Activator
1. Evaluate 
y^2 / ( 3ab + 2) if y = 4; a = -2; and b = -5 
2. Find the value: √17 =
0.25 x 0 =
6 : 10 =
Introduction to Algebra 2
Real Numbers and
Their Properties
CCSS:
N.RN.3
 Domain:
The Real Number System
 Clusters: Use properties of rational and irrational
numbers
 Standards:
N.RN.3 EXPLAIN why the sum or product of two
rational numbers is rational; that the sum of a rational
number and an irrational number is irrational; and
that the product of a nonzero rational number and an
irrational number is irrational.
N.RN.3
Write sets using set notation
A set is a collection of objects called the elements or members of
the set. Set braces { } are usually used to enclose the elements. In
Algebra, the elements of a set are usually numbers.



Example 1: 3 is an element of the set {1,2,3} Note: This is
referred to as a Finite Set since we can count the elements of the
set.
Example 2: N= {1,2,3,4,…} is referred to as a Natural Numbers
or Counting Numbers Set.
Example 3: W= {0,1,2,3,4,…} is referred to as a Whole Number
Set.
Write sets using set notation
A set is a collection of objects called the elements or members of
the set. Set braces { } are usually used to enclose the elements.
Example 4: A set containing no numbers is shown as { } Note:
This is referred to as the Null Set or Empty Set.
Caution: Do not write the {0} set as the null set. This set contains
one element, the number 0.
Example 5: To show that 3 “is a element of” the set {1,2,3}, use the
notation: 3  {1,2,3}. Note: This is also true: 3  N
Example 6: 0  N where  is read as “is not an element of set”

Write sets using set notation


Two sets are equal if they contain exactly the same elements.
(Order doesn’t matter)
Example 1: {1,12} = {12,1}
Example 2: {0,1,3}  {0,2,3}
Write sets using set notation
In Algebra, letters called variables are often used to represent
numbers or to define sets of numbers. (x or y). The notation
{x|x has property P}is an example of
“Set Builder Notation” and is read as:
{x  x has property P}
the set of

all elements x
such that x has a property P
Example 1: {x|x is a whole number less than 6}
Solution:
{0,1,2,3,4,5}
Write sets using set notation
In Algebra, letters called variables are often used to represent
numbers or to define sets of numbers. (x or y).
The notation {x|x has property P}is an example of
“Set Builder Notation” and is read as:
{x  x has property P}
the set of

all elements x
such that x has a property P
Example 2: {x|x is a natural number greater than 12}
Solution:
{13,14,15,…}
Using a number line
-2
 One
-1
0
1
2
3
4
5
way to visualize a set a numbers is to use a
“Number Line”.

Example 1: The set of numbers shown above includes positive
numbers, negative numbers and 0. This set is part of the set of
“Integers” and is written:
I = {…, -2, -1, 0, 1, 2, …}
Using a number line
Graph of -1
o
-2
-1
11
4
1
2
o
0
o
1
2
3
4
5
coordinate
Each
number on a number line is called the
coordinate of the point that it labels, while the
point is the graph of the number.

Example 1: The fractions shown above are examples of rational numbers. A
rational number is one than can be expressed as the quotient of two integers,
with the denominator not 0.
Using a number line
Graph of -1
o
-2
1
2
2
o
o
-1
0
1
11
4
16
o

oo o
2
7 3
4
4
o
5
coordinate
Decimal
numbers that neither terminate nor
repeat are called “irrational numbers”.

Example 1: Many square roots are irrational numbers, however some square
roots are rational.

Irrational:
Rational:
2
7 
4
16

Circumference
diameter
The common sets of numbers
The
following sets of numbers will be used for
this course:

N - Natural numbers or Counting Numbers: {1, 2, 3, …}

W – Whole numbers: {0, 1, 2, 3, …}

I – Integer numbers: {…,-2, -1, 0, 1, 2, …}

R - Rational numbers: 

