1. CHAPTER 1 : INTRODUCTION
EKT 121 / 4 ELEKTRONIK DIGIT 1
CHAPTER 1 : INTRODUCTION

2. 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

3. Signed numbers
Arithmetic operations with signed numbers
Binary-Coded-Decimal (BCD)
ASCII codes
Gray codes
Digital codes & parity

4. 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

5. 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

6. Analog Quantities
Continuous values

7. Digital Waveform

8. 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

9. Number Systems Decimal Binary Octal Hexadecimal 0 ~ 9 0 ~ 1 0 ~ 7
0 ~ F

10. Characteristics of Numbering Systems
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.

11.
A B C D E F
Binary
Octal
Hex
Dec N U M B E R S Y T

12. Most significant digit Least significant digit
Significant Digits
Binary:
Most significant digit Least significant digit
Hexadecimal: 1D63A7A

13. 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

14. Binary Numbering Scale
Base 2 Number
Base 10 Equivalent
Power
Positional Value
000 20 1
001 21 2
010 22 4
011 3 23 8
100 24 16
101 5 25 32
110 6 26 64
111 7 27
128

15. 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

16. 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

17. Number Conversion
Any Radix (base) to Decimal Conversion

18. Number Conversion
Binary to Decimal Conversion

19. Binary to Decimal Conversion
Convert ( )2 to its decimal equivalent:
Binary
Positional Values x x x x x x x x 27 26 25 24 23 22 21 20
Products
17310

20. Octal to Decimal Conversion
Convert 6538 to its decimal equivalent:
Octal Digits x x x
Positional Values
Products
42710

21. Hexadecimal to Decimal Conversion
Convert 3B4F16 to its decimal equivalent:
Hex Digits
3 B F x x x x
Positional Values
Products
15,18310

22. Number Conversion Decimal to Any Radix (Base) Conversion
INTEGER DIGIT:
Repeated division by the radix & record the remainder
FRACTIONAL DECIMAL:
Multiply the number by the radix until the answer is in integer
Example:
to Binary

23. Decimal to Binary Conversion
Remainder
2 5 = 2
1 2 =
6 =
3 =
1 =
MSB LSB
2510 =

24. Decimal to Binary Conversion
MSB
LSB
Carry
x 2 =
x 2 =
x 2 =
0.5 x 2 =
The Answer:

25. Decimal to Octal Conversion
Convert to its octal equivalent:
427 / 8 = 53 R3 Divide by 8; R is LSD
53 / 8 = 6 R5 Divide Q by 8; R is next digit
6 / 8 = 0 R6 Repeat until Q = 0
6538

26. Decimal to Hexadecimal Conversion
Convert to its hexadecimal equivalent:
830 / 16 = 51 R14
51 / 16 = 3 R3
3 / 16 = 0 R3
= E in Hex
33E16

27. Number Conversion
Binary to Octal Conversion (vice versa)
Grouping the binary position in groups of three starting at the least significant position.

28. Octal to Binary Conversion
Each octal number converts to 3 binary digits
To convert 6538 to binary, just substitute code:

29. Number Conversion
Example:
Convert the following binary numbers to their octal equivalent (vice versa).
b) 47.38
Answer:
11.748

30. Number Conversion
Binary to Hexadecimal Conversion (vice versa)
Grouping the binary position in 4-bit groups, starting from the least significant position.

31. 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

32. Number Conversion
Example:
Convert the following binary numbers to their hexadecimal equivalent (vice versa).
1F.C16
Answer:
10.816

33. Substitution Code 56AE6A16 0101 0110 1010 1110 0110 1010 5 6 A E 6 A
Convert to hex using the 4-bit substitution code :
A E A
56AE6A16

34. Substitution Code
Substitution code can also be used to convert binary to octal by using 3-bit groupings:

35. Binary Addition 0 + 0 = 0 Sum of 0 with a carry of 0
Example:
???

36. Simple Arithmetic Addition Example: Example: 100011002 5816 + 1011102
Substraction
101102
Example:
5816
7C16

37. Binary Subtraction 0 - 0 = 0 1 - 1 = 0 1 - 0 = 1
10 -1 = with a borrow of 1
Example:
???

38. Binary Multiplication
0 X 0 = 0
0 X 1 = 0 Example:
1 X 0 =
1 X 1 = X
100110
000000

39. Binary Division
Use the same procedure as decimal division

40. 1’s complements of binary numbers
Changing all the 1s to 0s and all the 0s to 1s
Example:
Binary number
’s complement

41. 2’s complements of binary numbers
Step 1: Find 1’s complement of the number
Binary #
1’s complement
Step 2: Add 1 to the 1’s complement

42. Signed Magnitude Numbers …
Sign bit
31 bits for magnitude
0 = positive
1 = negative
This is your basic
Integer format

43. Sign numbers Left most is the sign bit Sign-magnitude 1’s complement
0 is for positive, and 1 is for negative
Sign-magnitude
= +25
sign bit magnitude bits
1’s complement
The negative number is the 1’s complement of the corresponding positive number
Example:
+25 is is

44. Sign numbers 2’s complement Example Express +19 and -19 in
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

45. Digital Codes
BCD (Binary Coded Decimal) Code
Represent each of the 10 decimal digits (0~9) as a 4-bit binary code.
Example:
Convert 15 to BCD.
BCD
Convert 10 to binary and BCD.

46. Digital Codes
ASCII (American Standard Code for Information Interchange) Code
Used to translate from the keyboard characters to computer language

47. Digital Codes Decimal Binary Gray Code 0000 1 0001 2 0010 0011 3 4
0000 1
0001 2
0010
0011 3 4
0100
0110 5
0101
0111 6
The Gray Code
Only 1 bit changes
Can’t be used in arithmetic circuits
Binary to Gray Code and vice versa.

48. END OF Number & Codes