Number system logic gates

UNIT I
Number System
By Dr. Dhobale J V
Associate Professor
School of Engineering & Technology
RNB Global University, Bikaner
RNB Global University, Bikaner. 1Course Code - 19004000

Objectives
 Number Systems – Decimal, Binary, Octal,
Hexadecimal.
 1’s and 2’s Complements.
 Binary Arithmetic.
 Interconversions.
 Binary Coded Decimal (BCD), Gray code,
Excess Code, ASCII Code.
 Error detection and Correction. 2RNB Global University, Bikaner.Course Code - 19004000

Number System
 Decimal Number System : The decimal
numeral system has ten as its base. It is
the most widely used numerical base.
 When we write decimal (base 10)
numbers, we use a positional notation
system.
 Ex. 845= 8*102
+4*101
+5*100
= 8*100+4*10+5*1
= 800+40+5

Number System
 For whole numbers, the rightmost digit
position is the one’s position (100
= 1).
The numeral in that position indicates
how many ones are present in the
number.
 The next position to the left is ten’s, then
hundred’s, thousand’s, and so on. Each
digit position has a weight that is ten
times the weight of the position to its
right.

Number System
 The decimal numerals are the familiar
zero through nine (0, 1, 2, 3, 4, 5, 6, 7, 8,
9).
 Ex. 12710 - Radix or base is 10.

Number System
 The binary number system is also a
positional notation numbering system.
 Base of binary number system is two.
Each digit position in a binary number
represents a power of two.
 Ex.
101101=1*25
+0*24
+1*23
+1*22
+0*21
+1*20
= 32+0+8+4+0+1
= 32+8+4+1

Number System
 Only the numerals 0 and 1 are used in
binary numbers.
 When talking about binary numbers, it is
often necessary to talk of the number of
bits used to store or represent the
number.
 This merely describes the number of
binary digits that would be required to
write the number. The number in the
above example is a 6 bit number.

Number System
 Ex. 1011012 112 101102
 Conversion between decimal and binary:
(10110)2 = (?)10
=1*24+0*23+1*22+1*21+0*20
= 16+0+4+2+0
=16+4+2
=22

Number System
 Conversion between binary and decimal:
(22)10 = (?)2
22 / 2 = 11 remainder 0 (LSB)
11 / 2 = 5 remainder 1
(MSB)
Ans. 10110; (22)10 = (10110)2

Number System
 Hexadecimal Numbers:
 Base is 16.
 This number system is called
hexadecimal, and each digit position
represents a power of 16.
 For any number base greater than ten, a
problem occurs because there are more
than ten symbols needed to represent
the numerals for that number base.

Number System
 Hexadecimal Numbers: Since the
hexadecimal system is base 16, there
are sixteen numerals
 required. The following are the
hexadecimal numerals:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
Ex. 3FA16, 4716

Number System
 Hexadecimal to Decimal Conversion:
25H = (2*161) + (2*160)
 Decimal to Hexadecimal to
Conversion:
3710 = (?)16
37/16 = 2 R 5
= (25)16

Number System - Interconversions
 Binary to Hexadecimal Conversation:
1010000100111101
A 1 3 D
 Hexadecimal to Binary
A 1 3 D
1010 0001 0011 1101

 Octal Number System:

 Octal to Decimal Conversion:
37o = (3*81) + (7*80)
= 24 + 7
= 31
 Decimal to Octal Conversion:
3110 = (?)o
31 / 8 = 3 R 7
37

Number System- Interconversions
 Binary to Octal Conversion:
001010000100111101
1 2 0 4 7 5
 Octal to Binary:
1 2 0 4 7 5
001 010 000 100 111 101

 Convert 0.625 to binary equivalent.
Step1- 0.625 * 2 = 1.25 - The first
binary digit to the right of the point is 1.
Step 2- 0.25 * 2 = 0.50 - The second
binary digit to the right of the point is 0.
Step 3- 0.50 * 2 = 1.0 – The third binary
digit to the right of the point is 1.
 (0.625)10 = (0.101)2

