Real Numbers and Basic Arithmetic Operations in Algebra
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Intermediate Algebra
by Gustafson and Frisk
Chapter 1
A Review of Basic Algebra
Section 1.1: The Real Number System
SETS:
SETS:
collections of objects.
collections of objects.
Natural Numbers
Whole Numbers
Rational Numbers
Irrational Numbers
Real Numbers
Integers
Positive Numbers
Negative Numbers
Even Numbers
Odd Numbers
Use { } {x | x > 5}{x | x > 5}is read “the set of all x such that
is read “the set of all x such that
x is greater than 5”
x is greater than 5”
Section 1.1: The Real Number System
GRAPHS:
GRAPHS:
plot on the number line
plot on the number line
.
.
Individual numbers are dots
Section 1.1: The Real Number System
Section 1.1: The Real Number System
GRAPHS:
GRAPHS:
plot on the number line
plot on the number line
.
.
Intervals including end points
[
[
]
Section 1.1: The Real Number System
Section 1.1: The Real Number System
GRAPHS:
GRAPHS:
plot on the number line
plot on the number line
.
.
Intervals not including end points
(
(
)
Section 1.2: Arithmetic & Properties of Real Numbers
Section 1.2: Arithmetic & Properties of Real Numbers
OPERATIONS:
OPERATIONS:
Addition
Subtraction (the same as adding a
number with the opposite sign)
Multiplication
Division (the same as multiplying by
the reciprocal)
Section 1.2: Arithmetic & Properties of Real Numbers
Section 1.2: Arithmetic & Properties of Real Numbers
ADDITION:
ADDITION:
Addends that have opposite signs
Addends that have opposite signs
Subtract absolute values
Subtract absolute values
Keep the sign of the addend with the
Keep the sign of the addend with the
largest absolute value
largest absolute value
Addends that have the same signs
Addends that have the same signs
Add absolute values
Add absolute values
Keep the sign of the addends
Keep the sign of the addends
Section 1.2: Arithmetic & Properties of Real Numbers
Section 1.2: Arithmetic & Properties of Real Numbers
MULTIPLICATION:
MULTIPLICATION:
Multiply absolute values
Multiply absolute values
If the factors have the same signs,
If the factors have the same signs,
the product is positive
the product is positive
If the factors have opposite signs,
If the factors have opposite signs,
the product is negative
the product is negative
Section 1.2: Arithmetic & Properties of Real Numbers
Section 1.2: Arithmetic & Properties of Real Numbers
STATISTICS:
STATISTICS:
measures of central tendency
measures of central tendency
Mean
Mean
Median
Median
Mode
Mode
Section 1.2: Arithmetic & Properties of Real Numbers
Section 1.2: Arithmetic & Properties of Real Numbers
Properties:
Properties:
Associative – addition, multiplication
Associative – addition, multiplication
Commutative – addition, multiplication
Commutative – addition, multiplication
Distributive – multiplication is
Distributive – multiplication is
distributed over addition
distributed over addition
a (b + c) = ab + ac
a (b + c) = ab + ac
Section 1.2: Arithmetic & Properties of Real Numbers
Section 1.2: Arithmetic & Properties of Real Numbers
Identities:
Identities:
Addition – zero
Addition – zero
Multiplication – one
Multiplication – one
Inverses:
Inverses:
Addition – opposites
Addition – opposites
Multiplication – reciprocals
Multiplication – reciprocals
Section 1.3: Definition of Exponents
Section 1.3: Definition of Exponents
EXPONENTS:
EXPONENTS:
repeated multiplication
repeated multiplication
In the expression: a
In the expression: a
n
n
a is the base and n is the exponent
a is the base and n is the exponent
Exponents are
Exponents are
NOT
NOT
factors
factors
Means to multiply “a” n times
Means to multiply “a” n times
Section 1.3: Definition of Exponents
Section 1.3: Definition of Exponents
ORDER OF OPERATIONS:
ORDER OF OPERATIONS:
If an algebraic expression has more than one
If an algebraic expression has more than one
operation, the following order applies:
operation, the following order applies:
1.
