2. 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.
3. 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
7. 1. Number Systems
Location in
course textbook
Chapt. 1
ITEC 1011 Introduction to Information Technologies
8. 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
ITEC 1011 Introduction to Information Technologies
12. Conversion Among Bases
• The possibilities:
Decimal Octal
Binary Hexadecimal
pp. 40-46
ITEC 1011 Introduction to Information Technologies
13. Quick Example
2510 = 110012 = 318 = 1916
Base
ITEC 1011 Introduction to Information Technologies
14. Decimal to Decimal (just for fun)
Decimal Octal
Binary Hexadecimal
Next slide…
ITEC 1011 Introduction to Information Technologies
15. Weight
12510 => 5 x 100 = 5
2 x 101 = 20
1 x 102 = 100
125
Base
ITEC 1011 Introduction to Information Technologies
16. Binary to Decimal
Decimal Octal
Binary Hexadecimal
ITEC 1011 Introduction to Information Technologies
17. 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
ITEC 1011 Introduction to Information Technologies
18. 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
ITEC 1011 Introduction to Information Technologies
19. Octal to Decimal
Decimal Octal
Binary Hexadecimal
ITEC 1011 Introduction to Information Technologies
20. Octal to Decimal
• Technique
– Multiply each bit by 8n, 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
ITEC 1011 Introduction to Information Technologies
21. Example
7248 => 4 x 80 = 4
2 x 81 = 16
7 x 82 = 448
46810
ITEC 1011 Introduction to Information Technologies
22. Hexadecimal to Decimal
Decimal Octal
Binary Hexadecimal
ITEC 1011 Introduction to Information Technologies
23. Hexadecimal to Decimal
• Technique
– Multiply each bit by 16n, 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
ITEC 1011 Introduction to Information Technologies
24. Example
ABC16 => C x 160 = 12 x 1 = 12
B x 161 = 11 x 16 = 176
A x 162 = 10 x 256 = 2560
274810
ITEC 1011 Introduction to Information Technologies
25. Decimal to Binary
Decimal Octal
Binary Hexadecimal
ITEC 1011 Introduction to Information Technologies
26. 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.
ITEC 1011 Introduction to Information Technologies
28. Decimal to Octal
Decimal Octal
Binary Hexadecimal
ITEC 1011 Introduction to Information Technologies
29. Decimal to Octal
• Technique
– Divide by 8
– Keep track of the remainder
ITEC 1011 Introduction to Information Technologies
30. Example
123410 = ?8
8 1234
8 154 2
8 19 2
8 2 3
0 2
123410 = 23228
ITEC 1011 Introduction to Information Technologies
31. Decimal to Hexadecimal
Decimal Octal
Binary Hexadecimal
ITEC 1011 Introduction to Information Technologies
32. Decimal to Hexadecimal
• Technique
– Divide by 16
– Keep track of the remainder
ITEC 1011 Introduction to Information Technologies
33. Example
123410 = ?16
16 1234
16 77 2
16 4 13 = D
0 4
123410 = 4D216
ITEC 1011 Introduction to Information Technologies
34. Octal to Binary
Decimal Octal
Binary Hexadecimal
ITEC 1011 Introduction to Information Technologies
35. Octal to Binary
• Technique
– Convert each octal digit to a 3-bit equivalent
binary representation
ITEC 1011 Introduction to Information Technologies
36. Example
7058 = ?2
7 0 5
111 000 101
7058 = 1110001012
ITEC 1011 Introduction to Information Technologies
37. Hexadecimal to Binary
Decimal Octal
Binary Hexadecimal
ITEC 1011 Introduction to Information Technologies
38. Hexadecimal to Binary
• Technique
– Convert each hexadecimal digit to a 4-bit
equivalent binary representation
ITEC 1011 Introduction to Information Technologies
39. Example
10AF16 = ?2
1 0 A F
0001 0000 1010 1111
10AF16 = 00010000101011112
ITEC 1011 Introduction to Information Technologies
40. Binary to Octal
Decimal Octal
Binary Hexadecimal
ITEC 1011 Introduction to Information Technologies
41. Binary to Octal
• Technique
– Group bits in threes, starting on right
– Convert to octal digits
ITEC 1011 Introduction to Information Technologies
42. Example
10110101112 = ?8
1 011 010 111
1 3 2 7
10110101112 = 13278
ITEC 1011 Introduction to Information Technologies
43. Binary to Hexadecimal
Decimal Octal
Binary Hexadecimal
ITEC 1011 Introduction to Information Technologies
44. Binary to Hexadecimal
• Technique
– Group bits in fours, starting on right
– Convert to hexadecimal digits
ITEC 1011 Introduction to Information Technologies
45. Example
10101110112 = ?16
10 1011 1011
2 B B
10101110112 = 2BB16
ITEC 1011 Introduction to Information Technologies
46. Octal to Hexadecimal
Decimal Octal
Binary Hexadecimal
ITEC 1011 Introduction to Information Technologies
47. Octal to Hexadecimal
• Technique
– Use binary as an intermediary
ITEC 1011 Introduction to Information Technologies
48. Example
10768 = ?16
1 0 7 6
001 000 111 110
2 3 E
10768 = 23E16
ITEC 1011 Introduction to Information Technologies
49. Hexadecimal to Octal
Decimal Octal
Binary Hexadecimal
ITEC 1011 Introduction to Information Technologies
50. Hexadecimal to Octal
• Technique
– Use binary as an intermediary
ITEC 1011 Introduction to Information Technologies
51. Example
1F0C16 = ?8
1 F 0 C
0001 1111 0000 1100
1 7 4 1 4
1F0C16 = 174148
ITEC 1011 Introduction to Information Technologies
52. Exercise – Convert ...
Hexa-
Decimal Binary Octal decimal
33
1110101
703
1AF
Don’t use a calculator!
