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Questions and Answers
What is a real number?
What is a real number?
Which property of real numbers states that the order of numbers does not change their sum or product?
Which property of real numbers states that the order of numbers does not change their sum or product?
What is the definition of a rational number?
What is the definition of a rational number?
What is the result of dividing two real numbers, except for division by zero?
What is the result of dividing two real numbers, except for division by zero?
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What is the definition of the absolute value of a real number?
What is the definition of the absolute value of a real number?
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What is the purpose of interval notation in real numbers?
What is the purpose of interval notation in real numbers?
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Study Notes
Definition
- A real number is a value that can be represented on the number line.
- It is a member of the set of numbers that include all rational and irrational numbers.
Properties
- Commutative Property: The order of real numbers does not change their sum or product.
- Associative Property: The order in which real numbers are added or multiplied does not change their sum or product.
- Distributive Property: Real numbers can be multiplied and added in any order.
Classification
-
Rational Numbers:
- Can be expressed as the ratio of two integers (e.g., 3/4, 22/7).
- Can be expressed as a finite decimal or a repeating decimal.
-
Irrational Numbers:
- Cannot be expressed as the ratio of two integers (e.g., π, e, sqrt(2)).
- Can be expressed as an infinite non-repeating decimal.
Operations
- Addition: The sum of two real numbers is always a real number.
- Subtraction: The difference of two real numbers is always a real number.
- Multiplication: The product of two real numbers is always a real number.
- Division: The quotient of two real numbers is always a real number, except for division by zero.
Important Concepts
- Absolute Value: The distance of a real number from zero on the number line.
- Distance: The absolute value of the difference between two real numbers.
- Interval Notation: A way to represent a set of real numbers using interval notation (e.g., [a, b], (a, b], [a, b), (a, b)).
Real Numbers
- Represented on the number line, including rational and irrational numbers.
Properties of Real Numbers
- Commutative Property: Order of real numbers doesn't change their sum or product.
- Associative Property: Order in which real numbers are added or multiplied doesn't change their sum or product.
- Distributive Property: Real numbers can be multiplied and added in any order.
Classification of Real Numbers
Rational Numbers
- Expressed as the ratio of two integers (e.g., 3/4, 22/7).
- Can be expressed as a finite decimal or a repeating decimal.
Irrational Numbers
- Cannot be expressed as the ratio of two integers (e.g., π, e, sqrt(2)).
- Can be expressed as an infinite non-repeating decimal.
Operations on Real Numbers
- Addition: Sum of two real numbers is always a real number.
- Subtraction: Difference of two real numbers is always a real number.
- Multiplication: Product of two real numbers is always a real number.
- Division: Quotient of two real numbers is always a real number, except for division by zero.
Important Concepts
Absolute Value
- Distance of a real number from zero on the number line.
Distance
- Absolute value of the difference between two real numbers.
Interval Notation
- Way to represent a set of real numbers using interval notation (e.g., [a, b], (a, b], [a, b), (a, b)).
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Description
Explore the definition and properties of real numbers, including commutative, associative, and distributive properties.
