Download Algebra 2 - peacock

Algebra 2
Properties of Real Numbers
Lesson 1-2
Goals
Goal
• To graph and order real
numbers.
• To Identity properties of
real numbers.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Essential Question
Big Idea: Properties
• How can you use the properties of real numbers to
simplify algebraic expressions?
– Students will understand the set of real numbers has
several subsets related in particular ways.
– Students will understand that algebra involves
operations on and relations among numbers, including
real and imaginary numbers.
– Students will understand how rational and irrational
numbers form the set of real numbers.
Vocabulary
•
•
•
•
Opposite
Additive Inverse
Reciprocal
Multiplicative Inverse
Sets of Numbers
A set is a collection of items called
elements. The rules of 8-ball divide
the set of billiard balls into three
subsets:
solids (1 through 7), stripes
(9 through 15), and the 8 ball.
A subset is a set whose elements belong to another set.
The empty set, denoted , is a set containing no
elements.
Sets of Numbers
• Other useful set notation
–
–
–
–
U
∩
\
∈
Union (or statement)
Intersection (and statement)
excludes whatever follows
element of a set
Sets of Numbers
There are many ways to represent sets. For instance, you
can use words to describe a set. You can also use roster
notation, in which the elements in a set are listed between
braces, { }.
Words
Roster Notation
The set of billiard
balls is numbered
1 through 15.
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15}
Sets of Numbers
A Venn Diagram can also be used to display sets of numbers and
their relationships. The following sets of numbers are displayed:
Reals, Rationals, Irrationals, Integers, Wholes, and Naturals.
Reals
Rationals
2
3
.
-2 65
Integers
-3
-19
Wholes
0
Naturals
1, 2, 3
...
1
6
4
Irrationals

2
Sets of Numbers
Natural Numbers - ℕ
• The
numbers we use to count things are the set of natural
numbers, N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …}
–
–
–
–
The three dots, ellipses, show that the same pattern
continues in the same pattern indefinitely.
Each natural number other than 1 is either a prime
number or a composite number.
A prime number is a number greater than one that is
evenly divisible only by itself or 1.
A natural number other than 1 that is not a prime number
is a composite number.
Whole Numbers - 𝕎
• The set of whole numbers include 0 and the
natural numbers,
W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …}
Integers - ℤ
• The set of numbers including positive and
negative counting numbers and zero are
called the set of integers:
The negative integers are {…,–5,–4,–3,–2,–1}.
 The positive integers are {1, 2, 3, 4, 5, …}.

Z = {…, –5 , –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, …}
Rational - ℚ
• A rational number can be expressed as the
quotient of two integers (a fraction), with the
denominator not 0.
 Rational numbers can also be written in decimal
form, either as terminating; ¾=.75, -⅛ = -.125,
or 1¼= 2.75

or as repeating decimals; ⅓ = .33333… or ⅙
= .1666….
Irrational - ℚ
• Decimal numbers that neither terminate nor repeat
are not rational, and thus are called irrational
numbers.

The square root of any number that is not a perfect
square is an irrational number;
2  .1.41421356... or 

23  4.79583135...,
as well as numbers represented by familiar symbols;
 or e.
Caution!
A repeating decimal may not appear to repeat on a calculator,
because calculators show a finite number of digits.
Example: Classifying Real Numbers
Write all classifications that apply to each number.
A.
5
5 is a whole number that is not a perfect
square.
irrational, real
B.
–12.75
–12.75 is a terminating decimal.
rational, real
C.
16
16
2
2
=
4
=2
2
whole, integer, rational, real
Your Turn:
Write all classifications that apply to each number.
A.
9
9 =3
whole, integer, rational, real
B.
–35.9
–35.9 is a terminating decimal.
rational, real
C.
81
81
3
3
=
9
=3
3
whole, integer, rational, real
Example: Classifying Real Numbers
Consider the numbers
Classify each number by the subsets of the real
numbers to which it belongs.
Numbers
Real
Rational
2.3




Whole
Natural
Irrational



2.7652
Integer




Your Turn:
Consider the numbers –2, , –0.321, and
.
Classify each number by the subsets of the real
numbers to which it belongs.
Numbers
Real
Rational
Integer
–2