Skip answer Answer
ITEC 1011 Introduction to Information Technologies
54. 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
ITEC 1011 Introduction to Information Technologies
55. 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
ITEC 1011 Introduction to Information Technologies
56. Example
In the lab…
1. Double click on My Computer
2. Right click on C:
3. Click on Properties
/ 230 =
ITEC 1011 Introduction to Information Technologies
57. 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.
ITEC 1011 Introduction to Information Technologies
58. Review – multiplying powers
• For common bases, add powers
ab ac = ab+c
26 210 = 216 = 65,536
or…
26 210 = 64 210 = 64k
ITEC 1011 Introduction to Information Technologies
59. 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
ITEC 1011 Introduction to Information Technologies
60. 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
ITEC 1011 Introduction to Information Technologies
61. 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
ITEC 1011 Introduction to Information Technologies
62. 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
ITEC 1011 Introduction to Information Technologies
63. 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
ITEC 1011 Introduction to Information Technologies
64. 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
ITEC 1011 Introduction to Information Technologies
65. 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
ITEC 1011 Introduction to Information Technologies
66. Exercise – Convert ...
Hexa-
Decimal Binary Octal decimal
29.8
101.1101
3.07
C.82
Don’t use a calculator!
Skip answer Answer
ITEC 1011 Introduction to Information Technologies
68. 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
ITEC 1011 Introduction to Information Technologies
69. Binary Addition (2 of 2)
• Two n-bit values
– Add individual bits
– Propagate carries
– E.g.,
1 1
10101 21
+ 11001 + 25
101110 46
ITEC 1011 Introduction to Information Technologies
70. Multiplication (1 of 3)
• Decimal (just for fun)
35
x 105
175
000
35
3675
pp. 39
ITEC 1011 Introduction to Information Technologies
71. Multiplication (2 of 3)
• Binary, two 1-bit values
A B A B
0 0 0
0 1 0
1 0 0
1 1 1
ITEC 1011 Introduction to Information Technologies
72. Multiplication (3 of 3)
• Binary, two n-bit values
– As with decimal values
– E.g.,
1110
x 1011
1110
1110
0000
1110
10011010
ITEC 1011 Introduction to Information Technologies
73. 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
ITEC 1011 Introduction to Information Technologies
74. Binary Subtraction (2 of 2)
• Two n-bit values
– Subtract individual bits
– Propagate borrows
– E.g.,
10
11001
0 25
- 10101 - 21
00100 4
ITEC 1011 Introduction to Information Technologies
75. Binary Subtraction (2 of 2)
• Two n-bit values
– Subtract individual bits
– Propagate borrows
– E.g.,
10 10 0
10001 11001 25
- 10101 - 10101 - 21
00100 00100 4
ITEC 1011 Introduction to Information Technologies
76. 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.
ITEC 1011 Introduction to Information Technologies
77. 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.
ITEC 1011 Introduction to Information Technologies
78. 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
• The 1's complement of 0101101 is
• 1010010
ITEC 1011 Introduction to Information Technologies
79. 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.
ITEC 1011 Introduction to Information Technologies
80. 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.
• The 2's complement of 1101100 is
• 0010100
• The 2's complement of 0110111 is
• 1001001
ITEC 1011 Introduction to Information Technologies
81. 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.
ITEC 1011 Introduction to Information Technologies
82. 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.
ITEC 1011 Introduction to Information Technologies
83. X — Y= 1010100 —1000011
• X = 1010100
• 2's complement of Y = + 0111101
• Sum = 10010001
• Discard end carry = -10000000
• Answer: X — Y = 0010001
•
ITEC 1011 Introduction to Information Technologies
84. 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
ITEC 1011 Introduction to Information Technologies
85. Thank you
Next topic
ITEC 1011 Introduction to Information Technologies
86. 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
ITEC 1011 Introduction to Information Technologies
87. 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.
ITEC 1011 Introduction to Information Technologies
89. 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]
ITEC 1011 Introduction to Information Technologies
90. 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.
ITEC 1011 Introduction to Information Technologies
91. 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.
ITEC 1011 Introduction to Information Technologies
92. 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.
ITEC 1011 Introduction to Information Technologies
93. 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.
ITEC 1011 Introduction to Information Technologies
94. 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
ITEC 1011 Introduction to Information Technologies
95. Error Detecting and Correction Codes
• For reliable transmission and storage of digital
data, error detection and correction is
required.
ITEC 1011 Introduction to Information Technologies
96. 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.
ITEC 1011 Introduction to Information Technologies
97. 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
ITEC 1011 Introduction to Information Technologies
98. 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
ITEC 1011 Introduction to Information Technologies
99. 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
ITEC 1011 Introduction to Information Technologies
100. 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.
ITEC 1011 Introduction to Information Technologies
101. 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.
•
ITEC 1011 Introduction to Information Technologies
102. 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
ITEC 1011 Introduction to Information Technologies
103. 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.
ITEC 1011 Introduction to Information Technologies
104. 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).
ITEC 1011 Introduction to Information Technologies