1 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

2 8. Compound interest 9. Annuities 10. Amortisation of loans 11. Introduction to differentiation 12. Applications of differentiation Course Content (continued)

3 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

4 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

5 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

6 Integers can be written as rational numbers: Rational numbers with exact decimals:

7 · Rational numbers with repeating decimals: · Can use a ‘dot’ to indicate the digits repeat:

8 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

9 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

10 Order of operations: remember BOMDAS Brackets OfOf Multiplication Division Addition Subtraction Some examples: Operations with real numbers

11 Multiplication: is equivalent to Division: is equivalent to Expanding brackets: Operations with real numbers

12 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

13 Example Remember: 0  anything = 0 0  anything = 0 BUT you can’t divide by zero (the result is undefined).

14 Rounding When you have a nice round number, write 0.5 not and not 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.

15 1.Round to three decimal places. 2.Round to two decimal places. Example Answer Next digit (4 th one) is 3, which is four or less, 3 rd digit stays the same: Next digit (3 rd one) is 5, which is five or more, the 2 nd digit goes up by one: 4.16.

16 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.

17 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, but is better than 0.0!

18 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) (to 4 decimal places)

19 5 is called the numerator 9 is called the denominator Fractions

20 Equivalent fractions and are equivalent fractions since Multiplying numerator and denominator by the same number is equivalent to multiplying by 1.

21 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.

22 Simplify by cancelling. Start by dividing numerator and denominator by 2: Divide by 3: Divide by 7:

23 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

24 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

25 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

26 Find Multiplying the denominators together gives: Example

27 Find 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. Example

28 Find Example

29 To multiply fractions, multiply numerators and denominators together. To divide fractions, multiply by the reciprocal. Multiplying and dividing fractions

30 ‘of’ is equivalent to multiplication: Multiplying and dividing fractions

31 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 3 4 means 3333=81. Any number raised to the power of zero is equal to one:

32 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

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

34 The square root of a number is the reverse of squaring. Taking the root of a number (fractional powers)

35 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.

36 Roots with other bases: means “what number multiplied together three times gives 8?” is read as “the cube root of 8” or “the third root of 8” Since 2  2  2 =8,

37 is read “the fourth root of 81” since 3  3  3  3 = 81 Sometimes you may need to use a calculator: (to 6 decimal places)

38 Fractional powers Another way of writing roots is using fractional powers:

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

40 Examples of negative fractional exponents:

41 Percentages When we speak of 15% of a number, we mean Example Calculate 47% of 3092.

42 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? So the increase in salary is $5271. The new salary is $75,300 + $5271 = $80,571.

43 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

44 Mathematical functions (continued)

45 The memory keys Clear memory Put into memory Add to the contents of memory Subtract from the contents of memory

46 IMPORTANT Activity 1-1: using your calculator (ask for help in tutorials if you need to).

47 Accuracy and rounding Rounding intermediate results leads to loss of accuracy in final result.