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223 Reference Chapter

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

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223 Reference Chapter

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  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, -4π, 0, 2/5, 3, √7, 10} • List the elements from set V that belong to each set: • Natural numbers • Whole numbers • Integers • Rational • Irrational • 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, -4π, 0, 2/5, 3, √7, 10} • List the elements from set V that belong to each set: • Natural numbers 3, 10 • Whole numbers 0, 3, 10 • Integers -12, 0, 3, 10 • Rational -12, 0, 2/5, 3, 10 • Irrational -√5, -4π, √7 • 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 Addition a + b is a real number Closure Property of Multiplicationab is a real number Commutative Property of Addition a + 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 Addition a + 0 = a Identity Property of Multiplication a ∙1 = a Inverse Property of Addition a + -a = -a + a = 0 Inverse Property of Multiplication a ∙1/a = 1/a∙a = 1 Distributive Property a(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: • 3 + 4 = 4 + 3 __________________________________________ • 6∙ 1= 6 _______________________________________________ • 5x + 5y = 5(x + y) _______________________________________ • (7-y)∙1/(7-y) = 1 ________________________________________ • (6 + y) + 2 = 6 + (y + 2) __________________________________ • 7 + ¾ is a real number ___________________________________ • 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: • 3 + 4 = 4 + 3 commutative property of addition • 6∙ 1= 6 identity property of multiplication • 5x + 5y = 5(x + y) distributive property • (7-y)∙1/(7-y) = 1 inverse property of multiplication • (6 + y) + 2 = 6 + (y + 2) associative property of addition • 7 + ¾ is a real number closure property of addition • 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 | = 5 Example: | -8| = 8 Example: |0| = 0 Properties of Absolute Value • |a| ≥ 0 • |-a| = |a| • |a|∙|b| = |ab| • |a|/|b| = |a/b| (b ≠ 0) • |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 • 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 • 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

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