–0.321





Whole
Natural
Irrational


Example: Classifying a Variable
Your school is sponsoring a charity race. Which set
of numbers best describes the number of people p
who participate in the race?
The number of people p is a natural number. The correct answer is A.
Your Turn:
If each participant in a charity race makes a donation
d of $15.50 to a local charity, which subset of real
numbers best describes the amount of money they
will raise?
A. Natural numbers
B. Integers
C. Rational numbers
D. Irrational numbers
Sets of Numbers
A set can be finite like the set of billiard ball
numbers or infinite like the natural numbers
{1, 2, 3, 4 …}.
A finite set has a definite, or finite, number
of elements.
An infinite set has an unlimited, or infinite
number of elements.
The Density Property states that between any two numbers
there is another real number. So any interval that includes
more than one point contains infinitely many points.
The Number Line
Many infinite sets, such as the real numbers, cannot be
represented in roster notation. Therefore, we use a number
line. A number line is a line on which each point is
associated with a real number.
The number line represents the set of all real numbers. The
number line can be used to graph individual real numbers,
sets of real numbers both finite and infinite, and order real
numbers.
–5 –4 –3 –2 –1
0
1
2
3
4
5
The Number Line
To construct a number line,
1. Choose any point on a horizontal line and label it 0.
2. Choose a point to the right of 0 and label it 1.
3. The distance from 0 to 1 establishes a scale that can be used
to locate more points, with positive numbers to the right of 0
and negative numbers to the left of 0.

The number 0 is neither positive nor negative.
–5 –4 –3 –2 –1
– 4.8
Negative numbers
0
1
2
3
4
5
1.5
Positive numbers
The Number Line
You can use a number line to compare and order real
numbers. A number on a number line is greater
than all of the numbers to it’s left.
1. Approximate Irrational Numbers & fractions as a
decimal (Remember: a fraction is just division).
2. Graph the decimal numbers on the number line.
Determine if the number is positive or negative and
plot it by the whole number it is closest to.
3. To compare the numbers, determine the larger of
two numbers and place the appropriate inequality
between them.
Example: Graphing Numbers
on the Number Line
• What is the graph of the numbers
5
 , 2, and 2.6 ?
2
• Change the numbers to decimals.
5
2
   2.5,
2  1.4, 2.6  2.67
• Graph the decimal numbers on the number line.
-2.5
1.4
2.67
|
|
|
|
|
|
|
|
|
–4
–3
–2
–1
0
1
2
3
4
Your Turn:
What is the graph of the numbers
1
3, 1.4,
?
3
Example: Ordering Real Numbers
Consider the numbers
Order the numbers from least to greatest.
Write each number as a decimal to make it easier to compare them.
Use a decimal approximation for
 ≈ 3.14
.
Use a decimal approximation for .
Rewrite
in decimal form.
–5.5 < 2.23 < 2.3 < 2.7652 < 3.14
Use < to compare the numbers.
The numbers in order from least to great are
Your Turn:
Consider the numbers –2, , –0.321,
and
.
Order the numbers from least to greatest.
Write each number as a decimal to make it easier to compare them.
≈ –1.313
Use a decimal approximation for
= 1.5
Rewrite
 ≈ 3.14
.
in decimal form.
Use a decimal approximation for .
–2 < –1.313 < –0.321 < 1.50 < 3.14
Use < to compare the numbers.
The numbers in order from least to great are –2,
, and .
, –0.321,
Inequality Symbols
• An inequality symbol is used to compare numbers:
• Symbols include:
greater than:

greater than or equal to:


less than:

less than or equal to:

not equal to:
59
3 9  5
• Examples:
7 3  4
1 3  2
.
The Number Line
The set of real numbers between 3 and 5,
which is also an infinite set, can be
represented on a number line or by an
inequality.
-2
-1
0
1
2
3
4
3<x<5
5
6
7
8
Graphing a Set on the
Number Line
• If the end point of a set is included, then a
“closed circle” is used.
–x≥3
• If the end point of a set is excluded, then an
“open circle” is used.
–x>3
Example:
• Write the following intervals on the number line
as an inequality.
1)
|
|
|
|
|
|
|
|
|
–4
–3
–2
–1
0
1
2
3
4
 x > -2
 If the variable is on the left side of the inequality, then
the inequality sign is the same direction as the arrow.
Your Turn:
2)
|
|
|
|
|
|
|
|
|
–4
–3
–2
–1
0
1
2
3
4
 x≤1
Your Turn:
3)
|
|
|
|
|
|
|
|
|
–4
–3
–2
–1
0
1
2
3
4
 Conjunction “and statement”
 x ≥-3 and x < 2
 -3 ≤ x < 2 (compound inequality)
Your Turn:
4)
|
|
|
|
|
|
|
|
|
–4
–3
–2
–1
0
1
2
3
4
 Disjunction “or statement”
 x ≤ -3 or x > 0
 Disjunction must be written with the word
or (or symbol ∪).
 x ≤ -3 ∪ x > o (compound inequality)
Your Turn:
5)
|
|
|
|
|
|
|
|
|
–4
–3
–2
–1
0
1
2
3
4
 x equals all real numbers except 0
 x < 0 or x > 0
 x𝛜ℝ\0
Properties of Real
Numbers
The Rules of the Game
Properties of Real
Numbers
• Why are the properties of real numbers
important?
– Properties are the rules for the game involving
real numbers. To get the correct answer, you
must follow the rules.
– If the rules are changed a different kind of math
is played.
– You use the properties of real numbers for all
math problems that involve real numbers.
Definitions
• Opposite – the opposite or additive inverse of any
number a is –a. The sum of opposites is zero, the
additive identity.
– Example:
3 + (-3) = 0
5.2 + (-5.2) = 0
• Additive Inverse – the additive inverse or
opposite of any number a is –a. The sum of
additive inverses is zero, the additive identity.
– Example:
6.8 + (-6.8) = 0
Definitions
• Reciprocal – the reciprocal or multiplicative
inverse of any nonzero number a is 1/a. The
product of reciprocals is 1, the multiplicative
identity.
– Example: 5 • 1/5 = 1
• Multiplicative Inverse – the multiplicative
inverse or reciprocal of any nonzero number a is
1/a. The product of multiplicative inverses is 1,
the multiplicative identity.
– Example: 1/3 • 3 = 1
Example: Finding Inverses
Find the additive and multiplicative inverse of each
number. 12
additive inverse: –12
Check –12 + 12 = 0 
multiplicative inverse:
Check

The opposite of 12 is –12.
The Additive Inverse Property holds.
The reciprocal of 12 is
.
The Multiplicative Inverse Property
holds.
Example: Finding Inverses
Find the additive and multiplicative inverse of each
number.
additive inverse:
The opposite of
multiplicative inverse:
The reciprocal of
is
is
.
Your Turn:
Find the additive and multiplicative inverse of each
number. 500
additive inverse: –500
The opposite of 500 is –500.
Check 500 + (–500) = 0 
The Additive Inverse Property
holds.
multiplicative inverse:
Check