Clear up any grouping.
Clear up any grouping.
2.
Evaluate exponents.
Evaluate exponents.
3.
Do multiplication and division from left to
Do multiplication and division from left to
right.
right.
4.
Do addition and subtraction from left to
Do addition and subtraction from left to
right.
right.
Section 1.5: Solving Equations
Section 1.5: Solving Equations
Algebraic Expression vs. Equation
Algebraic Expression vs. Equation
Expressions: a combination of
Expressions: a combination of
numbers and operations
numbers and operations
Equation: a statement that two
Equation: a statement that two
expressions are equal
expressions are equal
Section 1.5: Solving Equations
Section 1.5: Solving Equations
EXPRESSIONS:
EXPRESSIONS:
Terms
Terms
Like terms
Like terms
When multiplying, the terms do not
When multiplying, the terms do not
need to be alike
need to be alike
Can only add like terms!
Can only add like terms!
Section 1.5: Solving Equations
Section 1.5: Solving Equations
TO SOLVE AN EQUATION IN ONE VARIABLE:
TO SOLVE AN EQUATION IN ONE VARIABLE:
If you see fractions,
If you see fractions,
multiply both sides by the LCD
multiply both sides by the LCD
.
.
This will eliminate the fractions.
This will eliminate the fractions.
Simplify
Simplify
the algebraic expressions on each side of the
the algebraic expressions on each side of the
equal sign (eliminate parentheses and combine like
equal sign (eliminate parentheses and combine like
terms).
terms).
Use the addition property of equality to
Use the addition property of equality to
isolate
isolate
the
the
variable terms from the constant terms
variable terms from the constant terms
on opposite
on opposite
sides of the equal sign.
sides of the equal sign.
Use the multiplication property to make the coefficient
Use the multiplication property to make the coefficient
of the variable equal to one.
of the variable equal to one.
Check your results by evaluating.
Check your results by evaluating.
Section 1.5: Solving Equations
Section 1.5: Solving Equations
TYPES OF EQUATIONS:
TYPES OF EQUATIONS:
CONDITIONAL: if x equals this, then y
CONDITIONAL: if x equals this, then y
equals that.
equals that.
IDENTITY: always true no matter what
IDENTITY: always true no matter what
numbers you use.
numbers you use.
CONTRADICTION: never true no matter
CONTRADICTION: never true no matter
what numbers you use.
what numbers you use.
FORMULAS: conditional equations that
FORMULAS: conditional equations that
model a relationship between the variables.
model a relationship between the variables.
Section 1.6 & 1.7: Solving Problems, Applications
Section 1.6 & 1.7: Solving Problems, Applications
TYPES OF PROBLEMS:
TYPES OF PROBLEMS:
Geometry
Geometry
Percent
Percent
Physics (forces)
Physics (forces)
Uniform motion
Uniform motion
Mixtures
Mixtures
Good ‘ole common sense analysis
Good ‘ole common sense analysis
Chapter 1: Basic Algebra Review
Chapter 1: Basic Algebra Review
SUMMARY:
SUMMARY:
KNOW YOUR VOCABULARY!
KNOW YOUR VOCABULARY!
You can’t follow
You can’t follow
directions if you don’t know what the words
directions if you don’t know what the words
in the instructions mean.
in the instructions mean.
Memorize the processes and the properties.
Memorize the processes and the properties.
I will provide formulas for your reference.
I will provide formulas for your reference.
Ask questions if you are unsure.
Ask questions if you are unsure.
Always check your work to make sure that
Always check your work to make sure that
you answered the question, and that your
you answered the question, and that your
answer is reasonable.
answer is reasonable.
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Review essential concepts from the first chapter of "Intermediate Algebra" by Gustafson and Frisk. Learn about the classification of real numbers, graphical representations on the number line, intervals, arithmetic operations such as addition, subtraction, multiplication, and division of real numbers. Understand the rules for adding and multiplying numbers with the same or opposite signs.