IR – Irrational numbers: {x| x is a real number that is not rational}

RN- Real numbers: {x| x is represented by a point on a number line}
p

p and q are integers, q  0
q

Real Numbers
Rational Numbers
Irrational Numbers
1/2
-2
3
3
0
15%
-0.7
1.456
3/18/2024
2/3
3
4
2
-5
2
Real Numbers
Rational Numbers
Irrational Numbers
Integers
1/2
3
-2
15%
2
2/3
0
- 0.7
3
3
4
1.456
-5
2
Real Numbers
Rational Numbers
Irrational Numbers
Integers
Whole
3
1/2
3
2/3
0
3
4
-2
- 0.7
15%
2
1.456
-5
2
Real Numbers
Rational Numbers
Irrational Numbers
Integers
Whole
3
Natural
1/2
0
2/3
3
3
4
-2
- 0.7
15%
2
1.456
-5
2
The common sets of numbers
Question:
Select all the words from the following
list that apply to the number:
4
Whole Number, Rational Number, Irrational Number,
Real Number, Undefined
Solution: Whole Number, Rational Number, Real
Number
42
Finding Additive inverses
any real number x, the number –x is the additive
inverse of x.
 For
Example 1:
Inverse
Number Additive
6
-4
2
3
- 8.7
0
-6
4
2

3
8.7
0
Using the Absolute Value

Examples: Find :
29  29
0 0
90  90
To find the absolute value of a signed number:
 if X > 0, then X = X 


 if X = 0, then X = 0 
X 

 if X < 0, then X = -X 




Caution: The absolute value of a number is always positive
Using Inequality Symbols
Equality/Inequality Symbols:
Caution: With the symbol  , if either the  or the
= part is true, then the inequality is true. This
is also the case for the  symbol.

Symbol
Meaning
Example
=
is equal to
4=4

is not equal to
4 5

is less than
45

is less than or equal
-4  -3

is greater than
-4  -5

is greater than or equal
-8  - 10
Graphing Sets of Real Numbers

A parenthesis ( or ) is used to indicate a number is not an
element of a set. A bracket [ or ] is used to indicate a number
is a member of a set.
Example 1: Write in interval notation and graph: {x|x  3}
Solution: Interval Notation (-,3)
)
-2
-1
0
1
2
3
4
5
Example 2: Write in interval notation and graph: {x|x  0}
Solution: Interval Notation [0, )
[
-2
-1
0
1
2
3
4
5
Graphing Sets of Real Numbers

A parenthesis ( or ) is used to indicate a number is not an
element of a set. A bracket [ or ] is used to indicate a number
is a member of a set.
Example 3: Write in interval notation and graph:
{x| -2  x  3}
Solution: Interval Notation [-2,3)
[
-2
)
-1
0
1
2
3
4
5
Adding Real Numbers

Example: Add (–5.6) + (-2.1) =
(-5.6) + (-2.1) = -( 5.6 + 2.1 ) = -7.7

Example:
2
5
ADD:
+ (- )
3
6
2
5
4
5
5 4
1
+ (- ) =
+ (- ) = -( - )  
3
6
6
6
6 6
6
To add signed numbers:
1)
If the numbers are alike, add their absolute values and use the common
sign.
2)
If the numbers are not alike, subtract the smaller absolute value from
the larger absolute value. Use the sign of the larger absolute value.
Subtracting Real Numbers


Example: Subtract (–56) - (-70) =
(-56) - (-70) = (-56) + (70 ) = +(70-56) = +14
Example: Subtract: - 3 - (- 3 ) - (- 7 ) - 7 
5
8
10 20
24
15
28
14
5
1
( ) + (+ ) + ( ) + (- ) =  ( )  
40
40
40
40
40
8
To subtract signed numbers:
1) Rewrite as an addition problem by adding the opposite of
the number to be subtracted. Find the sum.
Finding the distance between two points on a number line.

To find the distance between two points on a number line,
find the absolute value of the difference between the two
points.
Example 1: Find the distance between the points: –2 and 3
Solution: |(-2) – (3)| = |-2 -3| = |-5| = 5
or
|(3) – (-2)| = |3 +2| = |5| =5
o
-2
o
-1
0
1
2
3
4
5
Multiplying Real Numbers
To find the product of a positive and negative signed number: Find
the product of the absolute values. Make the sign negative.


Example: Multiply (–12)(5) =
(-12)  (5) = -(12  5) = -60
7
8
Example:
Multiply: (- ) (- ) =
12
25
7
8
7 8
14
(- ) (- ) = +(  ) = +
12
25
12 25
75
To find the product of two negative signed numbers:
Find the product of the absolute values. Make the sign positive.
Dividing Real Numbers
To find the quotient of a positive and negative signed number:
Find the quotient of the absolute values. Make the sign
negative.

