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# Class Note for ENGIN 112 at UMass(34)

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Date Created: 02/06/15

ENGIN 112 Intro to Electrical and Computer Engineering Lecture 2 Number Systems Russell Tessier KEB 309 G tessierecsumassedu ENGN112 L2 Number Systems September 5 2003 Overview The design of computers It all starts with numbers Building circuits Building computing machines Digital systems Understanding decimal numbers Binary and octal numbers The basis of computers Conversion between different number systems ENGN112 L2 Number Systems September 5 2003 Digital Computer Systems Digital systems consider discrete amounts of data Examples 26 letters in the alphabet 10 decimal digits Larger quantities can be built from discrete values Words made of letters Numbers made of decimal digits eg 23987532 Computers operate on binary values 0 and 1 Easy to represent binary values electrically Voltages and currents Can be implemented using circuits Create the building blocks of modern computers ENGN112 L2 Number Systems September 5 2003 Understanding Decimal Numbers Decimal numbers are made of decimal digits 0 1 2 3 4 5 6 7 8 9 But how many items does a decimal number represent 8653 8x103 6x102 5x101 3x10o What about fractions 9765435 9x104 7x103 6x102 5x1o1 4x100 3x10391 5x102 In formal notation gt 976543510 Why do we use 10 digits anyway ENGN112 L2 Number Systems September 5 2003 Understanding Octal Numbers Octal numbers are made of octal digits 01234567 How many items does an octal number represent 45358 4x83 5x82 3x816x8 136210 What about fractions 465278 4x82 6x81 5x80 2x8391 7x82 Octal numbers don t use digits 8 or 9 Who would use octal number anyway ENGN112 L2 Number Systems 7 September 5 2003 Understanding Binary Numbers Binary numbers are made of binary digits bits 0 and 1 How many items does an binary number represent 10112 1x23 0x221x211x2 1110 What about fractions 110102 1x22 1x210x20 1x21 0x22 Groups of eight bits are called a byte 11oo1oo12 Groups of four bits are called a nibble 1101 2 ENGN112 L2 Number Systems September 5 2003 Why Use Binary Numbers Volts 0 Easy to represent 0 and 1 using electrical values Possible to tolerate noise Range Easy to transmit data for logic1 0 Easy to build binary circuits Transition occurs between these limits AND Gate Range 0 for logic0 Fig 13 Example of binary signals ENGN112 L2 Number Systems September 5 2003 Conversion Between Number Bases Octalbase 8 Decimalbase 10 Binarybase 2 Hexadecimal base16 Learn to convert between bases Already demonstrated how to convert from binary to decimal Hexadecimal described in next lecture ENGN112 L2 Number Systems September 5 2003 Convert an Integer from Decimal to Another Base For each digit position 1 Divide decimal number by the base eg 2 2 The remainder is the lowestorder digit 3 Repeat first two steps until no divisor remains Example for 13 Integer Remainder Coefficient Quotient 132 6 12 a0 1 62 3 0 a1 O 32 1 12 a2 1 12 O 12 a3 1 Answer 131o a3 a2 a1a02 11012 ENGN112 L2 Number Systems September 5 2003 Convert an Fraction from Decimal to Another Base For each digit position 1 Multiply decimal number by the base eg 2 2 The integer is the highestorder digit 3 Repeat first two steps until fraction becomes zero Example for 0625 Integer Fraction Coefficient 0625 X 2 1 025 a1 1 0250 X 2 O 050 a2 O 0500X2 1 O a3 1 Answer 062510 0a1 a2 a3 2 01012 ENGN112 L2 Number Systems September 5 2003 The Growth of Binagy Numbers n 2quot n 2quot 0 201 8 28256 1 212 T 9 29512 2 224 10 21 1024 3 238 11 2112048 4 2416 12 2124096 5 2532 20 22 1 M Mega 6 2364 30 23 1G Giga 7 27128 1 40 24 1 T Tera ENGN112 L2 Number Systems September 5 2003 Binary Addition Binary addition is very simple This is best shown in an example of adding two binary numbers 1 1 1 1 1 1 carries 1 1 1 1 O 1 1 O 1 1 1 ENGN112 L2 Number Systems September 5 2003 Binary Subtraction We pan also perform subtraction with borrows in place of carrIes Let s subtract 101112 from 10011012 10 borrows ENGN112 L2 Number Systems September 5 2003 Binary Multiplication Binary multiplication is much the same as decimal multiplication except that the multIpIIcatIon operations are muc simpler X 1010 00000 10111 00000 10111 ENGN112 L2 Number Systems September 5 2003 Convert an Integer from Decimal to Octal For each digit position 1 Divide decimal number by the base 8 2 The remainder is the lowestorder digit 3 Repeat first two steps until no divisor remains Example for 17510 Integer Remainder Coefficient Quotient 1758 21 78 218 2 58 28 O 28 990 II II II Ixth Answer 1751o a2 a1 a02 2578 ENGN112 L2 Number Systems September 5 2003 Convert an Fraction from Decimal to Octal For each digit position 1 Multiply decimal number by the base eg 8 2 The integer is the highestorder digit 3 Repeat first two steps until fraction becomes zero Example for 03125 Integer Fraction Coefficient 03125X8 2 5 a12 05000 X 8 4 0 a2 4 Answer 0312510 024s ENGN112 L2 Number Systems September 5 2003 Summary Binary numbers are made of binary digits bits Binary and octal number systems Conversion between number systems Addition subtraction and multiplication in binary ENGN112 L2 Number Systems September 5 2003

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