1. CSC 110 – Intro to Computing Lecture 4: Arithmetic in other bases & Encoding Data

2. Announcements Copies of the slides are available on Blackboard and the course web page before and after each class Your slips selecting your service learning site are due next Tuesday

3. Announcements Homework #1 will be handed out today (and is on the course web page). It is due in my box by Monday at 4PM Quiz #1 on numerical systems will be given next Thursday in class

4. Review of Important Bases Binary (base-2) uses 2 digits: 0 & 1 Octal (base-8) uses 8 digits: 0, 1, 2, 3, 4, 5, 6, 7 Decimal (base-10) uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Hexadecimal (base-16) uses 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

5. Converting To Decimal We use positional notation to convert a number d n...d 3 d 2 d 1 d 0 from base-b to decimal: d 0 * b 0 d 1 * b 1 d 2 * b 2 d 3 * b 3 … + d n * b n

6. Converting From Decimal Converting a number from decimal to base-b While the decimal number is not 0 Divide the decimal number by b Move remainder to left end of answer Replace decimal number with quotient

7. Converting Binary To Octal Starting at the right, split into groupings of 3 bits For each group (working from right to left) Convert group from binary to decimal Add decimal number to left-end of solution

8. Converting Octal To Binary For each digit (working from right to left) Convert the digit from decimal to binary Add 0s to left of the binary to fill 3 bits Add binary number to left-end of solution Finally, remove any 0s padding leftmost digit

9. Binary to powers-of-two bases Conversion similar to binary ↔ octal  Octal is base 8 2 3 = 8 3 binary digits represent 1 octal digit  Hexadecimal is base 16 2 4 = 16 4 binary digits represent 1 hexadecimal digit When converting to hexadecimal, remember to use A, B, C, D, E, F and not 10, 11, 12, 13, 14, 15 for digits

10. Decimal Addition How do we add: 5763 +1263

11. Decimal Addition How do we add: 57633 ones +1263 +3 ones 6 ones

12. Decimal Addition How do we add: 57633 ones +1263 +3 ones 66 ones

13. Decimal Addition How do we add: 57636 tens +1263 +6 tens 612 tens

14. Decimal Addition How do we add: 57636 tens +1263 +6 tens 6 1 hundred 2 tens

15. Decimal Addition How do we add: 1 57639 tens +1233 +3 tens 26 1 hundred 2 tens

16. Decimal Addition How do we add: 1 1 1 hundred 57637 hundreds +1263 +2 hundreds 026 1 thousand 0 hundred

17. Decimal Addition How do we add: 1 1 1 thousand 57635 thousand +1263 +1 thousand 7026 7 thousand

18. Decimal Addition How do we add: 1 1 5763 +1263 7026

19. Oops… I forgot one detail I forgot this was supposed to be in octal 8383 8282 8181 8080 5763838 +1263838

20. Adding in Octal I forgot this was supposed to be in octal 8383 8282 8181 8080 5763838 +1263838 3 ones +3 ones 6 ones

21. Adding in Octal I forgot this was supposed to be in octal 8383 8282 8181 8080 5763838 +1263838 6868 3 ones +3 ones 6 ones Whew. 6 10 = 6 8

22. Adding in Octal 8 1 = 8 8 2 = 8 * 8 = 64 8383 8282 8181 8080 5763838 +1263838 6868 6 eights +6 eights 12 eights What do we do now?

23. Adding in Octal 8 1 = 8 8 2 = 8 * 8 = 64 8383 8282 8181 8080 5763838 +1263838 6868 6 eights +6 eights 12 eights 8 eights = 1 sixty-four

24. Adding in Octal 8 1 = 8 8 2 = 8 * 8 = 64 8383 8282 8181 8080 5763838 +1263838 6868 6 eights +6 eights 12 eights –or– 12-8 = 4 eights and 1 sixty-four

25. Adding in Octal 8 1 = 8 8 2 = 8 * 8 = 64 8383 8282 8181 8080 1 5763838 +1263838 46868 6 eights +6 eights 4 eights and 1 sixty-four

26. Adding in Octal 8 2 = 64 8383 8282 8181 8080 1 5763838 +1263838 46868 1 sixty-four 7 sixty-four +2 sixty-four 10 sixty-fours