Concept of Computer word &
Memory units
 Bit is an abbreviation of the term ‘binary
digit’ and is smallest unit of information.
 It is either 0 or 1.
 A byte is a string of eight bits, is the
basic unit of data operated upon as a
single unit in computer.
 A computer word is a string of bits whose
size, called the ‘word length’ or ‘word
size’, is fixed for a specified computer

Concept of Computer word &
Memory units

1’s Complements & 2’s Complement
 Complements are used in the digital
computers in order to simplify the
subtraction operation and for the logical
manipulations.
 Two types of Complements:
1. The radix complement is referred to as the
r's complement.
2. The diminished radix complement is
referred to as the r − 1's complement

1’s Complements & 2’s Complement
 Convert binary no 111001101 to 2’s
 1’s Complement - 000110010
 2’s Complement – 000110010
+ 000000001
000110011

Binary Arithmetic

Binary Arithmetic (Division)

Truth Table
 A truth table lists all possible combinations of
input binary variables and the corresponding
outputs of a logic system.
 The logic system output can be found from the
logic expression, often referred to as the
Boolean expression.
 When the number of input binary variables is
only one, then there are only two possible
inputs, i.e. ‘0’ and ‘1’.
30RNB Global University, Bikaner.Course Code - 19004000

Truth Table
 If the number of inputs is two, there can be
four possible input combinations, i.e. 00, 01,
10 and 11.
 This statement can be generalized to say that,
if a logic circuit has n binary inputs, its truth
table will have 2n possible input combinations.

Truth Table

Logic Gates
 The logic gate is the most basic building block
of any digital system, including computers.
 Each one of the basic logic gates is a piece of
hardware or an electronic circuit that can be
used to implement some basic logic
expression.

Logic Gates
 While laws of Boolean algebra could be used
to do manipulation with binary variables and
simplify logic expressions, these are actually
implemented in a digital system with the help
of electronic circuits called logic gates.
 The three basic logic gates are the OR gate,
the AND gate and the NOT gate.

Logic Gates – OR Gate
 An OR gate performs an ORing operation on
two or more than two logic variables.
 written as Y = A+B and reads as Y equals A
OR B and not as A plus B.
 An OR gate is a logic circuit with two or more
inputs and one output.

Logic Gates – AND Gate
 An AND gate is a logic circuit having two or
more inputs and one output.
 The output of an AND gate is HIGH only when
all of its inputs are in the HIGH state.
 The AND operation on two independent logic
variables A and B is written as Y = A.B and
reads as Y equals A AND B.

Logic Gates – NOT Gate
 A NOT gate is a one-input, one-output logic
circuit whose output is always the complement
of the input.
 That is, a LOW input produces a HIGH output,
and vice versa.
 It is also known as a ‘complementing circuit’ or
an ‘inverting circuit’.

Logic Gates – NOT Gate
 .

Logic Gates – NAND Gate
 NAND stands for NOT AND.
 An AND gate followed by a NOT circuit makes
it a NAND gate.
 The output of a NAND gate is a logic ‘0’ when
all its inputs are a logic ‘1’.
 Y = A.B

Logic Gates – NAND Gate

Logic Gates – NOR Gate
 NOR stands for NOT OR.
 An OR gate followed by a NOT circuit makes it
a NOR gate.
 The output of a NOR gate is a logic ‘1’ when
all its inputs are logic ‘0’; For all other input
combinations, the output is a logic ‘0’.
 Y = (A+B).

Logic Gates – NOR Gate

Logic Gates – EXCLUSIVE-OR Gate
 The EXCLUSIVE-OR gate, commonly written
as EX-OR gate, is a two-input, one-output
gate.
 As can be seen from the truth table, the output
of an EX-OR gate is a logic ‘1’ when the inputs
are unlike and a logic ‘0’ when the inputs are
like.

