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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, -4π, 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 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, -4π, 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, -4π, √7 f) Real -12, -√5, -4π, 0, 2/5, 3, √7, 10 223 Reference Chapter Section R2: Real Numbers and Their Properties Properties of Real Numbers Closure Property of Addition Closure Property of Multiplication Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication Identity Property of Addition Identity Property of Multiplication Inverse Property of Addition Inverse Property of Multiplication Distributive Property a + b is a real number ab is a real number a+b=b+a ab = ba (a + b) + c = a + (b + c) (ab)c = a(bc) a+0=a a ∙1 = a a + -a = -a + a = 0 a ∙1/a = 1/a∙a = 1 a(b + c) = ab + ac 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 ___________________________________________ 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 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 | = 5 Example: | -8| = 8 Example: |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) 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∙2 223 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