1. Number Systems

2. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No Hexa- decimal 160, 1, … 9, A, B, … F No

3. Quantities/Counting (1 of 3) DecimalBinaryOctal Hexa- decimal 0000 1111 21022 31133 410044 510155 611066 711177 p. 33

4. Quantities/Counting (2 of 3) DecimalBinaryOctal Hexa- decimal 81000108 91001119 10101012A 11101113B 12110014C 13110115D 14111016E 15111117F

5. Quantities/Counting (3 of 3) DecimalBinaryOctal Hexa- decimal 16100002010 17100012111 18100102212 19100112313 20101002414 21101012515 22101102616 23101112717 Etc.

6. Conversion Among Bases The possibilities: Hexadecimal DecimalOctal Binary pp. 40-46

7. Quick Example 25 10 = 11001 2 = 31 8 = 19 16 Base

8. Decimal to Decimal (just for fun) Hexadecimal DecimalOctal Binary Next slide…

9. 125 10 =>5 x 10 0 = 5 2 x 10 1 = 20 1 x 10 2 = 100 125 Base Weight

10. Binary to Decimal Hexadecimal DecimalOctal Binary

11. Binary to Decimal Technique Multiply each bit by 2 n, where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Add the results

12. Example 101011 2 => 1 x 2 0 = 1 1 x 2 1 = 2 0 x 2 2 = 0 1 x 2 3 = 8 0 x 2 4 = 0 1 x 2 5 = 32 43 10 Bit “0”

13. Octal to Decimal Hexadecimal DecimalOctal Binary

14. Octal to Decimal Technique Multiply each bit by 8 n, where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Add the results

15. Example 724 8 => 4 x 8 0 = 4 2 x 8 1 = 16 7 x 8 2 = 448 468 10

16. Hexadecimal to Decimal Hexadecimal DecimalOctal Binary

17. Hexadecimal to Decimal Technique Multiply each bit by 16 n, where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Add the results

18. Example ABC 16 =>C x 16 0 = 12 x 1 = 12 B x 16 1 = 11 x 16 = 176 A x 16 2 = 10 x 256 = 2560 2748 10

19. Decimal to Binary Hexadecimal DecimalOctal Binary

20. Decimal to Binary Technique Divide by two, keep track of the remainder First remainder is bit 0 (LSB, least-significant bit) Second remainder is bit 1 Etc.

