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Page 1
IMPORTANT FORMULAS ON DIGITAL ELECTRONICS
Number System and Codes
Fig. 1
Types of Number System:
The number can be represented in various ways to show the data and process it on the
processing devices.
Decimal Number
System
Hexadecimal Number
System
Octal Number
System
Binary Number System
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Table 1: Counting in different number system
A number system with base ‘r’, contains ‘r’ different digits and they are from 0 to r –1.
Decimal to other codes conversions:
To convert decimal number into other system with base ‘r’, divide integer part by r and
multiply fractional part with r.
Other codes to Decimal Conversions:
( ) ( ) ?
2 1 0 1 2
r 10
x x x .y y A ,
2 –1 –2
2 1 0 1 2
A x r x r x y r y r = + + + +
Hexadecimal to Binary:
Convert each Hexadecimal digit into 4 bits binary.
Page 2
IMPORTANT FORMULAS ON DIGITAL ELECTRONICS
Number System and Codes
Fig. 1
Types of Number System:
The number can be represented in various ways to show the data and process it on the
processing devices.
Decimal Number
System
Hexadecimal Number
System
Octal Number
System
Binary Number System
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Table 1: Counting in different number system
A number system with base ‘r’, contains ‘r’ different digits and they are from 0 to r –1.
Decimal to other codes conversions:
To convert decimal number into other system with base ‘r’, divide integer part by r and
multiply fractional part with r.
Other codes to Decimal Conversions:
( ) ( ) ?
2 1 0 1 2
r 10
x x x .y y A ,
2 –1 –2
2 1 0 1 2
A x r x r x y r y r = + + + +
Hexadecimal to Binary:
Convert each Hexadecimal digit into 4 bits binary.
( )
( )
2
16
1111 0101
1010
5AF
5 A F
?
Binary to Hexadecimal:
Grouping of 4 bits into one hex digit.
( ) ( )
2 16
110101.11 00110101.1100 35.C ??
Octal Binary and Binary to Octal:
Same procedure as discussed above but here group of 3 bits is made.
Codes:
Binary coded decimal (BCD):
• In BCD code each decimal digit is represented with 4 bit binary format.
( )
??
??
??
??
10
93 4
BCD
Eg: 943 1001 0100 0011
• It is also known as 8421 code.
• Invalid BCD codes are the codes whose decimal equivalent is more than 9. i.e. Valid
BCD codes ranges from 0 to 9.
Total number of codes possible
4
2 16 ??
Valid BCD codes 10 ?
Invalid BCD codes 16 10 6 -?
There 1010, 1011, 1100, 1110 and 1111
Excess-3 codes: (BCD + 0011)
• It can be derived from BCD by adding ‘3’ to each coded number.
• It is unweighted and self-complementing code.
Gray Code:
It is also known as minimum change codes or unit distance code or reflected code.
Binary code to Gray code:
In order to find Gray code from Binary code, XOR Gate is applied between the present
binary bit and the next binary bit starting from the MSB side(keeping MSB of Binary and
Gray as same) e.g.
Page 3
IMPORTANT FORMULAS ON DIGITAL ELECTRONICS
Number System and Codes
Fig. 1
Types of Number System:
The number can be represented in various ways to show the data and process it on the
processing devices.
Decimal Number
System
Hexadecimal Number
System
Octal Number
System
Binary Number System
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Table 1: Counting in different number system
A number system with base ‘r’, contains ‘r’ different digits and they are from 0 to r –1.
Decimal to other codes conversions:
To convert decimal number into other system with base ‘r’, divide integer part by r and
multiply fractional part with r.
Other codes to Decimal Conversions:
( ) ( ) ?
2 1 0 1 2
r 10
x x x .y y A ,
2 –1 –2
2 1 0 1 2
A x r x r x y r y r = + + + +
Hexadecimal to Binary:
Convert each Hexadecimal digit into 4 bits binary.
( )
( )
2
16
1111 0101
1010
5AF
5 A F
?
Binary to Hexadecimal:
Grouping of 4 bits into one hex digit.
( ) ( )
2 16
110101.11 00110101.1100 35.C ??
Octal Binary and Binary to Octal:
Same procedure as discussed above but here group of 3 bits is made.
Codes:
Binary coded decimal (BCD):
• In BCD code each decimal digit is represented with 4 bit binary format.
( )
??
??
??
??
10
93 4
BCD
Eg: 943 1001 0100 0011
• It is also known as 8421 code.
• Invalid BCD codes are the codes whose decimal equivalent is more than 9. i.e. Valid
BCD codes ranges from 0 to 9.
Total number of codes possible
4
2 16 ??
Valid BCD codes 10 ?
Invalid BCD codes 16 10 6 -?
There 1010, 1011, 1100, 1110 and 1111
Excess-3 codes: (BCD + 0011)
• It can be derived from BCD by adding ‘3’ to each coded number.
• It is unweighted and self-complementing code.
Gray Code:
It is also known as minimum change codes or unit distance code or reflected code.
Binary code to Gray code:
In order to find Gray code from Binary code, XOR Gate is applied between the present
binary bit and the next binary bit starting from the MSB side(keeping MSB of Binary and
Gray as same) e.g.
