1. Data Representation (in computer system)

2. Data Representation How do computers represent data? 1111111111 0000000000 b The computers are digital Recognize only two discrete states: on or off Computers are electronic devices powered by electricity, which has only two states, on or off on off

3. The digital computer is binary. Everything is represented by one of two states: 0, 1 on, off true, false voltage, no voltage In a computer, values are represented by sequences of binary digits or bits. How do computers represent data?

4. Data Storage Units Bit : An abbreviation for BIbary digiT, is the smallest unit data representation. Byte (B)= 8bits KiloByte (KB) = 1024B MegaByte (MB) = 1024KB GigaByte (GB) = 1024MB TeraByte (TB) = 1024GB

5. What is a byte? b Eight bits are grouped together to form a byte b 0s and 1s in each byte are used to represent individual characters such as letters of the alphabet, numbers, and punctuation

6. Data classification Quantitative Qualitative Not proportion to a value. Name, symbols... proportion to a value. Number Integer Non Integer

7. Data Representation What are two popular coding systems to represent data? b American Standard Code for Information Interchange (ASCII) b Extended Binary Coded Decimal Interchange Code (EBCDIC)

8. How is a character sent from the keyboard to the computer? Step 1: The user presses the letter T key on the keyboard Step 2: An electronic signal for the letter T is sent to the system unit Step 3: The signal for the letter T is converted to its ASCII binary code (01010100) and is stored in memory for processing Step 4: After processing, the binary code for the letter T is converted to an image on the output device

9. Number Systems

10. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No Hexa- decimal 16 0, 1, … 9, A, B, … F No

11. Quantities/Counting (1 of 2) DecimalBinaryOctal Hexa- decimal 0000 1111 21022 31133 410044 510155 611066 711177

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

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

14. 25 10 =>5 x 10 0 = 5 2 x 10 1 = 20 25 Base Weight

15. Number Base Conversion The possibilities: Hexadecimal DecimalOctal Binary

16. Binary to Decimal Hexadecimal DecimalOctal Binary

17. 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

18. 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”

19. Octal to Decimal Hexadecimal DecimalOctal Binary

20. 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

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

22. Hexadecimal to Decimal Hexadecimal DecimalOctal Binary

23. 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

24. 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

25. Decimal to Binary Hexadecimal DecimalOctal Binary

26. 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.

27. 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

28. Octal to Binary Hexadecimal DecimalOctal Binary

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

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

31. Hexadecimal to Binary Hexadecimal DecimalOctal Binary

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

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

34. Decimal to Octal Hexadecimal DecimalOctal Binary

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

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

37. Decimal to Hexadecimal Hexadecimal DecimalOctal Binary

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

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

40. Binary to Octal Hexadecimal DecimalOctal Binary

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

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

43. Binary to Hexadecimal Hexadecimal DecimalOctal Binary

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

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

46. Octal to Hexadecimal Hexadecimal DecimalOctal Binary

47. Octal to Hexadecimal Technique –Use binary as an intermediary

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

49. Hexadecimal to Octal Hexadecimal DecimalOctal Binary

50. Hexadecimal to Octal Technique –Use binary as an intermediary

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

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

53. Exercise – Convert … Deci mal BinaryOctal Hexa- decimal 331000014121 117111010116575 4511110000117031C3 4311101011116571AF Answer

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

55. Multiplication Binary, two n-bit values –As with decimal values –E.g., 1110 x 1011 1110 1110 0000 1110 10011010

56. Thank you