Digital Logic

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UNIT I: Number systems and Boolean algebra Number Systems -
Decimal, Binary, Octal, Hexadecimal and their inter conversions, - Binary
Arithmetic -1’s complement, 2’s complementand 9’s complement
.Binary codes - BCD, Excess-3, Graycode. Boolean Algebra: Boolean
Laws- Simplification of Boolean Functions - Logic gates and Truth Table
– Universal Gates (NANDand NOR ) - The K-map method up to five
variables, don’t care conditions, POS & SOP forms.
UNIT-II: Combinational and Sequential Circuits Combinational Logic:
Half/Full adder/subtractor, code conversion, Multiplexers, de
multiplexers, encoders, decoders, Combinational design using MUX &
DEMUX. BCD adder, magnitude comparator. Sequential logic: Flip flops
(RS, Clocked RS, D, JK, JK Master Slave)-Counters & types Synchronous
and Asynchronous counters- Registers, Shift registers and their types.
THIRUVALLUVAR UNIVERSITY
B.Sc. COMPUTER SCIENCE SYLLABUS UNDER CBCS
PAPER – 1 Digital Logic & Programming in C
Objective: Provide basic knowledge on Digital Electronics to understand the
working principles of Digital computer and to develop programming skill
using C language
SEMESTER I

What are Number Systems?
• Number systems are the technique to represent numbers
in the computer system architecture, every value that
you are saving into or getting from computer memory
has a defined number system. Computer architecture
supports following number systems.
• Binary Number System (2 digits)
• Octal Number System (8 digits)
• Decimal Number System (10 digits )
• Hexa-decimal Number System (16 digits)

The digit value in the number system is calculated using:
1.The digit
2.The index, where the digit is present in the number.
3.Finally, the base numbers, the total number of digits available in the
number system.

Decimal Binary Octal
Hexa-
Decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
8 1000 10 8
9 1001 11 9
Decimal Binary Octal
Hexa-
Decimal
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F

Binary Number System
• Digital computers represents all kinds of data and information in the
binary system.
• Binary Number System consists of two digits 0 and 1. Its base is 2.
Each digit or bit in binary number system can be 0 or 1.
• Binary to Decimal Conversion Techniques:
• Multiply each bit by 2n, where nis the “weight” of the bit.
• The weight is the position of the bit, starting from 0 on the right.
• Add the results.

Octal Number System
• Octal number system is the base 8 number system and uses the
digits from 0 to 7.
• This number system provides shortcut method to represent long
binary numbers
• The number after 7 is 10. the number after 17 is 20 and so forth.
• Octal to Decimal Conversion Techniques:
• The weight is the position of the bit, starting from 0 on the
right.

Decimal Number System
• Decimal number system is the base 10 number system and uses the
digits from 0 to
9. Using these digits you can express any quantity.
• It is what we most commonly use.
• Decimal to Binary Conversion Techniques:
• Divide each bit by 2, keep track of the remainder
• First remainder is bit 0 (LSB, least-significant bit)

Hexa-decimal Number System
• Hexa-decimal number system is the base 16 and uses the digits from 0
to 9 and Ato F.This number system provides shortcut method to
represent long binary numbers.
• Unlike binary and octal, hexa-decimal has six additional symbols that
it uses beyond the conventional ones found in decimal.
• Hexa-decimal to Decimal Conversion Techniques:
• The weight is the position of the bit, starting from 0 on the right.

Conversion Among Bases
Hexadecimal
Decimal Octal
Binary

•Group into 3's starting at least significant symbol (if the
number of bits is not evenly divisible by 3, then add 0's at
themostsignificantend)
•write 1 octal digit for each group
e.g.:(1010101)2 to ()8
001 010 101
1 2 5
Answer= 1258
Binary to Octal

Answer= (010101111)2
Octal to Binary
•Foreachof theOctaldigit writeitsbinaryequivalent
e.g.: (257)8to ()2
2 57
010
101
111

Binary to Hexadecimal
• Group into 4's starting at least significant symbol (if the
number of bits is not evenly divisible by 4, then add 0's at
themostsignificantend)
•write 1 hex digit for each group.
e.g.:(1010111011)2to ()16
10 1011 1011
B
2 B
Answer= (2BB)16

Hexadecimal to Binary
• For each of the Hex digit write its binary equivalent (use 4 bits to
represent).
e.g.: (25A0)16to ()2
25A0
0010
0101 1010
0000
Answer= (0010010110100000)2

• Steps:
1.Convertoctalnumberto itsbinary equivalent
2.Convertbinarynumberto itshexadecimalequivalent
e.g.: (635.27)8 to ()16
6 3 5 . 2 7
110 011 101 . 010 111
000 00
1 9 D . 5 C
Octal to Hexadecimal

