1

Chapter 2

Number Systems, Operations, and Codes

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

….105 104 103 102 101 100. 10-1 10-2 10-3 10-4 10-5…

14.2 = 1 101 + 4 100 + 2 10-1

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

….25 24 23 22 21 20 . 2-1 2-2 2-3 2-4 2-5 …

10111 = 1 24 + 0 23 +1 22 +1 21 + 1 20

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Table 2--1

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

6

Repeated division-by-2 method

7

Converting Decimal fractions to Binary

Using Sum-of-weights

Binary weights

64 32 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

8

Binary Addition

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10

11001

+1101

100110

9

Binary Subtraction

0 - 0 = 0

1 - 1 = 0

1 - 0 = 1

10 -1 = 1 0 -1 with a borrow of 1

1011

-111

100

10

Binary Multiplication0 0 = 0

0 1 = 0

1 0 = 0 100110

1 1 = 1 101

100110

000000

100110

10111110

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Binary Division

Use the same procedure as decimal division

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Hexadecimal Numbers

13

Binary-to-Hexadecimal Conversion

14

Hexadecimal-to-Binary Conversion

15

Hexadecimal-to-Decimal Conversion

16

17

Decimal-to-Hexadecimal Conversion

18

Binary Coded Decimal (BCD)

19

20

Digital Codes and Parity

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• ASCII (American Standard Code for Information Interchange)

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Parity Method for Error Detection

23

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

Digital System Application

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

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