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Number Systems and Codes
1
NUMBER SYSTEMS AND CODES
Topics to be covered:


Number systems
 Number notations
 Arithmetic
 Base conversions
 Signed number representation
Codes
 Decimal codes
 Error detection code
 Gray code
 ASCII code
1. Number Systems
The decimal (real), binary, octal hexadecimal number systems are used to represent
information in digital systems. Any number system consists of a set of digits and a set of
operators (+, , , ). The radix or base of the number system denotes the number of digits
used in the system.
Decimal (base 10)
Binary (base 2)
Octal (base 8)
0 1 2 3 4 5 6 7 8 9
0 1
0 1 2 3 4 5 6 7
Hexadecimal (base 16)
0 1 2 3 4 5 6 7 8 9 A B C D E F
Decimal
Binary
Octal
Hexadecimal
00
0000
00
0
01
0001
01
1
02
0010
02
2
03
0011
03
3
04
0100
04
4
05
0101
05
5
06
0110
06
6
07
0111
07
7
08
1000
10
8
09
1001
11
9
10
1010
12
A
11
1011
13
B
Number Systems and Codes
12
1100
14
C
13
1101
15
D
14
1110
16
E
15
1111
17
F
2
1.1 Number notations
A number can be represented in either positional notation or polynomial notation.
Positional notation
It is convenient to represent a number using positional notation. A positional notation is
written as a sequence of digits with a radix point separating the integer and fractional
part.
N r  an1an2 a1a0 . a1a2 am r
where r is the radix, n is the number of digits of the integer part, and m is the number
digits of the fractional part.
Polynomial notation
A number can be explicitly represented in polynomial notation.
N r  a n 1  r n 1  a n  2  r n  2    a1  r 1  a0  r 0  a 1  r 1  a  2  r 2    a  m  r  m
where rp is a weighted position and p is the position of a digit.
In decimal number system
7326.5910  7  10 3  3  10 2  2  101  6  10 0  5  10 1  9  10 2
In binary number system
11010.0112  1  2 4  1  2 3  0  2 2  1 21  0  2 0  0  2 1  1 2 2  1  2 3
In octal number system
673.1248  6  8 2  7  81  3  80  1  8 1  2  8 2  4  8 3
In hexadecimal number system
306.D 16  3  16 2  0  161  6  16 0  D  2 1
Number Systems and Codes
1.2 Arithmetic
Addition
In binary number system,
(101101)2 +(11101)2 :
1111 1
101101
+
11101
1001010
In octal system,
(6254)8 +(5173)8 :
1 1
6254
+
5173
13447
In hexadecimal system,
(9F1B)16 +(4A36)16 :
+
1 1
9F1B
4A36
D951
Subtraction
In binary number system,
(101101)2 -(11011)2 :
-
10 10
101101
11011
10010
In octal system,
(6254)8 -(5173)8 :
-
8
6254
5173
1061
In hexadecimal system,
(9F1B)16 -(4A36)16 :
-
Multiplication
In binary number system,
16
9F1B
4A36
54E5
3
Number Systems and Codes
(1101)2  (1001)2 :
1101
1001
1101
0000
0000
1101
1110101

Division
In binary number system,
1101
(1110111)2 (1001)2 :
1001 1110111
1001
1011
1001
1011
1001
10
1.3 Base conversions
Convert (100111010)2 to base 8
1001110102
or
 1  2 8  0  2 7  0  2 6  1  2 5  1  2 4  1  2 3  0  2 2  1  21  0  2 0
 4  8 2  4  81  2  81  1  81  2  8 0
 4  8 2  7  81  2  8 0
 4728
1001110102


100
4
111 0102
7
2
 4728
Convert (100111010)2 to base 10
1001110102
 1  2 8  0  2 7  0  2 6  1  2 5  1  2 4  1  2 3  0  2 2  1  21  0  2 0
 25610  32 10  1610  810  210
 314 10
Convert (100111010)2 to base 16
1001110102
 1  2 8  0  2 7  0  2 6  1  2 5  1  2 4  1  2 3  0  2 2  1  21  0  2 0
 1  16 2  2  161  1  161  8  16 0  2  16 0
 1  16 2  3  161  A  16 0
 13 A16
4
Number Systems and Codes
or
1001110102
0001


1
0011 10102
3
A
 13 A16
Convert (372)8 to base 2
3728
 3
7
2
 011 111 010  111110102
Convert (372)8 to base 10
3728
 3  8 2  7  81  2  8 0
 19210  5610  2 10
 25010
Convert (372)8 to base 16
3728


