Real Numbers Properties

Real Numbers

Definition

• A real number is a value that can be represented on the number line
• Includes all rational and irrational numbers

Properties

• Commutative Property: The order of real numbers does not change the result of addition and multiplication
  • a + b = b + a
  • a × b = b × a
• Associative Property: The order in which real numbers are added or multiplied does not change the result
  • (a + b) + c = a + (b + c)
  • (a × b) × c = a × (b × c)
• Distributive Property: Real numbers can be multiplied by a single value and distributed to all terms
  • a × (b + c) = a × b + a × c

Types of Real Numbers

• Rational Numbers: Can be expressed as a fraction (p/q), where p and q are integers and q ≠ 0
• Irrational Numbers: Cannot be expressed as a fraction, have infinite non-repeating decimals
  • Examples: π, e, √2

Operations on Real Numbers

• Addition: Combining two or more real numbers to get a total or a sum
• Subtraction: Finding the difference between two or more real numbers
• Multiplication: Combining two or more real numbers to get a product
• Division: Dividing one real number by another, as long as the divisor is not zero

Important Real Numbers

• Pi (π): An irrational number representing the ratio of a circle's circumference to its diameter
• Euler's Number (e): An irrational number approximately equal to 2.718, used in mathematical calculations

Definition of Real Numbers

• A real number is a value that can be represented on the number line
• Includes all rational and irrational numbers

Properties of Real Numbers

Commutative Property

• The order of real numbers does not change the result of addition and multiplication
• Examples:
  • a + b = b + a
  • a × b = b × a

Associative Property

• The order in which real numbers are added or multiplied does not change the result
• Examples:
  • (a + b) + c = a + (b + c)
  • (a × b) × c = a × (b × c)

Distributive Property

• Real numbers can be multiplied by a single value and distributed to all terms
• Example:
  • a × (b + c) = a × b + a × c

Types of Real Numbers

Rational Numbers

• Can be expressed as a fraction (p/q), where p and q are integers and q ≠ 0

Irrational Numbers

• Cannot be expressed as a fraction
• Have infinite non-repeating decimals
• Examples:
  • π
  • e
  • √2

Operations on Real Numbers

Addition

• Combining two or more real numbers to get a total or a sum

Subtraction

• Finding the difference between two or more real numbers

Multiplication

• Combining two or more real numbers to get a product

Division

• Dividing one real number by another, as long as the divisor is not zero

Important Real Numbers

Pi (π)

• An irrational number representing the ratio of a circle's circumference to its diameter

Euler's Number (e)

• An irrational number approximately equal to 2.718, used in mathematical calculations

Learn about the properties of real numbers, including commutative and associative properties, and how they apply to addition and multiplication.