Ch1.number systems

LOGIC DESIGN

Dr. Mahmoud Abo_elfetouh

Course objectives
• This course provides you with a basic understanding
of what digital devices are, how they operate, and
how they can be designed to perform useful
functions.
• The course is intended to give you an understanding of
Binary systems, Boolean algebra, digital design
techniques, logic gates, logic minimization, standard
combinational circuits, sequential circuits, flip-flops,
synthesis of synchronous sequential circuits, and
arithmetic circuits.

Contents
Week No. Topic Lecture Practical Total

1 Number Systems and Codes 3 2 5

2 Number Systems and Codes 2 5
3
Boolean Algebra and Logic
3 2 5
Simplification 3
Minimization Techniques-
4 2 5
Karnaugh Map 3
Minimization Techniques-
5 2 5
Karnaugh Map 3
6 Logic Gates 3 2 5

7
Arithmetic Circuits-Adders 3 2 5

8 Mid- Term Exam

Contents
Week No. Topic Lecture Practical Total

Arithmetic Circuits -Subtracter
9 3 2 5

10 Combinational Circuits 2 5
3

11 Combinational Circuits 3 2 5

12 Flip-Flops 3 2 5
Flip-Flops
13 3 2 5

14 Counters - Registers 3 2 5

15 Memory Devices 3 2 5
16 Final- Term Exam

Textbook

Logic and Computer Design Fundamentals, 4th
Edition by M. Morris Mano and Charles R. Kime,
Prentice Hall, 2008

1. Number Systems

Location in
course textbook

Chapt. 1
ITEC 1011 Introduction to Information Technologies

Common Number Systems

Used by Used in
System Base Symbols humans? computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexa- 16 0, 1, … 9, No No
decimal A, B, … F


Quantities/Counting (1 of 3)
Hexa-
Decimal Binary Octal 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
p. 33

Hexa-
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F


Hexa-
16 10000 20 10
17 10001 21 11
18 10010 22 12
19 10011 23 13
20 10100 24 14
21 10101 25 15
22 10110 26 16
23 10111 27 17 Etc.


Conversion Among Bases
• The possibilities:

Decimal Octal

Binary Hexadecimal

pp. 40-46

Quick Example

2510 = 110012 = 318 = 1916

Base


Decimal to Decimal (just for fun)

Decimal Octal

Binary Hexadecimal

Next slide…

Weight

12510 => 5 x 100 = 5
2 x 101 = 20
1 x 102 = 100
125

Base


Binary to Decimal

Decimal Octal

Binary Hexadecimal


Binary to Decimal
• Technique
– Multiply each bit by 2n, where n is the “weight”
of the bit
– The weight is the position of the bit, starting
from 0 on the right
– Add the results


Example
Bit “0”

1010112 => 1 x 20 = 1
1 x 21 = 2
0 x 22 = 0
1 x 23 = 8
0 x 24 = 0
1 x 25 = 32
4310


Octal to Decimal

Decimal Octal

Binary Hexadecimal


Octal to Decimal
• Technique
– Multiply each bit by 8n, where n is the “weight”
of the bit
from 0 on the right
– Add the results


Example

7248 => 4 x 80 = 4
2 x 81 = 16
7 x 82 = 448
46810


Hexadecimal to Decimal

Decimal Octal

Binary Hexadecimal


Hexadecimal to Decimal
• Technique
– Multiply each bit by 16n, where n is the
“weight” of the bit
from 0 on the right
– Add the results


Example

ABC16 => C x 160 = 12 x 1 = 12
B x 161 = 11 x 16 = 176
A x 162 = 10 x 256 = 2560
274810


Decimal to Binary

Decimal Octal

Binary Hexadecimal


Decimal to Binary
• Technique
– Divide by two, keep track of the remainder
– First remainder is bit 0 (LSB, least-significant
bit)
– Second remainder is bit 1
– Etc.


