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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
a n a n-1 an-2 …………… a 0. a -1 a -2 …………a - m
This will be interpreted as
Y = a (^) n x r n^ + a (^) n-1 x r n-1^ + an-2 x r n-2^ + ……… + a 0 x r^0 + a-1 x r -1^ + a (^) -2 x r -2^ +………. +a (^) -m x r –m
where Y = value of the entire number
a n = the value of the nth^ digit
r = radix
TYPES OF NUMBER SYSTEM:-
There are four types of number systems. They are
- Decimal number system
- Binary number system
- Octal number system
- 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 …………… d 0. d -1 d -2 …………d - m
is given by
(dn x 10 n) + (dn-1 x 10n-1) + (d (^) n-2 x 10n-2) + … + ( d 0 x 10^0 ) + ( 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 10 3 + 2 x 10 2 + 5 x 10^1 + 6 x 10 0 + 2 x 10-1^ + 6 x 10 -
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 …………… d 0. d -1 d -2 …………d – k
is given by
(dn x 2n) + (dn-1 x 2n-1) + (dn-2 x 2n-2) + ….+ ( d 0 x 2^0 ) + ( 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 = 2 3 , 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.
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 2 l 6 ― 1 2 l 3 ― 0 2 l 1 ― 1 0 ― 1
Result of (52) 10 is (110100) (^2) (ii) Convert (105.15) 10 into binary.
Solution: Integer part Fraction part 2 I 105 0.15 x 2 = 0. 2 l 52 ― 1 0.30 x 2 = 0. 2 l 26 ― 0 0.60 x 2 = 1. 2 l 13 ― 0 0.20 x 2 = 0. 2 l 6 ― 1 0.40 x 2 = 0. 2 l 3 ― 0 0.80 x 2 = 1. 2 l 1 ― 1 0 ― 1 Result of (105.15) 10 is (1101001.001001) (^2)
(b)Decimal to octal conversion:-
To convert the given decimal integer number to octal, successively divide the given number by 8 till the quotient is 0. To convert the given decimal fractions to octal successively multiply the decimal fraction and the subsequent decimal fractions by 8 till the product is 0 or till the required accuracy is obtained.
For example: (i) Convert (378.93) 10 into octal.
Solution:
8 I 378 0.93 x 8 = 7. 8 l 47 ― 2 0.44 x 8 = 3. 8 l 5 ― 7 0.52 x 8 = 4. 0 ― 5 0.16 x 8 = 1.
Result of (378.93) 10 is (572.7341) (^8)
(c)Decimal to hexadecimal conversion:-
The decimal to hexadecimal conversion is same as octal. For example: (i) Convert (2598.675) 10 into hexadecimal.
Solution: Remainder Decimal Hex Hex 16 I 2598 0.675 x 16 = 10.8 A 16 l 162 ― 6 6 0.800 x 16 = 12.8 C 16 l 10 ― 2 2 0.800 x 16 = 12.8 C 0 ― 10 A 0.800 x 16 = 12.8 C
Result of (2598.675) 10 is (A26.ACCC) (^16)
3. OCTAL NUMBER SYSTEM:-
(a)Octal to binary conversion:-
To convert a given a octal number to binary, replace each octal digit by its 3- bit binary equivalent.
(b)Hexadecimal to decimal conversion:-
For conversion of hexadecimal to decimal, multiply each digit in the hexadecimal number by its position weight and add all those product terms.
For example: - Convert (A0F9.0EB) 16 to decimal
Solution: (A0F9.0EB) 16 = (10 x 16^3 )+(0 x 16^2 )+(15 x 16^1 ) +( 9 x 16^0 ) +(0 x 16 – 1^ ) +(14 x 16- 2) +(11 x 16 -3^ ) = 40960 + 0 + 240 + 9 + 0 +0.0546 + 0. = (41209.0572) (^10)
Result is (41209.0572) (^10) (c) Hexadecimal to Octal conversion:-
For conversion of hexadecimal to octal, first convert the given hexadecimal number to binary and then binary number to octal.
For example :- Convert (B9F.AE) 16 to octal.
