A
number system is the set of symbols used to express quantities as the basis for
counting, determining order, comparing amounts, performing calculation and representing
value
There
are two kinds of number system: -
1) Non positional number system: - In
this number system, we have symbols such as I for 1, V for 5, X for 10,L for
50, C=100, D for 500 and M=1000. Each
value represents the same value regardless of its position of a number.
Example: III = 1+1+1=3
CX=100+10=101
2) Positional number system: - In a position
a number system, there are only few symbols a called digit, the symbols represents
different values depending on the position they occupy in a number.
Example, In decimal number system,
245
5 is in unit position 5
4 is in tenth position
40
2 is in hundred position 200
=
245
The position number systems are of
the following types
a)
Decimal
number system b) Binary Number system
b)
Octal
number system d) Hexadecimal
number system
Decimal
number system: - The number system that uses base or radix as 10 is called
decimal number system. The digits used are 0,1,2,3,4,5,6,7,8,9. Each position
in decimal number system represents a power of base 10.
Binary
number system: - The number system that uses base or radix as 2 is called
Binary number system. The digits used are 0 and 1. Each position in binary
number system represents a power of base 2.
Octal
number system- The number system that uses base or radix as 8 is called Octal
number system. The digits used are 0,1,2,3,4,5,6,7. Each position in octal
number system represents a power of base 8.
Hexadecimal
number system: - The number system that uses base or radix as 16 is called hexadecimal
number system. There are 16 digits or symbols. They are 0,1,2,3,4,5,6,7,8,9 and
A,B,C,D,E and F. Each position in hexadecimal number system represents a power
of base 16.
Binary
|
Decimal
|
Octal
|
Hexadecimal
|
0000
|
0
|
0
|
0
|
0001
|
1
|
1
|
1
|
0010
|
2
|
2
|
2
|
0011
|
3
|
3
|
3
|
0100
|
4
|
4
|
4
|
0101
|
5
|
5
|
5
|
0110
|
6
|
6
|
6
|
0111
|
7
|
7
|
7
|
1000
|
8
|
8
|
|
1001
|
9
|
9
|
|
1010
|
10
|
A
|
|
1011
|
11
|
B
|
|
1100
|
12
|
C
|
|
1101
|
13
|
D
|
|
1110
|
14
|
E
|
|
1111
|
15
|
F
|
Conversion
1)Decimal to binary
I.
Convert
(19)10 into binary number system answer
(10011)2
II.
Convert
(45)10 into (?)2 answer
(101101)2
III.
Convert
(87)10 into (?)2 answer (1010111)
2)Binary to decimal
I.
Convert
(1101)2 to (?)10
=1*23+
1*22+0*21+1*20
=1*8+1*4+0+1
=13
(1101)2
= (13)10
II.
Convert
()2 to (?)10
3) Decimal to octal
I.
Convert
(45)10 to (?)8
answer (55)
II.
Convert
(123)10 to (?)8 answer
(173)
4) Octal to decimal
I.
Convert
(456)8 to (?)10
=4*82+5*81+6*80
=4*64
+ 5*8+6*1
=256
+40+6
=302
(456)8
= (302)10
II.
Convert
(3176)8 to (?)10
=3*83+1*82+7*81+6*80
=3*512+64+7*8+6*1
=1536+64+56+6
=1662
5) Convert hexadecimal to decimal
number system
(3F7A)16
to (?)10
=3*163+F*162+7*161+A*160
=3*163+15*162+7*161+10*160
=3*4086+
15*256+7*16+10*1
=12288+3840+112+10
=16250
(3F7A)16
= (16250)10
(7DE)16
to (?)10
=7*162+D*161+E*160
=7*162+13*161+14*160
=7*256+13*16+14*1
=1792+208+14
=2014
(7DE)16=
(2014)10
6) Convert decimal to hexadecimal
i)
(333)10
into hexadecimal number. Answer (14D)
ii)
(123)10
into (?)16 ans= (7B)
7) Convert binary to octal
i)
Convert (1101010)2 into octal
Here
Binary number 001
101 010
Hexadecimal equivalent 1
5 2
(1101010)2= (152)8
8) Covert octal to binary
i)
Convert
(562)8 into binary
Here
Octal
number 5 6 2
Binary
number equivalent 101 110
010
(562)8
= (101110010)2
ii)
Convert
(375)8 into binary
Here
Octal
number 3 7
5
Binary
number equivalent 011 111
101
(375)8
= (011111101)2
9) Convert binary to hexadecimal
number system
i)
Convert
(11010011)2 into hexadecimal
Here
Binary
number 1101 0011
Octal
equivalent D 3
(1101001)2= (D3)16
ii)
Convert
(111011)2 into into hexadecimal
Here
Binary
number 0011 1011
Hexadecimal
equivalent 3 B
(1101001)2= (3B)16
10) Convert hexadecimal to binary
number system
i)
Convert(2AB)16
into binary
Hexadecimal
number 2 A B
Binary
equivalent 0010 1010
1011
(2AB)16=
(001010101011)2
ii)
(COFFEE)16
into binary
Here
Hexadecimal
number C 0
F F E E
Binary
equivalent 1100 0000 1111
1111 1110 1110
(COFFEE)16=
(110000001111111111101110)2
Arithmetic operation on binary number
Different arithmetic operations such
as additions, subtraction, multiplication and division can be performed on binary
numbers.
Adding binary numbers
Rules
0 + 0 = 0
|
0 +1= 1
|
1 +0=1
|
1 +1=10 ( carry 1)
|
Note :
1+1+carry=1(carry1)
Adding 11101 and 01101
11101 1001 1011 111
01101 1010 1110 110
101010 10011 11001 1101
Subtracting binary number
Rules:
0 -0 = 0
|
0 -1= 1 ( borrow 1 from the left
column.)
|
1 -0=1
|
1 -1=0
|
01011
111101 1011 1101001 11001010
1001
-00010 -100101 -0110 -
11110 -10011011 -
10
01001 011000 0101 1001011 00101111 0111
Multiplying
binary number
0 X 0 = 0
|
0 X 1= 0
|
1 X 0=0
|
1 X 1= 1
|
10101 1001
x101 x 1010
10101 0000
00000x 1001X
10101xx 0000XX
1101001 1001XXX
1011010
Dividing
binary numbers
Complement
of binary number system: complement of the binary number system is the reverse
or just opposite of given umber system. In binary number system, there are two
complement.
a) 1's
complement
b) 2's
complement
1's complement: - 1's
complement of
binary number is just reverse or opposite of given which is obtained by
replacing 1 into 0 and 0 into 1.
Example 101001=010110
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