1. CSC 110 – Intro to Computing Lecture 3: Converting between bases & Arithmetic in other bases

2. Announcements The end is near! All drop/add slips must be turned in by Jan. 24 th. I have a cool office with a full candy jar. Please stop by and get your sugar fix (you could also me questions you have at the same time).

3. Positional Notation Usually write numbers using positional notation  Add numbers from right-to-left  Digits (0, 1, 2, … 9) have a numerical value  Each position has an place value  E.g., 1275, 529, 3, 200 Makes understanding number and arithmetic easy

4. Positional Notation for 5862 2= 2 ones= 2 * 1 =2

5. Positional Notation for 5862 2= 2 ones= 2 * 1 =2 6= 6 tens= 6 * 10 =60

6. Positional Notation for 5862 2= 2 ones= 2 * 1 =2 6= 6 tens= 6 * 10 =60 8= 8 hundreds= 8 * 100 =800

7. Positional Notation for 5862 2= 2 ones= 2 * 1 =2 6= 6 tens= 6 * 10 =60 8= 8 hundreds= 8 * 100 =800 5= 5 thousands= 5 * 1000 =5000

8. Positional Notation for 5862 2= 2 ones= 2 * 1 =2 6= 6 tens= 6 * 10 =60 8= 8 hundreds= 8 * 100 =800 5= 5 thousands= 5 * 1000 =+ 5000 5862

9. Decimal Positional Notation Formal equation for a number d n...d 3 d 2 d 1 d 0  d 0 is digit in ones place, d 1 is in tens place, … d 0 * 10 0 d 1 * 10 1 d 2 * 10 2 d 3 * 10 3 … + d n * 10 n

10. Decimal Positional Notation 2= 2 * 1= 2 * 10 0 =2 6= 6 * 10= 6 * 10 1 =60 8= 8 * 100= 8 * 10 2 =800 5= 5 * 1000= 5 * 10 3 =+ 5000 5862

11. Bases Used With Computers Previous equation of positional notation assumes base-10 Computer use other bases  Binary (base-2)  Octal (base-8)  Hexadecimal (base-16) We use positional notation to convert from these bases into decimal

12. Positional Notation Convert a number d n...d 3 d 2 d 1 d 0 to decimal: From base-b d 0 * b 0 d 1 * b 1 d 2 * b 2 d 3 * b 3 … + d n * b n From base-10 d 0 * 10 0 d 1 * 10 1 d 2 * 10 2 d 3 * 10 3 … + d n * 10 n

13. Converting Binary to Decimal 101011 2 =1 * 2 0 = 1 * 2 1 = 0 * 2 2 = 1 * 2 3 = 0 * 2 4 = + 1 * 2 5 =

14. Converting Binary to Decimal 101011 2 =1 * 2 0 =1 * 1 1 * 2 1 =1 * 2 0 * 2 2 =0 * 4 1 * 2 3 =1 * 8 0 * 2 4 =0 * 16 1 * 2 5 = 1 * 32

15. Converting Binary to Decimal 101011 2 =1 * 2 0 =1 * 1= 1 1 * 2 1 =1 * 2= 2 0 * 2 2 =0 * 4= 0 1 * 2 3 =1 * 8= 8 0 * 2 4 =0 * 16= 0 1 * 2 5 =1 * 32+ 32 43 10

16. Converting Octal to Decimal 106 8 =6 * 8 0 = 0 * 8 1 = + 1 * 8 2 =

17. Converting Octal to Decimal 106 8 =6 * 8 0 =6 * 1 0 * 8 1 =0 * 8 1 * 8 2 =1 * 64

18. Converting Octal to Decimal 106 8 =6 * 8 0 =6 * 1= 6 0 * 8 1 =0 * 8= 0 1 * 8 2 =1 * 64 + 64 70 10

19. Digits In Other Bases Binary only needs 2 digits: 0 & 1 Octal needs 8 digits: 0, 1, 2, 3, 4, 5, 6, 7 Hexadecimal (base-16) uses 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F  A has value of 10 10  B has value of 11 10  C has value of 12 10  D has value of 13 10  E has value of 14 10  F has value of 15 10