1111
F
10102
A
 FA16
Convert (9F2)16 to base 2
9F 216

9
F
2
 1001 1111 0010  1001111100102
Convert (9F2)16 to base 8
9 F 216

9
F
2
 1001 1111 0010  (100
111 110
010) 2  47628


4 7 6 2
Convert (9F2)16 to base 10
9 F 216
 9  16 2  F  161  2  16 0
  230410  24010  210
 254610
Binomial expansion (series substitution)
To convert a number in base r to base p.
1) Represent the number in base p in binomial series.
2) Change the radix or base of each term to base p.
3) Simplify.
5
Number Systems and Codes
6
Convert base 10 to base r
Convert (174)10 to base 8
8
1 7
8 2
8
4
1
2
0
6
5
2
LSB
MSB
Therefore (174)10 = (256)8
Convert (0.275)10 to base 8
8
8
8
8
8





0.275
0.200
0.600
0.800
0.400





2.200
1.600
4.800
6.400
3.200





1. 3695
0.7390
1.4780
0.9560
1.9120
MSD
LSD
Therefore (0.275)10 = (0.21463)8
Convert (0.68475)10 to base 2
2
2
2
2
2





0.68475
0.3695
0.7390
0.4780
0.9560
MSD
LSD
Therefore (0.68475)10 = (0.10101)2
1.4 Signed Number Representation
There are 3 systems to represent signed numbers:



Signed-magnitude
1's complement
2's complement
In binary number system
Signed-magnitude system In signed-magnitude systems, the most significant bit represents
the number's sign, while the remaining bits represent its absolute value as an unsigned binary
magnitude.
 If the sign bit is a 0, the number is positive.
 If the sign bit is a 1, the number is negative.
Number Systems and Codes
7
1's Complement system A 1's complement system represents the positive numbers the same
way as in the signed-magnitude system. The only difference is negative number representations.
__
Let be N any positive integer number and N be a negative 1's complement integer of N. If the
number legnth is n bits, then N  (2 n  1)  N . For example in a 4-bit system, 0101 represents
+5 and
2 4  0001  0101  (10000  0001)  0101


 1111  0101
 1010
1010 represents 5
Number Systems and Codes
8
2's Complement System A 2's complement system is similar to 1's complement system, except
that there is only one representation for zero.
__
Let be N any positive integer number and N be a negative 2's complement integer of N. If the
length of the number is n bits, then N  2n  N. For example in a 4-bit system, 0101 represents
+5 and
2 4  0101  10000  0101
 1011
1011 represents 5
Adding and subtracting signed numbers
Signed-magnitude system
(a)
5
+2
7
0101
+0010
0111
(b)
-5
-2
-7
1101
+1010
1111
(c)
5
-2
3
0101
+1010
0011
(d)
-5
+2
-3
1101
+0010
1011
Number Systems and Codes
1's complement system
5
+2
7
(a)
0101
+0010
0111
(b)
-5
-2
-7
1010
+1101
1 0111
1
1000
(c)
5
-2
3
0101
+1101
1 0010
1
0011
(d)
-5
+2
-3
1010
+0010
1100
2's complement system
5
+2
7
(a)
0101
+0010
0111
(b)
-5
-2
-7
1011
+1110
1 1001
(c)
5
-2
3
0101
+1110
1 0011
(d)
-5
+2
-3
1011
+0010
1101
Overflow conditions
Carry-in  carry-out
5
+3
-8
0111
0101
+0011
1000
9
Number Systems and Codes
-5
-4
7
1000
1011
+1100
1 0111
5
+2
7
0000
0101
+0010
0111
-6
-2
-8
1110
1010
+1110
1 1000
Carry-in = carry-out
2. Codes
2.1. Decimal codes
Decimal
Digit
0
1
2
3
4
5
6
7
8
9
BCD
8421
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
Excess-3
2421
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
0000
0001
0010
0011
0100
1011
1100
1101
1110
1111
2.2. Error detection code
Parity bit
Odd Parity
P
Message
1
0000
0
0001
0
0010
1
0011
0
0100
1
0101
1
0110
Even Parity
P
Message
0
0000
1
0001
1
0010
0
0011
1
0100
0
0101
0
0110
10
Number Systems and Codes
0
0
1
1
0
1
0
0
1
0111
1000
1001
1010
1011
1100
1101
1110
1111
1
1
0
0
1
0
1
1
0
0111
1000
1001
1010
1011
1100
1101
1110
1111
2.3. Gray code
Decimal
Equivalent
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
2.4. ASCII code
Binary
Code
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Gray Code
0000
0001
0011
0010
0110
0111
0101
0100
1100
1101
1111
1110
1010
1011
1001
1000
11
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