Logic Gates – EXCLUSIVE-OR Gate

Logic Gates – Exclusive-NOR Gate
 EXCLUSIVE-NOR (commonly written as EX-
NOR) means NOT of EX-OR, i.e. the logic
gate that we get by complementing the output
of an EX-OR gate.
 The truth table of an EX-NOR gate is obtained
from the truth table of an EX-OR gate by
complementing the output entries. Logically,
Y = (A⊕B) = (A.B+A.B).

Logic Gates – Exclusive-NOR Gate

Logic Gates – Universal Gates
 OR, AND & NOT gates are the three basic
logic gates as they together can be used to
construct the logic circuit for any given
Boolean expression.
 NOR and NAND gates have the property that
they individually can be used to hardware
implement a logic circuit corresponding to any
given Boolean expression.

 Combination of NAND gates or a combination
of NOR gates can be used to perform
functions of any of the basic logic gates.

De Morgan’s Theorems
 De Morgan has suggested two theorems
which are extremely useful in Boolean
Algebra.
1. Theorem 1: A.B=A+B.
NAND = Bubbled OR
 The left hand side (LHS) of this theorem
represents a NAND gate with inputs A and B,
whereas the right hand side (RHS) of the
theorem represents an OR gate with inverted
inputs. 56RNB Global University, Bikaner.Course Code - 13004900

2. Theorem 2:
A+B = A.B
NOR = Bubbled AND
 The LHS of this theorem represents a NOR
gate with inputs A and B, whereas the RHS
represents an AND gate with inverted inputs.

Binary Codes
 In the coding, when numbers, letters or
words are represented by a specific
group of symbols, it is said that the
number, letter or word is being encoded
The group of symbols is called as a
code.
 The digital data is represented, stored
and transmitted as group of binary bits.
This group is also called as binary code.

Binary Codes
 Advantages of Binary Code
1. Binary codes are suitable for the
computer applications.
2. Binary codes are suitable for the digital
communications.
3. Binary codes make the analysis and
designing of digital circuits if we use the
binary codes.
4. Since only 0 & 1 are being used,
implementation becomes easy.RNB Global University, Bikaner. 61Course Code - 19004000

Binary Codes
 Classification of binary codes
 The codes are broadly categorized into
following four categories.
1. Weighted Codes & Non-Weighted
Codes
2. Binary Coded Decimal Code
3. Alphanumeric Codes
4. Error Detecting Codes & Error
Correcting Codes.

Binary Codes
 Weighted and Non Weighted code
 Weighted binary codes are those binary
codes which obey the positional weight
principle.
 Each position of the number represents
a specific weight.
 We can express any decimal number in
tens, hundreds, thousands and so on.
Ex.4327 = 4(103)+3(102)+2(101)+7(100)

Binary Codes
 Weighted and Non Weighted code
 Several systems of the codes are used
to express the decimal digits 0 through 9.
In these codes each decimal digit is
represented by a group of four bits.

Binary Codes
 Non-weighted code: In non-weighted
code, there is no positional weight i.e.
each position within the binary number is
not assigned a prefixed value.
 No specific weights are assigned to bit
position in non-weighted code.
 The non-weighted codes are:
1. The Excess-3 Code
2. The Gray Code

Binary Codes
 Non-weighted code:
1. The Excess-3 Code : The Excess-3
code is also called as XS-3 code. It is
non-weighted code used to express
decimal numbers.
 The Excess-3 code words are derived
from the 8421 BCD code words adding
00112 or 310 to each code word in 8421.

Binary Codes
1. The Excess-3 Code :

Binary Codes
2. Gray Code : It is the non-weighted code
and it is not arithmetic codes. That
means there are no specific weights
assigned to the bit position.
 It has a very special feature that, only
one bit will change each time the decimal
number is incremented as shown in fig.