Solution :- Given hexadecimal no.is B 9 F. A E Convert each hex. digit to binary = 1011 1001 1111. 1010 1110 Group of 3 bits are = 101 110 011 111. 101 011 100 Convert 3 bits group to octal. = 5 6 3 7. 5 3 4
Result is (5637.534) (^8)
BINARY ARITHEMATIC OPERATION :-
1. BINARY ADDITION:-
The binary addition rules are as follows 0 + 0 = 0 ; 0 + 1 = 1 ; 1 + 0 = 1 ; 1 + 1 = 10 , i.e 0 with a carry of 1
For example :-
Add (100101) 2 and (1101111) 2. Solution :-
1 0 0 1 0 1
- 1 1 0 1 1 1 1 1 0 0 1 0 1 0 0 Result is (10010100) 2
2. BINARY SUBTRACTION:-
The binary subtraction rules are as follows 0 - 0 = 0 ; 1 - 1 = 0 ; 1 - 0 = 1 ; 0 - 1 = 1 , with a borrow of 1
For example :- Substract (111.111) 2 from (1010.01) 2. Solution :-
1 0 1 0. 0 1 0
- 1 1 1. 1 1 1 0 0 1 0 .0 1 1 Result is (0010.011) (^2)
3. BINARY MULTIPLICATION:-
The binary multiplication rules are as follows 0 x 0 = 0 ; 1 x 1 = 1 ; 1 x 0 = 0 ; 0 x 1 = 0 For example :-
Multiply (1101) 2 by (110) 2. Solution :-
1 1 0 1 x 1 1 0___ 0 0 0 0 1 1 0 1
- 1 1 0 1_____ 1 0 0 1 1 1 0__ Result is (1001110) (^2)
4. BINARY DIVISION:-
The binary division is very simple and similar to decimal number system. The division by ‘0’ is meaningless. So we have only 2 rules 0 ÷ 1 = 0 1 ÷ 1 = 1 For example :- Divide (10110) 2 by (110) 2.
Solution :-
110 ) 101101 ( 111.
- 110___ 1010 110__ 1001 110__ 110 110___ 000___ Result is (111.1) (^2)
For example:- Subtract (10000) 2 from (11010) 2 using 1’s complement. Solution:- 1 1 0 1 0 1 1 0 1 0 = 26
- 1 0 0 0 0 => + 0 1 1 1 1 (1’s complement) = - 16
Carry → 1 0 1 0 0 1 + 10
Result is +
2’s COMPLEMENT:-
In 2’s complement subtraction, add the 2’s complement of subtrahend to the minuend. If there is a carry out, ignore it. If the MSB is 0, the result is positive. If the MSB is 1, the result is negative and is in its 2‘s complement form. Then take its 2’s complement to get the magnitude in binary.
For example:- Subtract (1010100) 2 from (1010100) 2 using 2’s complement. Solution:- 1 0 1 0 1 0 0 1 0 1 0 1 0 0 = 84
- 1 0 1 0 1 0 0 => + 0 1 0 1 1 0 0 (2’s complement) = - 84_
1 0 0 0 0 0 0 0 ( Ignore the carry) 0 = 0 (result = 0) Hence MSB is 0. The answer is positive. So it is +0000000 = 0 DIGITAL CODES:-
In practice the digital electronics requires to handle data which may be numeric, alphabets and special characters. This requires the conversion of the incoming data into binary format before it can be processed. There is various possible ways of doing this and this process is called encoding. To achieve the reverse of it, we use decoders.
WEIGHTED AND NON-WEIGHTED CODES:-
There are two types of binary codes
- Weighted binary codes
- Non- weighted binary codes In weighted codes, for each position ( or bit) ,there is specific weight attached. For example, in binary number, each bit is assigned particular weight 2n where ‘n’ is the bit number for n = 0,1,2,3,4 the weights are 1,2,4,8,16 respectively. Example :- BCD
Non-weighted codes are codes which are not assigned with any weight to each digit position, i.e., each digit position within the number is not assigned fixed value. Example:- Excess – 3 (XS -3) code and Gray codes
BINARY CODED DECIMAL (BCD):-
BCD is a weighted code. In weighted codes, each successive digit from right to left represents weights equal to some specified value and to get the equivalent decimal number add the products of the weights by the corresponding binary digit. 8421 is the most common because 8421 BCD is the most natural amongst the other possible codes.