Fig. 2
Gray code to Binary code:
In order to find Binary code from Gray code, XOR Gate is applied between the present
binary bit and the next gray bit starting from the MSB side (keeping MSB of Gray and
Binary as same) e.g.
Fig. 3
Alpha Numeric code:
EBCDIC (Extended BCD interchange code)
It is an 8-bit code. It can represent 128 possible characters.
• Parity method is most widely used schemes for error detection.
• Hamming code is most useful error correcting code.
• BCD code is used in calculators, counters.
Complements:
Its base is r then we can have two complements.
(i) (r – 1)’s complement
(ii) r’s complement
To determine(r – 1)’s complement: First write maximum possible number in the given
system and subtract the given number.
To determine r’s complements: (r – 1)’s complement + 1
i.e. First write (r – 1)’s complement and then add 1 to LSB
Example:
Q. Find 7’s and 8’s complement of 2456
Sol. 7’s Complement
7777
2456
5321
-
8’s Complement
5321
1
5322
+
Q. Find 2’s complement of 101.110
Sol. 1’s complements 010.001
For 2’s complement add 1 to the LSB
Page 4
IMPORTANT FORMULAS ON DIGITAL ELECTRONICS
Number System and Codes
Fig. 1
Types of Number System:
The number can be represented in various ways to show the data and process it on the
processing devices.
Decimal Number
System
Hexadecimal Number
System
Octal Number
System
Binary Number System
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Table 1: Counting in different number system
A number system with base ‘r’, contains ‘r’ different digits and they are from 0 to r –1.
Decimal to other codes conversions:
To convert decimal number into other system with base ‘r’, divide integer part by r and
multiply fractional part with r.
Other codes to Decimal Conversions:
( ) ( ) ?
2 1 0 1 2
r 10
x x x .y y A ,
2 –1 –2
2 1 0 1 2
A x r x r x y r y r = + + + +
Hexadecimal to Binary:
Convert each Hexadecimal digit into 4 bits binary.
( )
( )
2
16
1111 0101
1010
5AF
5 A F
?
Binary to Hexadecimal:
Grouping of 4 bits into one hex digit.
( ) ( )
2 16
110101.11 00110101.1100 35.C ??
Octal Binary and Binary to Octal:
Same procedure as discussed above but here group of 3 bits is made.
Codes:
Binary coded decimal (BCD):
• In BCD code each decimal digit is represented with 4 bit binary format.
( )
??
??
??
??
10
93 4
BCD
Eg: 943 1001 0100 0011
• It is also known as 8421 code.
• Invalid BCD codes are the codes whose decimal equivalent is more than 9. i.e. Valid
BCD codes ranges from 0 to 9.
Total number of codes possible
4
2 16 ??
Valid BCD codes 10 ?
Invalid BCD codes 16 10 6 -?
There 1010, 1011, 1100, 1110 and 1111
Excess-3 codes: (BCD + 0011)
• It can be derived from BCD by adding ‘3’ to each coded number.
• It is unweighted and self-complementing code.
Gray Code:
It is also known as minimum change codes or unit distance code or reflected code.
Binary code to Gray code:
In order to find Gray code from Binary code, XOR Gate is applied between the present
binary bit and the next binary bit starting from the MSB side(keeping MSB of Binary and
Gray as same) e.g.
Fig. 2
Gray code to Binary code:
In order to find Binary code from Gray code, XOR Gate is applied between the present
binary bit and the next gray bit starting from the MSB side (keeping MSB of Gray and
Binary as same) e.g.
Fig. 3
Alpha Numeric code:
EBCDIC (Extended BCD interchange code)
It is an 8-bit code. It can represent 128 possible characters.
• Parity method is most widely used schemes for error detection.
• Hamming code is most useful error correcting code.
• BCD code is used in calculators, counters.
Complements:
Its base is r then we can have two complements.
(i) (r – 1)’s complement
(ii) r’s complement
To determine(r – 1)’s complement: First write maximum possible number in the given
system and subtract the given number.
To determine r’s complements: (r – 1)’s complement + 1
i.e. First write (r – 1)’s complement and then add 1 to LSB
Example:
Q. Find 7’s and 8’s complement of 2456
Sol. 7’s Complement
7777
2456
5321
-
8’s Complement
5321
1
5322
+
Q. Find 2’s complement of 101.110
Sol. 1’s complements 010.001
For 2’s complement add 1 to the LSB
2’s complement
010.001
1
010.010
+
Data Representation:
Fig. 4
Unsigned Magnitude:
Range with n bit
n1
0 to 2 5 101
-
? + ?
5 Not possible -?
Signed Magnitude:
Range with n bit
sign bit sign bit
with 4 bits with 8 bits
6 1 110 1 0000110 -?
1’s complements:
Range with n bit
( ) ( )
n 1 n 1
2 1 to 2 1
--
? - - + -
+ ? - ? 6 0110 6 1001
2’s complements:
With n bit range –2
n-1
to (2
n-1
–1)
+ ? - ? 6 0110 6 1010
In any representation +ve numbers are represented similar to +ve number in sign
magnitude.
