Activator

Activator1 .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 2Introduction to Algebra 2Real Numbers and Real Numbers and

Their PropertiesTheir PropertiesCCSS:N.RN.3

N.RN.3N.RN.3

Domain: The Real Number System Clusters: Use properties of rational and irrational

numbersStandards: 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.

Write sets using set notationWrite 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.


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”


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}


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}


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 lineUsing a number line

One 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, …}

-2 -1 0 1 2 3 4 5


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.

-2 -1 0 1 2 3 4 5

coordinate

Graph of -1

o

1

2

11

4o o


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

-2 -1 0 1 2 3 4 5

coordinate

Graph of -1

o

1

2

11

4o o

2

7

4 16

o o oo o

Circumference

diameter

The common sets of numbersThe 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}

q are integers, q 0p

p andq

04/20/23

Real Numbers

Rational Numbers Irrational Numbers

3

1/2-2

15%

2/3

1.456

-0.7

0

3 2

5 2

34

Real Numbers


31/2 -2

15%

2/3

1.456- 0.7

0

3 2

5 2

34

Integers

Real Numbers


31/2

-2

15%

2/3

1.456- 0.7

0

3 2

5 2

34

Integers

Whole

Real Numbers


31/2

-2

15%

2/3

1.456- 0.7

0

3 2

5 2

34

Integers

Whole

Natural

The common sets of numbersThe common sets of numbers

Question: Select all the words from the

following list that apply to the number:

Whole Number, Rational Number, Irrational Number,

Real Number, Undefined

Solution: Whole Number, Rational Number, Real

Number

4

4 2

Finding Additive inversesFinding Additive inverses

For any real number x, the number –x is the additive

inverse of x.

Example 1:

NumberInverse

Additive6 - 6

- 4 4

- 8.7 8.70 0

2

32

3

Using the Using the Absolute ValueAbsolute Value

Examples: Find :

To find the absolute value of a signed number:

Caution: The absolute value of a number is always positive

X > 0, then = X

X = 0, then = 0

X < 0, then = -X

if X

if XX

if X

29 29

0 0

90 90

Using Inequality Symbols 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 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)

Example 2: Write in interval notation and graph: {x|x 0}Solution: Interval Notation [0, )

-2 -1 0 1 2 3 4 5)

-2 -1 0 1 2 3 4 5[

Graphing Sets of Real Numbers 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 NumbersAdding Real Numbers

Example: Add (–5.6) + (-2.1) =

Example:

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.

(-5.6) + (-2.1) = -( 5.6 + 2.1 ) = -7.7

2 5ADD: + (- )

3 6

2 5 4 5 5 4 1 + (- ) = + (- ) = -( - )

3 6 6 6 6 6 6

Subtracting Real NumbersSubtracting Real Numbers

Example: Subtract (–56) - (-70) =

Example:

To subtract signed numbers: 1) Rewrite as an addition problem by adding the opposite of

the number to be subtracted. Find the sum.

(-56) - (-70) = (-56) + (70 ) = +(70-56) = +143 3 7 7

Subtract: - - (- ) - (- ) -5 8 10 20

24 15 28 14 5 1( ) + (+ ) + ( ) + (- ) = ( )

40 40 40 40 40 8

Finding the distance between two points on a number line. 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

-2 -1 0 1 2 3 4 5o o

Multiplying Real NumbersMultiplying 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) =

Example:

To find the product of two negative signed numbers:Find the product of the absolute values. Make the sign positive.

(-12) (5) = -(12 5) = -60 7 8

Multiply: (- ) (- ) = 12 25

7 8 7 8 14(- ) (- ) = +( ) = +

12 25 12 25 75

Dividing Real NumbersDividing 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) =

Example:

To find the quotient of two negative signed numbers:Find the quotient of the absolute values. Make the sign positive.

Caution: Division by zero is “undefined”:

(-12) (5) = - 12 5 = - 2.4 2.4 7 8

Divide: (- ) (- ) = 12 25

7 8 7 8 7 25 175(- ) (- ) = +( ) = +( ) = +

12 25 12 25 12 8 96

6 0 , but = 0

0 61

if , x 2x-2

is undefined

then

Dividing Real NumbersDividing 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 Additive Inverse

6

0.05 20 -0.05

0 none 0

2

5

5

2 2

51

6

11

77

11

7

11

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 = 1101 = 10, 21 = 2, 81 = 8, #1 = #102 = 10 x 10 = 100, 22 = 2 x 2 = 4, 82 = 8 x 8 = 64103 = 10 x 10 x 10 = 1000, 23 = 2 x 2 x 2 = 8104 = 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

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 OperationsPerform left to right

HighestInside Parenthesis

ExponentiationMultiplication/DivisionAddition/Subtraction

Lowest

Order of Operations

•Example 1: Simplify

Solution:

Example 2: Simplify

2

110 6 9

25

12 3(2)6

2

110 6 3

5 6 3 22

110 6 9

25

15 10 12 212 3(4)6

12 3(2)6

2

10 6 2 9

11 4 3( 2)

2

10 6 2(3) 10 6 6 101

11 2 3(4)

10 6 2 9

11 4 3( 2 22 1) 12 0

Evaluating Algebraic Expressions

•Example 1: if w = 4, x = -12, y = 64, z = -3

Find:

Solution:

Example 2: Find:

5

1

x z y

x

2 32w z

5( 12) ( 3) 64 60 ( 3)8 60 24 84 84

( 12) 1 13 13 13 11 3

5x z y

x

2 3 2 34 2( 3) 16 2( 27) 16 54 32 8w z

Using the Distributive PropertyUsing the Distributive Property

For any real numbers a, b , and ca(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 PropertiesUsing 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 = 4Note: 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 = 4Note: This is referred to as the “multiplicative identity”

Using the Commutative and Associate PropertiesUsing 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) + cand a(bc) = (ab)c

Note: These are referred to as the “associative properties”

Using the Properties of Real NumbersUsing 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 0Using the Multiplication Property of 0

For any real number a: a • 0 = 0

Example 1: Simplify (6 – (2x + 7) –3)(0) =Solution: 0

Learning in the Learning in the 21st Century21st Century

Do you understandthat you are responsible

for your own success or failure?

Learning in the Learning in the 21st Century21st Century

Questions ?????

Activator

Documents

Transcript of Activator