For example:- (567) 10 is encoded in various 4 bit codes. Solution:- Decimal → 5 6 7 8421 code → 0101 0110 0111 6311 code → 0111 1000 1001 5421 code → 1000 0100 1010
BCD ADDITION:-
Addition of BCD (8421) is performed by adding two digits of binary, starting from least significant digit. In case if the result is an illegal code (greater than 9) or if there is a carry out of one then add 0110(6) and add the resulting carry to the next most significant. For example:- Add 679.6 from 536.8 using BCD addition. Solution:- 6 7 9. 6 0110 0111 1001. 0110 ( 679.6 in BCD)
- 5 3 6. 8 =>+ 0101 0011 0110. 1000 (536.8 in BCD)
1 2 1 6. 4 1011 1010 1111. 1110 ( All are illegal codes)
- 0110 +0110 +0110 .+0110 ( Add 0110 to each) 0001 0010 0001 0110. 0100 1 2 1 6. 4 ( corrected sum = 1216.4) Result is 1216.
BCD SUBTRACTION:-
The BCD subtraction is performed by subtracting the digits of each 4 – bit group of the subtrahend from corresponding 4 – bit group of the minuend in the binary starting from the LSD. If there is no borrow from the next higher group[ then no correction is required. If there is a borrow from the next group, then 6 10 (0110) is subtracted from the difference term of this group. For example:- Subtract 147.8 from 206.7 using 8421 BCD code.
Solution:- 2 0 6. 7 0010 0000 0110. 0111 ( 206.7 in BCD)
- 1 4 7. 8 =>- 0001 0100 0111. 1000 (147.8 in BCD)
5 8. 9 0000 1011 1110. 1111 ( Borrows are present)
- 0110 -0110 .- 0110 0101 1000. 1001 5 8. 9 ( corrected difference = 58.9) Result is (58.9) (^10)
EXCESS THREE(XS-3) CODE:-
The Excess-3 code, also called XS-3, is a non- weighted BCD code. This derives it name from the fact that each binary code word is the corresponding 8421 code word plus 0011(3). It is a sequential code. It is a self complementing code.
0100 EOT DC4 $ 4 D T d t
0101 ENQ NAK % 5 E U e u
0110 ACK SYN & 6 F V f v
0111 BEL ETB ‘ 7 G W g w
1000 BS CAN ( 8 H X h x
1001 HT EM ) 9 I Y i y
1010 LF SUB * : J Z j z
1011 VT ESC + ; K [ k {
1100 FF FS , < L \ l |
1101 CR GS - = M ] m }
1110 SO RS. > N ^ n ~
1111 SI US /? O _ o DLE
EBCDIC CODE:-
The Extended Binary Coded Decimal Interchange Code (EBCDIC) pronounced as ‘eb – si- dik’ is an 8 bit alphanumeric code. Since 28 = 256 bit patterns can be formed with 8 bits. It is used by most large computers to communicate in alphanumeric data. The table shown below shows the EBCDIC code.
The EBCDIC code
LSD (Hex)
MSD(Hex)
0 1 2 3 4 5 6 7 8 9 A B C D E F
0 NUL DLE DS SP & [ ] \ 0
1 SOH DC1 SOS / a j ~ A J 1
2 STX DC2 FS SYN b k s B K S 2
3 ETX DC3 c l t C L T 3
4 PF RES BYP PN d m u D M U 4
5 HT NL LF RS e n v E N V 5
6 LC BS EOB YC f o w F O W 6
7 DEL IL PRE EOT g p x G P X 7
8 CAN h q y H Q Y 8
9 EM i r z I R Z 9
A SMM CC SM Ø! I :
B VT. $ , #
C FF IFS DC4 < * % @
D CR IGS ENQ NAK ( ) _ ‘
E SO IRS ACK + ; > =
F SI IUS BEL SUB I ‘? ‘
GRAY CODE:-
The gray code is a non-weighted code. It is not a BCD code. It is cyclic code because successive words in this differ in one bit position only i.e it is a unit distance code.