Example: Divide (–12)(5) =
(-12)  (5) = - 12  5 = - 2.4  2.4
Divide: (-
7
8
)  ()=
12
25

Example:
7
8
7
8
7 25
175
(- )  (- ) = +(  ) = +(  ) = +
12
25
12 25
12 8
96
To find the quotient of two negative signed numbers:
Find the quotient of the absolute values. Make
the sign positive.
6
0
Caution: Division by zero is “undefined”:
0
if
is undefined , but
1
, then x  2
x-2
6
=0
Dividing Real Numbers
To find the Inverse of a real number:
Find the reciprocal of the number. Keep the same sign.
Caution:
A number and its “reciprocal” always have the same sign.
A number and its “additive inverse” have opposite signs.
Examples:
Number Reciprocal or Inverse

2
5

7
11
5
2
1

6
11
7
Additive Inverse
2
5

6

7
11
0.05
20
-0.05
0
none
0
Using Exponents
 If “a” is a real number and “n” is a natural number, then an =
a•a•a•••a•a (n factors of a).
where n is the exponent, a is the base, and an is an
exponential expression. Exponents are also called powers.
To find the value of a whole number exponent:
100 = 1, 20 = 1, 80 = 1, #0 = 1
101 = 10, 21 = 2, 81 = 8, #1 = #
102 = 10 x 10 = 100, 22 = 2 x 2 = 4, 82 = 8 x 8 = 64
103 = 10 x 10 x 10 = 1000, 23 = 2 x 2 x 2 = 8
104 = 10 x 10 x 10 x 10 = 10,000
24 = 2 x 2 x 2 x 2 = 16
(-10)3 = (-10)(-10)(-10)
(12).5 = 12
Order of Operations
Order of Operations
Perform left to right
Highest
Inside Parenthesis
Exponentiation
Multiplication/Division
Addition/Subtraction
Lowest
To evaluate an expression:
12 - 9 ÷ 3 = 12 – 3 = 9
(12 – 9) ÷ 3 = (3) ÷ 3 = 1
(43 – 120 ÷ 2)2 + 82 = (64 – 60)2 + 64 = 42 + 64 =
16 + 64 = 80
Order of Operations
•Example 1: Simplify
1
 10  6  9
2
5
 12  3(2) 2
6
Solution:
1
1
 10  6  9
 10  6  3
563 2
2
2



 1
5
5
10  12
2
 12  3(2) 2
 12  3(4)
6
6
10  6  2 9
Example 2: Simplify
10  6  2 9
11  4  3(2) 2
10  6  2(3) 10  6  6 10



1
2
11  2  3(4)
22  12
10
11  4  3(2)
Evaluating Algebraic Expressions
•Example 1: if w = 4, x = -12, y = 64, z = -3
Find:
5x  z y
x 1
Solution:
5x  z y 5(12)  (3) 64 60  (3)8 60  24 84 84





x 1
(12)  1
13
13
13 13
Example 2: Find:
w  2z
2
3
w2  2 z 3  42  2(3)3  16  2(27)  16  54  38
Using the Distributive Property

For any real numbers a, b , and c
a(b + c) = ab + ac or (b + c)a = ab +ac
Note: This is often referred to as “removing parenthesis”

Example 1: -4(p – 5) = -4p – 20

Example 2: -6m +2m = m(-6 + 2) = m(-4) = -4m
Note: This is often referred to as “factoring out m”
Using the Identity Properties

Zero is the only number that can be added to
any number to get that number.
0 is called the “identity element for addition”
a+0=a
Example 1: 4 + 0 = 4
Note: This is referred to as the “additive identity”

One is the only number that can be multiplied
by any number to get that number.
1 is called the “identity element for multiplication”
a•1=a
Example 2: 4 • 1 = 4
Note: This is referred to as the “multiplicative identity”
Using the Commutative and Associate Properties

For any real numbers a, b, and c,
a+b=b+a
and
ab = ba
Note: These are referred to as the “commutative
properties”

For any real numbers a, b, and c,
a + (b + c) = (a + b) + c
and
a(bc) = (ab)c
Note: These are referred to as the “associative properties”
Using the Properties of Real Numbers
Example
1: Simplify 12b – 9 + 4b – 7 b +1 =
Solution: 12b + 4b – 7b + (-9 + 1) =
b(12 +4 -7) + (-8) = 9b - 8
Example
2: Simplify 6 – (2x + 7) –3 =
Solution: -2x -7 + 6 – 3 = -2x -4
Using the Multiplication Property of 0
For
any real number a: a • 0 = 0
1: Simplify (6 – (2x + 7) –3)(0) =
Solution: 0
Example
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