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Converting Decimal Number system to Binary, Octal and Hexa Decimal with Examples and Vice Versa


Topics covered with examples: Converting Decimal Number system to Binary, Octal and Hexa Decimal with Examples and Vice Versa



 Number Systems in Computer Science and Conversion Method
Conversion of Decimal number system to other  number systems
Conversion of Decimal number system to binary number systems
Conversion of Decimal number system to octal number systems
Conversion of Decimal number system to hexadecimal number systems
Conversion of  non-decimal Number system to Other non-decimal number system

Conversion of other number systems to Decimal number system
Conversion of binary number systems to Decimal number system
Conversion of octal number systems to Decimal number system
Conversion of  hexadecimal number systems to Decimal number system

Q1. What is a Number System? What are different types of number system?

Number system
A number system is a set of digits, symbols and rules to express quantities in counting and calculations. Following are the most commonly used number systems:


Converting Decimal Number system to binary, octal and hex decimal with examples and vice versa
Number-systems-in-computer-science-easy-conversion-methods-with-pictures

Decimal Number System

It is most commonly used number system. It has 10 symbols{0,1,2,3,4,5,6,7,8,9}. It is also called base 10 number system. People normally use decimal number system to represent numbers.


Binary Number System

In binary number system there are only two symbols {0 and 1}. In a computer all data is represented by Binary number system. This is because computer’s electronic switches have only two states on and OFF. When a switch is ON it represents a 1 and when a switch is OFF, it represents a Zero.

Octal Number System

Octal number system consists of 8 digits{0,1,2,3,4,5,6,7}. Its base is 8. It is used as shorthand for long binary numbers. Each octal digit represents 3 binary digits.

Octal
Binary
0
000
1
001
2
010
3
011
4
100
5
101
6
110
7
111


Hexadecimal Number System

Hexadecimal number system is used to represent long binary numbers in an easy and short form. Its base is 16. It has sixteen symbols{0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}. Each hexadecimal digit represents 4 binary digits.
Hexadecimal
Binary
Hexadecimal
Binary
0
0000
8
1000
1
0001
9
1001
2
0010
A
1010
3
0011
B
1011
4
0100
C
1100
5
0101
D
1101
6
0110
E
1110
7
0111
F
1111



Q2. Conversion of Decimal (Integer part) to other systems

                                  
a) 4510 = (?)2   
b) 11910 =(?)8
c) 19010 =(?)16
2
45
2
22-1
2
11-0
2
5  -1
2
2  -1
2
1  -0

0  -1 
8
119
8
14-7
8
1  -6

0  -1
16
190
16
11-14(E)

0  -11(B)
   
           

=> 4510 = (101101)2                          11910=(167)8                       19010=(BE)16                       


Q3. Conversion of Decimal system (Fraction part) to other systems

a) (.23)10   = (?)2
b) (.225)10 = (?)8
c) (.225)10 = (?)16
Fraction
Integer
.23 X 2 = 0.46
.46
0
.46 X 2 = 0.92
.92
0
.92 X 2 = 1.84
.84
1
.84 X 2 = 1.68
.68
1
.68 X 2 = 1.36
.36
1

=> (.23)10   = (.00111)2        [Similar method is used for decimal (fraction) to Octal (multiply by 8)and hexadecimal(multiply by 16)]

Q4. Conversion of Other systems (Integer part) to Decimal


a)  (101101)2    =( ? )10     b)    (167)8  = (?)10                 (BE)16 =(?)10                      
   
  a) (101101)2                                                   
     =1x25+0x24+1x23+1x22+0x21+1x20 
    = 32   + 0   +   8   +  4   + 0     +1
    = 45 
b) (167)8          
     =1x82+6x81+7x80 
    = 64   + 48   +  7
    = 119 
c) (BE)2            
     =Bx161+Ex160 
    =  11x161+14x160 
    =  176   + 14
    =   190 


Q5. Conversion of Other systems(Fraction part) to Decimal number system

a)  (.110)2    =( ? )10     b)    (.75)8  = (?)10                 (.13)16 =(?)10                       
   
  a) (.110)2                                                        
     =.{1x2-1+1x2-2+0x2-3} 
     =.{ 1/2   + 1/4   +   0}
     =.{  .5    +  .25  +    0}
     =.75
b) (.75)8            
     =.{7x8-1+5x8-2  }
     =.{ 7/8   + 5/64 }
     = .{.875 + .0781}
     = .9531 
c) (.13)16           
     =.{1x16-1+3x16-2  }
     =.{1/16   +3/256} 
    =  .{.0625   + .01171}
    =   .0742 



      Conversion of  non-decimal Number system to Other non-decimal number system


      For example to convert OCTAL into HEXADECIMAL, use the follwing steps:

1)            Convert octal into decimal

2)            Convert decimal into hexadecimal

Topics covered with examples: Number Systems in Computer Science and Conversion Method

Conversion of Decimal number system to other  number systems

Conversion of Decimal number system to binary number systems
Conversion of Decimal number system to octal number systems
Conversion of Decimal number system to hexadecimal number systems
Conversion of  non-decimal Number system to Other non-decimal number system

Conversion of other number systems to Decimal number system
Conversion of binary number systems to Decimal number system
Conversion of octal number systems to Decimal number system
Conversion of  hexadecimal number systems to Decimal number system



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