Lecturer#1

Module Tit le: Mathematics 1 Module Type: Standard module Academic Year: 2009/10, Module Code: EM-0001D Module Occurrence: A, Module Credit: 20 Teaching Period: Semester 1 Level: Foundation

  

Reinforcement of basic numeracy and algebraic manipulation.

Learning Teaching & Assessment Strategy: A combination of lectures, seminars and tutorials is used to explain concepts and apply them through exercises

Study Hours

Lectures: 36.00 Directed Study: 138.00 Seminars/Tutorials: 24.00 Formal Exams: 2.00 Laboratory/Practical: 0.00 Other: 0.00 Total: 200

1 Assessment Type Duration (hours) Percentage

Classroom test - 25%

Description

2 classroom tests each lasting 1 hour

2 Assessment Type Duration (hours) Percentage

Examination - open book or seen paper 2 50%

Description

Examination

3 Assessment Type Duration (hours) Percentage

Coursework - 25%

Description

2 assignments consisting of Maths questions taking approx 2 hours to answer per assignment

900 Assessment Type Duration (hours) Percentage

Examination - open book or seen paper 2 100%

Description

Supplementary examination

Number is a mathematical concept used to describe and access quantity.

Here is an interesting and lovely way to look at the beauty of mathematics, and of God, the sum of all wonders.

The Beauty of Mathematics

Wonderful World

1 x 8 + 1 = 912 x 8 + 2 = 98

123 x 8 + 3 = 9871234 x 8 + 4 = 9876

12345 x 8 + 5 = 987 65123456 x 8 + 6 = 987654

1234567 x 8 + 7 = 987654312345678 x 8 + 8 = 98765432

123456789 x 8 + 9 = 987654321

9 x 9 + 7 = 8898 x 9 + 6 = 888

987 x 9 + 5 = 88889876 x 9 + 4 = 88888

98765 x 9 + 3 = 888888987654 x 9 + 2 = 8888888

9876543 x 9 + 1 = 8888888898765432 x 9 + 0 = 888888888

Brilliant, isn’t it?

1 x 9 + 2 = 1112 x 9 + 3 = 111

123 x 9 + 4 = 11111234 x 9 + 5 = 11111

12345 x 9 + 6 = 111111123456 x 9 + 7 = 1111111

1234567 x 9 + 8 = 1111111112345678 x 9 + 9 = 111111111

123456789 x 9 +10= 1111111111

1 x 1 = 111 x 11 = 121

111 x 111 = 123211111 x 1111 = 1234321

11111 x 11111 = 123454321111111 x 111111 = 12345654321

1111111 x 1111111 = 123456765432111111111 x 11111111 = 123456787654321

111111111 x 111111111 = 12345678987654321

And look at this symmetry:

The number system that we use today has taken thousand of years to develop. The Arabic system that we commonly use consists of exactly ten symbols:

0 1 2 3 4 5 6 7 8 9Each symbol is called a digit. Our system involves counting in tens. This type of system is called denary system, and 10 is called the base of the system.It is possible to use a number other than 10. For example, computer systems use base 2( the binary system)

Numbers are combined together, using the four arithmetic operations. addition (+), subtraction (-), multiplication (×) and division (÷)

Repeated multiplication by the same number is known as raising to a power. For example 8×8×8×8×8 is written 85 (8 to the power 5) Check your calculator for xy.

Once a number contains more then one digits, the idea of place value is used to tell us its worth. In number 2850 and 285, the 8 stands for something different. In 285, 8 stands for 8 ‘tens’. In 2850, the 8 stands for 8 ‘hundreds’. The following table show the names given to the first seven places.

The number shown is 4087026, which is 4 million eighty-seven thousands and twenty-six.

Millions Hundreds thousands

Ten thousands

Thousands Hundreds Tens units

4 0 8 7 0 2 6

The Arabic numbering system uses ten symbols and is a base ten numbering system. In any numbering system the base is raised to consecutive powers to determine positional (place) values.

103 102 101 100 basepowers

1000 100 10 1 place values

2 3 12 8 5

2 8 5 0

Real Numbers

Real numbers consist of all the rational and irrational numbers.

The real number system has many subsets: Natural Numbers Whole Numbers Integers

Natural Numbers Natural numbers are the set of counting

numbers.

{1, 2, 3,…}

Whole Numbers

Whole numbers are the set of numbers that include 0 plus the set of natural numbers.

{0, 1, 2, 3, 4, 5,…}

Integers Integers are the set of whole numbers and

their opposites.

{…,-3, -2, -1, 0, 1, 2, 3,…}

Rational Numbers

Rational numbers are any numbers that can be expressed in the form of , where a and b are integers, and b ≠ 0.

