Real Numbers Properties and Operations

Properties of Real Numbers

• Commutative Property: The order of real numbers does not change the result of addition or 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 distributed across addition and multiplication.
  • a × (b + c) = a × b + a × c
  • a + (b × c) = a + b × a + c

Irrational Numbers

• Definition: A real number that cannot be expressed as a finite decimal or fraction.
• Examples: π, e, √2, √3
• Properties:
  • Irrational numbers cannot be written in simplest form.
  • Irrational numbers have an infinite number of decimal places that never repeat.
  • Irrational numbers can be represented on a number line, but not exactly.

Real Number Operations

• Addition: Combining two real numbers to get a total or a sum.
  • a + b = c
• Subtraction: Finding the difference between two real numbers.
  • a - b = c
• Multiplication: Combining two real numbers to get a product.
  • a × b = c
• Division: Finding the quotient of two real numbers.
  • a ÷ b = c

Real Number Sets

• Natural Numbers (N): {1, 2, 3, ...}
• Whole Numbers (W): {0, 1, 2, 3, ...}
• Integers (Z): {..., -3, -2, -1, 0, 1, 2, 3, ...}
• Rational Numbers (Q): {a/b, where a and b are integers and b ≠ 0}
• Real Numbers (R): All rational and irrational numbers

Absolute Value

• Definition: The distance of a number from zero on a number line.
• Notation: |a|, where a is a real number.
• Properties:
  • |a| ≥ 0
  • |a| = 0 if and only if a = 0
  • |a × b| = |a| × |b|
  • |a + b| ≤ |a| + |b|