Download Chapter 1.0 - Number & Codes

EKT 121 / 4
ELEKTRONIK DIGIT 1
CHAPTER 1 : INTRODUCTION
1.0 Number & Codes








Digital and analog quantities
Decimal numbering system (Base 10)
Binary numbering system (Base 2)
Hexadecimal numbering system (Base 16)
Octal numbering system (Base 8)
Number conversion
Binary arithmetic
1’s and 2’s complements of binary numbers






Signed numbers
Arithmetic operations with signed numbers
Binary-Coded-Decimal (BCD)
ASCII codes
Gray codes
Digital codes & parity
Digital and analog quantities

Two ways of representing the numerical values of
quantities :
i) Analog (continuous)
ii) Digital (discrete)

Analog : a quantity represented by voltage, current
or meter movement that is proportional to the value
that quantity
Digital : the quantities are represented not by
proportional quantities but by symbols called digits

Digital and analog systems

Digital system:



combination of devices designed to manipulate logical
information or physical quantities that are represented in
digital forms
include digital computers and calculators, digital
audio/video equipments, telephone system.
Analog system:


contains devices manipulate physical quantities that are
represented in analog form
audio amplifiers, magnetic tape recording and playback
equipment, and simple light dimmer switch
Analog Quantities
• Continuous values
Digital Waveform
Introduction to Numbering Systems

We are all familiar with the decimal number
system (Base 10). Some other number systems
that we will work with are:



Binary  Base 2
Octal  Base 8
Hexadecimal  Base 16
Number Systems

Decimal

0~9

Binary

0~1

Octal

0~7

Hexadecimal

0~F
Characteristics of Numbering Systems
1)
2)
3)
4)
5)
The digits are consecutive.
The number of digits is equal to the size of the
base.
Zero is always the first digit.
When 1 is added to the largest digit, a sum of zero
and a carry of one results.
Numeric values determined by the implicit
positional values of the digits.
N
U
M
B
E
R
S
Y
S
T
E
M
S
Dec
Hex
Octal
Binary
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
000
001
002
003
004
005
006
007
010
011
012
013
014
015
016
017
00000000
00000001
00000010
00000011
00000100
00000101
00000110
00000111
00001000
00001001
00001010
00001011
00001100
00001101
00001110
00001111
Significant Digits
Binary: 11101101
Most significant digit
Least significant digit
Hexadecimal: 1D63A7A
Most significant digit
Least significant digit
Binary Number System




Also called the “Base 2 system”
The binary number system is used to model the
series of electrical signals computers use to
represent information
0 represents the no voltage or an off state
1 represents the presence of voltage or an
on state
Binary Numbering Scale
Base 2 Number
Base 10 Equivalent
Power
Positional Value
000
0
20
1
001
1
21
2
010
2
22
4
011
3
23
8
100
4
24
16
101
5
25
32
110
6
26
64
111
7
27
128
Octal Number System





Also known as the Base 8 System
Uses digits 0 - 7
Readily converts to binary
Groups of three (binary) digits can be used to
represent each octal digit
Also uses multiplication and division
algorithms for conversion to and from base
10
Hexadecimal Number System
Base 16 system
 Uses digits 0-9 &
letters A,B,C,D,E,F
 Groups of four bits
represent each
base 16 digit

Number Conversion

Any Radix (base) to Decimal Conversion
Number Conversion

Binary to Decimal Conversion
Binary to Decimal Conversion
Convert (10101101)2 to its decimal equivalent:
1
Binary
Positional Values
Products
0
1
0
1 1
0
1
x x x x x x x x
27 26 25 24 23 22 21 20
Octal to Decimal Conversion
Convert 6538 to its decimal equivalent:
Octal Digits
Positional Values
Products
6
5
3
82
81
80
x
x
x
Hexadecimal to Decimal Conversion
Convert 3B4F16 to its decimal equivalent:
Hex Digits
Positional Values
Products
3
x
B
x
4
x
F
x
163 162 161 160
Number Conversion

