Computation in Positional Systems

1. Computation in Positional Systems Section 4.3

2. ADDITION

3. ADDITION Example Add 334 and 134. 1 334 3 + 3 = 6 + 134 134 But 6 is not a base 4 number. 24 1 R 2 4 6 So 34 + 34 = 124

4. ADDITION Example Add 334 and 134. 1 334 1 + 3 + 1 = 5 + 134 But 5 is not a base 4 number. 11 24 1 R 1 4 5 So 14 + 34 + 14= 114

5. ADDITION Example Add 1112 and 1012. 1 1112 1 + 1 = 2 + 1012 1012 But 2 is not a base 2 number. 02 1 R 0 2 2 So 12 + 12 = 102

6. ADDITION Example Add 1112 and 1012. 1 1 1112 1 + 1 + 0 = 2 + 1012 But 2 is not a base 2 number. 0 02 1 R 0 2 2 So 12 + 12+ 02= 102

7. ADDITION Example Add 1112 and 1012. 1 1 1112 1 + 1 + 1 = 3 + 1012 But 3 is not a base 2 number. 11 0 02 1 R 1 2 3 So 12 + 12+ 12= 112

8. ADDITION Example Add 5246 and 2146. 1 5246 4 + 4 = 8 + 2146 2146 But 8 is not a base 6 number. 26 1 R 2 6 8 So 46 + 46 = 126

9. ADDITION Example Add 5246 and 2146. 1 5246 1 + 2 + 1 = 4 + 2146 And 4 is a base 6 number. 4 26

10. ADDITION Example Add 5246 and 2146. 1 5246 5 + 2 = 7 + 2146 But 7 is not a base 6 number. 11 4 26 1 R 1 6 7 So 56 + 26 = 116

11. SUBTRACTION

12. SUBTRACTION Example Subtract 124 from 314. 1 + 4 = 5 3 – 1 = 2 2 5 We cannot subtract 2 from 1, so we need to borrow. 314 – 124 124 1 34 Since we are in base 4, we will borrow 1 from column to left and ADD 4 where needed. Usually, we stick a 1 in front, but this is because we are in base 10 and we are adding 10. Now subtract.

13. SUBTRACTION Example Subtract 235 from 415. 1 + 5 = 6 4 – 1 = 3 3 6 We cannot subtract 3 from 1, so we need to borrow. 415 – 235 235 1 35 Since we are in base 5, we will borrow 1 from column to left and ADD 5 where needed. Now subtract.

14. SUBTRACTION Example Subtract 12425 from 34315. 3 – 1 = 2 1 + 5 = 6 2 6 We cannot subtract 2 from 1, so we need to borrow. 34315 – 12425 12425 45 Since we are in base 5, we will borrow 1 from column to left and ADD 5 where needed. Now subtract.

15. SUBTRACTION Example Subtract 12425 from 34315. 4 – 1 = 3 2 + 5 = 7 7 2 3 6 We cannot subtract 4 from 2, so we need to borrow. 34315 – 12425 45 3 Since we are in base 5, we will borrow 1 from column to left and ADD 5 where needed. Now subtract.

16. SUBTRACTION Example Subtract 12425 from 34315. 7 2 3 6 We can subtract 2 from 3. 34315 – 12425 45 3 1

17. SUBTRACTION Example Subtract 12425 from 34315. 7 2 3 6 We can subtract 1 from 3. 34315 – 12425 45 3 2 1

18. SUBTRACTION Example Subtract 32367 from 51447. 4 – 1 = 3 4 + 7 = 11 3 11 We cannot subtract 6 from 4, so we need to borrow. 51447 32367 – 32367 57 Since we are in base 7, we will borrow 1 from column to left and ADD 7 where needed. Now subtract.

19. SUBTRACTION Example Subtract 32367 from 51447. 3 11 We can subtract 3 from 3. 51447 – 32367 0 57

20. SUBTRACTION Example Subtract 32367 from 51447. 5 – 1 = 4 1 + 7 = 8 4 8 3 11 We cannot subtract 2 from 1, so we need to borrow. 51447 – 32367 0 57 6 Since we are in base 7, we will borrow 1 from column to left and ADD 7 where needed. Now subtract.

21. SUBTRACTION Example Subtract 32367 from 51447. 4 8 3 11 We can subtract 3 from 4. 51447 – 32367 0 1 57 6

22. SUBTRACTION Example Subtract 1E9A16from 3B2416. 11 First let’s substitute the values for the letters of base 16. 3B2416 3B2416 – 1E9A16 1E9A16 14 10 – 1E9A16 10 11 12 13 14 15 Base 16 digits: 0 1 2 3 4 5 6 7 8 9 A B C D E F

23. SUBTRACTION Example Subtract 1E9A16from 3B2416. 1 20 11 We cannot subtract 10 from 4, so we need to borrow. 3B2416 14 10 – 1E9A16 A16 Since we are in base 16, we will borrow 1 from column to left and ADD 16 where needed. 10 4 + 16 = 20 2 – 1 = 1 Now subtract. 10 11 12 13 14 15 Base 16 digits: 0 1 2 3 4 5 6 7 8 9 A B C D E F

