01 Number Systems

# 01 Number Systems - EE2000 Logic Circuit Design Number...

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1 EE2000 Logic Circuit Design ±

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2 Why Learning Number Systems? ± Simply because human and computers use different number systems ± Human: decimal system ± Computers: binary system ± The main outcome of this course is to design logic circuits ± Binary number system is a prerequisite to learn logic circuits
3 Outline ± 1.1 Number Systems ± Introduction to decimal, binary, octal, hexadecimal ± Conversion between these number systems ± 1.2 Binary Arithmetic ± Representation of negative numbers ± Addition and subtraction of binary numbers ± 1.3 Binary Codes for Decimal Digits ± BCD, Gray codes, ASCII ± 1.4 Parity and Error Correction

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4 1.1 Number Systems
5 Number Systems ± Decimal (base 10) ± E.g. 1, 2, 3, 4, 5, … 10, 11, 12, … ± Binary (base 2) ± E.g. 000, 001, 010, 011, … ± Octal (base 8) ± E.g. 00, 01, 02, …, 07, 10, 11, …17, 20, … ± Hexadecimal (base 16) ± E.g. 1, 2, 3, …, 9, A, B, C, D, E, F, 10, 11, …

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6 Number Decimal Binary Octal Hexadecimal Zero 0 0 0 0 One 1 1 1 1 Two 2 10 2 2 Three 3 11 3 3 Four 4 100 4 4 Five 5 101 5 5 Six 6 110 6 6 Seven 7 111 7 7 Eight 8 1000 10 8 Nine 9 1001 11 9 Ten 10 1010 12 A Eleven 11 1011 13 B Twelve 12 1100 14 C Thirteen 13 1101 15 D Fourteen 14 1110 16 E Fifteen 15 1111 17 F Sixteen 16 10000 20 10 Seventeen 17 10001 21 11 Question: What is “10”?
7 1.1.1 Decimal Number System ± Decimal number system employed in everyday arithmetic ± Represent numbers by strings of digits called positional notation ± Digits – the Latin word for fingers ± a n -1 a n -2 a 2 a 1 a 0 (0 a i < 10) ± Each digit is associated to a value depends on its position in the string

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8 Polynomial Form of Integer ± The no. can be expressed in power series ± N = a n -1 r n -1 + a n -2 r n -2 + … + a 2 r 2 + a 1 r + a 0 ± N : the decimal value of the integer ± n : number of digits ± r : radix (base) ± a i : coefficients (digit), 0 a i < r ± e.g. (7672) 10 ± 4 digits, so n = 4 ± decimal number, i.e. r = 10 ± a i : 0 a i < 10, i.e. one of the ten digits (0, 1, 2, …, 9) ± N = 7 x 10 3 + 6 x 10 2 + 7 x 10 + 2 These two 7 symbols have different meanings
9 Floating Point Number ± a n -1 a n -2 a 2 a 1 a 0 . a -1 a -2 a -( m -1) a - m (0 a i < 10) ± . ” is the radix point ± The left most digit a n -1 is the m ost s ignificant d igit ( MSD ) ± The right most digit a - m is the l east s ignificant d igit ( LSD ) ± General polynomial form ± N = a n -1 r n -1 + … + a 1 r 1 + a 0 r 0 + a -1 r -1 + … + a - m r - m ± Number of digits = n + m ± e.g. (767.2) 10 ± n = 3, m = 1 ± N = 7 x 10 2 + 6 x 10 1 + 7 + 2 x 10 -1

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10 Alternative Conversion Method ± (…(( a n -1 r + a n -2 ) r + a n -3 ) r + … + a 1 ) r + a 0 ± + ± ( a -1 + ( a -2 + (… + ( a - m +1 + a - m r -1 ) r -1 )… r -1 ) r -1 ± e.g. (767.295) 10 ± = (( 7 x 10 + 6 ) 10 + 7 ) ± + ( 2 + ( 9 + 5 x 10 -1 ) 10 -1 ) 10 -1
11 Other Number Systems ± Decimal (base 10) are for human ± Three number systems are commonly

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## This note was uploaded on 02/06/2011 for the course EE 2000 taught by Professor Vancwting during the Spring '07 term at City University of Hong Kong.

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01 Number Systems - EE2000 Logic Circuit Design Number...

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