NUMBER SYSTEM AND CODES
INTRODUCTION:-
The term digital refers to a process that is achieved by using discrete unit.
In number system there are different symbols and each symbol has an absolute value and also has
place value.
RADIX OR BASE:-
The radix or base of a number system is defined as the number of different digits which can occur in
each position in the number system.
RADIX POINT :-
The generalized form of a decimal point is known as radix point. In any positional number system the
radix point divides the integer and fractional part.
Nr = [ Integer part . Fractional part ]
↑
Radix point
NUMBER SYSTEM:-
In general a number in a system having base or radix ‘ r ’ can be written as
an an-1 an-2 …………… a0 . a -1 a -2 …………a - m
This will be interpreted as
Y = an x rn + an-1 x rn-1 + an-2 x rn-2 + ……… + a0 x r0 + a-1 x r -1 + a-2 x r -2 +………. +a -m x r –m
where Y = value of the entire number
an = the value of the nth digit
r = radix
TYPES OF NUMBER SYSTEM:-
There are four types of number systems. They are
1. Decimal number system
2. Binary number system
3. Octal number system
4. Hexadecimal number system
DECIMAL NUMBER SYSTEM:-
The decimal number system contain ten unique symbols 0,1,2,3,4,5,6,7,8 and 9.
In decimal system 10 symbols are involved, so the base or radix is 10.
It is a positional weighted system.
The value attached to the symbol depends on its location with respect to the decimal point.
In general,
, dn dn-1 dn-2 …………… d0 . d -1 d -2 …………d - m
is given by
(dn x 10n) + (dn-1 x 10n-1) + (dn-2 x 10n-2) + … + ( d0 x 100) + ( d-1 x 10 -1) + (d-2 x 10 -2) +…+(d -m x 10 –m)
For example:-
9256.26 = 9 x 1000 + 2 x 100 + 5 x 10 + 6 x 1 + 2 x (1/10) + 6 x ( 1/100)
= 9 x 103 + 2 x 102 + 5 x 101 + 6 x 100 + 2 x 10-1 + 6 x 10-2
BINARY NUMBER SYSTEM:-
The binary number system is a positional weighted system.
The base or radix of this number system is 2.
It has two independent symbols.
The symbols used are 0 and 1.
A binary digit is called a bit.
The binary point separates the integer and fraction parts.
In general,
dn dn-1 dn-2 …………… d0 . d -1 d -2 …………d – k
is given by
(dn x 2n) + (dn-1 x 2n-1) + (dn-2 x 2n-2) + ….+ ( d0 x 20) + ( d-1 x 2 -1) + (d-2 x 2 -2) +….+(d -k x 2 –k)
OCTAL NUMBER SYSTEM:-
It is also a positional weighted system.
Its base or radix is 8.
It has 8 independent symbols 0,1,2,3,4,5,6 and 7.
Its base 8 = 23 , every 3- bit group of binary can be represented by an octal digit.
HEXADECIMAL NUMBER SYSTEM:-
The hexadecimal number system is a positional weighted system.
The base or radix of this number system is 16.
The symbols used are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E and F
The base 16 = 24 , every 4 – bit group of binary can be represented by an hexadecimal digit.
CONVERSION FROM ONE NUMBER SYSTEM TO ANOTHER :-
1. BINARY NUMBER SYSTEM:-
(a) Binary to decimal conversion:-
In this method, each binary digit of the number is multiplied by its positional weight and the product terms
are added to obtain decimal number.
, For example:
(i) Convert (10101)2 to decimal.
Solution :
(Positional weight) 24 23 22 21 20
Binary number 10101
= (1 x 24) + (0 x 23) + ( 1x 22 ) + ( 0 x 21 ) + (1 x 20)
= 16 + 0+ 4+ 0+ 1
= (21)10
(ii) Convert (111.101)2 to decimal.
Solution:
(111.101)2 = (1 x 22) + (1 x 21) + ( 1x 20 ) + ( 1 x 2 -1 ) + (0 x 2 -2) + (1 x 2 -3)
= 4+ 2+ 1 + 0.5 + 0 + 0.125
= (7.625)10
(b) Binary to Octal conversion:-
For conversion binary to octal the binary numbers are divided into groups of 3 bits each, starting at the
binary point and proceeding towards left and right.
