1. 223 Reference Chapter Section R2: Real Numbers and Their Properties Natural Numbers: {1, 2, 3, 4, …} Whole Numbers: {0, 1, 2, …} Integers: {…, -3, -2, -1, 0, 1, 2, 3, …} Rational Numbers: {p/q | p and q are both integers and q ≠0 Examples of Rational Numbers: 9, -17, 3/5, -21/9, √4 Irrational Numbers: {x | x is real but not rational} Examples of Irrationals: √2, -√10, π Real Numbers: {x | x corresponds to a point on a number line} Example: Let set V = {-12, -√5, -4 π, 0, 2/5, 3, √7, 10} List the elements from set V that belong to each set: a)Natural numbers b)Whole numbers c)Integers d)Rational e)Irrational f)Real

2. 223 Reference Chapter Section R2: Real Numbers and Their Properties Natural Numbers: {1, 2, 3, 4, …} Whole Numbers: {0, 1, 2, …} Integers: {…, -3, -2, -1, 0, 1, 2, 3, …} Rational Numbers: {p/q | p and q are both integers and q ≠0 Examples of Rational Numbers: 9, -17, 3/5, -21/9, √4 Irrational Numbers: {x | x is real but not rational} Examples of Irrationals: √2, -√10, π Real Numbers: {x | x corresponds to a point on a number line} Example: Let set V = {-12, -√5, -4 π, 0, 2/5, 3, √7, 10} List the elements from set V that belong to each set: a)Natural numbers 3, 10 b)Whole numbers 0, 3, 10 c)Integers -12, 0, 3, 10 d)Rational -12, 0, 2/5, 3, 10 e)Irrational -√5, -4 π, √7 f)Real -12, -√5, -4π, 0, 2/5, 3, √7, 10

3. 223 Reference Chapter Section R2: Real Numbers and Their Properties Properties of Real Numbers Closure Property of Additiona + b is a real number Closure Property of Multiplicationab is a real number Commutative Property of Additiona + b = b + a Commutative Property of Multiplicationab = ba Associative Property of Addition(a + b) + c = a + (b + c) Associative Property of Multiplication(ab)c = a(bc) Identity Property of Additiona + 0 = a Identity Property of Multiplicationa ∙1 = a Inverse Property of Additiona + -a = -a + a = 0 Inverse Property of Multiplicationa ∙1/a = 1/a∙a = 1 Distributive Propertya(b + c) = ab + ac

4. 223 Reference Chapter Section R2: Real Numbers and Their Properties Properties of Real Numbers Write which property is being illustrated in each statement: a)3 + 4 = 4 + 3 __________________________________________ b)6∙ 1= 6 _______________________________________________ c)5x + 5y = 5(x + y) _______________________________________ d)(7-y)∙1/(7-y) = 1 ________________________________________ e)(6 + y) + 2 = 6 + (y + 2) __________________________________ f)7 + ¾ is a real number ___________________________________ g)15 + -15 = 0 ___________________________________________

5. 223 Reference Chapter Section R2: Real Numbers and Their Properties Properties of Real Numbers Write which property is being illustrated in each statement: a)3 + 4 = 4 + 3 commutative property of addition b)6∙ 1= 6 identity property of multiplication c)5x + 5y = 5(x + y) distributive property d)(7-y)∙1/(7-y) = 1 inverse property of multiplication e)(6 + y) + 2 = 6 + (y + 2) associative property of addition f)7 + ¾ is a real number closure property of addition g)15 + -15 = 0 inverse property of addition

6. 223 Reference Chapter Section R2: Real Numbers and Their Properties The Absolute Value of number is the distance on the number line from that place to 0. Example: | 5 | = 5Example: | -8| = 8Example: |0| = 0 Properties of Absolute Value 1. |a| ≥ 0 2. |-a| = |a| 3. |a|∙|b| = |ab| 4. |a|/|b| = |a/b| (b ≠ 0) 5. |a + b| ≤ |a| + |b| (this is the triangle inequality)

7. 223 Reference Chapter Section R2: Real Numbers and Their Properties Order of Operations (left to right for each) Parentheses Exponents Multiplication and Division Addition and Subtraction Example: Evaluate the following expressions a)5(3+1)^2-(9+10/2) b) 12/3 + (5-2)(4+1) (9-7)^3 - 7∙2

8. 223 Reference Chapter Section R2: Real Numbers and Their Properties Order of Operations (left to right for each) Parentheses Exponents Multiplication and Division Addition and Subtraction Example: Evaluate the following expressions a)5(3+1)^2-(9+10/2) 5(4)^2-(9+5) = 5*16 – 14 = 80 – 14 = 66 b) 12/3 + (5-2)(4+1) (9-7)^3 - 7∙2 4 + (3)(5) = 4 + 15 = 19 = -19/6 (2)^3 – 14 8 – 14 -6