21. Example 125 10 = ? 2 2 125 62 1 2 31 0 2 15 1 2 7 1 2 3 1 2 1 1 2 0 1 125 10 = 1111101 2

22. Octal to Binary Hexadecimal DecimalOctal Binary

23. Octal to Binary Technique Convert each octal digit to a 3-bit equivalent binary representation

24. Example 705 8 = ? 2 7 0 5 111 000 101 705 8 = 111000101 2

25. Hexadecimal to Binary Hexadecimal DecimalOctal Binary

26. Hexadecimal to Binary Technique Convert each hexadecimal digit to a 4-bit equivalent binary representation

27. Example 10AF 16 = ? 2 1 0 A F 0001 0000 1010 1111 10AF 16 = 0001000010101111 2

28. Decimal to Octal Hexadecimal DecimalOctal Binary

29. Decimal to Octal Technique Divide by 8 Keep track of the remainder

30. Example 1234 10 = ? 8 8 1234 154 2 8 19 2 8 2 3 8 0 2 1234 10 = 2322 8

31. Decimal to Hexadecimal Hexadecimal DecimalOctal Binary

32. Decimal to Hexadecimal Technique Divide by 16 Keep track of the remainder

33. Example 1234 10 = ? 16 1234 10 = 4D2 16 16 1234 77 2 16 4 13 = D 16 0 4

34. Binary to Octal Hexadecimal DecimalOctal Binary

35. Binary to Octal Technique Group bits in threes, starting on right Convert to octal digits

36. Example 1011010111 2 = ? 8 1 011 010 111 1 3 2 7 1011010111 2 = 1327 8

37. Binary to Hexadecimal Hexadecimal DecimalOctal Binary

38. Binary to Hexadecimal Technique Group bits in fours, starting on right Convert to hexadecimal digits

39. Example 1010111011 2 = ? 16 10 1011 1011 2 B B 1010111011 2 = 2BB 16

40. Octal to Hexadecimal Hexadecimal DecimalOctal Binary

41. Octal to Hexadecimal Technique Use binary as an intermediary

42. Example 1076 8 = ? 16 1 0 7 6 001 000 111 110 2 3 E 1076 8 = 23E 16

43. Hexadecimal to Octal Hexadecimal DecimalOctal Binary

44. Hexadecimal to Octal Technique Use binary as an intermediary

45. Example 1F0C 16 = ? 8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C 16 = 17414 8

46. Exercise – Convert... Don’t use a calculator! DecimalBinaryOctal Hexa- decimal 33 1110101 703 1AF Skip answer Answer

47. Exercise – Convert … DecimalBinaryOctal Hexa- decimal 331000014121 117111010116575 4511110000117031C3 4311101011116571AF Answer

48. Common Powers (1 of 2) Base 10 PowerPrefaceSymbol 10 -12 picop 10 -9 nanon 10 -6 micro  10 -3 millim 10 3 kilok 10 6 megaM 10 9 gigaG 10 12 teraT Value.000000000001.000000001.000001.001 1000 1000000 1000000000 1000000000000

49. Common Powers (2 of 2) Base 2 PowerPrefaceSymbol 2 10 kilok 2 20 megaM 2 30 GigaG Value 1024 1048576 1073741824 What is the value of “k”, “M”, and “G”? In computing, particularly w.r.t. memory, the base-2 interpretation generally applies

50. Example / 2 30 = In the lab… 1. Double click on My Computer 2. Right click on C: 3. Click on Properties

51. Exercise – Free Space Determine the “free space” on all drives on a machine in the lab Drive Free space BytesGB A: C: D: E: etc.

52. Review – multiplying powers For common bases, add powers 2 6  2 10 = 2 16 = 65,536 or… 2 6  2 10 = 64  2 10 = 64k a b  a c = a b+c

53. Binary Addition (1 of 2) Two 1-bit values pp. 36-38 ABA + B 000 011 101 1110 “two”

54. Binary Addition (2 of 2) Two n-bit values Add individual bits Propagate carries E.g., 10101 21 + 11001 + 25 101110 46 11

55. Multiplication (1 of 3) Decimal (just for fun) pp. 39 35 x 105 175 000 35 3675

56. Multiplication (2 of 3) Binary, two 1-bit values AB A  B 000 010 100 111

57. Multiplication (3 of 3) Binary, two n-bit values As with decimal values E.g., 1110 x 1011 1110 1110 0000 1110 10011010

58. Fractions Decimal to decimal (just for fun) pp. 46-50 3.14 =>4 x 10 -2 = 0.04 1 x 10 -1 = 0.1 3 x 10 0 = 3 3.14

59. Fractions Binary to decimal pp. 46-50 10.1011 => 1 x 2 -4 = 0.0625 1 x 2 -3 = 0.125 0 x 2 -2 = 0.0 1 x 2 -1 = 0.5 0 x 2 0 = 0.0 1 x 2 1 = 2.0 2.6875

60. Fractions Decimal to binary p. 50 3.14579.14579 x 2 0.29158 x 2 0.58316 x 2 1.16632 x 2 0.33264 x 2 0.66528 x 2 1.33056 etc. 11.001001...

61. Exercise – Convert... Don’t use a calculator! DecimalBinaryOctal Hexa- decimal 29.8 101.1101 3.07 C.82 Skip answer Answer

62. Exercise – Convert … DecimalBinaryOctal Hexa- decimal 29.811101.110011…35.63…1D.CC… 5.8125101.11015.645.D 3.10937511.0001113.073.1C 12.50781251100.1000001014.404C.82 Answer

63. http://www.ascii-code.com/ ascii-code

64. Binary Logic Truth Tables, Boolean Expressions, and Logic Gates xyz 000 010 100 111 xyz 000 011 101 111 xz 01 10 ANDORNOT z = x y = x yz = x + yz = x = x’

65. Switching Circuits ANDOR

66. Binary Logic Logic gates Example of binary signals 0 1 2 3 Logic 1 Logic 0 Un-define Figure 1.3 Example of binary signals

67. Binary Logic Logic gates Graphic Symbols and Input-Output Signals for Logic gates: Fig. 1.4 Symbols for digital logic circuits Fig. 1.5 Input-Output signals for gates

68. Binary Logic Logic gates Graphic Symbols and Input-Output Signals for Logic gates: Fig. 1.6 Gates with multiple inputs