Quantitative Analysis

1. Quantitative Analysis Half of Unit S101 Computing and Quantitative Analysis

2. Lecture 1 The Basics (What we may have forgotten from GCSE Maths !)

3. One initial point ... I will send you a text message about it !

4. Plz swch off ur mbl fone in lctres.

5. Plz swch off ur mbl fone in lctres. (Sorry to be so old-fashioned but it is very distracting to other participants if your phone goes off).

8. A Calculator If you do not already have one, I advise buying one .. Preferably one with a memory which can do exponentials and logarithms. The cheapest one with those features that I have seen recently was in Woolworths for about £4.

9. The assessment portfolio for this half of the unit will be based on:End Test (40 %); Assignment (60 %)

10. Helpful Books Mik Wisniewski, “Quantitative Methods for Decision-Makers” (Pitman Publishing) and the “Foundation” version of the same book.

11. The “Foundation” version covers most of the work in the Quantitative Analysis part of this unit and starts from “square one”; the “main” version covers more work than needed for this unit but assumes more prior knowledge of the basics.

12. Simple Mathematical ideas How would we work out 3  4 + 5  3 - 2  6 ? (other than “With a calculator” !) First question -- what does it mean ? What order should we do the operations in ?

13. The clue -- B O D M A S. This word does not (disappointingly) mean the birthday of the patron saint of Mathematics (St.Bod).

14. It actually means: Do any BRACKETS first -- that is the “B”.

15. Then any “of” -- for example, “25% of 60” (I think this one was only inserted so it fitted the easily-remembered BODMAS)

16. Then DIVISIONS .. then MULTIPLICATIONS .. then ADDITIONS .. then SUBTRACTIONS.

17. In fact the additions do NOT take precedence over the subtractions -- they are done together in order from left to right -- but BODMAS is still a good, easy-to-remember mnemonic.

18. In our example, there are no brackets and no “of”, so we start with the division. 5  3 = or .

19. In our example, there are no brackets and no “of”, so we start with the division. 5  3 = 1.66666... or 1.667 to three decimal places.

20. So our calculation is now3  4 + 1.667 - 2  6

21. Now we do the multiplications.3 x 4 = 122 x 6 = 12

22. Now the additions and subtractions ... Giving12 + 1.667 - 12 = 1.667

23. Where brackets come in handy We are buying ten computer disks at £0.50 each, a computer printer ribbon at £5.00, and a packet of printer paper at £6.00 ... all plus 17.5 % V.A.T. How do we work out the total we will have to pay ?

24. Is it (i) (0.50 x 10 + 5.00 + 6.00) x 1.175 or (ii) 0.50 x 10 + 5.00 + 6.00 x 1.175 ? (incidentally, where did the 1.175 come from?)

25. Based on The Words Ten disks at £0.50 = £5.00 Plus £5.00 for the ribbon and £6.00 for the paper all comes to £16.00

26. But we must not ignore the VAT inspector or we will be fined ! The VAT is 17.5% of £16.00 = £2.80. So we will have to pay £16.00 + £2.80 = £18.80.

27. Is it (0.50 x 10 + 5.00 + 6.00) x 1.175 ? My calculator makes the answer £18.80 which agrees with our other one.

28. Is it 0.50 x 10 + 5.00 + 6.00 x 1.175 ? Now the “Texas Neverite” gets £17.05, so this sum does not seem to be right. So the brackets are important !

29. We may argue that, if we are doing the sum, WE know what we mean even if we do not use the brackets. But what if we are “telling” a computer what the sum is ?

30. A relevant quote from M. Pittman, one of our researchers: “Computers do not do what you WANT them to do, only what you TELL them to do !” So we do need the brackets.

31. Powers ... 'squared', 'cubed' , etc. For example, 52 = 5 x 5 = 25; 53 = 5 x 5 x 5 = 125; etc. Anything2 = “squared” Anything3 = “cubed”

32. If the power is other than 2 or 3, we say "5 to the power 4" or "5 to the fourth" for 54. Use mental arithmetic or your calculator to work out 32, 23, 64, 33.5. (I think you will need the calculator for the last one).

33. What does 40.5 mean ? What is its value ?

34. What does 40.5 mean ? What is its value ? The square root of 4

35. What does 40.5 mean ? What is its value ? The square root of 4 Its value is 2

36. An important financial application of powers is interest (of the bank account variety !). This applies both to deposits and loans (including mortgages). Suppose we put £50 in a bank deposit account which pays 4 % per annum interest and pays it once a year .. and we do not withdraw any money.

37. After one year, the bank adds 4 % of £50 = £2.00, giving £52.00 in the account.

38. After one year, the bank adds 4 % of £50 = £2.00, giving £52.00 in the account. After a further year, the bank adds 4 % of £52 (the new PRINCIPAL) = £2.08, giving a total of £54.08 .. and so on.

39. After one year, the bank adds 4 % of £50 = £2.00, giving £52.00 in the account. After a further year, the bank adds 4 % of £52 (the new PRINCIPAL) = £2.08, giving a total of £54.08 .. and so on. We can alternatively look upon the bank as multiplying the amount in the account by 1.04 each year.

40. Confirm that the same amounts will result after one and two years.

41. Confirm that the same amounts will result after one and two years. • If we are looking several years ahead, this means of calculating the final amount is rather laborious. Powers will be easier, as the following will apply:

42. One year: Multiply the original amount by 1.04 Two years: Multiply by 1.042 (check this one -- is it right ?) Three years: Multiply by 1.043 and so on.

43. How long will it take before our money is doubled? (Calculator permitted !)

44. Banks often pay interest more often than once a year (they charge it usually at daily intervals if we are looking after their money instead of vice versa).

45. If, for example, the interest is paid monthly but the annual rate is still 4 %, it means that they pay 4 / 12 % = 0.3333 % each month. Is this arrangement better or worse than the interest being paid only once a year at 4% ?

46. (Better, but not much -- we have £52.04 after a year, and so on).

47. (Better, but not much -- we have £52.04 after a year, and so on). (Incidentally, it will still be £52.04 to the nearest penny if the interest is paid daily).

48. (Better, but not much -- we have £52.04 after a year, and so on). (Incidentally, it will still be £52.04 to the nearest penny if the interest is paid daily). A “Think for the bus !”

49. Logarithms These are the opposite of powers. The logarithm of a number to a particular base (another number) is the power to which the base number has to be raised to obtain our number.

50. Logarithms These are the opposite of powers. The logarithm of a number to a particular base (another number) is the power to which the base number has to be raised to obtain our number. For example, we have to raise 10 to the power 2 to obtain 100, so the logarithm of 100 to the base 10 is 2.