1. OTCQ 091509 Using [-11, 0) Write its associated: 1) inequality, or 2) set in braces, or 3) number line. (for integers only)

2. 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

3. 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

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.? B A

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. 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.

9. 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… }

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

11. Square Roots of integer perfect squares are always rational numbers. 1 = 1rational 4 =2rational 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.

12. 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

13. 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? {??????????????????}

14. 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).

15. 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. –12.75 rational, real 16 2 whole, integer, rational, real = = 2 4242 16 2 A. B. C.

16. Check It Out! Example 1 Write all classifications that apply to each number. 9 whole, integer, rational, real –35.9 is a terminating decimal. –35.9 rational, real 81 3 whole, integer, rational, real = = 3 9393 81 3 A. B. C. 9 = 3

17. 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.

18. State if each number is rational, irrational, or not a real number. 21 irrational 0303 rational 0303 = 0 Example 2: Determining the Classification of All Numbers A. B.

19. UNDEFINED. Example 3: Determining the Classification of All Numbers 4040 C. State if each number is rational, irrational, or not a real number.

20. 23 is a whole number that is not a perfect square. 23 irrational 9090 undefined, so not a real number Check It Out! Example 4 A. B. State if each number is rational, irrational, or not a real number.

21. 64 81 rational C. Check It Out! Example 5 State if each number is rational, irrational, or not a real number.

22. 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: 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.

23. 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!!!!!!!!

24. 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!!!!!

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

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

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

28. The Distributive Property Any factor outside of expression enclosed within grouping symbols, must be multiplied by each term inside the grouping symbols. Outside leftorOutside right a(b + c) = ab + ac(b + c)a = ba + ca a(b - c) = ab – ac(b - c)a = ba - ca

29. 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

30. Which property would justify rewriting the following expression without parentheses? 3(2x + 5y) 1.Associative property of multiplication 2.Distributive property 3.Addition property of zero 4.Commutative property of multiplication

31. Which property would justify the following statement? 8x + 4 = 4 + 8x 1.Associative property of addition 2.Distributive property 3.Addition property of zero 4.Commutative property of addition

32. 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