Mathematical Operators in Binary

Table of contents

  1. Mathematical Operators in Binary
    1. Addition
    2. Subtraction
    3. Multiplication
    4. Division
    5. Complements in Binary system
      1. 1’s complement
      2. 2’s complement
    6. Bitwise Operators

Addition

1. 0 + 0 = 0
2. 0 + 1 = 1
3. 1 + 0 = 1
4. 1 + 1 = 1

Subtraction

1. 0 - 0 = 0
2. 1 - 0 = 1
3. 1 - 1 = 0

Multiplication

          1  1  0       (6)
      *   1  0  1       (5)
      ------------
          1  1  0 
       0  0  0  x
    1  1  0  x  x
    --------------
    1  1  1  1  0       (30)
    --------------   

Division

          1 1 1 1 0 / 1 0 1
          
         -    1 0 1                     1st 
        -------------
          1 1 0 0 1
         -    1 0 1                     2nd
        -------------
          1 0 1 0 0
         -    1 0 1                     3rd
        -------------
          0 1 1 1 1
         -    1 0 1                     4th
        -------------
            1 0 1 0
         -    1 0 1                     5th
        -------------
              1 0 1
         -    1 0 1                     6th 
        -------------                 -------
                  0                   ans = 6 (110)
        -------------                 -------

Complements in Binary system

The Binary system has a base of r = 2. The base has r = 2 so the Binary system two types of complements. They are:

  1. 1’s complement
  2. 2’s complement.

1’s complement

To find the 1’s complement of a given number, we should change all the 0’s to 1’s and then all the 1’s to 0’s. This process is called as 1’s complement. An Example for 1’s complement is as follows:

Given number        1  0  1  0  1
1's complement      0  1  0  1  0 

2’s complement

To find the 2’s complemen of a given number, we should add 1 to the Least Significant Bit(LSB) or the last bit of the 1’s complement of the given number. 2’s complement = 1’s complement + 1. An Example for 2’s complement is as follows:

In the given number 1 0 1 0 **1**. **1** is the Least Significant Bit(LSB).

Given number        1  0  1  0  1
1's complement      0  1  0  1  0 

add 1               +           1
                   ---------------
2's complement      0  1  0  1  1             
                   --------------- 

Bitwise Operators

Operator Explanation    
bit1 & bit2 The AND(&) operator is used to compare two bits and gives a result equal to 1 if both the bits are 1, or it returns 0, if any one bit is 0.    
bit1 bit2 The OR(** **) operator is used to compare two bits and gives a result equal to 1 if any one of the bit is 1 or both the bits are 1, or if both are 0 it returns 0.
bit1 ^ bit2 The EXCLUSIVE-OR(^) or also called as XOR(^) operator is used to compare two bits and returns 1 if any one of the bit is 1 and it returns 0 if both the bits are 0 or both are 1.    
~bit1 The COMPLEMENT(~) operator is similar to 1’s complement and can be used to convert all the 1’s to 0’s and all the 1’ to 0’s of the operand.    
128
0
64
0
32
0
16
0
8
0
4
0
2
0
1
0
=

AND
AND
AND
AND
AND
AND
AND
AND
128
0
64
0
32
0
16
0
8
0
4
0
2
0
1
0
=

=
=
=
=
=
=
=
=

128
0
64
0
32
0
16
0
8
0
4
0
2
0
1
0
= 0