Gray code is used in instrumentation and data acquisition systems where linear or angular displacement is measured. They are also used in shaft encoders, I/O devices, A/D converters and other peripheral equipment.
BINARY- TO – GRAY CONVERSION:-
If an n-bit binary number is represented by B (^) n Bn- 1 - - - - - B 1 and its gray code equivalent by G (^) n Gn- 1 - - - - - G 1 , where Bn and G (^) n are the MSBs , then gray code bits are obtained from the binary code as follows G (^) n = Bn G (^) n-1 = Bn Bn- . . . . G 1 = B 2 B 1
Where the symbol stands for Exclusive OR (X-OR)
For example :- Convert the binary 1001 to the Gray code.
Solution :-`
Binary → 1 0 0 1
Gray → 1 1 0 1
The gray code is 1101
GRAY- TO - BINARY CONVERSION:-
If an n-bit gray number is represented by G (^) n G (^) n-1 ------- G 1 and its binary equivalent by B (^) n Bn- 1 - - - - - B 1 , then binary bits are obtained from Gray bits as follows : Bn = Gn Bn-1 = Bn G (^) n- . . . . B 1 = B 2 G (^1)
LOGIC GATES:-
Logic gates are the fundamental building blocks of digital systems. There are 3 basic types of gates AND, OR and NOT. Logic gates are electronic circuits because they are made up of a number of electronic devices and components. Inputs and outputs of logic gates can occur only in 2 levels. These two levels are termed HIGH and LOW, or TRUE and FALSE, or ON and OFF or si The table which lists all the possible combinations of input variables and the corresponding called a truth table.
LEVEL LOGIC:-
A logic in which the voltage levels represents logic 1 and logic 0. Level logic may be positive or neg Positive Logic:- A positive logic system is the one in which the higher of the two voltage levels represents the logic 1 and the lower of the two voltages level represents the logic 0. Negative Logic:- A negative logic system is the one in whi higher of the two voltages level represents the logic 0.
DIFFERENT TYPES OF LOGIC GATES
NOT GATE (INVERTER):-
A NOT gate, also called and inverter, has only one input and one output. It is a device whose output is always the complement of its input. The output of a NOT gate is the logic 1 state when its input is in logic 0 state and the logic 0 state when its inputs is in logic 1 state.
IC No. :- 7404
Logic Symbol
Timing Diagram 1 0 0 1
A
A
LOGIC GATES
Logic gates are the fundamental building blocks of digital systems. There are 3 basic types of gates AND, OR and NOT. Logic gates are electronic circuits because they are made up of a number of electronic devices and
Inputs and outputs of logic gates can occur only in 2 levels. These two levels are termed HIGH and LOW, or TRUE and FALSE, or ON and OFF or simply 1 and 0. The table which lists all the possible combinations of input variables and the corresponding
A logic in which the voltage levels represents logic 1 and logic 0. Level logic may be positive or neg
A positive logic system is the one in which the higher of the two voltage levels represents the logic 1 and the lower of the two voltages level represents the logic 0.
A negative logic system is the one in which the lower of the two voltage levels represents the logic 1 and the higher of the two voltages level represents the logic 0.
DIFFERENT TYPES OF LOGIC GATES:-
A NOT gate, also called and inverter, has only one input and one output. t is a device whose output is always the complement of its input. The output of a NOT gate is the logic 1 state when its input is in logic 0 state and the logic 0 state when
Truth table
INPUT
A
OUTPUT
A
Logic gates are electronic circuits because they are made up of a number of electronic devices and
Inputs and outputs of logic gates can occur only in 2 levels. These two levels are termed HIGH and
The table which lists all the possible combinations of input variables and the corresponding outputs is
A logic in which the voltage levels represents logic 1 and logic 0. Level logic may be positive or negative logic.
