W02-03_Numbers+Ch02CAO - 03 Numbers Systems CSC 230...

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03 Numbers Systems CSC 230 Department of Computer Science U i it f Vi t i University of Victoria 1
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Integer Number Systems Decimal Base: 10 Digits: 0,1,2,3,4,5,6,7,8,9 Binary Octal Base: 2 Base: 8 Digits: 0,1 Digits: 0,1,2,3,4,5,6,7 Hexadecimal Base: 16 Digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F 2
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Small Trivial Example 7423 in decimal = 3 + 3 + 3 x 10 0 + 20 + 400 + 2 x 10 + 4 x 100 + 2 x 10 1 + 4 x 10 2 + 7000 = 7 x 1000 7 x 10 3 3
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Integer Number Systems: Base 10 - Decimal = n 1 n 2 1 0 Integer D D D D Positional Number Systems e.g. 7423 in decimal Base 10: ( ) × n 1 n 1 D 10 ( ) + × 0 0 D 10 ( ) + × n 2 n 2 D 10 ( ) + + × 1 1 D 10 ( ) ( ) 1 0 ( ) 3 ( ) 2 + × + × 2 10 3 10 = × 10 7423 7 10 + × 4 10 4
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Integer Number Systems: Base 16 - Hexadecimal P iti l N b S t = n 1 n 2 1 0 Integer D D D D Positional Number Systems e.g. 8254 in hexadecimal Base 16: ( ) ( ) ( ) ( ) n 1 n 2 1 0 n 1 n 2 1 0 D 16 D 16 D 16 D 16 × + × + + × + × ( ) ( ) ( ) ( ) = × + × + × + × 3 2 1 0 16 8254 8 16 2 16 5 16 4 16 ( ) ( ) ( ) ( ) = × + × + × + × = 10 8 4096 2 256 5 16 4 1 33,364 5 NOTE: we have converted from hex to decimal!
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Integer Number Systems: Base 2 - Binary Positional Number Systems = n 1 n 2 1 0 Integer D D D D e.g. 011011 in binary ( ) ( ) ( ) ( ) n 1 n 2 1 0 n 1 n 2 1 0 D 2 D 2 D 2 D 2 × + × + + × + × Base 2: ( ) ( ) ( ) ( ) ( ) ( ) = 0 110 11 2 = × + × + × + × + × + × 5 4 3 2 1 0 0 2 1 2 1 2 0 2 1 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) = × + × + × + × + × + × = 10 0 32 1 16 1 8 0 4 1 2 1 1 27 6 NOTE: we have converted from binary to decimal!
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Weighted Positional Representation BASE: defines the range of values for digits (e.g. 0 – 9 for decimal; 0,1 for binary) GENERAL FORM AS AN n-BIT VECTOR: B = BASE n i i Integer Decimal Value d B = × 1 i = 0 d = DIGIT n i 1 i i m Decimal Value d B =− = × Include fractions . = × + × + × + × + × 2 1 0 1 2 10 145 52 1 10 4 10 5 10 5 10 2 10 Full example: 7 . . = + + + + 100 40 5 0 5 0 02
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Memorize This Table! Binary Decimal Hexadecimal 0000 0 0 0001 1 1 0010 2 2 0011 3 3 0100 4 4 0101 5 5 0110 6 6 0111 7 7 1000 8 8 1001 9 9 1010 10 A 1011 11 B 1100 12 C 1101 13 D 8 1110 14 E 1111 15 F
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Summary 1: Conversion from any Base “B” to Decimal B to Decimal Î Use the polynomial expansion in Base “B” as shown Base “” gives the powers of the positional systemB Base gives the powers of the positional systemB ( ) ( ) ( ) ( ) 0 1 2 3 ( ) ( ) ( ) ( ) = × + × + × + × = × + × + × + × 16 7423 3 16 2 16 4 16 7 16 3 1 2 16 4 256 7 4096 = 10 29,731 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = × + × + × + × + 0 1 2 3 2 4 5 6 7 11001011 1 2 1 2 0 2 1 2 9 × + × + × + × = 10 0 2 0 2 1 2 1 2 203
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Conversion from One Base to Another D i l t B “B” f iti i t Decimal to Base “B” for positive integers 1. Repeated division by base “B” 2. Collect remainders 3. Form result from right to left Example 1: from decimal to binary 35 10 = ??? 2 35/2 = 17 + remainder 1 35/2 17 + remainder 1 17/2 = 8 + remainder 1 8/2 = 4 + remainder 0 4/2 = 2 + remainder 0 2/2 = 1 + remainder 0 10 1/2 = 0 + remainder 1 answer: 100011 2
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Conversion from One Base to Another D i l t B “B” f iti i t Decimal to Base “B” for positive integers 1. Repeated division by base “B” 2. Collect remainders 3. Form result from right to left Example 2: from decimal to hexadecimal 35 10 = ???
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