• Steps:
1.Convert hexadecimal numberto itsbinaryequivalent
2.Convertbinarynumberto itsoctalequivalent
e.g.:
Hexadecimal to Octal
A 3 B . 7
1010 0011 1011 . 0111 00
5 0 7 3 . 3 4

Any Base to Decimal
Convertingfromanybaseto decimalisdone bymultiplying
eachdigit byitsweightand summing.
e.g.:
Binaryto Decimal
1011.112 = (1x23 ) + (0x22 )+ (1x21 ) + (1x20)+ (1x2-1) + (1x2-2)
= 8 + 0 + 2 + 1 + 0.5 + 0.25
= 11.7510

Binary to Decimal
(1010.01)2
1×23 + 0x22 + 1×21+ 0x20 + 0x2 -1 + 1×2 -2
= 8+0+2+0+0+0.25 = 10.25
(
(1010.01)2 = (10.25)10

Decimal to Any Base
Steps:
1. Convertintegerpart
( SuccessiveDivisionMethod )
2. Convertfractionalpart
( SuccessiveMultiplicationMethod )

e.g.:(125)10 to ( )2
Answer: (1111101)2

(10.25)10
Note: Keep multiplying the fractional part with 2 until decimal part 0.00 is obtained.
(0.25)10 = (0.01)2
Answer: (10.25)10 = (1010.01)2

Binary Arithmetic
• Binary arithmetic is essential part of all the digital computers and
many other digital system.
In the binary number system, there are only two digits 0 and 1, and
any number can be represented by these two digits. The arithmetic of
binary numbers means the operation of
• Binary Addition,
• Binary Subtraction,
• Binary Multiplication and
• Binary Division.

Binary Addition
consists of four possible elementary
The additin
operations:
Sr no. Operations
0. 0+0=0
1. 0+1=1
2. 1+0=1
3. 1+1=10 (0 with carry of 1)
Inthelastcase,sumisof twodigits:HigherSignificantbit is
calledCarryandlowersignificantbit iscalledSum.

Binary Addition
e.g.:
1
+ 0
1
1
0
1
0
0
1 0 0 1 0
Carr
y

Thesubtractionconsistsof four possibleelementary
operations:
Binary Subtraction
Incaseof secondoperationtheminuendbit issmaller
thanthesubtrahendbit,hence1isborrowed.
Sr no. Operations
0. 0-0=0
1. 0-1=1(borrow 1)
2. 1-0=1
3. 1-1=0

Binary Subtraction
e.g.:
0
- 0
1
1
0
1
1
0
1 1 1 1

Binary Multiplication
Rulesfor BinaryMultiplication are:
Sr no. Operations
0. 0*0=0
1. 0*1=0
2. 1*0=0
3. 1*1=1
*
e.g.: Multiply 110by 10
1 1
1
0
0
0 0 0
+ 1 1 0 0
1 1 0 0

Binary Division
Rulesfor BinaryDivision are:
Sr no. Operations
0. 0/0=0
1. 1/0=0
2. 0/1=0
3. 1/1=1
e.g.: Divide110by 10
1 1
1 0 1 1 0
1 0
0 1 0
1 0
0 0

complement
• Complements are used in the digital computers in order to
simplify the subtraction operation and for the logical
manipulations. For each radix-r system (radix r represents
base of number system) there are two types of complements.
S.N. Complement Description
1 Radix Complement The radix complement is referred to as
the r's complement
2 Diminished Radix
Complement
The diminished radix complement is
referred to as the (r-1)'s complement

1’sComplement
The 1’s complement of a binary number is the number that
resultswhenwechangeall1’sto zerosandthezeros to ones.
1 1 0 1 0 0 1 0
NOT OPEARATION
0 0 1 0 1 1 0 1

2’sComplement
The2’scomplementthebinarynumberthatresultswhen
add1to the1’scomplement.It’sgivenas,
2’s complement = 1’s complement + 1
Example:Express35in 8-bit 2’scomplementform.
Solution:
35in 8-bit form is 00100011
0 0 1 0 0 0 1 1
1 1 0 1 1 1 0 0
+ 1
1 1 0 1 1 1 0 1

9’sComplement
Thenines'complementof adecimaldigit isthe number that
mustbe added to it to produce 9. Thecomplementof 3is6,
thecomplementof 7is 2.
Example:Obtain9’scomplementof 7493
Solution:
9 9 99
- 749 3
2506 9’s complement

10’sComplement
The10’scomplementof thegivennumberisobtainedby
adding1to the9’scomplement.Itisgivenas,
10’s complement = 9’s complement + 1
Example:Obtain10’scomplementof 7493
Solution:
9999 2506
- 749 3 + 1
2506 2507 10’s complement

Digital Logic

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