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Course Content: Quantitative Methods A
1.
2.
3.
4.
5.
6.
7.
Introduction to mathematical thinking (today)
Introduction to algebra
Linear and quadratic equations
Applications of equations
Linear and quadratic functions
Exponential functions and an introduction to
logarithmic functions
Solving equations involving logarithmic and
exponential functions
1
Course Content (continued)
8.
9.
10.
11.
12.
Compound interest
Annuities
Amortisation of loans
Introduction to differentiation
Applications of differentiation
2
Introduction to Mathematical Thinking
On completion of this module you should be
able to:
• Understand real numbers, integers, rational
and irrational numbers
• Perform some mathematical operations
with real numbers
• Work with fractions
• Understand exponents and radicals
• Work with percentages
• Use your calculator to perform some simple
tasks
3
Numbers
Integers
• The counting numbers: 1, 2, 3, 4, … are
positive integers.
• The set of all integers is: …-3, -2, -1, 0,
1, 2, 3
• There are an infinite number of these.
• Integers are useful for counting objects
(eg people in a city, months a sum of
money is invested, units of product
needed to maximise profit etc).
4
Rational numbers
• These can be written as a ratio of
two integers.
eg 1 , 8
2
5
8
Note that 5 and 1.6 are the same
rational number.
• p/q is a rational number where p and
q are integers but q cannot be zero.
5
• Integers can be written as rational
numbers:
5
5 ,
1
9
 9=
1
• Rational numbers with exact
decimals:
1
 0.5,
2
7
 0.4375
16
6
 Rational numbers with repeating
decimals: 1
1
3
 0.3333
,
9
 0.1111
 Can use a ‘dot’ to indicate the digits
repeat: 0.3333  0.3

5
 0.416666   0.416
12
5 11  0.454545
 
 0.45
7
Irrational numbers
• A number whose decimal equivalent
repeats without any known pattern in the
digits or which has no known terminating
point is called an irrational number.
• Examples are:
  3.141592654
e  2.718281828
2  1.414213562
8
Real numbers
• Real numbers are all the rational and
irrational numbers combined.
• This can be illustrated using a number
line:
– 3.5
–5
–4
–3
2 1.6
–2
–1
0
1
2
e 
3
4
5
• Real numbers are used when we measure
something (height, weight, width, time,
distance etc) but also for interest rates,
cost, revenue, price, profit, marginal cost,
marginal revenue etc
9
Operations with real numbers
Order of operations: remember BODMAS
Brackets
Of
Multiplication
Division
Addition
Subtraction
Some examples:
4  2 3  4  6  2
3   4  2  3  2  6
 4  2  5  2  2  5  2  7  2  14
10
Operations with real numbers
Multiplication:
3   4  2  is equivalent to 3  4  2 
Division:
42
4

2

3
or  4  2 3

 is equivalent to
3
Expanding brackets:
3  4-2  =3  4  -3 2  =3×4-3×2
=12-6 =6
or 3  4  2  3 2  6
11
Rules for multiplying and dividing
negative numbers
• If a negative numbers is multiplied
or divided by a negative number, then
the answer is positive.
• If a negative numbers is multiplied
or divided by a positive number, then
the answer is negative.
• If a positive numbers is multiplied or
divided by a negative number, then
the answer is negative.
12
Example
9  8  72
9  8  72
9  8  72
9  8  72
8
2
4
8
 2
4
8
 2
4
8
2
4
Remember:
0  anything = 0
0  anything = 0
BUT you can’t
divide by zero
(the result is
undefined).
13
Rounding
• When you have a nice round number,
write 0.5 not 0.50000 and 0.4375
not 0.437500.
• If rounding, the last required digit
will round up one value if the next
digit is five or greater, but will stay
the same if the next digit is four or
less.
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Example
1. Round 0.1263 to three decimal places.
2. Round 4.15525 to two decimal places.
Answer
• Next digit (4th one) is 3, which is four
or less, 3rd digit stays the same:
0.126.
• Next digit (3rd one) is 5, which is five
or more, the 2nd digit goes up by one:
4.16.
15
Rounding guidelines
• Do what the questions asks…
• If your answer is people, cats, ball
bearings, pencils etc round to the
nearest whole number.
• If your answer is an intermediate
step in working, don’t round at all!!
• If your answer is money then round
to 2 decimal places – never more!!
eg $4.05.
16
Rounding guidelines (continued)
• If your answer is an interest rate,
keep at least two decimal places but
four to five may be wise.
• Use your common sense… Big
numbers usually need fewer decimal
places whereas small numbers need
more.
• So 1,056,900.3 is better than
1,056,900.27384034 but 0.013046 is
better than 0.0!
17
Rounding guidelines (continued)
• It would be wise to include the
number of decimal places you’ve used
with your answer:
1,056,900.3 (to 1 decimal place)
0.0130 (to 4 decimal places)
18
Fractions
numerator
fraction 
denominator
5
9
5 is called the numerator
9 is called the denominator
19
Equivalent fractions
•
1
4
2 and 8 are equivalent fractions
since
1 1 4 4


2 24 8
4
• Multiplying numerator and  1
4
denominator by the same number
is equivalent to multiplying by 1.
20
• If we divide both numerator and
denominator by the same factor
(called cancelling) we get an
equivalent fraction:
2 12 1


or
4 22 2
2
4
1
2
1

2
• If the numerator and denominator
have no factors in common, the
fraction is said to be in its lowest
terms.
21
42
Simplify
by cancelling.
168
• Start by dividing numerator and
21
42
denominator by 2: 42
168