27. Adding in Octal 8 2 = 648 3 = 64 * 8 = 256 8383 8282 8181 8080 11 5763838 +1263838 246868 1 sixty-four 7 sixty-four +2 sixty-four 10-8 = 2 sixty-fours 1 two fifty-six

28. Adding in Octal Finally, something makes sense again 8383 8282 8181 8080 11 5763838 +1263838 7246868 1 two fifty-six 5 two fifty-six +1 two fifty-six 7 two fity-six Yes! 7 is an octal digit

29. Adding in Octal That’s all, folks! 8383 8282 8181 8080 11 5763838 +1263838 7246868

30. Algorithm For each position from right to left 126868 +722828

31. Addition Algorithm To add 2 numbers in base-b: For each position from right to left Compute sum of the digits If sum > b then Record sum - b at position Carry the 1 If sum < b Record sum at position

32. Algorithm For each position from right to left 126868 +722828

33. Algorithm Compute sum of the digits6 + 2 = 8 126868 +722828

34. Algorithm If sum > b thenIf 8 > 8 then 126868 +722828

35. Algorithm Record sum - b for position8 – 8 = 0 126868 +722828 0808

36. Algorithm Carry the one 1 126868 +722828 0808

37. Algorithm For each position from right to left 1 126868 +722828 0808

38. Algorithm Compute sum of digits1 + 2 + 2 = 5 1 126868 +722828 0808

39. Algorithm If sum < b then 5 < 8 1 126868 +722828 0808

40. Algorithm Record sum for position 1 126868 +722828 50808

41. Algorithm For each position from right to left 1 126868 +722828 50808

42. Algorithm Compute sum of digits1 + 7 = 8 1 126868 +722828 50808

43. Algorithm If sum > b then8 > 8 1 126868 +722828 50808

44. Algorithm Record sum – b for position8 – 8 = 0 1 126868 +722828 050808

45. Algorithm Carry the one 11 126868 +722828 050808

46. Algorithm For each position from right to left 11 126868 +722828 050808

47. Algorithm Compute sum of digits1 = 1 11 126868 +722828 050808

48. Algorithm If sum < b then 1 < 8 11 126868 +722828 050808

49. Algorithm Record sum for position 11 126868 +722828 1050808

50. Algorithm We’re Done! 11 126868 +722828 1050808

51. Addition Algorithm To add numbers in base-b: For each position from right to left Compute sum of the digits If sum > b then Record sum - b at positionCarry the 1 If sum < b Record sum at position

52. Data Encoding In the real world, data (“information”) is analog  Falls along a infinite continuum Color changes when mixing paint Falling mercury levels as temperatures drop  Hard to capture numerically Is it 31.848174 o F or 31.848173 o F right now? Do you even care about this difference?

53. Data Encoding Normally, we work with discrete numbers  Round to nearest whole or rational number: 71 o F, 4.5 miles, 1 teaspoon  Limits the amount of space needed Imagine if the morning news had to put on screen: 71.23478923478234789234890234234789 o F or state the time as: 8:45:12.484589983890234890234890234890234 Computers follow this pattern  It takes a lot of space to be precise!

54. Digitizing Data Computers work in binary (0-1)  Good news: we know how to convert any number into binary!  Suppose we have a CD containing pictures of my daughter, how do we turn the 0s and 1s on a CD into the colors that make up the cutest girl ever?

55. Color by Number Most pictures rely upon the RGB standard  First 8 bits (“byte”) specify amount of red used  Next byte specifies amount of green used  Last byte specifies amount of blue used  Mixing these numbers gives us many colors

56. Data Storage Storing data can require lots of space  Each pixel (dot) in a color photo takes 4 bytes  5 megapixel (~million pixel) camera: 20MB per picture  32 pictures: 640MB (a CD holds 650MB)

57. Binary Representation 1 bit captures 2 states: 0 or 1 2 bits captures 4 states: 00, 01, 10, 11 3 bits capture 8 states: 000, 001, 010, 011, 100, 101, 110, 111

58. Binary Representation How many colors can 8 bits generate? How many different colors can n bits represent?

59. For Next Lecture Have Chapter 4 started Be ready to discuss:  Boolean logic  AND, OR, XOR, NOT, NAND, NOR gates