A positive logic system is the one in which the higher of the two voltage levels represents the logic 1 and the
ch the lower of the two voltage levels represents the logic 1 and the
The output of a NOT gate is the logic 1 state when its input is in logic 0 state and the logic 0 state when
OUTPUT
AND GATE:-
An AND gate has two or more inputs but only one output. The output is logic 1 state only when each one of its inputs is at logic 1 state. The output is logic 0 state even if one of its inputs is at logic 0 state. IC No.:- 7408 Logic Symbol
Timing Diagram
0 0 1 1
A
0 1 0 1
B
0 0 0 1
Q
OR GATE:-
An OR gate may have two or more inputs but only one output. The output is logic 1 state, even if on The output is logic 0 state, only when each one of its inputs is in logic state.
IC No.:- 7432 Logic Symbol
Timing Diagram
0 0 1 1
A
0 1 0 1
B
0 1 1 1
Q
An AND gate has two or more inputs but only one output. nly when each one of its inputs is at logic 1 state. The output is logic 0 state even if one of its inputs is at logic 0 state.
Truth Table
An OR gate may have two or more inputs but only one output. The output is logic 1 state, even if one of its input is in logic 1 state. The output is logic 0 state, only when each one of its inputs is in logic state.
Truth Table
OUTPUT
A B Q=A. B
INPUT OUTPUT
A B Q=A + B
OUTPUT
Q=A. B
OUTPUT
Q=A + B
Timing Diagram
0 0 1 1
A
0 1 0 1
B
1 0 0 0
Q
EXCLUSIVE – OR (X-OR) GATE
An X-OR gate is a two input, one output logic circuit. The output is logic 1 when one and only one of its two inputs is logic 1. When both the inputs is logic 0 or when both the inputs is logic 1, the output is logic 0.
IC No.:- 7486
Logic Symbol
INPUTS are A and B
OUTPUT is Q = A B
= A B + A B
Timing Diagram
0 0 1 1
A
0 1 0 1
B
0 1 1 0
Q
EXCLUSIVE – NOR (X-NOR) GATE
An X-NOR gate is the combination of an X An X-NOR gate is a two input, one output logic circuit. The output is logic 1 only when both the inputs The output is logic 0 when one of the inputs is logic 0 and other is 1.
OR) GATE:-
OR gate is a two input, one output logic circuit. The output is logic 1 when one and only one of its two inputs is logic 1. When both the inputs is logic 0 both the inputs is logic 1, the output is logic 0.
Truth Table
NOR) GATE:-
NOR gate is the combination of an X-OR gate and a NOT gate. NOR gate is a two input, one output logic circuit. The output is logic 1 only when both the inputs are logic 0 or when both the inputs is 1. The output is logic 0 when one of the inputs is logic 0 and other is 1.
INPUT OUTPUT A B Q = A
0 0 0 0 1 1 1 0 1 1 1 0
The output is logic 1 when one and only one of its two inputs is logic 1. When both the inputs is logic 0
are logic 0 or when both the inputs is 1.
OUTPUT A B
IC No.:- 74266
Logic Symbol
OUT =A B + A B
= A XNOR B
Timing Diagram
0 0 1 1
A
0 1 0 1
B
1 0 0 1
OUT
UNIVERSAL GATES:-
There are 3 basic gates AND, OR and NOT, there are two universal gates NAND and NOR, each of which can realize logic circuits single handedly. The NAND and NOR gates are called universal building blocks. Both NAND and NOR gates can perform all logic functions i.e. AND, OR, NOT, EXOR and EXNOR.
NAND GATE:- a) Inverter from NAND gate
Input = A Output Q = A
b) AND gate from NAND gate
Input s are A and B Output Q = A.B
c) OR gate from NAND gate
Inputs are A and B Output Q = A+B
There are 3 basic gates AND, OR and NOT, there are two universal gates NAND and NOR, each of which can dly. The NAND and NOR gates are called universal building blocks. Both NAND and NOR gates can perform all logic functions i.e. AND, OR, NOT, EXOR and EXNOR.
INPUT OUTPUT
A B OUT =A XNOR B
There are 3 basic gates AND, OR and NOT, there are two universal gates NAND and NOR, each of which can dly. The NAND and NOR gates are called universal building blocks. Both NAND and NOR gates can perform all logic functions i.e. AND, OR, NOT, EXOR and EXNOR.
XNOR B