Binary Codes
2. Gray Code : As only one bit changes at
a time, the gray code is called as a unit
distance code.
 The gray code is a cyclic code. Gray
code cannot be used for arithmetic
operation.

Binary Codes
2. Gray Code :

Binary Codes
2. Gray Code : Application of Gray Code –
 Gray code is popularly used in the shaft
position encoders.
 A shaft position encoder produces a
code word which represents the angular
position of the shaft.

Binary Coded Decimal (BCD) Code
 In this code each decimal digit is
represented by a 4-bit binary number.
 BCD is a way to express each of the
decimal digits with a binary code. In the
BCD, with four bits we can represent
sixteen numbers 0000to1111.
 But in BCD code only first ten of these
are used 0000to1001. The remaining six
code combinations i.e. 1010 to 1111 are
invalid in BCD.

 Advantages of BCD Codes:
1. It is very similar to decimal system.
2. We need to remember binary equivalent of
decimal numbers 0 to 9 only.

 Disadvantages of BCD Codes
1. The addition and subtraction of BCD have
different rules.
2. The BCD arithmetic is little more
complicated.
3. BCD needs more number of bits than
binary to represent the decimal number.
So BCD is less efficient than binary.

Alphanumeric codes
 A binary digit or bit can represent only
two symbols as it has only two states '0'
or '1'.
 But this is not enough for communication
between two computers because there
we need many more symbols for
communication.
 These symbols are required to represent
26 alphabets with capital and small
letters, numbers from 0 to 9, punctuation
marks and other symbols.RNB Global University, Bikaner. 75Course Code - 19004000

Alphanumeric codes
 The alphanumeric codes are the codes
that represent numbers and alphabetic
characters.
 Mostly such codes also represent other
characters such as symbol and various
instructions necessary for conveying
information.
 An alphanumeric code should at least
represent 10 digits and 26 letters of
alphabet i.e. total 36 items.

Alphanumeric codes
 The following three alphanumeric codes
are very commonly used for the data
representation.
1. American Standard Code for Information
Interchange ASCII.
2. Extended Binary Coded Decimal
Interchange Code EBCDIC.
3. Five bit Baudot Code.

 ASCII has 128 characters and symbols
represented by a 7-bit binary code.
 It can be considered an 8-bit code with
the MSB always 0. (00h-7Fh)
 00h-1Fh (the first 32) – control characters
 20h-7Fh – graphics symbols (can be
printed or displayed).
Alphanumeric codes

 There are an additional 128 characters
that were adopted by IBM for use in their
PCs.
 It’s popular and is used in applications
other than PCs, unofficial standard.
 The extended ASCII characters are
represented by an 8-bit code series from
80h-FFh
 7 Bit ASCII Characters
Alphanumeric codes

 Extended Binary Coded Decimal
Interchange Code (EBCDIC) is an
eight-bit character encoding used mainly
on IBM mainframe and IBM midrange
computer operating systems.
Alphanumeric codes

 The Baudot code, invented by Émile
Baudot, is a character set
predating EBCDIC and ASCII.
 It was the predecessor to the
International Telegraph Alphabet No. 2
(ITA2), the teleprinter code in use until
the advent of ASCII.
Alphanumeric codes

 Each character in the alphabet is
represented by a series of five bits, sent
over a communication channel such as a
telegraph wire or a radio signal.
Alphanumeric codes

Error Detection and Correction
 Error is a condition when the output
information does not match with the input
information.
 During transmission, digital signals suffer
from noise that can introduce errors in
the binary bits travelling from one system
to other.

 Error Detecting code: Whenever a
message is transmitted, it may get
scrambled by noise or data may get
corrupted.
 To avoid this, we use error-detecting
codes which are additional data added to
a given digital message to help us detect
if an error occurred during transmission
of the message.
 Ex- Parity Check.

