📚 Concepts

📚 Real Numbers = Rational Numbers + Irrational Numbers

📚 Rational Numbers: Can be expressed as p/q (where q ≠ 0)

📚 Irrational Numbers: Cannot be expressed as p/q (e.g., √2, π)

📚 Fundamental Theorem of Arithmetic:

• Every number > 1 has a unique prime factorisation

📚 HCF (Highest Common Factor): Greatest number that divides two numbers

📚 LCM (Lowest Common Multiple): Smallest number divisible by two numbers

📚 Relation between HCF and LCM: Product of numbers = HCF × LCM

📚 Euclid’s Division Lemma: Method to find HCF using repeated division

📚 Decimal Expansion of Rational Numbers:

• If denominator has only 2 and/or 5 → Terminating Decimal 

• Otherwise → Repeating Decimal

✍️ Formulas

✍️ Euclid’s Division Lemma: a = bq + r, where 0 ≤ r < b

✍️ To find HCF using Euclid’s Algorithm:  

• Apply a = bq + r  

• Replace a with b and b with r  

• Repeat until r = 0. The last non-zero b is the HCF

✍️ HCF and LCM Relationship:

• HCF(a, b) × LCM (a, b) = a × b

✍️ Fundamental Theorem of Arithmetic:

• Every positive integer > 1 has a unique prime factorization

• Example, 60 = 2² × 3 × 5

✍️ Decimal Expansion Rule:

If the denominator of a rational number (in simplest form):  

• has only 2 and/or 5 → Terminating decimal  

• has other primes → Non-terminating, repeating decimal