Octal Binary Octal Binary
0 000 4 100
1 001 5 101
2 010 6 110
3 011 7 111
For example:
(i) Convert (101111010110.110110011)2 into octal.
Solution :
Group of 3 bits are 101 111 010 110 . 110 110 011
Convert each group into octal = 5 7 2 6 . 6 6 3
The result is (5726.663)8
(ii) Convert (10101111001.0111)2 into octal.
Solution :
Binary number 10 101 111 001 . 011 1
Group of 3 bits are = 010 101 111 001 . 011 100
Convert each group into octal = 2 5 7 1 . 3 4
The result is (2571.34)8
(c) Binary to Hexadecimal conversion:-
For conversion binary to hexadecimal number the binary numbers starting from the binary point, groups are
made of 4 bits each, on either side of the binary point.
, 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
For example:
(i) Convert (1011011011)2 into hexadecimal.
Solution:
Given Binary number 10 1101 1011
Group of 4 bits are 0010 1101 1011
Convert each group into hex = 2 D B
The result is (2DB)16
(ii) Convert (01011111011.011111)2 into hexadecimal.
Solution:
Given Binary number 010 1111 1011 . 0111 11
Group of 3 bits are = 0010 1111 1011 . 0111 1100
Convert each group into octal = 2 F B . 7 C
The result is (2FB.7C)16
2. DECIMAL NUMBER SYSTEM:-
(a)Decimal to binary conversion:-
In the conversion the integer number are converted to the desired base using successive division by the
base or radix.
For example:
(i) Convert (52)10 into binary.
Solution:
Divide the given decimal number successively by 2 read the integer part remainder upwards to get
equivalent binary number. Multiply the fraction part by 2. Keep the integer in the product as it is and multiply
the new fraction in the product by 2. The process is continued and the integer are read in the products from
top to bottom.
2 I 52
2 l 26 ―0
2 l 13 ―0
2l6 ―1
2l3 ―0
2l1 ―1
0 ―1
INTRODUCTION:-
The term digital refers to a process that is achieved by using discrete unit.
In number system there are different symbols and each symbol has an absolute value and also has
place value.
RADIX OR BASE:-
The radix or base of a number system is defined as the number of different digits which can occur in
each position in the number system.
RADIX POINT :-
The generalized form of a decimal point is known as radix point. In any positional number system the
radix point divides the integer and fractional part.
Nr = [ Integer part . Fractional part ]
↑
Radix point
NUMBER SYSTEM:-
In general a number in a system having base or radix ‘ r ’ can be written as
an an-1 an-2 …………… a0 . a -1 a -2 …………a - m
This will be interpreted as
Y = an x rn + an-1 x rn-1 + an-2 x rn-2 + ……… + a0 x r0 + a-1 x r -1 + a-2 x r -2 +………. +a -m x r –m
where Y = value of the entire number
an = the value of the nth digit
r = radix
TYPES OF NUMBER SYSTEM:-
There are four types of number systems. They are
1. Decimal number system
2. Binary number system
3. Octal number system
4. Hexadecimal number system
DECIMAL NUMBER SYSTEM:-
The decimal number system contain ten unique symbols 0,1,2,3,4,5,6,7,8 and 9.
In decimal system 10 symbols are involved, so the base or radix is 10.
It is a positional weighted system.
The value attached to the symbol depends on its location with respect to the decimal point.
In general,
, dn dn-1 dn-2 …………… d0 . d -1 d -2 …………d - m
is given by
(dn x 10n) + (dn-1 x 10n-1) + (dn-2 x 10n-2) + … + ( d0 x 100) + ( d-1 x 10 -1) + (d-2 x 10 -2) +…+(d -m x 10 –m)
For example:-
9256.26 = 9 x 1000 + 2 x 100 + 5 x 10 + 6 x 1 + 2 x (1/10) + 6 x ( 1/100)
= 9 x 103 + 2 x 102 + 5 x 101 + 6 x 100 + 2 x 10-1 + 6 x 10-2
BINARY NUMBER SYSTEM:-
The binary number system is a positional weighted system.
