Real Number Systems
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Questions and Answers

What is the multiplicative identity in the real number system?

  • -1
  • 1 (correct)
  • 0
  • a

    • Which property of real numbers is demonstrated by the equation $(a imes b) imes c = a imes (b imes c)$?

  • Distributivity
  • Commutativity
  • Associativity (correct)
  • Existence of inverses

    • According to the trichotomy law, for any two real numbers $a$ and $b$:

  • At least one of the following is true: $a < b$, $a = b$, or $a > b$
  • Exactly one of the following is true: $a eq b$, or $a = b$
  • At least one of the following is true: $a = b$, or $a > b$
  • Exactly one of the following is true: $a < b$, $a = b$, or $a > b$ (correct)

    • Which of these is an example of the distributive property?

    <p>$a imes (b + c) = a imes b + a imes c$</p> Signup and view all the answers

    What does the completeness axiom guarantee in a set of real numbers?

    <p>Every nonempty set of real numbers that is bounded above has a least upper bound (supremum)</p> Signup and view all the answers

    Which of the following statements illustrates the total order property?

    <p>For any two real numbers $a$ and $b$, exactly one of these is true: $a &lt; b$, $a = b$, or $a &gt; b$</p> Signup and view all the answers

    What does transitivity mean in the context of real numbers?

    <p>If $a eq b$, then $a = c$</p> Signup and view all the answers

    Which statement best describes the additive inverse property?

    <p>For any $a$, there exists $-a$ such that $a + (-a) = 0$</p> Signup and view all the answers

    Study Notes

    Real Numbers Systems

    Definition

    • A real number system is a mathematical structure consisting of a set of real numbers, together with operations of addition and multiplication, and an ordering relation.

    Properties

    • Commutativity:
      • Addition: a + b = b + a
      • Multiplication: a × b = b × a
    • Associativity:
      • Addition: (a + b) + c = a + (b + c)
      • Multiplication: (a × b) × c = a × (b × c)
    • Distributivity: a × (b + c) = a × b + a × c
    • Existence of additive and multiplicative identities:
      • Additive identity: 0 (a + 0 = a)
      • Multiplicative identity: 1 (a × 1 = a)
    • Existence of additive inverses:
      • For each a, there exists -a such that a + (-a) = 0
    • Existence of multiplicative inverses:
      • For each a ≠ 0, there exists 1/a such that a × (1/a) = 1

    Ordering Relation

    • Total Order: For any two real numbers a and b, exactly one of the following is true: a < b, a = b, or a > b
    • Trichotomy Law: For any two real numbers a and b, a ≤ b or b ≤ a
    • Transitivity: If a ≤ b and b ≤ c, then a ≤ c

    Completeness Axiom

    • Least Upper Bound Property: Every nonempty set of real numbers that is bounded above has a least upper bound (supremum)
    • Greatest Lower Bound Property: Every nonempty set of real numbers that is bounded below has a greatest lower bound (infimum)

    Real Numbers Systems

    Definition

    • A real number system is a mathematical structure consisting of a set of real numbers, together with operations of addition and multiplication, and an ordering relation.

    Properties

    • Commutative Property: • Addition: the order of numbers does not change the result (a + b = b + a) • Multiplication: the order of numbers does not change the result (a × b = b × a)
    • Associative Property: • Addition: the order in which numbers are added does not change the result ((a + b) + c = a + (b + c)) • Multiplication: the order in which numbers are multiplied does not change the result ((a × b) × c = a × (b × c))
    • Distributive Property: multiplication is distributed over addition (a × (b + c) = a × b + a × c)
    • Existence of Identities: • Additive identity: 0 (a + 0 = a) • Multiplicative identity: 1 (a × 1 = a)
    • Existence of Inverses: • Additive inverse: for each a, there exists -a such that a + (-a) = 0 • Multiplicative inverse: for each a ≠ 0, there exists 1/a such that a × (1/a) = 1

    Ordering Relation

    • Total Order: exactly one of the following is true: a < b, a = b, or a > b
    • Trichotomy Law: for any two real numbers a and b, a ≤ b or b ≤ a
    • Transitivity: if a ≤ b and b ≤ c, then a ≤ c

    Completeness Axiom

    • Least Upper Bound Property: every nonempty set of real numbers that is bounded above has a least upper bound (supremum)
    • Greatest Lower Bound Property: every nonempty set of real numbers that is bounded below has a greatest lower bound (infimum)

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    Quiz Team

    Description

    This quiz covers the definition and properties of real number systems, including commutativity, associativity, and distributivity.

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