COLLEGE ALGEBRAMT 221

ISHRM, 1ST SEMESTER, 2014

Jeaneth BalabaLecturer/Instructor

Good morning! 1.Self-introduction2. ISHRM Vision and Mission3.Classroom policy4.Grading system5.Other clarifications6.Trivia information7.Lesson introduction8.Lesson proper

CLASSROOM POLICY

• Attendance & Punctuality• Classroom Behavior & Language• Grading System

• Attendance 10%• Quiz 15%• Activity 15%• Exams 60% [(100%/100%/30%/70%)]

• Overall Rating: Prelim Rating 100%/ Mid-Term Rating 100%/ Pre-final Rating 30% & Final Period Rating 70%

• Personal Profile (Index card) – Quiz #1

INDEX CARDName: 1x1 Photo

Subject and Section:

Course and Year:

Age/Birthday:

Home Location:

E-mail address:

Course expectation: (1-2 sentences)

*Note: Leave the back portion of the index card blank*

Making the connection…

Brainstorm on what algebra and algebraic thinking is…

DISCUSSION

• Review: What is algebra?

• How useful is algebra? What are its uses?

• What are its uses in the culinary arts practice? Hotel and restaurant management? Business administration?

Algebra – what is it?

• Fundamental language of mathematics

– Creates a mathematical model of a situation– Provides mathematical structure to use the

model– Links numerical and graphical representation– Condenses large amounts of data into efficient

statements

Algebraic Habits of Mind

• Analyze change

• Understand functions

• Variable (understand the idea and the variety of uses)

• Interpret, create and move fluently between multiple representations for data

Trivia: What is Algebra ?

Mathematics dictionary: “The branch of mathematics that deals with the general properties of numbers and the generalizations arising therefrom.

Make sense ?!?

Eulers' Complete Introduction to Algebra (1767): “Algebra has been defined as the science which teaches how to determine unknown quantities by means of those that are known.”

Trivia: The History of Algebra* Algebra has a long history even before Leonhard Euler (1707 – 1783) and Colin Maclaurin (1698-1746).

* The use of algebra predates the use of symbols which we often use for manipulation.

* Oldest text: by Muhammad ibn Musa al-Khwarizmi (c. 780- 850), ninth century Baghdad, The Condensed Book on the Calculation of al-Jabr and al-Muqabala - translation:

“One of the branches of knowledge needed in that division of philosophy known as mathematics is the science of al-jabr and al-muqabala which aims at the determination of numerical and geometric unknowns.”

“Algebra” is derived from “al-jabr” - about solving equations from first use of the word – but use is even older....

Stamp issued by Soviet Union, 1983 for 1200th birthday.

Using Al-jabr

Al-jabr: the transposition of a subtracted quantity from one side to the other side by adding it to both sides.

3x + 2 = 4 - 2x5x + 2 = 4

Al-muqabala: subtraction of equal amounts from both sides to reduce positive term. “Do to both sides equally”

5x + 2 = 45x = 2

How was Algebra Written ?

Algebraic operations were first performed before the use of symbols.

Al-Khwarizmi's text books were written as step-by-step instructions, in words! Describing how to solve each problem. This form of rhetorical algebra are known as algorithms (Algortmi dixit means “al-Khwarizmi says”)

Suppose the temperature is 20oC.. What is the temperature in degrees Fahrenheit ? The solution is this: You take the 20. This you multiply by nine; the product is 180. Divide this by five; the result is 36. To this you add 32; the sum is 68. This is the temperature in degrees Fahrenheit that you sought.

1. You can add, multiply, divide, double, halve, subtract, or perform an other operation you like, provided that you do exactly the same to both sides of an equation.

2. You can always replace an expression by any other expression which is equal to it

Tools for Equation Manipulation

LESSON 1 OUTLINE

• The Real Number System

• Properties of Real Numbers

I. The Real Number System

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.

• 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:36.80.1254.5

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

2

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.

Venn Diagram of the Real Number System

Irrational NumbersRational Numbers

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

rational, or irrational. List all that apply.

a. 117b. 0c. -12.64039…d. -½e. 6.36f. g. -3

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

diagram.

9

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.

9

4

2

1

FYI…For Your Information

• When taking the square root of any number that is not a perfect square, the resulting decimal will be non-terminating and non-repeating. Therefore, those numbers are always irrational.

II. Properties of Real Numbers

Commutative

Associative

Distributive

Identity + ×

Inverse + ×

Commutative Properties

• Changing the order of the numbers in addition or multiplication will not change the result.

• Commutative Property of Addition states: 2 + 3 = 3 + 2 or a + b = b + a.

• Commutative Property of Multiplication states: 4 • 5 = 5 • 4 or ab = ba.

Associative Properties

• Changing the grouping of the numbers in addition or multiplication will not change the result.

• Associative Property of Addition states: 3 + (4 + 5)= (3 + 4)+ 5 or a + (b + c)= (a + b)+ c

• Associative Property of Multiplication states: (2 • 3) • 4 = 2 • (3 • 4) or (ab)c = a(bc)

Distributive Property

• Multiplication distributes over addition.

acabcba

5323523

Additive Identity Property

• There exists a unique number 0 such that zero preserves identities under addition.

a + 0 = a and 0 + a = a• In other words adding zero to a

number does not change its value.

Multiplicative Identity Property

• There exists a unique number 1 such that the number 1 preserves identities under multiplication.

a ∙ 1 = a and 1 ∙ a = a• In other words multiplying a number

by 1 does not change the value of the number.

Additive Inverse Property

• For each real number a there exists a unique real number –a such that their sum is zero.

a + (-a) = 0• In other words opposites add to

zero.

Multiplicative Inverse Property

• For each real number a there exists a

unique real number such that their

product is 1.

1

a

11

a

a

Let’s play “Name that property!”

State the property or properties that justify the following.

3 + 2 = 2 + 3

Commutative Property

State the property or properties that justify the following.

10(1/10) = 1

Multiplicative Inverse Property

State the property or properties that justify the following.

3(x – 10) = 3x – 30

Distributive Property

State the property or properties that justify the following.

3 + (4 + 5) = (3 + 4) + 5 Associative Property

State the property or properties that justify the following.

(5 + 2) + 9 = (2 + 5) + 9

Commutative Property

3 + 7 = 7 + 3

Commutative Commutative Property of AdditionProperty of Addition

2.2.

8 + 0 = 8

Identity Property of Identity Property of AdditionAddition

3.3.

6 • 4 = 4 • 6

Commutative Property Commutative Property of Multiplicationof Multiplication

5.5.

5 • 1 = 5

Identity Property of Identity Property of MultiplicationMultiplication

11.11.

51/7 + 0 = 51/7

Identity Property of Identity Property of AdditionAddition

25.25.

a + (-a) = 0

Inverse Property of Inverse Property of AdditionAddition

40.40.

Properties of Real Numbers

Commutative

Associative

Distributive

Identity + ×

Inverse + ×

This session is finished.

Thank you for participating!