The base or radix of this number system is 2.
It has two independent symbols.
The symbols used are 0 and 1.
A binary digit is called a bit.
The binary point separates the integer and fraction parts.
In general,
dn dn-1 dn-2 …………… d0 . d -1 d -2 …………d – k
is given by
(dn x 2n) + (dn-1 x 2n-1) + (dn-2 x 2n-2) + ….+ ( d0 x 20) + ( d-1 x 2 -1) + (d-2 x 2 -2) +….+(d -k x 2 –k)
OCTAL NUMBER SYSTEM:-
It is also a positional weighted system.
Its base or radix is 8.
It has 8 independent symbols 0,1,2,3,4,5,6 and 7.
Its base 8 = 23 , every 3- bit group of binary can be represented by an octal digit.
HEXADECIMAL NUMBER SYSTEM:-
The hexadecimal number system is a positional weighted system.
The base or radix of this number system is 16.
The symbols used are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E and F
The base 16 = 24 , every 4 – bit group of binary can be represented by an hexadecimal digit.
CONVERSION FROM ONE NUMBER SYSTEM TO ANOTHER :-
1. BINARY NUMBER SYSTEM:-
(a) Binary to decimal conversion:-
In this method, each binary digit of the number is multiplied by its positional weight and the product terms
are added to obtain decimal number.
, For example:
(i) Convert (10101)2 to decimal.
Solution :
(Positional weight) 24 23 22 21 20
Binary number 10101
= (1 x 24) + (0 x 23) + ( 1x 22 ) + ( 0 x 21 ) + (1 x 20)
= 16 + 0+ 4+ 0+ 1
= (21)10
(ii) Convert (111.101)2 to decimal.
Solution:
(111.101)2 = (1 x 22) + (1 x 21) + ( 1x 20 ) + ( 1 x 2 -1 ) + (0 x 2 -2) + (1 x 2 -3)
= 4+ 2+ 1 + 0.5 + 0 + 0.125
= (7.625)10
(b) Binary to Octal conversion:-
For conversion binary to octal the binary numbers are divided into groups of 3 bits each, starting at the
binary point and proceeding towards left and right.
Octal Binary Octal Binary
0 000 4 100
1 001 5 101
2 010 6 110
3 011 7 111
For example:
(i) Convert (101111010110.110110011)2 into octal.
Solution :
Group of 3 bits are 101 111 010 110 . 110 110 011
Convert each group into octal = 5 7 2 6 . 6 6 3
The result is (5726.663)8
(ii) Convert (10101111001.0111)2 into octal.
Solution :
Binary number 10 101 111 001 . 011 1
Group of 3 bits are = 010 101 111 001 . 011 100
Convert each group into octal = 2 5 7 1 . 3 4
The result is (2571.34)8
(c) Binary to Hexadecimal conversion:-
For conversion binary to hexadecimal number the binary numbers starting from the binary point, groups are
made of 4 bits each, on either side of the binary point.
, 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
For example:
(i) Convert (1011011011)2 into hexadecimal.
Solution:
Given Binary number 10 1101 1011
Group of 4 bits are 0010 1101 1011
Convert each group into hex = 2 D B
The result is (2DB)16
(ii) Convert (01011111011.011111)2 into hexadecimal.
Solution:
Given Binary number 010 1111 1011 . 0111 11
Group of 3 bits are = 0010 1111 1011 . 0111 1100
Convert each group into octal = 2 F B . 7 C
The result is (2FB.7C)16
2. DECIMAL NUMBER SYSTEM:-
(a)Decimal to binary conversion:-
In the conversion the integer number are converted to the desired base using successive division by the
base or radix.
For example:
(i) Convert (52)10 into binary.
Solution:
Divide the given decimal number successively by 2 read the integer part remainder upwards to get
equivalent binary number. Multiply the fraction part by 2. Keep the integer in the product as it is and multiply
the new fraction in the product by 2. The process is continued and the integer are read in the products from
top to bottom.
2 I 52
2 l 26 ―0
2 l 13 ―0
2l6 ―1
2l3 ―0
2l1 ―1
0 ―1
