1. 1 Chapter 2 Number Systems, Operations, and Codes

2. 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

3. 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

4. 4 Table 2--1

5. 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. 6 Repeated division-by-2 method

7. 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

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

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

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

11. 11 Binary Division Use the same procedure as decimal division

12. 12 Hexadecimal Numbers

13. 13 Binary-to-Hexadecimal Conversion

14. 14 Hexadecimal-to-Binary Conversion

15. 15 Hexadecimal-to-Decimal Conversion

16. 16

17. 17 Decimal-to-Hexadecimal Conversion

18. 18 Binary Coded Decimal (BCD)

19. 19

20. 20 Digital Codes and Parity

21. 21 ASCII (American Standard Code for Information Interchange)

22. 22 Parity Method for Error Detection

23. 23

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

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

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