otcq 091509 using [-11, 0) write its associated: 1) inequality, or 2) set in braces, or 3) number...

OTCQ 091509

Using [-11, 0)

Write its associated:

1) inequality, or

2) set in braces, or

3) number line.

(for integers only)

OTCQ 091509

Using [-11, 0)

Write its associated:

1) inequality, or

2) set in braces, or

3) number line.

(for integers only)

Correct answers-11 < x < 0{-11, -10, … -1}

-11 0 0

Aim 1-2 How do we define the set of real numbers and their properties?

Performance Indicators

AA 29, AA 30, AN 1, AN 6

Venn Diagrams, Complements and Subsets

• Set B (blue area) is called a subset of set A (green area) if all of Set B is contained in Set A

• B⊂A A

• The complement of Set B within Set A means anything outside of Set B and still within set A.

BA

Venn Diagrams, Complements and Subsets

• Is set Set B (blue area) a subset of set A (green area)?

• B⊂A? A

• What is the complement of Set A on this screen?

• Is any of Set B in the complement of set A.?

BA

Union

• The union of two setsA and Bis the set of allelements thatare included in either set.

• Notation:A ∪ B

A B

A ∪ B

Intersection

• The intersectionof two setsA and Bis the set of allelements that are included in both sets.

• Notation:A ∩ B

A B

A ∩ B

Objectives: The students will be able to (“SWBAT”):

1. SWBAT recall the definition of the real numbers and its subsets.

2. SWBAT explain closure and why the real numbers are closed.

3. SWBAT state and apply the commutative and associative properties and the properties of equality.

The set of Real Numbers and its subsets

Real numbers

SUBSETS OF THE REAL NUMBERS

Natural numbers or counting numbers

The set of all rational and irrational numbers.

{1, 2, 3, 4, 5, 6 … }

Whole numbers {0, 1, 2, 3, 4, 5, 6… }

SUBSETS OF THE REAL NUMBERS

Integers {…, -3, -2, -1, 0, 1, 2, 3, … }

Rational numbers: Any number that may be written as a quotient/fraction of two integers or as repeating decimals.

Irrational numbers Any number that cannot be written as a quotient/fraction of two integers. Irrational numbers are non-repeating decimals.

Square Roots of integer perfect squares are always rational numbers.

1 = 1 rational

4 = 2 rational

5 = 2.23606…. irrational

6 = 2.44948. . . Irrational

How do you find square roots on the scientific/graphing calculator:

Press 5 and and then the number and enter.

Set of Perfect Squares using only integers: {1, 4, 9, 16, 25, 36 …}An integer perfect square is the product of any whole number multiplied by itself. Perfect Squares• 0*0= 0• 1*1 = 1• 2*2 = 4• 3*3= 9• 4*4= 16• 5*5= 25• 6*6= 36• 7*7= 49• 8*8= 64• 9*9= 81• 10*10 = 100

Perfect

Squares• 11*11 = 121• 12*12 = 144• 13*13= 169• 14*14= 196• 15*15= 225• 16*16= 256• 20*20= 400• 25*25= 625• 100*100= 10,000• 1000*1000 = 1,000,000

Set of Integer Perfect Squares: {0, 1, 4, 9, 16, 25, 36 …}

What integers are in the complement of the set of integer perfect squares?

{??????????????????}

Set of Integer Perfect Squares: {0, 1, 4, 9, 16, 25, 36 …}

{??????????????????}

{ . . . -5, -4, -3, -2, -1, 2, 3, 5,6,7,8,10…}

The square root of any integers in this complement set is either irrational (includes a decimal root)or imaginary (“error” on your calculator).

Additional Example 1: Classifying Real Numbers

Write all classifications that apply to each number.

35 is a whole number that is not a perfect square.

35

irrational, real

–12.75 is a terminating decimal.–12.75rational, real

16 2

whole, integer, rational, real

= = 24 2

16 2

A.

B.

C.

Check It Out! Example 1

Write all classifications that apply to each number.

9


–35.9 is a terminating decimal.–35.9rational, real

81 3


= = 39 3

81 3

A.

B.

C.

9 = 3

NEVER ZERO DENOMINATOR.

A fraction with a denominator of 0 is undefined because you cannot divide by zero.

A zero denominator is a big no no in math.

State if each number is rational, irrational, or not a real number.

21

irrational

0 3

rational

0 3

= 0

Example 2: Determining the Classification of All Numbers

A.

B.

UNDEFINED.

Example 3: Determining the Classification of All Numbers

4 0

C.


23 is a whole number that is not a perfect square.

23

irrational

9 0

undefined, so not a real number


A.

B.


64 81

rational

C.



Closure of Real NumbersClosure property of addition/subtraction:

If a and b are real numbers, then a + b will equal a real number. Examples: 4 + 11 = 15 and -20 + -11 = -31

Closure property of multiplication/division:

If a and b are real numbers, then ab will equal a real number. Examples: 4 * 4 = 16 and -2 ÷ -3 = .6666

In summary, anytime you add, subtract, multiply or divide real numbers, you get another real number. So we say you stay inside the closed set of real

numbers and that’s closure.

Commutative PropertyCommutative Property of Addition: a + b = b + a

Commutative Property of Multiplication: ab = ba

Examples

2 + 3 = 5 = 3 + 2

3• 4 = 12 = 4 • 3

The commutative property does not work for subtraction or division!!!!!!!!

Associative PropertyAssociative property of Addition:

(a + b) + c = a + (b + c)

Associative Property of Multiplication:

(ab) c = a (bc)

Examples

(1 + 2) + 3 = 1 + (2 + 3)

(2 • 3) • 4 = 2 • (3 • 4)

The associative property does not work for subtraction or division!!!!!

Identity Properties

1) Additive Identity

a + 0 = a

2) Multiplicative Identity

a • 1 = a

Inverse Properties

1) Additive Inverse (Opposite)

a + (-a) = 0

2) Multiplicative Inverse (Reciprocal)

a 1

a 1

Multiplicative Property of Zero

a • 0 = 0

(If you multiply by 0, the answer is 0.)

The Distributive PropertyAny factor outside of expression enclosed within

grouping symbols, must be multiplied by each term inside the grouping symbols.

Outside left or Outside right

a(b + c) = ab + ac (b + c)a = ba + ca

a(b - c) = ab – ac (b - c)a = ba - ca

Name the property

1) 5a + (6 + 2a) = 5a + (2a + 6)commutative (switching order)

2) 5a + (2a + 6) = (5a + 2a) + 6

associative (switching groups)

3) 2(3 + a) = 6 + 2a

distributive

Which property would justify rewriting the following expression without

parentheses? 3(2x + 5y)1. Associative property

of multiplication2. Distributive property3. Addition property of

zero4. Commutative property

of multiplication

Which property would justify the following statement?

8x + 4 = 4 + 8x1. Associative property of

addition

2. Distributive property

3. Addition property of zero

4. Commutative property of addition

Which property would justify the following statement?

8 + (2 + 6) = (8 + 2) + 6 1. Associative property of

addition

2. Distributive property

3. Addition property of zero

4. Commutative property of addition

otcq 091509 using [-11, 0) write its associated: 1) inequality, or 2) set in braces, or 3) number...

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