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Set_5-3

Set_5-3 - Positional Number System A number is represented...

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Positional Number System A number is represented by a string of digits where each digit position has an associated weight. The weight is based on the radix of the number system. Some common radices: Binary, Octal, Decimal, and Hexadecimal.

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Notation Decimal. W = 123 10 = 123d = 123 Binary. X = 10 2 = 10b Octal. Y = 45 8 = 45q = 45o Hexadecimal. Z = 0A3 16 = 0A3h = 0xA3
Radix 10 Numbers Decimal. Here the base or radix is 10. Digits used. 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. 2 1 0 217 200 10 7 2 10 1 10 7 10 D D D = = + + = × + × + ×

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Radix 2 Numbers Binary. Here the base or radix is 2. Digits used: 0, and 1. 5 . 5 2 1 2 1 2 0 2 1 1 . 101 1 0 1 2 2 = × + × + × + × = = - B B B
Radix 8 Numbers Octal. Here the base or radix is 8. Digits used: 0,1,2,3,4,5,6, and 7. 625 . 26 8 5 8 2 8 3 5 . 32 1 0 1 8 = × + × + × = = - O O O

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Radix 16 Numbers Hexadecimal. Here the base or radix is 16. Digits used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. 16 1 0 1 1 0 1 3 .8 3 16 16 8 16 3 16 12 16 8 16 60.5 H C H C H H - - = = × + × + × = × + × + × =
Radix r Numbers 1 0 1 1 0 1 1 0 1 . n m n m n m n i i i m A a a a a a A a r a r a r a r a r A a r - - - - - - = - = = + + + + + + = K K K K

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Conversion Table Decimal Binary Octal Hexadecimal 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 13 1101 15 D 14 1110 16 E 15 1111 17 F
Binary to Octal, Octal to Binary Conversions Substitute a three binary digit string with an octal digit. 2 8 8 2 100 101 111 754 4 . 11 1 . 001 1 = =

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Binary to Hexadecimal, Hexadecimal to Binary Conversions Substitute a four binary digit string, called a nibble, with a hexadecimal digit. 16 2 2 16 8A.C 1000 1010. 11 1 1110 1011 =1EB =
Simple Conversions The conversions just described are simple due to the fact that the radices are all powers of two. 2 1 = binary. 8 = 2 3 = octal. 16 = 2 4 = hexadecimal.

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Radix r to Decimal Conversions n i i i m n m n m A a r A a r a r =- - - = = + + K Ex: 101 2 = 1X2 2 + 0X2 1 + 1X2 0 = 5 432 5 = 4X5 2 + 3X5 1 + 2X5 0 = 117 75 8 = 7X8 1 + 5X8 0 = 61 1A 16 = 1X16 1 + 10X16 0 = 26
Decimal to Radix r Conversions 1 0 1 1 0 1 1 0 1 0 1 1 Integral part Fractional part n i i i m n n m n n m I F n n I n n m F m A a r A a r a r a r a r a r A A A A a r a r a r A a r a r =- - - - - - - - - - - - - = = + + + + + + = + = + + + = + + K K K K

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Decimal to Radix r Conversions A I / r = ( a n r+ a n-1 ) r+…+ a 1 as the quotient and a 0 as the remainder . Divide the result repeatedly until a zero quotients is reached. The remainders of the consecutive divisions form the numbers in base r. 1 1 0 1 1 0 1 1 0 Integral part (( ) ) n n I n n I n n A a r a r a r a r A a r a r a r a - - - = + + + + = + + + + K K
Decimal to Radix r Conversions Convert 26 decimal to binary. Quotient Remainder 26/2 13 0 LSB 13/2 6 1 6/2 3 0 3/2 1 1 1/2 0 1 MSB 26 = 11010 2

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Decimal to Radix r Conversions ( 29 ( 29 ( 29 m F m m F a r a r a r A r a r a r a A - - - - - - - - - - - - + + + = + + + = 1 2 1 1 1 2 2 1 1 Part Fractional K K A F * r = r -1 ( a -1 + r –1 ( a -2 +…)) * r = a -1 + r –1 ( a -2 +…) Multiplying the fractional part by r results in a mixed number.
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Set_5-3 - Positional Number System A number is represented...

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