Decimal to Any Radix (Base) Conversion
1. INTEGER DIGIT:
Repeated division by the radix & record the
remainder
2. FRACTIONAL DECIMAL:
Multiply the number by the radix until the
answer is in integer
Example:
25.3125 to Binary
Decimal to Binary Conversion
Remainder
2 5 = 12 +
2
1
12 = 6 +
2
0
6
2
= 3 +
0
3
2
= 1 +
1
1
2
= 0 +
1
MSB
LSB
2510 = 1 1 0 0 1 2
Decimal to Binary Conversion
Carry
0.3125 x 2 = 0.625
0
0.625 x 2 = 1.25
1
0.25 x 2 = 0.50
0
0.5 x 2 = 1.00
1
The Answer:
MSB
.0
1 1 0 0 1.0 1 0 1
LSB
1 0 1
Decimal to Octal Conversion
Convert 42710 to its octal equivalent:
427 / 8 = 53 R3
53 / 8 = 6 R5
6 / 8 = 0 R6
Divide by 8; R is LSD
Divide Q by 8; R is next digit
Repeat until Q = 0
6538
Decimal to Hexadecimal Conversion
Convert 83010 to its hexadecimal equivalent:
Number Conversion

Binary to Octal Conversion (vice versa)
1.
Grouping the binary position in groups
of three starting at the least significant
position.
Octal to Binary Conversion
Each octal number converts to 3 binary digits
To convert 6538 to binary, just
substitute code:
6
5
3
110 101 011
Number Conversion

Example:
 Convert the following binary numbers to their
octal equivalent (vice versa).
a)
c)

1001.11112
1010011.110112
Answer:
a)
b)
c)
b) 47.38
Number Conversion

Binary to Hexadecimal Conversion (vice
versa)
1.
Grouping the binary position in 4-bit
groups, starting from the least
significant position.
Binary to Hexadecimal Conversion


The easiest method for converting binary to
hexadecimal is to use a substitution code
Each hex number converts to 4 binary digits
Number Conversion

Example:
 Convert the following binary numbers to
their hexadecimal equivalent (vice versa).
a)
b)

10000.12
1F.C16
Answer:
a)
b)
Substitution Code
Convert 0101011010101110011010102 to hex using
the 4-bit substitution code :
0101 0110 1010 1110 0110 1010
Substitution Code
Substitution code can also be used to convert binary
to octal by using 3-bit groupings:
010 101 101 010 111 001 101 010
Binary Addition
0 + 0 = 0 Sum of 0 with a carry of 0
0 + 1 = 1 Sum of 1 with a carry of 0
1 + 0 = 1 Sum of 1 with a carry of 0
1 + 1 = 10 Sum of 1 with a carry of 1
Example:
11001
111
+ 1101
+ 11
100110
???
Simple Arithmetic


Addition
Example:

100011002
+ 1011102
101110102


Substraction
Example:
-
10001002
1011102
101102
Example:
5816
+ 2416
7C16
Binary Subtraction
0-0=0
1-1=0
1-0=1
10 -1 = 1
0 -1 with a borrow of 1
Example:
1011
101
- 111
- 11
100
???
Binary Multiplication
0X0=0
0X1=0
Example:
1X0=0
1X1=1
100110
X
101
100110
000000
+ 100110
10111110
Binary Division
Use the same procedure as decimal division
1’s complements of binary numbers

Changing all the 1s to 0s and all the 0s to 1s
Example:
110100101
Binary number
001011010
1’s complement
2’s complements of binary numbers

2’s complement


Step 1: Find 1’s complement of the number
Binary #
11000110
1’s complement
00111001
Step 2: Add 1 to the 1’s complement
00111001
+ 00000001
00111010
Signed Magnitude Numbers
110010..
Sign bit
0 = positive
1 = negative
…00101110010101
31 bits for magnitude
This is your basic
Integer format
Sign numbers

Left most is the sign bit


0 is for positive, and 1 is for negative
Sign-magnitude
00011001
= +25
sign bit magnitude bits


1’s complement
The negative number is the 1’s complement of
the corresponding positive number
 Example:
+25 is 00011001
-25 is 11100110

Sign numbers

2’s complement


The positive number – same as sign magnitude
and 1’s complement
The negative number is the 2’s complement of
the corresponding positive number.
Example
Express +19 and -19 in
i. sign magnitude
ii. 1’s complement
iii. 2’s complement
Digital Codes

BCD (Binary Coded Decimal) Code
1.

Represent each of the 10 decimal
digits (0~9) as a 4-bit binary code.
Example:
 Convert 15 to BCD.
1

5
0001 0101BCD
Convert 10 to binary and BCD.
Digital Codes

ASCII (American Standard Code for
Information Interchange) Code
1.
Used to translate from the keyboard
characters to computer language
Digital Codes

Decimal Binary
Gray
Code
Only 1 bit changes
Can’t be used in
arithmetic circuits
0
0000
0000
1
0001
0001
Binary to Gray Code
and vice versa.
2
0010
0011
3
0011
0010
4
0100
0110
5
0101
0111
6
0110
0101
The Gray Code



END OF
Number & Codes