Example
12510 = ?2 2 125
2 62 1
2 31 0
2 15 1
2 7 1
2 3 1
2 1 1
0 1

12510 = 11111012


Decimal to Octal

Decimal Octal

Binary Hexadecimal


Decimal to Octal
• Technique
– Divide by 8
– Keep track of the remainder


Example
123410 = ?8

8 1234
8 154 2
8 19 2
8 2 3
0 2

123410 = 23228


Decimal to Hexadecimal

Decimal Octal

Binary Hexadecimal


Decimal to Hexadecimal
• Technique
– Divide by 16
– Keep track of the remainder


Example
123410 = ?16

16 1234
16 77 2
16 4 13 = D
0 4

123410 = 4D216


Octal to Binary

Decimal Octal

Binary Hexadecimal


Octal to Binary
• Technique
– Convert each octal digit to a 3-bit equivalent
binary representation


Example
7058 = ?2

7 0 5

111 000 101

7058 = 1110001012


Hexadecimal to Binary

Decimal Octal

Binary Hexadecimal


Hexadecimal to Binary
• Technique
– Convert each hexadecimal digit to a 4-bit
equivalent binary representation


Example
10AF16 = ?2

1 0 A F

0001 0000 1010 1111

10AF16 = 00010000101011112


Binary to Octal

Decimal Octal

Binary Hexadecimal


Binary to Octal
• Technique
– Group bits in threes, starting on right
– Convert to octal digits


Example
10110101112 = ?8

1 011 010 111

1 3 2 7

10110101112 = 13278


Binary to Hexadecimal

Decimal Octal

Binary Hexadecimal


Binary to Hexadecimal
• Technique
– Group bits in fours, starting on right
– Convert to hexadecimal digits


Example
10101110112 = ?16

10 1011 1011

2 B B

10101110112 = 2BB16

Octal to Hexadecimal

Decimal Octal

Binary Hexadecimal


Octal to Hexadecimal
• Technique
– Use binary as an intermediary


Example
10768 = ?16

1 0 7 6

001 000 111 110

2 3 E

10768 = 23E16

Hexadecimal to Octal

Decimal Octal

Binary Hexadecimal


Hexadecimal to Octal
• Technique
– Use binary as an intermediary


Example
1F0C16 = ?8

1 F 0 C

0001 1111 0000 1100

1 7 4 1 4

1F0C16 = 174148

Exercise – Convert ...
Hexa-
33
1110101
703
1AF

Don’t use a calculator!

Skip answer Answer

Exercise – Convert …
Answer

Hexa-
33 100001 41 21
117 1110101 165 75
451 111000011 703 1C3
431 110101111 657 1AF


Common Powers (1 of 2)
• Base 10
Power Preface Symbol Value
10-12 pico p .000000000001

10-9 nano n .000000001

10-6 micro .000001

10-3 milli m .001

103 kilo k 1000

106 mega M 1000000

109 giga G 1000000000
1012 tera T 1000000000000


Common Powers (2 of 2)
• Base 2
Power Preface Symbol Value
210 kilo k 1024

220 mega M 1048576

230 Giga G 1073741824

• What is the value of “k”, “M”, and “G”?
• In computing, particularly w.r.t. memory,
the base-2 interpretation generally applies


Example
In the lab…
1. Double click on My Computer
2. Right click on C:
3. Click on Properties

/ 230 =


Exercise – Free Space
• Determine the “free space” on all drives on
a machine in the lab
Free space
Drive Bytes GB
A:
C:
D:
E:
etc.


Review – multiplying powers
• For common bases, add powers

ab ac = ab+c

26 210 = 216 = 65,536
or…
26 210 = 64 210 = 64k


Fractions
• Decimal to decimal (just for fun)

3.14 => 4 x 10-2 = 0.04
1 x 10-1 = 0.1
3 x 100 = 3
3.14

pp. 46-50

Fractions
• Binary to decimal

10.1011 => 1 x 2-4 = 0.0625
1 x 2-3 = 0.125
0 x 2-2 = 0.0
1 x 2-1 = 0.5
0 x 20 = 0.0
1 x 21 = 2.0
2.6875

pp. 46-50

Fractions
• Decimal to binary .14579
x 2
3.14579 0.29158
x 2
0.58316
x 2
1.16632
x 2
0.33264
x 2
0.66528
x 2
1.33056
11.001001... etc.

p. 50

Fractions
• Octal to decimal

15.42 => 2 x 8-2 = 0.03125
4 x 8-1 = 0.5
5 x 80 = 5.0
1 x 81 = 8.0
13.53125

pp. 46-50

Fractions
• Decimal to octal .14
x 8
3.14 1.12
x 8
0.96
x 8
7.68
x 8
5.44
x 8
3.52
x 8
4.16
3.107534... etc.

p. 50

Fractions
• Hexadecimal to decimal

2B.84 => 4 x 16-2 = 0.015625
8 x 16-1 = 0.5
B x 160 = 11.0
2 x 161 = 32.0
43.515625

pp. 46-50

Fractions
• Decimal to Hexadecima .1
x 16
3.1 1.6
x 16
9.6
x 16
9.6
x 16
9.6
x 16
9.6
x 16
9.6
3.199999... etc.

p. 50

Exercise – Convert ...

