1. Lecture 7 How computers process data (Number Systems) PRESENTED BY MD. MAHBUBUL ALAM, PHD 1

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 PRESENTED BY MD. MAHBUBUL ALAM, PHD 2

3. Quantities/Counting (1 of 3) DecimalBinaryOctal Hexa- decimal 0000 1111 21022 31133 410044 510155 611066 711177 PRESENTED BY MD. MAHBUBUL ALAM, PHD 3

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

5. Quantities/Counting (3 of 3) DecimalBinaryOctal Hexa- decimal 16100002010 17100012111 18100102212 19100112313 20101002414 21101012515 22101102616 23101112717 Etc. PRESENTED BY MD. MAHBUBUL ALAM, PHD 5

6. Conversion Among Bases The possibilities: Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 6

7. Quick Example 25 10 = 11001 2 = 31 8 = 19 16 Base PRESENTED BY MD. MAHBUBUL ALAM, PHD 7

8. Decimal to Decimal (just for fun) Hexadecimal DecimalOctal Binary Next slide… PRESENTED BY MD. MAHBUBUL ALAM, PHD 8

9. 125 10 =>5 x 10 0 = 5 2 x 10 1 = 20 1 x 10 2 = 100 125 Base Weight PRESENTED BY MD. MAHBUBUL ALAM, PHD 9

10. Binary to Decimal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 10

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 PRESENTED BY MD. MAHBUBUL ALAM, PHD 11

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” PRESENTED BY MD. MAHBUBUL ALAM, PHD 12

13. Octal to Decimal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 13

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 PRESENTED BY MD. MAHBUBUL ALAM, PHD 14

15. Example 724 8 => 4 x 8 0 = 4 2 x 8 1 = 16 7 x 8 2 = 448 468 10 PRESENTED BY MD. MAHBUBUL ALAM, PHD 15

16. Hexadecimal to Decimal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 16

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 PRESENTED BY MD. MAHBUBUL ALAM, PHD 17

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 PRESENTED BY MD. MAHBUBUL ALAM, PHD 18

19. Decimal to Binary Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 19

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. PRESENTED BY MD. MAHBUBUL ALAM, PHD 20

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 PRESENTED BY MD. MAHBUBUL ALAM, PHD 21

22. Octal to Binary Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 22

23. Octal to Binary Technique ◦Convert each octal digit to a 3-bit equivalent binary representation PRESENTED BY MD. MAHBUBUL ALAM, PHD 23

24. Example 705 8 = ? 2 7 0 5 111 000 101 705 8 = 111000101 2 PRESENTED BY MD. MAHBUBUL ALAM, PHD 24

25. Hexadecimal to Binary Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 25

26. Hexadecimal to Binary Technique ◦Convert each hexadecimal digit to a 4-bit equivalent binary representation PRESENTED BY MD. MAHBUBUL ALAM, PHD 26

27. Example 10AF 16 = ? 2 1 0 A F 0001 0000 1010 1111 10AF 16 = 0001000010101111 2 PRESENTED BY MD. MAHBUBUL ALAM, PHD 27

28. Decimal to Octal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 28

29. Decimal to Octal Technique ◦Divide by 8 ◦Keep track of the remainder PRESENTED BY MD. MAHBUBUL ALAM, PHD 29

30. Example 1234 10 = ? 8 8 1234 154 2 8 19 2 8 2 3 8 0 2 1234 10 = 2322 8 PRESENTED BY MD. MAHBUBUL ALAM, PHD 30

31. Decimal to Hexadecimal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 31

32. Decimal to Hexadecimal Technique ◦Divide by 16 ◦Keep track of the remainder PRESENTED BY MD. MAHBUBUL ALAM, PHD 32

33. Example 1234 10 = ? 16 1234 10 = 4D2 16 16 1234 77 2 16 4 13 = D 16 0 4 PRESENTED BY MD. MAHBUBUL ALAM, PHD 33

34. Binary to Octal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 34

35. Binary to Octal Technique ◦Group bits in threes, starting on right ◦Convert to octal digits PRESENTED BY MD. MAHBUBUL ALAM, PHD 35