24. SUBTRACTION Example Subtract 1E9A16from 3B2416. 17 10 1 20 11 We cannot subtract 9 from 1, so we need to borrow. 3B2416 14 10 – 1E9A16 A16 8 Since we are in base 16, we will borrow 1 from column to left and ADD 16 where needed. 10 1 + 16 = 17 11 – 1 = 10 Now subtract. 10 11 12 13 14 15 Base 16 digits: 0 1 2 3 4 5 6 7 8 9 A B C D E F

25. SUBTRACTION Example Subtract 1E9A16from 3B2416. 26 17 10 1 20 2 11 We cannot subtract 14 from 10, so we need to borrow. 3B2416 14 10 – 1E9A16 A16 8 C 1 Since we are in base 16, we will borrow 1 from column to left and ADD 16 where needed. 12 10 3 – 1 = 2 10 + 16 = 26 Now subtract. And subtract again. 10 11 12 13 14 15 Base 16 digits: 0 1 2 3 4 5 6 7 8 9 A B C D E F

26. MULTIPLICATION

27. MULTIPLICATION Example Multiply 346 by 26. Multiplication is repeated additions. Thus, many steps are the same. 1 346  26 26 26 4  2 = 8 But 8 is not a base 6 number. 1 R 2 6 8 So 46 26 = 126

28. MULTIPLICATION Example Multiply 346 by 26. 1 346 3  2 = 6, then adding the carried 1, 6 + 1 = 7  26 26 11 But 7 is not a base 6 number. 1 R 1 6 7 So 36 26+ 16 = 116

29. MULTIPLICATION Example Multiply 457 by 37. 2 457  37 37 17 5  3 = 15 But 15 is not a base 7 number. 2 R 1 7 15 So 57 37 = 217

30. MULTIPLICATION Example Multiply 457 by 37. 2 457 4  3 = 12, then adding the carried 2, 12 + 2 = 14  37 17 20 But 14 is not a base 7 number. The test will only cover multiplication by a single digit number, as here. 2 R 0 7 14 So 47 37+ 27 = 207

31. DIVISION

32. DIVISION Example Divide 2224 by 34. Division is multiplication done backwards. Since we have not memorized the multiplication tables for various bases, we will need to construct a multiplication table to help us. 2  2 = 4 not a base 4 number 1 R 0 4 4

33. DIVISION Example Divide 2224 by 34. Division is multiplication done backwards. Since we have not memorized the multiplication tables for various bases, we will need to construct a multiplication table to help us. 2  3 = 6 not a base 4 number 1 R 2 4 6

34. DIVISION Example Divide 2224 by 34. Division is multiplication done backwards. Since we have not memorized the multiplication tables for various bases, we will need to construct a multiplication table to help us. 3  3 = 9 not a base 4 number 2 R 1 4 9

35. DIVISION Example Divide 2224 by 34. Since 3434=214, write 3 on top and 21 below. Since 3424=124, write 2 on top and 12 below. Look for 12, or closest number, in the table using a 3. 3 will not divide into 2. 3 will divide into 22. Look for closest number in the table using a 3. Bring down next digit. 3 24 2 34 2 2 24 Subtract. Subtract. – 2 1 2 1 1 2 – 1 2 0

36. DIVISION Example Divide 1124 by 24. Since 2434=124, write 3 on top and 12 below. Since 2424=104, write 2 on top and 10 below. Bring down next digit. Look for 12, or closest number, in the table using a 2. 2 will divide into 11. Look for closest number in the table using a 2. 2 will not divide into 1. 34 2 3 24 1 1 24 Subtract. – 1 0 1 0 Subtract. 1 2 – 1 2 0

37. DIVISION Example Divide 23215 by 45. First we construct a multiplication table. We only really need the row for multiplication with 4, since that is the divisor. 4  1 = 4 4  2 = 8 not a base 5 number 1 R 3 5 8

38. DIVISION Example Divide 23215 by 45. First we construct a multiplication table. We only really need the row for multiplication with 4, since that is the divisor. 4  3 = 12 not a base 5 number 2 R 2 5 12

39. DIVISION Example Divide 23215 by 45. First we construct a multiplication table. We only really need the row for multiplication with 4, since that is the divisor. 4  4 = 16 not a base 5 number 3 R 1 5 16

40. DIVISION Example Divide 23215 by 45. Since 45  35 = 225, write 3 on top and 22 below. 4 will not divide into 2. 4 will divide into 23. Look for closest number in the table using a 4. 3 Subtract. Bring down next digit. 45 2 3 2 15 – 2 2 2 2 1 2

41. DIVISION Example Divide 23215 by 45. Since 45  15 = 45, write 1 on top and 4 below. Subtract. HOWEVER, we must borrow! Bring down next digit. Look for 12, or closest number, in the table using a 4. 1 3 45 2 3 2 15 – 2 2 7 0 1 2 – 4 1 3

42. DIVISION If a division problem has a remainder, simple note it (do not find decimal result). Example Divide 23215 by 45. Since 45  45 = 315, write 4 on top and 31 below. Look for 31, or closest number, in the table using a 4. Subtract. 4 45 1 3 45 2 3 2 15 – 2 2 7 0 1 2 – 4 1 3 – 3 1 0

43. Homework From the Cow book 4.1 pg 168 # 1 – 49 EOO, 61, 62 4.2 pg175 # 1 – 37 EOO, 49 4.3 pg181 # 1 – 33 EOO, 35, 37 NOTE: EOO means “every other odd”