Chapter 1 Analog vs. Digital Positional Number Systems Number - PDF document

Chapter Outline Chapter 1 � Analog vs. Digital � Positional Number Systems Number Systems And Codes � Positional Number System Conversions � Binary System operations Computer Application � Encoding Techniques Flaxer Eli - ComputerAppl Ch 1 - 1 Flaxer Eli - ComputerAppl Ch 1 - 2 Digital vs. Analog Digital vs. Analog � Natural world is analog � Digital characteristics v � Many devices better when digital: – Discrete signal levels (voltage usually) – Two levels: on/off, high/low 1/0 (binary) – Disjoint or quantized level changes t Digital Analog Compact Disc Magnetic Tape v Microcomputer-controlled Engine Mechanically-controlled Engine � Analog characteristics Telephone System Telephone System – Continuous signal levels Movie Special Effects Movie Special Effects – Very small, smooth level changes t Digital Computers: Analog Computer: PC, Mainframe, Supercomputer OpAmp, Res, Cap Flaxer Eli - ComputerAppl Flaxer Eli - ComputerAppl Ch 1 - 3 Ch 1 - 4 Positional Number System Digital vs. Analog � General form of a number : � Advantages of each technology: d d ...d d .d d ...d − − − − − p 1 p 2 1 0 1 2 n � The value of the number : Digital Analog − p 1 ∑ Reproducible results Less complex ? = ⋅ i D d r i Ease of design = − i n Higher speed ? Flexible and function Programmable � . : radix point High speed � r : base or radix Economical � dp-1 : the most significant digit � d-n : the least significant digit Flaxer Eli - ComputerAppl Ch 1 - 5 Flaxer Eli - ComputerAppl Ch 1 - 6 1

Binary System Decimal System The form of a binary number is: � d : 0 , 1 , 2 , . . . , 9 b b ... b b b . b ... b − − − − − p 1 p 2 1 0 1 2 n � r = 10 � Example : 2081.35 The decimal value of the number is: p = 4 , n = 2 − p 1 ∑ = ⋅ i B b r i = = = = = = d 2 , d 0 , d 8 , d 1 , d 3 , d 5 − − 3 2 1 0 1 2 = − i n 3 ∑ = ⋅ i � r = 2 ( binary radix ) 2 0 8 1 3 5 . d i 1 0 = − i 2 � . : binary point = ⋅ + ⋅ + ⋅ + ⋅ + ⋅ − + ⋅ − 3 2 1 0 1 2 208135 . 2 10 0 10 8 10 1 10 3 10 5 10 � bi ( binary digit = bit ) : 0 , 1 � bp-1 : the most significant bit ( MSB ) = ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ 208135 . 2 1000 0 100 8 10 1 1 3 01 . 5 0 01 . � b-n : the least significant bit ( LSB ) Flaxer Eli - ComputerAppl Ch 1 - 7 Flaxer Eli - ComputerAppl Ch 1 - 8 Exercise Example : � 11101.01 � Calculate the equivalent decimal numbers : 1 , 10 , 11 , 100 , 101, 10101.11 � p = 5 , n = 2 � Answer : 1 (2) = 1 (10) = = = = = = = b 1 , b 1 , b 1 , b 0 , b 1 , b 0 , b 1 − − 4 3 2 1 0 1 2 10 (2) = 2 (10) 4 ∑ 11 (2) = 3 (10) = ⋅ i B b i 2 100 (2) = 4 (10) = − i 2 101 (2) = 5 (10) − − B = ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ 4 3 2 1 0 1 2 1 2 1 2 1 2 0 2 1 2 0 2 1 2 10101.11 (2) = 21.75 (10) B = + + + + + + = 16 8 4 0 1 0 025 . 29 25 . Flaxer Eli - ComputerAppl Flaxer Eli - ComputerAppl Ch 1 - 9 Ch 1 - 10 Octal and Hexadecimal Numbers Binary, Decimal, Octal and Hexadecimal Numbers Decimal Binary Octal Hexadecimal � Octal number System : 0 0 0 0 - r = 8 1 1 1 1 - d : 0 , 1 , 2 , . . . , 7 2 10 2 2 3 11 3 3 � Hexadecimal number System : 4 100 4 4 - r = 16 5 101 5 5 6 110 6 6 - d : 0 , 1 , 2 , . . . , 9, A, B, C, D, E, F 7 111 7 7 � The radices are powers of 2 8 1000 10 8 9 1001 11 9 � Used for shorthand representations of long 10 1010 12 A binary numbers 11 1011 13 B 12 1100 14 C 13 1101 15 D 14 1110 16 E 15 1111 17 F Flaxer Eli - ComputerAppl Ch 1 - 11 2