Hexa-
29.8
101.1101
3.07
C.82
Don’t use a calculator!

Skip answer Answer

Exercise – Convert …
Answer

Hexa-
29.8 11101.110011… 35.63… 1D.CC…
5.8125 101.1101 5.64 5.D
3.109375 11.000111 3.07 3.1C
12.5078125 1100.10000010 14.404 C.82


Binary Addition (1 of 2)
• Two 1-bit values

A B A+B
0 0 0
0 1 1
1 0 1
1 1 10
“two”

pp. 36-38

Binary Addition (2 of 2)
• Two n-bit values
– Add individual bits
– Propagate carries
– E.g.,

1 1
10101 21
+ 11001 + 25
101110 46


Multiplication (1 of 3)
• Decimal (just for fun)

35
x 105
175
000
35
3675

pp. 39

• Binary, two 1-bit values

A B A B
0 0 0
0 1 0
1 0 0
1 1 1


• Binary, two n-bit values
– As with decimal values
– E.g.,
1110
x 1011
1110
1110
0000
1110
10011010

Binary Subtraction (1 of 2)
• Two 1-bit values

A B A- B
0 0 0
Borrow 1
0 1 1
1 0 1
1 1 0

pp. 36-38

– Subtract individual bits
– Propagate borrows
– E.g.,
10
11001
0 25
- 10101 - 21
00100 4


– Subtract individual bits
– Propagate borrows
– E.g.,
10 10 0
10001 11001 25
- 10101 - 10101 - 21
00100 00100 4


Subtraction with Complements
• Complements are used for simplifying
the subtraction operations.
• There are two types of complements for
each base-r system: the r's complement
and the (r — l)'s complement.
• 2's complement and 1's complement for
binary numbers, and the 10's
complement and 9's com-plement for
decimal numbers.

9's complement
• The 9's complement of a decimal number is
obtained by subtracting each digit from 9.
•
• The 9's complement of 546700 is:
• 999999 - 546700 = 453299.
• The 9's complement of 012398 is:
• 999999 - 012398 = 987601.


Binary numbers, the 1's complement
• The 1's complement of a binary number
is formed by changing 1's to 0's and 0's to
1's.
• Examples:
• The 1's complement of 1011000 is
• 0100111
• 1010010


10's complement

• The 10's complement can be formed by
leaving all least significant 0's un-changed,
subtracting the first nonzero least
significant digit from 10, and subtracting
all higher significant digits from 9.
• The 10's complement of 012398 is 987602.
• The 10's complement of 246700 is 753300.


2's complement
• The 2's complement can be formed by leaving
all least significant 0's and the first 1
unchanged, and replacing 1's with 0's and 0's
with 1's in all other higher significant digits.
• 0010100
• 1001001


Subtraction with Complements
• The subtraction of two n-digit unsigned numbers
M — N in base r can be done as follows:
• Add M to the r's complement of N.
• If M ≥ N, the sum will produce an end carry, r n,
which is discarded; what is left is the result M - N.
• If M < N, the sum does not produce an end carry.
To obtain the answer in a familiar form, take the
r's complement of the sum and place a negative
sign in front.


Examples to illustrate the procedure

• Given the two binary numbers;
• X = 1010100 and Y = 1000011,
• perform the subtraction:
• (a) X — Y
• (b) Y — X
• using 2's complements.


X — Y= 1010100 —1000011

• X = 1010100
• 2's complement of Y = + 0111101
• Sum = 10010001
• Discard end carry = -10000000
• Answer: X — Y = 0010001
•


Y — X= 1000011 — 1010100

• Y = 1000011
• 2's complement of X = + 0101100
• Sum = 1101111
• There is no end carry.
• Answer:
• Y - X = -(2's complement of 1101111)
• = - 0010001


Thank you

Next topic


Binary Codes
• Binary codes are codes which are represented
in binary system with modification from the
original ones.
• Binary codes are classified as:
– Weighted Binary Systems
– Non Weighted Codes


Weighted Binary Systems
• Weighted binary codes are those which
obey the positional weighting principles,
• Each position of the number represents a
specific weight.
• The codes 8421, 2421, 5421, and 5211 are
weighted binary codes.


