Chapter 1: Number Systems

Operations on Real Numbers

Key Rules to Remember

Rule 1: Rational + Irrational = IRRATIONAL

Examples:

  • 2 + √3 = irrational
  • 5 - √2 = irrational
  • 7 + π = irrational

Why? If rational + irrational were rational, then irrational = rational - rational = rational. Contradiction!

Rule 2: Rational × Irrational = IRRATIONAL (if rational ≠ 0)

Examples:

  • 3 × √5 = 3√5 (irrational)
  • 7π (irrational)
  • (2/3) × √2 = (2√2)/3 (irrational)

Rule 3: Irrational ± Irrational = COULD BE EITHER!

Examples:

  • √2 + √3 = irrational
  • √2 + (-√2) = 0 = rational!
  • √8 - √2 = 2√2 - √2 = √2 = irrational

Rule 4: Irrational × Irrational = COULD BE EITHER!

Examples:

  • √2 × √3 = √6 = irrational
  • √2 × √2 = 2 = rational!
  • √3 × √12 = √36 = 6 = rational!

Important Identities

For positive real numbers a and b:

IdentityExample
ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}√12 = √4 × √3 = 2√3
a/b=a/b\sqrt{a/b} = \sqrt{a} / \sqrt{b}√(9/16) = 3/4
(a+b)(a−b)=a−b(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b(√5 + √3)(√5 - √3) = 2
(a+b)2=a+2ab+b(\sqrt{a} + \sqrt{b})^2 = a + 2\sqrt{ab} + b(√3 + √2)² = 5 + 2√6