MATHEMATICS-1Lecturer#1

Module Title: Mathematics 1 Module Type: Standard module Academic Year: 2010/11, Module Code: EM-0001D Module Occurrence: A, Module Credit: 20 Teaching Period: Semester 1 Level: Foundation

  

AIMS

Reinforcement of basic numeracy and algebraic manipulation.

A combination of lectures, seminars and tutorials is used to explain concepts and apply them through exercises

Study Hours

Lectures: 48.00 Directed Study: 138.00

  Seminars/Tutorials: 32.00 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

NUMBERS

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

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

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 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:

NUMBER REPRESENTATIONThe 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 (÷)

POWERS

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.

PLACE VALUE 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

REAL NUMBERS

Real Numbers are any number on a number line. It is the combined set of the rational and irrational numbers.

RATIONAL NUMBERS

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

Example: 3/5, 1/3, -4/3, -25

IRRATIONAL NUMBERSIrrational Numbers are numbers that

cannot be written as a ratio of two integers. The decimal extensions of irrational numbers never terminate and never repeat.

Example: – 3.45455455545555…..

RATIO/QUOTIENT

A comparison of two numbers by division. The ratio of 2 to 3 can be stated as 2 out of 3, 2 to 3, 2:3 or 2/3.

WHOLE NUMBERS

Whole numbers are 0 and all positive numbers such as 1, 2, 3, 4 ………

INTEGERS

Any positive or negative whole numbers including zero. Integers are not decimal numbers are fractions.

. . .-3, -2, -1, 0, 1, 2, 3, …

The Real Number System

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Real Numbers

Rational Numbers Irrational Numbers

3

1/2-2

15%

2/3

1.456

-0.7

0

3 2

-5 2

34

The Real Number System

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Real Numbers

Rational Numbers Irrational Numbers

31/2 -2

15%

2/3

1.456- 0.7

0

3 2

-5 2

34

Integers

The Real Number System

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Real Numbers

Rational Numbers Irrational Numbers

31/2

-2

15%

2/3

1.456- 0.7

0

3 2

-5 2

34

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

2

12 2

every point on the number line represents one real number.

Properties of Real Numbers

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

Properties of Real Numbers

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:

. . . 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!

Properties of Real 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

The square root of any whole number is either whole or irrational.

x

0 1 32 4 5 6 7 98 10

For example, is a whole number, but , since it lies between 5 and 6, must be irrational.

36 30

36

. . . 477225575.5

25

30

Common Misconception:

Do not assume that a number is irrational just because it is expressed using the square root symbol. Find its value first!

Study Tip:

KNOW and recognize (at least) these numbers,

169644936251694 14412110081

Properties of Real Numbers

Any ?