CHAPTER 1 : INTRODUCTION 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 Binary Octal Hexadecimal 0 ~ 9 0 ~ 1 0 ~ 7 0 ~ F

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.

00000000 00000001 00000010 00000011 00000100 00000101 00000110 00000111 00001000 00001001 00001010 00001011 00001100 00001101 00001110 00001111 000 001 002 003 004 005 006 007 010 011 012 013 014 015 016 017 0 1 2 3 4 5 6 7 8 9 A B C D E F 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Binary Octal Hex Dec N U M B E R S Y T

Most significant digit Least significant digit Significant Digits Binary: 11101101 Most significant digit Least significant digit Hexadecimal: 1D63A7A

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 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

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: Binary 1 0 1 0 1 1 0 1 Positional Values x x x x x x x x 27 26 25 24 23 22 21 20 Products 128 + 32 + 8 + 4 + 1 17310

Octal to Decimal Conversion Convert 6538 to its decimal equivalent: Octal Digits 6 5 3 x x x Positional Values 82 81 80 Products 384 + 40 + 3 42710

Hexadecimal to Decimal Conversion Convert 3B4F16 to its decimal equivalent: Hex Digits 3 B 4 F x x x x Positional Values 163 162 161 160 Products 12288 +2816 + 64 +15 15,18310

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: 25.3125 to Binary

Decimal to Binary Conversion Remainder 2 5 = 12 + 1 2 1 2 = 6 + 0 6 = 3 + 0 3 = 1 + 1 1 = 0 + 1 2 MSB LSB 2510 = 1 1 0 0 1 2

Decimal to Binary Conversion MSB LSB Carry . 0 1 0 1 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: 1 1 0 0 1.0 1 0 1

Decimal to Octal Conversion Convert 42710 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

Decimal to Hexadecimal Conversion Convert 83010 to its hexadecimal equivalent: 830 / 16 = 51 R14 51 / 16 = 3 R3 3 / 16 = 0 R3 = E in Hex 33E16

Number Conversion Binary to Octal Conversion (vice versa) 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). 1001.11112 b) 47.38 1010011.110112 Answer: 11.748 100111.0112 123.668

Number Conversion Binary to Hexadecimal Conversion (vice versa) 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). 10000.12 1F.C16 Answer: 10.816 00011111.11002

Substitution Code 56AE6A16 0101 0110 1010 1110 0110 1010 5 6 A E 6 A Convert 0101011010101110011010102 to hex using the 4-bit substitution code : 0101 0110 1010 1110 0110 1010 5 6 A E 6 A 56AE6A16

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 2 5 5 2 7 1 5 2 255271528

Binary Addition 0 + 0 = 0 Sum of 0 with a carry of 0 Example: 11001 111 + 1101 + 11 100110 ???

Simple Arithmetic Addition Example: Example: 100011002 5816 + 1011102 + 1011102 101110102 Substraction 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 0 X 0 = 0 0 X 1 = 0 Example: 1 X 0 = 0 100110 1 X 1 = 1 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: 1 1 0 1 0 0 1 0 1 Binary number 0 0 1 0 1 1 0 1 0 1’s complement

2’s complements of binary numbers 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.. …00101110010101 Sign bit 31 bits for magnitude 0 = positive 1 = negative This is your basic Integer format

Sign numbers Left most is the sign bit Sign-magnitude 1’s complement 0 is for positive, and 1 is for negative Sign-magnitude 0 0 0 1 1 0 0 1 = +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 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

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. 1 5 0001 0101BCD Convert 10 to binary and BCD.

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

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.

END OF Number & Codes