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1. Introduction to mathematical thinking (today)

2. Introduction to algebra

3. Linear and quadratic equations

4. Applications of equations

5. Linear and quadratic functions

6. Exponential functions and an introduction to logarithmic functions

7. Solving equations involving logarithmic and exponential functions

Course Content: Quantitative Methods A

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8. Compound interest 9. Annuities10. Amortisation of loans11. Introduction to differentiation12. Applications of differentiation

Course Content (continued)

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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

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• 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).

Numbers

Integers

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• These can be written as a ratio of two integers.

eg

Note that 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.

Rational numbers

1 8,

2 5

85

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• Integers can be written as rational numbers:

• Rational numbers with exact decimals:

5 95 , 9=

1 1

1 7

0.5, 0.43752 16

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Rational numbers with repeating decimals:

Can use a ‘dot’ to indicate the digits repeat:

1 1

0.3333 , 0.11113 9

0.3333 0.3

5

0.416666 0.41612

5 11 0.454545 0.45

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• 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:

Irrational numbers

3.141592654

2.718281828e

2 1.414213562

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• Real numbers are all the rational and irrational numbers combined.• This can be illustrated using a number line:

• 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

Real numbers

1 2 3 4 5 0

0

1

1

2

2

3

3

4

4

5

5

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 3.5 2 1.6 e

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Order of operations: remember BOMDASBracketsOfMultiplicationDivisionAdditionSubtraction

Some examples:

Operations with real numbers

4 2 3 4 6 2 3 4 2 3 2 6

4 2 5 2 2 5 2 7 2 14

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Multiplication:is equivalent to

Division:is equivalent to

Expanding brackets:

Operations with real numbers

3 4 2 3 4 2

4 2 3

4 2 or 4 2 3

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3 4- 2 =3 4 - 3 2 =3×4- 3×2

=12- 6=6

or 3 4 2 3 2 6

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• 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.

Rules for multiplying and dividing negative numbers

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Example

9 8 72

9 8 72

9 8 72

9 8 72

8

24

82

4

8

24

8

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Remember:

0 anything = 0

0 anything = 0

BUT you can’t divide by zero (the result is undefined).

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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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1. Round 0.1263 to three decimal places.2. Round 4.15525 to two decimal places.

Example

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.

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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.

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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!

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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)

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numerator

fraction denominator

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5 is called the numerator

9 is called the denominator

Fractions

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Equivalent fractions

• and are equivalent fractions since

• Multiplying numerator and denominator by the same number is equivalent to multiplying by 1.

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48

1 1 4 42 2 4 8

4

14

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• If we divide both numerator and denominator by the same factor (called cancelling) we get an equivalent fraction:

• If the numerator and denominator have no factors in common, the fraction is said to be in its lowest terms.

2 1 2 1 2

or 4 2 2 2

1

42

12

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Simplify by cancelling.

• Start by dividing numerator and denominator by 2:

• Divide by 3:

• Divide by 7:

42168

42 42

168

21

16884

21 2184

7

8428

7 7

28

1

284

14

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• 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

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• To add or subtract fractions, they must be converted to equivalent fractions with the same denominator.

• For example can be calculated immediately since both denominators are 12.

Adding and subtracting fractions

5 1

12 12

5 1 5 1 6 112 12 12 12 2

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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.

Adding and subtracting fractions

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Find

Multiplying the denominators together gives:

Example

4 35 4

4 3 4 4 3 55 4 5 4 4 5

16 15 3120 20 2020 11 11 11

1 120 20 20 20

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Find

Multiplying the denominators together gives 548=160, but the lowest common denominator is actually 40 since 4, 5 and 8 all divide evenly into 40.

Example

4 3 15 4 8

4 3 1 4 8 3 10 1 55 4 8 40 40 40

32 30 5 67 271

40 40 40

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Find

Example

7 8

13 14

7 8 7 14 8 1313 14 13 14

98 104 6182 182391

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• To multiply fractions, multiply numerators and denominators together.

• To divide fractions, multiply by the reciprocal.

Multiplying and dividing fractions

2 4 2 4 2 4 83 5 3 5 3 5 15

2 4 2 5 10 53 5 3 4 12 6

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• ‘of’ is equivalent to multiplication:

Multiplying and dividing fractions

3 3 3 7 21 1

of 7 7 54 4 4 1 4 4

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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 3333=81.• Any number raised to the power of

zero is equal to one:

03 1 08 1 05 82 6 1

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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. If you are multiplying the same base with different exponents then add the exponents.

Multiplying powers

22 2 22 5 2 5 10

2 4 2 4 63 3 3 3

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3. If you are raising a number to an exponent and then to another exponent, multiply the exponents.

52 2 5 103 3 3

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The square root of a number is the reverse of squaring.

Taking the root of a number (fractional powers)

2

2

(2)4 4 2

( 2)

2

2

(4.5)20.25 20.25 4.5

( 4.5)

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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.

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Roots with other bases:

3 8 means “what number multiplied together three times gives 8?”

3 8 2.

3 8 is read as “the cube root of 8” or “the third root of 8”

Since 2 2 2 =8,

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is read “the fourth root of 81”4 81

4 81 3 since 3 3 3 3 = 81

Sometimes you may need to use a calculator:

6 805 3.049996 (to 6 decimal places)

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Fractional powers

1

24 4 1

3 327 27

1

12 12594 594

Another way of writing roots is using fractional powers:

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A negative power can be rewritten as one over the same number with a positive power:

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1 13

93 6

6

1 14

40964

Negative exponents

55

12 32

2 5

55

5

1 1 2Note that 1 2

1 122

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Examples of negative fractional exponents:

12

12

1 1 19

399

16

1 66

1 1 164

26464

1 3 31 3

127 27 3

27

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Percentages

When we speak of 15% of a number, we mean 15

number or 0.15 number100

4747% 3092 3092 or 0.47 3092

10047 3092 145324

100 1001453.24

ExampleCalculate 47% of 3092.

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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 75,3007% 75,300

100 17 75,300

5271100

So the increase in salary is $5271.The new salary is

$75,300 + $5271 = $80,571.

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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 2squares x

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Mathematical functions (continued)

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roots or yyx x

1 2square roots or x x

powers yx

logs to base 10 log

antilog to base 10 10x

logs to base lne

antilog to base xe e

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The memory keys

Clear memory

Put into memory

Add to the contents of memory

Subtract from the contents of memory

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IMPORTANT Activity 1-1: using your

calculator(ask for help in tutorials

if you need to).

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9913

(1.8571429)7

9913

(1.85)7

253.8315230

error 8 9578572 when truncated to 2D.9

913(1.86)

7

266.4504745

error 3.6610943 when rounded to 2D

Accuracy and rounding

Rounding intermediate results leads to loss of accuracy in final result.

262.7893802