 Error-Correcting codes: Along with error-
detecting code, we can also pass some
data to figure out the original message
from the corrupt message that we
received. This type of code is called an
error-correcting code.
 These codes detect the exact location of
the corrupt bit.
 In error-correcting codes, parity check
has a simple way to detect errors.

 How to Detect and Correct Errors?: To
detect and correct the errors, additional
bits are added to the data bits at the time
of transmission.
 The additional bits are called parity bits.
They allow detection or correction of the
errors.
 The data bits along with the parity bits form
a code word.

 Parity Checking of Error Detection:- It is
the simplest technique for detecting and
correcting errors.
 The MSB of an 8-bits word is used as
the parity bit and the remaining 7 bits are
used as data or message bits. The parity
of 8-bits transmitted word can be either
even parity or odd parity.

 Even parity -- Even parity means the
number of 1's in the given word including
the parity bit should be even (2,4,6,....).
 Odd parity -- Odd parity means the
number of 1's in the given word including
the parity bit should be odd (1,3,5,....).

Error Detection and Correction Codes
2. Repetition Code: The repetition code
makes use of repetitive transmission of
each data bit in the bit stream.
 Three fold repetition, 1 would be
transmitted as 111 and 0 as 000.
 The code becomes self-correcting if the
bit in the majority is taken as the correct
bit.

3.Cyclic Redundancy Checks Code:

3.Cyclic Redundancy Checks Code: This
Cyclic Redundancy Check is the most
powerful and easy to implement
technique.
CRC is based on binary division.
In CRC, a sequence of redundant bits,
called cyclic redundancy check bits, are
appended to the end of data unit so that
the resulting data unit becomes exactly
divisible by a second, predetermined
binary number.RNB Global University, Bikaner. 92Course Code - 19004000

3.Cyclic Redundancy Checks Code: At the
destination, the incoming data unit is
divided by the same number. If at this
step there is no remainder, the data unit
is assumed to be correct and is therefore
accepted
A remainder indicates that the data unit
has been damaged in transit and
therefore must be rejected.

4. Hamming Code:- an increase in the
number of redundant bits added to
message bits can enhance the
capability of the code to detect and
correct errors.

 If we have a sufficient number of
redundant bits, and if these bits can be
arranged such that different error bits
produce different error results, then it
should be possible not only to detect the
error bit but also to identify its location.
 The addition of redundant bits alters the
‘distance’ code parameter, which has
come to be known as the Hamming
distance.

 The Hamming distance is nothing but the
number of bit disagreements between
two code words.

Boolean Algebra
 Variable, complement, and literal are
terms used in Boolean algebra.
 A variable is a symbol used to represent
a logical quantity. Any single variable
can have a 1 or a 0 value.
 The complement is the inverse of a
variable and is indicated by a bar over
variable (over bar).

Boolean Algebra
 The complement of the variable A is read
as "not A" or "A bar“.
 Sometimes a prime symbol rather than
an over bar is used to denote the
complement of a variable; for example,
B' indicates the complement of B.
 A literal is a variable or the complement
of a variable.

Boolean Algebra
 Boolean Addition: Boolean addition is
equivalent to the OR operation.
 In Boolean algebra, a sum term is a sum
of literals.
 In logic circuits, a sum term is produced
by an OR operation with no AND
operations involved.
 Ex- A + B, A + B, A + B + C & A + B + C
+ D.

Boolean Algebra
 Boolean Addition: A sum term is equal to
1 when one or more of the literals in the
term are 1.
 A sum term is equal to 0 only if each of
the literals is 0.
 Example - Determine the values of A, B,
C and D that make the sum term
A + B + C + D equal to 0.

Boolean Algebra
 Boolean Multiplication: Boolean
multiplication is equivalent to the AND
operation.
 In Boolean algebra, a product term is the
product of literals.
 In logic circuits, a product term is
produced by an AND operation with no
OR operations involved.
 Ex.- AB, AB, ABC, and ABCD.

