# Perfect Computer Notes

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# Number Systems in Computer Science and Conversion Method

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

### 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 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: