Lecture 1

Dr. Nabil M. Eldakhly

Faculty of Computers and Information –
Department of CS-
CS-SAMS
&
The French University in Egypt (UFE)
Text Book
Digital Systems &
Binary Numbers
 Outline

1. Digital Systems
2. Binary Numbers
3. Number-base Conversions
4. Octal and Hexadecimal Numbers
5. Complements
6. Signed Binary Numbers
7. Binary Codes
8. Binary Storage and Registers
9. Binary Logic
 Outline

1. Digital Systems
2. Binary Numbers
3. Number-base Conversions
4. Octal and Hexadecimal Numbers
5. Complements
6. Signed Binary Numbers
7. Binary Codes
8. Binary Storage and Registers
9. Binary Logic
 Digital Systems & Binary Numbers
Introduction

The Information Age (also known as the Computer Age, Digital Age,
Internet Age, and Information Era) is an idea that the current age will
be characterized by the ability of individuals to transfer information
freely and to have instant access to knowledge that would have been
difficult or impossible to find previously.
‫رة‬ ‫ و ر ا و ت( ھو‬،‫ و ر ا ر ت‬، ‫ وا ر ا ر‬،‫و ر‬ ‫را‬ ‫م‬ ً ‫ر ا و ت )ا روف أ‬
‫ن‬ ‫ ول ا "وري ( ا ر * ا‬$ ‫ وا‬،* ‫ر‬$ ‫ز &درة ا' راد ( &ل ا و ت‬ ‫وف‬ $ ‫" دھ أن ا ر ا‬
. &ً , ‫ور‬0 ‫ ل ا‬$ ‫ أن ن ن ا ب أو ا‬, -.
The idea is linked to the concept of a "Digital Age" or "Digital
Revolution," and carries the ramifications of a shift from traditional
industry that the Industrial Revolution brought through
industrialization, to an economy based around the manipulation of
information.
, 4 ‫*ا & د*ا‬ ‫ول ن ا‬$ ‫ ل دا ت ا‬$ ‫ و‬،"* ‫ورة ا ر‬0 ‫وم "ا ر ا ر " أو "ا‬," ‫و ر ط ا " رة‬
6 .‫و ت‬ ‫ ب‬5 ‫م ( ا‬9 ‫د‬ ‫ إ ( ا‬،7 ‫ل ا‬56 ‫* ن‬ ‫ورة ا‬0 ‫ا‬
 Digital Systems & Binary Numbers
Introduction (Cont.)

Digital Age and Information Age

Digital Computers
– General purposes
– Many scientific, industrial and commercial applications
Digital Systems
– Telephone switching exchanges
– Digital camera
– Electronic calculators, Personal Digital Assistants (PDAs)
– Digital TV
Discrete Information-processing Systems
– Manipulate discrete elements of information
– For example, {1, 2, 3, …} and {A, B, C, …}…
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 Digital Systems & Binary Numbers
Introduction (Cont.)- Analog & Digital Signal

Analog System
– The physical quantities or signals may vary continuously over a specified
range.
Digital system
– The physical quantities or signals can assume only discrete values.
– Greater accuracy

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 Digital Systems & Binary Numbers
Binary Digital Signal

An information variable represented by physical quantity.

For digital systems, the variable takes on discrete values.
– Two level, or binary values are the most prevalent values.
Binary values are represented abstractly by:
– Digits: 0 and 1
– Words (symbols): False (F) and True (T)
– Words (symbols): Low (L) and High (H)
– words: On and Off
Binary values are represented by values or ranges of values of
physical quantities.
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 Digital Systems & Binary Numbers
Number Systems- 1. Decimal Number System

Base (also called radix) = 10

– 10 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }
Digit Position
2 1 0 -1 -2
– Integer & fraction
5 1 2 7 4
Digit Weight
102 101 100 10-1 10-2
Position
– Weight = (Base) 100 10 1 0.1 0.01
Magnitude
– Sum of “Digit x Weight”
5 *100 +1*10 + 2*1 + 7*0.1 + 4*0.01
Formal Notation
(512.74)10

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 Digital Systems & Binary Numbers
Number Systems- 2. Octal Number System

Base = 8
2 1 0 -1 -2
– 8 digits { 0, 1, 2, 3, 4, 5, 6, 7 }
5 1 2 7 4
Weights
82 81 80 8-1 8-2
Position
– Weight = (Base) 64 8 1 1/8 1/64
Magnitude 5 *64 +1*8 + 2*1+ 7/8+ 4/64

– Sum of “Digit x Weight” =(330.9375)10

Formal Notation (512.74)8

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 Digital Systems & Binary Numbers
Number Systems- 3. Binary Number System

Base = 2
– 2 digits { 0, 1 }, called binary digits or “bits”
2 1 0 -1 -2
Weights
Position 1 0 1 0 1
– Weight = (Base)
22 21 20 2-1 2-2
Magnitude 4 2 1 1/2 1/4
1 *4 + 0*2 + 1*1 + 0*1/2+ 1*1/4
– Sum of “Bit x Weight”
=(5.25)10
Formal Notation
(101.01)2

Groups of bits 4 bits = Nibble 1011

8 bits = Byte 11000101
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 Digital Systems & Binary Numbers
Number Systems- 4. Hexadecimal Number System

Base = 16
– 16 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F }
2 1 0 -1 -2
Weights
– Weight = (Base)
Position 1 E 5 7 A
162 161 160 16-1 16-2
Magnitude
256 16 1 1/16 1/256
– Sum of “Digit x Weight”
Formal Notation 1 *256+14 *16 + 5 *1+ 7 /16 +10 /256
=(485.4765625)10
(1E5.7A)16

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 Digital Systems & Binary Numbers
The Power of 2

n 2n n 2n
0 20=1 8 28=256
1 21=2 9 29=512
2 22=4 10 210=1024 Kilo

3 23=8 11 211=2048
4 24=16 12 212=4096
5 25=32 20 220=1M Mega

6 26=64 30 230=1G Giga

7 27=128 40 240=1T Tera

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 Digital Systems & Binary Numbers
Decimal Addition

1 1 Carry
5 5
+ 5 5

1 1 0
= Ten ≥ Base
Subtract a Base

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 Digital Systems & Binary Numbers
Binary Addition

Column Addition

1 1 1 1 1 1
1 1 1 1 0 1 = 61
+ 1 0 1 1 1 = 23

1 0 1 0 1 0 0 = 84

≥ (2)10

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 Digital Systems & Binary Numbers
Binary Subtraction

Borrow a “Base” when needed

1x25 2x22 = (10)2

0x26 2x25 2x24 0x23 0x22 2 x21

1x26 0x25 0x24 1x23 1x22 0x21 1x20

1 0 0 1 1 0 1 = 77
− 1x24 0x23 1x22 1x21 1x20

1 0 1 1 1 = 23

0 1 1 0 1 1 0 = 54

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 Digital Systems & Binary Numbers
Binary Subtraction (Cont.)

Borrow a “Base” when needed

1 2 = (10)2
0 2 2 0 0 2
1 0 0 1 1 0 1 = 77
− 1 0 1 1 1 = 23

0 1 1 0 1 1 0 = 54

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 Digital Systems & Binary Numbers
Binary Multiplication

Bit by bit
1 0 1 1 1
x 1 0 1 0
0 0 0 0 0
1 0 1 1 1
0 0 0 0 0
+ 1 0 1 1 1

1 1 1 0 0 1 1 0

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 Outline

1. Digital Systems
2. Binary Numbers
3. Number-base Conversions
4. Octal and Hexadecimal Numbers
5. Complements
6. Signed Binary Numbers
7. Binary Codes
8. Binary Storage and Registers
9. Binary Logic
 Digital Systems & Binary Numbers
Number Base Conversions

Evaluate
Magnitude
Octal
(Base 8)

Evaluate
Magnitude
Decimal Binary
(Base 10) (Base 2)

Hexadecimal
(Base 16)
Evaluate
Magnitude
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 Digital Systems & Binary Numbers
Decimal (Integer) To Binary Conversion

Divide the number by the ‘Base’ (=2)

Take the remainder (either 0 or 1) as a coefficient
Take the quotient and repeat the division

Example: (13)10
Quotient Remainder Coefficient
13/ 2 = 6 1 a0 = 1
6 /2= 3 0 a1 = 0
3 /2= 1 1 a2 = 1
1 /2= 0 1 a3 = 1
Answer: (13)10 = (a3 a2 a1 a0)2 = (1101)2

Most Significant Bit (MSB) MSB LSB

Least Significant Bit (LSB)
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 Digital Systems & Binary Numbers
Decimal (Fraction) To Binary Conversion

Multiply the number by the ‘Base’ (=2)

Take the integer (either 0 or 1) as a coefficient
Take the resultant fraction and repeat the division

Example: (0.625)10
Integer Fraction Coefficient
0.625 * 2 = 1 . 25 a-1 = 1
0.25 * 2 = 0 . 5 a-2 = 0
0.5 *2= 1 . 0 a-3 = 1
Answer: (0.625)10 = (0.a-1 a-2 a-3)2 = (0.101)2

MSB LSB

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 Digital Systems & Binary Numbers
Decimal To Octal Conversion

Example: (175)10
Quotient Remainder Coefficient
175 / 8 = 21 7 a0 = 7
21 / 8 = 2 5 a1 = 5
2 /8= 0 2 a2 = 2
Answer: (175)10 = (a2 a1 a0)8 = (257)8

Example: (0.3125)10
Integer Fraction Coefficient
0.3125 * 8 = 2 . 5 a-1 = 2
0.5 *8= 4 . 0 a-2 = 4
Answer: (0.3125)10 = (0.a-1 a-2)8 = (0.24)8
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 Outline

1. Digital Systems
2. Binary Numbers
3. Number-base Conversions
4. Octal and Hexadecimal Numbers
5. Complements
6. Signed Binary Numbers
7. Binary Codes
8. Binary Storage and Registers
9. Binary Logic
 Digital Systems & Binary Numbers
Binary To Octal Conversion

Octal Binary
8 = 23
Each group of 3 bits represents an octal digit 0 000
1 001

Assume Zeros 2 010

Example:
3 011
( 1 0 1 1 0 . 0 1 )2
4 100
5 101
6 110
( 2 6 . 2 )8
7 111
Works both ways (Binary to Octal & Octal to Binary)
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 Digital Systems & Binary Numbers
Binary To Hexadecimal Conversion
Hex Binary
16 = 24 0 0000
Each group of 4 bits represents a 1 0001
2 0010
hexadecimal digit 3 0011
4 0100
5 0101
Assume Zeros
Example: 6 0110
7 0111
( 1 0 1 1 0 . 0 1 )2 8 1000
9 1001
A 1010
B 1011
C 1100
(1 6 . 4 )16 D
E
1101
1110
F 1111
Works both ways (Binary to Hex & Hex to Binary)
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 Digital Systems & Binary Numbers
Octal To Hexadecimal Conversion

Convert to Binary as an intermediate step

Example:
( 2 6 . 2 )8

Assume Zeros Assume Zeros

( 0 1 0 1 1 0 . 0 1 0 )2

(1 6 . 4 )16

Works both ways (Octal to Hex & Hex to Octal)

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 Digital Systems & Binary Numbers
Decimal, Binary, Octal, & Hexadecimal Numbers
Decimal Binary Octal Hex
00 0000 00 0
01 0001 01 1
02 0010 02 2
03 0011 03 3
04 0100 04 4
05 0101 05 5
06 0110 06 6
07 0111 07 7
08 1000 10 8
09 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
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