168
84
7
21
21

• Divide by 3:
84 84 28
1
7
7
1


• Divide by 7:
4
28 28
4
22
• Try easy factors first (2, 3 etc).
• Sometimes a common factor is obvious.
• At other times, trial and error is
necessary to cancel and simplify
fractions successfully.
• A proper fraction has numerator less
than denominator.
• An improper fraction has numerator
greater than denominator.
• A mixed number has an integer and a
fraction. e.g. 2½
23
Adding and subtracting fractions
• To add or subtract fractions, they
must be converted to equivalent
fractions with the same denominator.
5
1

• For example 12 12 can be
calculated immediately since both
denominators are 12.
5
1
5 1
6
1




12 12
12
12 2
24
Adding and subtracting fractions
When the denominators differ we can
either:
• multiply denominators together to
find the common denominator or
• find the lowest common
denominator.
We will look at an example of each.
25
Example
4 3
Find 5  4
Multiplying the denominators together
gives: 4 3 4  4 3  5
5

4


5 4 45
16 15 31



20 20 20
20 11
11
11


1
1
20 20
20
20
26
Example
4 3 1
 
Find
5 4 8
Multiplying the denominators together
gives 548=160, but the lowest
common denominator is actually 40
since 4, 5 and 8 all divide evenly into
40. 4  3  1  4  8  3  10  1  5
5
4
8
40
40
40
32  30  5 67
27


1
40
40
40
27
Example
7
8

Find
13 14
7
8
7  14  8  13


13 14
13  14
98  104
6


182
182
3

91
28
Multiplying and dividing fractions
• To multiply fractions, multiply
numerators and denominators
together.
8
2 4 2 4 2  4
 3   5   3  5  3  5  15
  
• To divide fractions, multiply by the
reciprocal.
2 4 2 5 10 5
   

3 5 3 4 12 6
29
Multiplying and dividing fractions
• ‘of’ is equivalent to multiplication:
3
3
3 7 21
1
of 7   7   
5
4
4
4 1
4
4
30
Exponents and Radicals
• A number (called the base) raised
to a positive whole number (the
exponent) means multiply the base
by itself the number of times given
in the exponent.
• So 34 means 3333=81.
• Any number raised to the power of
zero is equal to one:
0
3 1
0
8 1
2
5
8
6

0
1
31
Multiplying powers
Rules for multiplying powers:
1. If you are multiplying bases with
the same exponent, then multiply
the bases and put them to this
exponent. 2 2
2
2
2  5  2  5  10
2. If you are multiplying the same
base with different exponents then
add the exponents.
32  34  32 4  36
32
3. If you are raising a number to an
exponent and then to another
exponent, multiply the exponents.
3 
2
5
 325  310
33
Taking the root of a number
(fractional powers)
The square root of a number is the
reverse of squaring.
(2)2 
4 4 2
2
(2) 
(4.5)2 
 20.25  20.25  4.5
2
(4.5) 
34
Note: When taking the square
root, the answer is usually taken to
be a positive number, although, as
can be seen in the examples on the
previous slide, either a positive or
negative number squared results in
the same answer.
35
Roots with other bases:
3
8 means “what number multiplied
together three times gives 8?”
Since 2  2  2 =8, 3 8  2.
3
8 is read as “the cube root of 8”
or “the third root of 8”
36
4
4
81 is read “the fourth root of 81”
81  3 since 3  3  3  3 = 81
Sometimes you may need to use a
calculator:
6
805  3.049996 (to 6 decimal places)
37
Fractional powers
Another way of writing roots is using
fractional powers:
1
44
2
12
3
1
27  27
3
1
12
594  594
38
Negative exponents
A negative power can be rewritten as
one over the same number with a
positive power:
2
3
1
1
 2 
9
3
1
5

2
 32
5
2
6
4
1
1
 6 
4096
4
1
1
25
Note that 5 
 1
 25
1
1
2
25
39
Examples of negative fractional
exponents:
9
1
64
2
1

6
1
1
9

2

1
1
64
1
1

9 3
1
6
1
 6

64 2
1
13
3

27

27  3
1 3
27
40
Percentages
When we speak of 15% of a number,
we mean
15
100
 number or 0.15  number
Example
Calculate 47% of 3092.
47
47%  3092 
 3092  or 0.47  3092 
100
47  3092  145324



100
100
 1453.24
41
Example
An account’s salary of $75 300 is going
to be increased by 7%. What are the
increase in salary and the new salary?
7
75300

100
1
7 75300 


 5271
100
7%  75300 
So the increase in salary is $5271.
The new salary is
$75 300 + $5271 = $80 571.
42
Using the calculator
You will require a scientific calculator for the
remainder of the course. The keys which
shall be required most frequently are:
Mathematical functions
add   
subtract -
multiply 
divide  
change of sign    squares  x 2 
43
Mathematical functions (continued)
x  or  x1 2 
powers  x y 
y

or x 
logs to base 10
square roots
1

roots  x y

log
antilog to base 10 10x 
logs to base e ln
antilog to base e e x 
44
The memory keys
Clear memory
Put into memory
Add to the contents of memory
Subtract from the contents of memory
45
IMPORTANT
Activity 1-1: using your
calculator
(ask for help in tutorials
if you need to).
46
Accuracy and rounding
9
 13 
9

(1.8571429)
 262.7893802
 7 


9
 13 
9

(1.85)
 253.8315230
 7 


error  8.9578572 when truncated to 2D
9
 13 
9

(1.86)
 266.4504745
 7 


error  3.6610943 when rounded to 2D
Rounding intermediate results leads
to loss of accuracy in final result.
47