EKT 121 / 4 ELEKTRONIK DIGIT 1 CHAPTER 1 : INTRODUCTION

1.0Number & 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  0 ~ 9  0 ~ 1  0 ~ 7  0 ~ F Decimal Binary Octal Hexadecimal

Characteristics of Numbering Systems 1) The digits are consecutive. 2) The number of digits is equal to the size of the base. 3) Zero is always the first digit. 4) When 1 is added to the largest digit, a sum of zero and a carry of one results. 5) Numeric values determined by the implicit positional values of the digits.

ABCDEF ABCDEF BinaryOctalHexDec NUMBER SYSTEMSNUMBER SYSTEMS

Significant Digits Binary: 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 NumberBase 10 EquivalentPowerPositional Value

Octal Number System Also known as the Base 8 System Uses digits 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

Convert ( ) 2 to its decimal equivalent: Binary Positional Values x x x xx x x x Products

Octal to Decimal Conversion Convert to its decimal equivalent: xx x Positional Values Products Octal Digits

Hexadecimal to Decimal Conversion Convert 3B4F16 to its decimal equivalent: Hex Digits 3 B 4 F xx x Positional Values Products x

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: to Binary

Decimal to Binary Conversion 2 5 = = = = = MSBLSB = Remainder

Decimal to Binary Conversion Carry x 2 = x 2 = x 2 = x 2 = The Answer: MSBLSB

Decimal to Octal Conversion Convert to its octal equivalent: 427 / 8 = 53 R3Divide by 8; R is LSD 53 / 8 = 6 R5Divide Q by 8; R is next digit 6 / 8 = 0 R6Repeat until Q =

Decimal to Hexadecimal Conversion Convert 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 to binary, just substitute code:

Number Conversion  Example:  Convert the following binary numbers to their octal equivalent (vice versa). a) b) c)  Answer: a) b) c)

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) 1F.C 16  Answer: a) b)

Convert to hex using the 4-bit substitution code : Substitution Code

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

Binary Addition = 0Sum of 0 with a carry of = 1Sum of 1 with a carry of = 1 Sum of 1 with a carry of = 10Sum of 1 with a carry of 1 Example: ???

Simple Arithmetic Addition  Example: Substraction  Example:  Example: C 16

Binary Subtraction = = = = with a borrow of 1 Example: ???

Binary Multiplication 0 X 0 = 0 0 X 1 = 0Example: 1 X 0 = X 1 = 1 X

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: Binary number ’s complement

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

Signed Magnitude Numbers Sign bit 0 = positive 1 = negative 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  = +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

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 BCD  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 The Gray Code  Only 1 bit changes  Can’t be used in arithmetic circuits Binary to Gray Code and vice versa. DecimalBinaryGray Code

END OF Number & Codes