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1 Chapter 2 Number Systems, Operations, and Codes

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2 Decimal Numbering System The decimal numbering system has 10 digits 0 through 9 The decimal numbering system has a base of 10 with each position weighted by a factor of 10 ….10 5 10 4 10 3 10 2 10 1 10 0. 10 -1 10 -2 10 -3 10 -4 10 -5 … 14.2 = 1 10 1 + 4 10 0 + 2 10 -1

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3 Binary Numbers The binary numbering system has 2 digits 0 and 1 The binary numbering system has a base of 2 with each position weighted by a factor of 2 ….2 5 2 4 2 3 2 2 2 1 2 0. 2 -1 2 -2 2 -3 2 -4 2 -5 … 10111 = 1 2 4 + 0 2 3 +1 2 2 +1 2 1 + 1 2 0

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4 Table 2--1

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5 Decimal-to-Binary Conversion Sum-of-weight method Binary weights 256 128 64 32 16 8 4 2 1 357 = 256 + 64 + 32 + 4 + 1 101100101 Binary weights 1024 512 256 128 64 32 16 8 4 2 1 1937 = 1024 + 512 + 256 + 128 + 16 + 111110010001

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6 Repeated division-by-2 method

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7 Converting Decimal fractions to Binary Using Sum-of-weights Binary weights 6432 16 8 4 2 1.5.25.125.0625 95.6875 = 64 + 16 + 8 + 4 + 2 + 1 +.5 +.125 +.0625 1011111.1011 Repeated division by 2 yields the whole number while repeated multiplication by 2 of the fraction yields the binary fraction

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8 Binary Addition 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 10 11001 +1101 100110

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9 Binary Subtraction 0 - 0 = 0 1 - 1 = 0 1 - 0 = 1 10 -1 = 10 -1 with a borrow of 1 1011 -111 100

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10 Binary Multiplication 0 0 = 0 0 1 = 0 1 0 = 0100110 1 1 = 1 101 100110 000000 100110 10111110

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11 Binary Division Use the same procedure as decimal division

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12 Hexadecimal Numbers

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13 Binary-to-Hexadecimal Conversion

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14 Hexadecimal-to-Binary Conversion

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15 Hexadecimal-to-Decimal Conversion

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17 Decimal-to-Hexadecimal Conversion

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18 Binary Coded Decimal (BCD)

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20 Digital Codes and Parity

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21 ASCII (American Standard Code for Information Interchange)

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22 Parity Method for Error Detection

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24 Figure 2--8 The system is in its initial state. Digital System Application

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25 Figure 2--9 The system has counted 50 bottles of tablets and is working on the next bottle.

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26 Figure 2--10 The system has just counted its fifty-first bottle of tablets.

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