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
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:
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