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Saturday, June 1, 2019

Number Systems in Computer Science and Conversion Method

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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:


Number-systems-in-computer-science-easy-conversion-methods-with-pictures
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                          4510=(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





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