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RIT EECC 341 - Positional Number Systems

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#1 Lec # 2 Winter 2001 12-5-2001EECC341 - ShaabanEECC341 - ShaabanPositional Number Systems• A number system consists of an order set of symbols (digits) with relationsdefined for +,-,*, /• The radix (or base) of the number system is the total number of digitsallowed in the the number system.– Example, for the decimal number system:• Radix, r = 10, Digits allowed = 0,1, 2, 3, 4, 5, 6, 7, 8, 9• In positional number systems, a number is represented by a string of digits,where each digit position has an associated weight.• The value of a number is the weighted sum of the digits.• The general representation of an unsigned number D with whole andfraction portions number in a number system with radix r: Dr = d p-1 d p-2 ….. d1 d0.d-1 d-2 …. D-n• The number above has p digits to the left of the radix point and n fractiondigits to the right.• A digit in position i has as associated weight ri• The value of the number is the sum of the digits multiplied by the associatedweight ri :rdi1pniiD ×=∑−−=#2 Lec # 2 Winter 2001 12-5-2001EECC341 - ShaabanEECC341 - ShaabanPositional Number Systems• For example in the decimal number system: 5185.6810 = 5x103 + 1x102 + 8x101 + 5x100 + 6 x 10-1 + 8 x 10-2 = 5x1000 + 1x100 + 8x10 + 5 x 1 + 6x.1 + 8x.01• For the binary number system with radix = 2, digits 0, 1 D2 = dp-1 ×× 2p-1 ….. d1 ×× 21 + d0 . 20 + d-1 ×× 2-1 + d-2 ×× 2-2 …..• For Example: 100112 = 1 ×× 16 + 0 ×× 8 + 0 ×× 4 + 1 ×× 2 + 1 ×× 1 = 1910 | | MSB LSB (least significant bit)(most significant bit) 101.0012 = 1x4 + 0x2 + 1x1 + 0x.5 + 0x.25 + 1x.125 = 5.12510rdi1pniiD×=∑−−=Number: Dr = d p-1 d p-2 ….. d1 d0.d-1 d-2 …. D-nValue: Binary Point#3 Lec # 2 Winter 2001 12-5-2001EECC341 - ShaabanEECC341 - ShaabanNumber Systems Used in ComputersNumber Systems Used in ComputersNameof RadixRadixSet of Digits ExampleDecimal r=10r=2r=16r= 8 {0,1,2,3,4,5,6,7,8,9} 25510Binary {0,1,2,3,4,5,6,7} 3778 {0,1} 111111112 {0,1,2,3,4,5,6,7,8,9,A, B, C, D, E, F} FF16OctalHexadecimalBinary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 Hex 0 1 2 3 4 5 6 7 8 9 A B C D E FDecimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15#4 Lec # 2 Winter 2001 12-5-2001EECC341 - ShaabanEECC341 - ShaabanRadix-r to Decimal ConversionRadix-r to Decimal Conversion• The decimal value of a number in any radix r is found by convertingeach digit to its radix 10 equivalent and expanding the value usingradix arithmetic:• Examples: 1101.1012 = 1××23 + 1××22 + 1××20 + 1××2-1 + 1××2-3 = 8 + 4 + 1 + 0.5 + 0.125 = 13.62510 572.68 = 5××82 + 7××81 + 2××80 + 6××8-1 = 320 + 56 + 16 + 0.75 = 392.7510 2A.816 = 2××161 + 10××160 + 8××16-1 = 32 + 10 + 0.5 = 42.510 132.34 = 1××42 + 3××41 + 2××40 + 3××4-1 = 16 + 12 + 2 + 0.75 = 30.7510 341.245 = 3××52 + 4××51 + 1××50 + 2××5-1 + 4××5-2 = 75 + 20 + 1 + 0.4 + 0.16 = 96.5610rdi1pniiD ×=∑−−=#5 Lec # 2 Winter 2001 12-5-2001EECC341 - ShaabanEECC341 - ShaabanDecimal-to-Binary ConversionDecimal-to-Binary Conversion• Separate the decimal number into whole and fraction portions.• To convert the whole number portion to binary, use successivedivision by 2 until the quotient is 0. The remainders form theanswer, with the first remainder as the least significant bit (LSB) andthe last as the most significant bit (MSB).• Example: Convert 17910 to binary: 179 / 2 = 89 remainder 1 (LSB) / 2 = 44 remainder 1 / 2 = 22 remainder 0 / 2 = 11 remainder 0 / 2 = 5 remainder 1 / 2 = 2 remainder 1 / 2 = 1 remainder 0 / 2 = 0 remainder 1 (MSB) 17910 = 101100112#6 Lec # 2 Winter 2001 12-5-2001EECC341 - ShaabanEECC341 - ShaabanDecimal-to-Binary ConversionDecimal-to-Binary Conversion• To convert decimal fractions to binary, repeated multiplication by 2 isused, until the fractional product is 0 (or until the desired number ofbinary places). The whole digits of the multiplication results producethe answer, with the first as the MSB, and the last as the LSB.• Example: Convert 0.312510 to binary Result Digit .3125 ×× 2 = 0.625 0 (MSB) .625 ×× 2 = 1.25 1 .25 ×× 2 = 0.50 0 .5 ×× 2 = 1.0 1 (LSB) 0.312510 = .01012#7 Lec # 2 Winter 2001 12-5-2001EECC341 - ShaabanEECC341 - ShaabanDecimal-to-Binary ConversionDecimal-to-Binary Conversion Sum-of-Weights MethodSum-of-Weights Method• Determine the set of binary weights whose sum is equal tothe decimal number.Examples: 910 = 8 + 1 = 23 + 20 = 10012 1810 = 16 + 2 = 24 + 21 = 100102 5810 = 32 + 16 + 8 + 2 = 25 + 24 + 23 + 21 = 1110102 0.62510 = 0.5 + 0.125 = 2-1 + 2-3 = 0.1012#8 Lec # 2 Winter 2001 12-5-2001EECC341 - ShaabanEECC341 - ShaabanDecimal to Radix-r ConversionDecimal to Radix-r Conversion• Separate the decimal number into whole and fraction portions.• To convert the whole number portion to binary, use successivedivision by r until the quotient is 0. The remainders form theanswer, with the first remainder as the least significant digit (LSD)and the last as the most significant digit (MSD).• To convert decimal fractions to radix-r, repeated multiplication by ris used, until the fractional product is 0 (or until the desired numberof binary places). The whole digits of the multiplication resultsproduce the answer, with the first as the MSD, and the last as theLSD.• Example: Convert 46710 to octal 467 / 8 = 58 remainder 3 (LSD) / 8 = 7 remainder 2 / 8 = 0 remainder 7 (MSD)


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