Ch1_Number_System

Ch1_Number_System - 1 2 A decimal number such as 7392...

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3 A decimal number such as 7392 represents 7392.34 = 7x10 3 + 3x10 2 + 9x10 1 + 2x10 0 +3x10 -1 +4x10 -2 It is practical to write only coefficients and deduce power of 10s from position In general, any radix (base) can be used Define coefficients a i in radix r 0 <= a i <r a n r n + a n-1 r n-1 . a 0 r 0 + .. a -m r -m Common radics r = 2, 4, 8, 10, 16
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4 General Radix Representation r = 10 (Dec.) r = 2 (Binary) r = 8 (Octal) r = 16 (Hex) 00 0000 00 0 01 0001 01 1 02 0010 02 2 03 0011 03 3 04 0100 04 4 05 0101 05 5 06 0110 06 6 07 0111 07 7 08 1000 10 8 09 1001 11 9 10 1010 12 A 11 1011 13 B 12 1100 14 C
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5 General Radix Representation r = 10 (Dec.) r = 2 (Binary) r = 8 (Octal) r = 16 (Hex) 13 1101 15 D 14 1110 16 E 15 1111 17 F Usually, radix is shown as subscript (1234.4) 5 = 1x5 3 + 2x5 2 + 3x5 1 4x5 0 + 4x5 -1 = (513.4) 10 (F75C.B) 16 = 15x16 3 + 7x16 2 + 5x16 1 + 12x16 0 + 11x16 -1 = (63,324.6875) 10
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6 Radix Conversion The integral part of a decimal number to radix r, repeatedly divide by r with reminders becoming a i Convert (77) 10 to binary Integer Remainder Coefficient 77 38 1 a0 19 0 a1 9 1 a2 4 1 a3 2 0 a4 1 0 a5 0 1 a6 (77) 10 = (1001101) 2
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7 Examples Convert (173) 10 to r = 7 Integer Remainder Coefficient 173 24 5 a0 3 3 a1 0 3 a2 (173) 10 = (335) 7 Converting from Binary to decimal (101101) 2 = 1x2 5 + 0x2 4 + 1x2 3 + 1x2 2 +0x2 1 +1x2 0 = (45) 10
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8 Fraction Conversion To convert the fractional part of a number to radix r, repeatedly multiply by r; integral parts of products becoming a i Convert (0.7215) 10 to binary 0.7215x2 = 1.443 a -1 = 1 0.443x2 = 0.866 a -2 = 0 0.866x2 = 1.772 a -3 =1 0.772x2 = 1.544 a -4 = 1 0.544 = 1.088 a -5 = 1 (0.7215) 10 = (0.10111..) 2
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9 Conversion between Binary/Octal/Hex Nice simple ways to convert between these three number systems, since all are a power of 2 Binary to Octal simply requires grouping bits into groups of 3-bits and converting Binary to Hex simply requires grouping bits into groups of 4-bits and converting
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