1. Ch 2. Number Systems and Codes 2.2 Octal and Hexadecimal Numbers 10 ~ 15 : Alphabet

2. p digit to the left of the point and n digits to the right of the point p n 2.3 General Positional-Number-System Conversions

3. Number conversion example

4. 123416 77 16 413 2 5678 70 8 86 7 8 10 Number conversion example (decimal to hexadecimal, octal)

5. 0.78 6.24 8 8 1.92 8 7.36 8 Number conversion example (decimal to octal)

8. Carry in 1 1+ 01 1 0+ 01 Carry out 1 0 1- 11 1 0 1- 11 Burrow out 1 Burrow in Burrow out

9. 2.4 Addition and Subtraction of Nondecimal Numbers

10. + Hexadecimal addition

11. Complement System –Negates a number by taking its complement –More difficult than changing the sign bit –Can be added or subtracted directly 2.5 Representation of Negative Numbers Sign bit

13. Conversion example Easy to complement

14. 2.6 Two’s-Complement Addition and Subtraction

16. [ -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7 ]

17. [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ]

18. 2.7 One’s-Complement Addition and Subtraction +6 (0110) -3 (1100)+ 10010 1 0011 End-around carry One’s complement

19. 2.8 Binary Multiplication

20. Shifted and negated multiplicand

21. 2.8 Binary Division

22. 2.10 Binary Codes for Decimal Numbers

23. 2.11 Gray Code

25. (0) 1 1 0 1 0 1 Binary to Gray Code If different, ‘1’ else (same) ‘0’ (0) 1 0 1 1 1 0 Gray Code to Binary If different, ‘1’ else (same) ‘0’ 1 2 3 1 2 3

27. 2.13 Codes for Actions, Conditions, and States

30. 2.14 n-Cubes and Distance

32. Hamming Distance –Distance between two vertices, the number of difference bits in each position EX) D(010, 111) = 2

33. 2.15 Codes for Detecting and Correcting Errors Parity-bit

34. At least two non codes between each pair of code words

35. If minimum distance = 2C+1, up to C-bits can be corrected If 2C+D+1, then C-bits can be corrected, and d bits can be detected 4= 2C+D+1, (a)C=1, D=1 (b)1 bit can be corrected (c)D=3, 3 bit errors can be detected

40. 111110101100011010001

41. LSB is 1 if all 7 bits are odd LSB is 0 if all 7 bits are even

42. k = # of parity bits m = # of info bits

44. Undetectable Error

45. An important application of 2-D codes

46. 2.16 Codes for Serial Data Transmission and Storage

47. NRZ : Non-Return to Zero NRZI : Non-Return to Zero Invert on 1s BPRZ : Bipolar Return to Zero