Binary - Octal/Hexadecimal Conversion Binary - Octal/Hexadecimal Conversion � Binary to Octal/Hexadecimal : � Octal/Hexadecimal to Binary: Starting from the decimal point : Convert each Octal/Hexadecimal digit into its corresponding - Separate the bits into groups of three/four 3 / 4 bit string - Replace each group with its corresponding Octal/Hexadecimal � Examples : digit - 621.3(8) = 110010001.011(2) � Examples : - F5A.2C(16) = 111101011010.00101100(2) - 100011110.10101 (2) = 100 011 110.101 010 = 436.52 (8) - 1011101000.001111 (2) = 0010 1110 1000.0011 1100 = 2E8.3C (16) Flaxer Eli - ComputerAppl Ch 1 - 13 Flaxer Eli - ComputerAppl Ch 1 - 14 Exercise General Positional Number Conversion � Convert 101100.101 into hexadecimal, octal and � radix-r to decimal : decimal. − p 1 ∑ = ⋅ i � Convert F4A into binary D d r i = − i n � Answers : - 101100.101 (2) = 2C.A (16) = 44.625 (10) � decimal to radix-r : - 101100.101 (2) = 54.5 (8) - Successive division of D by r - F4A (16 ) =111101001010 (2) - The remainder of the long divsion will give the digits starting from the least significant digit Flaxer Eli - ComputerAppl Flaxer Eli - ComputerAppl Ch 1 - 15 Ch 1 - 16 Example - Decimal to Binary : Exercise � 179 (10) � Convert 154 into binary. 179/2 = 89 ( 1 ) LSB � Answer : 10011010 89/2 = 44 ( 1 ) 44/2=22 ( 0 ) 22/2=11 ( 0 ) 11/2=5 ( 1 ) 5/2=2 ( 1 ) 2/2=1 ( 0 ) 1/2=0 (1)MSB � Result : 10110011 (2) Flaxer Eli - ComputerAppl Ch 1 - 17 Flaxer Eli - ComputerAppl Ch 1 - 18 3

Example - Decimal to Binary (Fraction): Binary Addition � .375 (10) Binary addition table : .375 * 2 = 0.75 ( 0 ) .75 * 2 = 1.5 ( 1 ) carry(in) x y x+y carry(out) .5 * 2 = 1.0 ( 1 ) 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 Fin � Result : .011 (2) 1 0 1 0 1 1 1 0 0 1 Example : 1 1 1 1 1 1 0 0 0 1 0 1 1 0 0 0 x 1 7 3 1 0 1 0 1 1 0 1 y + 4 4 0 0 1 0 1 1 0 0 _____ ____________ 2 1 7 1 1 0 1 1 0 0 1 Flaxer Eli - ComputerAppl Ch 1 - 19 Flaxer Eli - ComputerAppl Ch 1 - 20 Representation of Natural Numbers in Exercise Binary Systems (Unsigned Number) � There has 2 n parmutation in order n bits � 1 1 0 1 0 0 1 0 1 0 1 + 0 1 1 1 1 + 0 1 1 0 0 1 � The range for n-bit is : _________ __________ � 1 0 1 0 0 1 1 0 1 1 1 0 from 0 to + (2 n - 1) 26+15 = 41 21+25=46 Flaxer Eli - ComputerAppl Flaxer Eli - ComputerAppl Ch 1 - 21 Ch 1 - 22 Exercise Binary Subtraction � Binary Subtraction table : � 1 1 0 1 0 1 1 1 0 1 - 0 1 1 1 1 - 1 0 1 1 1 borrow(in) x y x-y borrow(out) ________ _________ 0 0 0 0 0 0 0 1 1 1 � 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 26 - 15 = 11 29 - 23 = 6 0 1 1 0 0 1 0 0 1 1 1 0 1 0 1 1 1 0 0 0 Example : 1 1 1 1 1 0 1 0 1 1 0 1 1 0 1 0 x 2 1 0 1 1 0 1 0 0 1 0 y - 1 0 9 0 1 1 0 1 1 0 1 _____ ____________ 1 0 1 0 1 1 0 0 1 0 1 Flaxer Eli - ComputerAppl Ch 1 - 23 Flaxer Eli - ComputerAppl Ch 1 - 24 4