1. What is the difference between a line segment and a line?
OTCQ
What is the difference between a line segment and a line?

2. Aim 1-2 How do we define a set and a subset and how do we define the set of real numbers and its subsets?
Performance Indicators
AA 29, AA 30, AN 1, AN 6
Homework read ch 1-2, problems 7- 16
rreidymath.wikispaces.com

3. Objectives
SWBAT define sets and subsets, intersections, unions, intersections.

4. 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. B A

5. 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.? A B

6. Union
The union of two sets A and B is the set of all elements that are included in either set.
Notation: A ∪ B A B
A ∪ B

7. Intersection
The intersection of two sets A and B is the set of all elements that are included in both sets.
Notation: A ∩ B A B
A ∩ B

8. The set of Real Numbers and its subsets
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, }

9. SUBSETS OF THE REAL NUMBERS Integers
{… -2, -1, 0, 1, 2, … }
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.

10. Square Roots of integer perfect squares are always rational numbers.
5 = …. irrational
6 = Irrational

11. 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
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= ,000
1000*1000 = 1,000,000
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

12. 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? {??????????????????}

13. Set of Integer Perfect Squares: {0, 1, 4, 9, 16, 25, 36 …} {. } {
Set of Integer Perfect Squares: {0, 1, 4, 9, 16, 25, 36 …} {??????????????????} { , -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).

14. Example: Classifying Real Numbers
Write all classifications that apply to each number. A. 35
35 is a whole number that is not a perfect square.
irrational, real B.
–12.75
–12.75 is a terminating decimal.
rational, real
16 2
= = 2
4 2
16 2 C.
whole, integer, rational, real

15. whole, integer, rational, real
Check It Out! Example 1
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
81 3
= = 3
9 3
81 3 C.
whole, integer, rational, real

16. 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.

17. Example: Classification of Numbers
State if each number is rational, irrational, or not a real number. A. 21
irrational
0 3
0 3
= 0 B.
rational

18. Example: Classification of Numbers
State if each number is rational, irrational, or not a real number.
4 0 C.
UNDEFINED.

19. State if each number is rational, irrational, or not a real number.23
23 is a whole number that is not a perfect square.
irrational
9 0 B.
undefined, so not a real number

20. State if each number is rational, irrational, or not a real number.
64 81 C.
rational

21. Closure of Real Numbers
Closure property of addition/subtraction:
If a and b are real numbers, then a + b will equal a real number. Examples: = 15 and = -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.

22. Commutative Property
Commutative 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!!!!!!!!

23. Associative Property Associative 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!!!!!

24. Identity Properties
1) Additive Identity
a + 0 = a
2) Multiplicative Identity
a • 1 = a

25. Inverse Properties 1) Additive Inverse (Opposite) a + (-a) = 0
2) Multiplicative Inverse (Reciprocal)

26. Multiplicative Property of Zero
(If you multiply by 0, the answer is 0.)

27. The Distributive Property
Any 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

28. Concept Check: 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

29. Which property would justify rewriting the following expression without parentheses? 3(2x + 5y)
Associative property of multiplication
Distributive property
Addition property of zero
Commutative property of multiplication

30. Which property would justify the following statement? 8x + 4 = 4 + 8x
Associative property of addition
Distributive property
Addition property of zero
Commutative property of addition

31. Which property would justify the following statement
Associative property of addition
Distributive property
Addition property of zero
Commutative property of addition