Transcript intro-algebra

COLLEGE ALGEBRA
MT 221
Jeaneth Balaba
Lecturer/Instructor
ISHRM, 1ST SEMESTER, 2014
Good morning!
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Self-introduction
ISHRM Vision and Mission
Classroom policy
Grading system
Other clarifications
Trivia information
Lesson introduction
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 CARD
Name:
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.
Stamp
issued by Soviet Union,
th
780- 1983 for 1200 birthday.
* Oldest text: by Muhammad ibn Musa al-Khwarizmi (c.
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....
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 - 2x
5x + 2 = 4
Al-muqabala: subtraction of equal amounts from both sides to
reduce positive term. “Do to both sides equally”
5x + 2 = 4
5x = 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.
Tools for Equation Manipulation
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
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 a , where a and b
are integers, and b ≠ 0.
b
• They can always be expressed by using
terminating decimals or repeating decimals.
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:
1.9
 0.333…

 7.689689…
FYI…The line above the decimals indicate that number
repeats.
Irrational Numbers
• Irrational numbers are any numbers that cannot be
expressed a as .
b
• They are expressed as non-terminating, nonrepeating decimals; decimals that go on forever
without repeating a pattern.
• Examples of irrational numbers:
– 0.34334333433334…
– 45.86745893…

– 2 (pi)
–
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
Rational Numbers
Irrational Numbers
Example
•
Classify all the following numbers as natural, whole, integer,
rational, or irrational. List all that apply.
a.
b.
c.
d.
e.
f.
g.
117
0
-12.64039…
-½
6.36

-3
To show how these number are classified, use the Venn
diagram. Place the number where it belongs on the Venn
diagram.
Rational Numbers
4
9
Irrational Numbers
Integers

Whole Numbers
Natural
Numbers
-3 0
117

1
2
6.36
4
9
-12.64039…
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.
–  12 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.
– 4 is a rational number.
9
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.
a b  c   ab  ac
32  5   3  2  3  5
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
product is 1.
1
a 1
a
1
a
such that their
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
2.
3+7=7+3
Commutative
Property of Addition
3.
8+0=8
Identity Property of
Addition
5.
6•4=4•6
Commutative Property
of Multiplication
11.
5•1=5
Identity Property of
Multiplication
25.
1
5/
+
0
=
7
1
5/
7
Identity Property of
Addition
40.
a + (-a) = 0
Inverse Property of
Addition
Properties of Real Numbers
Commutative
Associative
Distributive
Identity + ×
Inverse + ×
This session is finished.
Thank you for participating!