Binary Representation of Integers

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Binary Representation of Integers


Binary number can be represented only by using 0’s and 1’s, but cannot use the sign (-) to denote the negative number or sign (+) to denote the positive number. So, it must be either 0 or 1. There are three methods to represent binary number. They are
      1.     Sign and magnitude method
      2.    One’s complement method
     3.    Two’s complement method

·       Sign and magnitude method
In this method, first bit is considered as a sign bit. Here positive number starts with 0 and negative number starts with 1.
Example:            25

25/2 = 12    1
12/2 = 6      0
6/2 = 3        0
3/2 = 1        1
1/2 = 0        1
So the binary number is (11001)2. If we take the size of the word is 1 byte, then the number 25 will be represented as 00011001.
Suppose, if the number is -25, and then it will be represented as 10011001.

·       One’s Complement Method
In this method, the positive number is represented as same as the binary number. If the number is negative, then we need to find one’s complement of a binary number. The one’s complement of a binary number will replace every 0 with 1 and vice- versa.
Example
(i)                          Represent 86 in one’s complement method (1 byte representation)
86/2 =43  0
43/2 =21  1
21/2 =10  1
10/2 =5    0
5/2 =2      1
2/2 =1      0
1/2 =0      1
The binary number is 1010110
1 byte representation of number 86 is 01010110

(ii)                        Represent -55 in one’s complement method (1 byte representation)
55/2 =27  1
27/2 =13  1
13/2 =6    1
6/2 =3      0
3/2 =1      1
1/2 =0      1
The binary number is 110111
1 byte representation is 00110111
The given number is negative; hence we need to calculate one’s complement
One’s complement of 00110111 is 11001000 (convert 1 into 0 and 0 into 1)
Thus, the 1 byte representation of number -55 is 11001000.

·       Two’s Complement method
In this method, the positive number is represented as the binary number. If the number is negative, then we need to calculate two’s complement of a binary number. The two’s complement of a binary number is calculated by adding 1 to its one’s complement.
Example
(i) Represent 87 in two’s complement method (1 byte representation)
87/2 =43 1
43/2 =21 1
21/2 =10 1
10/2 =5   0
5/2 =2     1
2/2 =1     0
1/2 =0     1
The binary number is 1010111
Hence, the 1 byte representation of number 86 is 01010111

(ii) Represent -54 two’s complement method (1 byte representation)
54/2 =27 0
27/2 =13 1
13/2 =6   1
6/2 =3     0
3/2 =1     1
1/2 =0     1
The binary number is 110110
Hence, the 1 byte representation is 00110110
The given number is negative; hence we need to calculate two’s complement.
One’s complement of 00110110 is 11001001 (convert 1 into 0 and 0 into 1)

Add 1 to one’s complement
               1
11001001 (1+1=2, binary equivalent =10)
+               1
11001010
Thus, 1 byte representation of number -54 is 11001010

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