation.

Example:

To change to excess-7 representation, add the number $77$ with the 7.

$77+7=84$

Binary value of $84$ is $\text{1010100}$.

Hence, the result of excess-7 representation for $77$ is $\text{1010100}$.

Explanation of Solution

Representing the binary pattern using various formats in the table:

Unsigned Integer	4-bit binary value	Signed Magnitude	One’s complement	Two’s complement	Excess-7
0	0000	0	0	0	7
1	0001	1	1	1	8
2	0010	2	2	2	9
3	0011	3	3	3	10
4	0100	4	4	4	11
5	0101	5	5	5	12
6	0110	6	6	6	13
7	0111	7	7	7	14
8	1000	-0	-7	-8	15
9	1001	-1	-6	-7	16
10	1010	-2	-5	-6	17
11	1011	-3	-4	-5	18
12	1100	-4	-3	-4	19
13	1101	-5	-2	-3	20
14	1110	-6	-1	-2	21
15	1111	-7	-0	-1	22

Explanation:

First column represent the unsigned integer, and the corresponding 4-bit binary value is represented in the second column.

While representing the singed magnitude, if the Most Significant Bit is “+”, the given number is positive number and if the Most Significant Bit is “-”, the given number is negative number. For example, $1001$  the MSB bit is “1” and so the number is negative and the decimal equivalent for “$001$” is “1”.

While representing one’s complement, in binary number $1001$ the MSB bit is “1” and so the number is negative and the one’s complement is applied for the number $1001$ and the number becomes $0110$. So, the decimal equivalent for “$0110$” is 6”.

While representing two’s complement, in binary number $1001$ the MSB bit is “1” and so the number is negative. First one’s complement is applied for the number $1001$, the number becomes $0110$ and by adding “1” in the least significant bit, the equivalent two’s complement is attained. So, the number becomes $0111$. So, the decimal equivalent for $0111$ is “7”.

While representing excess-7, the given unsigned integer is added with the decimal number 7. Here, “1” is represented as “7”, “2” as “8”, and so on.