Podcast
Questions and Answers
What is the multiplicative identity in the real number system?
What is the multiplicative identity in the real number system?
Which property of real numbers is demonstrated by the equation $(a imes b) imes c = a imes (b imes c)$?
Which property of real numbers is demonstrated by the equation $(a imes b) imes c = a imes (b imes c)$?
According to the trichotomy law, for any two real numbers $a$ and $b$:
According to the trichotomy law, for any two real numbers $a$ and $b$:
Which of these is an example of the distributive property?
Which of these is an example of the distributive property?
Signup and view all the answers
What does the completeness axiom guarantee in a set of real numbers?
What does the completeness axiom guarantee in a set of real numbers?
Signup and view all the answers
Which of the following statements illustrates the total order property?
Which of the following statements illustrates the total order property?
Signup and view all the answers
What does transitivity mean in the context of real numbers?
What does transitivity mean in the context of real numbers?
Signup and view all the answers
Which statement best describes the additive inverse property?
Which statement best describes the additive inverse property?
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)
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This quiz covers the definition and properties of real number systems, including commutativity, associativity, and distributivity.