The reciprocal of 500 is
.
The Multiplicative Inverse Property
holds.
Your Turn:
Find the additive and multiplicative inverse of each
number.
–0.01
additive inverse: 0.01
The opposite of –0.01
is 0.01.
multiplicative inverse: –100
The reciprocal of –0.01
is –100.
Properties of Real
Numbers
The four basic math operations are
addition, subtraction, multiplication, and
division. Because subtraction is addition
of the opposite and division is
multiplication by the reciprocal, the
properties of real numbers focus on
addition and multiplication.
Properties of Real
Numbers
Identity Properties
of Addition & Multiplication
Identity = Same (no change)
What can you add to a number & get the same number back?
0 (zero)
What can you multiply a number by and get the number back?
1 (one)
Properties of Real
Numbers
Additive Identity
Properties Real Numbers
For all real numbers n,
WORDS
Additive Identity Property
The sum of a number and 0, the additive
identity, is the original number.
NUMBERS
3+0=0
ALGEBRA
n+0=0+n=n
Properties of Real
Numbers
Multiplicative Identity
Properties Real Numbers
For all real numbers n,
WORDS
Multiplicative Identity Property
The product of a number and 1, the
multiplicative identity, is the original
number.
NUMBERS
ALGEBRA
n1=1n=n
Properties of Real
Numbers
Inverse Properties
of Addition & Multiplication
Inverse = Opposite
What is the opposite (inverse) of addition?
What is the opposite of multiplication?
Subtraction (add the negative)
Division (multiply by reciprocal)
Properties of Real
Numbers
Additive Inverse
Properties Real Numbers
For all real numbers n,
WORDS
Additive Inverse Property
The sum of a number and its opposite,
or additive inverse, is 0.
NUMBERS
5 + (–5) = 0
ALGEBRA
n + (–n) = 0
Properties of Real
Numbers
Properties Real Numbers
Multiplicative Inverse
For all real numbers n,
WORDS
NUMBERS
ALGEBRA
Multiplicative Inverse Property
The product of a nonzero number and its
reciprocal, or multiplicative inverse, is 1.
Properties of Real
Numbers
Closure Properties
of Addition & Multiplication over the
Set of Real Numbers
A number set has closure under an operation
if performance of that operation on
members of the set always produces a
member of the same set.
Properties of Real
Numbers
Properties Real Numbers
Addition and Multiplication
For all real numbers a and b,
WORDS
NUMBERS
ALGEBRA
Closure Property
The sum or product of any two real
numbers is a real number
2+3=5
2(3) = 6
a+b
ab  
Properties of Real
Numbers
Commutative Properties
of Addition & Multiplication
Commute = Travel (move)
It doesn’t matter how you move addition or
multiplication around…the answer will be
the same!
Properties of Real
Numbers
Properties Real Numbers
Addition and Multiplication
For all real numbers a and b,
WORDS
NUMBERS
ALGEBRA
Commutative Property
You can add or multiply real numbers in
any order without changing the result.
7 + 11
7(11)
a+b
ab
=
=
=
=
11 + 7
11(7)
b+a
ba
Properties of Real
Numbers
Associative Properties
of Addition & Multiplication
Associate = Group
It doesn’t matter how you group (associate)
addition or multiplication…the answer will
be the same!
Properties of Real
Numbers
Properties Real Numbers
Addition and Multiplication
For all real numbers a and b,
WORDS
NUMBERS
ALGEBRA
Associative Property
The sum or product of three or more real
numbers is the same regardless of the
way the numbers are grouped.
(5 + 3) + 7
(5  3)7
a + (b + c)
(ab)c
=
=
=
=
5 + (3 + 7)
5(3  7)
a + (b + c)
a(bc)
Stop and think!
• Does the Associative Property hold true for
Subtraction and Division?
Is (5-2)-3 = 5-(2-3)?
Is (6/3)-2 the same as 6/(3-2)?
• Does the Commutative Property hold true for
Subtraction and Division?
Is 5-2 = 2-5?
Is 6/3 the same as 3/6?
Properties of real numbers are only for Addition and Multiplication
Properties of Real
Numbers
Distributive Property
of Multiplication over Addition/Subtraction
If something is sitting just outside a set of parenthesis,
you can distribute it through the parenthesis with
multiplication and remove the parenthesis.
Properties of Real
Numbers
Addition and Multiplication
Properties Real Numbers
For all real numbers a and b,
WORDS
NUMBERS
ALGEBRA
Distributive Property
When you multiply a sum by a number, the result
is the same whether you add and then multiply
or whether you multiply each term by the
number and add the products.
5(2 + 8) = 5(2)
(2 + 8)5 = (2)5
a(b + c) = ab
(b + c)a = ba
+
+
+
+
5(8)
(8)5
ac
ca
Let’s play “Name that property!”
State the property or
properties that justify the
following.
3+2=2+3
Commutative
Property of Addition
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
of Addition
State the property or
properties that justify the
following.
(5 + 2) + 9 = (2 + 5) + 9
Commutative
Property of Addition
State the property or
properties that justify the
following.
3+7=7+3
Commutative
Property of Addition
State the property or
properties that justify the
following.
8+0=8
Additive Identity
Property
State the property or
properties that justify the
following.
6•4=4•6
Commutative Property
of Multiplication
State the property or
properties that justify the
following.
5•1=5
Multiplicative Identity
Property
State the property or
properties that justify the
following.
1
5/
+
0
=
7
1
5/
7
Additive Identity
Property
State the property or
properties that justify the
following.
a + (-a) = 0
Additive Inverse
Property
Assignment
• Section 1-2, Pg 15 – 17; #1 – 9 all, 10 – 38
even, 42 – 50 even, 54 – 58 even