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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