Real Numbers

• Introduction
  • What are Real Numbers? (Rational + Irrational)
  • Number Line Representation
• Euclid's Division Lemma
  • Statement: $a = bq + r$, where $0 \le r < b$
  • Application to find HCF
• Fundamental Theorem of Arithmetic
  • Statement: Every composite number can be uniquely expressed as a product of primes
  • Prime Factorization Method
  • Finding HCF and LCM
    • Relation: $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$
• Revisiting Irrational Numbers
  • Proofs of Irrationality
    • Proof that $\sqrt{2}$ is irrational
    • Proof that $\sqrt{3}$, $\sqrt{5}$, etc., are irrational
  • Theorems on Rationals and Irrationals
    • Sum, difference, product, and quotient of rationals and irrationals
• Revisiting Rational Numbers & Their Decimal Expansions
  • Rational Number form: $\frac{p}{q}$ ($q \neq 0$)
  • Terminating Decimal Expansions
    • Prime factorization of $q$ is of the form $2^n 5^m$
  • Non-Terminating Repeating Decimal Expansions
    • Prime factorization of $q$ is NOT of the form $2^n 5^m$