20. Convert Hexadecimal To Decimal 3F 16 = 27 16 =

21. Convert Hexadecimal To Decimal 3F 16 =15 * 16 0 =15 * 1 =15 3 * 16 1 =3 * 16 =+ 48 63 10 27 16 =7 * 16 0 =7 * 1 =7 2 * 16 1 =2 * 16 =+ 32 39 10

22. Converting To Decimal Review Formula converting 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

23. Converting Decimal To Binary Algorithm converting from decimal to binary: While the decimal number is not 0 Divide the decimal number by 2 Move remainder to left end of answer Replace decimal number with quotient 34 10 =

24. Converting Decimal To Base-b More generally, converting 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 335 10 =

25. Converting Binary to Octal Special relationship between these bases  Octal uses 8 digits: 0, 1, 2, 3, 4, 5, 6, 7  3 binary digits (bits) can represent 8 numbers: 000 2 = 0 10 100 2 = 4 10 001 2 = 1 10 101 2 = 5 10 010 2 = 2 10 110 2 = 6 10 011 2 = 3 10 111 2 = 7 10  Every 3 bits is equal to one octal digit  Makes conversion easy

26. Converting Binary to Octal Starting from the right, split the number into groupings of 3 bits: 1110011010 2

27. Converting Binary to Octal Starting from the right, split the number into groupings of 3 bits: 1 110 011 010

28. Converting Binary to Octal Convert each group to decimal: 1 110 011 010 010 2 =

29. Converting Binary to Octal Convert each group to decimal: 1 110 011 010 2 010 2 =0 * 2 0 =0 * 1 =0 1 * 2 1 =1 * 2 =2 0 * 2 2 =0 * 4 =+ 0 2

30. Converting Binary to Octal Convert each group to decimal: 1 110 011 010 3 2 011 2 =1 * 2 0 =1 * 1 =1 1 * 2 1 =1 * 2 =2 0 * 2 2 =0 * 4 =+ 0 3

31. Converting Binary to Octal Convert each group to decimal: 1 110 011 010 6 3 2 110 2 =0 * 2 0 =0 * 1 =0 1 * 2 1 =1 * 2 =2 1 * 2 2 =1 * 4 =+ 4 6

32. Converting Binary to Octal Convert each group to decimal: 1 110 011 0101 6 3 2 1 2 =1 * 2 0 =1 * 1 =1

33. Converting Binary to Octal That’s all there us to it! 1110011010 2 = 1632 8

34. Converting Octal to Binary Starting from the right, convert each digit into binary: 235 8

35. Converting Octal to Binary Starting from the right, convert each digit into binary: 2 3 5 5 10

36. Converting Octal to Binary Starting from the right, convert each digit into binary: 2 3 5 101 5 10 = 101 2

37. Converting Octal to Binary Starting from the right, convert each digit into binary: 2 3 5 101 3 10 = 11 2

38. Converting Octal to Binary Add 0s to the left to pad solution to 3 digits 2 3 5 011 101 3 10 = 011 2

39. Converting Octal to Binary Starting from the right, convert each digit into binary: 2 3 5010 011 101 2 10 = 010 2

40. Converting Octal to Binary Remove any 0 padding the rightmost digit: 2 3 5 10 011 101

41. Converting Octal to Binary That’s it! 235 8 = 10011101 2

42. Binary to powers-of-two bases Special relationship between binary and any power-of-two base  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

43. Converting Between Binary & Hex Conversion from binary to hexadecimal is similar to binary to octal  Initially make groupings of 4 bits, not 3 bits Similarly, converting hexadecimal to binary is like converting octal to binary  But converting hex to binary means we pad each result out to 4 bits

44. For Next Lecture Read through Chapter 3 Be ready to discuss:  Arithmetic in bases other than 10  Ways computers representation information