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MATHEMATICS-1 Lecturer#1

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Page 1: Introduction

MATHEMATICS-1Lecturer#1

Page 2: Introduction

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

  

Page 3: Introduction

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

Page 4: Introduction

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

Page 5: Introduction

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

Page 6: Introduction

NUMBERS

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

Page 7: Introduction

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

http://darvish.wordpress.com/2008/03/16/the-beauty-of-mathematics-and-the-love-of-god/
http://darvish.wordpress.com/2008/03/16/the-beauty-of-mathematics-and-the-love-of-god/
Page 8: Introduction

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

Page 9: Introduction

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

Page 10: Introduction

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?

Page 11: Introduction

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:

Page 12: Introduction

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 (÷)

Page 13: Introduction

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.

Page 14: Introduction

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

Page 15: Introduction

REAL NUMBERS

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

Page 16: Introduction

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

Page 17: Introduction

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…..

Page 18: Introduction

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.

Page 19: Introduction

WHOLE NUMBERS

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

Page 20: Introduction

INTEGERS

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

. . .-3, -2, -1, 0, 1, 2, 3, …

Page 21: Introduction

The Real Number System

04/12/2023 jwaid 21

Real Numbers

Rational Numbers Irrational Numbers

3

1/2-2

15%

2/3

1.456

-0.7

0

3 2

-5 2

34

Page 22: Introduction

The Real Number System

04/12/2023 jwaid 22

Real Numbers

Rational Numbers Irrational Numbers

31/2 -2

15%

2/3

1.456- 0.7

0

3 2

-5 2

34

Integers

Page 23: Introduction

The Real Number System

04/12/2023 jwaid 23

Real Numbers

Rational Numbers Irrational Numbers

31/2

-2

15%

2/3

1.456- 0.7

0

3 2

-5 2

34

Integers

Whole

Page 24: Introduction

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

Page 25: Introduction

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

Page 26: Introduction

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

http://www.robertfant.com/Java/PI/
Page 27: Introduction

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

Page 28: Introduction

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

Page 29: Introduction

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