Page 5
IMPORTANT FORMULAS ON DIGITAL ELECTRONICS
Number System and Codes
Fig. 1
Types of Number System:
The number can be represented in various ways to show the data and process it on the
processing devices.
Decimal Number
System
Hexadecimal Number
System
Octal Number
System
Binary Number System
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Table 1: Counting in different number system
A number system with base ‘r’, contains ‘r’ different digits and they are from 0 to r –1.
Decimal to other codes conversions:
To convert decimal number into other system with base ‘r’, divide integer part by r and
multiply fractional part with r.
Other codes to Decimal Conversions:
( ) ( ) ?
2 1 0 1 2
r 10
x x x .y y A ,
2 –1 –2
2 1 0 1 2
A x r x r x y r y r = + + + +
Hexadecimal to Binary:
Convert each Hexadecimal digit into 4 bits binary.
( )
( )
2
16
1111 0101
1010
5AF
5 A F
?
Binary to Hexadecimal:
Grouping of 4 bits into one hex digit.
( ) ( )
2 16
110101.11 00110101.1100 35.C ??
Octal Binary and Binary to Octal:
Same procedure as discussed above but here group of 3 bits is made.
Codes:
Binary coded decimal (BCD):
• In BCD code each decimal digit is represented with 4 bit binary format.
( )
??
??
??
??
10
93 4
BCD
Eg: 943 1001 0100 0011
• It is also known as 8421 code.
• Invalid BCD codes are the codes whose decimal equivalent is more than 9. i.e. Valid
BCD codes ranges from 0 to 9.
Total number of codes possible
4
2 16 ??
Valid BCD codes 10 ?
Invalid BCD codes 16 10 6 -?
There 1010, 1011, 1100, 1110 and 1111
Excess-3 codes: (BCD + 0011)
• It can be derived from BCD by adding ‘3’ to each coded number.
• It is unweighted and self-complementing code.
Gray Code:
It is also known as minimum change codes or unit distance code or reflected code.
Binary code to Gray code:
In order to find Gray code from Binary code, XOR Gate is applied between the present
binary bit and the next binary bit starting from the MSB side(keeping MSB of Binary and
Gray as same) e.g.
Fig. 2
Gray code to Binary code:
In order to find Binary code from Gray code, XOR Gate is applied between the present
binary bit and the next gray bit starting from the MSB side (keeping MSB of Gray and
Binary as same) e.g.
Fig. 3
Alpha Numeric code:
EBCDIC (Extended BCD interchange code)
It is an 8-bit code. It can represent 128 possible characters.
• Parity method is most widely used schemes for error detection.
• Hamming code is most useful error correcting code.
• BCD code is used in calculators, counters.
Complements:
Its base is r then we can have two complements.
(i) (r – 1)’s complement
(ii) r’s complement
To determine(r – 1)’s complement: First write maximum possible number in the given
system and subtract the given number.
To determine r’s complements: (r – 1)’s complement + 1
i.e. First write (r – 1)’s complement and then add 1 to LSB
Example:
Q. Find 7’s and 8’s complement of 2456
Sol. 7’s Complement
7777
2456
5321
-
8’s Complement
5321
1
5322
+
Q. Find 2’s complement of 101.110
Sol. 1’s complements 010.001
For 2’s complement add 1 to the LSB
2’s complement
010.001
1
010.010
+
Data Representation:
Fig. 4
Unsigned Magnitude:
Range with n bit
n1
0 to 2 5 101
-
? + ?
5 Not possible -?
Signed Magnitude:
Range with n bit
sign bit sign bit
with 4 bits with 8 bits
6 1 110 1 0000110 -?
1’s complements:
Range with n bit
( ) ( )
n 1 n 1
2 1 to 2 1
--
? - - + -
+ ? - ? 6 0110 6 1001
2’s complements:
With n bit range –2
n-1
to (2
n-1
–1)
+ ? - ? 6 0110 6 1010
In any representation +ve numbers are represented similar to +ve number in sign
magnitude.
Logic Gates
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FAQs on Important Formulas : Digital Electronics - Digital Electronics - Electronics and Communication Engineering (ECE)
| 1. What are the basic logic gates in digital electronics? |  |
Ans. The basic logic gates in digital electronics are AND, OR, NOT, NAND, NOR, XOR, and XNOR gates.
| 2. How do you simplify Boolean expressions in digital electronics? | |
Ans. Boolean expressions can be simplified using algebraic laws, such as the distributive law, identity law, complement law, etc. Karnaugh maps can also be used for simplification.
| 3. What is the difference between combinational and sequential circuits? | |
Ans. Combinational circuits output solely depends on the current input, while sequential circuits have memory elements and their output is based on both current input and previous state.
| 4. How do you design a digital circuit using a truth table? | |
Ans. To design a digital circuit using a truth table, analyze the truth table to identify the combinations of inputs that result in the desired output and then use logic gates to implement the circuit accordingly.
| 5. What is the purpose of a multiplexer in digital electronics? | |
Ans. A multiplexer (MUX) is used to select one of many inputs and route it to a single output based on the select lines. It is often used to reduce the number of input/output lines in a circuit.
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