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Presentation Transcript
Intermediate Algebra by Gustafson and Frisk Chapter 1 A Review of Basic Algebra
Section 1.1: The Real Number System SETS: collections of objects. Natural Numbers Whole Numbers Rational Numbers Irrational Numbers Real Numbers Use { } {x | x > 5} Integers Positive Numbers Negative Numbers Even Numbers Odd Numbers is read the set of all x such that x is greater than 5
Section 1.1: The Real Number System GRAPHS: plot on the number line. Individual numbers are dots -3 -2 -1 0 1 2 3 4
Section 1.1: The Real Number System GRAPHS: plot on the number line. Intervals including end points [ -3 -2 -1 0 1 2 3 4 [ ] -3 -2 -1 0 1 2 3 4
Section 1.1: The Real Number System GRAPHS: plot on the number line. Intervals not including end points ( -3 -2 -1 0 1 2 3 4 ( ) -3 -2 -1 0 1 2 3 4
Section 1.2: Arithmetic & Properties of Real Numbers OPERATIONS: Addition Subtraction (the same as adding a number with the opposite sign) Multiplication Division (the same as multiplying by the reciprocal)
Section 1.2: Arithmetic & Properties of Real Numbers ADDITION: Addends that have the same signs Add absolute values Keep the sign of the addends Addends that have opposite signs Subtract absolute values Keep the sign of the addend with the largest absolute value
Section 1.2: Arithmetic & Properties of Real Numbers MULTIPLICATION: Multiply absolute values If the factors have the same signs, the product is positive If the factors have opposite signs, the product is negative
Section 1.2: Arithmetic & Properties of Real Numbers STATISTICS: measures of central tendency Mean Median Mode
Section 1.2: Arithmetic & Properties of Real Numbers Properties: Associative addition, multiplication Commutative addition, multiplication Distributive multiplication is distributed over addition a (b + c) = ab + ac
Section 1.2: Arithmetic & Properties of Real Numbers Identities: Addition zero Multiplication one Inverses: Addition opposites Multiplication reciprocals
Section 1.3: Definition of Exponents EXPONENTS: repeated multiplication In the expression: an a is the base and n is the exponent Exponents are NOT factors Means to multiply a n times
Section 1.3: Definition of Exponents ORDER OF OPERATIONS: If an algebraic expression has more than one operation, the following order applies: 1. Clear up any grouping. 2. Evaluate exponents. 3. Do multiplication and division from left to right. 4. Do addition and subtraction from left to right.
Section 1.5: Solving Equations Algebraic Expression vs. Equation Expressions: a combination of numbers and operations Equation: a statement that two expressions are equal
Section 1.5: Solving Equations EXPRESSIONS: Terms Like terms When multiplying, the terms do not need to be alike Can only add like terms!
Section 1.5: Solving Equations TO SOLVE AN EQUATION IN ONE VARIABLE: If you see fractions, multiply both sides by the LCD. This will eliminate the fractions. Simplify the algebraic expressions on each side of the equal sign (eliminate parentheses and combine like terms). Use the addition property of equality to isolate the variable terms from the constant terms on opposite sides of the equal sign. Use the multiplication property to make the coefficient of the variable equal to one. Check your results by evaluating.
Section 1.5: Solving Equations TYPES OF EQUATIONS: CONDITIONAL: if x equals this, then y equals that. IDENTITY: always true no matter what numbers you use. CONTRADICTION: never true no matter what numbers you use. FORMULAS: conditional equations that model a relationship between the variables.
Section 1.6 & 1.7: Solving Problems, Applications TYPES OF PROBLEMS: Geometry Percent Physics (forces) Uniform motion Mixtures Good ole common sense analysis
Chapter 1: Basic Algebra Review SUMMARY: KNOW YOUR VOCABULARY! You can t follow directions if you don t know what the words in the instructions mean. Memorize the processes and the properties. I will provide formulas for your reference. Ask questions if you are unsure. Always check your work to make sure that you answered the question, and that your answer is reasonable.
This powerpoint was kindly donated to www.worldofteaching.com http://www.worldofteaching.com Is home to well over a thousand powerpoints submitted by teachers. This a free site. Please visit and I hope it will help in your teaching