Rational Numbers are numbers that can be expressed as a fraction or ratio of two integers

They can always be expressed by using terminating decimals or repeating decimals.

b

a

Terminating Decimals

Terminating decimals are decimals that contain a finite number of digits.

Examples: 36.8 0.125 4.5

Repeating Decimals Repeating decimals are decimals that contain a infinite

number of digits. Examples:

0.333… 7.689689…

FYI…The line above the decimals indicate that number

repeats.

9.1

Irrational Numbers Irrational numbers are any numbers that cannot be

expressed as .

They are expressed as non-terminating, non-repeating decimals; decimals that go on forever without repeating a pattern.

Examples of irrational numbers: 0.34334333433334… 45.86745893… (pi)

b

a

π2

Other Vocabulary Associated with the Real Number System …(ellipsis)—continues without end { } (set)—a collection of objects or

numbers. Sets are notated by using braces { }.

Finite—having bounds; limited Infinite—having no boundaries or limits Venn diagram—a diagram consisting of

circles or squares to show relationships of a set of data.

Example Classify all the following numbers as natural, whole, integer,

rational, or irrational. List all that apply. 117 0 -12.64039… -½ 6.36 -3

π

To show how these number are classified, use the Venn diagram. Place the number where it belongs on the Venn diagram.

π9

4

2

1−

Rational Numbers

Integers

Whole Numbers

NaturalNumbers

Irrational Numbers

-12.64039…

117

0

6.369

4

-3

Solution Now that all the numbers are placed where they belong in

the Venn diagram, you can classify each number: 117 is a natural number, a whole number, an integer,

and a rational number. is a rational number. 0 is a whole number, an integer, and a rational number. -12.64039… is an irrational number. -3 is an integer and a rational number. 6.36 is a rational number. is an irrational number. is a rational number.

π9

4

2

1−

10/05/10 jwaid 29

Real Numbers

Rational Numbers Irrational Numbers

3

1/2-2

15%

2/3

1.456

-0.7

0

√3 2π

−√5 2

3π4

10/05/10 jwaid 30

Real Numbers

Rational Numbers Irrational Numbers

31/2 -2

15%

2/3

1.456- 0.7

0

√3 2π

−√5 2

3π4

Integers

10/05/10 jwaid 31

Real Numbers

Rational Numbers Irrational Numbers

31/2

-2

15%

2/3

1.456- 0.7

0

√3 2π

−√5 2

3π4

Integers

Whole

All of the numbers that you use in everyday life are real numbers.

Each real number corresponds to exactly one point on the number line, and

x

0 1 2 3 4 5-5 -4 -2 -1-3

π2

12− 2

every point on the number line represents one real number.

Real numbers can be classified as either _______ or ________.rational irrational

Rational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b

a

The decimal form of a rational number is either a terminating or repeating decimal.

Examples: ratio form decimal form

9 0.3

83

375.0

73

428571.0

or . . . 714285714285714285.0

zero

Real numbers can be classified a either _______ or ________.rational irrational

A real number that is not rational is irrational.

The decimal form of an irrational number neither __________ nor ________.terminates repeats

Examples:

≈ π . . . 141592654.3 More Digits of PI?

≈ e . . . 718281828.2

2 3 5 7 11 13

Do you notice a pattern within this group of numbers?

They’re all PRIME numbers!

Example 1

Classify each number as being real, rational, irrational, integer, whole, and/or natural numbers. Pick all that apply.

7

12 0

10.333

6

π−

−

Decimal places (d.p.) refer to the digits to right of the decimal point. For example ‘23.845’ have 3 decimal places.

The numbers we get from a computation often contains worthless digits that must be thrown away.

Whenever we do this we must round our answer to the required decimal places.

Rule 1

If the first digit to be discarded is greater than 5, or a 5 followed by a non zero in any of the decimal placed to the right, then round up by the last digit retained.

Number Rounded to Three Decimal Places

4.3654 4.365

4.3656 4.366

4.365501 4.366

1.764999 1.765

1.927499 1.927

Rule 2

If the first digit to be discarded is less than 5, round down by the last digit retained.

Write -6.0439 to 2 d.p. Here 3 is first discarded digit and 4 is the last retained digit. Since 3<5 so we round down and hence result will be -6.04.

Rule 3 (Important)

If the first digit to be discarded is equal to 5, always round to the nearest even number.

Numbers Rounded to 2 d.p.

4.365 4.36

4.355 4.36

7.76500 7.76

7.75500 7.76

Do not Forget to…

When you were asked to write a number to for example to 3 d.p., consider the first 4 d.p. places during the computation and write the end result to 3 decimal places.

For example, here is a series of addition and result is required to 2.d.p

85.18851.18911.294.15

94.15945.15375.857.7

57.7572.7894.3678.3

86.18858.18911.2375.8894.3678.3

≈=+≈=+≈=+

≈=+++

Any ?