Real Numbers Properties

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)).