Quantitative Analysis

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Quantitative Analysis

• Half of Unit S101 Computing and Quantitative
  Analysis

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Lecture 1

• The Basics
• (What we may have forgotten from GCSE Maths !)

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One initial point ...

• I will send you a text message about it !

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Plz swch off ur mbl fone in lctres.
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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).

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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.

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The assessment portfolio for this half of the
unit will be based onEnd Test (40 )
Assignment (60 )
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Helpful Books

• Mik Wisniewski, Quantitative Methods for
  Decision-Makers (Pitman Publishing)
• and the Foundation version of the same book.

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• 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.

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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
  ?

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The clue -- B O D M A S.

• This word does not (disappointingly) mean the
  birthday of the patron saint of Mathematics
  (St.Bod).

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It actually means

• Do any BRACKETS first -- that is the B.

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Then any of -- for example, 25 of 60

• (I think this one was only inserted so it fitted
  the easily-remembered BODMAS)

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Then DIVISIONS .. then MULTIPLICATIONS .. then
ADDITIONS .. then SUBTRACTIONS.
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• 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.

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• In our example, there are no brackets and no
  of, so we start with the division.
• 5 ? 3 or .

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• 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.

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So our calculation is now3 ? 4 1.667 - 2 ? 6
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Now we do the multiplications.3 x 4 122 x 6
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Now the additions and subtractions ... Giving12
1.667 - 12 1.667
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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
  ?

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• 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?)

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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

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• 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.

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• 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.

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• 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 !

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• 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 ?

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• 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.

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Powers ... 'squared', 'cubed' , etc.

• For example, 52 5 x 5 25
• 53 5 x 5 x 5 125 etc.
• Anything2 squared
• Anything3 cubed

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• 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).

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• What does 40.5 mean ?
• What is its value ?

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

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

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• 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.

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• After one year, the bank adds 4 of 50 2.00,
  giving 52.00 in the account.

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• 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.

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• 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.

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

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• 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

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• 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.

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

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• 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 ?

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

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• (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).

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• (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 !

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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.

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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.

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• In the days when calculators were mechanical,
  heavy, bulky and expensive, logarithms were a
  very useful method of doing multiplication and
  division because of the following rules to do
  with powers of numbers (sorry for introducing a
  little algebra notation early).

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• We will refer to a number y.
• ya x yb yab, so we add the powers when we
  multiply.

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• We will refer to a number y.
• ya x yb yab, so we add the powers when we
  multiply.
• (ya)/ yb) y(a-b), so when we divide, we
  subtract the power of the quantity by which we
  are dividing from that of the one we are dividing
  into.

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• So if we want to multiply m ya by n yb, we
  could look up (in "log tables") the values of a
  and b, add them together, and then look in
  "anti-log tables" to find the answer.

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• Small cheap electronic calculators have made this
  performance unnecessary, but logarithms still
  have their uses (especially for reducing many
  complicated-shaped graphs to straight-line ones
  which are easier to handle -- we will deal with
  this point in due course).

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• In terms of BODMAS, powers and logarithms take a
  priority between brackets and "of".

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We will do some supporting practice work (both
calculator and computer based) in your next
tutorial.