Boolean Algebra
 Boolean Multiplication: A product term is
equal to 1 only if each of the literals in
the term is 1.
 A product term is equal to 0 when one or
more of the literals are 0.
 Example - Determine the values of A, B,
C and D that make the product term
ABCD equal to 1.

Boolean Algebra - Laws
1. Commutative Law: The commutative
law of addition for two variables is
written as A+B = B+A.
 This law states that the order in which
the variables are ORed makes no
difference.
 In Boolean algebra as applied to logic
circuits, addition and the OR operation
are the same.

 Above figure illustrates the commutative
law as applied to the OR gate and shows
that it doesn't matter to which input each
variable is applied.

1. Commutative Law: The commutative
law of multiplication for two variables is
A.B = B.A
 This law states that the order in which
the variables are ANDed makes no
difference.

2. Associative Law: The associative law of
addition is written as follows for three
variables: A + (B + C) = (A + B) + C.
 This law states that when ORing more
than two variables, the result is the same
regardless of the grouping of the
variables.

2. Associative Law: The associative law of
multiplication is written as follows for
three variables: A(BC) = (AB)C.
 This law states that it makes no
difference in what order the variables are
grouped when ANDing more than two
variables.

3. Distributive Law: The distributive law is
written for three variables as follows:
A(B + C) = AB + AC.
 This law states that ORing two or more
variables and then ANDing the result
with a single variable is equivalent to
ANDing the single variable with each of
the two or more variables and then
ORing the products.

3. Distributive Law: The distributive law
also expresses the process of factoring
in which the common variable A is
factored out of the product terms, for
example, AB + AC = A(B + C).

Boolean Algebra - Rules
 Rules of Boolean Algebra: .

 Rules of Boolean Algebra: The proof is
shown in Table below, which shows the
truth table and the resulting logic circuit
simplification.

 Rules of Boolean Algebra:.

 Rules of Boolean Algebra: 12 basic rules
that are useful in manipulating and
simplifying Boolean expressions.
 Rules 1 through 9 will be viewed in terms
of their application to logic gates.
 Rules 10 through 12 will be derived in
terms of the simpler rules and the laws
previously discussed.

Boolean Algebra - Simplification
 Boolean Expression for the a logic
circuit: .

circuit: To derive the Boolean expression
for a given logic circuit, begin at the
leftmost inputs and work toward the final
output, writing the expression for each
gate.
 For the given circuit, the Boolean
expression is determined as follows:

circuit:
 The expression for the left-most AND gate
with inputs C and D is CD.
 The output of the left-most AND gate is one
of the inputs to the OR gate and B is the
other input. Therefore, the expression for
the OR gate is B + CD.

circuit:
 The output of the OR gate is one of the
inputs to the right-most AND gate and A is
the other input. Therefore, the expression
for this AND gate is A(B + CD), which is the
final output expression for the entire circuit.

 Constructing Truth table for a logic Circuit:-.
 Once the Boolean expression for a given logic
circuit has been determined, a truth table that
shows the output for all possible values of the
input variables can be developed.
 The procedure requires that you evaluate the
Boolean expression for all possible
combinations of values for the input variables.
 There are four input variables (A, B, C, and D)
and therefore sixteen (24 = 16) combinations
of values are possible.

 The first step is to list the sixteen input variable
combinations of 1s and 0s in a binary
sequence as shown in below table.
 Next, place a 1 in the output column for each
combination of input variables that was
determined in the evaluation.
 Finally, place a 0 in the output column for all
other combinations of input variables.

Review
 Number Systems – Decimal, Binary, Octal,
Hexadecimal.
 1’s and 2’s Complements.
 Binary Arithmetic.
 Interconversions.
 Binary Coded Decimal (BCD), Gray code,
Excess Code, ASCII Code.
 Error detection and Correction. 134RNB Global University, Bikaner.Course Code - 19004000

Thank You!

Number system logic gates

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