36. Example 1011010111 2 = ? 8 1 011 010 111 1 3 2 7 1011010111 2 = 1327 8 PRESENTED BY MD. MAHBUBUL ALAM, PHD 36

37. Binary to Hexadecimal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 37

38. Binary to Hexadecimal Technique ◦Group bits in fours, starting on right ◦Convert to hexadecimal digits PRESENTED BY MD. MAHBUBUL ALAM, PHD 38

39. Example 1010111011 2 = ? 16 10 1011 1011 2 B B 1010111011 2 = 2BB 16 PRESENTED BY MD. MAHBUBUL ALAM, PHD 39

40. Octal to Hexadecimal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 40

41. Octal to Hexadecimal Technique ◦Use binary as an intermediary PRESENTED BY MD. MAHBUBUL ALAM, PHD 41

42. Example 1076 8 = ? 16 1 0 7 6 001 000 111 110 2 3 E 1076 8 = 23E 16 PRESENTED BY MD. MAHBUBUL ALAM, PHD 42

43. Hexadecimal to Octal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 43

44. Hexadecimal to Octal Technique ◦Use binary as an intermediary PRESENTED BY MD. MAHBUBUL ALAM, PHD 44

45. Example 1F0C 16 = ? 8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C 16 = 17414 8 PRESENTED BY MD. MAHBUBUL ALAM, PHD 45

46. Exercise – Convert... Don’t use a calculator! DecimalBinaryOctal Hexa- decimal 33 1110101 703 1AF Skip answer Answer PRESENTED BY MD. MAHBUBUL ALAM, PHD 46

47. Exercise – Convert … DecimalBinaryOctal Hexa- decimal 331000014121 117111010116575 4511110000117031C3 4311101011116571AF Answer PRESENTED BY MD. MAHBUBUL ALAM, PHD 47

48. Binary Arithmetic: Addition & Subtraction XYX+Y 000 011 101 1110 PRESENTED BY MD. MAHBUBUL ALAM, PHD 48 XYX-Y 000 011 101 110

49. Binary Arithmetic: Multiplication & Division XYX*Y 000 010 100 111 PRESENTED BY MD. MAHBUBUL ALAM, PHD 49

50. Boolean Algebra The digital circuits present in a digital computer are designed using a mathematical discipline known as Boolean Algebra. It describes the relationship between the inputs and outputs of a digital circuit. Boolean Algebra was named in honor of Gorge Boole, an English Mathematician, who had proposed the basic principles of this. Objective: Boolean Algebra is used mainly by design engineers in order to obtain the required output by using least number of logic gates. PRESENTED BY MD. MAHBUBUL ALAM, PHD 50

51. Components Like any other algebra, Boolean Algebra also uses variables and operations. ◦A Boolean variable has only two possible values which is either true (1) or false (0) ◦Basic Boolean operations are: AND, OR and NOT PRESENTED BY MD. MAHBUBUL ALAM, PHD 51

52. Basic Logical Operations All these three basic logical operations can be represented symbolically as ◦A AND B = A. B ◦A OR B = A + B ◦NOT A = A’ These operations can be defined in a form known as Truth Table, which s a list of all possible input values and the output for each input combination. PRESENTED BY MD. MAHBUBUL ALAM, PHD 52

53. Truth Table for AND Operator Truth Table for a 2-input AND Operator is as follows ABY = A. B 000 010 100 111 PRESENTED BY MD. MAHBUBUL ALAM, PHD 53

54. Truth Table for OR Operator Truth Table for a 2-input OR Operator is as follows ABY = A + B 000 011 101 111 PRESENTED BY MD. MAHBUBUL ALAM, PHD 54

55. Truth Table for NOT Operator Truth Table for NOT Operator is as follows AY = A’ 01 10 PRESENTED BY MD. MAHBUBUL ALAM, PHD 55

56. Logic Gate In electronics, a logic gate is an idealized or physical device implementing a Boolean function; that is, it performs a logical operation on one or more logical inputs, and produces a single logical output. PRESENTED BY MD. MAHBUBUL ALAM, PHD 56

57. Thank you (Courtesy: Dept. of IT, York University) PRESENTED BY MD. MAHBUBUL ALAM, PHD 57