Binary Number System

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 Binary Number System:

The number system wit base(or radix) two is known  as the binary number system .Only two symbols   are used to represent numbers in this system and these are 0 and 1. These are known as bits. 


Decimal to binary conversion :

Any decimal number can be converted into its equivalent binary number. For integers ,the conversion is obtained by continuous division by 2 and keeping track of the remainder, while for fractional parts, the conversion is affected by continuous multiplication by 2 and keeping track of the integers generated .

Example: Convert (13)10 an equivalent base-2 number 


For Fractional number 

Example: Convert (0.65625)10 to an equivalent base-2 number 

Example: Express the 25.5 decimal numbers in the binary  from :


Binary to Decimal :

Any binary number can be converted into its equivalent decimal number using the weights assigned to each  bit position.

Example: find the decimal equivalent of the binary number (1 1 1 1 1)2



 

Binary  to octal:

Binary numbers can be converted into equivalent octal numbers by making groups of three bits starting from LSB and moving towards MSB for integer part of the number and then replacing each group of three bits by its octal representation.
Example:   Convert (1001110)2  to its octal equivalent 
Solution :                (1001110)2 = (001     001      110)2
                                                   = (1 1 6)8 


Binary to Hexadecimal :

Binary numbers can be converted into the equivalent hexadecimal numbers by making groups of four bits starting from LSB and moving toward MSB from integer part and replacing  each group of four bits by its hexadecimal representation.

Example: Convert the following binary numbers to their equivalent hex numbers.
(a) 10100110101111 

Solution: (10100110101111)2 = 0010    1001    1010    1111   
                                                    2            9            A        F

                  (10100110101111)2 =(29AF)16  


For fractional part:
For the fractional part, the above procedure is repeated starting from the bit next to the binary 
    point and moving towards the right.

(b) (0.00011110101101)2

Solution:  (0.00011110101101)2 = 0.0001    1110     1011    0100
                                                            1            E          B         4

                (0.00011110101101)2 = (0.1EB4)16    




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