Weighted Binary Systems


8421 Code/BCD Code
• The BCD (Binary Coded Decimal) is a straight
assignment of the binary equivalent.
• It is possible to assign weights to the binary bits
according to their positions.
• The weights in the BCD code are 8,4,2,1.
• Example: The bit assignment 1001, can be seen by
its weights to represent the decimal 9 because:
• 1x8+0x4+0x2+1x1 = 9
• Ex. number 12 is represented in BCD as [0001 0010]


2421 Code
• 2421 Code This is a weighted code, its weights
are 2, 4, 2 and 1.
• A decimal number is represented in 4-bit form and
the total four bits weight is 2 + 4 + 2 + 1 = 9.
• Hence the 2421 code represents the decimal
numbers from 0 to 9.


5211 Code
• 5211 Code This is a weighted code, its weights
are 5, 2, 1 and 1.
• A decimal number is represented in 4-bit form and
the total four bits weight is 5 + 2 + 1 + 1 = 9.
• Hence the 5211 code represents the decimal
numbers from 0 to 9.


Reflective Code
• Reflective Code A code is said to be
reflective when code for 9 is complement
for the code for 0, and so is for 8 and 1
codes, 7 and 2, 6 and 3, 5 and 4.
• Codes 2421, 5211, and excess-3 are
reflective, whereas the 8421 code is not.


Sequential Codes
• Sequential Codes A code is said to be sequential
when two subsequent codes, seen as numbers in
binary representation, differ by one.
• This greatly aids mathematical manipulation of
data.
• The 8421 and Excess-3 codes are sequential,
whereas the 2421 and 5211 codes are not.


Excess-3 Code
• Excess-3 Code Excess-3 is a non weighted code
used to express decimal numbers.
• The code derives its name from the fact that each
binary code is the corresponding 8421 code plus
0011(3).
• Example: 1000 of 8421 = 1011 in Excess-3


Error Detecting and Correction Codes
• For reliable transmission and storage of digital
data, error detection and correction is
required.


Error Detecting Codes
• When data is transmitted from one point to
another there are chances that data may get
corrupted.
• To detect these data errors, we use special
codes, which are error detection codes.


Parity check
• In parity codes, every binary message is checked if they
have even number of ones or even number of zeros.
• Based on this information an additional bit is appended
to the original data.
• At the receiver side, once again parity is calculated and
matched with the received parity, and if they match,
data is ok, otherwise data is corrupt.
• There are two types of parity: Even parity and Odd
Parity


Parity
• There are two types of parity:
• Even parity: Checks if there is an even
number of ones; if so, parity bit is zero.
When the number of ones is odd then parity
bit is set to 1.
Message Even parity code
xyz xyz p
000 000 0
001 001 1
011 011 0

Parity
• Odd Parity: Checks if there is an odd
number of ones; if so, parity bit is zero.
When number of ones is even then parity bit
is set to 1.
•
•

Message Odd parity code
xyz xyz p
000 000 1
001 001 0
011 011 1

Alphanumeric Codes
• The binary codes that can be used to represent all
the letters of the alphabet, numbers and
mathematical symbols, punctuation marks, are
known as alphanumeric codes or character codes.
• These codes enable us to interface the input-output
devices like the keyboard, printers, video displays
with the computer.


ASCII Code
• ASCII Code ASCII stands for American
Standard Code for Information Interchange.
• It has become a world standard alphanumeric code
for microcomputers and computers.
• It is a 7-bit code representing 27 = 128 different
characters.
• These characters represent 26 upper case letters (A
to Z), 26 lowercase letters (a to z), 10 numbers (0
to 9), 33 special characters and symbols and 33
control characters.
•

ASCII Code

• The 7-bit code is divided into two portions,
The leftmost 3 bits portion is called zone
bits and the 4-bit portion on the right is
called numeric bits.

Character 7-bit ASCII
A 100 0001
B 100 0010
3 011 0011


ASCII Code

• An 8-bit version of ASCII code is known
as ASCII-8.
• The 8-bit version can represent a maximum
of 256 characters.


EBCDIC Code
• EBCDIC Code EBCDIC stands for Extended
Binary Coded Decimal Interchange.
• It is mainly used with large computer systems like
mainframes.
• EBCDIC is an 8-bit code and thus accommodates
up to 256 characters.
• An EBCDIC code is divided into two portions: 4
zone bits (on the left) and 4 numeric bits (on the
right).


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